×

A consistent framework of material configurational mechanics in piezoelectric materials. (English) Zbl 1458.74043

Summary: The fundamental concepts of material configurational mechanics are formulated in piezoelectric materials. A consistent thermodynamic framework is outlined to develop the corresponding theory of material configurational stresses and forces associated with the \(J_k\)-, \(M\)- and \(L\)-integrals by the gradient, divergence and curl operation of the electric enthalpy density, respectively. The physical interpretation of material configurational stresses is explored, and they can be explained as the energy release rates due to the translation of material point along \(x_k\)-direction, the self-similar expansion, the rotation of material element, respectively. The path independence or path dependence of variant integrals such as the \(J_k\)-, \(M\)- and \(L\)-integrals is examined in piezoelectric material. As an application of material configurational mechanics in piezoelectric materials, an explicit method is derived to calculate the change of the \(J\)-integral as a dominant fracture parameter for a crack interaction with domain switching near the crack tip. It is concluded that domain switching has an obvious shielding effect on the fracture toughness in piezoelectrics by the present explicit forms. The present framework of material configurational mechanics is expected to provide an effective and convenient tool to deal with various crack or damage problems in piezoelectric materials.

MSC:

74F15 Electromagnetic effects in solid mechanics
74A15 Thermodynamics in solid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Herbert, J.: Ferroelectrics Transducers and Sensors. Gordon and Breach Science Publishes, New York (1982)
[2] Zuo, J.Z., Sih, G.C.: Energy density theory formulation and interpretation of cracking behavior for piezoelectric ceramics. Theor. Appl. Fract. Mech. 34, 17-33 (2000) · doi:10.1016/S0167-8442(00)00021-5
[3] Chen, Y.H., Lu, T.J.: Cracks and fracture in piezoelectrics. Adv. Appl. Mech. 39, 121-215 (2003) · doi:10.1016/S0065-2156(02)39003-3
[4] Zhang, T.Y., Gao, C.F.: Fracture behaviors of piezoelectric materials. Theor. Appl. Fract. Mech. 41, 339-379 (2004) · doi:10.1016/j.tafmec.2003.11.019
[5] Kuna, M.: Fracture mechanics of piezoelectric materials—where are we right now? Eng. Fract. Mech. 77, 309-326 (2010) · doi:10.1016/j.engfracmech.2009.03.016
[6] Fang, D.N., Liu, J.X.: Fracture Mechanics of Piezoelectric and Ferroelectric Solids. Springer, Berlin (2012) · Zbl 1303.74002
[7] Bayat, J., Ayatollahi, M., Bagheri, R.: Fracture analysis of an orthotropic strip with imperfect piezoelectric coating containing multiple defects. Theor. Appl. Fract. Mech. 77, 41-49 (2015) · doi:10.1016/j.tafmec.2015.01.009
[8] Hu, K.Q., Chen, Z.T.: Boundary effect on crack kinking in a piezoelectric strip with a central crack. Theor. Appl. Fract. Mech. 81, 11-24 (2016) · doi:10.1016/j.tafmec.2015.11.007
[9] Kienzler, R., Herrmann, G.: Mechanics of Material Space: With Applications to Defect and Fracture Mechanics. Springer, Berlin (2000) · Zbl 0954.74001 · doi:10.1007/978-3-642-57010-0
[10] Gurtin, M.E.: Configurational Forces as Basic Concepts of Continuum Physics. Springer, Berlin (2000) · Zbl 0951.74003
[11] Maugin, G.A.: Sixty years of configurational mechanics (1950-2010). Mech. Res. Commun. 50, 39-49 (2013) · doi:10.1016/j.mechrescom.2013.03.003
[12] Gommerstadt, B.Y.: The J and M integrals for a cylindrical cavity in a time-harmonic wave field. Int. J. Eng. Sci. 83, 76-84 (2014) · Zbl 06982784 · doi:10.1016/j.ijengsci.2014.03.007
[13] Kabil, B., Rohde, C.: The influence of surface tension and configurational forces on the stability of liquid-vapor interfaces. Nonlinear Anal. 107, 63-75 (2014) · Zbl 1453.76213 · doi:10.1016/j.na.2014.04.003
