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Hindawi Publishing Corporation Advances in Materials Science and Engineering Volume 2013, Article ID 810508, 11 pages http://dx.doi.org/10.1155/2013/810508 Research Article Thermo Elastic-Plastic Analysis of Rotating Functionally Graded Stainless Steel Composite Cylinder under Internal and External Pressure Using Finite Difference Method Sanjeev Sharma and Sanehlata Yadav Department of Mathematics, Jaypee Institute of Information Technology, A-10, Sector 62, Noida, Uttar Pradesh 201307, India Correspondence should be addressed to Sanjeev Sharma; [email protected] Received 22 July 2013; Accepted 26 October 2013 Academic Editor: Mohd Sapuan Salit Copyright © 2013 S. Sharma and S. Yadav. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. e thermal elastic-plastic stresses have been investigated for a rotating functionally graded stainless steel composite cylinder under internal and external pressure with general nonlinear strain hardening law and von Mises’ yield criterion using finite difference method. e modulus of elasticity in the rotating cylinder varies radially according to power law and the temperature distribution satisfies Laplace heat equation in radial direction. From the analysis, we can conclude that cylinder made of functionally graded stainless steel composite material with variable thickness and variable density under thermal loading for Swiſt’s strain hardening measure = 0.6 is better choice of the design as compared to homogeneous cylinder. is is because of the reason that circumferential stress is less for functionally graded stainless steel composite cylinder as compared to homogeneous cylinder for Swiſt’s strain hardening measure = 0.6 under internal and external pressure. is leads to the idea of “stress saving” which minimizes the possibility of fracture of cylinder. 1. Introduction ermo elastic-plastic analysis of functionally graded mate- rials under internal and external pressure is an active topic for engineering mechanics. e study of thick-walled cylinder is an interesting area of research and highly used due to vast utilization in the pressure vessels and pipes and so forth. Functionally graded materials are nothing but nonhomogeneous composite materials which are highly heat resistant and very promising in high-tech engineering fields [1, 2]. e demands of functionally graded materials are increasing under high rotation and temperature. ese materials find their applications in many areas, that is, flywheels, aerospace, nuclear reactors, compressors, and so forth. e problems of rotating thick-walled cylinder and plates can be found in many text books of elasticity [3, 4]. Obata et al. [5] calculated stresses for a hollow circular cylinder and hollow sphere under thermal loading using perturbation method. For these materials they found that the perturbation solutions had good convergence. Perry and Aboudi [6] calculated the residual stresses in a homogeneous cylinder using finite difference method and concluded that it is very effective in solving autofrettage problem numer- ically. Gao [7] investigated elastic-plastic stresses, strains, and displacements in a cylinder under internal pressure using elastic strain hardening in the plane strain conditions. Tutuncu and Ozturk [8] studied the closed form solution for stresses and displacement in cylindrical and spherical vessels made of functionally graded material subjected to internal pressure using infinitesimal theory of elasticity. ey compared stress distributions in functionally graded material with homogeneous material. Singh and Gupta [9] developed a mathematical model to describe steady-state creep in an isotropic functionally graded composite subjected to internal pressure which contains linearly varying silicon carbide particles in a matrix of pure aluminum. ey observed that the radial stress in the cylinder decreases throughout with the increase in reinforcement gradient, whereas the tangential,

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Page 1: Research Article Thermo Elastic-Plastic Analysis of Rotating …downloads.hindawi.com/journals/amse/2013/810508.pdf · 2019-07-31 · ermo elastic-plastic analysis of functionally

Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2013 Article ID 810508 11 pageshttpdxdoiorg1011552013810508

Research ArticleThermo Elastic-Plastic Analysis of Rotating FunctionallyGraded Stainless Steel Composite Cylinder under Internal andExternal Pressure Using Finite Difference Method

Sanjeev Sharma and Sanehlata Yadav

Department of Mathematics Jaypee Institute of Information Technology A-10 Sector 62 Noida Uttar Pradesh 201307 India

Correspondence should be addressed to Sanjeev Sharma sanjeevsharmajiitacin

Received 22 July 2013 Accepted 26 October 2013

Academic Editor Mohd Sapuan Salit

Copyright copy 2013 S Sharma and S Yadav This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The thermal elastic-plastic stresses have been investigated for a rotating functionally graded stainless steel composite cylinderunder internal and external pressure with general nonlinear strain hardening law and von Misesrsquo yield criterion using finitedifference method The modulus of elasticity in the rotating cylinder varies radially according to power law and the temperaturedistribution satisfies Laplace heat equation in radial direction From the analysis we can conclude that cylindermade of functionallygraded stainless steel composite material with variable thickness and variable density under thermal loading for Swiftrsquos strainhardening measure 119898 = 06 is better choice of the design as compared to homogeneous cylinder This is because of the reasonthat circumferential stress is less for functionally graded stainless steel composite cylinder as compared to homogeneous cylinderfor Swiftrsquos strain hardening measure 119898 = 06 under internal and external pressure This leads to the idea of ldquostress savingrdquo whichminimizes the possibility of fracture of cylinder

1 Introduction

Thermo elastic-plastic analysis of functionally graded mate-rials under internal and external pressure is an activetopic for engineering mechanics The study of thick-walledcylinder is an interesting area of research and highly useddue to vast utilization in the pressure vessels and pipesand so forth Functionally graded materials are nothingbut nonhomogeneous composite materials which are highlyheat resistant and very promising in high-tech engineeringfields [1 2] The demands of functionally graded materialsare increasing under high rotation and temperature Thesematerials find their applications in many areas that isflywheels aerospace nuclear reactors compressors and soforth The problems of rotating thick-walled cylinder andplates can be found in many text books of elasticity [3 4]Obata et al [5] calculated stresses for a hollow circularcylinder and hollow sphere under thermal loading usingperturbation method For these materials they found that

the perturbation solutions had good convergence Perry andAboudi [6] calculated the residual stresses in a homogeneouscylinder using finite difference method and concluded thatit is very effective in solving autofrettage problem numer-ically Gao [7] investigated elastic-plastic stresses strainsand displacements in a cylinder under internal pressureusing elastic strain hardening in the plane strain conditionsTutuncu and Ozturk [8] studied the closed form solutionfor stresses and displacement in cylindrical and sphericalvessels made of functionally graded material subjected tointernal pressure using infinitesimal theory of elasticityTheycompared stress distributions in functionally gradedmaterialwith homogeneous material Singh and Gupta [9] developeda mathematical model to describe steady-state creep in anisotropic functionally graded composite subjected to internalpressure which contains linearly varying silicon carbideparticles in a matrix of pure aluminum They observed thatthe radial stress in the cylinder decreases throughout with theincrease in reinforcement gradient whereas the tangential

2 Advances in Materials Science and Engineering

axial and effective stresses increase significantly near theinner radius but show significant decrease towards the outerradius Aggarwal et al [10] investigated safety factors interms of elastic-plastic stresses for functionally graded thick-walled circular cylinder under internal and external pressureand concluded that functionally graded thick-walled cylinderminimizes the possibility of fracture of the cylinder Aggarwalet al [11] calculated thermal creep stresses for nonhomo-geneous thick-walled cylinder under internal and externalpressure using Lebesgue strain measure and concluded thatnonhomogeneous cylinder is better choice of design ascompared to homogeneous cylinder Parvizi et al [12] studiedamathematicalmodel to predict the yielding in a functionallygraded Al A359SiCp cylinder and find closed form solutionfor plastic stresses using Trescarsquos yield criterion subjected tointernal pressure with thermal loading They observed thatthere is a point in cylinder where the hoop stress changesfrom compressive to tensile and the position of this pointis independent of the temperature gradient and depends onmaterial properties and geometry of the functionally gradedcylinder Eraslan and Akgul [13] calculated the numericalsolution for elastic-plastic stresses in a rotating disk withvon Misesrsquo yield criterion using general nonlinear strainhardening rule

In this paper thermal elastic-plastic stresses have beencalculated for cylinder made up of functionally gradedstainless steel composite material under internal and externalpressure using finite difference method In this problem ageneral nonlinear strain hardening law with von Misesrsquo yieldcriterion has been considered Results have been discussednumerically with the help of graphs and tables

2 Objective

For a rotating cylinder with varying material propertiescircumferential stress at the hub does not exceed the allow-able value which tells the designers little more than that thedesign of the cylinder is safe at the given pressure Thusour prime objective is to calculate allowable thermal elastic-plastic stresses in an open ended functionally graded stainlesssteel composite rotating cylinder under pressure for varyingexpansion to incorporate a ldquosafety factorrdquo that preventsthe cylinder from bursting under pressure and thermalloading

3 Mathematical Formulation

31 Distribution of Material Properties Consider a long openended axisymmetric cylindermade up of functionally gradedstainless steel composite material with inner and outer radii119886 and 119887 respectively and the cylinder is subjected to internalpressure 119901

119894and external pressure 119901

0as shown in Figure 1

The cylindrical polar coordinates (119903 120579) under plane stresscondition have been considered in this problem

In this study Poissonrsquos ratio (]) and thermal expansioncoefficient (120572 = 120572

0) are assumed to be constants The

other properties that is Youngrsquos modulus which is definedby power law temperature distribution that follows Laplace

b r

a

120579p0

pi

Figure 1 A functionally graded rotating cylinder with innerpressure 119901

119894and outer pressure 119901

0at the boundary

heat equation in radial direction thickness and density arevarying radially and are expressed as

119864 (119903) = 1198640(

119903

119887

)

1198901

120579 (119903) = 1205790log(119903

119887

)

120588 (119903) = 1205880(

119903

119887

)

119889

ℎ (119903) = ℎ0(

119903

119887

)

minus119897

(1)

where 1205790= 1205790(log(119886119887)) 119903 is the radius of the cylinder 119864

0

1205880 1205790 and ℎ

0are material constants and 119890

1 119889 and 119897 are the

geometric parameters

32 Basic Equations The equilibrium equation for the cylin-der in the absence of body forces is

119889

119889119903

(ℎ119903119879119903119903) minus ℎ119879

120579120579+ ℎ120588120596

21199032= 0 (2)

where 119879119903119903

and 119879120579120579

are radial and circumferential stressesrespectively

Using infinitesimal theory of elasticity the relationsbetween strains and radial displacements are

119890119903=

119889119906

119889119903

119890120579=

119906

119903

(3)

where 119890119903and 119890

120579are radial and circumferential strains

respectively and 119906 is the radial displacementThe equation of compatibility can be derived from (3) as

119889119890120579

119889119903

+

119890120579minus 119890119903

119903

= 0 (4)

From infinitesimal theory of elasticity the stress-strainrelations are

119890119890

119903=

1

119864

[119879119903119903minus ] (119879

120579120579+ 119879119911119911)] 119890

119890

120579=

1

119864

[119879120579120579minus ] (119879

119903119903+ 119879119911119911)]

119890119890

119911=

1

119864

[119879119911119911minus ] (119879

119903119903+ 119879120579120579)]

(5)

where 119890119890119903 119890119890120579 and 119890119890

119911are the elastic radial circumferential and

axial strains respectively

Advances in Materials Science and Engineering 3

Due to geometric symmetry of the cylinder circumfer-ential displacement shear stresses and strains are assumedto be zero

Using deformation theory of plasticity the relationbetween the stresses and plastic strains can be determined as

119890119901

119903=

119890119901

119890

119879119890119890

[119879119903119903minus

1

2

(119879120579120579+ 119879119911119911)]

119890119901

120579=

119890119901

119890

119879119890119890

[119879120579120579minus

1

2

(119879119903119903+ 119879119911119911)]

119890119901

119911=

119890119901

119890

119879119890119890

[119879119911119911minus

1

2

(119879119903119903+ 119879120579120579)]

(6)

where 119879119890119890is the equivalent stress 119890119901

119890is the equivalent plastic

strain and 119890119901119903 119890119901120579 and 119890119901

119911are the plastic radial circumferential

and axial strains respectivelyvon Misesrsquo yield criterion is given by

119879119890119890= radic(119879

119903119903minus 119879120579120579)2+ (119879120579120579minus 119879119911119911)2+ (119879119911119911minus 119879119903119903)2 (7)

The total radial circumferential and axial strains inthick-walled rotating cylinder are

119890119903= 119890119890

119903+ 119890119901

119903+ 120572120579 119890

120579= 119890119890

120579+ 119890119901

120579+ 120572120579

119890119911= 119890119890

119911+ 119890119901

119911+ 120572120579

(8)

The temperature field satisfying Laplace heat equation is

1198892120579

1198891199032+

1

119903

119889120579

119889119903

= 0 (9)

with 120579 = 1205790at 119903 = 119886 and 120579 = 0 at 119903 = 119887 where 120579

0is a constant

given by 120579(119903) = 1205790log(119903119887)

We define the stress function 120601(119903) for thick-walled rotat-ing cylinder which is related to radial and hoop stresses as

119879119903119903=

120601

ℎ119903

119879120579120579=

1

119889120601

119889119903

+ 12058812059621199032 119879

119911119911= 0 (10)

Since it has been assumed that the cylinder is long andopen ended and there is plane stress condition therefore axialstress is zero that is 119879

119911119911= 0

Substituting (10) and (5) into (8) we have

119890119903=

1

119864

(

120601

ℎ119903

minus ]1

119889120601

119889119903

minus ]12058812059621199032) + 119890119901119903+ 120572120579

119890120579=

1

119864

(

1

119889120601

119889119903

+ 12058812059621199032minus ]

120601

ℎ119903

) + 119890119901

120579+ 120572120579

119890119911=

minus]

119864

(

1

119889120601

119889119903

+ 12058812059621199032+

120601

ℎ119903

) + 119890119901

119911+ 120572120579

(11)

Substituting of (11) into compatibility (4) we have

119903212060110158401015840minus 1199031206011015840[1 + 119903(

ℎ1015840

+

1198641015840

119864

)] minus 120601[1 minus 119903](ℎ1015840

+

1198641015840

119864

)]

+ ℎ120588101584012059621199034+ (3 + ] minus 119903

1198641015840

119864

)ℎ12058812059621199033

minus 119864ℎ119903 (119890119901

119903minus 119890119901

120579) + 119864ℎ119903

2[(119890119901

120579)

1015840

+ 1205721205791015840+ 1205721015840120579] = 0

(12)where 1206011015840 = 119889120601119889119903 12060110158401015840 = 11988921206011198891199032 and 119890119901

120579

1015840

= 119889119890119901

120579119889119903

The relation between the yield stress 119879119890119890and the equiva-

lent plastic strain 119890119901119890for Swiftrsquos hardening law can be expressed

as

119890119890

119890=

119879119890119890

119864

119890119890le 1198900

119890119901

119890=

1

120578

[(

119879119890119890

1198790

)

119898

minus 1] 119890119890gt 1198900

(13)

where 120578 119898 1198790 119890119890 and 119890

0are hardening parameter material

parameter yield limit equivalent total strain and yield strainrespectively

Substituting 119890119901119890from (13) into (6) results in

119890119901

119903=

(1120578) [(1198791198901198901198790)119898minus 1]

119879119890119890

[119879119903119903minus 05 (119879

120579120579+ 119879119911119911)]

119890119901

120579=

(1120578) [(1198791198901198901198790)119898minus 1]

119879119890119890

[119879120579120579minus 05 (119879

119903119903+ 119879119911119911)]

119890119901

119911=

(1120578) [(1198791198901198901198790)119898minus 1]

119879119890119890

[119879119911119911minus 05 (119879

119903119903+ 119879120579120579)]

(14)

Substituting (14) into (12) we have

119903212060110158401015840[1 +

119864

21205781198792

119890119890

2119879119890119890[(

119879119890119890

1198790

)

119898

minus 1] +

1

2119879119890119890

(119879119903119903minus 2119879120579120579)2times (1 + (119898 minus 1) (

119879119890119890

1198790

)

119898

)]

+ 119864ℎ1199032[1205791205721015840+ 1205721205791015840] + ℎ120588120596

21199033(119864 + 4])

minus

119864

21205781198792

119890119890

[

[

[

[

[

[

[

[

[

[

[

119879119890119890times [(

119879119890119890

1198790

)

119898

minus 1] times (1 + 2119903

ℎ1015840

) 1199031206011015840minus (1 + 119903

ℎ1015840

)120601 minus 2ℎ120588101584012059621199034minus 4ℎ120588120596

21199033

+

1

2119879119890119890

(1 + (119898 minus 1) (

119879119890119890

1198790

)

119898

) (119879119903119903minus 2119879120579120579) times

(2119879120579120579minus 119879119903119903)

times [minus1199032 ℎ1015840

1206011015840+ ℎ120588101584012059621199034+ 2ℎ120588120596

21199033]

+ (2119879119903119903minus 119879120579120579) times [119903120601

1015840minus (1 + 119903

ℎ1015840

)120601]

]

]

]

]

]

]

]

]

]

]

]

4 Advances in Materials Science and Engineering

minus

3

2120578119879119890119890

119864ℎ119903 times [(

119879119890119890

1198790

)

119898

minus 1] times (119879119903119903minus 119879120579120579) + 119903120601

1015840[1 minus 119903

ℎ1015840

minus 119903(

1198641015840

119864

)] minus 120601[1 minus ]119903ℎ1015840

minus 119903(

1198641015840

119864

)]

= ℎ12059621199034[120588(

1198641015840

119864

) minus 1205881015840]

(15)

Equation (15) is the differential equation of the function-ally graded stainless steel composite rotating cylinder withnonlinear strain hardening subjected to thermal loading inthe plastic region in terms of stresses and stress function

Equation (15) can be described in the general form interms of stress function as

12060110158401015840= 119891 (119903 120601 120601

1015840) (16)

Equation (16) is a nonlinear two point boundary valueproblem and can be solved numerically subjected to theboundary conditions

119879119903119903= minus119901119894 at 119903 = 119886 119879

119903119903= minus1199010 at 119903 = 119887 119886 gt 0

(17)

where 119886 and 119887 are the inner and outer radii of the cylinder and119901119894and 119901

0are internal and external pressures respectively

Using finite difference method with central difference in(16) we get the following system of equations

120601119894+1minus 2120601119894+ 120601119894minus1

(Δ119903)2

= 119891(119903 120601119894

120601119894+1minus 120601119894minus1

2Δ119903

) 119894 = 2 3 119899

(18)

Equation (18) consists of algebraic system of (119899minus 1) equa-tions with the boundary conditions 120601(119886) = minus119901

119894ℎ119886 and 120601(119887) =

minus1199010ℎ119887 After solving (18) with boundary conditions we get a

stress function120601Then the radial and circumferential stressescan be obtained from (10) after substituting the value of stressfunction 120601

4 Numerical Discussion

The properties of a functionally graded stainless steel com-posite thick-walled rotating cylinder under internal andexternal pressure 119901

119894= 150 300 and 119901

0= 150 300MPa

respectively subjected to thermal loading (1205790= 0 400 800)

are defined as follows the radii of the cylinder are taken as119886 = 01m and 119887 = 05m Poissonrsquos ratio ] = 03 Youngrsquosmodulus 119864

0= 207GPa and thermal expansion coefficient

120572 = 1205720= 178 times 10

minus6∘

Cminus1 The geometric parameters of thecylinder are taken as 119890

1= 0 1 2 in Youngrsquos modulus function

and119898 = 04 06 is nonlinear strain hardening measureTo show the effect of internal and external pressure

on a functionally graded stainless steel composite rotatingcylinder with strain hardening measure 119898 = 04 06 havingconstant thickness and constant density Tables 1 and 2show the circumferential stresses with different parametersof Youngrsquos modulus 119890

1= 0 1 2

It has been observed from Table 1 that when externalpressure is greater than the internal pressure circumferen-tial stresses approaches tensile to compressible Also these

stresses aremaximum at external surface for homogeneous aswell as functionally graded stainless steel composite cylinderThese stresses are less for functionally graded stainless steelcomposite cylinder as compared to homogeneous cylinderAs nonhomogeneity changes from 119890

1= 1 to 119890

1= 2

circumferential stresses decrease significantlyWith the intro-duction of thermal effects circumferential stresses increasefor homogeneous as well as for functionally graded stainlesssteel composite cylinder (119890

1= 1) but decrease for functionally

graded stainless steel composite cylinder with 1198901= 2 It

has also been noticed from Table 1 that with the increasein thermal effects these stresses decrease significantly forhomogeneous as well as for functionally graded stainlesssteel composite cylinder and are less for the cylinder withnonhomogeneity parameter 119890

1= 2 With the increase in

angular speed circumferential stresses increase significantlyWhen external pressure is less than the internal pressurecircumferential stresses are maximum at internal surface forhomogeneous cylinder while maximum at external surfacefor functionally graded stainless steel composite cylinder andthese stresses decreased with the change in nonhomogeneitymeasure from 119890

1= 1 to 119890

1= 2 With the introduc-

tion of thermal effects circumferential stresses increase forhomogenous as well as for functionally graded stainless steelcomposite measure 119890

1= 1 while decrease for functionally

graded stainless steel composite cylinder with 1198901= 2 With

the increase in thermal effects these circumferential stressesdecrease significantly while increase with the increase inangular speed It has been observed from Table 2 that thebehavior of homogeneous and functionally graded stainlesssteel composite cylinder same as discussed in Table 1 but ithas been observed that with increase in strain hardeningmeasure from 119898 = 04 to 119898 = 06 these stresses decreasesignificantly for functionally graded stainless steel compositecylinder

Tables 3 and 4 have beenmade for circumferential stressesin rotating cylinders with variable thickness and variabledensity with different parameters of Youngrsquos modulus 119890

1=

0 1 2 and strain hardening measure119898 = 04 06It has been observed from Table 3 that for cylinder with

varying thickness and density whose external pressure isgreater than the internal pressure circumferential stressesapproach from tensile to compressible and are maximumat external surface for homogeneous as well as functionallygraded stainless steel composite cylinder These stressesare less for functionally graded stainless steel compositecylinder as compared to homogeneous cylinder with varyingthickness and density as well as cylinder with constant thick-ness and density With increase in strain hardening measurefrom 119898 = 04 to 119898 = 06 these circumferential stressesdecrease significantly for functionally graded stainless steel

Advances in Materials Science and Engineering 5

Table 1 Circumferential stresses for rotating cylinder with constant thickness and constant density 120596 = 300 500 nonlinear strain hardeningmeasure119898 = 04 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 30001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 83454530141 34064915847 00677187483 minus39596784280 minus896024571031 33799641785 27171534961 08969909345 minus26654005915 minus855489788262 11503024505 18118357165 12081310260 minus15371690373 minus78899736228

119901119894= 300

1199010= 150

1205790= 0

0 88188655176 36471590677 02649872301 minus37776104911 minus878521510291 36049642996 29421535689 11219909709 minus24404004448 minus832989835602 11869707579 19774764390 14492915544 minus12424185539 minus75536537230

119901119894= 150

1199010= 300

1205790= 400

0 83454529134 34064914474 00677186071 minus39596781512 minus896024633301 33799641163 27171535146 08969910927 minus26654006381 minus855489857932 11503024507 18118357152 12081310262 minus15371690375 minus78899736082

119901119894= 300

1199010= 150

1205790= 400

0 88188661275 36471591256 02649873453 minus37776106334 minus878521459301 36049644528 29421536040 11219910130 minus24404005339 minus832989848932 11869707578 19774764389 14492915543 minus12424185538 minus75536537227

119901119894= 150

1199010= 300

1205790= 800

0 83454532713 34064916133 00677188786 minus39596783083 minus896024607391 03799644417 27171537078 08969909208 minus26654005810 minus855489848852 11503024510 18118357165 12081310255 minus15371690384 minus78899735899

119901119894= 300

1199010= 150

1205790= 800

0 88188658525 36471597197 02649871677 minus37776107003 minus878521600471 36049639860 29421535165 11219910039 minus24404004285 minus832989834522 11869707577 19774764387 14492915543 minus12424185537 minus75536537224

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 50001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 11199643681 04548287897 00269123600 minus04863252783 minus121916263931 06272813080 05095768873 01910926579 minus04339086008 minus153519940132 03263905498 05038039436 03586769803 minus03274307737 minus19402396397

119901119894= 300

1199010= 150

1205790= 0

0 11512903744 04761073745 00479630332 minus04638575699 minus119486110581 06497506092 05320656463 02136358300 minus04115308637 minus151265455362 03354452568 05269344433 03869603568 minus03022694774 minus19278605585

119901119894= 150

1199010= 300

1205790= 400

0 11199644787 04548287386 00269122948 minus04863252189 minus121916251551 06272812978 05095769102 01910926887 minus04339085404 minus153519934162 03263905755 05038038300 03586769873 minus03274306978 minus19402449592

119901119894= 300

1199010= 150

1205790= 400

0 11512904788 04761073218 00479628568 minus04638573696 minus119486136701 06497506075 05320656332 02136358153 minus04115307488 minus151265453652 03354452679 05269344696 03869603369 minus03022695403 minus19278600529

119901119894= 150

1199010= 300

1205790= 800

0 11199642715 04548288045 00269123491 minus04863251255 minus121916247691 11199644787 04548287386 00269122948 minus04863252189 minus121916251552 03263905007 05038038190 03586771087 minus03274306558 minus19402425453

119901119894= 300

1199010= 150

1205790= 800

0 11512903765 04761073125 00479629909 minus04638574341 minus119486128581 06497505961 05320656017 02136358897 minus04115308258 minus151265454652 03354452477 05269344295 03869603736 minus03022695113 minus19278604854

composite cylinder with varying thickness and density as canbe seen from Table 4

Figures 2ndash4 have been drawn to discuss the effect ofinternal and external pressure on stresses in rotating cylin-der made of functionally graded stainless steel composite

material with constant thickness and constant density withnonlinear strain hardening measure

It has been observed from Figure 2 that circumferentialstress approaches towards compressive from tensile It hasalso been observed thatwhen external pressure is greater than

6 Advances in Materials Science and Engineering

Table 2 Circumferential stresses for rotating cylinder with constant thickness and constant density with 120596 = 300 500 nonlinear strainhardening measure119898 = 06 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa

119903

1198901

119898 = 06 120596 = 30001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 83453827289 34064900899 00677286065 minus39596681544 minus896024143091 33799456580 27171435351 08969901053 minus26653896254 minus855488042282 11502969495 18118279175 12081251826 minus15371621151 minus78899506975

119901119894= 300

1199010= 150

1205790= 0

0 88187936296 36471591317 02649975386 minus37776011860 minus878521216671 36049445586 29421431766 11219899984 minus24403890591 minus832987989572 11869649855 19774683014 14492853591 minus12424112423 minus75536293888

119901119894= 150

1199010= 300

1205790= 400

0 83453833202 34064903276 00677284767 minus39596684114 minus896024179131 33799454509 27171434189 08969901711 minus26653895895 minus855488050342 11502969501 18118279178 12081251821 minus15371621158 minus78899507229

