# zbMATH — the first resource for mathematics

Size-dependent piezoelectricity: a 2D finite element formulation for electric field-mean curvature coupling in dielectrics. (English) Zbl 1406.74213
Summary: The classical theory of piezoelectricity defines linear size-independent electromechanical response in non-centrosymmetric dielectrics that involves coupling between the electric field and the mechanical strains. However, with the continuing push to develop novel micro- and nano-scale materials, structures and devices, there is a need to refine and explore size-dependent electro-mechanical coupling phenomena, which have been observed in experiments on centrosymmetric dielectrics. Here a finite element variational formulation is developed based upon a recent consistent size-dependent theory that incorporates the interactions between the electric field and the mechanical mean curvatures in dielectrics, including those with centrosymmetric structure. The underlying formulation is theoretically consistent in several important aspects. In particular, the electric field equations are consistent with Maxwell’s equations, while the mechanical field equations are based upon the recent consistent couple stress theory, involving skew-symmetric mean curvature and couple stress tensors. This, in turn, permits the development of a fully-consistent finite element method for the solution of size-dependent piezoelectric boundary value problems. In this paper, an overview of size-dependent piezoelectricity is first provided, followed by the development of the variational formulation and finite element representation specialized for the planar response of centrosymmetric cubic and isotropic materials. The new formulation is then applied to several illustrative examples to bring out important characteristics predicted by this consistent size-dependent piezoelectric theory.

##### MSC:
 74F15 Electromagnetic effects in solid mechanics 74S05 Finite element methods applied to problems in solid mechanics
Matlab
Full Text:
##### References:
 [1] Allik, H.; Hughes, T. J.R., Finite element method for piezo-electric vibration, Int. J. Numer. Method Eng., 2, 151-157, (1970) [2] Baskaran, S.; He, X. T.; Chen, Q.; Fu, J. Y., Experimental studies on the direct flexoelectric effect in α-phase polyvinylidene fluoride films, Appl. Phys. Lett., 98, 242901, (2011) [3] Bathe, K. J., Finite element procedures, (1996), Prentice Hall Englewood Cliffs, N.J [4] Bechmann, R., Elastic, piezoelectric, and dielectric constants of polarized barium titanate ceramics and some applications of the piezoelectric equations, J. Acoust. Soc. Am., 28, 347-350, (1956) [5] Benjeddou, A., Advances in piezoelectric finite element modeling of adaptive structural elements: a survey, Comput. Struct., 76, 347-363, (2000) [6] Buhlmann, S.; Dwir, B.; Baborowski, J.; Muralt, P., Size effects in mesoscopic epitaxial ferroelectric structures: increase of piezoelectric response with decreasing feature-size, Appl. Phys. Lett., 80, 3195-3197, (2002) [7] Cady, W. G., Piezoelectricity: an introduction to the theory and applications of electro-mechanical phenomena in crystals, (1964), Dover New York [8] Catalan, G.; Lubk, A.; Vlooswijk, A. H.G.; Snoeck, E.; Magen, C.; Janssens, A.; Rispens, G.; Rijnders, G.; Blank, D. H.A.; Noheda, B., Flexoelectric rotation of polarization in ferroelectric thin films, Nat. Mater., 10, 963-967, (2011) [9] Cross, L. E., Flexoelectric effects: charge separation in insulating solids subjected to elastic strain gradients, J. Mater. Sci., 41, 53-63, (2006) [10] Curie, J.; Curie, P., (Comptes rendus hebdomadaires des séances de l’Académie des sciences, vol. 91, (1880)), 294-295 [11] Darrall, B. T.; Dargush, G. F.; Hadjesfandiari, A. R., Finite element Lagrange multiplier formulation for size-dependent skew-symmetric couple-stress planar elasticity, Acta Mech., 225, 195-212, (2014) · Zbl 1401.74269 [12] Davis, T. A., A column pre-ordering strategy for the unsymmetric-pattern multifrontal method, ACM Trans. Math. Softw., 30, 165-195, (2004) · Zbl 1072.65036 [13] Davis, T. A.; Duff, I. S., An unsymmetric-pattern multifrontal method for sparse LU factorization, SIAM J. Matrix Anal. Appl., 18, 140-158, (1997) · Zbl 0884.65021 [14] Eliseev, E. A.; Morozovska, A. N.; Glinchuk, M. D.; Blinc, R., Spontaneous flexoelectric/flexomagnetic effect in nanoferroics, Phys. Rev. B, 79, 165433, (2009) [15] Gaudenzi, P.; Bathe, K. J., An iterative finite-element procedure for the analysis of piezoelectric