[14] Baxevanakis, K.P., Giannakopoulos, A.E.: Finite element analysis of discrete edge dislocations: configurational forces and conserved integrals. Int. J. Solids Struct. 62, 52-65 (2015) · doi:10.1016/j.ijsolstr.2015.01.025
[15] Wang, J.: Accurate evaluation of the configurational forces in single-crystalline NiMnGa alloys under mechanical loading conditions. Acta Mater. 105, 306-316 (2016) · doi:10.1016/j.actamat.2015.12.018
[16] Eshelby, J.D.: The force on an elastic singularity. Philos. Trans. R. Soc. A. 244, 87-112 (1951) · Zbl 0043.44102 · doi:10.1098/rsta.1951.0016
[17] Knowles, J.K., Sternberg, E.: On a class of conservation laws in linearized and finite elastostatics. Arch. Ration. Mech. Anal. 44, 187-211 (1972) · Zbl 0232.73017 · doi:10.1007/BF00250778
[18] Budiansky, B., Rice, J.R.: Conservation laws and energy-release rates. J. Appl. Mech. 40, 201-203 (1973) · Zbl 0261.73059 · doi:10.1115/1.3422926
[19] Cherepanov, G.P.: The propagation of cracks in a continuous medium. J. Appl. Math. Mech. 31, 503-512 (1967) · Zbl 0288.73078 · doi:10.1016/0021-8928(67)90034-2
[20] Rice, J.R.: A path independent integral and the approximate analysis of strain concentration by notch and cracks. J. Appl. Mech.: Trans. ASME 35, 379-386 (1968) · doi:10.1115/1.3601206
[21] Wang, S.S., Yau, J.F., Corten, H.T.: A mixed-mode crack analysis of rectilinear anisotropic solids using conservation laws of elasticity. Int. J. Fract. 16, 247-259 (1980) · Zbl 0463.73103 · doi:10.1007/BF00013381
[22] Kienzler, R., Kordisch, H.: Calculation of \[\text{ J }_1\] J1 and \[\text{ J }_2\] J2 using the L and M integrals. Int. J. Fract. 43, 213-225 (1990) · doi:10.1007/BF00018343
[23] Ozenc, K., Chinaryan, G., Kaliske, M.: A configurational force approach to model the branching phenomenon in dynamic brittle fracture. Eng. Fract. Mech. 157, 26-42 (2016) · doi:10.1016/j.engfracmech.2016.02.017
[24] Zhao, L.G., Chen, Y.H.: Interaction of multiple interface cracks. Int. J. Fract. 70, 53-62 (1995) · doi:10.1007/BF00012941
[25] Zhao, L.G., Chen, Y.H.: On the contribution of subinterface microcracks near the tip of an interface macrocrack to the J-integral in bimaterial solids. Int. J. Eng. Sci. 35, 387-407 (1997) · Zbl 0926.74011 · doi:10.1016/S0020-7225(96)00076-6
[26] Zhao, L.G., Chen, Y.H.: Further investigation of subinterface cracks. Arch. Appl. Mech. 67, 393-406 (1997) · Zbl 0893.73048 · doi:10.1007/s004190050126
[27] Chen, Y.H.: M-integral analysis for two-dimensional solids with strongly interacting microcracks. Part I: in an infinite brittle solid. Int. J. Solids Struct. 38, 3193-3212 (2001) · Zbl 1058.74616 · doi:10.1016/S0020-7683(00)00242-0
[28] Chang, J.H., Peng, D.J.: Use of M-integral for rubbery material problems containing defects. J. Eng. Mech. 130, 589-598 (2004) · doi:10.1061/(ASCE)0733-9399(2004)130:5(589)
[29] Yu, N.Y., Li, Q.: Failure theory via the concept of material configurational forces associated with the M-integral. Int. J. Solids Struct. 50, 4320-4332 (2013) · doi:10.1016/j.ijsolstr.2013.09.001
[30] Judt, P.O., Ricoeur, A.: A new application of M- and L-integrals for the numerical loading analysis of two interacting cracks. In: Hutter, G., Zybell, L. (eds.) Recent Trends in Fracture and Damage Mechanics. Springer, ISBN:978-3-319-21466-5 (2015) · Zbl 1335.74003
[31] Pak, Y.E.: Crack extension force in a piezoelectric material. J. Appl. Mech.: Trans. ASME 57, 647-653 (1990a) · Zbl 0724.73191 · doi:10.1115/1.2897071
[32] Pak, Y.E.: Force on a piezoelectric screw dislocation. J. Appl. Mech.: Trans. ASME 57, 863-869 (1990b) · Zbl 0735.73072 · doi:10.1115/1.2897653
[33] McMeeking, R.M.: A J-integral for the analysis of electrically induced mechanical stress at cracks in elastic dielectrics. Int. J. Eng. Sci. 28, 605-613 (1990) · Zbl 0715.73058 · doi:10.1016/0020-7225(90)90089-2
[34] Goy, O., Mueller, R., Gross, D.: Configurational forces on point defects in ferroelectric materials. ZAMM J. Appl. Math. Mech. 89, 641-650 (2009) · Zbl 1168.74023 · doi:10.1002/zamm.200800159
