Abstract
Based on the fundamental equations of piezoelasticity of quasicrystals (QCs), with the symmetry operations of point groups, the plane piezoelasticity theory of onedimensional (1D) QCs with all point groups is investigated systematically. The governing equations of the piezoelasticity problem for 1D QCs including monoclinic QCs, orthorhombic QCs, tetragonal QCs, and hexagonal QCs are deduced rigorously. The general solutions of the piezoelasticity problem for these QCs are derived by the operator method and the complex variable function method. As an application, an antiplane crack problem is further considered by the semi-inverse method, and the closed-form solutions of the phonon, phason, and electric fields near the crack tip are obtained. The path-independent integral derived from the conservation integral equals the energy release rate.
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References
Shechtman, D., Blech, I., Gratias, D., and Cahn, J. W. Metallic phase with long-range orientational order and no translational symmetry. Physical Review Letters, 53, 1951–1953 (1984)
Elser, V. Comment on QCs: a new class of ordered structures. Physical Review Letters, 54, 1730 (1985)
Kramer, P. and Neri, R. On periodic and non-periodic space fillings obtained by projection. Acta Crystallographica, A40, 580–587 (1984)
Ding, D. H., Yang, W. G., Hu, C. Z., and Wang, R. H. Generalized elasticity theory of QCs. Physical Review B, 48, 7003–7009 (1993)
Yang, W. G., Wang, R. H., Ding, D. H., and Hu, C. Z. Linear elasticity theory of cubic QCs. Physical Review B, 48, 6999–7002 (1993)
Hu, C. Z., Yang, W. G., Wang, R. H., and Ding, D. H. Point groups and elastic properties of two-dimensional QCs. Acta Crystallographica, A52, 251–256 (1996)
Hu, C. Z., Wang, R. H., and Ding, D. H. Symmetry groups, physical property tensors, elasticity and dislocations in QCs. Reports on Progress in Physics, 63, 1–39 (2000)
Fan, T. Y. and Mai, Y. W. Elasticity theory, fracture mechanics, and some relevant thermal properties of quasi-crystalline materials. Applied Mechanics Reviews, 57, 325–343 (2004)
Liu, G. T., Fan, T. Y., and Guo, R. P. Governing equations and general solutions of plane elasticity of one-dimensional QCs. International Journal of Solids and Structures, 41, 3949–3959 (2004)
Chen, W. Q., Ma, Y. L., and Ding, H. J. On three-dimensional elastic problems of one-dimensional hexagonal quasicrystal bodies. Mechanics Research Communications, 31, 633–641 (2004)
Wang, X. The general solution of one-dimensional hexagonal quasicrystal. Mechanics Research Communications, 33, 576–580 (2006)
Wang, X. and Pan, E. Analytical solutions for some defect problems in 1D hexagonal and 2D octagonal quasicrystals. Pramana - Journal of Physics, 70, 911–933 (2008)
Gao, Y., Xu, S. P., and Zhao, B. S. General solutions of equilibrium equations for 1D hexagonal QCs. Mechanics Research Communications, 36, 302–308 (2009)
Liu, G. T., Guo, R. P., and Fan, T. Y. On the interaction between dislocations and cracks in one-dimensional hexagonal QCs. Chinese Physics, 2, 1149–1155 (2003)
Gao, Y., Zhao, B. S., and Xu, S. P. A theory of general solutions of plane problems in twodimensional octagonal QCs. Journal of Elasticity, 93, 263–277 (2008)
Gao, Y. and Zhao, B. S. General solutions of three-dimensional problems for two-dimensional QCs. Applied Mathematical Model, 33, 3382–3391 (2009)
Fan, T. Y. and Guo, L. H. The final governing equation and fundamental solution of plane elasticity of icosahedral QCs. Physics Letters A, 341, 235–239 (2005)
Li, L. H. and Fan, T. Y. Final governing equation of plane elasticity of icosahedral QCs and general solution based on stress potential function. Chinese Physics Letters, 23, 2519–2521 (2006)
Gao, Y. and Zhao, B. S. A general treatment of three-dimensional elasticity of QCs by an operator method. Physica Status Solidi, B, Basic Solid State Physics, 243, 4007–4019 (2006)
Gao, Y. Governing equations and general solutions of plane elasticity of cubic QCs. Physics Letters A, 373, 885–889 (2009)
Thiel, P. A. and Dubois, J. M. QCs reaching maturity for technological applications. Materials Today, 2, 3–7 (1999)
Athanasiou, N. S., Politis, C., Spirlet, J. C., Baskoutas, S., and Kapaklis, V. The significance of valence electron concentration on the formation mechanism of some ternary aluminum-based QCs. International Journal of Modern Physics, B16, 4665–4683 (2002)