119901119894= 300

1199010= 150

1205790= 400

0 88187922056 36471584895 02649978782 minus37776012451 minus878521236811 36049444470 29421430233 11219900447 minus24403890386 minus832988022922 11869649854 19774683013 14492853591 minus12424112422 minus75536293885

119901119894= 150

1199010= 300

1205790= 800

0 83453820024 34064896339 00677286501 minus39596681965 minus896024157281 33799459533 27171437210 08969899701 minus26653898309 minus855488077952 11502969490 18118279176 12081251818 minus15371621155 minus78899507012

119901119894= 300

1199010= 150

1205790= 800

0 88187929120 36471586248 02649979578 minus37776012681 minus878521057051 36049446223 29421429798 11219900968 minus24403889573 minus832987897072 11869649853 19774683011 14492853590 minus12424112421 minus75536293882

1119864minus4lowast 119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 50001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 11199583152 04548258695 00269146492 minus04863265526 minus121916543001 06272799179 05095770638 01910931884 minus04339078950 minus153520394292 03263903739 05038038750 03586772209 minus03274306376 minus19402344736

119901119894= 300

1199010= 150

1205790= 0

0 11512567236 04761058146 00480074980 minus04638915755 minus119483525601 06496608008 05320325649 02137603620 minus04118738110 minus151255087832 03354423760 05269311856 03869645881 minus03022635649 minus19278571554

119901119894= 150

1199010= 300

1205790= 400

0 11199579631 04548258318 00269147731 minus04863261541 minus121916558021 06272799298 05095770743 01910931873 minus04339077941 minus153520404932 03263904649 05038039194 03586771853 minus03274308913 minus19402307798

119901119894= 300

1199010= 150

1205790= 400

0 11512569402 04761057447 00480074391 minus04638917626 minus119483523931 06496607027 05320325183 02137605230 minus04118740006 minus151255088322 03354423676 05269312306 03869645451 minus03022636353 minus19278567951

119901119894= 150

1199010= 300

1205790= 800

0 11199579102 04548258849 00269148341 minus04863259137 minus121916561241 06272799292 05095770557 01910931905 minus04339078242 minus153520389162 03263903269 05038037303 03586773178 minus03274301493 minus19402403280

119901119894= 300

1199010= 150

1205790= 800

0 11512563866 04761057469 00480078293 minus04638922712 minus119483505711 06496607065 05320325064 02137604900 minus04118740845 minus151255078652 03354423734 05269312334 03869645540 minus03022636247 minus19278571251

the internal pressure these stresses are maximum at externalsurface for homogeneous as well as functionally gradedstainless steel composite cylinder Also it has been observedthat circumferential stress is maximum at internal surface forhomogeneous cylinder while maximum at external surface

for functionally graded stainless steel composite cylinderwhen external pressure is less than the internal pressure Alsowith the increase in angular speed circumferential stressesincrease significantly From Figure 3 it can be seen that ascircumferential stresses increase for homogeneous cylinder

Advances in Materials Science and Engineering 7

Table 3 Circumferential stresses for rotating cylinder with variable thickness and variable density 119897 = 07 119889 = 05 120596 = 300 500 strainhardening measure119898 = 04 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 300 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 4078610121 1692083635 minus0362927882 minus3599668076 minus83881223761 1844023846 1536347566 0332616455 minus2596999032 minus81860251262 0698588556 1166718627 0723851443 minus1632455674 minus7703313292

119901119894= 300

1199010= 150

1205790= 0

0 4468498659 1869943306 minus0216047724 minus3455691317 minus82392598161 2039946535 1721858320 0520942472 minus2403128817 minus79859401892 0725452105 1314126561 0957769537 minus1332711442 minus7351503187

119901119894= 150

1199010= 300

1205790= 400

0 4078610926 1692083796 minus0362928200 minus3599667939 minus83881241561 1844023759 1536347360 0332616655 minus2596998868 minus81860264052 0698588485 1166718653 0723851463 minus1632455697 minus7703312785

119901119894= 300

1199010= 150

1205790= 400

0 4468498329 1869943517 minus0216047300 minus3455691093 minus82392517301 2039946605 1721858238 0520942423 minus2403128796 minus79859403832 0725452128 1314126566 0957769488 minus1332711458 minus7351503426

119901119894= 150

1199010= 300

1205790= 800

0 4078610278 1692083456 minus0362927548 minus3599667594 minus83881250271 1844023766 1536347565 0332616556 minus2596998868 minus81860262912 0698588473 1166718654 0723851445 minus1632455676 minus7703312834

119901119894= 300

1199010= 150

1205790= 800

0 4468498296 1869943321 minus0216047655 minus3455691489 minus82392377231 2039946200 1721858288 0520942491 minus2403128819 minus79859400642 0725452082 1314126521 0957769539 minus1332711402 minus7351503219

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 500 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 0502344054 0209118575 minus0032085608 minus0411840852 minus10757559141 0301690685 0253918828 0067872877 minus0387759274 minus13375609912 0170338387 0272622555 0177093634 minus0328795174 minus1672186161

119901119894= 300

1199010= 150

1205790= 0

0 0520602572 0223194821 minus0013327208 minus0385714307 minus10416636041 0314858338 0269151250 0087139893 minus0362225697 minus13033628312 0175533196 0289517104 0201213547 minus0301959686 minus1648231058

119901119894= 150

1199010= 300

1205790= 400

0 0502344076 0209118532 minus0032085631 minus0411840853 minus10757560931 0301690546 0253918731 0067873081 minus0387757210 minus13375620512 0170338398 0272622541 0177093558 minus0328795106 minus1672185933

119901119894= 300

1199010= 150

1205790= 400

0 0520602619 0223194813 minus0013327161 minus0385714149 minus10416645331 0314858326 0269151199 0087139890 minus0362225687 minus13033627732 0175533191 0289517093 0201213591 minus0301959717 minus1648231090

119901119894= 150

1199010= 300

1205790= 800

0 0502344041 0209118525 minus0032085593 minus0411841019 minus10757557561 0301690548 0253918749 0067873075 minus0387757974 minus13375601512 0170338413 0272622528 0177093598 minus0328795120 minus1672185805

119901119894= 300

1199010= 150

1205790= 800

0 0520602532 0223194802 minus0013327191 minus0385714162 minus10416644831 0314858370 0269151215 0087139921 minus0362225740 minus13033627232 0175533168 0289517159 0201213488 minus0301959669 minus1648230944

as well as for nonhomogenous cylinder but with the changein nonhomogeneity from 119890

1= 1 to 119890

1= 2 circumferential

stresses decrease when external pressure is greater than theinternal pressure with thermal effects Also these stresses

increase for homogenous cylinder as well as for functionallygraded stainless steel composite cylinder (119890

1= 1) while

decrease for functionally graded stainless steel compositecylinder with 119890

1= 2 With the increase in temperature these

8 Advances in Materials Science and Engineering

Table 4 Circumferential stresses for rotating cylinder with variable thickness and variable density 119897 = 07 119889 = 05 120596 = 300 500 strainhardening measure119898 = 06 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 300 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 4078550787 1692092827 minus0362917634 minus3599662505 minus83881243031 1844008854 1536344037 0332619755 minus2596989139 minus81860118602 0698583918 1166713498 0723849363 minus1632447308 minus7703291360

119901119894= 300

1199010= 150

1205790= 0

0 4468438868 1869957168 minus0216034432 minus3455686049 minus82394923451 2039930194 1721854313 0520946730 minus2403118071 minus79859271952 0725447053 1314120776 0957767141 minus1332701862 minus7351479020

119901119894= 150

1199010= 300

1205790= 400

0 4078550758 1692093003 minus0362917541 minus3599662684 minus83881247581 1844009034 1536344170 0332619845 minus2596989055 minus81860133232 0698583989 1166713507 0723849336 minus1632447294 minus7703291352

119901119894= 300

1199010= 150

1205790= 400

0 4468440841 1869957567 minus0216033626 minus3455685275 minus82395880401 2039930222 1721854440 0520946513 minus2403117933 minus79859260012 0725447107 1314120882 0957767050 minus1332701999 minus7351479230

119901119894= 150

1199010= 300

1205790= 800

0 4078551089 1692092847 minus0362917694 minus3599662435 minus83881264791 1844008698 1536343957 0332619859 minus2596988916 minus81860122392 0698583989 1166713524 0723849328 minus1632447317 minus7703291542

119901119894= 300

1199010= 150

1205790= 800

0 4468438120 1869956228 minus0216035341 minus3455687051 minus82393686321 2039930039 1721854569 0520946538 minus2403117966 minus79859257312 0725447049 1314120768 0957767137 minus1332701833 minus7351479029

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 500 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 0502339190 0209119023 minus0032080434 minus0411843003 minus10757515751 0301689441 0253919137 0067872897 minus0387758158 minus13375674432 0170338021 0272622433 0177094183 minus0328793821 minus1672191050

119901119894= 300

1199010= 150

1205790= 0

0 0520600260 0223196743 minus0013328691 minus0385729358 minus10416359501 0314857828 0269151955 0087140006 minus0362234515 minus13033506742 0175532879 0289517002 0201213980 minus0301960040 minus1648233514

119901119894= 150

1199010= 300

1205790= 400

0 0502339173 0209119009 minus0032080335 minus0411843048 minus10757520131 0301689029 0253918998 0067873493 minus0387754850 minus13375668252 0170338061 0272622405 0177094107 minus0328793799 minus1672190560

119901119894= 300

1199010= 150

1205790= 400

0 0520600268 0223196732 minus0013328653 minus0385729330 minus10416363341 0314857844 0269151936 0087139999 minus0362234465 minus13033507222 0175532856 0289516980 0201214076 minus0301960031 minus1648233768

119901119894= 150

1199010= 300

1205790= 800

0 0502339102 0209118939 minus0032080279 minus0411842835 minus10757521591 0301689252 0253919092 0067873202 minus0387759066 minus13375656632 0170338011 0272622432 0177094145 minus0328793808 minus1672190610

119901119894= 300

1199010= 150

1205790= 800

0 0520600345 0223196689 minus0013328705 minus0385729338 minus10416358761 0314857846 0269151949 0087139963 minus0362234522 minus13033507592 0175532843 0289516995 0201214043 minus0301960031 minus1648233661

stresses decrease significantly for homogeneous as well as forfunctionally graded stainless steel composite cylinder as canbe seen from Figure 4

Figures 5ndash7 have been drawn to discuss the effect ofinternal and external pressure on stresses in rotating cylin-der made of functionally graded stainless steel composite

Advances in Materials Science and Engineering 9

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 150 and p0 = 300

120596 = 300 e1 = 0

120596 = 300 e1 = 2

120596 = 500 e1 = 1

120596 = 300 e1 = 1

120596 = 500 e1 = 0

120596 = 500 e1 = 2

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 300 and p0 = 150

120596 = 300 e1 = 0

120596 = 300 e1 = 2

120596 = 500 e1 = 1

120596 = 300 e1 = 1

120596 = 500 e1 = 0

120596 = 500 e1 = 2

(b)

Figure 2 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density without thermal effects (1205790= 0) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

pi = 150 and p0 = 3001205721 = 0 1205790 = 400 m = 04

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 300 and p0 = 150

(b)

Figure 3 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density with thermal effects (1205790= 400) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 300 and p0 = 150

(b)

Figure 4 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density with thermal effects (1205790= 800) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

10 Advances in Materials Science and Engineering

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 300 and p0 = 150

(b)

Figure 5 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density without thermal effects (1205790= 0) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 300 and p0 = 150

(b)

Figure 6 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 400) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 300 and p0 = 150

(b)

Figure 7 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 800) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

Advances in Materials Science and Engineering 11

material with variable thickness and variable density withnonlinear strain hardening measure

It has been observed from Figure 5 that circumferentialstresses are maximum at external surface for homogeneousas well as functionally graded stainless steel compositecylinder It has also been observed that with the increase inangular speed circumferential stresses increase significantlyWith the introduction of thermal effects circumferentialstresses increase for homogeneous cylinder as well as forfunctionally graded stainless steel composite cylinder (119890

1=

1) but with the change in nonhomogeneity from 1198901= 1

to 1198901= 2 circumferential stresses decrease when external

pressure is greater than the internal pressure as can beseen from Figure 6 Circumferential stresses decrease forhomogeneous cylinder while increase for functionally gradedstainless steel composite cylinder when external pressure isless than the internal pressure It has also been observedfrom Figure 7 that with the increase in temperature thesestresses increase significantly for homogeneous cylinder aswell for functionally graded stainless steel composite cylinderwith 119890

1= 1 while decrease for functionally graded stainless

steel composite cylinder with 1198901= 2 when external pressure

is greater than the internal pressure while these stressesdecrease significantly for homogeneous cylinder as well forfunctionally graded stainless steel composite cylinder whenexternal pressure is less than the internal pressure

5 Conclusion

From the analysis we can conclude that rotating cylin-der made of functionally graded stainless steel compositematerial having variable thickness and variable density withSwiftrsquos strain hardening measure 119898 = 06 and thermalloading is better choice for designers as compared to rotatingcylinder with constant thickness and constant density Thisis because of the reason that circumferential stress is less forfunctionally graded stainless steel composite cylinder withvariable thickness and variable density as compared to othercases which leads to the idea of stress saving that minimizesthe possibility of fracture of cylinder

Conflict of Interests

The authors declare that they have no conflict of interest

References

[1] S Suresh and A Mortensen Fundamentals of FunctionallyGradedMaterials IOM3 Maney Publishing London UK 1998

[2] J N Reddy ldquoAnalysis of functionally graded platesrdquo Interna-tional Journal for Numerical Methods in Engineering vol 47 no1ndash3 pp 663ndash684 2000

[3] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill NewYork NY USA 3rd edition 1970

[4] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998

[5] Y Obata T Oji and N Noda ldquoSteady thermal stresses in a hol-low circular cylinder and a hollow sphere made of functionallygraded material (analysis with perturbation method)rdquo Reports

of the National Industrial Research Institute of Nagoya vol 47no 4 pp 317ndash338 1998

[6] J Perry and J Aboudi ldquoElasto-plastic stresses in thick walledcylindersrdquo Journal of Pressure Vessel Technology vol 125 no 3pp 248ndash252 2003

[7] X-L Gao ldquoElasto-plastic analysis of an internally pressurizedthick-walled cylinder using a strain gradient plasticity theoryrdquoInternational Journal of Solids and Structures vol 40 no 23 pp6445ndash6455 2003

[8] N Tutuncu and M Ozturk ldquoExact solutions for stresses infunctionally graded pressure vesselsrdquo Composites B vol 32 no8 pp 683ndash686 2001

[9] T Singh and V K Gupta ldquoCreep analysis of an internallypressurized thick cylinder made of a functionally graded com-positerdquo Journal of Strain Analysis for Engineering Design vol 44no 7 pp 583ndash594 2009

[10] A K Aggarwal Sharma Richa and Sharma Sanjeev ldquoSafetyanalysis using lebesgue strain measure of thick-walled cylinderfor functionally graded material under internal and externalpressurerdquo The Scientific World Journal vol 2013 Article ID676190 10 pages 2013

[11] A K Aggarwal S Richa and S Sanjeev ldquoSafety analysis ofthermal creep non-homogeneous thick-walled circular cylinderunder internal and external pressure using lebesgue strainmeasurerdquoMultidiscipline Modeling in Materials and Structuresvol 9 no 4 2013

[12] A Parvizi R Naghdabadi and J Arghavani ldquoAnalysis of AlA359SiCp functionally graded cylinder subjected to internalpressure and temperature gradient with elastic-plastic deforma-tionrdquo Journal of Thermal Stresses vol 34 no 10 pp 1054ndash10702011

[13] A N Eraslan and F Akgul ldquoYielding and elastoplastic deforma-tion of annular disks of a parabolic section subject to externalcompressionrdquo Turkish Journal of Engineering and Environmen-tal Sciences vol 29 no 1 pp 51ndash60 2005

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Journal ofNanomaterials

Page 2: Research Article Thermo Elastic-Plastic Analysis of Rotating …downloads.hindawi.com/journals/amse/2013/810508.pdf · 2019-07-31 · ermo elastic-plastic analysis of functionally

2 Advances in Materials Science and Engineering

axial and effective stresses increase significantly near theinner radius but show significant decrease towards the outerradius Aggarwal et al [10] investigated safety factors interms of elastic-plastic stresses for functionally graded thick-walled circular cylinder under internal and external pressureand concluded that functionally graded thick-walled cylinderminimizes the possibility of fracture of the cylinder Aggarwalet al [11] calculated thermal creep stresses for nonhomo-geneous thick-walled cylinder under internal and externalpressure using Lebesgue strain measure and concluded thatnonhomogeneous cylinder is better choice of design ascompared to homogeneous cylinder Parvizi et al [12] studiedamathematicalmodel to predict the yielding in a functionallygraded Al A359SiCp cylinder and find closed form solutionfor plastic stresses using Trescarsquos yield criterion subjected tointernal pressure with thermal loading They observed thatthere is a point in cylinder where the hoop stress changesfrom compressive to tensile and the position of this pointis independent of the temperature gradient and depends onmaterial properties and geometry of the functionally gradedcylinder Eraslan and Akgul [13] calculated the numericalsolution for elastic-plastic stresses in a rotating disk withvon Misesrsquo yield criterion using general nonlinear strainhardening rule

In this paper thermal elastic-plastic stresses have beencalculated for cylinder made up of functionally gradedstainless steel composite material under internal and externalpressure using finite difference method In this problem ageneral nonlinear strain hardening law with von Misesrsquo yieldcriterion has been considered Results have been discussednumerically with the help of graphs and tables

2 Objective

For a rotating cylinder with varying material propertiescircumferential stress at the hub does not exceed the allow-able value which tells the designers little more than that thedesign of the cylinder is safe at the given pressure Thusour prime objective is to calculate allowable thermal elastic-plastic stresses in an open ended functionally graded stainlesssteel composite rotating cylinder under pressure for varyingexpansion to incorporate a ldquosafety factorrdquo that preventsthe cylinder from bursting under pressure and thermalloading

3 Mathematical Formulation

31 Distribution of Material Properties Consider a long openended axisymmetric cylindermade up of functionally gradedstainless steel composite material with inner and outer radii119886 and 119887 respectively and the cylinder is subjected to internalpressure 119901

119894and external pressure 119901

0as shown in Figure 1

The cylindrical polar coordinates (119903 120579) under plane stresscondition have been considered in this problem

In this study Poissonrsquos ratio (]) and thermal expansioncoefficient (120572 = 120572

0) are assumed to be constants The

other properties that is Youngrsquos modulus which is definedby power law temperature distribution that follows Laplace

b r

a

120579p0

pi

Figure 1 A functionally graded rotating cylinder with innerpressure 119901

119894and outer pressure 119901

0at the boundary

heat equation in radial direction thickness and density arevarying radially and are expressed as

119864 (119903) = 1198640(

119903

119887

)

1198901

120579 (119903) = 1205790log(119903

119887

)

120588 (119903) = 1205880(

119903

119887

)

119889

ℎ (119903) = ℎ0(

119903

119887

)

minus119897

(1)

where 1205790= 1205790(log(119886119887)) 119903 is the radius of the cylinder 119864

0

1205880 1205790 and ℎ

0are material constants and 119890

1 119889 and 119897 are the

geometric parameters

32 Basic Equations The equilibrium equation for the cylin-der in the absence of body forces is

119889

119889119903

(ℎ119903119879119903119903) minus ℎ119879

120579120579+ ℎ120588120596

21199032= 0 (2)

where 119879119903119903

and 119879120579120579

are radial and circumferential stressesrespectively

Using infinitesimal theory of elasticity the relationsbetween strains and radial displacements are

119890119903=

119889119906

119889119903

119890120579=

119906

119903

(3)

where 119890119903and 119890

120579are radial and circumferential strains

respectively and 119906 is the radial displacementThe equation of compatibility can be derived from (3) as

119889119890120579

119889119903

+

119890120579minus 119890119903

119903

= 0 (4)

From infinitesimal theory of elasticity the stress-strainrelations are

119890119890

119903=

1

119864

[119879119903119903minus ] (119879

120579120579+ 119879119911119911)] 119890

119890

120579=

1

119864

[119879120579120579minus ] (119879

119903119903+ 119879119911119911)]

119890119890

119911=

1

119864

[119879119911119911minus ] (119879

119903119903+ 119879120579120579)]

(5)

where 119890119890119903 119890119890120579 and 119890119890

119911are the elastic radial circumferential and

axial strains respectively

Advances in Materials Science and Engineering 3

Due to geometric symmetry of the cylinder circumfer-ential displacement shear stresses and strains are assumedto be zero

Using deformation theory of plasticity the relationbetween the stresses and plastic strains can be determined as

119890119901

119903=

119890119901

119890

119879119890119890

[119879119903119903minus

1

2

(119879120579120579+ 119879119911119911)]

119890119901

120579=

119890119901

119890

119879119890119890

[119879120579120579minus

1

2

(119879119903119903+ 119879119911119911)]

119890119901

119911=

119890119901

119890

119879119890119890

[119879119911119911minus

1

2

(119879119903119903+ 119879120579120579)]

(6)

where 119879119890119890is the equivalent stress 119890119901

119890is the equivalent plastic

strain and 119890119901119903 119890119901120579 and 119890119901

119911are the plastic radial circumferential

and axial strains respectivelyvon Misesrsquo yield criterion is given by

119879119890119890= radic(119879

119903119903minus 119879120579120579)2+ (119879120579120579minus 119879119911119911)2+ (119879119911119911minus 119879119903119903)2 (7)

The total radial circumferential and axial strains inthick-walled rotating cylinder are

119890119903= 119890119890

119903+ 119890119901

119903+ 120572120579 119890

120579= 119890119890

120579+ 119890119901

120579+ 120572120579

119890119911= 119890119890

119911+ 119890119901

119911+ 120572120579

(8)

The temperature field satisfying Laplace heat equation is

1198892120579

1198891199032+

1

119903

119889120579

119889119903

= 0 (9)

with 120579 = 1205790at 119903 = 119886 and 120579 = 0 at 119903 = 119887 where 120579

0is a constant

given by 120579(119903) = 1205790log(119903119887)

We define the stress function 120601(119903) for thick-walled rotat-ing cylinder which is related to radial and hoop stresses as

119879119903119903=

120601

ℎ119903

119879120579120579=

1

119889120601

119889119903

+ 12058812059621199032 119879

119911119911= 0 (10)

Since it has been assumed that the cylinder is long andopen ended and there is plane stress condition therefore axialstress is zero that is 119879

119911119911= 0

Substituting (10) and (5) into (8) we have

119890119903=

1

119864

(

120601

ℎ119903

minus ]1

119889120601

119889119903

minus ]12058812059621199032) + 119890119901119903+ 120572120579

119890120579=

1

119864

(

1

119889120601

119889119903

+ 12058812059621199032minus ]

120601

ℎ119903

) + 119890119901

120579+ 120572120579

119890119911=

minus]

119864

(

1

119889120601

119889119903

+ 12058812059621199032+

120601

ℎ119903

) + 119890119901

119911+ 120572120579

(11)

Substituting of (11) into compatibility (4) we have

119903212060110158401015840minus 1199031206011015840[1 + 119903(

ℎ1015840

+

1198641015840

119864

)] minus 120601[1 minus 119903](ℎ1015840

+

1198641015840

119864

)]

+ ℎ120588101584012059621199034+ (3 + ] minus 119903

1198641015840

119864

)ℎ12058812059621199033

minus 119864ℎ119903 (119890119901

119903minus 119890119901

120579) + 119864ℎ119903

2[(119890119901

120579)

1015840

+ 1205721205791015840+ 1205721015840120579] = 0

(12)where 1206011015840 = 119889120601119889119903 12060110158401015840 = 11988921206011198891199032 and 119890119901

120579

1015840

= 119889119890119901

120579119889119903

The relation between the yield stress 119879119890119890and the equiva-

lent plastic strain 119890119901119890for Swiftrsquos hardening law can be expressed

as

119890119890

119890=

119879119890119890

119864

119890119890le 1198900

119890119901

119890=

1

120578

[(

119879119890119890

1198790

)

119898

minus 1] 119890119890gt 1198900

(13)

where 120578 119898 1198790 119890119890 and 119890

0are hardening parameter material

parameter yield limit equivalent total strain and yield strainrespectively

Substituting 119890119901119890from (13) into (6) results in

119890119901

119903=

(1120578) [(1198791198901198901198790)119898minus 1]

119879119890119890

[119879119903119903minus 05 (119879

120579120579+ 119879119911119911)]

119890119901

120579=

(1120578) [(1198791198901198901198790)119898minus 1]

119879119890119890

[119879120579120579minus 05 (119879

119903119903+ 119879119911119911)]

119890119901

119911=

(1120578) [(1198791198901198901198790)119898minus 1]

119879119890119890

[119879119911119911minus 05 (119879

119903119903+ 119879120579120579)]

(14)

Substituting (14) into (12) we have

119903212060110158401015840[1 +

119864

21205781198792

119890119890

2119879119890119890[(

119879119890119890

1198790

)

119898

minus 1] +

1

2119879119890119890

(119879119903119903minus 2119879120579120579)2times (1 + (119898 minus 1) (

119879119890119890

1198790

)

119898

)]

+ 119864ℎ1199032[1205791205721015840+ 1205721205791015840] + ℎ120588120596

21199033(119864 + 4])

minus

119864

21205781198792

119890119890

[

[

[

[

[

[

[

[

[

[

[

119879119890119890times [(

119879119890119890

1198790

)

119898

minus 1] times (1 + 2119903

ℎ1015840

) 1199031206011015840minus (1 + 119903

ℎ1015840

)120601 minus 2ℎ120588101584012059621199034minus 4ℎ120588120596

21199033

+

1

2119879119890119890

(1 + (119898 minus 1) (

119879119890119890

1198790

)

119898

) (119879119903119903minus 2119879120579120579) times

(2119879120579120579minus 119879119903119903)

times [minus1199032 ℎ1015840

1206011015840+ ℎ120588101584012059621199034+ 2ℎ120588120596

21199033]

+ (2119879119903119903minus 119879120579120579) times [119903120601

1015840minus (1 + 119903

ℎ1015840

)120601]

]

]

]

]

]

]

]

]

]

]

]

4 Advances in Materials Science and Engineering

minus

3

2120578119879119890119890

119864ℎ119903 times [(

119879119890119890

1198790

)

119898

minus 1] times (119879119903119903minus 119879120579120579) + 119903120601

1015840[1 minus 119903

ℎ1015840

minus 119903(

1198641015840

119864

)] minus 120601[1 minus ]119903ℎ1015840

minus 119903(

1198641015840

119864

)]

= ℎ12059621199034[120588(

1198641015840

119864

) minus 1205881015840]

(15)

