Green’s functions of two-dimensional piezoelectric quasicrystal in half-space and bimaterials

doi:10.1007/s10483-023-2955-9

Abstract

Abstract: In this paper, we obtain Green’s functions of two-dimensional (2D) piezoelectric quasicrystal (PQC) in half-space and bimaterials. Based on the elastic theory of QCs, the Stroh formalism is used to derive the general solutions of displacements and stresses. Then, we obtain the analytical solutions of half-space and bimaterial Green’s functions. Besides, the interfacial Green’s function for bimaterials is also obtained in the analytical form. Before numerical studies, a comparative study is carried out to validate the present solutions. Typical numerical examples are performed to investigate the effects of multi-physics loadings such as the line force, the line dislocation, the line charge, and the phason line force. As a result, the coupling effect among the phonon field, the phason field, and the electric field is prominent, and the butterfly-shaped contours are characteristic in 2D PQCs. In addition, the changes of material parameters cause variations in physical quantities to a certain degree.

Key words: Green's function, two-dimensional (2D) piezoelectric quasicrystal (PQC), Stroh formalism, half-space, bimaterial

2010 MSC Number:

74B05

Xiaoyu FU, Xiang MU, Jinming ZHANG, Liangliang ZHANG, Yang GAO. Green’s functions of two-dimensional piezoelectric quasicrystal in half-space and bimaterials. Applied Mathematics and Mechanics (English Edition), 2023, 44(2): 237-254.

