Closed-form solution for elliptical inclusion problem in antiplane piezoelectricity with far-field loading at an arbitrary angle. (English) Zbl 1406.74222
Summary: An elliptical piezoelectric inclusion embedded in an infinite piezoelectric matrix is analyzed in the framework of linear piezoelectricity. Using the conformal mapping technique, a closed-form solution is obtained for the case of a far-field antiplane mechanical load, \(\tau_0\), and an inplane electrical load, \(E_0\), at an arbitrary angle \(\beta\). The stress and electric field distribution patterns for different defect shapes, loading angles, and material constants are studied. The energy release rates of self-similarly expanding and rotating defects in the presence of an electric field are obtained using the generalized \(M\)- and \(L\)-integrals as a function of the loading angle. The physical significance of these results is discussed in terms of the stress and electric field distributions as well as the energy release rates.
MSC:
| 74F15 | Electromagnetic effects in solid mechanics |
| 74E30 | Composite and mixture properties |
| 78A30 | Electro- and magnetostatics |
Keywords:
piezoelectric elliptical inclusion; antiplane loading at an arbitrary angle; generalized path-independent \(L\)- and \(M\)-integralsReferences:
| [1] | Berlincourt, D. A.; Curran, D. R.; Jaffe, H., Piezoelectric and piezomagnetic materials and their function in transducers, (. Mason, W. P., Phycl. Acoust. 1A, (1964), Academic Press New York), 177 |
| [2] | Chao, L. P.; Huang, J. H., On a piezoelectric material containing a permeable elliptical crack, Int. J. Solids Struct., 37, 5161-5176, (2000) · Zbl 0977.74022 |
| [3] | Chen, Y. Z.; Lee, K. Y., Analysis of the M-integral in plane elasticity, ASME J. Appl. Mech., 71, 572-574, (2004) · Zbl 1111.74363 |
| [4] | Chung, M. Y.; Ting, T. C.T., Piezoelectric solid with elliptical inclusion or hole, Int. J. Solids Struct., 33, 23, 3341-3361, (1996) · Zbl 0901.73064 |
| [5] | Crawley, E. F.; de Luis, J., Use of piezoelectric actuators as elements of intelligent structures, AIAA J., 25, 1373-1385, (1987) |
| [6] | Deeg, W.F., 1980. The Analysis of Dislocation, Crack, and Inclusion Problems in Piezoelectric Solids. Ph.D. thesis, Stanford University, Stanford, California. |
| [7] | Eischen, J. W.; Herrmann, G., Energy release rates and related balance laws in linear elastic defect mechanics, J. Appl. Mech., 54, 2, 388-392, (1987) · Zbl 0613.73003 |
| [8] | Freiman, S. W.; Pohanka, R. C., Review of mechanically related failures of ceramic capacitors and capacitor materials, J. Am. Ceram. Soc., 72, 12, 2258-2263, (1989) |
| [9] | Freund, L. B., Stress-intensity factor calculations based on a conservational integral, Int. J. Solids Struct., 14, 241-249, (1978) |
| [10] | Fuller, C. R., Active control of sound transmission through elastic plates using piezoelectric actuators, AIAA J., 29, 1771-1777, (1992) · Zbl 0729.76560 |
| [11] | Gong, S. X.; Meguid, S. A., A general treatment of the elastic field of an elliptical inhomogeneity under antiplane shear, ASME J. Appl. Mech., 59, S131-S135, (1992) · Zbl 0760.73007 |
| [12] | Gunther, W., Uber einige randintegrale der elastomechanik, in adhandlungen der braunschweigischen wisssenschaftlichen. gesellschaft, XIV, (1962), Verlag Friedr Vieweg, Braunschweig · Zbl 0149.42703 |
| [13] | Hui, T.; Chen, Y. H., The M-integral analysis for a nano-inclusion in plane elastic materials under uni-axial or bi-axial loadings, J. Appl. Mech., 77, 210-219, (2010) |
