zbMATH — the first resource for mathematics

Coupled variational principles of piezoelectricity. (English) Zbl 1210.74175
Summary: A family of generalized variational principles of piezoelectricity can be obtained straightforwardly from the field equations and boundary conditions via the semi-inverse method of establishing variational principles proposed by He without using Lagrange multipliers. The present theory provides a quite straightforward tool to search for various variational principles for physical problems. This paper aims at providing a more complete theoretical basis for the finite element applications and other direct variational methods such as Ritz’s, Trefftz’s and Kantorovitch’s methods.

74S30 Other numerical methods in solid mechanics (MSC2010)
74F15 Electromagnetic effects in solid mechanics
Full Text: DOI
[1] Zhou, S.A.; Hsien, R.K.T.; Maugin, G.A., Electric and elastic multipole defects in finite piezoelectric media, Int. J. solids struct., 22, 12, 1411-1422, (1986) · Zbl 0603.73111
[2] ()
[3] Maugin, G.A., Continuum mech. electromagnetic solids, (1991), Elsevier Amsterdam
[4] G.L. Liu, A systematic approach to the search and transformation for variational principles in fluid mechanics with emphasis on inverse and hybrid problems, in: First International Symposium on Experimental and Computational Aerothermodynamics of Internal Flows, Beijing, 1990, pp. 128-135
[5] J.H. He, Modified Lagrange multiplier method and generalized variational principles in fluid mechanics, J. Shanghai University (English Edition), 2(2) (1997) 117-122. (Find the paper in the Internet: http//www.shu.edu.cn) · Zbl 0905.76066
[6] Santilli, R.M., Foundations of theoretical mechanics I: the inverse problem in Newtonian mechanics, (1978), Springer Berlin · Zbl 0401.70015
[7] Santilli, R.M., Foundations of theoretical mechanics II: birkhoffian generalization of Hamiltonian mechanics, (1983), Springer Berlin · Zbl 0536.70001
[8] Tonti, E., Variational principles, (1968), Tamburini Milano · Zbl 0194.42501
[9] He, J.H., Semi-inverse method of establishing generalized variational principles for fluid mechanics with emphasis on turbomachinery aerodynamics, Int. J. turbo & jet-engines, 4, 1, 23-28, (1997)
[10] He, J.H., A generalized variational principle for 3-D unsteady transonic rotational flow in rotor using Clebsch variables, Int. J. turbo & jet-engines, 14, 1, 17-22, (1997)
[11] He, J.H., A variational theory for one-dimensional unsteady compressible flow: an image plane approach, Appl. math. modell., 22, 395-403, (1998)
[12] He, J.H., A family of variational principles for compressible rotational blade-to-blade flow using semi-inverse method, Int. J. turbo and jet-engines, 15, 2, 95-100, (1998)
[13] He, J.H., Generalized variational principle for compressible S2-flow in mixed-flow turbomachinery using semi-inverse method, Int. J. turbo & jet-engines, 15, 2, 101-107, (1998)
[14] J.H. He, Generalized Hellinger-Reissner principle, ASME J. Appl. Mech. (accepted) · Zbl 1110.74473
[15] Chien, W.Z., Method of high-order Lagrange multiplier and generalized variational principles of elasticity with more general forms of functionals, Applied math. & mech., 4, 2, 137-150, (1983)
[16] Washizu, K., Variational methods in elasticity and plasticity, (1982), Pergamon Press Oxford · Zbl 0164.26001
[17] J.H. He, Variational crisis in elasticity and its removal, Shanghai J. Mech., 18 (4) (1997) 305-310 (in Chinese)
[18] G.L. Liu, The variational crisis and generalized variational principles in elasticity, J. Shanghai University, 4 (6) (1998) 591-599 (Natural Science, in Chinese) · Zbl 1082.74503
[19] Felippa, C., Parametrized multifield variational principles in elasticity, Comm. appl. num. engr., 5, 79-99, (1989) · Zbl 0659.73015
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.