Screw dislocations in piezoelectric laminates with four or more phases. (English) Zbl 1425.74131

Wang, X.; Chen, L.; Schiavone, P.

Arch. Mech. 71, No. 3, 263-283 (2019).

Summary: We present an analytical solution to the problem of a screw dislocation in a four-phase piezoelectric laminate composed of two piezoelectric layers of equal thickness sandwiched between two semi-infinite piezoelectric media. A new version of the complex variable formulation is proposed such that the \(2 \times 2\) real symmetric matrix appearing in the formulation becomes dimensionless. Using analytic continuation, the original boundary value problem is reduced to the identification of a single 2D analytic vector function which is completely determined following rigorous solution of the resulting linear recurrence relations in matrix form. An explicit expression for the image force acting on the piezoelectric screw dislocation is obtained once the single \(2 \times 2\) real matrix function is identified. We also discuss the solution for a screw dislocation in an N-phase piezoelectric laminate composed of \(N - 2\) piezoelectric layers of equal thickness sandwiched between two semi-infinite piezoelectric media.

MSC:

74E30	Composite and mixture properties
74F15	Electromagnetic effects in solid mechanics

Keywords:

four-phase piezoelectric laminate; screw dislocation; analytical solution; analytic continuation; recurrence relations of matrix form

References:

[1]	Y.T. Chou, Screw dislocations in and near lamellar inclusions, Physica Status Solidi (B), 17, 509-516, 1966.
[2]	S.N.G. Chu, Screw dislocation in a two-phase isotropic thin film, Journal of Applied Physics, 53, 3019-3023, 1982
[3]	S.V. Kamat, J.P. Hirth, B. Carnahan, Image forces on screw dislocations in multilayer structures, Scripta Metallurgica, 21, 1587-1592, 1987.
[4]	M.L. Öveçoğlu, M.F. Doerner, W.D. Nix, Elastic interactions of screw dislocations in thin films on substrates, Acta Metallurgica, 35, 2947-2957, 1987.
[5]	L. Stagni, R. Lizzio, Interaction of an edge dislocation with a lamellar inhomogeneity, Mechanics of Materials, 6, 17-25, 1987.
[6]	N. Fares, V.C. Li, General image method in a plane-layered elastostatic medium, Journal of Applied Mechanics, 55, 781-785, 1988. · Zbl 0673.73025
[7]	C.C. Ma, H.T. Lu, Theoretical analysis of screw dislocations and image forces in anisotropic multilayered media, Physical Review B, 73, 14, 144102, 2006.
[8]	H.Y. Wang, M.S. Wu, H. Fan, Image decomposition method for the analysis of a mixed dislocation in a general multilayer, International Journal of Solids and Structures, 44, 1563-1581, 2007. · Zbl 1155.74323
[9]	C.Q. Ru, Analytic solution for Eshelby’s problem of an inclusion of arbitrary shape in a plane or half-plane, Journal of Applied Mechanics, 66, 315-322, 1999.
[10]	Y.E. Pak, Force on a piezoelectric screw dislocation, Journal of Applied Mechanics, 57, 863-869, 1990. · Zbl 0735.73072
[11]	K.Y. Lee, W.G. Lee, Y.E. Pak, Interaction between a semi-infinite crack and a screw dislocation in a piezoelectric material, Journal of Applied Mechanics, 67, 165-170, 2000. · Zbl 1110.74537
[12]	J.X. Liu, S.Y. Du, B. Wang, A screw dislocation interacting with a piezoelectric bimaterial interface, Mechanics Research Commununications, 26, 415-420, 1999. · Zbl 0973.74032
[13]	X. Wang, H. Fan, A piezoelectric screw dislocation in a bimaterial with surface piezoelectricity, Acta Mechanica, 226, 3317-3331, 2015. · Zbl 1329.74089
[14]	B.J. Chen, K.M. Liew, Z.M. Xiao, Green’s functions for anti-plane problems in piezoelectric media with a finite crack, International Journal of Solids and Structures, 41, 5285-5300, 2004. · Zbl 1179.74039
[15]	B.J. Chen, Z.M. Xiao, K.M. Liew, A screw dislocation interacting with a finite crack in a piezoelectric medium, International Journal of Engineering Science, 42, 1325-1345, 2004.