Equation (15) is the differential equation of the function-ally graded stainless steel composite rotating cylinder withnonlinear strain hardening subjected to thermal loading inthe plastic region in terms of stresses and stress function

Equation (15) can be described in the general form interms of stress function as

12060110158401015840= 119891 (119903 120601 120601

1015840) (16)

Equation (16) is a nonlinear two point boundary valueproblem and can be solved numerically subjected to theboundary conditions

119879119903119903= minus119901119894 at 119903 = 119886 119879

119903119903= minus1199010 at 119903 = 119887 119886 gt 0

(17)

where 119886 and 119887 are the inner and outer radii of the cylinder and119901119894and 119901

0are internal and external pressures respectively

Using finite difference method with central difference in(16) we get the following system of equations

120601119894+1minus 2120601119894+ 120601119894minus1

(Δ119903)2

= 119891(119903 120601119894

120601119894+1minus 120601119894minus1

2Δ119903

) 119894 = 2 3 119899

(18)

Equation (18) consists of algebraic system of (119899minus 1) equa-tions with the boundary conditions 120601(119886) = minus119901

119894ℎ119886 and 120601(119887) =

minus1199010ℎ119887 After solving (18) with boundary conditions we get a

stress function120601Then the radial and circumferential stressescan be obtained from (10) after substituting the value of stressfunction 120601

4 Numerical Discussion

The properties of a functionally graded stainless steel com-posite thick-walled rotating cylinder under internal andexternal pressure 119901

119894= 150 300 and 119901

0= 150 300MPa

respectively subjected to thermal loading (1205790= 0 400 800)

are defined as follows the radii of the cylinder are taken as119886 = 01m and 119887 = 05m Poissonrsquos ratio ] = 03 Youngrsquosmodulus 119864

0= 207GPa and thermal expansion coefficient

120572 = 1205720= 178 times 10

minus6∘

Cminus1 The geometric parameters of thecylinder are taken as 119890

1= 0 1 2 in Youngrsquos modulus function

and119898 = 04 06 is nonlinear strain hardening measureTo show the effect of internal and external pressure

on a functionally graded stainless steel composite rotatingcylinder with strain hardening measure 119898 = 04 06 havingconstant thickness and constant density Tables 1 and 2show the circumferential stresses with different parametersof Youngrsquos modulus 119890

1= 0 1 2

It has been observed from Table 1 that when externalpressure is greater than the internal pressure circumferen-tial stresses approaches tensile to compressible Also these

stresses aremaximum at external surface for homogeneous aswell as functionally graded stainless steel composite cylinderThese stresses are less for functionally graded stainless steelcomposite cylinder as compared to homogeneous cylinderAs nonhomogeneity changes from 119890

1= 1 to 119890

1= 2

circumferential stresses decrease significantlyWith the intro-duction of thermal effects circumferential stresses increasefor homogeneous as well as for functionally graded stainlesssteel composite cylinder (119890

1= 1) but decrease for functionally

graded stainless steel composite cylinder with 1198901= 2 It

has also been noticed from Table 1 that with the increasein thermal effects these stresses decrease significantly forhomogeneous as well as for functionally graded stainlesssteel composite cylinder and are less for the cylinder withnonhomogeneity parameter 119890

1= 2 With the increase in

angular speed circumferential stresses increase significantlyWhen external pressure is less than the internal pressurecircumferential stresses are maximum at internal surface forhomogeneous cylinder while maximum at external surfacefor functionally graded stainless steel composite cylinder andthese stresses decreased with the change in nonhomogeneitymeasure from 119890

1= 1 to 119890

1= 2 With the introduc-

tion of thermal effects circumferential stresses increase forhomogenous as well as for functionally graded stainless steelcomposite measure 119890

1= 1 while decrease for functionally

graded stainless steel composite cylinder with 1198901= 2 With

the increase in thermal effects these circumferential stressesdecrease significantly while increase with the increase inangular speed It has been observed from Table 2 that thebehavior of homogeneous and functionally graded stainlesssteel composite cylinder same as discussed in Table 1 but ithas been observed that with increase in strain hardeningmeasure from 119898 = 04 to 119898 = 06 these stresses decreasesignificantly for functionally graded stainless steel compositecylinder

Tables 3 and 4 have beenmade for circumferential stressesin rotating cylinders with variable thickness and variabledensity with different parameters of Youngrsquos modulus 119890

1=

0 1 2 and strain hardening measure119898 = 04 06It has been observed from Table 3 that for cylinder with

varying thickness and density whose external pressure isgreater than the internal pressure circumferential stressesapproach from tensile to compressible and are maximumat external surface for homogeneous as well as functionallygraded stainless steel composite cylinder These stressesare less for functionally graded stainless steel compositecylinder as compared to homogeneous cylinder with varyingthickness and density as well as cylinder with constant thick-ness and density With increase in strain hardening measurefrom 119898 = 04 to 119898 = 06 these circumferential stressesdecrease significantly for functionally graded stainless steel

Advances in Materials Science and Engineering 5

Table 1 Circumferential stresses for rotating cylinder with constant thickness and constant density 120596 = 300 500 nonlinear strain hardeningmeasure119898 = 04 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 30001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 83454530141 34064915847 00677187483 minus39596784280 minus896024571031 33799641785 27171534961 08969909345 minus26654005915 minus855489788262 11503024505 18118357165 12081310260 minus15371690373 minus78899736228

119901119894= 300

1199010= 150

1205790= 0

0 88188655176 36471590677 02649872301 minus37776104911 minus878521510291 36049642996 29421535689 11219909709 minus24404004448 minus832989835602 11869707579 19774764390 14492915544 minus12424185539 minus75536537230

119901119894= 150

1199010= 300

1205790= 400

0 83454529134 34064914474 00677186071 minus39596781512 minus896024633301 33799641163 27171535146 08969910927 minus26654006381 minus855489857932 11503024507 18118357152 12081310262 minus15371690375 minus78899736082

119901119894= 300

1199010= 150

1205790= 400

0 88188661275 36471591256 02649873453 minus37776106334 minus878521459301 36049644528 29421536040 11219910130 minus24404005339 minus832989848932 11869707578 19774764389 14492915543 minus12424185538 minus75536537227

119901119894= 150

1199010= 300

1205790= 800

0 83454532713 34064916133 00677188786 minus39596783083 minus896024607391 03799644417 27171537078 08969909208 minus26654005810 minus855489848852 11503024510 18118357165 12081310255 minus15371690384 minus78899735899

119901119894= 300

1199010= 150

1205790= 800

0 88188658525 36471597197 02649871677 minus37776107003 minus878521600471 36049639860 29421535165 11219910039 minus24404004285 minus832989834522 11869707577 19774764387 14492915543 minus12424185537 minus75536537224

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 50001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 11199643681 04548287897 00269123600 minus04863252783 minus121916263931 06272813080 05095768873 01910926579 minus04339086008 minus153519940132 03263905498 05038039436 03586769803 minus03274307737 minus19402396397

119901119894= 300

1199010= 150

1205790= 0

0 11512903744 04761073745 00479630332 minus04638575699 minus119486110581 06497506092 05320656463 02136358300 minus04115308637 minus151265455362 03354452568 05269344433 03869603568 minus03022694774 minus19278605585

119901119894= 150

1199010= 300

1205790= 400

0 11199644787 04548287386 00269122948 minus04863252189 minus121916251551 06272812978 05095769102 01910926887 minus04339085404 minus153519934162 03263905755 05038038300 03586769873 minus03274306978 minus19402449592

119901119894= 300

1199010= 150

1205790= 400

0 11512904788 04761073218 00479628568 minus04638573696 minus119486136701 06497506075 05320656332 02136358153 minus04115307488 minus151265453652 03354452679 05269344696 03869603369 minus03022695403 minus19278600529

119901119894= 150

1199010= 300

1205790= 800

0 11199642715 04548288045 00269123491 minus04863251255 minus121916247691 11199644787 04548287386 00269122948 minus04863252189 minus121916251552 03263905007 05038038190 03586771087 minus03274306558 minus19402425453

119901119894= 300

1199010= 150

1205790= 800

0 11512903765 04761073125 00479629909 minus04638574341 minus119486128581 06497505961 05320656017 02136358897 minus04115308258 minus151265454652 03354452477 05269344295 03869603736 minus03022695113 minus19278604854

composite cylinder with varying thickness and density as canbe seen from Table 4

Figures 2ndash4 have been drawn to discuss the effect ofinternal and external pressure on stresses in rotating cylin-der made of functionally graded stainless steel composite

material with constant thickness and constant density withnonlinear strain hardening measure

It has been observed from Figure 2 that circumferentialstress approaches towards compressive from tensile It hasalso been observed thatwhen external pressure is greater than

6 Advances in Materials Science and Engineering

Table 2 Circumferential stresses for rotating cylinder with constant thickness and constant density with 120596 = 300 500 nonlinear strainhardening measure119898 = 06 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa

119903

1198901

119898 = 06 120596 = 30001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 83453827289 34064900899 00677286065 minus39596681544 minus896024143091 33799456580 27171435351 08969901053 minus26653896254 minus855488042282 11502969495 18118279175 12081251826 minus15371621151 minus78899506975

119901119894= 300

1199010= 150

1205790= 0

0 88187936296 36471591317 02649975386 minus37776011860 minus878521216671 36049445586 29421431766 11219899984 minus24403890591 minus832987989572 11869649855 19774683014 14492853591 minus12424112423 minus75536293888

119901119894= 150

1199010= 300

1205790= 400

0 83453833202 34064903276 00677284767 minus39596684114 minus896024179131 33799454509 27171434189 08969901711 minus26653895895 minus855488050342 11502969501 18118279178 12081251821 minus15371621158 minus78899507229

119901119894= 300

1199010= 150

1205790= 400

0 88187922056 36471584895 02649978782 minus37776012451 minus878521236811 36049444470 29421430233 11219900447 minus24403890386 minus832988022922 11869649854 19774683013 14492853591 minus12424112422 minus75536293885

119901119894= 150

1199010= 300

1205790= 800

0 83453820024 34064896339 00677286501 minus39596681965 minus896024157281 33799459533 27171437210 08969899701 minus26653898309 minus855488077952 11502969490 18118279176 12081251818 minus15371621155 minus78899507012

119901119894= 300

1199010= 150

1205790= 800

0 88187929120 36471586248 02649979578 minus37776012681 minus878521057051 36049446223 29421429798 11219900968 minus24403889573 minus832987897072 11869649853 19774683011 14492853590 minus12424112421 minus75536293882

1119864minus4lowast 119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 50001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 11199583152 04548258695 00269146492 minus04863265526 minus121916543001 06272799179 05095770638 01910931884 minus04339078950 minus153520394292 03263903739 05038038750 03586772209 minus03274306376 minus19402344736

119901119894= 300

1199010= 150

1205790= 0

0 11512567236 04761058146 00480074980 minus04638915755 minus119483525601 06496608008 05320325649 02137603620 minus04118738110 minus151255087832 03354423760 05269311856 03869645881 minus03022635649 minus19278571554

119901119894= 150

1199010= 300

1205790= 400

0 11199579631 04548258318 00269147731 minus04863261541 minus121916558021 06272799298 05095770743 01910931873 minus04339077941 minus153520404932 03263904649 05038039194 03586771853 minus03274308913 minus19402307798

119901119894= 300

1199010= 150

1205790= 400

0 11512569402 04761057447 00480074391 minus04638917626 minus119483523931 06496607027 05320325183 02137605230 minus04118740006 minus151255088322 03354423676 05269312306 03869645451 minus03022636353 minus19278567951

119901119894= 150

1199010= 300

1205790= 800

0 11199579102 04548258849 00269148341 minus04863259137 minus121916561241 06272799292 05095770557 01910931905 minus04339078242 minus153520389162 03263903269 05038037303 03586773178 minus03274301493 minus19402403280

119901119894= 300

1199010= 150

1205790= 800

0 11512563866 04761057469 00480078293 minus04638922712 minus119483505711 06496607065 05320325064 02137604900 minus04118740845 minus151255078652 03354423734 05269312334 03869645540 minus03022636247 minus19278571251

the internal pressure these stresses are maximum at externalsurface for homogeneous as well as functionally gradedstainless steel composite cylinder Also it has been observedthat circumferential stress is maximum at internal surface forhomogeneous cylinder while maximum at external surface

for functionally graded stainless steel composite cylinderwhen external pressure is less than the internal pressure Alsowith the increase in angular speed circumferential stressesincrease significantly From Figure 3 it can be seen that ascircumferential stresses increase for homogeneous cylinder

Advances in Materials Science and Engineering 7

Table 3 Circumferential stresses for rotating cylinder with variable thickness and variable density 119897 = 07 119889 = 05 120596 = 300 500 strainhardening measure119898 = 04 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 300 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 4078610121 1692083635 minus0362927882 minus3599668076 minus83881223761 1844023846 1536347566 0332616455 minus2596999032 minus81860251262 0698588556 1166718627 0723851443 minus1632455674 minus7703313292

119901119894= 300

1199010= 150

1205790= 0

0 4468498659 1869943306 minus0216047724 minus3455691317 minus82392598161 2039946535 1721858320 0520942472 minus2403128817 minus79859401892 0725452105 1314126561 0957769537 minus1332711442 minus7351503187

119901119894= 150

1199010= 300

1205790= 400

0 4078610926 1692083796 minus0362928200 minus3599667939 minus83881241561 1844023759 1536347360 0332616655 minus2596998868 minus81860264052 0698588485 1166718653 0723851463 minus1632455697 minus7703312785

119901119894= 300

1199010= 150

1205790= 400

0 4468498329 1869943517 minus0216047300 minus3455691093 minus82392517301 2039946605 1721858238 0520942423 minus2403128796 minus79859403832 0725452128 1314126566 0957769488 minus1332711458 minus7351503426

119901119894= 150

1199010= 300

1205790= 800

0 4078610278 1692083456 minus0362927548 minus3599667594 minus83881250271 1844023766 1536347565 0332616556 minus2596998868 minus81860262912 0698588473 1166718654 0723851445 minus1632455676 minus7703312834

119901119894= 300

1199010= 150

1205790= 800

0 4468498296 1869943321 minus0216047655 minus3455691489 minus82392377231 2039946200 1721858288 0520942491 minus2403128819 minus79859400642 0725452082 1314126521 0957769539 minus1332711402 minus7351503219

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 500 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 0502344054 0209118575 minus0032085608 minus0411840852 minus10757559141 0301690685 0253918828 0067872877 minus0387759274 minus13375609912 0170338387 0272622555 0177093634 minus0328795174 minus1672186161

119901119894= 300

1199010= 150

1205790= 0

0 0520602572 0223194821 minus0013327208 minus0385714307 minus10416636041 0314858338 0269151250 0087139893 minus0362225697 minus13033628312 0175533196 0289517104 0201213547 minus0301959686 minus1648231058

119901119894= 150

1199010= 300

1205790= 400

0 0502344076 0209118532 minus0032085631 minus0411840853 minus10757560931 0301690546 0253918731 0067873081 minus0387757210 minus13375620512 0170338398 0272622541 0177093558 minus0328795106 minus1672185933

119901119894= 300

1199010= 150

1205790= 400

0 0520602619 0223194813 minus0013327161 minus0385714149 minus10416645331 0314858326 0269151199 0087139890 minus0362225687 minus13033627732 0175533191 0289517093 0201213591 minus0301959717 minus1648231090

119901119894= 150

1199010= 300

1205790= 800

0 0502344041 0209118525 minus0032085593 minus0411841019 minus10757557561 0301690548 0253918749 0067873075 minus0387757974 minus13375601512 0170338413 0272622528 0177093598 minus0328795120 minus1672185805

119901119894= 300

1199010= 150

1205790= 800

0 0520602532 0223194802 minus0013327191 minus0385714162 minus10416644831 0314858370 0269151215 0087139921 minus0362225740 minus13033627232 0175533168 0289517159 0201213488 minus0301959669 minus1648230944

as well as for nonhomogenous cylinder but with the changein nonhomogeneity from 119890

1= 1 to 119890

1= 2 circumferential

stresses decrease when external pressure is greater than theinternal pressure with thermal effects Also these stresses

increase for homogenous cylinder as well as for functionallygraded stainless steel composite cylinder (119890

1= 1) while

decrease for functionally graded stainless steel compositecylinder with 119890

1= 2 With the increase in temperature these

8 Advances in Materials Science and Engineering

Table 4 Circumferential stresses for rotating cylinder with variable thickness and variable density 119897 = 07 119889 = 05 120596 = 300 500 strainhardening measure119898 = 06 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 300 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 4078550787 1692092827 minus0362917634 minus3599662505 minus83881243031 1844008854 1536344037 0332619755 minus2596989139 minus81860118602 0698583918 1166713498 0723849363 minus1632447308 minus7703291360

119901119894= 300

1199010= 150

1205790= 0

0 4468438868 1869957168 minus0216034432 minus3455686049 minus82394923451 2039930194 1721854313 0520946730 minus2403118071 minus79859271952 0725447053 1314120776 0957767141 minus1332701862 minus7351479020

119901119894= 150

1199010= 300

1205790= 400

0 4078550758 1692093003 minus0362917541 minus3599662684 minus83881247581 1844009034 1536344170 0332619845 minus2596989055 minus81860133232 0698583989 1166713507 0723849336 minus1632447294 minus7703291352

119901119894= 300

1199010= 150

1205790= 400

0 4468440841 1869957567 minus0216033626 minus3455685275 minus82395880401 2039930222 1721854440 0520946513 minus2403117933 minus79859260012 0725447107 1314120882 0957767050 minus1332701999 minus7351479230

119901119894= 150

1199010= 300

1205790= 800

0 4078551089 1692092847 minus0362917694 minus3599662435 minus83881264791 1844008698 1536343957 0332619859 minus2596988916 minus81860122392 0698583989 1166713524 0723849328 minus1632447317 minus7703291542

119901119894= 300

1199010= 150

1205790= 800

0 4468438120 1869956228 minus0216035341 minus3455687051 minus82393686321 2039930039 1721854569 0520946538 minus2403117966 minus79859257312 0725447049 1314120768 0957767137 minus1332701833 minus7351479029

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 500 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 0502339190 0209119023 minus0032080434 minus0411843003 minus10757515751 0301689441 0253919137 0067872897 minus0387758158 minus13375674432 0170338021 0272622433 0177094183 minus0328793821 minus1672191050

119901119894= 300

1199010= 150

1205790= 0

0 0520600260 0223196743 minus0013328691 minus0385729358 minus10416359501 0314857828 0269151955 0087140006 minus0362234515 minus13033506742 0175532879 0289517002 0201213980 minus0301960040 minus1648233514

119901119894= 150

1199010= 300

1205790= 400

0 0502339173 0209119009 minus0032080335 minus0411843048 minus10757520131 0301689029 0253918998 0067873493 minus0387754850 minus13375668252 0170338061 0272622405 0177094107 minus0328793799 minus1672190560

119901119894= 300

1199010= 150

1205790= 400

0 0520600268 0223196732 minus0013328653 minus0385729330 minus10416363341 0314857844 0269151936 0087139999 minus0362234465 minus13033507222 0175532856 0289516980 0201214076 minus0301960031 minus1648233768

119901119894= 150

1199010= 300

1205790= 800

0 0502339102 0209118939 minus0032080279 minus0411842835 minus10757521591 0301689252 0253919092 0067873202 minus0387759066 minus13375656632 0170338011 0272622432 0177094145 minus0328793808 minus1672190610

119901119894= 300

1199010= 150

1205790= 800

0 0520600345 0223196689 minus0013328705 minus0385729338 minus10416358761 0314857846 0269151949 0087139963 minus0362234522 minus13033507592 0175532843 0289516995 0201214043 minus0301960031 minus1648233661

stresses decrease significantly for homogeneous as well as forfunctionally graded stainless steel composite cylinder as canbe seen from Figure 4

Figures 5ndash7 have been drawn to discuss the effect ofinternal and external pressure on stresses in rotating cylin-der made of functionally graded stainless steel composite

Advances in Materials Science and Engineering 9

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 150 and p0 = 300

120596 = 300 e1 = 0

120596 = 300 e1 = 2

120596 = 500 e1 = 1

120596 = 300 e1 = 1

120596 = 500 e1 = 0

120596 = 500 e1 = 2

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 300 and p0 = 150

120596 = 300 e1 = 0

120596 = 300 e1 = 2

120596 = 500 e1 = 1

120596 = 300 e1 = 1

120596 = 500 e1 = 0

120596 = 500 e1 = 2

(b)

Figure 2 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density without thermal effects (1205790= 0) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

pi = 150 and p0 = 3001205721 = 0 1205790 = 400 m = 04

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 300 and p0 = 150

(b)

Figure 3 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density with thermal effects (1205790= 400) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 300 and p0 = 150

(b)

Figure 4 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density with thermal effects (1205790= 800) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

10 Advances in Materials Science and Engineering

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 300 and p0 = 150

(b)

Figure 5 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density without thermal effects (1205790= 0) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 300 and p0 = 150

(b)

Figure 6 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 400) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 300 and p0 = 150

(b)

Figure 7 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 800) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

Advances in Materials Science and Engineering 11

material with variable thickness and variable density withnonlinear strain hardening measure

It has been observed from Figure 5 that circumferentialstresses are maximum at external surface for homogeneousas well as functionally graded stainless steel compositecylinder It has also been observed that with the increase inangular speed circumferential stresses increase significantlyWith the introduction of thermal effects circumferentialstresses increase for homogeneous cylinder as well as forfunctionally graded stainless steel composite cylinder (119890

1=

1) but with the change in nonhomogeneity from 1198901= 1

to 1198901= 2 circumferential stresses decrease when external

pressure is greater than the internal pressure as can beseen from Figure 6 Circumferential stresses decrease forhomogeneous cylinder while increase for functionally gradedstainless steel composite cylinder when external pressure isless than the internal pressure It has also been observedfrom Figure 7 that with the increase in temperature thesestresses increase significantly for homogeneous cylinder aswell for functionally graded stainless steel composite cylinderwith 119890

1= 1 while decrease for functionally graded stainless

steel composite cylinder with 1198901= 2 when external pressure

is greater than the internal pressure while these stressesdecrease significantly for homogeneous cylinder as well forfunctionally graded stainless steel composite cylinder whenexternal pressure is less than the internal pressure

5 Conclusion

From the analysis we can conclude that rotating cylin-der made of functionally graded stainless steel compositematerial having variable thickness and variable density withSwiftrsquos strain hardening measure 119898 = 06 and thermalloading is better choice for designers as compared to rotatingcylinder with constant thickness and constant density Thisis because of the reason that circumferential stress is less forfunctionally graded stainless steel composite cylinder withvariable thickness and variable density as compared to othercases which leads to the idea of stress saving that minimizesthe possibility of fracture of cylinder

Conflict of Interests

The authors declare that they have no conflict of interest

References

[1] S Suresh and A Mortensen Fundamentals of FunctionallyGradedMaterials IOM3 Maney Publishing London UK 1998

[2] J N Reddy ldquoAnalysis of functionally graded platesrdquo Interna-tional Journal for Numerical Methods in Engineering vol 47 no1ndash3 pp 663ndash684 2000

[3] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill NewYork NY USA 3rd edition 1970

[4] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998

[5] Y Obata T Oji and N Noda ldquoSteady thermal stresses in a hol-low circular cylinder and a hollow sphere made of functionallygraded material (analysis with perturbation method)rdquo Reports

of the National Industrial Research Institute of Nagoya vol 47no 4 pp 317ndash338 1998

[6] J Perry and J Aboudi ldquoElasto-plastic stresses in thick walledcylindersrdquo Journal of Pressure Vessel Technology vol 125 no 3pp 248ndash252 2003

[7] X-L Gao ldquoElasto-plastic analysis of an internally pressurizedthick-walled cylinder using a strain gradient plasticity theoryrdquoInternational Journal of Solids and Structures vol 40 no 23 pp6445ndash6455 2003

[8] N Tutuncu and M Ozturk ldquoExact solutions for stresses infunctionally graded pressure vesselsrdquo Composites B vol 32 no8 pp 683ndash686 2001

[9] T Singh and V K Gupta ldquoCreep analysis of an internallypressurized thick cylinder made of a functionally graded com-positerdquo Journal of Strain Analysis for Engineering Design vol 44no 7 pp 583ndash594 2009

[10] A K Aggarwal Sharma Richa and Sharma Sanjeev ldquoSafetyanalysis using lebesgue strain measure of thick-walled cylinderfor functionally graded material under internal and externalpressurerdquo The Scientific World Journal vol 2013 Article ID676190 10 pages 2013

[11] A K Aggarwal S Richa and S Sanjeev ldquoSafety analysis ofthermal creep non-homogeneous thick-walled circular cylinderunder internal and external pressure using lebesgue strainmeasurerdquoMultidiscipline Modeling in Materials and Structuresvol 9 no 4 2013

[12] A Parvizi R Naghdabadi and J Arghavani ldquoAnalysis of AlA359SiCp functionally graded cylinder subjected to internalpressure and temperature gradient with elastic-plastic deforma-tionrdquo Journal of Thermal Stresses vol 34 no 10 pp 1054ndash10702011

[13] A N Eraslan and F Akgul ldquoYielding and elastoplastic deforma-tion of annular disks of a parabolic section subject to externalcompressionrdquo Turkish Journal of Engineering and Environmen-tal Sciences vol 29 no 1 pp 51ndash60 2005

Submit your manuscripts athttpwwwhindawicom

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Journal ofNanomaterials

Page 3: Research Article Thermo Elastic-Plastic Analysis of Rotating …downloads.hindawi.com/journals/amse/2013/810508.pdf · 2019-07-31 · ermo elastic-plastic analysis of functionally

Advances in Materials Science and Engineering 3

Due to geometric symmetry of the cylinder circumfer-ential displacement shear stresses and strains are assumedto be zero

Using deformation theory of plasticity the relationbetween the stresses and plastic strains can be determined as

119890119901

119903=

119890119901

119890

119879119890119890

[119879119903119903minus

1

2

(119879120579120579+ 119879119911119911)]

119890119901

120579=

119890119901

119890

119879119890119890

[119879120579120579minus

1

2

(119879119903119903+ 119879119911119911)]

119890119901

119911=

119890119901

119890

119879119890119890

[119879119911119911minus

1

2

(119879119903119903+ 119879120579120579)]

(6)

where 119879119890119890is the equivalent stress 119890119901

119890is the equivalent plastic

strain and 119890119901119903 119890119901120579 and 119890119901

119911are the plastic radial circumferential

and axial strains respectivelyvon Misesrsquo yield criterion is given by

119879119890119890= radic(119879

119903119903minus 119879120579120579)2+ (119879120579120579minus 119879119911119911)2+ (119879119911119911minus 119879119903119903)2 (7)

The total radial circumferential and axial strains inthick-walled rotating cylinder are

