A refined theory of transversely isotropic piezoelectric plates

2008

Arch Appl Mech

Based on elasticity theory, various two-dimensional (2D) equations and solutions for extensional deformation have been deduced systematically and directly from the three-dimensional (3D) theory of thick rectangular plates by using the Papkovich-Neuber solution and the Lur'e method without ad hoc assumptions. These equations and solutions can be used to construct a refined theory of thick plates for extensional deformation. It is shown that the displacements and stresses of the plate can be represented by the displacements and transverse normal strain of the midplane. In the case of homogeneous boundary conditions, the exact solutions for the plate are derived, and the exact equations consist of three governing differential equations: the biharmonic equation, the shear equation, and the transcendental equation. With the present theory a solution of these can satisfy all the fundamental equations of 3D elasticity. Moreover, the refined theory of thick plate for bending deformation constructed by Cheng is improved, and some physical or mathematical explanations and proof are provided to support our justification. It is important to note that the refined theory is consistent with the decomposition theorem by Gregory. In the case of nonhomogeneous boundary conditions, the approximate governing differential equations and solutions for the plate are accurate up to the second-order terms with respect to plate thickness. The correctness of the stress assumptions in the classic plane-stress problems is revised. In an example it is shown that the exact or accurate solutions may be obtained by applying the refined theory deduced herein.

“…(27) and (28) and the shear solution of Eqs. (32) and (33), we arrive at a second-order refined theory with the two differential governing equations (26) and (34).…”

Section: The Shear Equation and The Shear Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A refined theory of elastic thick plates for extensional deformation

2008

Arch Appl Mech

“…The great complexity of the governing equations of piezoelectric materials makes it extremely difficult to obtain the solutions analytically. In recent years, however, great advances have been made in various aspects such as fracture [1,2] , inclusion [3,4] , beam and plate structures [5,6] .…”

Section: Introductionmentioning

confidence: 99%

The Remarkable Nature of Radially Symmetric Deformation of Anisotropic Piezoelectric Inclusion

Wang

2008

Acta Mech. Solida Sin.

The present paper deals with spherically symmetric deformation of an inclusionmatrix problem, which consists of an infinite isotropic matrix and a spherically uniform anisotropic piezoelectric inclusion. The interface between the two phases is supposed to be perfect and the system is subjected to uniform loadings at infinity. Exact solutions are obtained for solid spherical piezoelectric inclusion and isotropic matrix. When the system is subjected to a remote traction, analytical results show that remarkable nature exists in the spherical inclusion. It is demonstrated that an infinite stress appears at the center of the inclusion. Furthermore, a cavitation may occur at the center of the inclusion when the system is subjected to uniform tension, while a black hole may be formed at the center of the inclusion when the applied traction is uniform pressure. The appearance of different remarkable nature depends only on one non-dimensional material parameter and the type of the remote traction, while is independent of the magnitude of the traction.

“…Besides, other scholars, such as Lee [6] , Bisegna and Caruso [7] , and Krommer [8] , performed similar analyses based on classical plate theories or refined plate theories in combination with a gross linear, quadratic or biquadratic through-the-thickness distribution of the electric potential. Recently, by utilizing the general solution of piezoelasticity and Lur'e method, a refined theory for piezoelectric plates was deduced systematically and directly without ad hoc assumptions [9] .By an application of the Betti-Rayleigh reciprocal theorem, Gregory and Wan developed a decay analysis technique determining the interior solution successfully and effectively. They provided the results for several plate problems, and derived a set of correct boundary conditions for arbitrarily prescribed admissible edgedata [10][11][12][13][14][15][16][17] .…”

mentioning

confidence: 99%

mentioning

confidence: 99%

Mixed boundary conditions for piezoelectric plates

Sci. China Ser. G-Phys. Mech. Astron.

2009

For plate bending and stretching problems in piezoelectric materials, the reciprocal theorem and the general solution of piezoelasticity are applied in a novel way to obtain the appropriate mixed boundary conditions accurate to all order. A decay analysis technique is used to establish necessary conditions that the prescribed data on the edge of the plate must satisfy in order that it should generate a decaying state within the plate. For the case of axisymmetric bending and stretching of a circular plate, these decaying state conditions are obtained explicitly for the first time when the mixed conditions are imposed on the plate edge. They are then used for the correct formulation of boundary conditions for the interior solution.boundary conditions, piezoelectric plates, mixed data, bending problems, stretching problems As piezoelectric patches used as actuator or sensor are usually considered as plates because of their plate-like geometry, many scientists paid more attention to the study of piezoelectric plates. Tiersten [1] and Mindlin [2,3] initiated the study based on power series expansions of the mechanical displacements and the electric potential along the thickness of the plate and the variational principle, while Bugdayci and Bogy [4] and Lee et al. [5] applied trigonometric series representation to the research of piezoelectric plates. Besides, other scholars, such as Lee [6] , Bisegna and Caruso [7] , and Krommer [8] , performed similar analyses based on classical plate theories or refined plate theories in combination with a gross linear, quadratic or biquadratic through-the-thickness distribution of the electric potential. Recently, by utilizing the general solution of piezoelasticity and Lur'e method, a refined theory for piezoelectric plates was deduced systematically and directly without ad hoc assumptions [9] .By an application of the Betti-Rayleigh reciprocal theorem, Gregory and Wan developed a decay analysis technique determining the interior solution successfully and effectively. They provided the results for several plate problems, and derived a set of correct boundary conditions for arbitrarily prescribed admissible edgedata [10][11][12][13][14][15][16][17] . From these results, they have now explicit examples showing that the higher order accuracy offered by the governing differential equations of a higher order plate theory may not be attained unless commensurate boundary conditions are developed and used for these equations. These general results also show that, to be strictly correct, Saint-Venant's principle should generally be applied only to the leading term outer solution, i.e. the classical plate theory.Recently, relevant boundary conditions for elastic beams and piezoelectric beams have been attempted [18,19] . Moreover, Xu et al. [20] extended the model and method suggested by Gregory and Wan [11] for elastic plates to piezoelectric plates, which enables us to formulate the correct edge conditions for two-dimensional piezoelectric plate theories with the stress data. When str...