| [14] | Knowles, J. K.; Sternberg, E., On a class of conservation laws in linearized and finite elastostatics, Arch. Ratnl. Mech. Anls, 44, 187-211, (1972) · Zbl 0232.73017 |
| [15] | Koepke, B. G.; McHenry, K. D.; Seifried, L. M.; Stokes, R. J., Mechanical integrity of piezoelectric and dielectric ceramics, (1986), IEEE Ultrasonic Symposium, Williamsburg, Virginia |
| [16] | Levin, V. M.; Michelitsch, Th.; Sevostianov, I., Spheroidal inhomogeneity in a transversely isotropic piezoelectric medium, Arch. Appl. Mech., 70, 673-693, (2000) · Zbl 1013.74027 |
| [17] | Li, Q.; Chen, Y. H., Surface effect and size dependent on the energy release due to nanosized hole expansion in plane elastic materials, ASME J. Appl. Mech., 75, (2008), 061008(1-5) |
| [18] | McMeeking, R. M., Towards a fracture mechanics for brittle piezoelectric and dielectric materials, Int. J. Fract., 108, 25-41, (2001) |
| [19] | Meguid, S. A.; Zhong, Z., On the elliptical inhomogeneity problem in pieozelectric materials under antiplane shear and inplane electric field, Int. J. Eng. Sci., 36, 3, 329-344, (1998) |
| [20] | Mishra, D.; Yoo, S. H.; Park, C. Y.; Pak, Y. E., Elliptical inclusion problem in antiplane piezoelectricity: stress concentrations and energy release rates, Int. J. Fract., 179, 213-220, (2013) |
| [21] | Pak, Y. E., Crack extension force in a piezoelectric material, ASME J. Appl. Mech., 57, 647-653, (1990) · Zbl 0724.73191 |
| [22] | Pak, Y. E., Circular inclusion problem in antiplane piezoelectricity, Int. J. Solids Struct., 29, 2403-2419, (1992) · Zbl 0764.73069 |
| [23] | Pak, Y. E., Piezoelectric elliptical inclusion under longitudinal shear. mechanics and materials for electronic packaging: coupled field behavior in materials, ASME-AMD, 193, 263-268, (1994) |
| [24] | Pak, Y. E., Elliptical inclusion problem in antiplane piezoelectricity: implications for fracture mechanics, Int. J. Eng. Sci., 48, 209-222, (2010) |
| [25] | Pak, Y. E.; Mishra, D.; Yoo, S. H., Closed-form solution for a coated circular inclusion under uniaxial tension, Act. Mech., 223, 5, 937-951, (2012) · Zbl 1401.74060 |
| [26] | Reay, N. K.; Pietraszewski, K. A.R. B., Liquid-nitrogen cooled servo-stabilized Fabry-Perot interferometer for the infrared, Opt. Eng., 31, 1667-1670, (1992) |
| [27] | Rice, J. R., A path-independent integral and the approximate analysis of strain concentration by notches and cracks, J. Appl. Mech., 35, 379, (1968) |
| [28] | Shen, M. H.; Chen, S. N.; Chen, F. M., Piezoelectric study on confocally multicoated elliptical inclusion, Int. J. Eng. Sci., 43, 1299-1312, (2005) |
| [29] | Sosa, H., Plane problems in piezoelectric media with defects, Int. J. Solids Struct., 28, 491-505, (1991) · Zbl 0749.73070 |
| [30] | Sosa, H.; Khutoryansky, N., New development concerning piezoelectric materials with defects, Int. J. Solids Struct., 33, 3399-3414, (1996) · Zbl 0924.73198 |
| [31] | Sudak, L. J., Effect of an interphase layer on the electroelastic stresses within a three-phase elliptic inclusion, Int. J. Eng. Sci., 41, 1019-1039, (2003) |
| [32] | Tobin, A.; Pak, Y. E., Effect of electric fields on fracture behavior of PZT ceramics, Procd. SPIE Conf. on Smart Struct. Mat., vol. 1916, (1993), pp. 78-86 |
| [33] | Wang, J.; Landis, C., Effects of in-plane electric fields on the toughening behavior of ferroelectric ceramics, J. Mech. Mat. Struct., 1, 6, 1075-1095, (2006) |
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