Abstract
For crystals, the compliance (s ij ) and the stiffness (c ij ) matrices are specified in the orthogonal coordinate systems (Y i ), which do not coincide with the crystal axes (X i ) commonly used except for cubic and orthorhombic crystal systems. Transformations have been done in this paper and the general compliance transformation relations from the orthogonal coordinate systems (Y i ) to the measurement systems (M i ) are given for all seven crystal systems. Accordingly, useful expressions for Young’s modulus E and Poisson’s ratio υ are also derived.
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References
Zhang J M, Zhang Y, Xu K W. Anisotropy analysis of the thermal stresses and strain energies in BCC metal films. Physica B, 2005, 368: 215–222
Zhang J M, Zhang Y, Xu K W. Dependence of stresses and strain energies on grain orientations in FCC metal films. J Cryst Growth, 2005, 285: 427–435
Zhang J M, Zhang Y, Xu K W, et al. Dependence of the strain energies on grain orientations in HCP metal films. Appl Surf Sci, 2006, 253: 2432–2436
Zhang J M, Zhang Y, Xu K W, et al. Dependence of the stresses on grain orientations in hexagonal films. Physica B, 2007, 388: 261–265
Zhang Y, Zhang J M, Ji V. Anisotropy analysis of stresses and strain energies in diamond-cubic films. Physica B, 2007, 389: 372–376
Zhang J M, Zhang Y, Xu K W, et al. Anisotropic elasticity in textured cubic film plane. Physica B, 2008, 403: 3379–3383
Ohmi T, Saito T, Otsuki M, et al. Formation of copper thin films by a low kinetic energy particle process. J Electrochem Soc, 1991, 138: 1089–1097
Huang H H, Chang H P, Chien Y T, et al. Influence of annealing temperature on the grain growth of samarium-doped ceria. J Cryst Growth, 2006, 287: 458–462
Chen K N, Fan A, Tan C S, et al. Microstructure evolution and abnormal grain growth during copper wafer bonding. Appl Phys Lett, 2002, 81: 3774–3776
Zhang J M, Xu K W, Ji V. Strain-energy-driven abnormal grain growth in copper films on silicon substrates. J Cryst Growth, 2001, 226: 168–174
Zhang J M, Xu K W, Ji V. Competition between surface and strain energy during grain growth in free-standing and attached Ag and Cu films on Si substrates. Appl Surf Sci, 2002, 187: 60–67
Kočka J, Mates T, Stuchlíková H, et al. Characterization of grain growth, nature and role of grain boundaries in microcrystalline silicon-review of typical features. Thin Solid Films, 2006, 501: 107–112
Brini R, Schmerber G, Kanzari M, et al. Study of the growth of CuAlS2 thin films on oriented silicon (111). Thin Solid Films, 2009, 517: 2191–2194
Sasajima Y, Kageyama J, Khoo K, et al. Grain coarsening mechanism of Cu thin films by rapid annealing. Thin Solid Films, 2010, 518: 6883–6890
Dölle H. The influence of multiaxial stress states, stress gradients and elastic anisotropy on the evaluation of (residual) stresses by X-rays. J Appl Crystallogr, 1979, 12: 489–501
Murarka S P. Multilevel interconnections for ULSI and GSI. Mater Sci Eng R-Rep, 1997, 19: 87–151
Nye J F. Physical Properties of Crystals. Oxford: Clarendon Press, 1985. 131–149
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Zhang, Y., Ji, V. General compliance transformation relations for all seven crystal systems. Sci. China Phys. Mech. Astron. 56, 694–700 (2013). https://doi.org/10.1007/s11433-013-5025-5
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DOI: https://doi.org/10.1007/s11433-013-5025-5