Acta Mechanica Sinica

, Volume 31, Issue 5, pp 601–626 | Cite as

Some recent advances in 3D crack and contact analysis of elastic solids with transverse isotropy and multifield coupling

Review Paper
First Online:
Received:
Revised:
Accepted:

Abstract

Significant progress has been made in mixed boundary-value problems associated with three-dimensional (3D) crack and contact analyses of advanced materials featuring more complexities compared to the conventional isotropic elastic materials. These include material anisotropy and multifield coupling, two typical characteristics of most current multifunctional materials. In this paper we try to present a state-of-the-art description of 3D exact/analytical solutions derived for crack and contact problems of elastic solids with both transverse isotropy and multifield coupling in the latest decade by the potential theory method in the spirit of V. I. Fabrikant, whose ingenious breakthrough brings new vigor and vitality to the old research subject of classical potential theory. We are particularly interested in crack and contact problems with certain nonlinear features. Emphasis is also placed on the coupling between the temperature field (or the like) and other physical fields (e.g., elastic, electric, and magnetic fields). We further highlight the practical significance of 3D contact solutions, in particular in applications related to modern scanning probe microscopes.

Keywords

Crack Contact Mixed boundary-value problem Transverse isotropy Multifield coupling Potential theory  Exact solution Scanning probe microscope 
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Notes

Acknowledgments

This project was supported by the National Natural Science Foundation of China (Grant 11321202) and the Specialized Research Fund for the Doctoral Program of Higher Education (Grant  20130101110120).

References

  1. 1.
    Cheng, A.H.D., Cheng, D.T.: Heritage and early history of the boundary element method. Eng. Anal. Bound. Elem. 29, 268–302 (2005)MATHCrossRefGoogle Scholar
  2. 2.
    Brelot, M.: Potential Theory. Springer, Berlin (2010)Google Scholar
  3. 3.
    Kellogg, O.D.: Foundations of Potential Theory. Dover, Berlin (1929)MATHCrossRefGoogle Scholar
  4. 4.
    Muskhelishvili, N.I.: Singular Integral Equations. Noordhoff, Groningen (1953)MATHGoogle Scholar
  5. 5.
    Sneddon, I.N.: Mixed Boundary Value Problems in Potential Theory. North-Holland, Amsterdam (1966)MATHGoogle Scholar
  6. 6.
    Sneddon, I.N., Lowengrub, M.: Crack Problems in the Classical Theory of Elasticity. Wiley, New York (1969)MATHGoogle Scholar
  7. 7.
    Fabrikant, V.I.: Applications of Potential Theory in Mechanics: A Selection of New Results. Kluwer, Dordrecht (1989)MATHGoogle Scholar
  8. 8.
    Fabrikant, V.I.: Mixed Boundary Value Problem of Potential Theory and Their Applications in Engineering. Kluwer, Dordrecht (1991)MATHGoogle Scholar
  9. 9.
    Fabrikant, V.I.: Crack and Contact Problems in Linear Theory of Elasticity. Bentham Science Publishers, Sharjah (2010)Google Scholar
  10. 10.
    Chen, W.Q., Ding, H.J.: Potential theory method for 3D crack and contact problems of multi-field coupled media: A survey. J. Zhejiang Univ. Sci. 5, 1009–1021 (2004)CrossRefGoogle Scholar
  11. 11.
    Hanson, M.T.: The elastic field for spherical Hertzian contact including sliding friction for transversely isotropy. J. Tribol. 114, 606–611 (1992)CrossRefGoogle Scholar
  12. 12.
    Yong, Z., Hanson, M.T.: Three-dimensional crack and contact problems with a general geometric configuration. Int. J. Solids Struct. 31, 215–239 (1994)MATHCrossRefGoogle Scholar
  13. 13.
    Chen, W.Q., Ding, H.J.: A penny-shaped crack in a transversely isotropic piezoelectric solid: modes II and III problems. Acta Mech. Sin. 15, 52–58 (1999)CrossRefGoogle Scholar
  14. 14.
    Chen, W.Q., Ding, H.J.: Indentation of a transversely isotropic piezoelectric half-space by a rigid sphere. Acta Mech. Solida Sin. 12, 114–120 (1999)Google Scholar
  15. 15.
    Chen, W.Q.: On piezoelastic contact problem for a smooth punch. Int. J. Solids Struct. 37, 2331–2340 (2000)MATHCrossRefGoogle Scholar
  16. 16.
    Kalinin, S.V., Karapetian, E., Kachanov, M.: Nanoelectromechanics of piezoresponse force microscopy. Phys. Rev. B 70, 184101 (2004)CrossRefGoogle Scholar
  17. 17.
    Karapetian, E., Kachanov, M., Kalinin, S.V.: Nanoelectromechanics of piezoelectric indentation and applications to scanning probe microscopies of ferroelectric materials. Philos. Mag. 85, 1017–1051 (2005)CrossRefGoogle Scholar
  18. 18.
    Pan, E., Chen, W.Q.: Static Green’s Functions in Anisotropic Media. Cambridge University Press, New York (2015)MATHGoogle Scholar
  19. 19.
    Selvadurai, A.P.S.: The analytical method in geomechanics. Appl. Mech. Rev. 60, 87–106 (2007)CrossRefGoogle Scholar
  20. 20.
    Ding, H.J., Chen, W.Q., Zhang, L.C.: Elasticity of Transversely Isotropic Materials. Springer, Dordrecht (2006)MATHGoogle Scholar
  21. 21.
    Ding, H.J., Chen, W.Q.: Three Dimensional Problems of Piezoelasticity. Nova Science Publishers, New York (2001)Google Scholar
  22. 22.
    Ding, H.J., Chen, B., Liang, J.: General solutions for coupled equations for piezoelectric media. Int. J. Solids Struct. 33, 2283–2298 (1996)MATHCrossRefGoogle Scholar
  23. 23.
    Chen, W.Q.: On the application of potential theory in piezoelasticity. J. Appl. Mech. 66, 808–810 (1999)CrossRefGoogle Scholar
  24. 24.
    Chen, W.Q., Lee, K.Y., Ding, H.J.: General solution for transversely isotropic magneto-electro-thermo-elasticity and the potential theory method. Int. J. Eng. Sci. 42, 1361–1379 (2004)MATHCrossRefGoogle Scholar
  25. 25.
    Ding, H.J., Chen, B., Liang, J.: On the Green’s functions for two-phase transversely isotropic piezoelectric media. Int. J. Solids Struct. 34, 3041–3057 (1997)MATHCrossRefGoogle Scholar
  26. 26.
    Chen, W.Q., Lim, C.W.: 3D point force solution for a permeable penny-shaped crack embedded in an infinite transversely isotropic piezoelectric medium. Int. J. Fract. 131, 231–246 (2005)MATHCrossRefGoogle Scholar
  27. 27.
    Gao, C.F., Wang, M.Z.: Generalized 2D problem of piezoelectric media containing collinear cracks. Acta Mech. Sin. 15, 235–244 (1999)CrossRefGoogle Scholar
  28. 28.
    Qi, H., Fang, D.N., Yao, Z.H.: Analysis of electric boundary condition effects on crack propagation in piezoelectric ceramics. Acta Mech. Sin. 17, 59–70 (2001)CrossRefGoogle Scholar
  29. 29.
    Chen, Y.H., Lu, T.J.: Cracks and fracture in piezoelectrics. Adv. Appl. Mech. 39, 121–215 (2003)CrossRefGoogle Scholar
  30. 30.
    Huang, Z.Y., Kuang, Z.B.: A mixed electric boundary value problem for an anti-plane piezoelectric crack. Acta Mech. Solida Sin. 16, 110–115 (2003)Google Scholar
  31. 31.
    Wang, B.L., Han, J.C., Du, S.Y.: Applicability of the crack face electrical boundary conditions in piezoelectric mechanics. Acta Mech. Solida Sin. 17, 290–296 (2004)Google Scholar
  32. 32.
    Li, F.X., Sun, Y., Rajapakse, R.K.N.D.: Effect of electric boundary conditions on crack propagation in ferroelectric ceramics. Acta Mech. Sin. 30, 153–160 (2014)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Zhang, T.Y., Tong, P.: Fracture mechanics for a mode-III crack in a piezoelectric material. Int. J. Solids Struct. 33, 343–359 (1996)MATHCrossRefGoogle Scholar
  34. 34.