[35] Steinmann, P.: Application of material forces to hyper elastostatic fracture mechanics. I. Continuum mechanical setting. Int. J. Solids Struct. 37, 7371-7391 (2000) · Zbl 0992.74008 · doi:10.1016/S0020-7683(00)00203-1
[36] Li, Q., Kuna, M.: Evaluation of electromechanical fracture behavior by configurational forces in cracked ferroelectric polycrystals. Comput. Mater. Sci. 57, 94-101 (2012a) · doi:10.1016/j.commatsci.2011.01.050
[37] Li, Q., Kuna, M.: Inhomogeneity and material configurational forces in three dimensional ferroelectric polycrystals. Eur. J. Mech. A Solids 31, 77-89 (2012b) · Zbl 1278.74052 · doi:10.1016/j.euromechsol.2011.07.004
[38] Kienzler, R., Herrmann, G.: Fracture criteria based on local properties of the Eshelby tensor. Mech. Res. Commu. 29, 521-527 (2002) · Zbl 1094.74541 · doi:10.1016/S0093-6413(02)00299-9
[39] Noether, E.: Invariant variational problems. Transp. Theor. Stat. 1, 183-207 (1971) · Zbl 0291.49035 · doi:10.1080/00411457108231445
[40] Eischen, J.W., Herrmann, G.: Energy release rates and related balance laws in linear elastic defect mechanics. J. Appl. Mech.: Trans. ASME 54, 388-392 (1987) · Zbl 0613.73003 · doi:10.1115/1.3173024
[41] Kienzler, R., Herrmann, G.: On the properties of the Eshelby tensor. Acta Mech. 125, 73-91 (1997) · Zbl 0896.73007 · doi:10.1007/BF01177300
[42] Li, Q., Lv, J.N.: Invariant integrals of crack interaction with an inhomogeneity. Eng. Fract. Mech. 171, 76-84 (2017) · doi:10.1016/j.engfracmech.2016.12.013
[43] Lv, J.N., Fan, X.L., Li, Q.: The impact of the growth of thermally grown oxide layer on the propagation of surface cracks within thermal barrier coatings. Surf. Coat. Technol. 309, 1033-1044 (2017) · doi:10.1016/j.surfcoat.2016.10.039
[44] Li, Q., Lv, J.N., Hou, J.L., Zuo, H.: Crack-tip shielding by the dilatant transformation of particles/fibers embedded in composite materials. Theor. Appl. Fract. Mech. 80, 242-252 (2015) · doi:10.1016/j.tafmec.2015.06.005
[45] Kessler, H., Balke, H.J.: On the local and average energy release in polarization switching phenomena. J. Mech. Phys. Solids 49, 953-978 (2001) · Zbl 0980.74020 · doi:10.1016/S0022-5096(00)00073-9
[46] Ricoeur, A., Kuna, M.: A micromechanical model for the fracture process zone in ferroelectrics. Comput. Mater. Sci. 27, 235-249 (2003) · doi:10.1016/S0927-0256(02)00360-9
[47] Fang, D.N., Jiang, Y.J., Li, S., Sun, C.T.: Interactions between domain switching and crack propagation in poled \[{\text{ BaTiO }_3}\] BaTiO3 single crystal under mechanical loading. Acta Mater. 55, 5758-5767 (2007) · doi:10.1016/j.actamat.2007.06.024
[48] Zhang, Y.H., Li, J.Y., Fang, D.N.: Fracture analysis of ferroelectric single crystals: domain switching near crack tip and electric field induced crack propagation. J. Mech. Phys. Solids 61, 114-130 (2013) · doi:10.1016/j.jmps.2012.08.008
[49] Li, F.X., Rajapakse, R.K.N.D.: A constrained domain-switching model for polycrystalline ferroelectric ceramics. Part I: model formulation and application to tetragonal materials. Acta Mater. 55, 6472-6480 (2007) · doi:10.1016/j.actamat.2007.08.002
[50] Hwang, S.C., McMeeking, R.M.: A finite element model of ferroelectric polycrystals. Int. J. Solids Struct. 36, 1541-1556 (1999) · Zbl 0958.74065 · doi:10.1016/S0020-7683(98)00051-1
[51] Xu, X.L., Rajapakse, R.K.N.D.: On a plane crack in piezoelectric solids. Int. J. Solids Struct. 38, 7643-7658 (2001) · Zbl 1020.74035 · doi:10.1016/S0020-7683(01)00029-4
[52] Hao, T.H., Shen, Z.Y.: A new electric boundary condition of electric fracture mechanics and its applications. Eng. Fract. Mech. 47, 793-802 (1994) · doi:10.1016/0013-7944(94)90243-7
[53] McMeeking, R.M.: The energy release rate for a Griffith crack in a piezoelectric material. Eng. Fract. Mech. 71, 1149-1163 (2004) · doi:10.1016/S0013-7944(03)00135-8
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.