119890119903= 119890119890

119903+ 119890119901

119903+ 120572120579 119890

120579= 119890119890

120579+ 119890119901

120579+ 120572120579

119890119911= 119890119890

119911+ 119890119901

119911+ 120572120579

(8)

The temperature field satisfying Laplace heat equation is

1198892120579

1198891199032+

1

119903

119889120579

119889119903

= 0 (9)

with 120579 = 1205790at 119903 = 119886 and 120579 = 0 at 119903 = 119887 where 120579

0is a constant

given by 120579(119903) = 1205790log(119903119887)

We define the stress function 120601(119903) for thick-walled rotat-ing cylinder which is related to radial and hoop stresses as

119879119903119903=

120601

ℎ119903

119879120579120579=

1

119889120601

119889119903

+ 12058812059621199032 119879

119911119911= 0 (10)

Since it has been assumed that the cylinder is long andopen ended and there is plane stress condition therefore axialstress is zero that is 119879

119911119911= 0

Substituting (10) and (5) into (8) we have

119890119903=

1

119864

(

120601

ℎ119903

minus ]1

119889120601

119889119903

minus ]12058812059621199032) + 119890119901119903+ 120572120579

119890120579=

1

119864

(

1

119889120601

119889119903

+ 12058812059621199032minus ]

120601

ℎ119903

) + 119890119901

120579+ 120572120579

119890119911=

minus]

119864

(

1

119889120601

119889119903

+ 12058812059621199032+

120601

ℎ119903

) + 119890119901

119911+ 120572120579

(11)

Substituting of (11) into compatibility (4) we have

119903212060110158401015840minus 1199031206011015840[1 + 119903(

ℎ1015840

+

1198641015840

119864

)] minus 120601[1 minus 119903](ℎ1015840

+

1198641015840

119864

)]

+ ℎ120588101584012059621199034+ (3 + ] minus 119903

1198641015840

119864

)ℎ12058812059621199033

minus 119864ℎ119903 (119890119901

119903minus 119890119901

120579) + 119864ℎ119903

2[(119890119901

120579)

1015840

+ 1205721205791015840+ 1205721015840120579] = 0

(12)where 1206011015840 = 119889120601119889119903 12060110158401015840 = 11988921206011198891199032 and 119890119901

120579

1015840

= 119889119890119901

120579119889119903

The relation between the yield stress 119879119890119890and the equiva-

lent plastic strain 119890119901119890for Swiftrsquos hardening law can be expressed

as

119890119890

119890=

119879119890119890

119864

119890119890le 1198900

119890119901

119890=

1

120578

[(

119879119890119890

1198790

)

119898

minus 1] 119890119890gt 1198900

(13)

where 120578 119898 1198790 119890119890 and 119890

0are hardening parameter material

parameter yield limit equivalent total strain and yield strainrespectively

Substituting 119890119901119890from (13) into (6) results in

119890119901

119903=

(1120578) [(1198791198901198901198790)119898minus 1]

119879119890119890

[119879119903119903minus 05 (119879

120579120579+ 119879119911119911)]

119890119901

120579=

(1120578) [(1198791198901198901198790)119898minus 1]

119879119890119890

[119879120579120579minus 05 (119879

119903119903+ 119879119911119911)]

119890119901

119911=

(1120578) [(1198791198901198901198790)119898minus 1]

119879119890119890

[119879119911119911minus 05 (119879

119903119903+ 119879120579120579)]

(14)

Substituting (14) into (12) we have

119903212060110158401015840[1 +

119864

21205781198792

119890119890

2119879119890119890[(

119879119890119890

1198790

)

119898

minus 1] +

1

2119879119890119890

(119879119903119903minus 2119879120579120579)2times (1 + (119898 minus 1) (

119879119890119890

1198790

)

119898

)]

+ 119864ℎ1199032[1205791205721015840+ 1205721205791015840] + ℎ120588120596

21199033(119864 + 4])

minus

119864

21205781198792

119890119890

[

[

[

[

[

[

[

[

[

[

[

119879119890119890times [(

119879119890119890

1198790

)

119898

minus 1] times (1 + 2119903

ℎ1015840

) 1199031206011015840minus (1 + 119903

ℎ1015840

)120601 minus 2ℎ120588101584012059621199034minus 4ℎ120588120596

21199033

+

1

2119879119890119890

(1 + (119898 minus 1) (

119879119890119890

1198790

)

119898

) (119879119903119903minus 2119879120579120579) times

(2119879120579120579minus 119879119903119903)

times [minus1199032 ℎ1015840

1206011015840+ ℎ120588101584012059621199034+ 2ℎ120588120596

21199033]

+ (2119879119903119903minus 119879120579120579) times [119903120601

1015840minus (1 + 119903

ℎ1015840

)120601]

]

]

]

]

]

]

]

]

]

]

]

4 Advances in Materials Science and Engineering

minus

3

2120578119879119890119890

119864ℎ119903 times [(

119879119890119890

1198790

)

119898

minus 1] times (119879119903119903minus 119879120579120579) + 119903120601

1015840[1 minus 119903

ℎ1015840

minus 119903(

1198641015840

119864

)] minus 120601[1 minus ]119903ℎ1015840

minus 119903(

1198641015840

119864

)]

= ℎ12059621199034[120588(

1198641015840

119864

) minus 1205881015840]

(15)

Equation (15) is the differential equation of the function-ally graded stainless steel composite rotating cylinder withnonlinear strain hardening subjected to thermal loading inthe plastic region in terms of stresses and stress function

Equation (15) can be described in the general form interms of stress function as

12060110158401015840= 119891 (119903 120601 120601

1015840) (16)

Equation (16) is a nonlinear two point boundary valueproblem and can be solved numerically subjected to theboundary conditions

119879119903119903= minus119901119894 at 119903 = 119886 119879

119903119903= minus1199010 at 119903 = 119887 119886 gt 0

(17)

where 119886 and 119887 are the inner and outer radii of the cylinder and119901119894and 119901

0are internal and external pressures respectively

Using finite difference method with central difference in(16) we get the following system of equations

120601119894+1minus 2120601119894+ 120601119894minus1

(Δ119903)2

= 119891(119903 120601119894

120601119894+1minus 120601119894minus1

2Δ119903

) 119894 = 2 3 119899

(18)

Equation (18) consists of algebraic system of (119899minus 1) equa-tions with the boundary conditions 120601(119886) = minus119901

119894ℎ119886 and 120601(119887) =

minus1199010ℎ119887 After solving (18) with boundary conditions we get a

stress function120601Then the radial and circumferential stressescan be obtained from (10) after substituting the value of stressfunction 120601

4 Numerical Discussion

The properties of a functionally graded stainless steel com-posite thick-walled rotating cylinder under internal andexternal pressure 119901

119894= 150 300 and 119901

0= 150 300MPa

respectively subjected to thermal loading (1205790= 0 400 800)

are defined as follows the radii of the cylinder are taken as119886 = 01m and 119887 = 05m Poissonrsquos ratio ] = 03 Youngrsquosmodulus 119864

0= 207GPa and thermal expansion coefficient

120572 = 1205720= 178 times 10

minus6∘

Cminus1 The geometric parameters of thecylinder are taken as 119890

1= 0 1 2 in Youngrsquos modulus function

and119898 = 04 06 is nonlinear strain hardening measureTo show the effect of internal and external pressure

on a functionally graded stainless steel composite rotatingcylinder with strain hardening measure 119898 = 04 06 havingconstant thickness and constant density Tables 1 and 2show the circumferential stresses with different parametersof Youngrsquos modulus 119890

1= 0 1 2

It has been observed from Table 1 that when externalpressure is greater than the internal pressure circumferen-tial stresses approaches tensile to compressible Also these

stresses aremaximum at external surface for homogeneous aswell as functionally graded stainless steel composite cylinderThese stresses are less for functionally graded stainless steelcomposite cylinder as compared to homogeneous cylinderAs nonhomogeneity changes from 119890

1= 1 to 119890

1= 2

circumferential stresses decrease significantlyWith the intro-duction of thermal effects circumferential stresses increasefor homogeneous as well as for functionally graded stainlesssteel composite cylinder (119890

1= 1) but decrease for functionally

graded stainless steel composite cylinder with 1198901= 2 It

has also been noticed from Table 1 that with the increasein thermal effects these stresses decrease significantly forhomogeneous as well as for functionally graded stainlesssteel composite cylinder and are less for the cylinder withnonhomogeneity parameter 119890

1= 2 With the increase in

angular speed circumferential stresses increase significantlyWhen external pressure is less than the internal pressurecircumferential stresses are maximum at internal surface forhomogeneous cylinder while maximum at external surfacefor functionally graded stainless steel composite cylinder andthese stresses decreased with the change in nonhomogeneitymeasure from 119890

1= 1 to 119890

1= 2 With the introduc-

tion of thermal effects circumferential stresses increase forhomogenous as well as for functionally graded stainless steelcomposite measure 119890

1= 1 while decrease for functionally

graded stainless steel composite cylinder with 1198901= 2 With

the increase in thermal effects these circumferential stressesdecrease significantly while increase with the increase inangular speed It has been observed from Table 2 that thebehavior of homogeneous and functionally graded stainlesssteel composite cylinder same as discussed in Table 1 but ithas been observed that with increase in strain hardeningmeasure from 119898 = 04 to 119898 = 06 these stresses decreasesignificantly for functionally graded stainless steel compositecylinder

Tables 3 and 4 have beenmade for circumferential stressesin rotating cylinders with variable thickness and variabledensity with different parameters of Youngrsquos modulus 119890

1=

0 1 2 and strain hardening measure119898 = 04 06It has been observed from Table 3 that for cylinder with

varying thickness and density whose external pressure isgreater than the internal pressure circumferential stressesapproach from tensile to compressible and are maximumat external surface for homogeneous as well as functionallygraded stainless steel composite cylinder These stressesare less for functionally graded stainless steel compositecylinder as compared to homogeneous cylinder with varyingthickness and density as well as cylinder with constant thick-ness and density With increase in strain hardening measurefrom 119898 = 04 to 119898 = 06 these circumferential stressesdecrease significantly for functionally graded stainless steel

Advances in Materials Science and Engineering 5

Table 1 Circumferential stresses for rotating cylinder with constant thickness and constant density 120596 = 300 500 nonlinear strain hardeningmeasure119898 = 04 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 30001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 83454530141 34064915847 00677187483 minus39596784280 minus896024571031 33799641785 27171534961 08969909345 minus26654005915 minus855489788262 11503024505 18118357165 12081310260 minus15371690373 minus78899736228

119901119894= 300

1199010= 150

1205790= 0

0 88188655176 36471590677 02649872301 minus37776104911 minus878521510291 36049642996 29421535689 11219909709 minus24404004448 minus832989835602 11869707579 19774764390 14492915544 minus12424185539 minus75536537230

119901119894= 150

1199010= 300

1205790= 400

0 83454529134 34064914474 00677186071 minus39596781512 minus896024633301 33799641163 27171535146 08969910927 minus26654006381 minus855489857932 11503024507 18118357152 12081310262 minus15371690375 minus78899736082

119901119894= 300

1199010= 150

1205790= 400

0 88188661275 36471591256 02649873453 minus37776106334 minus878521459301 36049644528 29421536040 11219910130 minus24404005339 minus832989848932 11869707578 19774764389 14492915543 minus12424185538 minus75536537227

119901119894= 150

1199010= 300

1205790= 800

0 83454532713 34064916133 00677188786 minus39596783083 minus896024607391 03799644417 27171537078 08969909208 minus26654005810 minus855489848852 11503024510 18118357165 12081310255 minus15371690384 minus78899735899

119901119894= 300

1199010= 150

1205790= 800

0 88188658525 36471597197 02649871677 minus37776107003 minus878521600471 36049639860 29421535165 11219910039 minus24404004285 minus832989834522 11869707577 19774764387 14492915543 minus12424185537 minus75536537224

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 50001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 11199643681 04548287897 00269123600 minus04863252783 minus121916263931 06272813080 05095768873 01910926579 minus04339086008 minus153519940132 03263905498 05038039436 03586769803 minus03274307737 minus19402396397

119901119894= 300

1199010= 150

1205790= 0

0 11512903744 04761073745 00479630332 minus04638575699 minus119486110581 06497506092 05320656463 02136358300 minus04115308637 minus151265455362 03354452568 05269344433 03869603568 minus03022694774 minus19278605585

119901119894= 150

1199010= 300

1205790= 400

0 11199644787 04548287386 00269122948 minus04863252189 minus121916251551 06272812978 05095769102 01910926887 minus04339085404 minus153519934162 03263905755 05038038300 03586769873 minus03274306978 minus19402449592

119901119894= 300

1199010= 150

1205790= 400

0 11512904788 04761073218 00479628568 minus04638573696 minus119486136701 06497506075 05320656332 02136358153 minus04115307488 minus151265453652 03354452679 05269344696 03869603369 minus03022695403 minus19278600529

119901119894= 150

1199010= 300

1205790= 800

0 11199642715 04548288045 00269123491 minus04863251255 minus121916247691 11199644787 04548287386 00269122948 minus04863252189 minus121916251552 03263905007 05038038190 03586771087 minus03274306558 minus19402425453

119901119894= 300

1199010= 150

1205790= 800

0 11512903765 04761073125 00479629909 minus04638574341 minus119486128581 06497505961 05320656017 02136358897 minus04115308258 minus151265454652 03354452477 05269344295 03869603736 minus03022695113 minus19278604854

composite cylinder with varying thickness and density as canbe seen from Table 4

Figures 2ndash4 have been drawn to discuss the effect ofinternal and external pressure on stresses in rotating cylin-der made of functionally graded stainless steel composite

material with constant thickness and constant density withnonlinear strain hardening measure

It has been observed from Figure 2 that circumferentialstress approaches towards compressive from tensile It hasalso been observed thatwhen external pressure is greater than

6 Advances in Materials Science and Engineering

Table 2 Circumferential stresses for rotating cylinder with constant thickness and constant density with 120596 = 300 500 nonlinear strainhardening measure119898 = 06 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa

119903

1198901

119898 = 06 120596 = 30001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 83453827289 34064900899 00677286065 minus39596681544 minus896024143091 33799456580 27171435351 08969901053 minus26653896254 minus855488042282 11502969495 18118279175 12081251826 minus15371621151 minus78899506975

119901119894= 300

1199010= 150

1205790= 0

0 88187936296 36471591317 02649975386 minus37776011860 minus878521216671 36049445586 29421431766 11219899984 minus24403890591 minus832987989572 11869649855 19774683014 14492853591 minus12424112423 minus75536293888

119901119894= 150

1199010= 300

1205790= 400

0 83453833202 34064903276 00677284767 minus39596684114 minus896024179131 33799454509 27171434189 08969901711 minus26653895895 minus855488050342 11502969501 18118279178 12081251821 minus15371621158 minus78899507229

119901119894= 300

1199010= 150

1205790= 400

0 88187922056 36471584895 02649978782 minus37776012451 minus878521236811 36049444470 29421430233 11219900447 minus24403890386 minus832988022922 11869649854 19774683013 14492853591 minus12424112422 minus75536293885

119901119894= 150

1199010= 300

1205790= 800

0 83453820024 34064896339 00677286501 minus39596681965 minus896024157281 33799459533 27171437210 08969899701 minus26653898309 minus855488077952 11502969490 18118279176 12081251818 minus15371621155 minus78899507012

119901119894= 300

1199010= 150

1205790= 800

0 88187929120 36471586248 02649979578 minus37776012681 minus878521057051 36049446223 29421429798 11219900968 minus24403889573 minus832987897072 11869649853 19774683011 14492853590 minus12424112421 minus75536293882

1119864minus4lowast 119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 50001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 11199583152 04548258695 00269146492 minus04863265526 minus121916543001 06272799179 05095770638 01910931884 minus04339078950 minus153520394292 03263903739 05038038750 03586772209 minus03274306376 minus19402344736

119901119894= 300

1199010= 150

1205790= 0

0 11512567236 04761058146 00480074980 minus04638915755 minus119483525601 06496608008 05320325649 02137603620 minus04118738110 minus151255087832 03354423760 05269311856 03869645881 minus03022635649 minus19278571554

119901119894= 150

1199010= 300

1205790= 400

0 11199579631 04548258318 00269147731 minus04863261541 minus121916558021 06272799298 05095770743 01910931873 minus04339077941 minus153520404932 03263904649 05038039194 03586771853 minus03274308913 minus19402307798

119901119894= 300

1199010= 150

1205790= 400

0 11512569402 04761057447 00480074391 minus04638917626 minus119483523931 06496607027 05320325183 02137605230 minus04118740006 minus151255088322 03354423676 05269312306 03869645451 minus03022636353 minus19278567951

119901119894= 150

1199010= 300

1205790= 800

0 11199579102 04548258849 00269148341 minus04863259137 minus121916561241 06272799292 05095770557 01910931905 minus04339078242 minus153520389162 03263903269 05038037303 03586773178 minus03274301493 minus19402403280

119901119894= 300

1199010= 150

1205790= 800

0 11512563866 04761057469 00480078293 minus04638922712 minus119483505711 06496607065 05320325064 02137604900 minus04118740845 minus151255078652 03354423734 05269312334 03869645540 minus03022636247 minus19278571251

the internal pressure these stresses are maximum at externalsurface for homogeneous as well as functionally gradedstainless steel composite cylinder Also it has been observedthat circumferential stress is maximum at internal surface forhomogeneous cylinder while maximum at external surface

for functionally graded stainless steel composite cylinderwhen external pressure is less than the internal pressure Alsowith the increase in angular speed circumferential stressesincrease significantly From Figure 3 it can be seen that ascircumferential stresses increase for homogeneous cylinder

Advances in Materials Science and Engineering 7

Table 3 Circumferential stresses for rotating cylinder with variable thickness and variable density 119897 = 07 119889 = 05 120596 = 300 500 strainhardening measure119898 = 04 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 300 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 4078610121 1692083635 minus0362927882 minus3599668076 minus83881223761 1844023846 1536347566 0332616455 minus2596999032 minus81860251262 0698588556 1166718627 0723851443 minus1632455674 minus7703313292

119901119894= 300

1199010= 150

1205790= 0

0 4468498659 1869943306 minus0216047724 minus3455691317 minus82392598161 2039946535 1721858320 0520942472 minus2403128817 minus79859401892 0725452105 1314126561 0957769537 minus1332711442 minus7351503187

119901119894= 150

1199010= 300

1205790= 400

0 4078610926 1692083796 minus0362928200 minus3599667939 minus83881241561 1844023759 1536347360 0332616655 minus2596998868 minus81860264052 0698588485 1166718653 0723851463 minus1632455697 minus7703312785

119901119894= 300

1199010= 150

1205790= 400

0 4468498329 1869943517 minus0216047300 minus3455691093 minus82392517301 2039946605 1721858238 0520942423 minus2403128796 minus79859403832 0725452128 1314126566 0957769488 minus1332711458 minus7351503426

119901119894= 150

1199010= 300

1205790= 800

0 4078610278 1692083456 minus0362927548 minus3599667594 minus83881250271 1844023766 1536347565 0332616556 minus2596998868 minus81860262912 0698588473 1166718654 0723851445 minus1632455676 minus7703312834

119901119894= 300

1199010= 150

1205790= 800

0 4468498296 1869943321 minus0216047655 minus3455691489 minus82392377231 2039946200 1721858288 0520942491 minus2403128819 minus79859400642 0725452082 1314126521 0957769539 minus1332711402 minus7351503219

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 500 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 0502344054 0209118575 minus0032085608 minus0411840852 minus10757559141 0301690685 0253918828 0067872877 minus0387759274 minus13375609912 0170338387 0272622555 0177093634 minus0328795174 minus1672186161

119901119894= 300

1199010= 150

1205790= 0

0 0520602572 0223194821 minus0013327208 minus0385714307 minus10416636041 0314858338 0269151250 0087139893 minus0362225697 minus13033628312 0175533196 0289517104 0201213547 minus0301959686 minus1648231058

119901119894= 150

1199010= 300

1205790= 400

0 0502344076 0209118532 minus0032085631 minus0411840853 minus10757560931 0301690546 0253918731 0067873081 minus0387757210 minus13375620512 0170338398 0272622541 0177093558 minus0328795106 minus1672185933

119901119894= 300

1199010= 150

1205790= 400

0 0520602619 0223194813 minus0013327161 minus0385714149 minus10416645331 0314858326 0269151199 0087139890 minus0362225687 minus13033627732 0175533191 0289517093 0201213591 minus0301959717 minus1648231090

119901119894= 150

1199010= 300

1205790= 800

0 0502344041 0209118525 minus0032085593 minus0411841019 minus10757557561 0301690548 0253918749 0067873075 minus0387757974 minus13375601512 0170338413 0272622528 0177093598 minus0328795120 minus1672185805

119901119894= 300

1199010= 150

1205790= 800

0 0520602532 0223194802 minus0013327191 minus0385714162 minus10416644831 0314858370 0269151215 0087139921 minus0362225740 minus13033627232 0175533168 0289517159 0201213488 minus0301959669 minus1648230944

as well as for nonhomogenous cylinder but with the changein nonhomogeneity from 119890

1= 1 to 119890

1= 2 circumferential

stresses decrease when external pressure is greater than theinternal pressure with thermal effects Also these stresses

increase for homogenous cylinder as well as for functionallygraded stainless steel composite cylinder (119890

1= 1) while

decrease for functionally graded stainless steel compositecylinder with 119890

1= 2 With the increase in temperature these

8 Advances in Materials Science and Engineering

Table 4 Circumferential stresses for rotating cylinder with variable thickness and variable density 119897 = 07 119889 = 05 120596 = 300 500 strainhardening measure119898 = 06 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 300 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 4078550787 1692092827 minus0362917634 minus3599662505 minus83881243031 1844008854 1536344037 0332619755 minus2596989139 minus81860118602 0698583918 1166713498 0723849363 minus1632447308 minus7703291360

119901119894= 300

1199010= 150

1205790= 0

0 4468438868 1869957168 minus0216034432 minus3455686049 minus82394923451 2039930194 1721854313 0520946730 minus2403118071 minus79859271952 0725447053 1314120776 0957767141 minus1332701862 minus7351479020

119901119894= 150

1199010= 300

1205790= 400

0 4078550758 1692093003 minus0362917541 minus3599662684 minus83881247581 1844009034 1536344170 0332619845 minus2596989055 minus81860133232 0698583989 1166713507 0723849336 minus1632447294 minus7703291352

119901119894= 300

1199010= 150

1205790= 400

0 4468440841 1869957567 minus0216033626 minus3455685275 minus82395880401 2039930222 1721854440 0520946513 minus2403117933 minus79859260012 0725447107 1314120882 0957767050 minus1332701999 minus7351479230

119901119894= 150

1199010= 300

1205790= 800

0 4078551089 1692092847 minus0362917694 minus3599662435 minus83881264791 1844008698 1536343957 0332619859 minus2596988916 minus81860122392 0698583989 1166713524 0723849328 minus1632447317 minus7703291542

119901119894= 300

1199010= 150

1205790= 800

0 4468438120 1869956228 minus0216035341 minus3455687051 minus82393686321 2039930039 1721854569 0520946538 minus2403117966 minus79859257312 0725447049 1314120768 0957767137 minus1332701833 minus7351479029

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 500 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 0502339190 0209119023 minus0032080434 minus0411843003 minus10757515751 0301689441 0253919137 0067872897 minus0387758158 minus13375674432 0170338021 0272622433 0177094183 minus0328793821 minus1672191050

119901119894= 300

1199010= 150

1205790= 0

0 0520600260 0223196743 minus0013328691 minus0385729358 minus10416359501 0314857828 0269151955 0087140006 minus0362234515 minus13033506742 0175532879 0289517002 0201213980 minus0301960040 minus1648233514

119901119894= 150

1199010= 300

1205790= 400

0 0502339173 0209119009 minus0032080335 minus0411843048 minus10757520131 0301689029 0253918998 0067873493 minus0387754850 minus13375668252 0170338061 0272622405 0177094107 minus0328793799 minus1672190560

119901119894= 300

1199010= 150

1205790= 400

0 0520600268 0223196732 minus0013328653 minus0385729330 minus10416363341 0314857844 0269151936 0087139999 minus0362234465 minus13033507222 0175532856 0289516980 0201214076 minus0301960031 minus1648233768

119901119894= 150

1199010= 300

1205790= 800

0 0502339102 0209118939 minus0032080279 minus0411842835 minus10757521591 0301689252 0253919092 0067873202 minus0387759066 minus13375656632 0170338011 0272622432 0177094145 minus0328793808 minus1672190610

119901119894= 300

1199010= 150

1205790= 800

0 0520600345 0223196689 minus0013328705 minus0385729338 minus10416358761 0314857846 0269151949 0087139963 minus0362234522 minus13033507592 0175532843 0289516995 0201214043 minus0301960031 minus1648233661

stresses decrease significantly for homogeneous as well as forfunctionally graded stainless steel composite cylinder as canbe seen from Figure 4

Figures 5ndash7 have been drawn to discuss the effect ofinternal and external pressure on stresses in rotating cylin-der made of functionally graded stainless steel composite

Advances in Materials Science and Engineering 9

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 150 and p0 = 300

120596 = 300 e1 = 0

120596 = 300 e1 = 2

120596 = 500 e1 = 1

120596 = 300 e1 = 1

120596 = 500 e1 = 0

120596 = 500 e1 = 2

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 300 and p0 = 150

120596 = 300 e1 = 0

120596 = 300 e1 = 2

120596 = 500 e1 = 1

120596 = 300 e1 = 1

120596 = 500 e1 = 0

120596 = 500 e1 = 2

(b)

Figure 2 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density without thermal effects (1205790= 0) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

pi = 150 and p0 = 3001205721 = 0 1205790 = 400 m = 04

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 300 and p0 = 150

(b)

Figure 3 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density with thermal effects (1205790= 400) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 300 and p0 = 150

(b)

Figure 4 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density with thermal effects (1205790= 800) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

10 Advances in Materials Science and Engineering

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 300 and p0 = 150

(b)

Figure 5 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density without thermal effects (1205790= 0) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 300 and p0 = 150

(b)

Figure 6 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 400) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 300 and p0 = 150

(b)

Figure 7 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 800) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

Advances in Materials Science and Engineering 11

material with variable thickness and variable density withnonlinear strain hardening measure

It has been observed from Figure 5 that circumferentialstresses are maximum at external surface for homogeneousas well as functionally graded stainless steel compositecylinder It has also been observed that with the increase inangular speed circumferential stresses increase significantlyWith the introduction of thermal effects circumferentialstresses increase for homogeneous cylinder as well as forfunctionally graded stainless steel composite cylinder (119890