    Benveniste, Y.: On the decay of end effects in conduction phenomena: A sandwich strip with imperfect interfaces of low or high conductivity. J. Appl. Phys. 86, 1273–1279 (1999)CrossRefGoogle Scholar
  35. 35.
    Chen, W.Q., Shioya, T.: Fundamental solution for a penny-shaped crack in a piezoelectric medium. J. Mech. Phys. Solids 47, 1459–1475 (1999)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Li, X.F., Lee, K.Y.: Three-dimensional electroelastic analysis of a piezoelectric material with a penny-shaped dielectric crack. J. Appl. Mech. 71, 866–878 (2005)MATHCrossRefGoogle Scholar
  37. 37.
    Li, X.F., Lee, K.Y.: Electro-elastic behavior induced by an external circular crack in a piezoelectric material. Int. J. Fract. 126, 17–38 (2004)MATHCrossRefGoogle Scholar
  38. 38.
    Li, X.Y.: Fundamental electro-elastic field in an infinite transversely isotropic piezoelectric medium with a permeable external circular crack. Smart Mater. Struct. 21, 065019 (2012)CrossRefGoogle Scholar
  39. 39.
    Chen, W.Q.: Exact solution of a semi-infinite crack in an infinite piezoelectric body. Arch. Appl. Mech. 69, 309–316 (1999)MATHCrossRefGoogle Scholar
  40. 40.
    Chen, W.Q., Pan, E.N., Wang, H.M., Zhang, C.Z.: Theory of indentation on multiferroic composite materials. J. Mech. Phys. Solids 58, 1524–1551 (2010)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Chen, W.Q., Shioya, T., Ding, H.J.: The elasto-electric field for a rigid conical punch on a transversely isotropic piezoelectric half-space. J. Appl. Mech. 66, 764–771 (1999)CrossRefGoogle Scholar
  42. 42.
    Gao, H.J., Zhang, T.Y., Tong, P.: Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramic. J. Mech. Phys. Solids 45, 491–510 (1997)CrossRefGoogle Scholar
  43. 43.
    Beom, H.G., Atluri, S.N.: Effect of electric fields on fracture behavior of ferroelectric ceramics. J. Mech. Phys. Solids 51, 1107–1125 (2003)MATHCrossRefGoogle Scholar
  44. 44.
    Zhang, T.Y., Zhao, M.H., Gao, C.F.: The strip dielectric breakdown model. Int. J. Fract. 132, 311–327 (2005)MATHCrossRefGoogle Scholar
  45. 45.
    Dugdale, D.S.: Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8, 100–104 (1960)CrossRefGoogle Scholar
  46. 46.
    Li, X.Y., Yang, D., Chen, W.Q., Kang, G.Z.: Penny-shaped Dugdale crack in a transversely isotropic medium. Int. J. Fract. 176, 207–214 (2012)CrossRefGoogle Scholar
  47. 47.
    Li, X.Y., Guo, S.T., He, Q.C., Chen, W.Q.: Penny-shaped Dugdale crack in a transversely isotropic medium and under axisymmetric loading. Mech. Math. Solids 18, 246–263 (2013)CrossRefGoogle Scholar
  48. 48.
    Zhao, M.H., Shen, Y.P., Liu, G.N., Liu, Y.J.: Dugdale model solutions for a penny-shaped crack in three-dimensional transversely isotropic piezoelectric media by boundary-integral equation method. Eng. Anal. Bound. Elem. 23, 573–576 (1999)MATHCrossRefGoogle Scholar
  49. 49.
    Maugis, D.: Contact, Adhesion and Rupture of Elastic Solids. Springer, Berlin (2000)MATHCrossRefGoogle Scholar
  50. 50.
    Chen, S.H., Gao, H.J.: Bio-inspired mechanics of reversible adhesion: orientation-dependent adhesion strength for non-slipping adhesive contact with transversely isotropic elastic materials. J. Mech. Phys. Solids 55, 1001–1015 (2005)MATHCrossRefGoogle Scholar
  51. 51.
    Wu, J., Kim, S., Carlson, A., Lu, C.F., Hwang, K.C., Huang, Y.G., Rogers, J.A.: Contact radius of stamps in reversible adhesion. Theor. Appl. Mech. Lett. 1, 011001 (2011)CrossRefGoogle Scholar
  52. 52.
    Wang, J.Z., Yao, J.Y., Gao, H.J.: Specific adhesion of a soft elastic body on a wavy surface. Theor. Appl. Mech. Lett. 2, 014002 (2012)CrossRefGoogle Scholar
  53. 53.
    Chen, Z.R., Yu, S.W.: Micro-scale adhesive contact of a spherical rigid punch on a piezoelectric half-space. Compos. Sci. Technol. 65, 1372–1381 (2005)Google Scholar
  54. 54.
    Johnson, K.L., Kendall, K., Roberts, A.D.: Surface energy and the contact of elastic solids. Proc. R. Soc. Lond. A 324, 301–313 (1971)CrossRefGoogle Scholar
  55. 55.
    Chen, W.Q.: Adhesive contact between a rigid indenter and a piezoelectric half-space. In: Yang, W., Feng, X.Q., Qin, Q.H. (eds.) Advances in Damage, Fracture and Nanomechanics, pp. 58–65. Tsinghua University Press, Beijing (2009). (in Chinese)Google Scholar
  56. 56.
    Maugis, D.: Adhesion of spheres: the JKR-DMT transition using a Dugdale model. J. Colloid Interface Sci. 150, 243–269 (1992)CrossRefGoogle Scholar
  57. 57.
    Borodich, F.M., Galanov, B.A., Keer, L.M., Suarez-Alvarez, M.M.: The JKR-type adhesive contact problems for transversely isotropic elastic solids. Mech. Mater. 75, 33–44 (2014)CrossRefGoogle Scholar
  58. 58.
    Bui, H.D.: An integral equations method for solving the problem of a plane crack of arbitrary shape. J. Mech. Phys. Solids 25, 29–39 (1977)MathSciNetMATHCrossRefGoogle Scholar
  59. 59.
    Vlassak, J.J., Ciavarella, M., Barber, J.R., Wang, X.: The indentation modulus of elastically anisotropic materials for indenters of arbitrary shape. J. Mech. Phys. Solids 51, 1701–1721 (2003)MATHCrossRefGoogle Scholar
  60. 60.
    Wang, B.: Three-dimensional analysis of a flat elliptical crack in a piezoelectric material. Int. J. Eng. Sci. 30, 781–791 (1992)MATHCrossRefGoogle Scholar
  61. 61.
    Fabrikant, V.I., Rubin, B.S., Karapetian, E.N.: Half-plane crack under normal load: complete solution. J. Eng. Mech. 119, 2238–2251 (1993)MATHCrossRefGoogle Scholar
  62. 62.
    Huang, Z.Y., Bao, R.H., Bian, Z.G.: The potential theory method for a half-plane crack and contact problems of piezoelectric materials. Compos. Struct. 78, 596–601 (2007)CrossRefGoogle Scholar
  63. 63.
    Fabrikant, V.I., Karapetian, E.N.: Elementary exact method for solving boundary-value problems of potential theory with application to half-plane crack and contact problems. Q. J. Mech. Appl. Math. 47, 159–174 (1994)MathSciNetMATHCrossRefGoogle Scholar
  64. 64.
    Zhang, N., Gao, C.F., Jiang, Q.: Solution of a flat elliptical crack in an electrostrictive solid. Int. J. Solids Struct. 51, 786–793 (2014)CrossRefGoogle Scholar
  65. 65.
    Zhao, M.H., Zhang, Q.Y., Pan, E., Fan, C.Y.: Fundamental solutions and numerical modeling of an elliptical crack with polarization saturation in a transversely isotropic piezoelectric medium. Eng. Fract. Mech. 131, 627–642 (2014)CrossRefGoogle Scholar
  66. 66.
    Kassir, M.K., Sih, G.C.: Three-Dimensional Crack Problems. Noordhoff, Leyden (1975)MATHGoogle Scholar
  67. 67.
    Nuller, B., Karapetian, E., Kachanov, M.: On the stress intensity factor for the elliptical crack. Int. J. Fract. 92, L17–L20 (1998)CrossRefGoogle Scholar
  68. 68.
