Solutions to a limited-permeable crack or two limited-permeable collinear cracks in piezoelectric/piezomagnetic materials. (English) Zbl 1161.74481
Summary: The solutions of a limited-permeable crack (case \(\mathbf I\)) or two collinear limited-permeable cracks (case \(\mathbf {II}\)) in piezoelectric/piezomagnetic materials subjected to a uniform tension loading were investigated in this paper using the generalized Almansi’s theorem. At the same time, the electric permittivity and the magnetic permeability of air in crack were firstly considered. Through the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables were jumps of displacements across crack surfaces, not the dislocation density functions or the complex variable functions. To solve the dual integral equations, the jumps of displacements across crack surfaces were directly expanded in a series of Jacobi polynomials to obtain the relations among electric displacement intensity factors, magnetic flux intensity factors and stress intensity factors at crack tips.
References:
| [1] | Wu T.L., Huang J.H. (2000). Closed-form solutions for the magnetoelectric coupling coefficients in fibrous composites with piezoelectric and piezomagnetic phases. Int. J. Solids Struct. 37: 2981–3009 · Zbl 0997.74019 · doi:10.1016/S0020-7683(99)00116-X |
| [2] | Wippler K., Ricoeur A., Kuna M. (2004). Towards the computation of electrically permeable cracks in piezoelectrics. Eng. Fracture Mech. 71: 2567–2587 · doi:10.1016/j.engfracmech.2004.03.003 |
| [3] | Fujimoto Y., Shintaku E., Pirker G., Liu G. (2003). Piezoelectric sensor for stress intensity factor measurement of two dimensional cracks. Eng. Fracture Mech. 70: 1203–1218 · doi:10.1016/S0013-7944(02)00091-7 |
| [4] | Sih G.C., Song Z.F. (2003). Magnetic and electric poling effects associated with crack growth in BaTiO3 - CoFe2O4 composite. Theor. Appl. Fracture Mech. 39: 209–227 · doi:10.1016/S0167-8442(03)00003-X |
| [5] | Song Z.F., Sih G.C. (2003). Crack initiation behavior in magnetoelectroelastic composite under in-plane deformation. Theor. Appl. Fracture Mech. 39: 189–207 · doi:10.1016/S0167-8442(03)00002-8 |
| [6] | Wang B.L., Mai Y.W. (2004). Fracture of piezoelectromagnetic materials. Mech. Res. Commun. 31(1): 65–73 · Zbl 1045.74586 · doi:10.1016/j.mechrescom.2003.08.002 |
| [7] | Gao C.F., Kessler H., Balke H. (2003). Fracture analysis of electromagnetic thermoelastic solids. Eur. J. Mech. Solid 22(3): 433–442 · Zbl 1032.74633 |
| [8] | Gao C.F., Tong P., Zhang T.Y. (2003). Interfacial crack problems in magneto-electroelastic solids. Int. J. Eng. Sci. 41(18): 2105–2121 · doi:10.1016/S0020-7225(03)00206-4 |
| [9] | Spyropoulos C.P., Sih G.C., Song Z.F. (2003). Magnetoelectroelastic composite with poling parallel to plane of line crack under out-of-plane deformation. Theor. Appl. Fracture Mech. 39(3): 281–289 · doi:10.1016/S0167-8442(03)00021-1 |
| [10] | Liu J.X., Liu X.L., Zhao Y.B. (2001). Green’s functions for anisotropic magnetoelectroelastic solids with an elliptical cavity or a crack. Int. J. Eng. Sci. 39(12): 1405–1418 · Zbl 1210.74043 · doi:10.1016/S0020-7225(01)00049-0 |
| [11] | Chung M.Y., Ting T.C.T. (1995). The Green function for a piezoelectric piezomagnetic anisotropic elastic medium with an elliptic hole or rigid inclusion. Philos. Mag. Lett. 72: 405–410 · doi:10.1080/09500839508242480 |
