m.e. biancolini, c. brutti, g. cappellini & m. d'ulisse...equations for displacement field:...

Fatigue life prediction for a cracked notched

element under symmetric load condition

M.E. Biancolini, C. Brutti, G. Cappellini & M. D'Ulisse

Dep. of Mechanical Engineering Rome University "Tor Vergata", Italy

Abstract

In this paper the crack growth emerging from a notch is studied for round barunder symmetric load condition.Modifying and synthesising some models found in literature, a method forfatigue life prediction is outlined and verified. The model takes into account theelastic-plastic crack growth and the change of crack shape during propagation.Overall life is divided in two parts: first a surface emerging crack propagates inthe 3 D stressed region, then a Paris law stable propagation is assumed.A detailed 3D FEM model was developed in order to evaluate SIF vs. cracklength, an elliptical crack shape was assumed imposing the proper eccentricitygrowth. The same FEM model was then exploited to evaluate the extension ofplastic zone at notch tip.Fatigue life prediction for a given geometry was then carried out by means of asimple numerical framework, showing a good agreement with experimental dataranging from low cycle to high cycle fatigue.

1 Introduction

In structural engineering, in general, and in machine design, in particular, it'svery important to evaluate the life under fatigue loads. In the traditional approachthis is performed using limit stress and safety coefficient. More recently manyresearch efforts were devoted to define reliable procedures able to evaluateinitiation and growth of cracks. As the machine elements are generally notched,this procedure has to take into account not only the material properties and theexternal loads but also the stress gradient and its features.

Damage & Fracture Mechanics VI, C.A. Brebbia, A.P.S. Selvadurai, (Editors) © 2000 WIT Press, www.witpress.com, ISBN 1-85312-812-0

334 Damage and Fracture Mechanics VI

This approach normally named "Damage tolerance design" is a very powerfultool to make more reliable structures and to know, during their operating life, thereal state regarding to the remaining cycles before the collapse, in order toschedule the periodical inspections necessary for monitoring the defectsevolution.Linear Elastic Fracture Mechanics (LEFM) is a very powerful tool to predictfatigue crack growth and critical crack size but for notched components onlyelastic-plastic fracture Mechanics (EPFM) gives corrects results in the first stageof crack growth. To apply this approach to actual component design is verydifficult, because closed form solution to evaluate critical dimension andgrowing law, are available only for simple geometry.Therefore numerical methods for stress and strain fields prediction (FEM, BEM)are widely exploited together with crack law model [9].Lin and Smith [1] have proposed a numerical method suitable to plate withelliptical crack, successively improved for arbitrary crack in notched and smoothbar [2].Ahmad and Yates proposed an elastic-plastic model for notched bar withcircumferential crack [8], based on a closed form solution for the stress intensityfactor proposed by Yates [6].A detailed numerical and experimental analysis was carried out by Carpinteri [3]in order to investigate the shape evolution of elliptical cracks in notched andsmooth specimens.In this paper an elastic-plastic model is proposed for fatigue life prediction ofnotched bar with elliptical crack under symmetric load condition. We used themodel of Ahmad and Yates for the estimation of the first part of fatigue life,when the crack propagates in the notch influenced region. The extension ofplastic zone was evaluated numerically by a simplified axisymmetric FEMmodel. The first step of long crack propagation study was performed achievingthe curve AK vs. crack length by a detailed 3D FEM model. Shape evolution wasimposed as observed experimentally by Carpinteri. Then basing on this result theParis law was integrated.The results are then compared with the experimental data published by Ahmad etal. [7] showing a good agreement.

2 Numerical model

2.1 Short crack growth

Ahmad and Yates [8] observed that a little crack immersed in the plastic zone atthe notch root, behaves as a crack linked to the microstucture of the material.Even with little applied load, a plastic zone arises at notch root; then in the firstpart of operating life of the notched specimen, the crack belongs completely tothis plastic zone. In this condition of full plasticity around the crack, LEFM isnot yet a reliable tool; therefore Paris-Erdogan law, tailored for smoothspecimens and long cracks, predicts inaccurately the growth rate of the crack.