    Fabrikant, V.I.: The stress intensity factor for an external elliptical crack. Int. J. Solids Struct. 23, 465–467 (1987)MathSciNetMATHCrossRefGoogle Scholar
  69. 69.
    Hanson, M.T., Puja, I.W.: The elastic field resulting from elliptical Hertzian contact of transversely isotropic bodies: closed form solutions for normal and shear loading. J. Appl. Mech. 64, 457–465 (1997)MATHCrossRefGoogle Scholar
  70. 70.
    Ding, H.J., Hou, P.F., Guo, F.L.: The elastic and electric fields for elliptical contact for transversely isotropic piezoelectric bodies. J. Appl. Mech. 66, 560–562 (1999)CrossRefGoogle Scholar
  71. 71.
    Ding, H.J., Hou, P.F., Guo, F.L.: The elastic and electric fields for three-dimensional contact for transversely isotropic piezoelectric materials. Int. J. Solids Struct. 37, 3201–3229 (2000)MATHCrossRefGoogle Scholar
  72. 72.
    Fabrikant, V.I.: A new symbolism for solving the Hertz contact problem. Q. J. Mech. Appl. Math. 58, 367–381 (2005)MathSciNetMATHCrossRefGoogle Scholar
  73. 73.
    Dyson, F.W.: The potentials of ellipsoids of variable densities. Q. J. Pure Appl. Math. Oxford Ser. 25, 259–288 (1891)MATHGoogle Scholar
  74. 74.
    Rahman, M.: Some problems of a rigid elliptical disk-inclusion bonded inside a transversely isotropic space: Part I. J. Appl. Mech. 66, 612–620 (1999)CrossRefGoogle Scholar
  75. 75.
    Fabrikant, V.I.: Utilization of divergent integrals and a new symbolism in crack and contact analysis. IMA J. Appl. Math. 72, 180–190 (2007)MathSciNetMATHCrossRefGoogle Scholar
  76. 76.
    Li, X.Y., Wu, F., Jin, X., Chen, W.Q.: 3D coupled field in a transversely isotropic magneto-electro-elastic half space punched by an elliptic indenter. J. Mech. Phys. Solids 75, 1–44 (2015)Google Scholar
  77. 77.
    Lü, C.F., Chen, W., Zhou, J.X., Qu, S.X., Chen, W.Q.: Editorial: mechanics of soft materials, structures and systems. Theor. Appl. Mech. Lett. 3, 054001 (2013)CrossRefGoogle Scholar
  78. 78.
    Shi, W.D., Feng, X.Q., Gao, H.J.: Two-dimensional model of vesicle adhesion on curved substrates. Acta Mech. Sin. 22, 529–535 (2006)MATHCrossRefGoogle Scholar
  79. 79.
    Peng, X.L., Huang, J.Y., Qin, L., Xiong, C.Y., Fang, J.: A method to determine Young’s modulus of soft gels for cell adhesion. Acta Mech. Sin. 25, 565–570 (2009)CrossRefGoogle Scholar
  80. 80.
    Suo, Z.: Theory of dielectric elastomers. Acta Mech. Solida Sin. 23, 549–578 (2010)CrossRefGoogle Scholar
  81. 81.
    Dorfmann, A., Ogden, R.W.: Nonlinear electroelastic deformations. J. Elasticity 82, 99–127 (2006)MathSciNetMATHCrossRefGoogle Scholar
  82. 82.
    Dorfmann, A., Ogden, R.W.: Nonlinear electroelastostatics: incremental equations and stability. Int. J. Eng. Sci. 48, 1–14 (2010)MathSciNetMATHCrossRefGoogle Scholar
  83. 83.
    Zhang, W.L., Qian, J., Chen, W.Q.: Indentation of a compressible soft electroactive half-space: some theoretical aspects. Acta Mech. Sin. 28, 1133–1142 (2012)MathSciNetMATHCrossRefGoogle Scholar
  84. 84.
    Chen, W.Q., Dai, H.H.: Waves in pre-stretched incompressible soft electroactive cylinders: exact solution. Acta Mech. Solida Sin. 25, 530–541 (2012)CrossRefGoogle Scholar
  85. 85.
    Chen, W.Q.: The renaissance of continuum mechanics. J. Zhejiang Univ. Sci. A 15, 231–240 (2014)CrossRefGoogle Scholar
  86. 86.
    Nan, C.W., Bichurin, M.I., Dong, S.X., Viehland, D., Srinivasan, G.: Multiferroic magnetoelectric composites: historical perspective, status, and future directions. J. Appl. Phys. 103, 031101 (2008)CrossRefGoogle Scholar
  87. 87.
    Ma, J., Hu, J., Li, Z., Nan, C.W.: Recent progress in multiferroic magnetoelectric composites: from bulk to thin films. Adv. Mater. 23, 1062–1087 (2011)CrossRefGoogle Scholar
  88. 88.
    Wang, X., Shen, Y.P.: The general solution of three-dimensional problems in magnetoelectroelastic media. Int. J. Eng. Sci. 40, 1069–1080 (2002)MathSciNetMATHCrossRefGoogle Scholar
  89. 89.
    Liu, J.X., Liu, X.G., Zhao, Y.B.: Green’s functions for anisotropic magnetoelectroelastic solids with an elliptical cavity or a crack. Int. J. Eng. Sci. 39, 1405–1418 (2001)MATHCrossRefGoogle Scholar
  90. 90.
    Du, J.K., Shen, Y.P., Gao, B.: Scattering of anti-plane shear waves by a single crack in an unbounded transversely isotropic electro-magneto-elastic medium. Appl. Math. Mech. Eng. Ed. 25, 1344–1353 (2004)MATHCrossRefGoogle Scholar
  91. 91.
    Zhou, Z.G., Wang, B.: Dynamic behavior of two parallel symmetry cracks in magneto-electro-elastic composites under harmonic anti-plane waves. Appl. Math. Mech. Eng. Ed. 27, 583–591 (2006)MATHCrossRefGoogle Scholar
  92. 92.
    Zhang, P.W., Zhou, Z.G., Wang, B.: Dynamic behavior of two collinear interface cracks between two dissimilar functionally graded piezoelectric/ piezomagnetic material strips. Appl. Math. Mech. Eng. Ed. 28, 615–625 (2007)MATHCrossRefGoogle Scholar
  93. 93.
    Feng, W.J., Nie, H., Han, X.: A penny-shaped crack in a magnetoelectroelastic layer under radial shear impact loading. Acta Mech. Solida Sin. 20, 275–282 (2007)CrossRefGoogle Scholar
  94. 94.
    Fan, C.Y., Zhou, Y.H., Wang, H., Zhao, M.H.: Singular behaviors of interfacial cracks in 2D magnetoelectroelastic bimaterials. Acta Mech. Solida Sin. 22, 232–239 (2009)CrossRefGoogle Scholar
  95. 95.
    Pan, S.D., Zhou, Z.G., Wu, L.Z.: Basic solutions of multiple parallel symmetric mode-III cracks in functionally graded piezoelectric/piezomagnetic material plane. Appl. Math. Mech. Eng. Ed. 34, 1201–1224 (2013)MathSciNetMATHCrossRefGoogle Scholar
  96. 96.
    Tang, Y.L., Zhou, Z.G., Wu, L.Z.: The basic solution of a 3-D rectangular permeable crack in a piezoelectric/piezomagnetic composite material. Acta Mech. Solida Sin. 26, 403–418 (2013)CrossRefGoogle Scholar
  97. 97.
    Chen, W.Q.: Exact 3D thermoelastic solutions for a penny-shaped crack in an infinite magnetoelectric medium. Trans. Nanjing Univ. Aeronaut. Astronaut. 31, 109–117 (2014)Google Scholar
  98. 98.
    Gao, C.F., Kessler, H., Balke, H.: Fracture analysis of electromagnetic thermoelastic solids. Eur. J. Mech. A Solids 22, 433–442 (2003)MATHCrossRefGoogle Scholar
  99. 99.
    Wang, B.L., Han, J.C.: Discussion on electromagnetic crack face boundary conditions for the fracture mechanics of magneto-electro-elastic materials. Acta Mech. Sin. 22, 233–242 (2006)MATHCrossRefGoogle Scholar
  100. 100.