Park, J. Y., Ogletree, D. F., Salmeron, M., Ribeiro, R. A., Canfield, P. C., Jenks, C. J., and Thiel, P. A. High frictional anisotropy of periodic and aperiodic directions on a quasicrystal surface. Science, 309, 1354–1356 (2005)
Park, J. Y., Sacha, G. M., Enachescu, M., Ogletree, D. F., Ribeiro, R. A., Canfield, P. C., Jenks, C. J., Thiel, P. A., Sáenz, J. J., and Salmeron, M. Sensing dipole fields at atomic steps with combined scanning tunneling and force microscopy. Physical Review Letters, 95, 136802 (2005)
Lipp, H., Engel, M., and Trebin, H. R. Phason dynamics in one-dimensional lattices. Physical Review B, 81, 064302 (2010)
Engel, M., Umezaki, M., and Trebin, H. R. Dynamics of particle flips in two-dimensional QCs. Physical Review B, 82, 134206 (2010)
Guo, J. H., Yu, J., and Si, R. A semi-inverse method of a Griffith crack in one-dimensional hexagonal QCs. Applied Mathematics and Computation, 219, 7445–7449 (2013)
Guo, J. H., Yu, J., and Xing, Y. M. Anti-plane analysis on a finite crack in a one-dimensional hexagonal quasicrystal strip. Mechanics Research Communications, 52, 40–45 (2013)
Jazbec, S., Vrtnik, S., Jagličić, Z., Kashimoto, S., Ivkov, J., Popčcević, P., Smontara, A., Kim, H. J., Kim, J. G., and Dolinšek, J. Electronic density of states and metastability of icosahedral Au-Al-Yb quasicrystal. Journal of Alloys and Compounds, 586, 343–348 (2014)
Yang, W. G., Wang, R., Ding, D. H., and Hu, C. Z. Elastic strains induced by electric fields in QCs. Journal of Physics: Condensed Matter, 7, L499–L502 (1995)
Hu, C. Z., Wang, R., Ding, D. H., and Yang, W. Piezoelectric effects in QCs. Physical Review B, 56, 2463–2468 (1997)
Rao, R. M. K., Rao, H. P., and Chaitanya, B. S. K. Piezoelectricity in QCs: a group-theoretical study. Pramana - Journal of Physics, 68, 481–487 (2007)
Grimmer, H. The piezoelectric effect of second order in stress or strain: its form for crystals and QCs of any symmetry. Acta Crystallographica A, 63, 441–446 (2007)
Fujiwara, T. and Ishii, Y. Quasicrystals, Handbook of Metal Physics, Elsevier, Amsterdam (2008)
Suck, J. B., Schreiber, M., and Haussler, P. Quasicrystals: An Introduction to Structure, Physical Properties and Applications, Springer-Verlag, Berlin (2010)
Fan, T. Y. Mathematical Theory of Elasticity of QCs and Its Applications, Science Press, Beijing (2011)
Altay, G. and Dökmeci, M. C. On the fundamental equations of piezoelasticity of quasicrystal media. International Journal of Solids and Structures, 49, 3255–3262 (2012)
Li, X. Y., Li, P. D., Wu, T. H., Shi, M. X., and Zhu, Z. W. Three-dimensional fundamental solutions for one-dimensional hexagonal quasicrystal with piezoelectric effect. Physics Letters A, 378, 826–834 (2014)
Zhang, L. L., Zhang, Y. M., and Gao, Y. General solutions of plane elasticity of one-dimensional orthorhombic quasicrystals with piezoelectric effect. Physics Letters A, 378, 2768–2776 (2014)
Merlin, R., Bajema, K., Clarke, R., Yuang, F., and Bhattacharya, P. K. Quasiperiodic GaAs-AlAs heterostructures. Physical Review Letters, 55, 1768–1770 (1985)
Wang, R. H., Yang, W. G., Hu, C. Z., and Ding, D. H. Point and space groups and elastic behaviours of one-dimensional quasicrystals. Journal of Physics: Condensed Matter, 9, 2411–2422 (1997)
Wang, M. Z. and Wang, W. Completeness and nonuniqueness of general solutions of transversely isotropic elasticity. International Journal of Solids and Structures, 32, 501–513 (1995)
Wang, W. and Shi, M. X. On the general solutions of transversely isotropic elasticity. International Journal of Solids and Structures, 35, 3283–3297 (1998)
Pak, Y. E. Crack extension force in a piezoelectric material. Journal of Applied Physics, 57, 647–653 (1990)
Mariano, P. M. and Planas, J. Phason self-actions in quasicrystals. Physica D: Nonlinear Phenomena, 249, 46–57 (2013)
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Project supported by the National Nature Science Foundation of China (Nos. 11262012, 11262017, 11462020, and 10761005) and the Scientific Research Key Program of Inner Mongolia University of Technology (No. ZD201219)
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Yu, J., Guo, J., Pan, E. et al. General solutions of plane problem in one-dimensional quasicrystal piezoelectric materials and its application on fracture mechanics. Appl. Math. Mech.-Engl. Ed. 36, 793–814 (2015). https://doi.org/10.1007/s10483-015-1949-6
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DOI: https://doi.org/10.1007/s10483-015-1949-6