Stress intensity factor analysis for an interface crack between dissimilar isotropic materials

Stress intensity factor analysis for an interface crack between dissimilar isotropic materials under thermal stress T. Ikeda* & C. T. Sun* I Chemical Engineering Group, Department of Materials Process Engineering, Graduate School of Engineering, Kyushu University, Japan 2 School of Aeronautics and Astronautics, Purdue University, USA Abstract Thermal stresses, one of the most important causes of interfacial failure between dissimilar materials, arise from different coefficients of linear thermal expansion. An efficient numerical procedure in conjunction with the finite element method (FEM) for the stress intensity factor (SIF) analysis of interface cracks under thermal stresses is presented. The crack closure integral method is modified using the superposition method. The SIF analyses of some interface crack problems under mechanical and thermal loads are demonstrated. Very accurate mode separated SIF's are obtained. 1 Introduction The stress intensity factors (SIF's) of a crack between two dissimilar materials are important parameters for evaluating delamination strength. For practical applications, the SIFs of bimaterial interface cracks most likely would be obtained by numerical analyses such as the finite element method (FEM) or the boundary element method (BEM). Energy approaches such as the crack closure integral method [1], the J-integral method [2] and the virtual crack extension method (VCE) [3, 4] are reliable methods for calculating the energy release rate using FEM and BEM.

64 Damage and Fracture Mechanics VI Unlike cracks in homogeneous materials, a bimaterial interface crack always induces both opening and shearing modes of stress in the vicinity of the crack tip. In order to obtain the respective modes of SIF's, Yau and Wang [5] applied the M/- integral which is a combination of the J-integral and the superposition method with FEM to bimaterial interface crack problems. Matos et al. [6] applied the virtual crack extension method to the calculation of the mixed mode SIF's for interface cracks in conjunction with the superposition method and FEM. Miyazaki et al. [7, 8] also applied the M^-integral method and the virtual crack extension method to the SIF analysis for interface cracks with BEM. The modified crack closure integral method (MCCI) [9], which is a modified version of Irwin's original, is one of the most popular techniques for the SIF analysis. Many researchers [10-12] recognized the inadequacy in using MCCI for fracture mode separation for interface cracks because of the nonconvergence nature of individual mode of the strain energy release rate. Sun and Qian [13] proposed a mode separation technique in conjunction with the MCCI, but solution of a nonlinear equation containing complex numbers is required. Yuuki and Cho [14] applied the displacement extrapolation method (DE) to the SIF analysis with the BEM. However, the accuracy of DE is worse than the energy method. Sun and Qian [13] recommended the use of the displacement ratio method (DR), which is a similar method to DE without extrapolation, for only the mode separation of the SIF's of interface cracks. Great residual stress is often caused near an interface between dissimilar materials because of the difference in the coefficient of linear thermal expansion between the two jointed materials. For this reason, thermal stresses are more important for an interface crack than for a crack in a homogeneous material. However, none of the techniques for the SIF analysis mentioned above can analyze the SIF of interface cracks in the presence of thermal stresses. In this paper, we modified the MCCI for an interface crack under thermal stresses. The obtained SIF's using this method were compared with analytical solutions. Excellent accuracy of this method is demonstrated. 2 Stress intensity factors of an interface crack A definition of the SIF's of an interface crack was originally proposed by Erdogan [15]. In the coordinates system shown in Figure 1, stresses along the x- axis near an interface crack tip is where Kj and K^ are SIF's of an interface crack respectively, a^. + /o^, is complex stress, i is the imaginary number, f=-l, and 8 is the bielastic constant given by r e = (l/27i) In (Ki/jaj + l/p^)/^/!^ "*" ^M-i) (2)

Damage and Fracture Mechanics VI 65 Material Figure 1: Coordinate system around an interface crack tip., = 3-4v, (Plane Strain) (Plane Stress) (3) The mode angle of SIF's, y, is defined as y = sign (KJ cos- \K,IK,} (-n< y < n) sign (K,j) = 1 (for > 0) = -1 (forff,, < 0) (4) (5) where K^ = \JK] + K]J. We used the definition by Erdogan [15]. The relationship between the energy release rate G and the SIF's is expressed as the following equations by Malyshev and Salganik [16]. G=P[JT/+ //J (6) 3 Analysis of stress intensity facors 3.1 Mode Separation of the Stress Intensity Factors by the Principle of Superposition The stress distribution around an interface crack tip is essentially mixed mode. Thus, it is necessary to evaluate both mode I and mode II SIF's in order to characterize the fracture of a bimaterial interface crack. The distributions of stress and displacement around a crack tip in homogeneous material is symmetric for mode I and skew-symmetric for mode II to the jc-axis in Figure 1. This relationship is often utilized for the mode separation of the SIF's. However, the distributions of stress and displacement around an interface crack do not have this feature. Yau and Wang [5] and Matos et al [6] used the superposition method for the mode separation of the SEF for interface cracks in conjunction with the J-integral method and the VCE, respectively. The concept of the superposition method is briefly explained as follows.

66 Damage and Fracture Mechanics VI Consider two independent equilibrium states with field variables denoted by superscripts (1) and (2) for a region surrounding a crack tip. The superscript (1) indicates the 'target problem' which we try to solve, and the superscript (2) indicates 'the reference problem' whose distributions of displacement, stress, temperature and stress intensity factors are already known. The superposition of the two equilibrium states leads to another equilibrium state, the 'superposed problem', (1+2). The distributions of displacement, stress and temperature in the superposed problem are obtained by the superposition of these two equilibrium states (1) and (2), f. e., where % and a are displacement and stress vectors respectively, and T is temperature. The SIF's can _ also be superposed as, -,, /-(I). r/-(2) v/ = ^// // v/ +*// (9) Substituting eqn (9) into eqn (6), the energy release rate of the superposed state (1+2) is obtained as Hence, (10) Any known problem can be used as the reference problem. The asymptotic solutions [15] near a bimaterial interface crack tip are most convenient for the reference problem. If the asymptotic solutions for K = 1, K = 0 and K = 0, Kj = 1 are selected, we can separate the SIF's using eqn (11). The accuracies of K and K^ calculated according to eqn (11) depend on the accuracies of G<^> and G<". The MCCI is utilized for calculating G<^*> and G in this study. 3.2 MCCI for an interface crack under thermal stresses (11) Asymptotic Solution Asymptotic Solution (Displacement) (Displacement) / P! JP2 Pi' P2 C P3 P4 Pa' P4* ^ c PS P6 P? Small FEM model (MCCI-FEM1) Figure 2 Modified crack closure integral method. Small FEM model (MCCI - FEM2)

Damage and Fracture Mechanics VI 67 The modified crack closure integral method (MCCI) proposed by Rybicki and Kanninen [9] for 8-nodes isoparametric finite elements is shown as 1 - (12) r = lim T*-?f (a? where Gj and G^ are mode I and mode II components of energy release rate respectively; / is nodal force; u, nodal displacement; and c is the crack extension. Subscripts PJ - p^ and p^' - p^ indicate the nodes near a crack tip as shown in Figure 3. The lengths of the finite elements immediately ahead and behind the crack tip are usually taken the same as the crack extension. Total strain energy release rate is the sum of G, and G^. G = G/ + G//= lim JH(F^)^ AfM + (F^Y A(7^ d3) ' " c-$o 2c / ' J where F and U are nodal force vector and nodal displacement vector. Each mode of the strain energy release rate in eqn (12) for an interface crack depends on the crack extension c. Sun and Jih [10] discovered that both G/ and G// do not converge as c» 0, so that the mode separation using eqn (12) is meaningless for an interface crack between dissimilar materials. We utilized the superposition technique for the mode separation. The GO) and GO+2) ^ eqn (11) must be obtained for the superposition method. The GO) can be easily obtained using eqn (13). The MCCI for GO+2) js expressed as where subscripts (1+2) denote the sum of the target problem and the reference problem for the superposition method. F and At7^ in eqn (14) can be obtained as follows. The boundary of the small finite element mesh around a crack tip (see Fig 4), the MCCI-FEM1 or the MCCI-FEM2, is subjected to the asymptotic solution for displacement. This small mesh should be taken to be identical to the mesh of the target problem. F** and F^ are obtained as the reaction force of the nodes of small FEM analysis. The &W* and Af/^' are also obtained as the solution of the small FEM analysis. The MCCI cannot be applied to the thermal stress problem in the strict sense; however, it is expected that the obtained values by the MCCI should approach the exact values with decreasing values of c. 3.3 Displacement extrapolation method Yuuki and Cho [14] applied the stress extrapolation method (SE) and the displacement extrapolation method (DE) to interface cracks between dissimilar materials. The DE method extrapolates the SIF's using the near tip relations (14)

68 Damage and Fracture Mechanics VI Material 1 a\a I atepai I 2 / * * * * * * * Mesh 1 (c/a=0.5) Figure 3 : A crack along the interface of jointed dissimilar semi-infinite plates subjected to tension and shear. Table 1. Exact solutions of the stress intensity factors of an interface _ crack subjected to tension and shear for e= - 0.07582. * /(-) &W F(-) y(deg.) Oo=1.0, Y= 0.0 LO 0.15162 1.01143-8.62172 ^=0.0, T^=1.0 0.15162 1.0 1.01143 81.3783 between the SIF's and relative crack surface displacements. This method is very simple, but its accuracy is worse than that of energy methods like the VCE or the MCCI. Sun and Qian [13] obtained K^ using the MCCI and obtained g using the displacement ratio method (DR). Theoretically, DE or DR cannot be applied for a crack under thermal stresses. However, noting that the thermal displacement in the vicinity of a crack tip can be ignored, the DE or DR method is applicable in the presence of thermal stresses if the crack surface displacement very close to the crack tip are used. 4 Numerical Results The accuracies of the MCCI and the DE for interface cracks under mechanical and thermal loads were examined for several examples. 4.1 An Interface Crack Subjected to Tension and Shear The first example is a crack along the interface of jointed dissimilar semiinfinite plates subjected to tension and shear as shown in Figure 3. Salganik [17] presented the exact solution of the stress distribution on the %-axis in the vicinity of the crack tip. The SIF's of this problem are shown as K, ± re J for L= 2a (15)

i 4 ^ Tension 1 3: MCCI Damage and Fracture Mechanics VI 4 3 69 ^ 1 0 1 _UJ ^-1 ' -2 S* o J "3 k, ^ -4 ~ -5 Mesh 1 Mesh 2 Mesh 3 Mesh 4 Mesh 1 Mesh 2 Mesh 3 Mesh 4 5," - 1 's -2 j-3 ^-4 ~ -5 Shear MCCI Mesh 1 Mesh 2 Mesh 3 Mesh 4 7 'present -Y 'Exact Weg.) 4 - y = sign (Kjj) arccos 1 Kj I K\ 3 2 1 0-1 -2-3 -4 : -#- DE -A- MCCI-FEM1 F _e_ MCCI-FEM2 Mesh 1 Mesh 2 Mesh 3 Mesh 4 Figure 4: The relative errors of F, and the errors of y obtained by several techniques for an interface crack in jointed semi-infinite plate under uniform tension. where the + and the - correspond to the right and left crack tips, respectively. The Young's moduli and Poisson's ratios were set to be E= IX 10 (MPa), E^= IX 10& (MPa) and \\= ^= 0.3, respectively. The corresponding bimaterial constant is 8 = 0.07582. The SIFs can be nondimensionalized as (J = I U and e) (16) The exact solutions for separate form tension and uniform shear are shown in Table 1. Large jointed panels with a center interface crack whose total size is 200a* 200a were modeled by the FEM. The width and the height of the plate were taken to be 100 times crack length a. Four types of meshes with different fineness around the crack tip were used for the analyses. The coasest mesh (Mesh 1) is shown in Figure 3. The da in Figure 3 is set in 0.5 for Mesh 1, 0.25 for Mesh 2, 0.125 for Mesh 3 and 0.0625 for Mesh 4, respectively. Plane strain was assumed for all FEM analyses in this study. The SIF's of this problem were analyzed using the MCCI and the DE. The relative errors of F, obtained by the MCCI, and the errors of y obtained by the

70 Damage and Fracture Mechanics VI Mesh 1 (c/fe=0.5) Figure 5: Residual stress in jointed dissimilar semi-infinite plates with double edge cracks. MCCI and the DE for four types of meshes are shown in Fig. 4. The F obtained by the DE is not shown, because this method has been used only for calculating y. The F, obtained by DE is not used because of its poor accuracy [13]. The MCCI provide accurate F\ The relative errors of F\ obtained by Mesh 3 and Mesh 4 are within 1 % for both uniform tension and uniform shear. The MCCI-FEM1 and MCCI-FEM2 provided an excellent accuracy in y even using Mesh 1. Notice that Mesh 1 is a very coarse mesh whose element size c is 50% of a, and even Mesh 4 whose element size is still 6.25% of a, is not very fine. Sun and Qian [13] recommended to use 1% of a for the mode separation using the DR method. 4.2 Residual stress in jointed dissimilar semi-infinite plates with double edge cracks The second example is double edge cracks in jointed dissimilar semi-infinite plates subjected to a uniform change of temperature as shown in Figure 5. Erdogan [18] presented the exact solution of stress along the X-axis. The SIF's of this problem are Kj + i KU = Go(a2T 2 - ait JATVn-F (2e -1) for l^= 2b (17) T, = 1 (for plane stress), = 1 + v, (for plane strain) (%g) where cc, and oc^ are coefficients of linear thermal expansion for material 1 and material 2, respectively, AT is the change of temperature. We assumed the coefficients of linear thermal expansion, a, and a,,, to be 1.0 X 10? (l/ c ) and 1.0 X 10'G (1/ C ), respectively, and assumed the change of temperature AT to be 100 C. Other material properties, E\, E^ Vj, v^ and E are assumed to be the same as in the previous example. The SIF's can be nondimensionalized as

-5 MCC! Mesh 1 Mesh 2 Mesh 3 Mesh 4 1 i J3?>-c J 2 1 Damage and Fracture Mechanics VI 71 0 -% -#- DE -1 : -^r- MCCI-FEM1?- ; -a- MCCI-FEM2-2 Meshl Mesh 2 Mesh 3 Mesh 4 Figure 6: The relative errors of F, and the errors of y obtained by several techniques for the residual stress in jointed dissimilar semiinfinite plates with double edge cracks. F,= - air] (j = /, // and e) (20) The exact solutions of this problem are F= - 0.1516, F,,= - 1, F^ = 1.0114 and ^= - 98.623 (deg.). The left symmetrical half of a large jointed panel with double edge interface cracks was modeled by thefem. The size of the entire jointed panel was 200/7 X 2006. Four meshes with increasing refinements around the crack tip are similar to the previous example. The relative errors in F^ and y obtained by different procedures with these four finite element meshes are shown in Figure 6. The relative errors in F, obtained by the MCCI are within 1% for Mesh 3 and within 0.5% for Mesh 4. The MCCI-FEM1 and the MCCI-FEM2 can yield accurate y for all meshes. However, the error in y obtained by the DE is more than 1% even by Mesh 4. 5 Conclusion The MCCI was modified for the SIF analysis of interface cracks in the presence of thermal stresses. These methods in conjunction with the FEM were applied to three jointed semi-infinite plates with interface cracks, a center crack subjected to tension and shear, double edge cracks under uniform change of temperature, and the numerical solutions were compared with the exact solutions. The modified MCCI proposed in this study provided excellent accuracy for mechanical and uniform temperature loads using much coarser mesh than that used by the DE for the same accuracy. Another advantage in this technique is its ease to implement. References [ 1 ] Irwin, G.R., Analysis of stresses and strains near the end of a crack traversing

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