    Zhao, M.H., Yang, F., Liu, T.: Analysis of a penny-shaped crack in a magneto-electro-elastic medium. Philos. Mag. 86, 4397–4416 (2006)CrossRefGoogle Scholar
  101. 101.
    Hou, P.F., Leung, A.Y.T., Ding, H.J.: The elliptical Hertzian contact of transversely isotropic magnetoelectroelastic bodies. Int. J. Solids Struct. 40, 2833–2850 (2003)MATHCrossRefGoogle Scholar
  102. 102.
    Li, X.Y., Zheng, R.F., Chen, W.Q.: Fundamental solutions to contact problems of a magneto-electro-elastic half-space indented by a semi-infinite punch. Int. J. Solids Struct. 51, 164–178 (2014)CrossRefGoogle Scholar
  103. 103.
    Rogowski, B., Kaliński, W.: Indentation of piezoelectromagneto-elastic half- space by a truncated conical punch. Int. J. Eng. Sci. 60, 77–93 (2012)MathSciNetCrossRefGoogle Scholar
  104. 104.
    Wang, H.M., Pan, E., Sangghaleh, A., Wang, R., Han, X.: Circular loadings on the surface of an anisotropic and magnetoelectroelastic half-space. Smart Mater. Struct. 21, 075003 (2012)CrossRefGoogle Scholar
  105. 105.
    Zhou, Y.T., Lee, K.Y.: Theory of sliding contact for multiferroic materials indented by a rigid punch. Int. J. Mech. Sci. 66, 156–167 (2013)CrossRefGoogle Scholar
  106. 106.
    Elloumia, R., Guler, M.A., Kallel-Kamoun, I., El-Borgi, S.: Closed-form solutions of the frictional sliding contact problem for a magneto-electro-elastic half-plane indented by a rigid conducting punch. Int. J. Solids Struct. 50, 3778–3792 (2013)CrossRefGoogle Scholar
  107. 107.
    Zhou, Y.T., Zhong, Z.: Frictional indentation of anisotropic magneto-electro- elastic materials by a rigid indenter. J. Appl. Mech. 81, 071001 (2014)CrossRefGoogle Scholar
  108. 108.
    Suck, J.B., Schreiber, M., Häussler, P.: Quasicrystals: An Introduction to Structure, Physical Properties and Applications. Springer, Berlin (2010)Google Scholar
  109. 109.
    Dubois, J.M.: Useful Quasicrystals. World Scientific, Singapore (2005)CrossRefGoogle Scholar
  110. 110.
    Fan, T.Y.: Mathematical Theory of Elasticity of Quasicrystals and Its Applications. Springer, Berlin (2011)CrossRefGoogle Scholar
  111. 111.
    Guo, L.H., Fan, T.Y.: Solvability on boundary-value problems of elasticity of three-dimensional quasicrystals. Appl. Math. Mech. Eng. Ed. 28, 1061–1070 (2007)MathSciNetMATHCrossRefGoogle Scholar
  112. 112.
    Guo, Y.C., Fan, T.Y.: A mode- II Griffith crack in decagonal quasicrystals. Appl. Math. Mech. Eng. Ed. 22, 1311–1317 (2001)MATHCrossRefGoogle Scholar
  113. 113.
    Fan, T.Y., Tang, Z.Y., Chen, W.Q.: Theory of linear, nonlinear and dynamic fracture for quasicrystals. Eng. Fract. Mech. 82, 185–194 (2012)CrossRefGoogle Scholar
  114. 114.
    Zhou, W.M., Fan, T.Y.: Axisymmetric elasticity problem of cubic quasicrystal. Chin. Phys. 9, 294–303 (2000)CrossRefGoogle Scholar
  115. 115.
    Zhou, W.M., Fan, T.Y., Yin, S.Y.: Crack problem under shear loading in cubic quasicrystal. Appl. Math. Mech. Eng. Ed. 24, 720–726 (2003)MATHCrossRefGoogle Scholar
  116. 116.
    Zhou, W.M., Fan, T.Y., Yin, S.Y.: Axisymmetric contact problem of cubic quasicrystalline materials. Acta Mech. Solida Sin. 15, 68–74 (2002)Google Scholar
  117. 117.
    Chen, W.Q., Ma, Y.L., Ding, H.J.: On three-dimensional elastic problems of one-dimensional hexagonal quasicrystal bodies. Mech. Res. Commun. 31, 633–641 (2004)MathSciNetMATHCrossRefGoogle Scholar
  118. 118.
    Wang, X.: The general solution of one-dimensional hexagonal quasicrystal. Mech. Res. Commun. 33, 576–580 (2006)MathSciNetMATHCrossRefGoogle Scholar
  119. 119.
    Peng, Y.Z., Fan, T.Y.: Crack and indentation problems for one-dimensional hexagonal quasicrystals. Eur. Phys. J. B 21, 39–44 (2001)CrossRefGoogle Scholar
  120. 120.
    Li, X.Y., Li, P.D., Wu, T.H., Shi, M.X., Zhu, Z.W.: Three-dimensional fundamental solutions for one-dimensional hexagonal quasicrystal with piezoelectric effect. Phys. Lett. A 378, 826–834 (2014)MathSciNetMATHCrossRefGoogle Scholar
  121. 121.
    Wu, Y.F., Chen, W.Q., Li, X.Y.: Indentation on one-dimensional hexagonal quasicrystals: general theory and complete exact solutions. Philos. Mag. 93, 858–882 (2013)CrossRefGoogle Scholar
  122. 122.
    Li, X.Y.: Elastic field in an infinite medium of one-dimensional hexagonal quasicrystal with a planar crack. Int. J. Solids Struct. 51, 1442–1455 (2014)CrossRefGoogle Scholar
  123. 123.
    Gao, Y., Zhao, B.S.: A general treatment of three-dimensional elasticity of quasicrystals by an operator method. Phys. Stat. Sol. (b) 243, 4007–4019 (2006)CrossRefGoogle Scholar
  124. 124.
    Gao, Y., Zhao, B.S.: General solutions of three-dimensional problems for two-dimensional quasicrystals. Appl. Math. Mod. 33, 3382–3391 (2009)MathSciNetMATHCrossRefGoogle Scholar
  125. 125.
    Gao, Y., Ricoeur, A.: Three-dimensional analysis of a spheroidal inclusion in a two-dimensional quasicrystal body. Philos. Mag. 92, 4334–4353 (2012)CrossRefGoogle Scholar
  126. 126.
    Li, X.Y., Wu, F., Wu, Y.F., Chen, W.Q.: Indentation on two-dimensional hexagonal quasicrystals. Mech. Mater. 76, 121–136 (2014)CrossRefGoogle Scholar
  127. 127.
    Wang, T.C., Han, X.L.: Crack problems of piezoelectric materials. Acta Mech. Solida Sin. 12, 95–105 (1999)Google Scholar
  128. 128.
    Fang, D.N., Soh, A.K., Liu, J.X.: Electromechanical deformation and fracture of piezoelectric-ferroelectric materials. Acta Mech. Sin. 17, 193–213 (2001)CrossRefGoogle Scholar
  129. 129.
    Gao, C.F., Balke, H.: Green’s functions of internal electrodes between two dissimilar piezoelectric media. Appl. Math. Mech. Eng. Ed. 26, 234–241 (2005)MATHCrossRefGoogle Scholar
  130. 130.
    Li, Q., Chen, Y.H.: Analysis of crack-tip singularities for an interfacial permeable crack in metal-piezoelectric bimaterials. Acta Mech. Solida Sin. 20, 247–257 (2007)CrossRefGoogle Scholar
  131. 131.
    Li, Q., Chen, Y.H.: Analysis of a permeable interface crack in elastic dielectric-piezoelectric bimaterials. Acta Mech. Sin. 23, 681–687 (2007)MATHCrossRefGoogle Scholar
  132. 132.
    Wang, B.L., Noda, N., Han, J.C., Du, S.Y.: A penny-shaped crack in a transversely isotropic piezoelectric layer. Eur. J. Mech. A Solids 20, 997–1005 (2001)MATHCrossRefGoogle Scholar
  133. 133.
    Yang, J.H., Lee, K.Y.: Penny shaped crack in a three-dimensional piezoelectric strip under in-plane normal loadings. Acta Mech. 148, 187–197 (2001)MATHCrossRefGoogle Scholar
  134. 134.
