Conservation laws in linear piezoelectricity

Yang, Jun; Batra, R.C.

doi:10.1016/0013-7944(94)00271-i

Cited by 23 publications

(10 citation statements)

References 8 publications

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“…Shock waves were investigated in [25]. Conservation laws for the polarization gradient theory were derived [26] from the invariance of the variational integral.…”

Section: Polarization Gradient Theorymentioning

confidence: 99%

Nonlocal and Gradient Effects

Yang

2009

Special Topics in the Theory of Piezoelectricity

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Classical continuum theories such as elasticity, electrostatics, and piezoelectricity are the long-wave and low-frequency limits of the equations of lattice dynamics. The partial differential equations of these continuum theories can be obtained from the finite difference equations of lattice dynamics by Taylor expansions and truncations. The equations of classical continuum theories are accurate for phenomena with a characteristic length much larger than microscopic characteristic lengths, for example, the distance between neighboring atoms in a lattice. When the characteristic length of a problem is not much larger than the microscopic characteristic length, classical continuum theories do not predict results consistent with lattice dynamics, and hence are no longer valid. For example, lattice waves are dispersive but the theory of elasticity only predicts nondispersive plane waves which are the long wave limit of lattice waves.There are different ways to modify the classical continuum theories so that their range of applicability can be extended to problems with smaller characteristic lengths, with results closer to lattice dynamics in a wider range of wave lengths.One is to generalize the constitutive relations in classical continuum theories from local to nonlocal [1][2][3][4][5]. In classical continuum theories, the constitutive relations are for stress and strain (or electric field and polarization) at the same material point, with a zero interaction distance (local theories). These constitutive relations neglect the microstructure of real materials and lead to macroscopic theories for large characteristic lengths. In nonlocal constitutive relations, a finite microscopic interaction distance is

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“…Shock waves were investigated in [25]. Conservation laws for the polarization gradient theory were derived [26] from the invariance of the variational integral.…”

Section: Polarization Gradient Theorymentioning

confidence: 99%

Nonlocal and Gradient Effects

Yang

2009

Special Topics in the Theory of Piezoelectricity

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“…Throughout this paper the Einstein summation convention on repeated indices is adopted. We consider the following stationary linear piezoelectric system [15][16][17] :…”

Section: Two-scale Asymptotic Expansionsmentioning

confidence: 99%

Two-scale finite element method for piezoelectric problem in periodic structure

Deng

Yong

2011

Appl. Math. Mech.-Engl. Ed.

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The prediction of the mechanical and electric properties of piezoelectric fibre composites has become an active research area in recent years. By means of introducing a boundary layer problem, some new kinds of two-scale finite element methods for solutions to the electric potential and the displacement for composite material in periodic structure under the coupled piezoelectricity are derived. The coupled two-scale relation of the electric potential and the displacement is set up, and some finite element approximate estimates and numerical examples which show the effectiveness of the method are presented.

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“…Iesan [15] established the reciprocal theorem and uniqueness theorem for quasi-static and dynamic piezoelectricity avoiding the use of the Laplace transform. Yang and Batra [16] obtained a class of conservation laws for linear piezoelectric materials by using the Nother theorem in conjunction with four variational formulations of quasi-static piezoelectricity. Attempts to derive the general theorems in the linear theory of piezoelectric micropolar thermoelectricity have also been made by some authors [17][18][19] .…”

Section: Introductionmentioning

confidence: 99%

Uniqueness theorem, theorem of reciprocity, and eigenvalue problems in linear theory of porous piezoelectricity

Vashishth

Gupta

2011

Appl. Math. Mech.-Engl. Ed.

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The uniqueness theorem and the theorem of reciprocity in the linearized porous piezoelectricity are established under the assumption of positive definiteness of elastic and electric fields. General theorems in the linear theory of porous piezoelectric materials are proved for the quasi-static electric field approximation. The uniqueness theorem is also proved without using the positive definiteness of the elastic field. An eigenvalue problem associated with free vibrations of a porous piezoelectric body is studied using the abstract formulation. Some properties of operators are also proved. The problem of frequency shift due to small disturbances, based on an abstract formulation, is studied using a variational and operator approach. A perturbation analysis of a special case is also given.

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Conservation laws in linear piezoelectricity

Cited by 23 publications

References 8 publications

Nonlocal and Gradient Effects

Nonlocal and Gradient Effects

Two-scale finite element method for piezoelectric problem in periodic structure

Uniqueness theorem, theorem of reciprocity, and eigenvalue problems in linear theory of porous piezoelectricity

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