1=

1) but with the change in nonhomogeneity from 1198901= 1

to 1198901= 2 circumferential stresses decrease when external

pressure is greater than the internal pressure as can beseen from Figure 6 Circumferential stresses decrease forhomogeneous cylinder while increase for functionally gradedstainless steel composite cylinder when external pressure isless than the internal pressure It has also been observedfrom Figure 7 that with the increase in temperature thesestresses increase significantly for homogeneous cylinder aswell for functionally graded stainless steel composite cylinderwith 119890

1= 1 while decrease for functionally graded stainless

steel composite cylinder with 1198901= 2 when external pressure

is greater than the internal pressure while these stressesdecrease significantly for homogeneous cylinder as well forfunctionally graded stainless steel composite cylinder whenexternal pressure is less than the internal pressure

5 Conclusion

From the analysis we can conclude that rotating cylin-der made of functionally graded stainless steel compositematerial having variable thickness and variable density withSwiftrsquos strain hardening measure 119898 = 06 and thermalloading is better choice for designers as compared to rotatingcylinder with constant thickness and constant density Thisis because of the reason that circumferential stress is less forfunctionally graded stainless steel composite cylinder withvariable thickness and variable density as compared to othercases which leads to the idea of stress saving that minimizesthe possibility of fracture of cylinder

Conflict of Interests

The authors declare that they have no conflict of interest

References

[1] S Suresh and A Mortensen Fundamentals of FunctionallyGradedMaterials IOM3 Maney Publishing London UK 1998

[2] J N Reddy ldquoAnalysis of functionally graded platesrdquo Interna-tional Journal for Numerical Methods in Engineering vol 47 no1ndash3 pp 663ndash684 2000

[3] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill NewYork NY USA 3rd edition 1970

[4] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998

[5] Y Obata T Oji and N Noda ldquoSteady thermal stresses in a hol-low circular cylinder and a hollow sphere made of functionallygraded material (analysis with perturbation method)rdquo Reports

of the National Industrial Research Institute of Nagoya vol 47no 4 pp 317ndash338 1998

[6] J Perry and J Aboudi ldquoElasto-plastic stresses in thick walledcylindersrdquo Journal of Pressure Vessel Technology vol 125 no 3pp 248ndash252 2003

[7] X-L Gao ldquoElasto-plastic analysis of an internally pressurizedthick-walled cylinder using a strain gradient plasticity theoryrdquoInternational Journal of Solids and Structures vol 40 no 23 pp6445ndash6455 2003

[8] N Tutuncu and M Ozturk ldquoExact solutions for stresses infunctionally graded pressure vesselsrdquo Composites B vol 32 no8 pp 683ndash686 2001

[9] T Singh and V K Gupta ldquoCreep analysis of an internallypressurized thick cylinder made of a functionally graded com-positerdquo Journal of Strain Analysis for Engineering Design vol 44no 7 pp 583ndash594 2009

[10] A K Aggarwal Sharma Richa and Sharma Sanjeev ldquoSafetyanalysis using lebesgue strain measure of thick-walled cylinderfor functionally graded material under internal and externalpressurerdquo The Scientific World Journal vol 2013 Article ID676190 10 pages 2013

[11] A K Aggarwal S Richa and S Sanjeev ldquoSafety analysis ofthermal creep non-homogeneous thick-walled circular cylinderunder internal and external pressure using lebesgue strainmeasurerdquoMultidiscipline Modeling in Materials and Structuresvol 9 no 4 2013

[12] A Parvizi R Naghdabadi and J Arghavani ldquoAnalysis of AlA359SiCp functionally graded cylinder subjected to internalpressure and temperature gradient with elastic-plastic deforma-tionrdquo Journal of Thermal Stresses vol 34 no 10 pp 1054ndash10702011

[13] A N Eraslan and F Akgul ldquoYielding and elastoplastic deforma-tion of annular disks of a parabolic section subject to externalcompressionrdquo Turkish Journal of Engineering and Environmen-tal Sciences vol 29 no 1 pp 51ndash60 2005

Submit your manuscripts athttpwwwhindawicom

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Journal ofNanomaterials

Page 4: Research Article Thermo Elastic-Plastic Analysis of Rotating …downloads.hindawi.com/journals/amse/2013/810508.pdf · 2019-07-31 · ermo elastic-plastic analysis of functionally

4 Advances in Materials Science and Engineering

minus

3

2120578119879119890119890

119864ℎ119903 times [(

119879119890119890

1198790

)

119898

minus 1] times (119879119903119903minus 119879120579120579) + 119903120601

1015840[1 minus 119903

ℎ1015840

minus 119903(

1198641015840

119864

)] minus 120601[1 minus ]119903ℎ1015840

minus 119903(

1198641015840

119864

)]

= ℎ12059621199034[120588(

1198641015840

119864

) minus 1205881015840]

(15)

Equation (15) is the differential equation of the function-ally graded stainless steel composite rotating cylinder withnonlinear strain hardening subjected to thermal loading inthe plastic region in terms of stresses and stress function

Equation (15) can be described in the general form interms of stress function as

12060110158401015840= 119891 (119903 120601 120601

1015840) (16)

Equation (16) is a nonlinear two point boundary valueproblem and can be solved numerically subjected to theboundary conditions

119879119903119903= minus119901119894 at 119903 = 119886 119879

119903119903= minus1199010 at 119903 = 119887 119886 gt 0

(17)

where 119886 and 119887 are the inner and outer radii of the cylinder and119901119894and 119901

0are internal and external pressures respectively

Using finite difference method with central difference in(16) we get the following system of equations

120601119894+1minus 2120601119894+ 120601119894minus1

(Δ119903)2

= 119891(119903 120601119894

120601119894+1minus 120601119894minus1

2Δ119903

) 119894 = 2 3 119899

(18)

Equation (18) consists of algebraic system of (119899minus 1) equa-tions with the boundary conditions 120601(119886) = minus119901

119894ℎ119886 and 120601(119887) =

minus1199010ℎ119887 After solving (18) with boundary conditions we get a

stress function120601Then the radial and circumferential stressescan be obtained from (10) after substituting the value of stressfunction 120601

4 Numerical Discussion

The properties of a functionally graded stainless steel com-posite thick-walled rotating cylinder under internal andexternal pressure 119901

119894= 150 300 and 119901

0= 150 300MPa

respectively subjected to thermal loading (1205790= 0 400 800)

are defined as follows the radii of the cylinder are taken as119886 = 01m and 119887 = 05m Poissonrsquos ratio ] = 03 Youngrsquosmodulus 119864

0= 207GPa and thermal expansion coefficient

120572 = 1205720= 178 times 10

minus6∘

Cminus1 The geometric parameters of thecylinder are taken as 119890

1= 0 1 2 in Youngrsquos modulus function

and119898 = 04 06 is nonlinear strain hardening measureTo show the effect of internal and external pressure

on a functionally graded stainless steel composite rotatingcylinder with strain hardening measure 119898 = 04 06 havingconstant thickness and constant density Tables 1 and 2show the circumferential stresses with different parametersof Youngrsquos modulus 119890

1= 0 1 2

It has been observed from Table 1 that when externalpressure is greater than the internal pressure circumferen-tial stresses approaches tensile to compressible Also these

stresses aremaximum at external surface for homogeneous aswell as functionally graded stainless steel composite cylinderThese stresses are less for functionally graded stainless steelcomposite cylinder as compared to homogeneous cylinderAs nonhomogeneity changes from 119890

1= 1 to 119890

1= 2

circumferential stresses decrease significantlyWith the intro-duction of thermal effects circumferential stresses increasefor homogeneous as well as for functionally graded stainlesssteel composite cylinder (119890

1= 1) but decrease for functionally

graded stainless steel composite cylinder with 1198901= 2 It

has also been noticed from Table 1 that with the increasein thermal effects these stresses decrease significantly forhomogeneous as well as for functionally graded stainlesssteel composite cylinder and are less for the cylinder withnonhomogeneity parameter 119890

1= 2 With the increase in

angular speed circumferential stresses increase significantlyWhen external pressure is less than the internal pressurecircumferential stresses are maximum at internal surface forhomogeneous cylinder while maximum at external surfacefor functionally graded stainless steel composite cylinder andthese stresses decreased with the change in nonhomogeneitymeasure from 119890

1= 1 to 119890

1= 2 With the introduc-

tion of thermal effects circumferential stresses increase forhomogenous as well as for functionally graded stainless steelcomposite measure 119890

1= 1 while decrease for functionally

graded stainless steel composite cylinder with 1198901= 2 With

the increase in thermal effects these circumferential stressesdecrease significantly while increase with the increase inangular speed It has been observed from Table 2 that thebehavior of homogeneous and functionally graded stainlesssteel composite cylinder same as discussed in Table 1 but ithas been observed that with increase in strain hardeningmeasure from 119898 = 04 to 119898 = 06 these stresses decreasesignificantly for functionally graded stainless steel compositecylinder

Tables 3 and 4 have beenmade for circumferential stressesin rotating cylinders with variable thickness and variabledensity with different parameters of Youngrsquos modulus 119890

1=

0 1 2 and strain hardening measure119898 = 04 06It has been observed from Table 3 that for cylinder with

varying thickness and density whose external pressure isgreater than the internal pressure circumferential stressesapproach from tensile to compressible and are maximumat external surface for homogeneous as well as functionallygraded stainless steel composite cylinder These stressesare less for functionally graded stainless steel compositecylinder as compared to homogeneous cylinder with varyingthickness and density as well as cylinder with constant thick-ness and density With increase in strain hardening measurefrom 119898 = 04 to 119898 = 06 these circumferential stressesdecrease significantly for functionally graded stainless steel

Advances in Materials Science and Engineering 5

Table 1 Circumferential stresses for rotating cylinder with constant thickness and constant density 120596 = 300 500 nonlinear strain hardeningmeasure119898 = 04 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 30001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 83454530141 34064915847 00677187483 minus39596784280 minus896024571031 33799641785 27171534961 08969909345 minus26654005915 minus855489788262 11503024505 18118357165 12081310260 minus15371690373 minus78899736228

119901119894= 300

1199010= 150

1205790= 0

0 88188655176 36471590677 02649872301 minus37776104911 minus878521510291 36049642996 29421535689 11219909709 minus24404004448 minus832989835602 11869707579 19774764390 14492915544 minus12424185539 minus75536537230

119901119894= 150

1199010= 300

1205790= 400

0 83454529134 34064914474 00677186071 minus39596781512 minus896024633301 33799641163 27171535146 08969910927 minus26654006381 minus855489857932 11503024507 18118357152 12081310262 minus15371690375 minus78899736082

119901119894= 300

1199010= 150

1205790= 400

0 88188661275 36471591256 02649873453 minus37776106334 minus878521459301 36049644528 29421536040 11219910130 minus24404005339 minus832989848932 11869707578 19774764389 14492915543 minus12424185538 minus75536537227

119901119894= 150

1199010= 300

1205790= 800

0 83454532713 34064916133 00677188786 minus39596783083 minus896024607391 03799644417 27171537078 08969909208 minus26654005810 minus855489848852 11503024510 18118357165 12081310255 minus15371690384 minus78899735899

119901119894= 300

1199010= 150

1205790= 800

0 88188658525 36471597197 02649871677 minus37776107003 minus878521600471 36049639860 29421535165 11219910039 minus24404004285 minus832989834522 11869707577 19774764387 14492915543 minus12424185537 minus75536537224

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 50001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 11199643681 04548287897 00269123600 minus04863252783 minus121916263931 06272813080 05095768873 01910926579 minus04339086008 minus153519940132 03263905498 05038039436 03586769803 minus03274307737 minus19402396397

119901119894= 300

1199010= 150

1205790= 0

0 11512903744 04761073745 00479630332 minus04638575699 minus119486110581 06497506092 05320656463 02136358300 minus04115308637 minus151265455362 03354452568 05269344433 03869603568 minus03022694774 minus19278605585

119901119894= 150

1199010= 300

1205790= 400

0 11199644787 04548287386 00269122948 minus04863252189 minus121916251551 06272812978 05095769102 01910926887 minus04339085404 minus153519934162 03263905755 05038038300 03586769873 minus03274306978 minus19402449592

119901119894= 300

1199010= 150

1205790= 400

0 11512904788 04761073218 00479628568 minus04638573696 minus119486136701 06497506075 05320656332 02136358153 minus04115307488 minus151265453652 03354452679 05269344696 03869603369 minus03022695403 minus19278600529

119901119894= 150

1199010= 300

1205790= 800

0 11199642715 04548288045 00269123491 minus04863251255 minus121916247691 11199644787 04548287386 00269122948 minus04863252189 minus121916251552 03263905007 05038038190 03586771087 minus03274306558 minus19402425453

119901119894= 300

1199010= 150

1205790= 800

0 11512903765 04761073125 00479629909 minus04638574341 minus119486128581 06497505961 05320656017 02136358897 minus04115308258 minus151265454652 03354452477 05269344295 03869603736 minus03022695113 minus19278604854

composite cylinder with varying thickness and density as canbe seen from Table 4

Figures 2ndash4 have been drawn to discuss the effect ofinternal and external pressure on stresses in rotating cylin-der made of functionally graded stainless steel composite

material with constant thickness and constant density withnonlinear strain hardening measure

It has been observed from Figure 2 that circumferentialstress approaches towards compressive from tensile It hasalso been observed thatwhen external pressure is greater than

6 Advances in Materials Science and Engineering

Table 2 Circumferential stresses for rotating cylinder with constant thickness and constant density with 120596 = 300 500 nonlinear strainhardening measure119898 = 06 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa

119903

1198901

119898 = 06 120596 = 30001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 83453827289 34064900899 00677286065 minus39596681544 minus896024143091 33799456580 27171435351 08969901053 minus26653896254 minus855488042282 11502969495 18118279175 12081251826 minus15371621151 minus78899506975

119901119894= 300

1199010= 150

1205790= 0

0 88187936296 36471591317 02649975386 minus37776011860 minus878521216671 36049445586 29421431766 11219899984 minus24403890591 minus832987989572 11869649855 19774683014 14492853591 minus12424112423 minus75536293888

119901119894= 150

1199010= 300

1205790= 400

0 83453833202 34064903276 00677284767 minus39596684114 minus896024179131 33799454509 27171434189 08969901711 minus26653895895 minus855488050342 11502969501 18118279178 12081251821 minus15371621158 minus78899507229

119901119894= 300

1199010= 150

1205790= 400

0 88187922056 36471584895 02649978782 minus37776012451 minus878521236811 36049444470 29421430233 11219900447 minus24403890386 minus832988022922 11869649854 19774683013 14492853591 minus12424112422 minus75536293885

119901119894= 150

1199010= 300

1205790= 800

0 83453820024 34064896339 00677286501 minus39596681965 minus896024157281 33799459533 27171437210 08969899701 minus26653898309 minus855488077952 11502969490 18118279176 12081251818 minus15371621155 minus78899507012

119901119894= 300

1199010= 150

1205790= 800

0 88187929120 36471586248 02649979578 minus37776012681 minus878521057051 36049446223 29421429798 11219900968 minus24403889573 minus832987897072 11869649853 19774683011 14492853590 minus12424112421 minus75536293882

1119864minus4lowast 119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 50001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 11199583152 04548258695 00269146492 minus04863265526 minus121916543001 06272799179 05095770638 01910931884 minus04339078950 minus153520394292 03263903739 05038038750 03586772209 minus03274306376 minus19402344736

119901119894= 300

1199010= 150

1205790= 0

0 11512567236 04761058146 00480074980 minus04638915755 minus119483525601 06496608008 05320325649 02137603620 minus04118738110 minus151255087832 03354423760 05269311856 03869645881 minus03022635649 minus19278571554

119901119894= 150

1199010= 300

1205790= 400

0 11199579631 04548258318 00269147731 minus04863261541 minus121916558021 06272799298 05095770743 01910931873 minus04339077941 minus153520404932 03263904649 05038039194 03586771853 minus03274308913 minus19402307798

119901119894= 300

1199010= 150

1205790= 400

0 11512569402 04761057447 00480074391 minus04638917626 minus119483523931 06496607027 05320325183 02137605230 minus04118740006 minus151255088322 03354423676 05269312306 03869645451 minus03022636353 minus19278567951

119901119894= 150

1199010= 300

1205790= 800

0 11199579102 04548258849 00269148341 minus04863259137 minus121916561241 06272799292 05095770557 01910931905 minus04339078242 minus153520389162 03263903269 05038037303 03586773178 minus03274301493 minus19402403280

119901119894= 300

1199010= 150

1205790= 800

0 11512563866 04761057469 00480078293 minus04638922712 minus119483505711 06496607065 05320325064 02137604900 minus04118740845 minus151255078652 03354423734 05269312334 03869645540 minus03022636247 minus19278571251

the internal pressure these stresses are maximum at externalsurface for homogeneous as well as functionally gradedstainless steel composite cylinder Also it has been observedthat circumferential stress is maximum at internal surface forhomogeneous cylinder while maximum at external surface

for functionally graded stainless steel composite cylinderwhen external pressure is less than the internal pressure Alsowith the increase in angular speed circumferential stressesincrease significantly From Figure 3 it can be seen that ascircumferential stresses increase for homogeneous cylinder

Advances in Materials Science and Engineering 7

Table 3 Circumferential stresses for rotating cylinder with variable thickness and variable density 119897 = 07 119889 = 05 120596 = 300 500 strainhardening measure119898 = 04 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 300 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 4078610121 1692083635 minus0362927882 minus3599668076 minus83881223761 1844023846 1536347566 0332616455 minus2596999032 minus81860251262 0698588556 1166718627 0723851443 minus1632455674 minus7703313292

119901119894= 300

1199010= 150

1205790= 0

0 4468498659 1869943306 minus0216047724 minus3455691317 minus82392598161 2039946535 1721858320 0520942472 minus2403128817 minus79859401892 0725452105 1314126561 0957769537 minus1332711442 minus7351503187

119901119894= 150

1199010= 300

1205790= 400

0 4078610926 1692083796 minus0362928200 minus3599667939 minus83881241561 1844023759 1536347360 0332616655 minus2596998868 minus81860264052 0698588485 1166718653 0723851463 minus1632455697 minus7703312785

119901119894= 300

1199010= 150

1205790= 400

0 4468498329 1869943517 minus0216047300 minus3455691093 minus82392517301 2039946605 1721858238 0520942423 minus2403128796 minus79859403832 0725452128 1314126566 0957769488 minus1332711458 minus7351503426

119901119894= 150

1199010= 300

1205790= 800

0 4078610278 1692083456 minus0362927548 minus3599667594 minus83881250271 1844023766 1536347565 0332616556 minus2596998868 minus81860262912 0698588473 1166718654 0723851445 minus1632455676 minus7703312834

119901119894= 300

1199010= 150

1205790= 800

0 4468498296 1869943321 minus0216047655 minus3455691489 minus82392377231 2039946200 1721858288 0520942491 minus2403128819 minus79859400642 0725452082 1314126521 0957769539 minus1332711402 minus7351503219

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 500 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 0502344054 0209118575 minus0032085608 minus0411840852 minus10757559141 0301690685 0253918828 0067872877 minus0387759274 minus13375609912 0170338387 0272622555 0177093634 minus0328795174 minus1672186161

119901119894= 300

1199010= 150

1205790= 0

0 0520602572 0223194821 minus0013327208 minus0385714307 minus10416636041 0314858338 0269151250 0087139893 minus0362225697 minus13033628312 0175533196 0289517104 0201213547 minus0301959686 minus1648231058

119901119894= 150

1199010= 300

1205790= 400

0 0502344076 0209118532 minus0032085631 minus0411840853 minus10757560931 0301690546 0253918731 0067873081 minus0387757210 minus13375620512 0170338398 0272622541 0177093558 minus0328795106 minus1672185933

119901119894= 300

1199010= 150

1205790= 400

0 0520602619 0223194813 minus0013327161 minus0385714149 minus10416645331 0314858326 0269151199 0087139890 minus0362225687 minus13033627732 0175533191 0289517093 0201213591 minus0301959717 minus1648231090

119901119894= 150

1199010= 300

1205790= 800

0 0502344041 0209118525 minus0032085593 minus0411841019 minus10757557561 0301690548 0253918749 0067873075 minus0387757974 minus13375601512 0170338413 0272622528 0177093598 minus0328795120 minus1672185805

119901119894= 300

1199010= 150

1205790= 800

0 0520602532 0223194802 minus0013327191 minus0385714162 minus10416644831 0314858370 0269151215 0087139921 minus0362225740 minus13033627232 0175533168 0289517159 0201213488 minus0301959669 minus1648230944

as well as for nonhomogenous cylinder but with the changein nonhomogeneity from 119890

1= 1 to 119890

1= 2 circumferential

stresses decrease when external pressure is greater than theinternal pressure with thermal effects Also these stresses

increase for homogenous cylinder as well as for functionallygraded stainless steel composite cylinder (119890

1= 1) while

decrease for functionally graded stainless steel compositecylinder with 119890

1= 2 With the increase in temperature these

8 Advances in Materials Science and Engineering

Table 4 Circumferential stresses for rotating cylinder with variable thickness and variable density 119897 = 07 119889 = 05 120596 = 300 500 strainhardening measure119898 = 06 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 300 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 4078550787 1692092827 minus0362917634 minus3599662505 minus83881243031 1844008854 1536344037 0332619755 minus2596989139 minus81860118602 0698583918 1166713498 0723849363 minus1632447308 minus7703291360

119901119894= 300

1199010= 150

1205790= 0

0 4468438868 1869957168 minus0216034432 minus3455686049 minus82394923451 2039930194 1721854313 0520946730 minus2403118071 minus79859271952 0725447053 1314120776 0957767141 minus1332701862 minus7351479020

119901119894= 150

1199010= 300

1205790= 400

0 4078550758 1692093003 minus0362917541 minus3599662684 minus83881247581 1844009034 1536344170 0332619845 minus2596989055 minus81860133232 0698583989 1166713507 0723849336 minus1632447294 minus7703291352

119901119894= 300

1199010= 150

1205790= 400

0 4468440841 1869957567 minus0216033626 minus3455685275 minus82395880401 2039930222 1721854440 0520946513 minus2403117933 minus79859260012 0725447107 1314120882 0957767050 minus1332701999 minus7351479230

119901119894= 150

1199010= 300

1205790= 800

0 4078551089 1692092847 minus0362917694 minus3599662435 minus83881264791 1844008698 1536343957 0332619859 minus2596988916 minus81860122392 0698583989 1166713524 0723849328 minus1632447317 minus7703291542

119901119894= 300

1199010= 150

1205790= 800

0 4468438120 1869956228 minus0216035341 minus3455687051 minus82393686321 2039930039 1721854569 0520946538 minus2403117966 minus79859257312 0725447049 1314120768 0957767137 minus1332701833 minus7351479029

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 500 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 0502339190 0209119023 minus0032080434 minus0411843003 minus10757515751 0301689441 0253919137 0067872897 minus0387758158 minus13375674432 0170338021 0272622433 0177094183 minus0328793821 minus1672191050

119901119894= 300

1199010= 150

1205790= 0

0 0520600260 0223196743 minus0013328691 minus0385729358 minus10416359501 0314857828 0269151955 0087140006 minus0362234515 minus13033506742 0175532879 0289517002 0201213980 minus0301960040 minus1648233514

119901119894= 150

1199010= 300

1205790= 400

0 0502339173 0209119009 minus0032080335 minus0411843048 minus10757520131 0301689029 0253918998 0067873493 minus0387754850 minus13375668252 0170338061 0272622405 0177094107 minus0328793799 minus1672190560

119901119894= 300

1199010= 150

1205790= 400

0 0520600268 0223196732 minus0013328653 minus0385729330 minus10416363341 0314857844 0269151936 0087139999 minus0362234465 minus13033507222 0175532856 0289516980 0201214076 minus0301960031 minus1648233768

119901119894= 150

1199010= 300

1205790= 800

0 0502339102 0209118939 minus0032080279 minus0411842835 minus10757521591 0301689252 0253919092 0067873202 minus0387759066 minus13375656632 0170338011 0272622432 0177094145 minus0328793808 minus1672190610

119901119894= 300

1199010= 150

1205790= 800

0 0520600345 0223196689 minus0013328705 minus0385729338 minus10416358761 0314857846 0269151949 0087139963 minus0362234522 minus13033507592 0175532843 0289516995 0201214043 minus0301960031 minus1648233661

stresses decrease significantly for homogeneous as well as forfunctionally graded stainless steel composite cylinder as canbe seen from Figure 4

Figures 5ndash7 have been drawn to discuss the effect ofinternal and external pressure on stresses in rotating cylin-der made of functionally graded stainless steel composite

Advances in Materials Science and Engineering 9

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 150 and p0 = 300

120596 = 300 e1 = 0

120596 = 300 e1 = 2

120596 = 500 e1 = 1

120596 = 300 e1 = 1

120596 = 500 e1 = 0

120596 = 500 e1 = 2

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 300 and p0 = 150

120596 = 300 e1 = 0

120596 = 300 e1 = 2

120596 = 500 e1 = 1

120596 = 300 e1 = 1

120596 = 500 e1 = 0

120596 = 500 e1 = 2

(b)

Figure 2 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density without thermal effects (1205790= 0) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

pi = 150 and p0 = 3001205721 = 0 1205790 = 400 m = 04

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 300 and p0 = 150

(b)

Figure 3 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density with thermal effects (1205790= 400) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 300 and p0 = 150

(b)

Figure 4 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density with thermal effects (1205790= 800) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

10 Advances in Materials Science and Engineering

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 300 and p0 = 150

(b)

Figure 5 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density without thermal effects (1205790= 0) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 300 and p0 = 150

(b)

Figure 6 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 400) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 300 and p0 = 150

(b)

Figure 7 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 800) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

Advances in Materials Science and Engineering 11

material with variable thickness and variable density withnonlinear strain hardening measure

It has been observed from Figure 5 that circumferentialstresses are maximum at external surface for homogeneousas well as functionally graded stainless steel compositecylinder It has also been observed that with the increase inangular speed circumferential stresses increase significantlyWith the introduction of thermal effects circumferentialstresses increase for homogeneous cylinder as well as forfunctionally graded stainless steel composite cylinder (119890

1=

1) but with the change in nonhomogeneity from 1198901= 1

to 1198901= 2 circumferential stresses decrease when external

pressure is greater than the internal pressure as can beseen from Figure 6 Circumferential stresses decrease forhomogeneous cylinder while increase for functionally gradedstainless steel composite cylinder when external pressure isless than the internal pressure It has also been observedfrom Figure 7 that with the increase in temperature thesestresses increase significantly for homogeneous cylinder aswell for functionally graded stainless steel composite cylinderwith 119890

1= 1 while decrease for functionally graded stainless

steel composite cylinder with 1198901= 2 when external pressure

is greater than the internal pressure while these stressesdecrease significantly for homogeneous cylinder as well forfunctionally graded stainless steel composite cylinder whenexternal pressure is less than the internal pressure