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References

[1] 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(20), 1951–1953(1984)
[2] JARIC, M. V. and NELSON, D. R. Introduction to quasicrystals. Physics Today, 43(3), 77–79(1990)
[3] FAN, T. Y. Mathematical theory and methods of mechanics of quasicrystalline materials. Engineering, 5(4), 407–448(2013)
[4] DE BOISSIEU, M. Atomic structure of quasicrystals. Structural Chemistry, 23(4), 965–976(2012)
[5] DUBOIS, J. M. Properties and applications of quasicrystals and complex metallic alloys. Chemical Society Reviews, 41(20), 6760–6777(2012)
[6] LI, X. Y., WANG, T., ZHENG, R. F., and KANG, G. Z. Fundamental thermo-electro-elastic solutions for 1D hexagonal QC. Zeitschrift für Angewandte Mathematik und Mechanik, 95(5), 457–468(2015)
[7] HUANG, Y. Z., LI, Y., ZHANG, L. L., ZHANG, H., and GAO, Y. Dynamic analysis of a multilayered piezoelectric two-dimensional quasicrystal cylindrical shell filled with compressible fluid using the state-space approach. Acta Mechanica, 231(6), 2351–2368(2020)
[8] FAN, T. Y. Mathematical Theory of Elasticity of Quasicrystals and Its Applications, Springer, Heidelberg (2011)
[9] LIU, G. T. Complex Method of Quasicrystal Elasticity and Explicit Solution of Nonlinear Equation (in Chinese), Inner Mongolia Peoples Publishing House, Hohhot (2005)
[10] GAO, Y., RICOEUR, A., and ZHANG, L. L. Plane problems of cubic quasicrystal media with an elliptic hole or a crack. Physics Letters A, 375(28-29), 2775–2781(2011)
[11] GAO, Y. and RICOEUR, A. Green’s functions for infinite bi-material planes of cubic quasicrystals with imperfect interface. Physics Letters A, 374(42), 4354–4358(2010)
[12] LI, L. H. and LIU, G. T. Stroh formalism for icosahedral quasicrystal and its application. Physics Letters A, 376(8-9), 987–990(2012)
[13] LI, Y., YANG, L. Z., ZHANG, L. L., and GAO, Y. Static response of functionally graded multilayered one-dimensional quasicrystal cylindrical shells. Mathematics and Mechanics of Solids, 24(6), 1908–1921(2019)
[14] LI, Y., YANG, L. Z., ZHANG, L. L., and GAO, Y. Exact thermoelectroelastic solution of layered one-dimensional quasicrystal cylindrical shells. Journal of Thermal Stresses, 41(10-12), 1450–1467(2018)
[15] PAN, E. and CHEN, W. Static Green’s Functions in Anisotropic Media, Cambridge University Press, Cambridge (2015)
[16] TING, T. C. T. and HORGAN, C. O. Anisotropic elasticity: theory and applications. Journal of Applied Mechanic, 63(4), 1056(1996)
[17] HWU, C. Anisotropic Elastic Plates, Springer Science & Business Media, New York (2010)
[18] PAN, E. Three-dimensional Green’s functions in anisotropic magneto-electro-elastic bimaterials. Zeitschrift für Angewandte Mathematik und Physik, 53(5), 815–838(2002)
[19] QIN, Q. 2D Green’s functions of defective magnetoelectroelastic solids under thermal loading. Engineering Analysis with Boundary Elements, 29(6), 577–585(2005)
[20] PAN, E. and YUAN, F. G. Three-dimensional Green’s functions in anisotropic bimaterials. International Journal of Solids and Structures, 37(38), 5329–5351(2000)
[21] GAO, C. F. and WANG, M. Z. Green’s functions of an interfacial crack between two dissimilar piezoelectric media. International Journal of Solids and Structures, 38(30-31), 5323–5334(2001)
[22] GAO, Y. and RICOEUR, A. Three-dimensional Green’s functions for two-dimensional quasicrystal bimaterials. Proceedings of the Royal Society A-Mathematical Physical and Engineering Sciences, 467(2133), 2622–2642(2011)
[23] CHEN, W. Q., MA, Y. L., and DING, H. J. On three-dimensional elastic problems of onedimensional hexagonal quasicrystal bodies. Mechanics Research Communications, 31(6), 633–641(2004)
[24] WANG, W. S., LI, C. Q., and ZHOU, Y. T. Thermo-electric response in 2D hexagonal QC exhibiting piezoelectric effect. Zeitschrift für Angewandte Mathematik und Mechanik, 101(2), e201900212(2021)
[25] XU, W. S., WU, D., and GAO, Y. Fundamental elastic field in an infinite plane of two-dimensional piezoelectric quasicrystal subjected to multi-physics loads. Applied Mathematical Modelling, 52, 186–196(2017)
[26] MU, X., XU, W. S., ZHU, Z. W., ZHANG, L. L., and GAO, Y. Multi-field coupling solutions of functionally graded two-dimensional piezoelectric quasicrystal wedges and spaces. Applied Mathematical Modelling, 109, 251–264(2022)
[27] HU, C. Z., WANG, R. H., DING, D. H., and YANG, W. G. Piezoelectric effects in quasicrystals. Physical Review B, 56(5), 2463–2468(1997)
[28] LI, L. H. and LIU, G. T. Study on a straight dislocation in an icosahedral quasicrystal with piezoelectric effects. Applied Mathematics and Mechanics (English Edition), 39(9), 1259–1266(2018) https://doi.org/10.1007/s10483-018-2363-9
[29] WU, D., ZHANG, L., XU, W., YANG, L., and GAO, Y. Electroelastic Green’s function of onedimensional piezoelectric quasicrystals subjected to multi-physics loads. Journal of Intelligent Material Systems and Structures, 28(12), 1651–1661(2016)
[30] ZHANG, L., WU, D., XU, W., YANG, L., RICOEUR, A., and WANG, Z. Green’s functions of one-dimensional quasicrystal bi-material with piezoelectric effect. Physics Letters A, 380(39), 3222–3228(2016)
[31] GAO, Y. and ZHAO, B. S. General solutions of three-dimensional problems for two-dimensional quasicrystals. Applied Mathematical Modelling, 33(8), 3382–3391(2009)
[32] YANG, L. Z., LI, Y., GAO, Y., PAN, E., and WAKSMANSKI, N. Three-dimensional exact electricelastic analysis of a multilayered two-dimensional decagonal quasicrystal plate subjected to patch loading. Composite Structures, 171, 198–216(2017)
[33] LI, Y., YANG, L. Z., GAO, Y., and PAN, E. N. Cylindrical bending analysis of a layered two-dimensional piezoelectric quasicrystal nanoplate. Journal of Intelligent Material Systems and Structures, 29(12), 2660–2676(2018)
[34] LI, C. Q. and ZHOU, Y. T. Fundamental solutions and frictionless contact problem in a semiinfinite space of 2D hexagonal piezoelectric QCs. Zeitschrift für Angewandte Mathematik und Mechanik, 99(5), e201800132(2019)
[35] MU, X., XU, W., ZHU, Z., ZHUANG, L., and GAO, Y. Multi-field coupling solutions of functionally graded two-dimensional piezoelectric quasicrystal wedges and spaces. Applied Mathematical Modelling, 109, 251–264(2022)