    Li, X.F., Lee, K.Y.: Effects of electric field on crack growth for a penny-shaped dielectric crack in a piezoelectric layer. J. Mech. Phys. Solids 52, 2079–2100 (2004)MathSciNetMATHCrossRefGoogle Scholar
  135. 135.
    Wang, B.L., Sun, Y.G., Zhu, Y.: Fracture of a finite piezoelectric layer with a penny-shaped crack. Int. J. Fract. 172, 19–39 (2011)Google Scholar
  136. 136.
    Zhao, M.H., Li, D.X., Shen, Y.P.: Interfacial crack analysis in three-dimensional transversely isotropic bi-materials by boundary integral equation method. Appl. Math. Mech. Eng. Ed. 26, 1539–1546 (2005)MATHCrossRefGoogle Scholar
  137. 137.
    Wang, J.H., Chen, C.Q., Lu, T.J.: Indentation response of piezoelectric films. J. Mech. Phys. Solids 56, 3331–3351 (2008)MATHCrossRefGoogle Scholar
  138. 138.
    Wu, Y.F., Yu, H.Y., Chen, W.Q.: Mechanics of indentation for piezoelectric thin films on elastic substrate. Int. J. Solids Struct. 49, 95–110 (2012)CrossRefGoogle Scholar
  139. 139.
    Wu, Y.F., Yu, H.Y., Chen, W.Q.: Indentation responses of piezoelectric layered half-space. Smart Mater. Struct. 22, 015007 (2013)CrossRefGoogle Scholar
  140. 140.
    Fabrikant, V.I.: Application of the generalized images method to contact problems for a transversely isotropic elastic layer. J. Strain Anal. 39, 55–70 (2004)CrossRefGoogle Scholar
  141. 141.
    Fabrikant, V.I.: Tangential contact problem for a transversely isotropic elastic layer bonded to a rigid foundation. Math. Proc. Camb. Philos. Soc. 138, 173–191 (2005)MathSciNetMATHCrossRefGoogle Scholar
  142. 142.
    Fabrikant, V.I.: Elementary solution of contact problems for a transversely isotropic elastic layer bonded to a rigid foundation. Z. Angew. Math. Phys. 57, 464–490 (2006)MathSciNetMATHCrossRefGoogle Scholar
  143. 143.
    Fabrikant, V.I.: Solution of contact problems for a transversely isotropic elastic layer bonded to an elastic half-space. Proc. IMechE Part C. J. Mech. Eng. Sci. 223, 2487–2499 (2009)CrossRefGoogle Scholar
  144. 144.
    Fabrikant, V.I.: Application of generalized images method to contact problems for a transversely isotropic elastic layer on a smooth half-space. Arch. Appl. Mech. 81, 957–974 (2011)MATHCrossRefGoogle Scholar
  145. 145.
    Fabrikant, V.I.: Contact problems for several transversely isotropic elastic layers on a smooth elastic half-space. Meccanica 46, 1239–1263 (2011)MathSciNetMATHCrossRefGoogle Scholar
  146. 146.
    Fabrikant, V.I.: Tangential contact problems for several transversely isotropic elastic layers bonded to an elastic foundation. J. Eng. Math. 81, 93–126 (2013)MathSciNetCrossRefGoogle Scholar
  147. 147.
    Fabrikant, V.I.: Generalized method of images in the crack analysis. Int. J. Eng. Sci. 35, 1159–1184 (1997)MathSciNetMATHCrossRefGoogle Scholar
  148. 148.
    Hu, K.Q., Zhong, Z., Jin, B.: Electroelastic intensification near anti-plane crack in a functionally gradient piezoelectric ceramic strip. Acta Mech. Solida Sin. 16, 197–204 (2003)Google Scholar
  149. 149.
    Feng, W.J., Li, X.G., Wang, S.D.: Torsional impact response of a penny-shaped crack in a functional graded strip. Appl. Math. Mech. Eng. Ed. 25, 1398–1404 (2004)MATHCrossRefGoogle Scholar
  150. 150.
    Hao, T.H.: Crack tip field in functionally gradient material with exponential variation of elastic constants in two directions. Acta Mech. Sin. 21, 601–607 (2005)MATHCrossRefGoogle Scholar
  151. 151.
    Volkov, S., Aizikovich, S., Wang, Y.S., Fedotov, I.: Analytical solution of axisymmetric contact problem about indentation of a circular indenter into a soft functionally graded elastic layer. Acta Mech. Sin. 29, 196–201 (2013)MathSciNetCrossRefGoogle Scholar
  152. 152.
    Ma, J., Ke, L.L., Wang, Y.S.: Frictionless contact of a functionally graded magneto-electro-elastic layered half-plane under a conducting punch. Int. J. Solids Struct. 51, 2791–2806 (2014)CrossRefGoogle Scholar
  153. 153.
    Sankar, T.S., Fabrikant, V.I.: Asymmetric contact problem including wear for nonhomogeneous half space. J. Appl. Math. Mech. 49, 43–46 (1982)MathSciNetMATHGoogle Scholar
  154. 154.
    Fabrikant, V.I., Sankar, T.S.: On contact problems in an inhomogeneous half-space. Int. J. Solids Struct. 20, 159–166 (1984)MATHCrossRefGoogle Scholar
  155. 155.
    Li, X.Y., Chen, W.Q., Wang, H.Y., Wang, G.D.: Crack tip plasticity of a penny-shaped Dugdale crack in a power-law graded elastic infinite medium. Eng. Fract. Mech. 88, 1–14 (2012)MathSciNetCrossRefGoogle Scholar
  156. 156.
    Martin, P.A.: Exact solution of some integral equations over a circular disc. J. Integral Equ. Appl. 18, 39–58 (2006)MathSciNetMATHCrossRefGoogle Scholar
  157. 157.
    Tao, F.M., Tang, R.J.: The crack-inclusion interaction and the analysis of singularity for the horizontal contact. Appl. Math. Mech. Eng. Ed. 22, 547–556 (2001)MATHCrossRefGoogle Scholar
  158. 158.
    Zhong, Z.: Analysis of a partially debonded elliptic inhomogeneity in piezoelectric materials. Appl. Math. Mech. Eng. Ed. 25, 445–457 (2004)MATHCrossRefGoogle Scholar
  159. 159.
    Hu, Y.T., Li, G.Q., Jiang, S.N., Hu, H.P., Yang, J.S.: Interaction of electric charges in a piezoelectric with rigid external cracks. Appl. Math. Mech. Eng. Ed. 26, 996–1006 (2005)MATHCrossRefGoogle Scholar
  160. 160.
    Fang, Q.H., Liu, Y.W.: Elastic interaction between wedge disclination dipole and internal crack. Appl. Math. Mech. Eng. Ed. 27, 1239–1247 (2006)MATHCrossRefGoogle Scholar
  161. 161.
    Zhou, Z.G., Wang, B.: Basic solution of two parallel non-symmetric permeable cracks in piezoelectric materials. Appl. Math. Mech. Eng. Ed. 28, 417–428 (2007)MathSciNetMATHCrossRefGoogle Scholar
  162. 162.
    Xiao, W.S., Xie, C., Liu, Y.W.: Interaction between heat dipole and circular interfacial crack. Appl. Math. Mech. Eng. Ed. 30, 1221–1232 (2009)MathSciNetMATHCrossRefGoogle Scholar
  163. 163.
    Xu, C.H., Qin, T.Y., Yuan, L., Noda, N.A.: Analysis of multiple interfacial cracks in three-dimensional bimaterials using hypersingular integro-differential equation method. Appl. Math. Mech. Eng. Ed. 30, 293–301 (2009)MATHCrossRefGoogle Scholar
  164. 164.
    Karapetian, E., Hanson, T.: Crack opening displacements and stress intensity factors caused by a concentrated load outside a circular crack. Int. J. Solids Struct. 31, 2035–2052 (1994)MATHCrossRefGoogle Scholar
  165. 165.
    Karapetian, E., Kachanov, M.: Three-dimensional interactions of a circular crack with dipoles, centers of dilatation and moments. Int. J. Solids Struct. 33, 3951–3967 (1996)MATHCrossRefGoogle Scholar
  166. 166.
    Kachanov, M., Karapetian, E.: Three-dimensional interactions of a half-plane crack with point forces, dipoles and moments. Int. J. Solids Struct. 34, 4101–4125 (1997)MATHCrossRefGoogle Scholar
  167. 167.