5 Conclusion

From the analysis we can conclude that rotating cylin-der made of functionally graded stainless steel compositematerial having variable thickness and variable density withSwiftrsquos strain hardening measure 119898 = 06 and thermalloading is better choice for designers as compared to rotatingcylinder with constant thickness and constant density Thisis because of the reason that circumferential stress is less forfunctionally graded stainless steel composite cylinder withvariable thickness and variable density as compared to othercases which leads to the idea of stress saving that minimizesthe possibility of fracture of cylinder

Conflict of Interests

The authors declare that they have no conflict of interest

References

[1] S Suresh and A Mortensen Fundamentals of FunctionallyGradedMaterials IOM3 Maney Publishing London UK 1998

[2] J N Reddy ldquoAnalysis of functionally graded platesrdquo Interna-tional Journal for Numerical Methods in Engineering vol 47 no1ndash3 pp 663ndash684 2000

[3] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill NewYork NY USA 3rd edition 1970

[4] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998

[5] Y Obata T Oji and N Noda ldquoSteady thermal stresses in a hol-low circular cylinder and a hollow sphere made of functionallygraded material (analysis with perturbation method)rdquo Reports

of the National Industrial Research Institute of Nagoya vol 47no 4 pp 317ndash338 1998

[6] J Perry and J Aboudi ldquoElasto-plastic stresses in thick walledcylindersrdquo Journal of Pressure Vessel Technology vol 125 no 3pp 248ndash252 2003

[7] X-L Gao ldquoElasto-plastic analysis of an internally pressurizedthick-walled cylinder using a strain gradient plasticity theoryrdquoInternational Journal of Solids and Structures vol 40 no 23 pp6445ndash6455 2003

[8] N Tutuncu and M Ozturk ldquoExact solutions for stresses infunctionally graded pressure vesselsrdquo Composites B vol 32 no8 pp 683ndash686 2001

[9] T Singh and V K Gupta ldquoCreep analysis of an internallypressurized thick cylinder made of a functionally graded com-positerdquo Journal of Strain Analysis for Engineering Design vol 44no 7 pp 583ndash594 2009

[10] A K Aggarwal Sharma Richa and Sharma Sanjeev ldquoSafetyanalysis using lebesgue strain measure of thick-walled cylinderfor functionally graded material under internal and externalpressurerdquo The Scientific World Journal vol 2013 Article ID676190 10 pages 2013

[11] A K Aggarwal S Richa and S Sanjeev ldquoSafety analysis ofthermal creep non-homogeneous thick-walled circular cylinderunder internal and external pressure using lebesgue strainmeasurerdquoMultidiscipline Modeling in Materials and Structuresvol 9 no 4 2013

[12] A Parvizi R Naghdabadi and J Arghavani ldquoAnalysis of AlA359SiCp functionally graded cylinder subjected to internalpressure and temperature gradient with elastic-plastic deforma-tionrdquo Journal of Thermal Stresses vol 34 no 10 pp 1054ndash10702011

[13] A N Eraslan and F Akgul ldquoYielding and elastoplastic deforma-tion of annular disks of a parabolic section subject to externalcompressionrdquo Turkish Journal of Engineering and Environmen-tal Sciences vol 29 no 1 pp 51ndash60 2005

Submit your manuscripts athttpwwwhindawicom

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Journal ofNanomaterials

Page 5: Research Article Thermo Elastic-Plastic Analysis of Rotating …downloads.hindawi.com/journals/amse/2013/810508.pdf · 2019-07-31 · ermo elastic-plastic analysis of functionally

Advances in Materials Science and Engineering 5

Table 1 Circumferential stresses for rotating cylinder with constant thickness and constant density 120596 = 300 500 nonlinear strain hardeningmeasure119898 = 04 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 30001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 83454530141 34064915847 00677187483 minus39596784280 minus896024571031 33799641785 27171534961 08969909345 minus26654005915 minus855489788262 11503024505 18118357165 12081310260 minus15371690373 minus78899736228

119901119894= 300

1199010= 150

1205790= 0

0 88188655176 36471590677 02649872301 minus37776104911 minus878521510291 36049642996 29421535689 11219909709 minus24404004448 minus832989835602 11869707579 19774764390 14492915544 minus12424185539 minus75536537230

119901119894= 150

1199010= 300

1205790= 400

0 83454529134 34064914474 00677186071 minus39596781512 minus896024633301 33799641163 27171535146 08969910927 minus26654006381 minus855489857932 11503024507 18118357152 12081310262 minus15371690375 minus78899736082

119901119894= 300

1199010= 150

1205790= 400

0 88188661275 36471591256 02649873453 minus37776106334 minus878521459301 36049644528 29421536040 11219910130 minus24404005339 minus832989848932 11869707578 19774764389 14492915543 minus12424185538 minus75536537227

119901119894= 150

1199010= 300

1205790= 800

0 83454532713 34064916133 00677188786 minus39596783083 minus896024607391 03799644417 27171537078 08969909208 minus26654005810 minus855489848852 11503024510 18118357165 12081310255 minus15371690384 minus78899735899

119901119894= 300

1199010= 150

1205790= 800

0 88188658525 36471597197 02649871677 minus37776107003 minus878521600471 36049639860 29421535165 11219910039 minus24404004285 minus832989834522 11869707577 19774764387 14492915543 minus12424185537 minus75536537224

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 50001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 11199643681 04548287897 00269123600 minus04863252783 minus121916263931 06272813080 05095768873 01910926579 minus04339086008 minus153519940132 03263905498 05038039436 03586769803 minus03274307737 minus19402396397

119901119894= 300

1199010= 150

1205790= 0

0 11512903744 04761073745 00479630332 minus04638575699 minus119486110581 06497506092 05320656463 02136358300 minus04115308637 minus151265455362 03354452568 05269344433 03869603568 minus03022694774 minus19278605585

119901119894= 150

1199010= 300

1205790= 400

0 11199644787 04548287386 00269122948 minus04863252189 minus121916251551 06272812978 05095769102 01910926887 minus04339085404 minus153519934162 03263905755 05038038300 03586769873 minus03274306978 minus19402449592

119901119894= 300

1199010= 150

1205790= 400

0 11512904788 04761073218 00479628568 minus04638573696 minus119486136701 06497506075 05320656332 02136358153 minus04115307488 minus151265453652 03354452679 05269344696 03869603369 minus03022695403 minus19278600529

119901119894= 150

1199010= 300

1205790= 800

0 11199642715 04548288045 00269123491 minus04863251255 minus121916247691 11199644787 04548287386 00269122948 minus04863252189 minus121916251552 03263905007 05038038190 03586771087 minus03274306558 minus19402425453

119901119894= 300

1199010= 150

1205790= 800

0 11512903765 04761073125 00479629909 minus04638574341 minus119486128581 06497505961 05320656017 02136358897 minus04115308258 minus151265454652 03354452477 05269344295 03869603736 minus03022695113 minus19278604854

composite cylinder with varying thickness and density as canbe seen from Table 4

Figures 2ndash4 have been drawn to discuss the effect ofinternal and external pressure on stresses in rotating cylin-der made of functionally graded stainless steel composite

material with constant thickness and constant density withnonlinear strain hardening measure

It has been observed from Figure 2 that circumferentialstress approaches towards compressive from tensile It hasalso been observed thatwhen external pressure is greater than

6 Advances in Materials Science and Engineering

Table 2 Circumferential stresses for rotating cylinder with constant thickness and constant density with 120596 = 300 500 nonlinear strainhardening measure119898 = 06 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa

119903

1198901

119898 = 06 120596 = 30001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 83453827289 34064900899 00677286065 minus39596681544 minus896024143091 33799456580 27171435351 08969901053 minus26653896254 minus855488042282 11502969495 18118279175 12081251826 minus15371621151 minus78899506975

119901119894= 300

1199010= 150

1205790= 0

0 88187936296 36471591317 02649975386 minus37776011860 minus878521216671 36049445586 29421431766 11219899984 minus24403890591 minus832987989572 11869649855 19774683014 14492853591 minus12424112423 minus75536293888

119901119894= 150

1199010= 300

1205790= 400

0 83453833202 34064903276 00677284767 minus39596684114 minus896024179131 33799454509 27171434189 08969901711 minus26653895895 minus855488050342 11502969501 18118279178 12081251821 minus15371621158 minus78899507229

119901119894= 300

1199010= 150

1205790= 400

0 88187922056 36471584895 02649978782 minus37776012451 minus878521236811 36049444470 29421430233 11219900447 minus24403890386 minus832988022922 11869649854 19774683013 14492853591 minus12424112422 minus75536293885

119901119894= 150

1199010= 300

1205790= 800

0 83453820024 34064896339 00677286501 minus39596681965 minus896024157281 33799459533 27171437210 08969899701 minus26653898309 minus855488077952 11502969490 18118279176 12081251818 minus15371621155 minus78899507012

119901119894= 300

1199010= 150

1205790= 800

0 88187929120 36471586248 02649979578 minus37776012681 minus878521057051 36049446223 29421429798 11219900968 minus24403889573 minus832987897072 11869649853 19774683011 14492853590 minus12424112421 minus75536293882

1119864minus4lowast 119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 50001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 11199583152 04548258695 00269146492 minus04863265526 minus121916543001 06272799179 05095770638 01910931884 minus04339078950 minus153520394292 03263903739 05038038750 03586772209 minus03274306376 minus19402344736

119901119894= 300

1199010= 150

1205790= 0

0 11512567236 04761058146 00480074980 minus04638915755 minus119483525601 06496608008 05320325649 02137603620 minus04118738110 minus151255087832 03354423760 05269311856 03869645881 minus03022635649 minus19278571554

119901119894= 150

1199010= 300

1205790= 400

0 11199579631 04548258318 00269147731 minus04863261541 minus121916558021 06272799298 05095770743 01910931873 minus04339077941 minus153520404932 03263904649 05038039194 03586771853 minus03274308913 minus19402307798

119901119894= 300

1199010= 150

1205790= 400

0 11512569402 04761057447 00480074391 minus04638917626 minus119483523931 06496607027 05320325183 02137605230 minus04118740006 minus151255088322 03354423676 05269312306 03869645451 minus03022636353 minus19278567951

119901119894= 150

1199010= 300

1205790= 800

0 11199579102 04548258849 00269148341 minus04863259137 minus121916561241 06272799292 05095770557 01910931905 minus04339078242 minus153520389162 03263903269 05038037303 03586773178 minus03274301493 minus19402403280

119901119894= 300

1199010= 150

1205790= 800

0 11512563866 04761057469 00480078293 minus04638922712 minus119483505711 06496607065 05320325064 02137604900 minus04118740845 minus151255078652 03354423734 05269312334 03869645540 minus03022636247 minus19278571251

the internal pressure these stresses are maximum at externalsurface for homogeneous as well as functionally gradedstainless steel composite cylinder Also it has been observedthat circumferential stress is maximum at internal surface forhomogeneous cylinder while maximum at external surface

for functionally graded stainless steel composite cylinderwhen external pressure is less than the internal pressure Alsowith the increase in angular speed circumferential stressesincrease significantly From Figure 3 it can be seen that ascircumferential stresses increase for homogeneous cylinder

Advances in Materials Science and Engineering 7

Table 3 Circumferential stresses for rotating cylinder with variable thickness and variable density 119897 = 07 119889 = 05 120596 = 300 500 strainhardening measure119898 = 04 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 300 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 4078610121 1692083635 minus0362927882 minus3599668076 minus83881223761 1844023846 1536347566 0332616455 minus2596999032 minus81860251262 0698588556 1166718627 0723851443 minus1632455674 minus7703313292

119901119894= 300

1199010= 150

1205790= 0

0 4468498659 1869943306 minus0216047724 minus3455691317 minus82392598161 2039946535 1721858320 0520942472 minus2403128817 minus79859401892 0725452105 1314126561 0957769537 minus1332711442 minus7351503187

119901119894= 150

1199010= 300

1205790= 400

0 4078610926 1692083796 minus0362928200 minus3599667939 minus83881241561 1844023759 1536347360 0332616655 minus2596998868 minus81860264052 0698588485 1166718653 0723851463 minus1632455697 minus7703312785

119901119894= 300

1199010= 150

1205790= 400

0 4468498329 1869943517 minus0216047300 minus3455691093 minus82392517301 2039946605 1721858238 0520942423 minus2403128796 minus79859403832 0725452128 1314126566 0957769488 minus1332711458 minus7351503426

119901119894= 150

1199010= 300

1205790= 800

0 4078610278 1692083456 minus0362927548 minus3599667594 minus83881250271 1844023766 1536347565 0332616556 minus2596998868 minus81860262912 0698588473 1166718654 0723851445 minus1632455676 minus7703312834

119901119894= 300

1199010= 150

1205790= 800

0 4468498296 1869943321 minus0216047655 minus3455691489 minus82392377231 2039946200 1721858288 0520942491 minus2403128819 minus79859400642 0725452082 1314126521 0957769539 minus1332711402 minus7351503219

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 500 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 0502344054 0209118575 minus0032085608 minus0411840852 minus10757559141 0301690685 0253918828 0067872877 minus0387759274 minus13375609912 0170338387 0272622555 0177093634 minus0328795174 minus1672186161

119901119894= 300

1199010= 150

1205790= 0

0 0520602572 0223194821 minus0013327208 minus0385714307 minus10416636041 0314858338 0269151250 0087139893 minus0362225697 minus13033628312 0175533196 0289517104 0201213547 minus0301959686 minus1648231058

119901119894= 150

1199010= 300

1205790= 400

0 0502344076 0209118532 minus0032085631 minus0411840853 minus10757560931 0301690546 0253918731 0067873081 minus0387757210 minus13375620512 0170338398 0272622541 0177093558 minus0328795106 minus1672185933

119901119894= 300

1199010= 150

1205790= 400

0 0520602619 0223194813 minus0013327161 minus0385714149 minus10416645331 0314858326 0269151199 0087139890 minus0362225687 minus13033627732 0175533191 0289517093 0201213591 minus0301959717 minus1648231090

119901119894= 150

1199010= 300

1205790= 800

0 0502344041 0209118525 minus0032085593 minus0411841019 minus10757557561 0301690548 0253918749 0067873075 minus0387757974 minus13375601512 0170338413 0272622528 0177093598 minus0328795120 minus1672185805

119901119894= 300

1199010= 150

1205790= 800

0 0520602532 0223194802 minus0013327191 minus0385714162 minus10416644831 0314858370 0269151215 0087139921 minus0362225740 minus13033627232 0175533168 0289517159 0201213488 minus0301959669 minus1648230944

as well as for nonhomogenous cylinder but with the changein nonhomogeneity from 119890

1= 1 to 119890

1= 2 circumferential

stresses decrease when external pressure is greater than theinternal pressure with thermal effects Also these stresses

increase for homogenous cylinder as well as for functionallygraded stainless steel composite cylinder (119890

1= 1) while

decrease for functionally graded stainless steel compositecylinder with 119890

1= 2 With the increase in temperature these

8 Advances in Materials Science and Engineering

Table 4 Circumferential stresses for rotating cylinder with variable thickness and variable density 119897 = 07 119889 = 05 120596 = 300 500 strainhardening measure119898 = 06 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 300 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 4078550787 1692092827 minus0362917634 minus3599662505 minus83881243031 1844008854 1536344037 0332619755 minus2596989139 minus81860118602 0698583918 1166713498 0723849363 minus1632447308 minus7703291360

119901119894= 300

1199010= 150

1205790= 0

0 4468438868 1869957168 minus0216034432 minus3455686049 minus82394923451 2039930194 1721854313 0520946730 minus2403118071 minus79859271952 0725447053 1314120776 0957767141 minus1332701862 minus7351479020

119901119894= 150

1199010= 300

1205790= 400

0 4078550758 1692093003 minus0362917541 minus3599662684 minus83881247581 1844009034 1536344170 0332619845 minus2596989055 minus81860133232 0698583989 1166713507 0723849336 minus1632447294 minus7703291352

119901119894= 300

1199010= 150

1205790= 400

0 4468440841 1869957567 minus0216033626 minus3455685275 minus82395880401 2039930222 1721854440 0520946513 minus2403117933 minus79859260012 0725447107 1314120882 0957767050 minus1332701999 minus7351479230

119901119894= 150

1199010= 300

1205790= 800

0 4078551089 1692092847 minus0362917694 minus3599662435 minus83881264791 1844008698 1536343957 0332619859 minus2596988916 minus81860122392 0698583989 1166713524 0723849328 minus1632447317 minus7703291542

119901119894= 300

1199010= 150

1205790= 800

0 4468438120 1869956228 minus0216035341 minus3455687051 minus82393686321 2039930039 1721854569 0520946538 minus2403117966 minus79859257312 0725447049 1314120768 0957767137 minus1332701833 minus7351479029

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 500 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 0502339190 0209119023 minus0032080434 minus0411843003 minus10757515751 0301689441 0253919137 0067872897 minus0387758158 minus13375674432 0170338021 0272622433 0177094183 minus0328793821 minus1672191050

119901119894= 300

1199010= 150

1205790= 0

0 0520600260 0223196743 minus0013328691 minus0385729358 minus10416359501 0314857828 0269151955 0087140006 minus0362234515 minus13033506742 0175532879 0289517002 0201213980 minus0301960040 minus1648233514

119901119894= 150

1199010= 300

1205790= 400

0 0502339173 0209119009 minus0032080335 minus0411843048 minus10757520131 0301689029 0253918998 0067873493 minus0387754850 minus13375668252 0170338061 0272622405 0177094107 minus0328793799 minus1672190560

119901119894= 300

1199010= 150

1205790= 400

0 0520600268 0223196732 minus0013328653 minus0385729330 minus10416363341 0314857844 0269151936 0087139999 minus0362234465 minus13033507222 0175532856 0289516980 0201214076 minus0301960031 minus1648233768

119901119894= 150

1199010= 300

1205790= 800

0 0502339102 0209118939 minus0032080279 minus0411842835 minus10757521591 0301689252 0253919092 0067873202 minus0387759066 minus13375656632 0170338011 0272622432 0177094145 minus0328793808 minus1672190610

119901119894= 300

1199010= 150

1205790= 800

0 0520600345 0223196689 minus0013328705 minus0385729338 minus10416358761 0314857846 0269151949 0087139963 minus0362234522 minus13033507592 0175532843 0289516995 0201214043 minus0301960031 minus1648233661

stresses decrease significantly for homogeneous as well as forfunctionally graded stainless steel composite cylinder as canbe seen from Figure 4

Figures 5ndash7 have been drawn to discuss the effect ofinternal and external pressure on stresses in rotating cylin-der made of functionally graded stainless steel composite

Advances in Materials Science and Engineering 9

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 150 and p0 = 300

120596 = 300 e1 = 0

120596 = 300 e1 = 2

120596 = 500 e1 = 1

120596 = 300 e1 = 1

120596 = 500 e1 = 0

120596 = 500 e1 = 2

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 300 and p0 = 150

120596 = 300 e1 = 0

120596 = 300 e1 = 2

120596 = 500 e1 = 1

120596 = 300 e1 = 1

120596 = 500 e1 = 0

120596 = 500 e1 = 2

(b)

Figure 2 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density without thermal effects (1205790= 0) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

pi = 150 and p0 = 3001205721 = 0 1205790 = 400 m = 04

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 300 and p0 = 150

(b)

Figure 3 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density with thermal effects (1205790= 400) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 300 and p0 = 150

(b)

Figure 4 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density with thermal effects (1205790= 800) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

10 Advances in Materials Science and Engineering

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 300 and p0 = 150

(b)

Figure 5 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density without thermal effects (1205790= 0) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 300 and p0 = 150

(b)

Figure 6 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 400) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 300 and p0 = 150

(b)

Figure 7 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 800) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

Advances in Materials Science and Engineering 11

material with variable thickness and variable density withnonlinear strain hardening measure

It has been observed from Figure 5 that circumferentialstresses are maximum at external surface for homogeneousas well as functionally graded stainless steel compositecylinder It has also been observed that with the increase inangular speed circumferential stresses increase significantlyWith the introduction of thermal effects circumferentialstresses increase for homogeneous cylinder as well as forfunctionally graded stainless steel composite cylinder (119890

1=

1) but with the change in nonhomogeneity from 1198901= 1

to 1198901= 2 circumferential stresses decrease when external

pressure is greater than the internal pressure as can beseen from Figure 6 Circumferential stresses decrease forhomogeneous cylinder while increase for functionally gradedstainless steel composite cylinder when external pressure isless than the internal pressure It has also been observedfrom Figure 7 that with the increase in temperature thesestresses increase significantly for homogeneous cylinder aswell for functionally graded stainless steel composite cylinderwith 119890

1= 1 while decrease for functionally graded stainless

steel composite cylinder with 1198901= 2 when external pressure

is greater than the internal pressure while these stressesdecrease significantly for homogeneous cylinder as well forfunctionally graded stainless steel composite cylinder whenexternal pressure is less than the internal pressure

5 Conclusion

From the analysis we can conclude that rotating cylin-der made of functionally graded stainless steel compositematerial having variable thickness and variable density withSwiftrsquos strain hardening measure 119898 = 06 and thermalloading is better choice for designers as compared to rotatingcylinder with constant thickness and constant density Thisis because of the reason that circumferential stress is less forfunctionally graded stainless steel composite cylinder withvariable thickness and variable density as compared to othercases which leads to the idea of stress saving that minimizesthe possibility of fracture of cylinder

Conflict of Interests

The authors declare that they have no conflict of interest

References

[1] S Suresh and A Mortensen Fundamentals of FunctionallyGradedMaterials IOM3 Maney Publishing London UK 1998

[2] J N Reddy ldquoAnalysis of functionally graded platesrdquo Interna-tional Journal for Numerical Methods in Engineering vol 47 no1ndash3 pp 663ndash684 2000

[3] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill NewYork NY USA 3rd edition 1970

[4] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998

[5] Y Obata T Oji and N Noda ldquoSteady thermal stresses in a hol-low circular cylinder and a hollow sphere made of functionallygraded material (analysis with perturbation method)rdquo Reports

of the National Industrial Research Institute of Nagoya vol 47no 4 pp 317ndash338 1998

[6] J Perry and J Aboudi ldquoElasto-plastic stresses in thick walledcylindersrdquo Journal of Pressure Vessel Technology vol 125 no 3pp 248ndash252 2003

[7] X-L Gao ldquoElasto-plastic analysis of an internally pressurizedthick-walled cylinder using a strain gradient plasticity theoryrdquoInternational Journal of Solids and Structures vol 40 no 23 pp6445ndash6455 2003

[8] N Tutuncu and M Ozturk ldquoExact solutions for stresses infunctionally graded pressure vesselsrdquo Composites B vol 32 no8 pp 683ndash686 2001

[9] T Singh and V K Gupta ldquoCreep analysis of an internallypressurized thick cylinder made of a functionally graded com-positerdquo Journal of Strain Analysis for Engineering Design vol 44no 7 pp 583ndash594 2009

[10] A K Aggarwal Sharma Richa and Sharma Sanjeev ldquoSafetyanalysis using lebesgue strain measure of thick-walled cylinderfor functionally graded material under internal and externalpressurerdquo The Scientific World Journal vol 2013 Article ID676190 10 pages 2013

[11] A K Aggarwal S Richa and S Sanjeev ldquoSafety analysis ofthermal creep non-homogeneous thick-walled circular cylinderunder internal and external pressure using lebesgue strainmeasurerdquoMultidiscipline Modeling in Materials and Structuresvol 9 no 4 2013

[12] A Parvizi R Naghdabadi and J Arghavani ldquoAnalysis of AlA359SiCp functionally graded cylinder subjected to internalpressure and temperature gradient with elastic-plastic deforma-tionrdquo Journal of Thermal Stresses vol 34 no 10 pp 1054ndash10702011

[13] A N Eraslan and F Akgul ldquoYielding and elastoplastic deforma-tion of annular disks of a parabolic section subject to externalcompressionrdquo Turkish Journal of Engineering and Environmen-tal Sciences vol 29 no 1 pp 51ndash60 2005

Submit your manuscripts athttpwwwhindawicom

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Journal ofNanomaterials

Page 6: Research Article Thermo Elastic-Plastic Analysis of Rotating …downloads.hindawi.com/journals/amse/2013/810508.pdf · 2019-07-31 · ermo elastic-plastic analysis of functionally

6 Advances in Materials Science and Engineering

Table 2 Circumferential stresses for rotating cylinder with constant thickness and constant density with 120596 = 300 500 nonlinear strainhardening measure119898 = 06 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa

119903

1198901

119898 = 06 120596 = 30001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 83453827289 34064900899 00677286065 minus39596681544 minus896024143091 33799456580 27171435351 08969901053 minus26653896254 minus855488042282 11502969495 18118279175 12081251826 minus15371621151 minus78899506975

119901119894= 300

1199010= 150

1205790= 0

0 88187936296 36471591317 02649975386 minus37776011860 minus878521216671 36049445586 29421431766 11219899984 minus24403890591 minus832987989572 11869649855 19774683014 14492853591 minus12424112423 minus75536293888

119901119894= 150

1199010= 300

1205790= 400

0 83453833202 34064903276 00677284767 minus39596684114 minus896024179131 33799454509 27171434189 08969901711 minus26653895895 minus855488050342 11502969501 18118279178 12081251821 minus15371621158 minus78899507229

119901119894= 300

1199010= 150

1205790= 400

0 88187922056 36471584895 02649978782 minus37776012451 minus878521236811 36049444470 29421430233 11219900447 minus24403890386 minus832988022922 11869649854 19774683013 14492853591 minus12424112422 minus75536293885

119901119894= 150

1199010= 300

1205790= 800

0 83453820024 34064896339 00677286501 minus39596681965 minus896024157281 33799459533 27171437210 08969899701 minus26653898309 minus855488077952 11502969490 18118279176 12081251818 minus15371621155 minus78899507012

119901119894= 300

1199010= 150

1205790= 800

0 88187929120 36471586248 02649979578 minus37776012681 minus878521057051 36049446223 29421429798 11219900968 minus24403889573 minus832987897072 11869649853 19774683011 14492853590 minus12424112421 minus75536293882

1119864minus4lowast 119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 50001 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 11199583152 04548258695 00269146492 minus04863265526 minus121916543001 06272799179 05095770638 01910931884 minus04339078950 minus153520394292 03263903739 05038038750 03586772209 minus03274306376 minus19402344736

119901119894= 300

1199010= 150

1205790= 0

0 11512567236 04761058146 00480074980 minus04638915755 minus119483525601 06496608008 05320325649 02137603620 minus04118738110 minus151255087832 03354423760 05269311856 03869645881 minus03022635649 minus19278571554