| [12] | Pan E. (2002). Three-dimensional Green’s functions in anisotropic magneto-electro-elastic bimaterails. Zeitschrift fur Angewandte Mathematik und Physik 53: 815–838 · Zbl 1031.74024 · doi:10.1007/s00033-002-8184-1 |
| [13] | Gao C.F., 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 · Zbl 1211.74187 · doi:10.1016/S0020-7225(02)00323-3 |
| [14] | Gao C.F., Kessler H., Balke H. (2003). Crack problems in magnetoelectroelastic solids. Part II: general solution of collinear cracks. Int. J. Eng. Sci. 41(9): 983–994 · Zbl 1211.74188 · doi:10.1016/S0020-7225(02)00324-5 |
| [15] | Wang B.L., Mai Y.W. (2003). Crack tip field in piezoelectric/piezomagnetic media. Eur. J. Mech. Solid 22(4): 591–602 · Zbl 1032.74641 · doi:10.1016/S0997-7538(03)00062-7 |
| [16] | Chen W.Q., Lee K.Y., Ding H.J. (2004). General solution for transversely isotropic magneto-electro-thermo-elasticity and the potential theory method. Int. J. Eng. Sci. 42: 1361–1379 · Zbl 1211.74062 · doi:10.1016/j.ijengsci.2004.04.002 |
| [17] | Wang X., Shen Y.P. (2002). The general solution of three-dimensional problems in magnetoelectroelastic media. Int. J. Eng. Sci. 40: 1069–1080 · Zbl 1211.74103 |
| [18] | Van Suchtelen J. (1972). Product properties: a new application of composite materials. Phillips Research Reports 27: 28–37 |
| [19] | Harshe G., Dougherty J.P., Newnham R.E. (1993). Theoretical modeling of 3-0/0-3 magnetoelectric composites. Int. J. Appl. Electromag. Mat. 4: 161–171 |
| [20] | Avellaneda M., Harshe G. (1994). Magnetoelectric effect in piezoelectric/magnetostrictive multiplayer (2–2) composites. J. Intell. Mat. Syst. Struct. 5: 501–513 · doi:10.1177/1045389X9400500406 |
| [21] | Nan C.W. (1994). Magnetoelectric effect in composites of piezoelectric and piezomagnetic phases. Phys. Rev. B 50: 6082–6088 · doi:10.1103/PhysRevB.50.6082 |
| [22] | Benveniste Y. (1995). Magnetoelectric effect in fibrous composites with piezoelectric and magnetostrictive phases. Phys. Rev. B 51: 16424–16427 · doi:10.1103/PhysRevB.51.16424 |
| [23] | Huang J.H., Kuo W.S. (1997). The analysis of piezoelectric/piezomagnetic composite materials containing ellipsoidal inclusions. J. Appl. Phys. 81(3): 1378–1386 · doi:10.1063/1.363874 |
| [24] | Li J.Y. (2000). Magnetoelectroelastic multi-inclusion and inhomogeneity problems and their applications in composite materials. Int. J. Eng. Sci. 38: 1993–2011 · doi:10.1016/S0020-7225(00)00014-8 |
| [25] | Zhou Z.G., Wang B. (2004). Two parallel symmetry permeable cracks in functionally graded piezoelectric/piezomagnetic materials under anti-plane shear loading. Int. J. Solids Struct. 41: 4407–4422 · Zbl 1079.74621 · doi:10.1016/j.ijsolstr.2004.03.004 |
| [26] | Zhou Z.G., Wang B., Sun Y.G. (2004). Two collinear interface cracks in magneto-electro-elastic composites. Int. J. Eng. Sci. 42: 1157–1167 · Zbl 1211.76107 |
| [27] | Zhou Z.G., Wu L.Z., Wang B. (2005). The dynamic behavior of two collinear interface cracks in magneto-electro-elastic composites. Eur. J. Mech. Solids 24(2): 253–262 · Zbl 1069.74047 · doi:10.1016/j.euromechsol.2004.10.006 |
| [28] | Zhou Z.G., Wu L.Z., Wang B. (2005). The behavior of a crack in functionally graded piezoelectric/piezomagnetic materials under anti-plane shear loading. Archive Appl. Mech. 74(8): 526–535 · Zbl 1100.74052 · doi:10.1007/s00419-004-0369-y |