    Karapetian, E., Kachanov, M.: Green’s functions for the isotropic or transversely isotropic space containing a circular crack. Acta Mech. 126, 169–187 (1998)MATHCrossRefGoogle Scholar
  168. 168.
    Xiao, Z.M., Fan, H., Zhang, T.L.: Stress intensity factors of two skew-parallel penny-shaped cracks in a 3-D transversely isotropic solid. Mech. Mater. 20, 261–272 (1995)CrossRefGoogle Scholar
  169. 169.
    Zhan, S.G., Wang, T.C.: Interactions of penny-shaped cracks in three- dimensional solids. Acta Mech. Sin. 22, 341–353 (2006)MATHCrossRefGoogle Scholar
  170. 170.
    Fabrikant, V.I.: Interaction of an arbitrary force with a flexible punch or with a penny-shaped crack. Q. J. Mech. Appl. Math. 50, 303–319 (1997)MathSciNetMATHCrossRefGoogle Scholar
  171. 171.
    Hou, P.F., Ding, H.J., Guan, F.L.: Circular crack in a transversely isotropic piezoelectric space under point forces and point charges. Acta Mech. Sin. 18, 159–169 (2002)CrossRefGoogle Scholar
  172. 172.
    Hou, P.F., Pan, X.P., Ding, H.J.: Three-dimensional interactions of a half-plane crack in a transversely isotropic piezoelectric space with resultant sources. Acta Mech. Solida Sin. 18, 265–271 (2005)Google Scholar
  173. 173.
    Hou, P.F., Ding, H.J., Leung, A.Y.T.: Three-dimensional interactions of circular crack in transversely isotropic piezoelectric space with resultant sources. Appl. Math. Mech. Eng. Ed. 27, 1439–1449 (2006)MATHCrossRefGoogle Scholar
  174. 174.
    Goryacheva, I.G.: Mechanics of discrete contact. Tribol. Int. 39, 381–386 (2006)CrossRefGoogle Scholar
  175. 175.
    Bedoidze, M.V., Pozharskii, D.A.: The interaction of punches on a transversely isotropic half-space. J. Appl. Math. Mech. 78, 409–414 (2014)MathSciNetCrossRefGoogle Scholar
  176. 176.
    Hetnarski, R.B., Eslami, M.R.: Thermal Stresses—Advanced Theory and Applications. Springer, Berlin (2009)MATHGoogle Scholar
  177. 177.
    Williams, W.E.: A solution of the steady-state thermoelastic equations. Z. Angew. Math. Phys. 12, 452–455 (1961)MathSciNetMATHCrossRefGoogle Scholar
  178. 178.
    Barber, J.R.: Elasticity, 3rd revised ed. Springer, Dordrecht (2010)Google Scholar
  179. 179.
    Chen, W.Q., Ding, H.J., Ling, D.S.: Thermoelastic field of a transversely isotropic elastic medium containing a penny-shaped crack: exact fundamental solution. Int. J. Solids Struct. 41, 69–83 (2004)MATHCrossRefGoogle Scholar
  180. 180.
    Chen, W.Q.: On the general solution for piezothermoelasticity for transverse isotropy with application. J. Appl. Mech. 67, 705–711 (2000)MathSciNetMATHCrossRefGoogle Scholar
  181. 181.
    Chen, W.Q., Lim, C.W., Ding, H.J.: Point temperature solution for a penny- shaped crack in an infinite transversely isotropic thermo-piezo-elastic medium. Eng. Anal. Bound. Elem. 29, 524–532 (2005)MATHCrossRefGoogle Scholar
  182. 182.
    Barber, J.R.: Steady-state thermal stresses caused by an imperfectly conducting penny-shaped crack in an elastic solid. J. Therm. Stresses 3, 77–83 (1980)CrossRefGoogle Scholar
  183. 183.
    Shen, S.P., Kuang, Z.B.: Interface crack in bi-piezothermoelastic media. Acta Mech. Solida Sin. 9, 13–26 (1996)Google Scholar
  184. 184.
    Xu, C.H., Qin, T.Y., Hua, Y.L.: Singular integral equations and boundary element method of cracks in thermally stressed planar solids. Appl. Math. Mech. Eng. Ed. 21, 399–406 (2000)MATHCrossRefGoogle Scholar
  185. 185.
    Niraula, O.P., Wang, B.L.: A magneto-electro-elastic material with a penny-shaped crack subjected to temperature loading. Acta Mech. 187, 151–168 (2006)MATHCrossRefGoogle Scholar
  186. 186.
    Niraula, O.P., Wang, B.L.: Thermal stress analysis in magneto-electro-thermo-elasticity with a penny-shaped crack under uniform heat flow. J. Therm. Stresses 29, 423–437 (2006)CrossRefGoogle Scholar
  187. 187.
    Yang, J., Jin, X.Y., Jin, N.G.: A penny-shaped crack in transversely isotropic magneto-electro-thermo-elastic medium subjected to uniform symmetric heat flux. Int. J. Solids Struct. 51, 1792–1808 (2014)CrossRefGoogle Scholar
  188. 188.
    Yang, J., Jin, X.Y., Jin, N.G.: A penny-shaped crack in an infinite linear transversely isotropic medium subjected to uniform anti-symmetric heat flux: Closed-form solution. Eur. J. Mech. A Solids 47, 254–270 (2014)MathSciNetCrossRefGoogle Scholar
  189. 189.
    Li, X.Y., Chen, W.Q., Wang, H.Y.: General steady state solutions for transversely isotropic thermoporoelastic media in three dimensions and its application. Eur. J. Mech. A Solids 29, 317–326 (2010)CrossRefGoogle Scholar
  190. 190.
    Li, X.Y., Wu, J., Chen, W.Q., Wang, H.Y., Zhou, Z.Q.: Exact and complete fundamental solutions for penny-shaped crack in an infinite transversely isotropic thermoporoelastic medium: Mode I problem. Struct. Eng. Mech. 42, 313–334 (2012)CrossRefGoogle Scholar
  191. 191.
    Barber, J.R.: Contact problems involving a cooled punch. J. Elasticity 8, 409–423 (1978)MATHCrossRefGoogle Scholar
  192. 192.
    Chen, P.J., Chen, S.H.: Thermo-mechanical contact behavior of a finite graded layer under a sliding punch with heat generation. Int. J. Solids. Struct. 50, 1108–1119 (2013)CrossRefGoogle Scholar
  193. 193.
    Karapetian, E., Kalinin, S.V.: Indentation of a punch with chemical or heat distribution at its base into transversely isotropic half-space: Application to local thermal and electrochemical probes. J. Appl. Phys. 113, 187201 (2013)CrossRefGoogle Scholar
  194. 194.
    Yang, J., Jin, X.Y.: Indentation of a flat circular punch with uniform heat flux at its base into transversely isotropic magneto-electro-thermo-elastic half space. J. Appl. Phys. 115, 083516 (2014)CrossRefGoogle Scholar
  195. 195.
    Fan, T.Y., Mai, Y.W.: Elasticity theory, fracture mechanics, and some relevant thermal properties of quasi-crystalline materials. Appl. Mech. Rev. 57, 325–343 (2004)CrossRefGoogle Scholar
  196. 196.
    Li, X.Y.: Fundamental solutions of penny-shaped and half-infinite plane cracks embedded in an infinite space of one-dimensional quasicrystal under thermal loading. Proc. R. Soc. A 469, 20130023 (2013)CrossRefGoogle Scholar
  197. 197.
    Li, X.Y., Li, P.D.: Three-dimensional thermo-elastic general solutions of one-dimensional hexagonal quasi-crystal and fundamental solutions. Phys. Lett. A 376, 2004–2009 (2012)MATHCrossRefGoogle Scholar
  198. 198.
    Yang, L.Z., Zhang, L.L., Song, F., Gao, Y.: General solutions for three-dimensional thermoelasticity of two-dimensional hexagonal quasicrystals and an application. J. Therm. Stresses 37, 363–379 (2014)CrossRefGoogle Scholar
  199. 199.
    Chen, W.Q., Shioya, T., Ding, H.J.: Integral equations for mixed boundary value problem of a piezoelectric half-space and the applications. Mech. Res. Commun. 26, 583–590 (1999)MathSciNetMATHCrossRefGoogle Scholar
  200. 200.
