Zhen-Qiang Cheng

Three-dimensional asymptotic approach to inhomogeneous and laminated piezoelectric plates

An asymptotic theory is developed for anisotropic inhomogeneous and laminated piezoelectric plates on the basis of three-dimensional linear piezoelectricity. The inhomogeneity is assumed in the thickness direction and includes the important piezoelectric laminates as a special case. Through asymptotic expansions, the resulting two-dimensional differential equations are of the same form for each order, with different nonhomogeneous terms being determined systematically by preceding-order solutions. The governing equations of the leading-order, when degenerated to pure elasticity, are shown to be the same as those for equivalent classical thin elastic plates. The proposed methodology is illustrated by considering a rectangular piezoelectric plate subject to both mechanical and electric loadings with its edges simply supported and grounded. A three-dimensional solution for the fully electromechanically coupled problem is obtained by successively solving the two-dimensional field equations from the leading order to higher orders. Excellent agreement is observed with established results and new results are presented, from which significant physical insights are discussed.

Micropolar elastic fields due to a spherical inclusion

Abstract After introducing two concepts of asymmetric eigenstrain and eigentorsion, a general theory of homogeneous isotropic centrosymmetric micropolar media with defects is derived with the help of Greens function technique. In particular, an inclusion problem for micropolar elasticity is investigated. By use of Greens functions an exact closed-form solution is presented for the case of a spherical inclusion embedded in an infinite Cosserat medium. The Eshelby tensors for the inside and outside micropolar fields of the spherical inclusion are defined and determined. It is confirmed that the classical Eshelby tensor is obtained as a special case of the micropolar Eshelby tensors.

Influence of imperfect interfaces on bending and vibration of laminated composite shells

This paper is devoted to modeling elastic behavior of laminated composite shells, with special emphasis on incorporating interfacial imperfection. The conditions of imposing traction continuity and displacement jump across each interface are used to model imperfect interfaces. Vanishing transverse shear stresses on two free surfaces of a shell eliminate the need for shear correction factors. A linear theory underlying elastostatics and kinetics of laminated composite shells in a general configuration is presented from Hamiltons principle. In the special case of vanishing interfacial parameters, this theory reduces to the conventional third-order zigzag theory for perfectly bonded laminated shells. Numerical results for bending and vibration problems of laminated circular cylindrical panels are tabulated and plotted to indicate the influence of the interfacial imperfection

Micropolar elastic fields due to a circular cylindrical inclusion

As a fundamental element of a systematic study on micromechanics of micropolar media with defects, this paper is concerned with a micropolar inclusion problem for the typical case of an infinitely long circular cylindrical inclusion. The micropolar Eshelby tensors, as previously defined by Cheng and He [Int. J. Engng Sci., 1995, 33, 389] are obtained in an exact closed form for the problem. It is observed that the micropolar Eshelby tensors are size-dependent both for the inside and for the outside of the circular cylinder. As a limit, where classical elasticity is degenerated from micropolar elasticity, the classical Eshelby tensor is recovered as a special case of the micropolar Eshelby tensors. The Colonnettis theorem in classical elasticity is extended to micropolar elasticity and the elastic strain energy caused by a circular cylindrical inclusion is presented.

Three-dimensional exact solution for inhomogeneous and laminated piezoelectric plates

This paper presents an exact three-dimensional method of solution for the distribution of mechanical and electric quantities in interior points of a symmetrically inhomogeneous and laminated piezoelectric plate in the framework of linear theory of piezoelectricity. A transfer matrix method and asymptotic expansions are used as the elements of the formulation. The full three-dimensional electroelasticity solution is generated from a set of two-dimensional differential equations on the midplane which, when degenerated to pure elasticity, are found to be the same as those for an equivalent classical elastic plate model. As an illustrative example, the analysis considers a rigidly clamped, elliptic monoclinic piezoelectric plate which is inhomogeneous through the thickness but symmetric with respect to the midplane and under uniform normal loading both mechanically and electrically. Excluding boundary layer effects, an exact closed-form analytical solution for the fully electromechanically coupled problem is developed as a consequence of the asymptotic expansions termination after a few terms. Apart from some important properties already noted for a purely elastic plate in the same configuration, this exact solution reveals that certain significant electric quantities have been unjustifiably oversimplified by some of the existing piezoelectric plate models.

