H.J. Ding

The transient responses of piezoelectric hollow cylinders for axisymmetric plane strain problems

By virtue of the separation of variables technique, the axisymmetric plane strain electroelastic dynamic problem of hollow cylinder is transferred to an integral equation about a function with respect to time, which can be solved successfully by means of the interpolation method. Then the solution of the displacements, stresses, electric displacements and electric potentials are finally obtained. The present method is suitable for the hollow cylinder with arbitrary thickness subjected to arbitrary mechanical and electrical loads. Numerical results are also presented.

A solution of a non-homogeneous orthotropic cylindrical shell for axisymmetric plane strain dynamic thermoelastic problems

A solution of a non-homogeneous orthotropic elastic cylindrical shell for axisymmetric plane strain dynamic thermoelastic problems is developed. Firstly, a new dependent variable is introduced to rewrite the governing equation, the boundary conditions as well as the initial conditions. Secondly, a special function is introduced to transform the inhomogeneous boundary conditions to the homogeneous ones. Then by virtue of the orthogonal expansion technique, the equation with respect to the time variable is derived, of which the solution can be obtained. The displacement solution is finally presented, which can degenerate in a rather straightforward way to the solution for a homogeneous orthotropic cylindrical shell and isotropic solid cylinder as well as that for a non-homogeneous isotropic cylindrical shell. Using the present method, integral transform can be avoided. It is fit for a cylindrical shell with arbitrary thickness subjected to arbitrary thermal loads. It is also very convenient to deal with dynamic thermoelastic problems for different boundary conditions. Besides, the numerical calculation involved is very easy to be performed. Several examples are presented.

Piezoelasticity solutions for functionally graded piezoelectric beams

This paper considers the plane stress problem of generally anisotropic piezoelectric beams with the coefficients of elastic compliance, piezoelectric and dielectric impermeability being arbitrary functions of the thickness coordinate. Firstly, the partial differential equations for the plane problem of anisotropic functionally graded piezoelectric materials are derived, which the stress function and electric displacement function satisfy. Secondly, the stress and electric displacement functions are assumed in forms of polynomials of the longitudinal coordinate, so that the stress and electric displacement functions can be acquired through successive integrations. The analytical expressions of axial force, bending moment, shear force, displacements, electric displacements and electric potential are then deduced. Thirdly, the stress and electric displacement functions are employed to solve problems of functionally graded piezoelectric plane beams, with the integral constants completely determined from boundary conditions. Two piezoelasticity solutions are thus obtained, for cantilever beams subjected to shear force and point charge applied at the free end, for cantilever beams subjected to uniform load. These solutions can be easily degenerated into the piezoelasticity solutions for homogeneous anisotropic piezoelectric beams. Finally, a numerical example is presented to show the application of the proposed method to a specific case.

The exact elasto-electric field of a rotating piezoceramic spherical shell with a functionally graded property

A displacement separation technique is employed to simplify the basic equations of a piezoceramic body with radial inhomogeneity. It is shown that the controlling equations are finally reduced to an uncoupled second-order ordinary differential equation and a coupled system of three second-order ordinary differential equations. Solutions to these differential equations are given for the case that material constants are of power functions of the radial coordinate. The static analysis of a steadily rotating spherical shell is then presented.

Free vibration of a fluid-filled hollow sphere of a functionally graded material with spherical isotropy

An exact, three-dimensional method is developed in the paper to analyze the free vibration of a spherically isotropic hollow sphere made of a functionally graded material and filled with a compressible fluid medium. The material is assumed to be inhomogeneous along the radial direction. By introducing three displacement functions and employing the function expansion method, the governing equations are simplified to an uncoupled second-order ordinary differential equation, and a coupled system of two such equations. Solutions to these equations are given when the elastic constants and the mass density are power functions of the radial direction. To investigate the effect of material gradient on the natural frequencies, numerical calculations are finally performed.

Analytical solution for the axisymmetric plane strain electroelastic dynamics of a special non-homogeneous piezoelectric hollow cylinder

By virtue of the introduction of a dependent variable and the separation of variables technique, the axisymmetric plane strain electroelastic dynamic problem of a special non-homogeneous piezoelectric hollow cylinder is transformed to a Volterra integral equation of the second kind about a function with respect to time, which can be solved successfully by means of the interpolation method. Then the solutions of displacements, stresses, electric displacements and electric potential are obtained. The present method is suitable for a piezoelectric hollow cylinder with an arbitrary thickness subjected to arbitrary mechanical and electrical loads. Numerical results are finally presented.

Analytical solution of a special non-homogeneous pyroelectric hollow cylinder for piezothermoelastic axisymmetric plane strain dynamic problems

By virtue of the introduction of a dependant variable and the separation of variables technique, the piezothermoelastic axisymmetric plane strain dynamic problem of a special non-homogeneous pyroelectric hollow cylinder is transformed to a Volterra integral equation of the second kind about a function with respect to time, which can be solved successfully by means of the interpolation method. Then the solutions of displacements, stresses, electric displacements and electric potential are obtained. The present method is suitable for a non-homogeneous pyroelectric hollow cylinder with an arbitrary thickness subjected to arbitrary axisymmetric thermal loads. Numerical results are presented graphically.

Natural frequencies of a fluid-filled anisotropic spherical shell

In this paper, the general nonaxisymmetric free vibration of a spherically isotropic elastic spherical shell filled with a compressible fluid medium is investigated. To this end, the three-dimensional elasticity solution method recently developed by the authors [H. J. Ding and W. Q. Chen, Int. J. Solids Struct. 33, 2575–2590 (1996)] is employed. The effect of fluid is considered by introducing a relation between the normal displacement and the normal stress of the shell at the inner spherical surface. It is shown that the coupled vibration can be divided into two independent classes as in the case of empty spherical shell. The exact three-dimensional frequency equations are then derived. As the exact elasticity solution can serve as a benchmark to check various approximate theories, frequency equations of three typical shell theories are also presented. Numerical examples are given and comparisons between four theories are made.

Analysis of functionally graded and laminated piezoelectric cantilever actuators subjected to constant voltage

Functionally graded and laminated piezoelectric cantilever actuators are investigated. Each material parameter of the functionally graded actuator can be an arbitrary continuous function of the thickness coordinate of the beam, while the property of each layer in the laminated actuator is uniform. Piezoelectricity solutions for the two actuators subjected to a constant electric potential difference are presented. Firstly, the partial differential equations for the plane problem of functionally graded piezoelectric materials, which govern the stress function and electric displacement function, are derived. Secondly, the stress function is assumed to be an undetermined function of the thickness coordinate, and the electric displacement function is assumed as a linear function of the longitudinal coordinate. In such a case, the stress and electric displacement function can be acquired through successive integrations. The analytical expressions of axial force, bending moment, shear force, displacements, electric displacements and electric potential are then deduced. The analytical solutions are finally obtained, with the integral constants completely determined from the boundary conditions. Comparisons of the present analytical solutions with beam theory, finite element method and experiments indicate that the analytical solutions are effective and exact, while certain deviations of the beam theory can be found.

Dynamic response of a pyroelectric hollow sphere under radial deformation

The dynamic response of a pyroelectric hollow sphere under radial deformation is considered. By virtue of the separation of variables method, it is transformed to a second kind Volterra integral equation about a function with respect to time, which can be solved by using the interpolation method. Then the expressions for displacements, stresses, electric displacements and electric potential can be obtained. The present method is suitable for the hollow sphere with an arbitrary thickness subjected to arbitrary symmetric thermal loads. Numerical results of thermally shocked hollow sphere are also presented.