    Hou, P.F., Zhou, X.H., He, Y.J.: Green’s functions for a semi-infinite transversely isotropic piezothermoelastic material. Smart Mater. Struct. 16, 1915–1923 (2007)CrossRefGoogle Scholar
  201. 201.
    Hou, P.F., Luo, W., Leung, A.Y.T.: A point heat source on the surface of a semi-infinite transversely isotropic piezothermoelastic material. J. Appl. Mech. 75, 011013 (2008)CrossRefGoogle Scholar
  202. 202.
    Hou, P.F., Leung, A.Y.T., Ding, H.J.: A point heat source on the surface of a semi-infinite transversely isotropic electro-magneto-thermo-elastic material. Int. J. Eng. Sci. 46, 273–285 (2008)MATHCrossRefGoogle Scholar
  203. 203.
    Hou, P.F., Yi, T., Leung, A.Y.T.: Green’s functions for semi-infinite transversely isotropic electro-magneto-thermo-elastic material. Int. J. Appl. Electromagnet. Mech. 29, 83–100 (2009)Google Scholar
  204. 204.
    Hou, P.F., Leung, A.Y.T.: Three-dimensional Green’s functions for two-phase transversely isotropic piezothermoelastic media. J. Intell. Mater. Syst. Struct. 20, 11–21 (2009)CrossRefGoogle Scholar
  205. 205.
    Hou, P.F., Li, Q.H., Jiang, H.Y.: Three-dimensional steady-state general solution for isotropic thermoelastic materials with applications II: Green’s functions for two-phase infinite body. J. Therm. Stresses 36, 851–867 (2013)CrossRefGoogle Scholar
  206. 206.
    Hou, P.F., Zhao, M., Ju, J.W.: Three-dimensional Green’s functions for transversely isotropic thermoporoelastic bimaterials. J. Appl. Geophys. 95, 36–46 (2013)CrossRefGoogle Scholar
  207. 207.
    Hou, P.F., Zhao, M., Tong, J., Fu, B.: Three-dimensional steady-state Green’s functions for fluid-saturated, transversely isotropic, poroelastic bimaterials. J. Hydrol. 496, 217–224 (2013)CrossRefGoogle Scholar
  208. 208.
    Hou, P.F., Yuan, K., Tian, W.: Three-dimensional Green’s functions for a fluid and pyroelectric two-phase material. Appl. Math. Comput. 249, 303–319 (2014)MathSciNetGoogle Scholar
  209. 209.
    Hou, P.F., Li, Z.S., Zhang, Y.: Three-dimensional quasi-static Green’s function for an infinite transversely isotropic pyroelectric material under a step point heat source. Mech. Res. Commun. 62, 66–76 (2014)CrossRefGoogle Scholar
  210. 210.
    Karapetian, E., Kalinin, S.V.: Point force and generalized point source on the surface of semi-infinite transversely isotropic material. J. Appl. Phys. 110, 052020 (2011)CrossRefGoogle Scholar
  211. 211.
    Oliver, W.C., Pharr, G.M.: Improved technique for determining hardness and elastic modulus using load and displacement sensing indentation experiments. J. Mater. Res. 7, 1564–1583 (1992)CrossRefGoogle Scholar
  212. 212.
    Fischer-Cripps, A.C.: Introduction to Contact Mechanics, 2nd edn. Springer, Berlin (2007)MATHCrossRefGoogle Scholar
  213. 213.
    Giannakopoulos, A.E., Suresh, S.: Theory of indentation of piezoelectric materials. Acta Mater. 47, 2153–2164 (1999)CrossRefGoogle Scholar
  214. 214.
    Sridhar, S., Giannakopoulos, A.E., Suresh, S., Ramamurty, U.: Electrical response during indentation of piezoelectric materials: a new method for material characterization. J. Appl. Phys. 85, 380–387 (1999)CrossRefGoogle Scholar
  215. 215.
    Ramamurty, U., Sridhar, S., Giannakopoulos, A.E., Suresh, S.: An experimental study of spherical indentation on piezoelectric materials. Acta Mater. 47, 2417–2430 (1999)CrossRefGoogle Scholar
  216. 216.
    Sridhar, S., Giannakopoulos, A.E., Suresh, S.: Mechanical and electrical responses of piezoelectric solids to conical indentation. J. Appl. Phys. 87, 8451–8456 (2000)CrossRefGoogle Scholar
  217. 217.
    Giannakopoulos, A.E.: Strength analysis of spherical indentation of piezoelectric materials. J. Appl. Mech. 67, 409–416 (2000)MATHCrossRefGoogle Scholar
  218. 218.
    Giannakopoulos, A.E., Parmaklis, A.Z.: The contact problem of a circular rigid punch on piezomagnetic materials. Int. J. Solids Struct. 44, 4593–4612 (2007)MATHCrossRefGoogle Scholar
  219. 219.
    Kalinin, S.V., Bonnell, D.A.: Imaging mechanism of piezoresponse force microscopy of ferroelectric surfaces. Phys. Rev. B 65, 125408 (2002)CrossRefGoogle Scholar
  220. 220.
    Rar, A., Pharr, G.M., Oliver, W.C., Karapetian, E., Kalinin, S.V.: Piezoelectric nanoindentation. J. Mater. Res. 21, 552–556 (2006)CrossRefGoogle Scholar
  221. 221.
    Kalinin, S.V., Rodriguez, B.J., Jesse, S., Karapetian, E., Mirman, B., Eliseev, E.A., Morozovska, A.N.: Nanoscale electromechanics of ferroelectric and biological systems: a new dimension in scanning probe microscopy. Annu. Rev. Mater. Res. 37, 189–238 (2007)CrossRefGoogle Scholar
  222. 222.
    Makagon, A., Kachanov, M., Karapetian, E., Kalinin, S.V.: Indentation of spherical and conical punches into piezoelectric half-space with frictional sliding: applications to scanning probe microscopy. Phys. Rev. B 76, 040511 (2007)CrossRefGoogle Scholar
  223. 223.
    Karapetian, E., Kachanov, M., Kalinin, S.V.: Stiffness relations for piezoelectric indentation of flat and non-flat punches of arbitrary planform: applications to probing nanoelectromechanical properties of materials. J. Mech. Phys. Solids 57, 673–688 (2009)MathSciNetMATHCrossRefGoogle Scholar
  224. 224.
    Makagon, A., Kachanov, M., Karapetian, E., Kalinin, S.V.: Piezoelectric indentation of a flat circular punch accompanied by frictional sliding and applications to scanning probe microscopy. Int. J. Eng. Sci. 47, 221–229 (2009)CrossRefGoogle Scholar
  225. 225.
    Pan, K., Liu, Y.Y., Xie, S.H., Liu, Y.M., Li, J.Y.: The electromechanics of piezoresponse force microscopy for a transversely isotropic piezoelectric medium. Acta Mater. 61, 7020–7033 (2013)CrossRefGoogle Scholar
  226. 226.
    Kalinin, S.V., Mirman, B., Karapetian, E.: Relationship between direct and converse piezoelectric effect in a nanoscaled electromechanical contact. Phys. Rev. B 76, 212102 (2007)CrossRefGoogle Scholar
  227. 227.
    Prashanthi, K., Mandal, M., Duttagupta, S.P., Ramgopal Rao, V., Pant, P., Dhale, K., Palkar, V.R.: Nanomechanical characterization of multiferroic thin films for micro-electromechanical systems. Int. J. Nanosci. 10, 1039–1042 (2011)CrossRefGoogle Scholar
  228. 228.
    Nelson, B.A., King, W.P.: Measuring material softening with nanoscale spatial resolution using heated silicon probes. Rev. Sci. Instrum. 78, 023702 (2007)CrossRefGoogle Scholar
  229. 229.
    Nikiforov, M.P., Jesse, S., Morozovska, A.N., Eliseev, E.A., Germinario, L.T., Kalinin, S.V.: Probing the temperature dependence of the mechanical properties of polymers at the nanoscale with band excitation thermal scanning probe microscopy. Nanotechnology 20, 395709 (2009)CrossRefGoogle Scholar
  230. 230.
    Balke, N., Jesse, S., Kim, Y., Adamczyk, L., Tselev, A., Ivanov, I.N., Dudney, N.J., Kalinin, S.V.: Real space mapping of Li-Ion transport in amorphous Si anodes with nanometer resolution. Nano Lett. 10, 3420–3425 (2010)CrossRefGoogle Scholar
  231. 231.