119901119894= 150

1199010= 300

1205790= 400

0 11199579631 04548258318 00269147731 minus04863261541 minus121916558021 06272799298 05095770743 01910931873 minus04339077941 minus153520404932 03263904649 05038039194 03586771853 minus03274308913 minus19402307798

119901119894= 300

1199010= 150

1205790= 400

0 11512569402 04761057447 00480074391 minus04638917626 minus119483523931 06496607027 05320325183 02137605230 minus04118740006 minus151255088322 03354423676 05269312306 03869645451 minus03022636353 minus19278567951

119901119894= 150

1199010= 300

1205790= 800

0 11199579102 04548258849 00269148341 minus04863259137 minus121916561241 06272799292 05095770557 01910931905 minus04339078242 minus153520389162 03263903269 05038037303 03586773178 minus03274301493 minus19402403280

119901119894= 300

1199010= 150

1205790= 800

0 11512563866 04761057469 00480078293 minus04638922712 minus119483505711 06496607065 05320325064 02137604900 minus04118740845 minus151255078652 03354423734 05269312334 03869645540 minus03022636247 minus19278571251

the internal pressure these stresses are maximum at externalsurface for homogeneous as well as functionally gradedstainless steel composite cylinder Also it has been observedthat circumferential stress is maximum at internal surface forhomogeneous cylinder while maximum at external surface

for functionally graded stainless steel composite cylinderwhen external pressure is less than the internal pressure Alsowith the increase in angular speed circumferential stressesincrease significantly From Figure 3 it can be seen that ascircumferential stresses increase for homogeneous cylinder

Advances in Materials Science and Engineering 7

Table 3 Circumferential stresses for rotating cylinder with variable thickness and variable density 119897 = 07 119889 = 05 120596 = 300 500 strainhardening measure119898 = 04 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 300 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 4078610121 1692083635 minus0362927882 minus3599668076 minus83881223761 1844023846 1536347566 0332616455 minus2596999032 minus81860251262 0698588556 1166718627 0723851443 minus1632455674 minus7703313292

119901119894= 300

1199010= 150

1205790= 0

0 4468498659 1869943306 minus0216047724 minus3455691317 minus82392598161 2039946535 1721858320 0520942472 minus2403128817 minus79859401892 0725452105 1314126561 0957769537 minus1332711442 minus7351503187

119901119894= 150

1199010= 300

1205790= 400

0 4078610926 1692083796 minus0362928200 minus3599667939 minus83881241561 1844023759 1536347360 0332616655 minus2596998868 minus81860264052 0698588485 1166718653 0723851463 minus1632455697 minus7703312785

119901119894= 300

1199010= 150

1205790= 400

0 4468498329 1869943517 minus0216047300 minus3455691093 minus82392517301 2039946605 1721858238 0520942423 minus2403128796 minus79859403832 0725452128 1314126566 0957769488 minus1332711458 minus7351503426

119901119894= 150

1199010= 300

1205790= 800

0 4078610278 1692083456 minus0362927548 minus3599667594 minus83881250271 1844023766 1536347565 0332616556 minus2596998868 minus81860262912 0698588473 1166718654 0723851445 minus1632455676 minus7703312834

119901119894= 300

1199010= 150

1205790= 800

0 4468498296 1869943321 minus0216047655 minus3455691489 minus82392377231 2039946200 1721858288 0520942491 minus2403128819 minus79859400642 0725452082 1314126521 0957769539 minus1332711402 minus7351503219

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 500 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 0502344054 0209118575 minus0032085608 minus0411840852 minus10757559141 0301690685 0253918828 0067872877 minus0387759274 minus13375609912 0170338387 0272622555 0177093634 minus0328795174 minus1672186161

119901119894= 300

1199010= 150

1205790= 0

0 0520602572 0223194821 minus0013327208 minus0385714307 minus10416636041 0314858338 0269151250 0087139893 minus0362225697 minus13033628312 0175533196 0289517104 0201213547 minus0301959686 minus1648231058

119901119894= 150

1199010= 300

1205790= 400

0 0502344076 0209118532 minus0032085631 minus0411840853 minus10757560931 0301690546 0253918731 0067873081 minus0387757210 minus13375620512 0170338398 0272622541 0177093558 minus0328795106 minus1672185933

119901119894= 300

1199010= 150

1205790= 400

0 0520602619 0223194813 minus0013327161 minus0385714149 minus10416645331 0314858326 0269151199 0087139890 minus0362225687 minus13033627732 0175533191 0289517093 0201213591 minus0301959717 minus1648231090

119901119894= 150

1199010= 300

1205790= 800

0 0502344041 0209118525 minus0032085593 minus0411841019 minus10757557561 0301690548 0253918749 0067873075 minus0387757974 minus13375601512 0170338413 0272622528 0177093598 minus0328795120 minus1672185805

119901119894= 300

1199010= 150

1205790= 800

0 0520602532 0223194802 minus0013327191 minus0385714162 minus10416644831 0314858370 0269151215 0087139921 minus0362225740 minus13033627232 0175533168 0289517159 0201213488 minus0301959669 minus1648230944

as well as for nonhomogenous cylinder but with the changein nonhomogeneity from 119890

1= 1 to 119890

1= 2 circumferential

stresses decrease when external pressure is greater than theinternal pressure with thermal effects Also these stresses

increase for homogenous cylinder as well as for functionallygraded stainless steel composite cylinder (119890

1= 1) while

decrease for functionally graded stainless steel compositecylinder with 119890

1= 2 With the increase in temperature these

8 Advances in Materials Science and Engineering

Table 4 Circumferential stresses for rotating cylinder with variable thickness and variable density 119897 = 07 119889 = 05 120596 = 300 500 strainhardening measure119898 = 06 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 300 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 4078550787 1692092827 minus0362917634 minus3599662505 minus83881243031 1844008854 1536344037 0332619755 minus2596989139 minus81860118602 0698583918 1166713498 0723849363 minus1632447308 minus7703291360

119901119894= 300

1199010= 150

1205790= 0

0 4468438868 1869957168 minus0216034432 minus3455686049 minus82394923451 2039930194 1721854313 0520946730 minus2403118071 minus79859271952 0725447053 1314120776 0957767141 minus1332701862 minus7351479020

119901119894= 150

1199010= 300

1205790= 400

0 4078550758 1692093003 minus0362917541 minus3599662684 minus83881247581 1844009034 1536344170 0332619845 minus2596989055 minus81860133232 0698583989 1166713507 0723849336 minus1632447294 minus7703291352

119901119894= 300

1199010= 150

1205790= 400

0 4468440841 1869957567 minus0216033626 minus3455685275 minus82395880401 2039930222 1721854440 0520946513 minus2403117933 minus79859260012 0725447107 1314120882 0957767050 minus1332701999 minus7351479230

119901119894= 150

1199010= 300

1205790= 800

0 4078551089 1692092847 minus0362917694 minus3599662435 minus83881264791 1844008698 1536343957 0332619859 minus2596988916 minus81860122392 0698583989 1166713524 0723849328 minus1632447317 minus7703291542

119901119894= 300

1199010= 150

1205790= 800

0 4468438120 1869956228 minus0216035341 minus3455687051 minus82393686321 2039930039 1721854569 0520946538 minus2403117966 minus79859257312 0725447049 1314120768 0957767137 minus1332701833 minus7351479029

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 500 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 0502339190 0209119023 minus0032080434 minus0411843003 minus10757515751 0301689441 0253919137 0067872897 minus0387758158 minus13375674432 0170338021 0272622433 0177094183 minus0328793821 minus1672191050

119901119894= 300

1199010= 150

1205790= 0

0 0520600260 0223196743 minus0013328691 minus0385729358 minus10416359501 0314857828 0269151955 0087140006 minus0362234515 minus13033506742 0175532879 0289517002 0201213980 minus0301960040 minus1648233514

119901119894= 150

1199010= 300

1205790= 400

0 0502339173 0209119009 minus0032080335 minus0411843048 minus10757520131 0301689029 0253918998 0067873493 minus0387754850 minus13375668252 0170338061 0272622405 0177094107 minus0328793799 minus1672190560

119901119894= 300

1199010= 150

1205790= 400

0 0520600268 0223196732 minus0013328653 minus0385729330 minus10416363341 0314857844 0269151936 0087139999 minus0362234465 minus13033507222 0175532856 0289516980 0201214076 minus0301960031 minus1648233768

119901119894= 150

1199010= 300

1205790= 800

0 0502339102 0209118939 minus0032080279 minus0411842835 minus10757521591 0301689252 0253919092 0067873202 minus0387759066 minus13375656632 0170338011 0272622432 0177094145 minus0328793808 minus1672190610

119901119894= 300

1199010= 150

1205790= 800

0 0520600345 0223196689 minus0013328705 minus0385729338 minus10416358761 0314857846 0269151949 0087139963 minus0362234522 minus13033507592 0175532843 0289516995 0201214043 minus0301960031 minus1648233661

stresses decrease significantly for homogeneous as well as forfunctionally graded stainless steel composite cylinder as canbe seen from Figure 4

Figures 5ndash7 have been drawn to discuss the effect ofinternal and external pressure on stresses in rotating cylin-der made of functionally graded stainless steel composite

Advances in Materials Science and Engineering 9

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 150 and p0 = 300

120596 = 300 e1 = 0

120596 = 300 e1 = 2

120596 = 500 e1 = 1

120596 = 300 e1 = 1

120596 = 500 e1 = 0

120596 = 500 e1 = 2

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 300 and p0 = 150

120596 = 300 e1 = 0

120596 = 300 e1 = 2

120596 = 500 e1 = 1

120596 = 300 e1 = 1

120596 = 500 e1 = 0

120596 = 500 e1 = 2

(b)

Figure 2 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density without thermal effects (1205790= 0) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

pi = 150 and p0 = 3001205721 = 0 1205790 = 400 m = 04

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 300 and p0 = 150

(b)

Figure 3 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density with thermal effects (1205790= 400) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 300 and p0 = 150

(b)

Figure 4 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density with thermal effects (1205790= 800) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

10 Advances in Materials Science and Engineering

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 300 and p0 = 150

(b)

Figure 5 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density without thermal effects (1205790= 0) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 300 and p0 = 150

(b)

Figure 6 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 400) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 300 and p0 = 150

(b)

Figure 7 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 800) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

Advances in Materials Science and Engineering 11

material with variable thickness and variable density withnonlinear strain hardening measure

It has been observed from Figure 5 that circumferentialstresses are maximum at external surface for homogeneousas well as functionally graded stainless steel compositecylinder It has also been observed that with the increase inangular speed circumferential stresses increase significantlyWith the introduction of thermal effects circumferentialstresses increase for homogeneous cylinder as well as forfunctionally graded stainless steel composite cylinder (119890

1=

1) but with the change in nonhomogeneity from 1198901= 1

to 1198901= 2 circumferential stresses decrease when external

pressure is greater than the internal pressure as can beseen from Figure 6 Circumferential stresses decrease forhomogeneous cylinder while increase for functionally gradedstainless steel composite cylinder when external pressure isless than the internal pressure It has also been observedfrom Figure 7 that with the increase in temperature thesestresses increase significantly for homogeneous cylinder aswell for functionally graded stainless steel composite cylinderwith 119890

1= 1 while decrease for functionally graded stainless

steel composite cylinder with 1198901= 2 when external pressure

is greater than the internal pressure while these stressesdecrease significantly for homogeneous cylinder as well forfunctionally graded stainless steel composite cylinder whenexternal pressure is less than the internal pressure

5 Conclusion

From the analysis we can conclude that rotating cylin-der made of functionally graded stainless steel compositematerial having variable thickness and variable density withSwiftrsquos strain hardening measure 119898 = 06 and thermalloading is better choice for designers as compared to rotatingcylinder with constant thickness and constant density Thisis because of the reason that circumferential stress is less forfunctionally graded stainless steel composite cylinder withvariable thickness and variable density as compared to othercases which leads to the idea of stress saving that minimizesthe possibility of fracture of cylinder

Conflict of Interests

The authors declare that they have no conflict of interest

References

[1] S Suresh and A Mortensen Fundamentals of FunctionallyGradedMaterials IOM3 Maney Publishing London UK 1998

[2] J N Reddy ldquoAnalysis of functionally graded platesrdquo Interna-tional Journal for Numerical Methods in Engineering vol 47 no1ndash3 pp 663ndash684 2000

[3] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill NewYork NY USA 3rd edition 1970

[4] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998

[5] Y Obata T Oji and N Noda ldquoSteady thermal stresses in a hol-low circular cylinder and a hollow sphere made of functionallygraded material (analysis with perturbation method)rdquo Reports

of the National Industrial Research Institute of Nagoya vol 47no 4 pp 317ndash338 1998

[6] J Perry and J Aboudi ldquoElasto-plastic stresses in thick walledcylindersrdquo Journal of Pressure Vessel Technology vol 125 no 3pp 248ndash252 2003

[7] X-L Gao ldquoElasto-plastic analysis of an internally pressurizedthick-walled cylinder using a strain gradient plasticity theoryrdquoInternational Journal of Solids and Structures vol 40 no 23 pp6445ndash6455 2003

[8] N Tutuncu and M Ozturk ldquoExact solutions for stresses infunctionally graded pressure vesselsrdquo Composites B vol 32 no8 pp 683ndash686 2001

[9] T Singh and V K Gupta ldquoCreep analysis of an internallypressurized thick cylinder made of a functionally graded com-positerdquo Journal of Strain Analysis for Engineering Design vol 44no 7 pp 583ndash594 2009

[10] A K Aggarwal Sharma Richa and Sharma Sanjeev ldquoSafetyanalysis using lebesgue strain measure of thick-walled cylinderfor functionally graded material under internal and externalpressurerdquo The Scientific World Journal vol 2013 Article ID676190 10 pages 2013

[11] A K Aggarwal S Richa and S Sanjeev ldquoSafety analysis ofthermal creep non-homogeneous thick-walled circular cylinderunder internal and external pressure using lebesgue strainmeasurerdquoMultidiscipline Modeling in Materials and Structuresvol 9 no 4 2013

[12] A Parvizi R Naghdabadi and J Arghavani ldquoAnalysis of AlA359SiCp functionally graded cylinder subjected to internalpressure and temperature gradient with elastic-plastic deforma-tionrdquo Journal of Thermal Stresses vol 34 no 10 pp 1054ndash10702011

[13] A N Eraslan and F Akgul ldquoYielding and elastoplastic deforma-tion of annular disks of a parabolic section subject to externalcompressionrdquo Turkish Journal of Engineering and Environmen-tal Sciences vol 29 no 1 pp 51ndash60 2005

Submit your manuscripts athttpwwwhindawicom

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Journal ofNanomaterials

Page 7: Research Article Thermo Elastic-Plastic Analysis of Rotating …downloads.hindawi.com/journals/amse/2013/810508.pdf · 2019-07-31 · ermo elastic-plastic analysis of functionally

Advances in Materials Science and Engineering 7

Table 3 Circumferential stresses for rotating cylinder with variable thickness and variable density 119897 = 07 119889 = 05 120596 = 300 500 strainhardening measure119898 = 04 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 300 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 4078610121 1692083635 minus0362927882 minus3599668076 minus83881223761 1844023846 1536347566 0332616455 minus2596999032 minus81860251262 0698588556 1166718627 0723851443 minus1632455674 minus7703313292

119901119894= 300

1199010= 150

1205790= 0

0 4468498659 1869943306 minus0216047724 minus3455691317 minus82392598161 2039946535 1721858320 0520942472 minus2403128817 minus79859401892 0725452105 1314126561 0957769537 minus1332711442 minus7351503187

119901119894= 150

1199010= 300

1205790= 400

0 4078610926 1692083796 minus0362928200 minus3599667939 minus83881241561 1844023759 1536347360 0332616655 minus2596998868 minus81860264052 0698588485 1166718653 0723851463 minus1632455697 minus7703312785

119901119894= 300

1199010= 150

1205790= 400

0 4468498329 1869943517 minus0216047300 minus3455691093 minus82392517301 2039946605 1721858238 0520942423 minus2403128796 minus79859403832 0725452128 1314126566 0957769488 minus1332711458 minus7351503426

119901119894= 150

1199010= 300

1205790= 800

0 4078610278 1692083456 minus0362927548 minus3599667594 minus83881250271 1844023766 1536347565 0332616556 minus2596998868 minus81860262912 0698588473 1166718654 0723851445 minus1632455676 minus7703312834

119901119894= 300

1199010= 150

1205790= 800

0 4468498296 1869943321 minus0216047655 minus3455691489 minus82392377231 2039946200 1721858288 0520942491 minus2403128819 minus79859400642 0725452082 1314126521 0957769539 minus1332711402 minus7351503219

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 04 120596 = 500 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 0502344054 0209118575 minus0032085608 minus0411840852 minus10757559141 0301690685 0253918828 0067872877 minus0387759274 minus13375609912 0170338387 0272622555 0177093634 minus0328795174 minus1672186161

119901119894= 300

1199010= 150

1205790= 0

0 0520602572 0223194821 minus0013327208 minus0385714307 minus10416636041 0314858338 0269151250 0087139893 minus0362225697 minus13033628312 0175533196 0289517104 0201213547 minus0301959686 minus1648231058

119901119894= 150

1199010= 300

1205790= 400

0 0502344076 0209118532 minus0032085631 minus0411840853 minus10757560931 0301690546 0253918731 0067873081 minus0387757210 minus13375620512 0170338398 0272622541 0177093558 minus0328795106 minus1672185933

119901119894= 300

1199010= 150

1205790= 400

0 0520602619 0223194813 minus0013327161 minus0385714149 minus10416645331 0314858326 0269151199 0087139890 minus0362225687 minus13033627732 0175533191 0289517093 0201213591 minus0301959717 minus1648231090

119901119894= 150

1199010= 300

1205790= 800

0 0502344041 0209118525 minus0032085593 minus0411841019 minus10757557561 0301690548 0253918749 0067873075 minus0387757974 minus13375601512 0170338413 0272622528 0177093598 minus0328795120 minus1672185805

119901119894= 300

1199010= 150

1205790= 800

0 0520602532 0223194802 minus0013327191 minus0385714162 minus10416644831 0314858370 0269151215 0087139921 minus0362225740 minus13033627232 0175533168 0289517159 0201213488 minus0301959669 minus1648230944

as well as for nonhomogenous cylinder but with the changein nonhomogeneity from 119890

1= 1 to 119890

1= 2 circumferential

stresses decrease when external pressure is greater than theinternal pressure with thermal effects Also these stresses

increase for homogenous cylinder as well as for functionallygraded stainless steel composite cylinder (119890

1= 1) while

decrease for functionally graded stainless steel compositecylinder with 119890

1= 2 With the increase in temperature these

8 Advances in Materials Science and Engineering

Table 4 Circumferential stresses for rotating cylinder with variable thickness and variable density 119897 = 07 119889 = 05 120596 = 300 500 strainhardening measure119898 = 06 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 300 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 4078550787 1692092827 minus0362917634 minus3599662505 minus83881243031 1844008854 1536344037 0332619755 minus2596989139 minus81860118602 0698583918 1166713498 0723849363 minus1632447308 minus7703291360

119901119894= 300

1199010= 150

1205790= 0

0 4468438868 1869957168 minus0216034432 minus3455686049 minus82394923451 2039930194 1721854313 0520946730 minus2403118071 minus79859271952 0725447053 1314120776 0957767141 minus1332701862 minus7351479020

119901119894= 150

1199010= 300

1205790= 400

0 4078550758 1692093003 minus0362917541 minus3599662684 minus83881247581 1844009034 1536344170 0332619845 minus2596989055 minus81860133232 0698583989 1166713507 0723849336 minus1632447294 minus7703291352

119901119894= 300

1199010= 150

1205790= 400

0 4468440841 1869957567 minus0216033626 minus3455685275 minus82395880401 2039930222 1721854440 0520946513 minus2403117933 minus79859260012 0725447107 1314120882 0957767050 minus1332701999 minus7351479230

119901119894= 150

1199010= 300

1205790= 800

0 4078551089 1692092847 minus0362917694 minus3599662435 minus83881264791 1844008698 1536343957 0332619859 minus2596988916 minus81860122392 0698583989 1166713524 0723849328 minus1632447317 minus7703291542

119901119894= 300

1199010= 150

1205790= 800

0 4468438120 1869956228 minus0216035341 minus3455687051 minus82393686321 2039930039 1721854569 0520946538 minus2403117966 minus79859257312 0725447049 1314120768 0957767137 minus1332701833 minus7351479029

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 500 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 0502339190 0209119023 minus0032080434 minus0411843003 minus10757515751 0301689441 0253919137 0067872897 minus0387758158 minus13375674432 0170338021 0272622433 0177094183 minus0328793821 minus1672191050

119901119894= 300

1199010= 150

1205790= 0

0 0520600260 0223196743 minus0013328691 minus0385729358 minus10416359501 0314857828 0269151955 0087140006 minus0362234515 minus13033506742 0175532879 0289517002 0201213980 minus0301960040 minus1648233514

119901119894= 150

1199010= 300

1205790= 400

0 0502339173 0209119009 minus0032080335 minus0411843048 minus10757520131 0301689029 0253918998 0067873493 minus0387754850 minus13375668252 0170338061 0272622405 0177094107 minus0328793799 minus1672190560

119901119894= 300

1199010= 150

1205790= 400

0 0520600268 0223196732 minus0013328653 minus0385729330 minus10416363341 0314857844 0269151936 0087139999 minus0362234465 minus13033507222 0175532856 0289516980 0201214076 minus0301960031 minus1648233768

119901119894= 150

1199010= 300

1205790= 800

0 0502339102 0209118939 minus0032080279 minus0411842835 minus10757521591 0301689252 0253919092 0067873202 minus0387759066 minus13375656632 0170338011 0272622432 0177094145 minus0328793808 minus1672190610

119901119894= 300

1199010= 150

1205790= 800

0 0520600345 0223196689 minus0013328705 minus0385729338 minus10416358761 0314857846 0269151949 0087139963 minus0362234522 minus13033507592 0175532843 0289516995 0201214043 minus0301960031 minus1648233661

stresses decrease significantly for homogeneous as well as forfunctionally graded stainless steel composite cylinder as canbe seen from Figure 4

Figures 5ndash7 have been drawn to discuss the effect ofinternal and external pressure on stresses in rotating cylin-der made of functionally graded stainless steel composite

Advances in Materials Science and Engineering 9

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 150 and p0 = 300

120596 = 300 e1 = 0

120596 = 300 e1 = 2

120596 = 500 e1 = 1

120596 = 300 e1 = 1

120596 = 500 e1 = 0

120596 = 500 e1 = 2

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 300 and p0 = 150

120596 = 300 e1 = 0

120596 = 300 e1 = 2

120596 = 500 e1 = 1

120596 = 300 e1 = 1

120596 = 500 e1 = 0

120596 = 500 e1 = 2

(b)

Figure 2 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density without thermal effects (1205790= 0) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

pi = 150 and p0 = 3001205721 = 0 1205790 = 400 m = 04

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 300 and p0 = 150

(b)

Figure 3 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density with thermal effects (1205790= 400) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 300 and p0 = 150

(b)

Figure 4 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density with thermal effects (1205790= 800) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

10 Advances in Materials Science and Engineering

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 300 and p0 = 150

(b)

Figure 5 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density without thermal effects (1205790= 0) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 300 and p0 = 150

(b)

Figure 6 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 400) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 300 and p0 = 150

(b)

Figure 7 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 800) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

Advances in Materials Science and Engineering 11

material with variable thickness and variable density withnonlinear strain hardening measure

It has been observed from Figure 5 that circumferentialstresses are maximum at external surface for homogeneousas well as functionally graded stainless steel compositecylinder It has also been observed that with the increase inangular speed circumferential stresses increase significantlyWith the introduction of thermal effects circumferentialstresses increase for homogeneous cylinder as well as forfunctionally graded stainless steel composite cylinder (119890

1=

1) but with the change in nonhomogeneity from 1198901= 1

to 1198901= 2 circumferential stresses decrease when external

pressure is greater than the internal pressure as can beseen from Figure 6 Circumferential stresses decrease forhomogeneous cylinder while increase for functionally gradedstainless steel composite cylinder when external pressure isless than the internal pressure It has also been observedfrom Figure 7 that with the increase in temperature thesestresses increase significantly for homogeneous cylinder aswell for functionally graded stainless steel composite cylinderwith 119890

1= 1 while decrease for functionally graded stainless

steel composite cylinder with 1198901= 2 when external pressure

is greater than the internal pressure while these stressesdecrease significantly for homogeneous cylinder as well forfunctionally graded stainless steel composite cylinder whenexternal pressure is less than the internal pressure

5 Conclusion

From the analysis we can conclude that rotating cylin-der made of functionally graded stainless steel compositematerial having variable thickness and variable density withSwiftrsquos strain hardening measure 119898 = 06 and thermalloading is better choice for designers as compared to rotatingcylinder with constant thickness and constant density Thisis because of the reason that circumferential stress is less forfunctionally graded stainless steel composite cylinder withvariable thickness and variable density as compared to othercases which leads to the idea of stress saving that minimizesthe possibility of fracture of cylinder

Conflict of Interests

The authors declare that they have no conflict of interest

References

[1] S Suresh and A Mortensen Fundamentals of FunctionallyGradedMaterials IOM3 Maney Publishing London UK 1998

[2] J N Reddy ldquoAnalysis of functionally graded platesrdquo Interna-tional Journal for Numerical Methods in Engineering vol 47 no1ndash3 pp 663ndash684 2000

[3] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill NewYork NY USA 3rd edition 1970

[4] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998

[5] Y Obata T Oji and N Noda ldquoSteady thermal stresses in a hol-low circular cylinder and a hollow sphere made of functionallygraded material (analysis with perturbation method)rdquo Reports

of the National Industrial Research Institute of Nagoya vol 47no 4 pp 317ndash338 1998

[6] J Perry and J Aboudi ldquoElasto-plastic stresses in thick walledcylindersrdquo Journal of Pressure Vessel Technology vol 125 no 3pp 248ndash252 2003

[7] X-L Gao ldquoElasto-plastic analysis of an internally pressurizedthick-walled cylinder using a strain gradient plasticity theoryrdquoInternational Journal of Solids and Structures vol 40 no 23 pp6445ndash6455 2003

[8] N Tutuncu and M Ozturk ldquoExact solutions for stresses infunctionally graded pressure vesselsrdquo Composites B vol 32 no8 pp 683ndash686 2001

[9] T Singh and V K Gupta ldquoCreep analysis of an internallypressurized thick cylinder made of a functionally graded com-positerdquo Journal of Strain Analysis for Engineering Design vol 44no 7 pp 583ndash594 2009

[10] A K Aggarwal Sharma Richa and Sharma Sanjeev ldquoSafetyanalysis using lebesgue strain measure of thick-walled cylinderfor functionally graded material under internal and externalpressurerdquo The Scientific World Journal vol 2013 Article ID676190 10 pages 2013