Damage and Fracture Mechanics VI 335

Basing on experimental results obtained for smooth medium steel specimenunder low cycle fatigue load, Hobson [4] proposed a correlation for flaw growthvalid for the first stage influenced by material microstructure:

^ = C(d_a,) (1)dNwhere a, is the surface crack length, d is the distance between microstructuralbarriers, C is a function of applied load and material properties.NFFM theory proposed by Ahmad and Yates is derived from Hobson theory forshort cracks with two simple transformations. The surface crack length a^ isreplaced with the crack length in the specimen c, and microstructural barrierdistance d is replaced with An, the extent of plastic zone at notch root.Equation (1) becomes:

c) (2)

To take into account that the growth rate is greater in the notch influencedregion, Ahmad and Yates proposed to use the Smith and Miller correction forcrack length [5]:

e = 1.69\c I— (3)V P

valid when the crack is in the region influenced by the stress field at the notchfor

e ~ D when the crack go out from this region c > Q.llJDp .

This further correction lead to:

^ -- _a);witho=c+e (4)

The coefficient C can be derived experimentally by high strain fatigue testing onsmooth specimens. Experimental results for an HY80 steel are well correlated(95% confidence) with the following formula for C:

C = %" (5)

where C/ and /// are material dependent parameters, &<, is the notch root strain.



2.2 Long crack growth

The Paris-Erdogan law (eq. 6) was assumed valid when the crack become longenough to emerge from the notch plasticity influenced zone. Furthermore thisequation can be applied if AK is greater than the threshold value; otherwise thecrack does not propagate. When the crack length reaches the critic value thepropagation becomes unstable producing the failure of the structure.The parameters C, m and AKu, are material features, together with AK% thatmeasures the fracture toughness.

" (6)

As it is well known in the LEFM AK depends from the load, the crack shape andsize and from the specimen geometry according with the following equations:

AX"y=yAo-V^7 (7)

where a is a reference crack length, Ac is the amplitude of applied cyclic stress;Y is the dimensionless stress intensity factor and is a function of geometry and ofthe type of applied load.Normally the crack initiation yields an elliptic crack generated from the surfaceand propagating through the specimen. In order to predict elliptic crack growthtwo approach are available. The first one considers directly the variation of AKalong the crack front, the greater are the local values of AK the greater are thelocal speed of crack front. For this reason fatigue growth produces a variation incrack shape, the ellipse becomes flat due to the fact that in radial direction Kvalues are lower than near the boundary. To predict this variation it is necessaryto develop a FEM model able to compute the K along the crack front and tomodify the elements shapes according to real crack propagation. This can beachieved remeslung the model at every load step.The second method available has a lower numerical cost, for this reason wasadopted in this work, and it is based on an assumed evolution of ellipseeccentricity. To predict crack evolution, Paris law has to be integrated only inradial direction, because the evolution in other directions is implicitly assumed.

2.3 Plastic zone extent

As in the model chosen a parameter very important is the extent of plastic zonenear the notch, a specific study was performed in order to evaluate the plasticzone size for the notched bar. An axisymmetric FEM model was developed; it isenough detailed because the non symmetric crack shape was taken into accountin the calculation of the stress intensity factor. In figure 1 the FEM model andthe strain resulting for a particular configuration are shown. For the sameconfiguration the evolution of plastic zone size A/? for different loads are plottedin figure 2. The results show a good agreement with values found in literature.



Figure 1: Axisymmetric FEM model, and plastic strain contour map.

2.4 Numerical evaluation of SIF vs. crack length

In order to calculate the stress intensity factor along the crack boundary fordifferent crack lengths, a detailed 3D FEM model was developed. The physicalmodel has two symmetry plane, for this only a quarter of the bar was modelledas shown in figure 3. The mesh is optimised in order to manage different cracklengths and shapes. Remeshing is not necessary because the shape evolution wasassumed a priori.

An(C)

0.8

0.6

0.4

0.2

:00 250 300 350 400 450 500 550C,Load .

~*~ FEM analisys""*~ Reference values

Figure 2: Extent of the plastic zone vs. applied load.



Figure 3: 3D FEM model

In order to choose the better method of calculation of KI by means of the FEManalysis, a preliminary comparison was performed for a simple geometry. Threemethods were tested using the same mesh; the first one consists in an energeticapproach in which the Griffith energy is obtained by the closure work as theproduct of closure reaction at the crack tip and the displacement of the samepoints computed after an increment da of the crack length.

- 1 Fu (g)

If LEFM hypothesis are satisfied K is then correlated to G as follow:

(9)The second method consists in the substitution of nodal displacement in theWestergaard equation to obtain KI. Also the third method is based onsubstitution of nodal displacement, but using the quarter point shifting technique.This technique consists in the shifting of the midside node in the quarter pointposition, in such a manner the shape functions of the isoparametric elementsbecome singular and the order of the singularity is the same of the Westergaardfunctions. This artifice changes the element in a special one that captures verywell, even with a coarse mesh, the high gradient in displacement field [9]. K canbe evaluated by substitution of quarter point displacement in the Westergaardequations for displacement field:

(10)

The errors relative to theoretical value for the numerical test are about 16% forsimple substitution, 7% using quarter point element and 4% using the energeticapproach; all the calculation are performed using the same mesh.



The energetic method is the most suitable when the crack shape evolution isassumed a priori, because the only price to pay to the high precision of thismethod is the request of performing two analyses; if the evolution fromminimum to maximum crack length is divided in N step, only N+1 calculationare required. In fact the complete curve KI vs crack length was computed bymeans of the energetic approach post processing a series of analysis performedwith different crack advance.To reach the same accuracy with a single run method, a substitution methodcould be used but with a finer mesh: on the contrary a better result can beachieved with special element and the quarter point technique.The local approach is able to predict the value of K for each nodal position onthe crack front but only the value in the radial direction was taken into accountfor the growing law, the growing in other directions derives directly from theassumed shape. The mesh used for calculation is shown in figure 3.FEM model results are computed for a greater than 0.5 mm, the values for thezero limit of a are taken from Erjian Si [10] as Y=0.74K, ,values near zero anda=0.5 are connected by a straight line.The results are exposed in figure 4 together with circumferential crack ones,assuming the geometric data of the application described in the last paragraphand imposing that the expression adopted for dimensionless FIS is:

(H)

where a is the crack length, for circumferential crack or ellipse semi axis forelliptical crack, <j^ is the nominal stress referred to the net area of the uncrackedspecimen.

0.5 0.6 0.7 0.8

Figure 4: FIS vs. crack length. 7 refers to circumferential crack, 7/ for ellipticcrack, c crack length or minor ellipse semi axis, Jnet section diameter



2.5 Overall procedure

Starting from the pre-processed value of SIF vs. crack length and from theplastic zone size vs. load, the crack evolution was divided in two parts.Assuming for each part the proper crack growth law. The initiation stage ishandled with the theory for short cracks previously exposed, while for thepropagation stage the Paris law was integrated.

^ total1* initiation 1*propagation \ *-£)

^ A/7 - a (

Integration limits as chosen assuming that in the first part of fatigue life, theinitial crack length corresponds to total roughness value; the transition cracklength between initiation and propagation stage was assumed equal to the plasticzone extent; the final crack length correspond to the critical crack size anddepends on the applied load.

3 Results and Discussion

The procedure now exposed was applied, in order to compare the results withexperimental data, the experimental results were published by Ahmad et al. [7].

3.1 Geometry features and load condition

The application regards a circular notched bar, with nominal diameter d^l2 mm.The semi circular notch deep is D=J mm, notch radius is p=l mm, the overallFEM model length is 1=25 nun and correspond, due to the symmetry condition,to a 50 mm bar. Net section elastic stress concentration factor (theoretic) isAy=2.3. The bar is subjected to a fatigue load, applied as a symmetric push pullcycle.

3.2 Material

The test was performed for a medium strength steel with a bainitic structureHY80, Standard Number 1.6780. The nominal composition is 2.8 % Ni, 1.4%Cr, 0.4% Mo, 0.15% C. Stress-strain behaviour under cyclic load wasrepresented by the Ramberg-Osgood equation:

v \l N(14)

with E=2070 MPa, =77 0 MPa, A =0.072The parameter C in eq. (5) become C = 4.1 • 10 6 ^ where is the strain ( %)

at notch root.For fatigue life in the stage of stable propagation, Paris-Erdogan law wasassumed with the following parameters

— = OA/C",C = I



3.3 Results

The numerical model was applied for this example, first performing a series ofFEM 3D linear analysis, in order to obtain FIS vs. Crack length for thisgeometry, and recording the results; then performing a series of non linear FEMaxisymmetric analysis, this geometry was studied in order to obtain the plasticzone extent vs. applied load.

100

10

s•§

0.0110 WO 1'10" 1-10* 1'10" t'10* I'lO/~**~ Elastic-plastic model~* Elastic model°™ Experimental

Figure 5: Fatigue cycle life vs. notch root strain amplitude

This results were then imported in a worksheet where the growth law wasintegrated for different load values.The results are shown in figure 5 together with the experimental results, showinga very good agreement; in the same figure the results of a full elastic model areplotted too; it is clear a slower evolution predicted by such model, comparedwith the ones elastic-plastic. Furthermore the elastoplastic procedure proposed inthis paper seems to get results in good agreement with the experimental dataavailable.

4 Conclusions

In this paper was exposed a numerical approach for the study of crackpropagation for round notched bar under symmetric load condition.



Overall life is divided in two parts: an initiation stage in which a surface crackpropagates in the notch root, followed by a stable propagation stage governed bythe Paris law.For the first stage the elastoplastic model proposed by Ahmad and Yates wasimproved and generalised performing a numerical evaluation of the plasticregion at notch root by a non-linear FEM analysis, easily applicable to differentnotch shapes for which a closed form solution is not available.For the propagation stage, a detailed 3D FEM model was developed in order toevaluate SIF vs. crack length; an elliptical crack shape was assumed imposingthe proper eccentricity growth. Starting from this preprocessed results, theintegration of Paris law can easily performed.The numerical example developed, using the proposed procedure, is in goodagreement with the experimental data available in literature. This confirms thatthe way chosen is very promising in order to evaluate correctly the fatigue life ofnotched elements with small cracks.

References

[1] X. B. Lin and R.A. Smith, A numerical prediction of fatigue crack growth fora surface defect Fa fig. and Ft-act. ofEng. Mat. and Struct., 1995, 18, 247.

[2] X. B. Lin and R.A. Smith, Fatigue growth simulation for cracks in notchedand unnotched round bars Int. J. Mech, Sci., 1998, 40, 405.

[3] A. Carpinteri Elliptical-arc surface cracks in round bars Fatigue and Fractureof Engineriing Materials and Structures, 1992, 15, 1141.

[4] P.O. Hobson, M.W. Brown, E.R. de los Rios (1986) Two phases of shortcrack growth in a medium carbon steel Behaviour of short fatigue crack(edited by K.J. Miller and E.R. de los Rios) EOF publ.l, MEP, Inst. Mech.Eng. London pp.441-459

[5] R. A. Smith and K. J. Miller. The growth of fatigue cracks from circularnotches. Int. Journ. of Fracture 9, 1973, pp.101-104

[6] J. R. Yates (1991). A simple approximation for the stress intensity factor ofcrack at notch. Journal of strain analysis. 1991 26, 1.

[7] H. Y. Ahmad, M. P. Clode and J. R. Yates. Predicting Notch FatigueLifetimes. Fatigue '96.

[8] H. Y. Ahamad , J. R. Yates (1994). An elastic-plastic model for fatigue crackgrowth at notches. Fatig. Fract. Engng. Mater. Struct. 17(6) pp. 651-660.

[9] "Fracture Mechanics". T.L. Anderson, CRC Press Boca Ratmn, 1995.[10] Erjian Si (1990). Stress intensity factors for surface cracks emanating from

the circumferential notch root in notched round bars. Eng. Fract. Mech. 37, 4,pp. 813-816.


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