pastrama2015.pdf

Procedia Engineering 114 ( 2015 ) 140 148

1877-7058 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineeringdoi: 10.1016/j.proeng.2015.08.052

ScienceDirectAvailable online at www.sciencedirect.com

1st International Conference on Structural Integrity

Practical applications of numerical procedures for structural integrity assessment

Stefan Dan Pastramaa*, Pedro M.G.P. Moreirab, Paulo M.S.T. de Castroc aDepartment of Strength of Materials, University Politehnica, Splaiul Independentei nr. 313, Sector 6, 060042, Bucharest, Romania

bLaboratory of Optics and Experimental Mechanics, Institute of Mechanical Engineering and Industrial Management, Rua Dr. Roberto Frias, Campus da FEUP, 400 4200-465, Porto, Portugal

cDepartment of Mechanical Engineering, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465, Porto, Portugal

Abstract

The paper presents two practical applications in which the weight function method is used to determine the stress intensity factor as the main structural integrity assessment parameter in linear elastic fracture mechanics. In this method, one may obtain the solution for the stress intensity factor for a given crack configuration and a certain loading, provided a complete solution (the Mode I stress intensity factor KIr(a) and the displacements of the crack faces uIr(x,a)) for the same crack problem is known in another loading case called the reference case. Three different cases can be encountered when the weight function method is used: i. The stress intensity factor KIr(a) and the displacement field uIr(x,a) are known; ii. Only the stress intensity factor KIr(a) is known and iii. Both KIr(a) and uIr(x,a) are unknown. In the first case, the method can be applied directly. This paper presents two practical applications for the second and the third case: Stress intensity factors for a strip with a cracked hole subjected to point loads (case 2) and stress intensity factors for an axially cracked thick walled cylinder subjected to internal pressure (case 3). For case 2, the reference stress intensity factors were first calculated using the compounding technique in a reference case (uniform remote traction). Then, an approximate expression of the displacements field found in the literature was used to calculate the weight function. In case 3, the authors propose a solution based on curve fitting of the crack face displacements obtained through finite element analysis. The obtained results were compared with the ones from the literature (where available). In all cases the agreement was very good, showing the reliability of the proposed solutions. 2015 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineering.

Keywords: weight function; crack; displacement; stress intensity factor

* Corresponding author. Tel.: +40-21-4029206; fax: +40-21-4029477

E-mail address: [email protected]

2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineering

141 Stefan Dan Pastrama et al. / Procedia Engineering 114 ( 2015 ) 140 148

1. Introduction

Most structural components have cracks that initiate near regions of stress concentrations, and, in favorable loading condition, may propagate, leading to the final failure. In linear elastic fracture mechanics, the parameter used for the prediction of crack propagation is the stress-intensity factor (SIF). This parameter depends on the geometry of the structure and the applied load. In the case of complex structures, exact solutions for SIF are not always available. Thus, approximate analytical, numerical or experimental procedures should be used in order to obtain the stress intensity factor and received great attention from researchers. A very efficient method is the weight function method, proposed by Bueckner [1]. In order to use it, it is necessary to know a complete solution (the stress intensity factor and the displacements of the crack faces) to a crack problem for one loading system. With these results, one may obtain the stress intensity factor for the same crack configuration with any other loading.

Rice [2] showed that, if one knows the stress intensity factor KIr(a), and the displacement field uIr(x,a) for a cracked body under a symmetrical loading, called the reference case, then the Mode I weight function for an edge crack can be determined from:

,', 2 IrI Ir

u x aEh x aK a a

ww (1)

where Ec = E for the case of plane stress or Ec = E/(1 Q2) in plane strain, E being the Youngs modulus and Q - Poissons ratio. The weight function depends only on the geometry of the body and the crack and is independent on the loading. Thus, if one knows the weight function for a given cracked body, then the stress intensity factor at the tip of the crack in the investigated structure can be computed directly for any other loading as:

0 0

( , )'( ) ( , )d ( ) da a

Ir

Ir

u x aEK x h x a x x xK a

wV V w (2) In equation (2), V(x) are the stresses on the crack line that appear in the uncracked body due to the considered

loading and the system of axes is the one in Fig. 1. There are three different cases when applying the weight function method: 1. The stress intensity factor KIr(a), and the displacement field uIr(x,a) are known, case in which the method can

be applied directly, by using equation (2); 2. Only the stress intensity factor KIr(a) is known; 3. Both KIr(a) and uIr(x,a) are unknown. For the second case, several methods were proposed, especially based on finite or boundary element analyses.

Paris and McMeeking [3] obtained bidimensional weight functions for a single edge cracked strip using a procedure based on the crack tip displacements field and finite element analyses.

Fig. 1: The co-ordinate system.

a

u (x,a)

xO

y

x

Ir


Parks and Kamenetzki [4] and Vanderglas [5] used the stiffness-derivative method for calculating the linear elastic crack tip stress intensity factor [6], combined with the procedure proposed by Rice [2], the virtual crack extension technique and finite element analyses to determine weight functions. The weight functions derived by the virtual crack extension technique are used by Shi et al. [7] to calculate the stress intensity factors in cracked plates made of functionally graded materials. Labbens et al. [8] use relations between bidimensional weight functions and compliance, established through finite element analysis to obtain curves of nondimensional weight functions for cylindrical geometries currently used in engineering. Sham [9] uses the finite element method based on a specialized form of a proposed variational principle and calculates the weight functions for any combination of the three fracture modes in a finite, linear elastic and homogeneous cracked body. Chen and Atluri [10] used the virtual crack extension technique and the equivalent domain integral method to obtain weight functions from two reference load cases. A numerical procedure based on the line spring elements to determine weight functions for semi-elliptical surface cracks in a plate was proposed by Akimkin and Nikishkov [11]. A new general form of point load weight function is suggested by Wang and Glinka [12] based on the properties of weight functions and the available weight functions for two-dimensional cracks. The authors obtained the unknown parameters in the general form from one reference stress intensity factor solution and the method was used to derive the weight functions for embedded elliptical cracks in an infinite and a semi-infinite body. Mixed mode weight functions in the form proposed by Wang and Glinka [12] were derived by Ghajar and Kaklar [13] for elliptical subsurface cracks under shear loadings using reference mixed mode stress intensity factors calculated by finite element analysis.

Other methods use approximate expressions for the displacements field uIr(x,a), together with the known values of the reference stress intensity factor KIr, either available in the literature or obtained with different techniques. Petroski and Achenbach [14] and Fett et al. [15] propose approximate expressions of the displacements field for different geometries of plates and shells. Fett [16] analyses additional geometrical and equilibrium conditions that can be derived in order to obtain approximate displacement fields. Ojdrovic and Petroski [17] considered the derivative of crack opening displacements wu(x,a)/wa instead of the crack profile in the form of a series and computed the unknown coefficients in the series from one or more known stress intensity factors in order to determine weight functions for plates with edge cracks. An extension of the Petroski-Achenbach displacement equation was proposed by Fett et al. [18] for functionally graded materials and used by Santare et al. [19] to obtain stress intensity factors in a plate made from a functionally graded material and having an edge crack, for three different types of loading - uniform traction, fixed grip and pure bending. The crack opening displacement function proposed by Petroski and Achenbach was used also by Shahani and Nabavi [20] to obtain stress intensity factors for semi-elliptical cracks in a thick-walled cylinder under thermal stress. In their approach, the semi-elliptical crack was replaced by an edge crack, which for the reference load, has the reference stress intensity factors of the semi-elliptical crack at its deepest and surface points. A new methodology called Weight function Complex Taylor Series Expansion was proposed by Wagner and Millwater [21] to develop the weight function for an arbitrary geometry and loading scenario from a single complex variable finite element solution without other reference solutions or auxiliary information. Here, the complex Taylor series expansion method is embedded within the finite element method to determine the required partial derivative wu(x,a)/wa at the nodes along the crack line, and, subsequently the weight function, directly from a single finite element analysis.

Pastrama and de Castro [22] have shown that the weight function can be obtained even if both the stress intensity factors and crack face displacements in a reference loading are unknown. The solution is based on curve fitting of the crack face displacements obtained through finite element analysis for several crack lengths and was used also by Jahed et al. [23] to obtain stress intensity factors in cracked autofrettaged cylinders subjected both to autofrettage residual stresses and thermal loads.

This paper reviews some earlier results of the authors, adding some new insights in the case of two practical application of the weight function method. For the case when only the reference SIF is known, the calculation of the stress intensity factor in the case of a strip with cracked hole subjected to point load is described. This structure is an approximation of a thin sheet with rows of rivet holes, often used in the aeronautical industry. The point load can model the action of the rivet. For the case when both SIF and crack face displacements are unknown, the method proposed in [22] to obtain these parameters is described. Finally, the practical case of the stress intensity factor determination in a thick walled tube with external crack is presented. Conclusions on the reliability of the calculations are drawn by comparing the obtained results with the ones available in the scientific literature.


2. Crack face approximate displacement using Petroski-Achenbach approach

Petroski and Achenbach [14] proposed an approximate expression of the displacement field around the crack tip in the form of a series expansion whose first term is the displacement around the crack tip in an infinite cracked plate and the other terms tend to zero in the vicinity of the crack tip:

1 12 2, n nin

u x a C a a x (3) From this series expansion, they used only the first two terms, written in the form:

1/2 3/21/2 1/20, 42Ira au a x F a a x G a a xL LE

V c (4) in which F(a/L) and G(a/L) are functions of the crack length and the characteristic dimension of the studied structure. The function F(a/L) = K/[V(Sa)1/2] can be calculated from the solutions for the stress intensity factor taken from handbooks or obtained using other analytical or numerical methods, and G(a/L) is obtained from equation (2) written for the reference case K = KIr (self-consistency). Details on the equations used to obtain the coefficient G(a/L) can be found in [14]. Once the two-term expression of the displacement field is obtained, the weight function can be determined, and, finally, the stress intensity factor can be calculated for any other loading case V(x), as:

1/2 3/21/2 1/200 0

( , )' ( ) ( ) 42a a

Ir

Ir Ir

u x aE a aK x dx x F a a x G a a x dxK a a L LK

VwV Vw w w (5)

3. Practical application of Case 2: Stress intensity factors for a strip with a cracked hole subjected to point loads

Thin sheets with rows of rivet holes are currently encountered in the aeronautical industry and in many other fields. Due to fatigue loadings, cracks may appear and develop, especially at the rivet holes, which are places of high stress concentration. An infinite plate with a row of rivet holes can be viewed as a succession of strips having a width equal to the distance between the holes. That is why the study of a strip of finite width, having a central hole is equivalent to the study of a plate with equally spaced holes.

This study presents the determination of stress intensity factors for a crack starting from the rivet hole subjected to a point load, in an finite width plate (Fig. 2, a). The crack length, hole radius and width of the sheet were chosen close to those encountered in the aeronautical industry. That is why three hole diameter to sheet width ratios were chosen: 2R/W = 0.1, 0.16 and 0.2. Since the usual cracks that appear at the rivet holes in these structures are small, 12 ratios a/R, from 1.01 to 1.5, were studied, namely a/R = 1.01, 1.02, 1.04, 1.06, 1.08, 1.1, 1.15, 1.2, 1.25, 1.3, 1.4 and 1.5

Values of the stress intensity factor for the strip with a cracked hole were obtained by Pastrama and de Castro [24] with the compounding technique [25] in the case of remote uniform traction (Fig. 2, b). The values are presented in the nondimensional form KIr /[V(Sac)]1/2, where the crack length ac=aR is considered in the calculations since the co-ordinate system has the origin at the crack mouth (Fig. 1). These results are further used as the reference case in the weight function approach together with the approximate displacement expression (4).

With the hole radius R as the characteristic parameter, the function F(ac/L) = F(ac/R) is obtained from the reference stress intensity factors for each crack length. A fourth degree polynomial fit was used to obtain a general expression of F(ac/R), the values calculated using this fit being different from those obtained with the compounding method with less than 0.3%. Then, the function G(ac/R) was calculated with the relationships given in [14]. For applying these equations, the stress variation Vr(x) on the crack line in the uncracked structure for the reference case is needed and was obtained by finite element analyses.


a. b.

Fig. 2. a. Strip with a cracked hole subjected to point load; b. Strip with a cracked hole subjected to remote traction (reference case).

Once the function G(a/R) was determined, then the approximate equation of the crack face displacement (4) was established. Further, the stress intensity factor in the case of point load, was calculated, using equation (5). To apply this relationship, the variation of the crack line stress in the uncracked strip should be known in this loading case and was also established using the finite element method. The stress intensity factors obtained through this methodology are plotted in Fig. 3 in the normalized form KI /[V0(Sac)]1/2 where V0 is the remote stress due to the force P.

No results for comparison were found in the literature for the calculated SIFs. But, since for 2R/W=0.1 the boundary of the strip is far from the crack, results for this case can be checked with the ones listed for a point load applied on the hole surface in an infinite plate, [26]. Nevertheless, the boundaries should have a small effect on the stress intensity factor and so, the values for 2R/W = 0.1 should be slightly greater than those for an infinite plate.The obtained results for a single crack in this case are listed in Table 1.

One can see that the results obtained through the weight function method differ with less than 5% from those listed in [26]. As it was underlined above, they are slightly greater than the values for an infinite strip. Since for this ratio 2R/W the boundary is far enough to approach the case of an infinite strip, the results could be considered as accurate. Clearly, for other ratios 2R/W the values of the stress intensity factor are greater, the boundary being closer to the crack.

Fig. 3: Nondimensional SIFs for a strip with a cracked hole subjected to point load.

a

W

R P

Pa

W

V

V

a' R a'


Table 1 Comparison of the obtained results for 2R/W=0.1 with the case of a point load on the hole surface

ac/R K/V0(Sac)1/2 This paper K/V0(Sac)1/2

Wu and Carlsson [26] Difference

[%] 0.01 1.4483 1.400 3.45 0.02 1.4206 0.04 1.3679 0.06 1.3186 0.08 1.2724 0.10 1.2294 1.194 2.96 0.15 1.1401 0.20 1.0495 1.016 3.30 0.25 0.9762 0.30 0.9110 0.878 3.76 0.40 0.7987 0.768 4.00 0.50 0.7077 0.680 4.07

4. Stress intensity factors and crack face displacement from finite element analyses

In the case when neither the stress intensity factor nor the crack face displacements are known for a chosen geometry and different crack lengths, one can rewrite equation (4) in the following general form:

1/2 1/2 1/2 3/21 2, a au x a C a a x C a a xL L (6)

For determining the coefficients C1 and C2, the finite element method was used in [22] to obtain values of the

crack face displacements for several crack lengths. An expression for uIr(x) can be obtained by curve fitting of the finite element results and then, a general expression uIr(x,a) can be established by polynomial interpolation of the C1 and C2 values from the previous expressions. Then, by writing equation (2) for the case KI = KIr (self-consistency), it results that the loading for which the stress intensity factor is calculated with (2) is in fact the reference loading. In this case, taking into account relationships (2) and(6), it yields:

1/2 1/2 1/2 3/22 1 20 0

,' 'a a

IrIr

u x a a aK E dx E C a a x C a a x dxa a L L

w ww w

(7)

Equation (7) can be used to obtain reference stress intensity factors KIr for different crack lengths. Then, the stress intensity factors for any other loading case can be calculated by simply using relation (2), provided that the crack line stresses V(x) in the studied case are known, either from analytical expressions or from finite element analyses.

5. Practical application of Case 3: Thick walled tube with external axial crack, subjected to internal pressure

The method described above is applied to a thick walled cylinder having the ratio Re/Ri =2, subjected to uniform internal pressure. An external axial crack is considered, with seven different crack length to wall thickness ratios a/t: 0.2, 0.3, 0.4, 0.5, 0.6, 0.7 and 0.8. The geometry is depicted in Fig. 4. The material of the tube has the following elastic constants: Youngs modulus E = 21104MPa and Poissons ratio Q = 0.3.

The reference loading case was considered uniform pressure on the crack faces. Because the stress intensity factor can be expressed in the nondimensional form Kr/V(Sa)1/2, it is convenient to choose a pressure of 1MPa for the reference case. For each crack length, finite element analyses were performed in order to obtain the crack face displacements in the reference case. The values of the coefficients C1 and C2 from relation (6) were obtained by appropriate curve fitting of the finite element displacements, and are listed in Table 2.


Fig. 4: Thick walled tube with axial external crack.