| [29] | Morse P.M., Feshbach H.: Methods of theoretical physics. vol.1, McGraw-Hill, New York (1958) · Zbl 0051.40603 |
| [30] | Gao H.J., Zhang T.Y., Tong P. (1997). Local and global energy release rates for an electrically yielded crack in a piezoelectric ceramics. J. Mech. Phys. Solids 45(4): 491–510 · doi:10.1016/S0022-5096(96)00108-1 |
| [31] | Hao T.H., Shen Z.Y. (1994). A new electric boundary condition of electric fracture mechanics and its applications. Eng. Fracture Mech. 47(6): 793–802 · doi:10.1016/0013-7944(94)90243-7 |
| [32] | Govorukha V.B. (2006). On the influence of the electric permeability on an interface crack in a piezoelectric biomaterial compound. Int. J. Solids Struct. 43: 1979–1990 · Zbl 1121.74363 · doi:10.1016/j.ijsolstr.2005.04.009 |
| [33] | Suo Z., Kuo C.M., Barnett D.M. (1992). Willis J R Fracture mechanics for piezoelectric ceramics. J. Mech. Phys. Solids 40(4): 739–765 · Zbl 0825.73584 · doi:10.1016/0022-5096(92)90002-J |
| [34] | Zhang T.Y., Tong P. (1996). Fracture mechanics for a mode III crack in a piezoelectric material. Int. J. Solids Struct. 33: 343–359 · Zbl 0900.73061 · doi:10.1016/0020-7683(95)00073-9 |
| [35] | Ueda S. (2006). Transient response of a center crack in a functionally graded piezoelectric strip under electromechanical impact. Eng. Fracture Mech. 73: 1455–1471 · doi:10.1016/j.engfracmech.2006.01.025 |
| [36] | Zhong Z., Meguid S.A. (1997). Analysis of a circular arc-crack in piezoelectric materials. Int. J. Fracture 84: 143–158 · doi:10.1023/A:1007327931730 |
| [37] | McMeeking R.M. (1989). On mechanical stress at cracks in dielectrics with application to dielectric breakdown. J. Appl. Phys. 62: 3316–3122 |
| [38] | Parton V.S. (1976) Fracture mechanics of piezoelectric materials, ACTA Astronautra 3, 671–683 · Zbl 0351.73115 |
| [39] | Mikhailov G.K., Parton V.S.: Electromagnetoelasticity. Hemisphere, New York (1990) |
| [40] | Pak Y.E. (1990). Crack extension force in a piezoelectric material. J. Appl. Mech. 57: 647–653 · Zbl 0724.73191 · doi:10.1115/1.2897071 |
| [41] | Deeg W.E.F.: The analysis of dislocation, crack and inclusion problems in piezoelectric solids, Ph.D. thesis. Stanford University (1980) |
| [42] | Soh A.K., Fang D.N., Lee K.L. (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. Solid 19: 961–977 · Zbl 1001.74044 · doi:10.1016/S0997-7538(00)01107-4 |
| [43] | Hao T.H. (2001). Multiple collinear cracks in a piezoelectric material. Int. J. Solids Struct. 38(50–51): 9201–9208 · Zbl 0999.74099 · doi:10.1016/S0020-7683(01)00069-5 |
| [44] | Yang F.Q. (2001). Fracture mechanics for a Mode I crack in piezoelectric materials. Int. J. Solids Struct. 38: 3813–3830 · Zbl 0981.74019 · doi:10.1016/S0020-7683(00)00244-4 |
| [45] | Ding H.J., Chen B., Liang J. (1996). General solutions for coupled equations for piezoelectric media. Int. J. Solids Struct. 33(16): 2283–2296 · Zbl 0900.73390 · doi:10.1016/0020-7683(95)00171-9 |
| [46] | Gradshteyn I.S. (1980). Ryzhik IM. Table of integral, series and products. Academic · Zbl 0521.33001 |
| [47] | Erdelyi A. (1954). Tables of integral transforms, vol. 1. McGraw-Hill, New York |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