[11] A K Aggarwal S Richa and S Sanjeev ldquoSafety analysis ofthermal creep non-homogeneous thick-walled circular cylinderunder internal and external pressure using lebesgue strainmeasurerdquoMultidiscipline Modeling in Materials and Structuresvol 9 no 4 2013

[12] A Parvizi R Naghdabadi and J Arghavani ldquoAnalysis of AlA359SiCp functionally graded cylinder subjected to internalpressure and temperature gradient with elastic-plastic deforma-tionrdquo Journal of Thermal Stresses vol 34 no 10 pp 1054ndash10702011

[13] A N Eraslan and F Akgul ldquoYielding and elastoplastic deforma-tion of annular disks of a parabolic section subject to externalcompressionrdquo Turkish Journal of Engineering and Environmen-tal Sciences vol 29 no 1 pp 51ndash60 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 8: Research Article Thermo Elastic-Plastic Analysis of Rotating …downloads.hindawi.com/journals/amse/2013/810508.pdf · 2019-07-31 · ermo elastic-plastic analysis of functionally

8 Advances in Materials Science and Engineering

Table 4 Circumferential stresses for rotating cylinder with variable thickness and variable density 119897 = 07 119889 = 05 120596 = 300 500 strainhardening measure119898 = 06 and different parameters of Youngrsquos modulus under internal and external pressure

1119864minus3lowast 119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 300 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 4078550787 1692092827 minus0362917634 minus3599662505 minus83881243031 1844008854 1536344037 0332619755 minus2596989139 minus81860118602 0698583918 1166713498 0723849363 minus1632447308 minus7703291360

119901119894= 300

1199010= 150

1205790= 0

0 4468438868 1869957168 minus0216034432 minus3455686049 minus82394923451 2039930194 1721854313 0520946730 minus2403118071 minus79859271952 0725447053 1314120776 0957767141 minus1332701862 minus7351479020

119901119894= 150

1199010= 300

1205790= 400

0 4078550758 1692093003 minus0362917541 minus3599662684 minus83881247581 1844009034 1536344170 0332619845 minus2596989055 minus81860133232 0698583989 1166713507 0723849336 minus1632447294 minus7703291352

119901119894= 300

1199010= 150

1205790= 400

0 4468440841 1869957567 minus0216033626 minus3455685275 minus82395880401 2039930222 1721854440 0520946513 minus2403117933 minus79859260012 0725447107 1314120882 0957767050 minus1332701999 minus7351479230

119901119894= 150

1199010= 300

1205790= 800

0 4078551089 1692092847 minus0362917694 minus3599662435 minus83881264791 1844008698 1536343957 0332619859 minus2596988916 minus81860122392 0698583989 1166713524 0723849328 minus1632447317 minus7703291542

119901119894= 300

1199010= 150

1205790= 800

0 4468438120 1869956228 minus0216035341 minus3455687051 minus82393686321 2039930039 1721854569 0520946538 minus2403117966 minus79859257312 0725447049 1314120768 0957767137 minus1332701833 minus7351479029

1119864minus4lowast119879120579120579MPa 119890

1

119903

119898 = 06 120596 = 500 119897 = 07 119889 = 0501 02 03 04 05

119901119894= 150

1199010= 300

1205790= 0

0 0502339190 0209119023 minus0032080434 minus0411843003 minus10757515751 0301689441 0253919137 0067872897 minus0387758158 minus13375674432 0170338021 0272622433 0177094183 minus0328793821 minus1672191050

119901119894= 300

1199010= 150

1205790= 0

0 0520600260 0223196743 minus0013328691 minus0385729358 minus10416359501 0314857828 0269151955 0087140006 minus0362234515 minus13033506742 0175532879 0289517002 0201213980 minus0301960040 minus1648233514

119901119894= 150

1199010= 300

1205790= 400

0 0502339173 0209119009 minus0032080335 minus0411843048 minus10757520131 0301689029 0253918998 0067873493 minus0387754850 minus13375668252 0170338061 0272622405 0177094107 minus0328793799 minus1672190560

119901119894= 300

1199010= 150

1205790= 400

0 0520600268 0223196732 minus0013328653 minus0385729330 minus10416363341 0314857844 0269151936 0087139999 minus0362234465 minus13033507222 0175532856 0289516980 0201214076 minus0301960031 minus1648233768

119901119894= 150

1199010= 300

1205790= 800

0 0502339102 0209118939 minus0032080279 minus0411842835 minus10757521591 0301689252 0253919092 0067873202 minus0387759066 minus13375656632 0170338011 0272622432 0177094145 minus0328793808 minus1672190610

119901119894= 300

1199010= 150

1205790= 800

0 0520600345 0223196689 minus0013328705 minus0385729338 minus10416358761 0314857846 0269151949 0087139963 minus0362234522 minus13033507592 0175532843 0289516995 0201214043 minus0301960031 minus1648233661

stresses decrease significantly for homogeneous as well as forfunctionally graded stainless steel composite cylinder as canbe seen from Figure 4

Figures 5ndash7 have been drawn to discuss the effect ofinternal and external pressure on stresses in rotating cylin-der made of functionally graded stainless steel composite

Advances in Materials Science and Engineering 9

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 150 and p0 = 300

120596 = 300 e1 = 0

120596 = 300 e1 = 2

120596 = 500 e1 = 1

120596 = 300 e1 = 1

120596 = 500 e1 = 0

120596 = 500 e1 = 2

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 300 and p0 = 150

120596 = 300 e1 = 0

120596 = 300 e1 = 2

120596 = 500 e1 = 1

120596 = 300 e1 = 1

120596 = 500 e1 = 0

120596 = 500 e1 = 2

(b)

Figure 2 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density without thermal effects (1205790= 0) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

pi = 150 and p0 = 3001205721 = 0 1205790 = 400 m = 04

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 300 and p0 = 150

(b)

Figure 3 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density with thermal effects (1205790= 400) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 300 and p0 = 150

(b)

Figure 4 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density with thermal effects (1205790= 800) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

10 Advances in Materials Science and Engineering

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 300 and p0 = 150

(b)

Figure 5 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density without thermal effects (1205790= 0) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 300 and p0 = 150

(b)

Figure 6 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 400) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 300 and p0 = 150

(b)

Figure 7 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 800) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

Advances in Materials Science and Engineering 11

material with variable thickness and variable density withnonlinear strain hardening measure

It has been observed from Figure 5 that circumferentialstresses are maximum at external surface for homogeneousas well as functionally graded stainless steel compositecylinder It has also been observed that with the increase inangular speed circumferential stresses increase significantlyWith the introduction of thermal effects circumferentialstresses increase for homogeneous cylinder as well as forfunctionally graded stainless steel composite cylinder (119890

1=

1) but with the change in nonhomogeneity from 1198901= 1

to 1198901= 2 circumferential stresses decrease when external

pressure is greater than the internal pressure as can beseen from Figure 6 Circumferential stresses decrease forhomogeneous cylinder while increase for functionally gradedstainless steel composite cylinder when external pressure isless than the internal pressure It has also been observedfrom Figure 7 that with the increase in temperature thesestresses increase significantly for homogeneous cylinder aswell for functionally graded stainless steel composite cylinderwith 119890

1= 1 while decrease for functionally graded stainless

steel composite cylinder with 1198901= 2 when external pressure

is greater than the internal pressure while these stressesdecrease significantly for homogeneous cylinder as well forfunctionally graded stainless steel composite cylinder whenexternal pressure is less than the internal pressure

5 Conclusion

From the analysis we can conclude that rotating cylin-der made of functionally graded stainless steel compositematerial having variable thickness and variable density withSwiftrsquos strain hardening measure 119898 = 06 and thermalloading is better choice for designers as compared to rotatingcylinder with constant thickness and constant density Thisis because of the reason that circumferential stress is less forfunctionally graded stainless steel composite cylinder withvariable thickness and variable density as compared to othercases which leads to the idea of stress saving that minimizesthe possibility of fracture of cylinder

Conflict of Interests

The authors declare that they have no conflict of interest

References

[1] S Suresh and A Mortensen Fundamentals of FunctionallyGradedMaterials IOM3 Maney Publishing London UK 1998

[2] J N Reddy ldquoAnalysis of functionally graded platesrdquo Interna-tional Journal for Numerical Methods in Engineering vol 47 no1ndash3 pp 663ndash684 2000

[3] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill NewYork NY USA 3rd edition 1970

[4] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998

[5] Y Obata T Oji and N Noda ldquoSteady thermal stresses in a hol-low circular cylinder and a hollow sphere made of functionallygraded material (analysis with perturbation method)rdquo Reports

of the National Industrial Research Institute of Nagoya vol 47no 4 pp 317ndash338 1998

[6] J Perry and J Aboudi ldquoElasto-plastic stresses in thick walledcylindersrdquo Journal of Pressure Vessel Technology vol 125 no 3pp 248ndash252 2003

[7] X-L Gao ldquoElasto-plastic analysis of an internally pressurizedthick-walled cylinder using a strain gradient plasticity theoryrdquoInternational Journal of Solids and Structures vol 40 no 23 pp6445ndash6455 2003

[8] N Tutuncu and M Ozturk ldquoExact solutions for stresses infunctionally graded pressure vesselsrdquo Composites B vol 32 no8 pp 683ndash686 2001

[9] T Singh and V K Gupta ldquoCreep analysis of an internallypressurized thick cylinder made of a functionally graded com-positerdquo Journal of Strain Analysis for Engineering Design vol 44no 7 pp 583ndash594 2009

[10] A K Aggarwal Sharma Richa and Sharma Sanjeev ldquoSafetyanalysis using lebesgue strain measure of thick-walled cylinderfor functionally graded material under internal and externalpressurerdquo The Scientific World Journal vol 2013 Article ID676190 10 pages 2013

[11] A K Aggarwal S Richa and S Sanjeev ldquoSafety analysis ofthermal creep non-homogeneous thick-walled circular cylinderunder internal and external pressure using lebesgue strainmeasurerdquoMultidiscipline Modeling in Materials and Structuresvol 9 no 4 2013

[12] A Parvizi R Naghdabadi and J Arghavani ldquoAnalysis of AlA359SiCp functionally graded cylinder subjected to internalpressure and temperature gradient with elastic-plastic deforma-tionrdquo Journal of Thermal Stresses vol 34 no 10 pp 1054ndash10702011

[13] A N Eraslan and F Akgul ldquoYielding and elastoplastic deforma-tion of annular disks of a parabolic section subject to externalcompressionrdquo Turkish Journal of Engineering and Environmen-tal Sciences vol 29 no 1 pp 51ndash60 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 9: Research Article Thermo Elastic-Plastic Analysis of Rotating …downloads.hindawi.com/journals/amse/2013/810508.pdf · 2019-07-31 · ermo elastic-plastic analysis of functionally

Advances in Materials Science and Engineering 9

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 150 and p0 = 300

120596 = 300 e1 = 0

120596 = 300 e1 = 2

120596 = 500 e1 = 1

120596 = 300 e1 = 1

120596 = 500 e1 = 0

120596 = 500 e1 = 2

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 300 and p0 = 150

120596 = 300 e1 = 0

120596 = 300 e1 = 2

120596 = 500 e1 = 1

120596 = 300 e1 = 1

120596 = 500 e1 = 0

120596 = 500 e1 = 2

(b)

Figure 2 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density without thermal effects (1205790= 0) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

pi = 150 and p0 = 3001205721 = 0 1205790 = 400 m = 04

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 300 and p0 = 150

(b)

Figure 3 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density with thermal effects (1205790= 400) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

005

115times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 300 and p0 = 150

(b)

Figure 4 Elastic-plastic stresses in a rotating cylinder with constant thickness and constant density with thermal effects (1205790= 800) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

10 Advances in Materials Science and Engineering

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 300 and p0 = 150

(b)

Figure 5 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density without thermal effects (1205790= 0) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 300 and p0 = 150

(b)

Figure 6 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 400) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 300 and p0 = 150

(b)

Figure 7 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 800) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

Advances in Materials Science and Engineering 11

material with variable thickness and variable density withnonlinear strain hardening measure

It has been observed from Figure 5 that circumferentialstresses are maximum at external surface for homogeneousas well as functionally graded stainless steel compositecylinder It has also been observed that with the increase inangular speed circumferential stresses increase significantlyWith the introduction of thermal effects circumferentialstresses increase for homogeneous cylinder as well as forfunctionally graded stainless steel composite cylinder (119890

1=

1) but with the change in nonhomogeneity from 1198901= 1

to 1198901= 2 circumferential stresses decrease when external

pressure is greater than the internal pressure as can beseen from Figure 6 Circumferential stresses decrease forhomogeneous cylinder while increase for functionally gradedstainless steel composite cylinder when external pressure isless than the internal pressure It has also been observedfrom Figure 7 that with the increase in temperature thesestresses increase significantly for homogeneous cylinder aswell for functionally graded stainless steel composite cylinderwith 119890

1= 1 while decrease for functionally graded stainless

steel composite cylinder with 1198901= 2 when external pressure

is greater than the internal pressure while these stressesdecrease significantly for homogeneous cylinder as well forfunctionally graded stainless steel composite cylinder whenexternal pressure is less than the internal pressure

5 Conclusion

From the analysis we can conclude that rotating cylin-der made of functionally graded stainless steel compositematerial having variable thickness and variable density withSwiftrsquos strain hardening measure 119898 = 06 and thermalloading is better choice for designers as compared to rotatingcylinder with constant thickness and constant density Thisis because of the reason that circumferential stress is less forfunctionally graded stainless steel composite cylinder withvariable thickness and variable density as compared to othercases which leads to the idea of stress saving that minimizesthe possibility of fracture of cylinder

Conflict of Interests

The authors declare that they have no conflict of interest

References

[1] S Suresh and A Mortensen Fundamentals of FunctionallyGradedMaterials IOM3 Maney Publishing London UK 1998

[2] J N Reddy ldquoAnalysis of functionally graded platesrdquo Interna-tional Journal for Numerical Methods in Engineering vol 47 no1ndash3 pp 663ndash684 2000

[3] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill NewYork NY USA 3rd edition 1970

[4] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998

[5] Y Obata T Oji and N Noda ldquoSteady thermal stresses in a hol-low circular cylinder and a hollow sphere made of functionallygraded material (analysis with perturbation method)rdquo Reports

of the National Industrial Research Institute of Nagoya vol 47no 4 pp 317ndash338 1998

[6] J Perry and J Aboudi ldquoElasto-plastic stresses in thick walledcylindersrdquo Journal of Pressure Vessel Technology vol 125 no 3pp 248ndash252 2003

[7] X-L Gao ldquoElasto-plastic analysis of an internally pressurizedthick-walled cylinder using a strain gradient plasticity theoryrdquoInternational Journal of Solids and Structures vol 40 no 23 pp6445ndash6455 2003

[8] N Tutuncu and M Ozturk ldquoExact solutions for stresses infunctionally graded pressure vesselsrdquo Composites B vol 32 no8 pp 683ndash686 2001

[9] T Singh and V K Gupta ldquoCreep analysis of an internallypressurized thick cylinder made of a functionally graded com-positerdquo Journal of Strain Analysis for Engineering Design vol 44no 7 pp 583ndash594 2009

[10] A K Aggarwal Sharma Richa and Sharma Sanjeev ldquoSafetyanalysis using lebesgue strain measure of thick-walled cylinderfor functionally graded material under internal and externalpressurerdquo The Scientific World Journal vol 2013 Article ID676190 10 pages 2013

[11] A K Aggarwal S Richa and S Sanjeev ldquoSafety analysis ofthermal creep non-homogeneous thick-walled circular cylinderunder internal and external pressure using lebesgue strainmeasurerdquoMultidiscipline Modeling in Materials and Structuresvol 9 no 4 2013

[12] A Parvizi R Naghdabadi and J Arghavani ldquoAnalysis of AlA359SiCp functionally graded cylinder subjected to internalpressure and temperature gradient with elastic-plastic deforma-tionrdquo Journal of Thermal Stresses vol 34 no 10 pp 1054ndash10702011

[13] A N Eraslan and F Akgul ldquoYielding and elastoplastic deforma-tion of annular disks of a parabolic section subject to externalcompressionrdquo Turkish Journal of Engineering and Environmen-tal Sciences vol 29 no 1 pp 51ndash60 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 10: Research Article Thermo Elastic-Plastic Analysis of Rotating …downloads.hindawi.com/journals/amse/2013/810508.pdf · 2019-07-31 · ermo elastic-plastic analysis of functionally

10 Advances in Materials Science and Engineering

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 0 m = 04 pi = 300 and p0 = 150

(b)

Figure 5 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density without thermal effects (1205790= 0) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 400 m = 04 pi = 300 and p0 = 150

(b)

Figure 6 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 400) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 150 and p0 = 300

(a)

01 015 02 025 03 035 04 045 05minus2

minus15

minus1

minus05

0

05

1times10

10

r

Stre

sses

1205721 = 0 1205790 = 800 m = 04 pi = 300 and p0 = 150

(b)

Figure 7 Elastic-plastic stresses in a rotating cylinder with variable thickness and variable density with thermal effects (1205790= 800) with

parameters 1198901= 0 1 2 and strain hardening measure119898 = 04 under internal and external pressure

Advances in Materials Science and Engineering 11

material with variable thickness and variable density withnonlinear strain hardening measure

It has been observed from Figure 5 that circumferentialstresses are maximum at external surface for homogeneousas well as functionally graded stainless steel compositecylinder It has also been observed that with the increase inangular speed circumferential stresses increase significantlyWith the introduction of thermal effects circumferentialstresses increase for homogeneous cylinder as well as forfunctionally graded stainless steel composite cylinder (119890

1=

1) but with the change in nonhomogeneity from 1198901= 1

to 1198901= 2 circumferential stresses decrease when external

pressure is greater than the internal pressure as can beseen from Figure 6 Circumferential stresses decrease forhomogeneous cylinder while increase for functionally gradedstainless steel composite cylinder when external pressure isless than the internal pressure It has also been observedfrom Figure 7 that with the increase in temperature thesestresses increase significantly for homogeneous cylinder aswell for functionally graded stainless steel composite cylinderwith 119890

1= 1 while decrease for functionally graded stainless

steel composite cylinder with 1198901= 2 when external pressure

is greater than the internal pressure while these stressesdecrease significantly for homogeneous cylinder as well forfunctionally graded stainless steel composite cylinder whenexternal pressure is less than the internal pressure

5 Conclusion

From the analysis we can conclude that rotating cylin-der made of functionally graded stainless steel compositematerial having variable thickness and variable density withSwiftrsquos strain hardening measure 119898 = 06 and thermalloading is better choice for designers as compared to rotatingcylinder with constant thickness and constant density Thisis because of the reason that circumferential stress is less forfunctionally graded stainless steel composite cylinder withvariable thickness and variable density as compared to othercases which leads to the idea of stress saving that minimizesthe possibility of fracture of cylinder

Conflict of Interests

The authors declare that they have no conflict of interest

References

[1] S Suresh and A Mortensen Fundamentals of FunctionallyGradedMaterials IOM3 Maney Publishing London UK 1998

[2] J N Reddy ldquoAnalysis of functionally graded platesrdquo Interna-tional Journal for Numerical Methods in Engineering vol 47 no1ndash3 pp 663ndash684 2000

[3] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill NewYork NY USA 3rd edition 1970

[4] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998

[5] Y Obata T Oji and N Noda ldquoSteady thermal stresses in a hol-low circular cylinder and a hollow sphere made of functionallygraded material (analysis with perturbation method)rdquo Reports

of the National Industrial Research Institute of Nagoya vol 47no 4 pp 317ndash338 1998

[6] J Perry and J Aboudi ldquoElasto-plastic stresses in thick walledcylindersrdquo Journal of Pressure Vessel Technology vol 125 no 3pp 248ndash252 2003

[7] X-L Gao ldquoElasto-plastic analysis of an internally pressurizedthick-walled cylinder using a strain gradient plasticity theoryrdquoInternational Journal of Solids and Structures vol 40 no 23 pp6445ndash6455 2003

[8] N Tutuncu and M Ozturk ldquoExact solutions for stresses infunctionally graded pressure vesselsrdquo Composites B vol 32 no8 pp 683ndash686 2001

[9] T Singh and V K Gupta ldquoCreep analysis of an internallypressurized thick cylinder made of a functionally graded com-positerdquo Journal of Strain Analysis for Engineering Design vol 44no 7 pp 583ndash594 2009

[10] A K Aggarwal Sharma Richa and Sharma Sanjeev ldquoSafetyanalysis using lebesgue strain measure of thick-walled cylinderfor functionally graded material under internal and externalpressurerdquo The Scientific World Journal vol 2013 Article ID676190 10 pages 2013

[11] A K Aggarwal S Richa and S Sanjeev ldquoSafety analysis ofthermal creep non-homogeneous thick-walled circular cylinderunder internal and external pressure using lebesgue strainmeasurerdquoMultidiscipline Modeling in Materials and Structuresvol 9 no 4 2013

[12] A Parvizi R Naghdabadi and J Arghavani ldquoAnalysis of AlA359SiCp functionally graded cylinder subjected to internalpressure and temperature gradient with elastic-plastic deforma-tionrdquo Journal of Thermal Stresses vol 34 no 10 pp 1054ndash10702011

[13] A N Eraslan and F Akgul ldquoYielding and elastoplastic deforma-tion of annular disks of a parabolic section subject to externalcompressionrdquo Turkish Journal of Engineering and Environmen-tal Sciences vol 29 no 1 pp 51ndash60 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 11: Research Article Thermo Elastic-Plastic Analysis of Rotating …downloads.hindawi.com/journals/amse/2013/810508.pdf · 2019-07-31 · ermo elastic-plastic analysis of functionally

Advances in Materials Science and Engineering 11

material with variable thickness and variable density withnonlinear strain hardening measure

It has been observed from Figure 5 that circumferentialstresses are maximum at external surface for homogeneousas well as functionally graded stainless steel compositecylinder It has also been observed that with the increase inangular speed circumferential stresses increase significantlyWith the introduction of thermal effects circumferentialstresses increase for homogeneous cylinder as well as forfunctionally graded stainless steel composite cylinder (119890

1=

1) but with the change in nonhomogeneity from 1198901= 1

to 1198901= 2 circumferential stresses decrease when external

pressure is greater than the internal pressure as can beseen from Figure 6 Circumferential stresses decrease forhomogeneous cylinder while increase for functionally gradedstainless steel composite cylinder when external pressure isless than the internal pressure It has also been observedfrom Figure 7 that with the increase in temperature thesestresses increase significantly for homogeneous cylinder aswell for functionally graded stainless steel composite cylinderwith 119890

1= 1 while decrease for functionally graded stainless

steel composite cylinder with 1198901= 2 when external pressure

is greater than the internal pressure while these stressesdecrease significantly for homogeneous cylinder as well forfunctionally graded stainless steel composite cylinder whenexternal pressure is less than the internal pressure

5 Conclusion

From the analysis we can conclude that rotating cylin-der made of functionally graded stainless steel compositematerial having variable thickness and variable density withSwiftrsquos strain hardening measure 119898 = 06 and thermalloading is better choice for designers as compared to rotatingcylinder with constant thickness and constant density Thisis because of the reason that circumferential stress is less forfunctionally graded stainless steel composite cylinder withvariable thickness and variable density as compared to othercases which leads to the idea of stress saving that minimizesthe possibility of fracture of cylinder

Conflict of Interests

The authors declare that they have no conflict of interest

References

[1] S Suresh and A Mortensen Fundamentals of FunctionallyGradedMaterials IOM3 Maney Publishing London UK 1998

[2] J N Reddy ldquoAnalysis of functionally graded platesrdquo Interna-tional Journal for Numerical Methods in Engineering vol 47 no1ndash3 pp 663ndash684 2000

[3] S P Timoshenko and J N Goodier Theory of ElasticityMcGraw-Hill NewYork NY USA 3rd edition 1970

[4] RHillTheMathematicalTheory of Plasticity OxfordUniversityPress Oxford UK 1998

[5] Y Obata T Oji and N Noda ldquoSteady thermal stresses in a hol-low circular cylinder and a hollow sphere made of functionallygraded material (analysis with perturbation method)rdquo Reports

of the National Industrial Research Institute of Nagoya vol 47no 4 pp 317ndash338 1998

[6] J Perry and J Aboudi ldquoElasto-plastic stresses in thick walledcylindersrdquo Journal of Pressure Vessel Technology vol 125 no 3pp 248ndash252 2003

[7] X-L Gao ldquoElasto-plastic analysis of an internally pressurizedthick-walled cylinder using a strain gradient plasticity theoryrdquoInternational Journal of Solids and Structures vol 40 no 23 pp6445ndash6455 2003

[8] N Tutuncu and M Ozturk ldquoExact solutions for stresses infunctionally graded pressure vesselsrdquo Composites B vol 32 no8 pp 683ndash686 2001

[9] T Singh and V K Gupta ldquoCreep analysis of an internallypressurized thick cylinder made of a functionally graded com-positerdquo Journal of Strain Analysis for Engineering Design vol 44no 7 pp 583ndash594 2009

[10] A K Aggarwal Sharma Richa and Sharma Sanjeev ldquoSafetyanalysis using lebesgue strain measure of thick-walled cylinderfor functionally graded material under internal and externalpressurerdquo The Scientific World Journal vol 2013 Article ID676190 10 pages 2013

[11] A K Aggarwal S Richa and S Sanjeev ldquoSafety analysis ofthermal creep non-homogeneous thick-walled circular cylinderunder internal and external pressure using lebesgue strainmeasurerdquoMultidiscipline Modeling in Materials and Structuresvol 9 no 4 2013

[12] A Parvizi R Naghdabadi and J Arghavani ldquoAnalysis of AlA359SiCp functionally graded cylinder subjected to internalpressure and temperature gradient with elastic-plastic deforma-tionrdquo Journal of Thermal Stresses vol 34 no 10 pp 1054ndash10702011

[13] A N Eraslan and F Akgul ldquoYielding and elastoplastic deforma-tion of annular disks of a parabolic section subject to externalcompressionrdquo Turkish Journal of Engineering and Environmen-tal Sciences vol 29 no 1 pp 51ndash60 2005

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials

Page 12: Research Article Thermo Elastic-Plastic Analysis of Rotating …downloads.hindawi.com/journals/amse/2013/810508.pdf · 2019-07-31 · ermo elastic-plastic analysis of functionally

Submit your manuscripts athttpwwwhindawicom

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CorrosionInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Polymer ScienceInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CeramicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CompositesJournal of

NanoparticlesJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Biomaterials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

NanoscienceJournal of

TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Journal of

NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

CrystallographyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CoatingsJournal of

Advances in

Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Smart Materials Research

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MetallurgyJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

BioMed Research International

MaterialsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Nano

materials

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofNanomaterials