本文探討雙材料界面裂紋受熱負載作用下，使用有限元素分析配合虛擬裂紋閉合積分式計算，裂紋前緣之破壞力學參數分佈，包括應變能釋放率及應力強度因子。一般三維破壞力學分析中，為了配合虛擬裂紋積分式之自我相似假設，須將裂紋尖端前後元素在界面上之網格建立為長方形且尺寸相同，可得到較正確的計算結果，但此方法應用於曲線型之裂紋前緣將造成有限元素網格建立不易。為此，本文提出一搭配自由網格使用之破壞力學參數計算方法，可減少建立網格所需時間，有利於複雜結構中之裂紋成長分析。另外，由於界面裂紋尖端的奇異彈性應力場具有震盪的行為，此震盪之特性會造成直接使用虛擬裂紋閉合法求解界面裂紋之應力混合模式相位角的困難。對此問題，可藉由特定之材料特徵長度來修正震盪的應力強度因子，及對應之相位角。針對於本文所發展之破壞力學參數求解方法，首先透過與界面裂紋問題的理論解析解，同時也利用三維奇異有限元素模型比對驗證；而後，本文討論應用此方法分析半導體電子元件中之矽穿孔(Through-Silicon Via, TSV)載板界面裂紋受熱負載情況下，界面脫層成長之問題。

A numerical procedure based on the virtual crack closure technique (VCCT) is developed to post-process finite element solutions for calculating the fracture mechanics parameters including the strain energy release rate and the stress intensity factors. A basic requirement in applying the VCCT is the satisfaction of the self-similar growth assumption. For the cases of the three-dimensional problem, the requirement is typically met by restricting the elements in front of and behind the crack tip to be rectangular and of the same size, for which the implementation is not straightforward when a curvilinear crack front is involved. In this thesis, a procedure involving calculating the crack closure integrals along a line path perpendicular to the crack front is presented. The procedure can be used with finite element model with free meshes. The proposed methodology was applied to study the fatigue growth of an embedded interface crack in a Si interposer under thermal cycling condition.

[1] X. Wu, K. W. Paik and S. N. Bhandarkar, “To cut or not to cut: a thermomechanical stress analysis of polyimide thin-film on ceramic structures”, IEEE Transactions on Components, Packaging, and Manufacturing Technology, Vol. 18, pp. 150-153, 1995.
[2] J. L. Beuth and S. H. Narayan, “Residual stress-driven delamination in deposited multi-layers”, International Journal of Solids and Structures, Vol. 33, pp. 65-78, 1996.
[3] J. W. Hutchinson, Z. Suo, “Mixed mode cracking in layered materials.”, Advances in Applied Mechanics, Vol. 29, pp. 63-191, 1992.
[4] V. I. Mossakovskii and M. T. Rybka, “Generalization of the Griffith-Sneddon criterion for the case of a nonhomogeneous body”, Journal of Applied Mathematics and Mechanics, Vol. 28, pp. 1277-1286, 1964.
[5] T.-C. Chiu and H.-C. Lin, “Analysis of stress intensity factors for three-dimensional interface crack problems in electronic packages using the virtual crack closure technique”, International Journal of Fracture, Vol. 156, pp. 75-96, 2009.
[6] G. R. Irwin, “Analysis of stresses and strains near the end of a crack traversing a plate”, Journal of Applied Mechanics, Vol. 24, pp. 361-364, 1957.
[7] E. F. Rybicki and M. F. Kanninen, “A finite element calculation of stress intensity factors by a modified crack closure integral”, Engineering Fracture Mechanics, Vol. 9, pp. 931-938, 1977.
[8] K. N. Shivakumar, P. W. Tan and J. C. Newman, Jr., “A virtual crack-closure technique for calculating stress intensity factors for cracked three dimensional bodies”, International Journal of Fracture, Vol. 36, pp. R43-R50, 1988.
[9] I. S. Raju, “Simple formulas for strain-energy release rates with higher order and singular finite elements”, NASA/CR-1986-178186.
[10] G. De Roeck and M. M. Abdel Wahab, “Strain energy release rate formulae for 3D finite element”, Engineering Fracture Mechanics, Vol. 50, pp. 569-580, 1995.
[11] I. S. Raju, R. Sistla and T. Krishnamurthy, “Fracture Mechanics analyses for skin-stiffener debonding”, Engineering Fracture Mechanics, Vol. 54, pp. 371-385, 1996.
