Abstract
The non-local theory solution of a mode-I permeable crack in a piezoelectric/piezomagnetic composite material plane was given by using the generalized Almansi’s theorem and the Schmidt method in this paper. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of the crack length and the lattice parameter on the stress field, the electric displacement field and the magnetic flux field near the crack tips. Unlike the classical elasticity solutions, it is found that no stress, electric displacement and magnetic flux singularities are present at the crack tips in piezoelectric/piezomagnetic composite materials. The non-local elastic solution yields a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion.
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References
Chen WQ, Lee KY, Ding HJ (2004) General solution for transversely isotropic magneto-electro-thermo-elasticity and the potential theory method. Int J Eng Sci 42(13–14): 1361–1379
Erdelyi A (1954) Tables of integral transforms, vol 1. McGraw-Hill, New York pp, pp 34–89
Edelen DGB (1976) Non-local field theory. In: Eringen AC (eds) Continuum physics, vol 4. Academic Press, New York, pp 75–204
Eringen AC (1976) Non-local polar field theory. In: Eringen AC (eds) Continuum physics, vol 4. Academic Press, New York, pp 205–267
Eringen AC, Speziale CG, Kim BS (1977) Crack tip problem in non-local elasticity. J Mech Phys Solids 25(5): 339–355
Eringen AC (1978) Linear crack subject to shear. Int J Fract 14(4): 367–379
Eringen AC (1979) Linear crack subject to anti-plane shear. Eng Fract Mech 12(2): 211–219
Eringen AC (1983) Interaction of a dislocation with a crack. J Appl Phys 54(12): 6811–6817
Eringen AC, Kim BS (1977) Relation between non-local elasticity and lattice dynamics. Cryst Lattice Defects 7(1): 51–57
Gao CF, Kessler H, Balke H (2003) Fracture analysis of electromagnetic thermoelastic solids. Eur J Mech A Solid 22(3): 433–442
Gao CF, Kessler H, Balke H (2003) Crack problems in magnetoelectroelastic solids. Part I: exact solution of a crack. Int J Eng Sci 41(9): 969–981
Gradshteyn IS, Ryzhik IM (1980) Table of integral, series and products. Academic Press, New York, p.361, p.333, p.483
Green AE, Rivilin RS (1965) Multipolar continuum mechanics: functional theory. I. In: Proceeding of the royal society of London A, vol 284, pp 303–315
Hou PF, Teng GH, Chen HR (2009) Three-dimensional Green’s function for a point heat source in two-phase transversely isotropic magneto-electro-thermo-elastic material. Mech Mater 41(3): 329–338
Itou S (1978) Three dimensional waves propagation in a cracked elastic solid. J Appl Mech 45(4): 807–811
Liu JX, Liu XL, Zhao YB (2001) Green’s functions for anisotropic magnetoelectroelastic solids with an elliptical cavity or a crack. Int J Eng Sci 39(12): 1405–1418
Morse PM, Feshbach H (1958) Methods of theoretical physics. McGraw-Hill, New York, pp 828–930
Pan KL, Takeda N (1998) Non-local stress field of interface dislocations. Archive Appl Mech 68(3–4): 179–184
Parton VS (1976) Fracture mechanics of piezoelectric materials. ACTA Astronautra 3(9–10): 671–683
Rice JR (1968) A path independent integral and the approximate analysis of strain concentrations by notches and cracks. J Appl Mech 35(3): 379–386
Sih GC, Song ZF (2003) Magnetic and electric poling effects associated with crack growth in BaTiO3-CoFe2O4 composite. Theor Appl Fract Mech 39(3): 209–227
Soh AK, Fang DN, Lee KL (2000) Analysis of a bi-piezoelectric ceramic layer with an interfacial crack subjected to anti-plane shear and in-plane electric loading. Eur J Mech A Solid 19(6): 961–977
Song ZF, Sih GC (2003) Crack initiation behavior in magnetoelectroelastic composite under in-plane deformation. Theor Appl Fract Mech 39(3): 189–207
Spyropoulos CP, Sih GC, Song ZF (2003) Magnetoelectroelastic composite with poling parallel to plane of line crack under out-of-plane deformation. Theor Appl Fract Mech 39(3): 281–289
Tian WY, Gabbert U (2004) Multiple crack interaction problem in magnetoelectroelastic solids. Eur J Mech A Solid 23(4): 599–614
Wang BL, Mai YW (2004) Fracture of piezoelectromagnetic materials. Mech Res Commun 31(1): 65–73
Wang X, Shen YP (2002) The general solution of three-dimensional problems in magnetoelectroelastic media. Int J Eng Sci 40(10): 1069–1080
Wu TL, Huang JH (2000) Closed-form solutions for the magnetoelectric coupling coefficients in fibrous composites with piezoelectric and piezomagnetic phases. Int J Solids Struct 37(21): 2981–3009
Yan WF (1967) Axisymmetric slipless indentation of an infinite elastic cylinder. SIAM J Appl Math 15(2): 219–227
Yang FQ (2001) Fracture mechanics for a mode I crack in piezoelectric materials. Int J Solids Struct 38(21): 3813–3830
Zhao MH, Fan CY, Yang F, Liu T (2007) Analysis method of planar cracks of arbitrary shape in the isotropic plane of a three-dimensional transversely isotropic magnetoelectroelastic medium. Int J Solids Struct 44(13): 4505–4523
Zhou ZG, Wu LZ, Du SY (2006) Non-local theory solution for a Mode I crack in piezoelectric materials. Eur J Mech A Solids 25(5): 793–807
Zhou ZG, Wu LZ, Du SY (2007) Non-local theory solution of two Mode-I collinear cracks in the piezoelectric materials. Mech Adv Mater Struct 14(3): 191–201
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Zhang, PW., Zhou, ZG. & Wu, LZ. Non-local theory solution of a mode-I crack in a piezoelectric/piezomagnetic composite material plane. Int J Fract 164, 213–229 (2010). https://doi.org/10.1007/s10704-010-9477-6
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DOI: https://doi.org/10.1007/s10704-010-9477-6