Table 2: Values of the coeffients C1 and C2 obtained by curve fitting of the finite element results

a/t C1 C2 0.2 15.361410-6 -3.032410-7 0.3 17.013310-6 2.563510-7 0.4 19.066510-6 11.224410-7 0.5 21.571310-6 24.454410-7 0.6 24.663110-6 45.187110-7 0.7 28.578810-6 74.346710-7 0.8 34.434510-6 107.96610-7

By using a sixth degree polynomial interpolation of the values from Table 2, the following expressions were

obtained for C1 and C2: 2 3 4 5 6

61

2 3 4

2

18.7116 80.1886 569.12 1779.74 3126.17 2840.38 1057.13 10

20.6358 408.522 2407.6 6313.34 8609.46

a a a a a aCt t t t t t

a a a aCt t t t

5 674954.57 773.611 10a a

t t

(8) The values of the dimensionless stress intensity factor for the reference case, calculated with equation (7) are

presented in Table 3 togther with results found in the literature, [27]. A very good agreement can be noticed, the differences being, with only one exception, less than 1%.

Table 3: Dimensionless stress intensity factors KIr/V(Sa)1/2 in the reference case for Re/Ri = 2

a/t This paper Andrasic and Parker [27] Difference

[%] 0.2 1.279 1.282 -0.23 0.3 1.421 1.418 0.21 0.4 1.580 1.581 -0.06 0.5 1.783 1.787 -0.22 0.6 2.033 2.033 0.0 0.7 2.339 2.345 -0.26 0.8 2.792 2.606 7.0

The stress intensity factors from Table 3 and the expression (6) of the crack face displacements with the

coefficients listed in Table 2 were inserted in relationship (2) to obtain the stress intensity factors for the the thick walled cylinder with axial external crack, subjected to uniform internal pressure. In this equation, the stress V(x) in the uncracked body on the prospective crack faces is in fact the hoop stress. For a thick walled cylinder subjected to

a

pi

y

x

RR

tie


uniform internal and external pressure, this stress is given by [28]:

2 22 2

2 2 2 2 2i e e ii i e e

he i e i

p p R RR p R pR R R R r

V (9) where pe and pi are the external and respectively internal pressure acting on the tube. Since for the loading case studied herein pe = 0 and the radius r can be expressed in the system of axes from Fig. 4 as r = Re x, one obtains:

2 22

2 2 22 2i e ii i

e i e i e

p R RR pxR R R R R x

V (10) The values of the stress intensity factor for this loading case, obtained with relation (2), are presented in

nondimensional form in Table 4. Results from Delale and Erdogan [29] for the same configuration are also listed for comparison, a very good agreement being noticed.

Table 4: Dimensionless stress intensity factors for internal pressure

a/t This paper Delale and Erdogan [29] Difference

[%] 0.2 0.909 0.909 0.0 0.3 1.043 1.040 0.29 0.4 1.200 1.202 -0.17 0.5 1.402 1.402 0.00 0.6 1.655 1.654 0.06 0.7 1.977 1.982 -0.25 0.8 2.459 2.446 0.53

6. Conclusions

The paper presents two practical applications of the weight function method in the cases when one or both the parameters necessary to apply this method (reference stress intensity factors and crack face displacements) are unknown.

A methodology based on an approximate expression for the crack face displacement proposed by Petroski and Achenbach is applied in order to obtain the stress intensity factors for a strip with a cracked hole, subjected to a point load on the whole surface, normal to the crack line. Reference stress intensity factors were obtained using the compounding method and then, the coefficients in the expression of the crack face displacements were computed. Only results for the case 2R/W = 0.1 were compared with those found in the literature for an infinite plate with a hole, since results for other studied ratios were not found. The case of an infinite plate provides stress intensity factors slightly smaller than for 2R/W = 0.1, a result that was expected and shows the accuracy of the calculated values.

The second case presented in this paper is the one when neither the reference stress intensity factor nor the crack face displacements are known. The method is based on the determination of the crack face displacements using the finite element method and on curve fitting techniques for obtaining a general expression of these displacements. The resulting variation law is used for calculating the stress intensity factor KIr in a reference case. Then, the weight function, which depends only on the studied geometry, can be established. With this function, the stress intensity factor for any other loading case can be easily calculated. The method is applied for a thick walled tube with an external axial crack and subjected to uniform internal pressure. The obtained results were compared to those found in the literature. Again, a very good agreement was noticed, the differences being less than 1%. The advantage of the proposed method is that it can be applied for cracked bodies for which stress intensity factors solutions are not available. General solutions, for any geometry and any crack aspect ratios can be obtained.

The two methods used in this research can be utilized for calculating the stress intensity factor for many other loading cases and many values of the crack length, thus providing important information for subsequent studies, especially for fatigue loads, where SIFs are necessary for the crack growth rate determination.


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