continua, J. Intell. Mater. Sys. Struct., 6, 266-273, (1995) [16] Griffiths, D. J., Introduction to electrodynamics, (1989), Prentice Hall Englewood Cliffs, N.J [17] Hadjesfandiari, A. R., Size-dependent piezoelectricity, Int. J. Solids Struct., 50, 2781-2791, (2013) [18] Hadjesfandiari, A. R., Size-dependent theories of piezoelectricity: comparisons and further developments for centrosymmetric dielectrics, (2014), arXiv:1409-1082 [physics.class-ph] [19] Hadjesfandiari, A. R.; Dargush, G. F., Couple stress theory for solids, Int. J. Solids Struct., 48, 2496-2510, (2011) [20] Hadjesfandiari, A. R.; Dargush, G. F., Fundamental solutions for isotropic size-dependent couple stress elasticity, Int. J. Solids Struct., 50, 1253-1265, (2013) [21] Harden, J.; Mbanga, B.; Eber, N.; Fodor-Csorba, K.; Sprunt, S.; Gleeson, J. T.; Jakli, A., Giant flexoelectricity of bent-core nematic liquid crystals, Phys. Rev. Lett., 97, 157802, (2006) [22] Hwang, W. S.; Park, H. C.; Ha, S. K., Finite element modeling of piezoelectric sensors and actuators, AIAA J., 31, 930-937, (1993) [23] Jaffe, B.; Cook, W. R.; Jaffe, H., Piezoelectric ceramics, (1971), Academic Press New York, N.Y [24] Kogan, S. M., Piezoelectric effect during inhomogeneous deformation and acoustic scattering of carriers in crystals, Sov. Phys. Solid State, 5, 2069-2070, (1964) [25] Koiter, W. T., Couple stresses in the theory of elasticity, I and II, Proc. Kon. Neder. Akad. Weten. B, 67, 17-44, (1964) · Zbl 0124.17405 [26] Li, A.; Zhou, S.; Zhou, S.; Wang, B., Size-dependent analysis of a three-layer microbeam including electromechanical coupling, Compos. Struct., 116, 120-127, (2014) [27] Ma, W.; Cross, L. E., Flexoelectricity of barium titanate, Appl. Phys. Lett., 88, 232902, (2006) [28] Majdoub, M. S.; Sharma, P.; Cagin, T., Enhanced size-dependent piezoelectricity and elasticity in nanostructures due to the flexoelectric effect, Phys. Rev. B, 77, 125424, (2008) [29] Maranganti, R.; Sharma, P., Atomistic determination of flexoelectric properties of crystalline dielectrics, Phys. Rev. B, 80, 054109, (2009) [30] Maranganti, R.; Sharma, N. D.; Sharma, P., Electromechanical coupling in nonpiezoelectric materials due to nanoscale nonlocal size effects: Green’s function solutions and embedded inclusions, Phys. Rev. B, 74, 014110, (2006) [31] MATLAB, Release 2014a, (2014), The MathWorks, Inc. Natick, MA [32] Meyer, R. B., Piezoelectric effects in liquid crystals, Phys. Rev. Lett., 22, 918-921, (1969) [33] Mindlin, R. D.; Tiersten, H. F., Effects of couple-stresses in linear elasticity, Arch. Ration. Mech. Anal., 11, 415-448, (1962) · Zbl 0112.38906 [34] Mishima, T.; Fujioka, H.; Nagakari, S.; Kamigaki, K.; Nambu, S., Lattice image observations of nanoscale ordered regions in pb (mg1/3nb2/3)O-3, Jpn. J. App. Phys., 36, 6141-6144, (1997) [35] Resta, R., Towards a bulk theory of flexoelectricity, Phys. Rev. Lett., 105, 127601, (2010) [36] Sharma, N. D.; Maranganti, R.; Sharma, P., On the possibility of piezoelectric nanocomposites without using piezoelectric materials, J. Mech. Phys. Solids, 55, 2328-2350, (2007) · Zbl 1171.74016 [37] Shvartsman, V. V.; Emelyanov, A. Y.; Kholkin, A. L.; Safari, A., Local hysteresis and grain size effects in pb(mg1/3nb2/3)O-sbtio3, Appl. Phys. Lett., 81, 117-119, (2002) [38] Tagantsev, A. K., Piezoelectricity and flexoelectricity in crystalline dielectrics, Phys. Rev. B, 34, 5883-5889, (1986) [39] Toupin, R. A., Elastic materials with couple-stresses, Arch. Ration. Mech. Anal., 11, 385-414, (1962) · Zbl 0112.16805 [40] Voigt, W., Lehrbuch der kristallphysik, (1910), BG Teubner Berlin · JFM 54.0929.03 [41] Wang, G. F.; Yu, S. W.; Feng, X. Q., A piezoelectric constitutive theory with rotation gradient effects, Eur. J. Mech. A/Solids, 23, 455-466, (2004) · Zbl 1060.74542 [42] Yudin, P. V.; Tagantsev, A. K., Fundamentals of flexoelectricity in solids, Nanotechnology, 24, 432001, (2013) [43] Zhu, W.; Fu, J. Y.; Li, N.; Cross, L. E., Piezoelectric composite based on the enhanced flexoelectric effects, Appl. Phys. Lett., 89, 192904, (2006) [44] Zienkiewicz, O. C.; Taylor, R. L., The finite element method, (2000), Butterworth-Heinemann Oxford · Zbl 0991.74002 [45] Zubko, P.; Catalan, G.; Buckley, A.; Welche, P. R.L.; Scott, J. F., Strain-gradient-induced polarization in srtio3 single crystals, Phys. Rev. Lett., 99, 167601, (2007) [46] Zubko, P.; Catalan, G.; Tagantsev, A. K., Flexoelectric effect in solids, Ann. Rev. Mater. Res., 43, 387-421, (2013)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.