Consistency and recovery in electroelasticity. I: Standard finite elements. (English) Zbl 1025.74030

Summary: Developing finite elements for electroelastic analysis requires a special care as the coupling in the discrete equations depends on the matching between the approximations assumed for mechanical and electrical variables. To provide a formal basis to this intuitive remark, a notion of consistency is established and a rigorous consistency analysis is presented for the standard finite element model based on assumed displacement and electric potential. In this way, specific analytical requirements are obtained which serve as a guide to select the interpolation functions for primary variables. Moreover, the analysis shows that violating consistency can be reflected as spurious outcomes upsetting the local distributions of secondary variables. Indeed, this undesirable effect is shown to be typical of the standard approach if stress and electric flux density are computed via the constitutive equations. To cure the trouble, an alternative recovery procedure is devised based on the consistency analysis. The procedure is variationally correct and reconstitutes in a consistent manner the distributions of stress and electric flux density.


74S05 Finite element methods applied to problems in solid mechanics
74F15 Electromagnetic effects in solid mechanics
Full Text: DOI


[1] Eringen, A.C.; Maugin, G.A., Electrodynamics of continua I. foundations and solid media, (1990), Springer-Verlag New York
[2] Ikeda, T., Fundamentals of piezoelectricity, (1996), Oxford University Press New York
[3] Gandhi, M.V.; Thompson, B.S., Smart materials and structures, (1992), Chapman & Hall London
[4] Tiersten, H.F., Hamilton’s principle for linear piezoelectric media, Proc. IEEE, 1523-1524, (1967)
[5] EerNisse, E.P., Variational method for electroelastic vibration analysis, IEEE trans. sonics ultrasonics, SU-14, 153-160, (1967)
[6] Allik, H.; Hughes, T.J.R., Finite element method for piezoelectric vibration, Internat. J. numer. methods engrg., 2, 151-157, (1970)
[7] Oden, J.T.; Kelley, B.E., Finite element formulation of general electro – thermoelasticity problems, Internat. J. numer. methods engrg., 3, 161-179, (1971) · Zbl 0251.73062
[8] Lee, H.-J.; Saravanos, D.A., Generalized finite element formulation for smart multilayered thermal piezoelectric composite plates, Internat. J. solid structures, 34, 3355-3371, (1997) · Zbl 0942.74647
[9] Wang, J.; Yong, Y.-K.; Imai, T., Finite element analysis of the piezoelectric vibrations of quartz plate resonators with higher-order plate theory, Internat. J. solid structures, 36, 2303-2319, (1999) · Zbl 0956.74066
[10] Prathap, G., The finite element method in structural mechanics, (1993), Kluwer Academic Dordrecht · Zbl 0849.73001
[11] Patel, B.P.; Ganapathi, M.; Saravanan, J., Shear flexible field-consistent curved spline beam element for vibration analysis, Internat. J. numer. methods engrg., 46, 387-407, (1999) · Zbl 0981.74068
[12] Marur, S.R.; Prathap, G., Consistency and correctness evaluation of shear deformable anisoparametric formulations, Internat. J. solid structures, 37, 701-713, (2000) · Zbl 0966.74068
[13] Prathap, G.; Naganarayana, B.P., Consistent force resultant distributions in displacement elements with varying sectional properties, Internat. J. numer. methods engrg., 29, 775-783, (1990)
[14] Prathap, G.; Naganarayana, B.P., Consistent thermal stress evaluation in finite elements, Computers and structures, 54, 415-426, (1995) · Zbl 0875.73267
[15] Naganarayana, B.P.; Rama Mohan, P.; Prathap, G., Accurate thermal stress predictions using C0-continuous higher-order shear deformable elements, Comput. methods appl. mech. engrg., 144, 61-75, (1997) · Zbl 0892.73067
[16] de Miranda, S.; Ubertini, F., On the consistency of finite element models in thermoelastic analysis, Comput. methods appl. mech. engrg., 190, 2411-2427, (2001) · Zbl 1048.74047
[17] de Miranda, S.; Ubertini, F., Recovery of consistent stresses for compatible finite elements, Comput. methods appl. mech. engrg., 191, 1595-1609, (2002) · Zbl 1141.74371
[18] Yeo, S.T.; Ko, J.H.; Lee, B.C., Consistent thermal stress calculation by using a new approach of the enhanced assumed strain method, Comput. methods appl. mech. engrg., 188, 331-345, (2000) · Zbl 0974.74071
[19] J.H. Argyris, Continua and discontinua, in: Proceedings of the First Conference on Matrix Methods in Structural Mechanics, AFFDL. TR., Dayton, 1966, pp. 66-88
[20] Corradi, L., On stress computation in displacement finite element models, Comput. methods appl. mech. engrg., 54, 325-339, (1986) · Zbl 0566.73057
[21] Oden, J.T.; Brauchli, H.J., On the calculation of consistent stress distributions in finite element approximations, Internat. J. numer. methods engrg., 3, 317-325, (1971) · Zbl 0251.73056
[22] Yang, J.S., Mixed variational principles for piezoelectric elasticity, (), II.1.31-38
[23] Sosa, H., Plane problems in piezoelectric media with defects, Internat. J. solids structures, 28, 491-505, (1991) · Zbl 0749.73070
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.