    Kumar, A., Ciucci, F., Morozovska, A.N., Kalinin, S.V., Jesse, S.: Measuring oxygen reduction/evolution reactions on the nanoscale. Nat. Chem. 3, 707–713 (2011)CrossRefGoogle Scholar
  232. 232.
    Oliver, W.C., Pharr, G.M.: Measurement of hardness and elastic modulus by instrumented indentation: advances in understanding and refinements to methodology. J. Mater. Res. 19, 3–20 (2004)CrossRefGoogle Scholar
  233. 233.
    Chan, E.P., Hu, Y.H., Johnson, P.M., Suo, Z.G., Stafford, C.M.: Spherical indentation testing of poroelastic relaxations in thin hydrogel layers. Soft Matter 8, 1492–1498 (2012)CrossRefGoogle Scholar
  234. 234.
    Yang, L., Tu, Y.S., Tan, H.L.: Influence of atomic force microscope (AFM) probe shape on adhesion force measured in humidity environment. Appl. Math. Mech. Eng. Ed. 33, 829–844 (2014)MATHGoogle Scholar
  235. 235.
    Borodich, F.M., Keer, L.M.: Evaluation of elastic modulus of materials by adhesive (no-slip) nano-indentation. Proc. R. Soc. Lond. A 460, 507–514 (2004)MathSciNetMATHCrossRefGoogle Scholar
  236. 236.
    Borodich, F.M.: The Hertz-type and adhesive contact problems for depth- sensing indentation. Adv. Appl. Mech. 47, 225–366 (2014)CrossRefGoogle Scholar
  237. 237.
    Rogowski, B., Kaliński, W.: The adhesive contact problem for a piezoelectric half-space. Int. J. Press. Vessels Pip. 84, 502–511 (2007)CrossRefGoogle Scholar
  238. 238.
    Dundurs, J., Markenscoff, X.: A Green’s function formulation of anticracks and their interaction with load-induced singularities. J. Appl. Mech. 56, 550–555 (1989)MathSciNetMATHCrossRefGoogle Scholar
  239. 239.
    Rahman, M.: Some problems of a rigid elliptical disk-inclusion bonded inside a transversely isotropic space, Part II: Solutions of the integral equations. J. Appl. Mech. 66, 621–630 (1999)CrossRefGoogle Scholar
  240. 240.
    Rahman, M.: The normal shift of a rigid elliptical disk in a transversely isotropic solid. Int. J. Solids Struct. 38, 3965–3977 (2001)MATHCrossRefGoogle Scholar
  241. 241.
    Kaczyński, A.: On 3D anticrack problems in a transversely isotropic solid. Eur. J. Mech. A Solids 43, 142–151 (2014)MathSciNetCrossRefGoogle Scholar
  242. 242.
    Kaczyński, A.: Thermal stress analysis of a three-dimensional anticrack in a transversely isotropic solid. Int. J. Solids Struct. 51, 2382–2389 (2014)CrossRefGoogle Scholar
  243. 243.
    Gouldstone, A., Chollacoop, N., Dao, M., Li, J., Minor, A.M., Shen, Y.L.: Indentation across size scales and disciplines: recent developments in experimentation and modeling. Acta Mater. 55, 4015–4039 (2007)Google Scholar
  244. 244.
    Wang, J.X., Huang, Z.P., Duan, H.L., Yu, S.W., Feng, X.Q., Wang, G.F., Zhang, W.X., Wang, T.J.: Surface stress effect in mechanics of nanostructured materials. Acta Mech. Solida Sin. 24, 52–81 (2011)Google Scholar
  245. 245.
    Chen, W.Q.: Surface effect on Bleustein–Gulyaev wave in a piezoelectric half-space. Theor. Appl. Mech. Lett. 1, 041001 (2011)CrossRefGoogle Scholar
  246. 246.
    Qin, J., Qu, S.X., Feng, X., Huang, Y.G., Xiao, J.L., Hwang, K.C.: A numerical study of indentation with small spherical indenters. Acta Mech. Solida Sin. 22, 18–26 (2009)CrossRefGoogle Scholar
  247. 247.
    Wei, Y.G., Wang, X.Z., Zhao, M.H., Cheng, C.M., Bai, Y.L.: Size effect and geometrical effect of solids in micro-indentation test. Acta Mech. Sin. 19, 59–70 (2003)CrossRefGoogle Scholar
  248. 248.
    Zhou, H., Zhang, H.L., Pei, Y.M., Chen, H.S., Zhao, H.W., Fang, D.N.: Scaling relationship among indentation properties of electromagnetic materials at micro- and nanoscale. Appl. Phys. Lett. 106, 081904 (2015)CrossRefGoogle Scholar
  249. 249.
    Zhao, M.H., Cheng, C.J., Liu, Y.J., Liu, G.N., Zhang, S.S.: The method of analysis of crack problem in three-dimensional non-local elasticity. Appl. Math. Mech. Eng. Ed. 20, 469–475 (1999)MATHCrossRefGoogle Scholar
  250. 250.
    Dai, T.M.: The mixed boundary-value problem for non-local asymmetric elasticity. Appl. Math. Mech. Eng. Ed. 21, 27–32 (2000)MATHCrossRefGoogle Scholar
  251. 251.
    Willis, J.R.: Hertzian contact of anisotropic bodies. J. Mech. Phys. Solids 14, 163–176 (1966)MathSciNetMATHCrossRefGoogle Scholar
  252. 252.
    Willis, J.R.: The stress field around an elliptical crack in an anisotropic elastic medium. Int. J. Eng. Sci. 6, 253–263 (1968)MATHCrossRefGoogle Scholar
  253. 253.
    Barik, S.P., Kanoria, M., Chaudhuri, P.K.: Effect of anisotropy on thermoelastic contact problem. Appl. Math. Mech. Eng. Ed. 29, 501–510 (2008)MATHCrossRefGoogle Scholar
  254. 254.
    Fabrikant, V.I.: Non-traditional contact problem for transversely isotropic half-space. Q. J. Mech. Appl. Math. 64, 151–170 (2011)MathSciNetMATHCrossRefGoogle Scholar
  255. 255.
    Fabrikant, V.I.: Non-traditional crack problem for transversely-isotropic body. Eur. J. Mech. A Solids 30, 902–912 (2011)MathSciNetMATHCrossRefGoogle Scholar
  256. 256.
    Sevostianov, I., Paulo da Silva, U., Aguiar, A.R.: Green’s function for piezoelectric 622 hexagonal crystals. Int. J. Eng. Sci. 84, 18–28 (2014)CrossRefGoogle Scholar
  257. 257.
    Li, X.Y., Wang, M.Z.: Hertzian contact of anisotropic piezoelectric bodies. J. Elasticity 84, 153–166 (2006)MathSciNetMATHCrossRefGoogle Scholar
  258. 258.
    Tian, J.Y., Xie, Z.M.: Dynamic contact stiffness of vibrating rigid sphere contacting semi-infinite transversely isotropic viscoelastic solid. Acta Mech. Solida Sin. 21, 580–588 (2008)CrossRefGoogle Scholar
  259. 259.
    Zhang, T.Y.: Effects of static electric field on the fracture behavior of piezoelectric ceramics. Acta Mech. Sin. 18, 537–550 (2002)CrossRefGoogle Scholar
  260. 260.
    Yang, J.S.: An Introduction to the Theory of Piezoelectricity. Springer, New York (2005)MATHGoogle Scholar
  261. 261.
    Dorfmann, L., Ogden, R.W.: Nonlinear Theory of Electroelastic and Magnetoelastic Interactions. Springer, New York (2014)MATHCrossRefGoogle Scholar
  262. 262.
    Wang, Q.M., Mohan, A.C., Oyen, M.L., Zhao, X.H.: Separating viscoelasticity and poroelasticity of gels with different length and time scales. Acta Mech. Sin. 30, 20–27 (2014)MathSciNetCrossRefGoogle Scholar
  263. 263.
    Touzaline, A.: Analysis of a quasistatic contact problem with adhesion and nonlocal friction for viscoelastic materials. Appl. Math. Mech. Eng. Ed. 31, 623–634 (2010)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© The Chinese Society of Theoretical and Applied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Engineering MechanicsZhejiang UniversityHangzhouChina

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