[12] S. A. Fawaz, “Application of the virtual crack closure technique to calculate stress intensity factors for through cracks with an elliptical crack front”, Engineering Fracture Mechanics, Vol. 59, pp. 327-342, 1998.
[13] D. Xie and J. S. Biggers, “Strain energy release rate calculation for a moving delamination front of arbitrary shape based on virtual crack closure technique”, Part I: Formulation and validation, Engineering Fracture Mechanics, Vol. 73, pp. 771-785, 2006.
[14] A. Miravete and M. A. Jimenez, “Application of the finite element method to prediction of onset of delamination growth”, Applied Mechanics Reviews, Vol. 55, pp. 89-106, 2002.
[15] R. Krueger, “The virtual crack closure technique: history, approach and applications”, NASA/CR-2002-211628.
[16] M. L. Williams, “The stresses around a fault or crack in dissimilar media”, Bulletin of the Seismological Society of America, Vol. 49, pp. 199-204, 1959.
[17] P. P. L. Matos, R. M. McMeeking, P. G. Charalambides and M. D. Drory, “A method for calculating stress intensities in bimaterial fracture”, International Journal of Fracture, Vol. 40, pp. 235-254, 1989.
[18] W. T. Chow and S. N. Atluri, “Finite element calculation of stress intensity factors for interfacial crack using virtual crack closure integral”, Computational Mechanics, Vol. 16, pp. 417-425, 1995.
[19] C. T. Sun and W. Qian, “The use of finite extension strain energy release rates in fracture of interfacial cracks”, International Journal of Solids and Structures, Vol. 34, pp. 2595-2609, 1997.
[20] A. Agrawal and A. M. Karlsson, “Obtaining mode mixity for a bimaterial interface crack using the virtual crack closure technique”, International Journal of Fracture, Vol. 141, pp. 75-98, 2006.
[21] C. Bjerken and C. Persson, “A numerical method for calculating stress intensity factors for interface cracks in bimaterials”, Engineering Fracture Mechanics, Vol. 68, pp. 235-246, 2001.
[22] H. Okada, H. Kawai, T. Tokuda, Y. Fukui, “Fully Automated Mixed Mode Crack Propagation Analyses using VCCM (Virtual Crack Closure-Integral Method) for Tetrahedral Finite Element”, Key Engineering Materials, Vols. 462-463, pp. 900-905, 2011.
[23] H. Pathak, A. Singh, I. V. Singh, “Fatigue crack growth simulations of 3-D problems using XFEM”, International Journal of Mechanical Sciences, Vol. 76, pp. 112-131, 2013.
[24] J. R. Rice, “Elastic fracture mechanics concepts for interfacial cracks”, Journal of Applied Mechanics, Vol. 55, pp. 98-103, 1988.
[25] J. R. Rice, Z. Suo, J. S. Wang, “Mechanics and thermodynamics of brittle interfacial failure in bimaterial systems”, Metal-Ceramics Interfaces, Vol. 4, pp. 269-294, 1990.
[26] J. Dundurs, “Edge bonded dissimilar orthogonal elastic wedges”, Journal of Applied Mechanics, Vol. 36, pp.650-652, 1969.
[27] P. C. Paris and F. Erdogan, “A critical analysis of crack propagation laws”, Journal of Basic Engineering, Transactions ASME, Vol. 85, pp. 528-534, 1963.
[28] J. M. Snodgrass, D. Pantelidis, M. L. Jenkins, J. C. Bravman and R. H. Dauskardt, “Subcritical Debonding of Polymer/Silica Interfaces Under Monotonic and Cyclic Loading”, Acta Materialia, Vol. 50, pp. 2395-2411, 2002.
[29] A. M. Bregman and M. K. Kassir, “Thermal fracture of bonded dissimilar media containing a penny-shaped crack”, International Journal of Fracture, Vol. 10, pp. 87-98, 1974.
[30] X. F. Wu, A. D. Yuris, “Closed-form solution for a mode-III interfacial edge crack between two bonded dissimilar elastic strips.”, Mechanics Research Communication, Vol. 29, pp. 407-412, 2002.
[31] C. D. Hartfield, E. T. Ogawa, Y.-J. Park, T.-C. Chiu and H. Guo, “Interface reliability assessments for copper/low-k products”, IEEE Trans. Device Mater. Rel., Vol. 4, No. 2, pp. 129-141, Jun. 2004.