Modelling of weakly bonded laminated composite plates at large deflections

Abstract Based on a general representation of displacement variation through the thickness of laminated plates, a Karman type nonlinear theory of laminated composite plates with weakened interfacial bonding is developed. Each weakly bonded interface is modelled by a spring-layer model which has recently been used efficiently in the field of micromechanics of composites. This spring-layer model allows for a discontinuous distribution of displacements, but requires the tractions to be continuous across each interface of adjacent layers. The set of governing equations has variable coefficients in the most general form of bonding and includes conventional third-order zigzag nonlinear theory of Karman type for laminated composite plates as a special case when extreme values of interface parameters are used. Some simple numerical examples allowing for a closed-form solution are presented to give an understanding of how a small amount of interfacial weakness affects the overall and local behaviour of laminated composite plates. These include the important practical problem of reduced interface stresses due to weakened interfacial bonding, which can be predicted by the theory presented herein.

Nonlinear Theory for Composite Laminated Shells With Interfacial Damage

Interfacial damage is incorporated in the proposed nonlinear theory for composite laminated shells. A spring-layer model is employed to characterize damaged interfaces spanning from perfect bonding to different degrees of imperfect bonding in shear. By enforcing compatibility conditions for transverse shear stresses both at interfaces and on two bounding surfaces of a laminated shell, only five unknowns are needed for modeling its behavior. The principle of virtual work is used to derive the governing equations, which are of 14th order in lines of curvature coordinates, variationally self-consistent with seven prescribed boundary conditions. This theory includes the conventional higher-order zigzag model for a perfectly bonded shell as a special case. Numerical results provide a physical understanding of the effect of interracial damage on bending and buckling responses of composite laminated shells.

Prestressed composite laminates featuring interlaminar imperfection

The third-order zigzag displacement model is improved to include inplane displacement jumps across each layer interface of composite laminates to enable interlaminar imperfection to be incorporated. The imperfection is characterized by a linear spring-layer model which includes perfect bonding as a special case. The principle of virtual work is used to derive a boundary value formulation for laminated composite plates initially in a prestress state. Bending, buckling and vibration problems are studied for the case of rectangular cross-ply laminated plates for illustrative purpose.

Nonlinear flexural vibration of rectangular moderately thick plates and sandwich plates

Abstract Nonlinear flexural vibration is investigated for rectangular Reissner moderately thick plates and sandwich plates. The fundamental equations and boundary conditions are expressed in unified dimensionless form for rectangular moderately thick plates and sandwich plates. Highly accurate solutions of series form with many different movable and immovable boundary conditions, especially with unsymmetrical boundary conditions, are obtained by means of the method of harmonic balance and by developing a new technique of mixed Fourier series in nonlinear analysis. The nonlinear partial differential equations are reduced to an infinite set of simultaneous nonlinear algebraic equations, which are truncated in numerical computations. Solutions of the nonlinear fundamental frequency of rectangular plates are obtained by iteration. The multimode approach includes not only the influences of transverse shearing deformation and rotatory inertia, but also the coupling effect of vibrating modes on the nonlinear fundamental frequency. The present solutions are satisfactory in comparison with other available results.

Exact bending solution of inhomogeneous plates from homogeneous thin-plate deflection

Based on the first-order shear deformation theory, an exact solution for sandwich plates with dissimilar facings by the use of the potential function technique is presented. The solution is expressed in terms of the deflection of a homogeneous Kirchhoff thin plate and is valid for simply supported polygonal plates made of isotropic materials. Because of its mathematical similarity, the solution for functionally graded plates is also presented.