Green's functions for transversely isotropic piezoelectric multilayered half-spaces

This paper presents Green's functions for transversely isotropic piezoelectric and layered half-spaces. The surface of the half-space can be under general boundary conditions and a point source (point-force/point-charge) can be applied to the layered structure at any location. The Green's functions are obtained in terms of two systems of vector functions, combined with the propagator-matrix method. The most noticeable feature is that the homogeneous solution and propagator matrix are independent of the choice of the system of vector functions, and can therefore be treated in a unified manner. Since the physical-domain Green's functions involve improper integrals of Bessel functions, an adaptive Gauss-quadrature approach is applied to accelerate the convergence of the numerical integral. Typical numerical examples are presented for four different half-space models, and for both the spring-like and general traction-free boundary conditions. While the four half-space models are used to illustrate the effect of material stacking sequence and anisotropy, the spring-like boundary condition is chosen to show the effect of the spring constant on the Green's function solutions. In particular, it is observed that, when the spring constant is relatively large, the response curve can be completely different to that when it is small or when it is equal to zero, with the latter corresponding to the traction-free boundary condition.     

A refined theory of transversely isotropic piezoelectric plates

Summary. A refined theory for transversely isotropic piezoelectric plates is derived from the general solution of three-dimensional transversely isotropic piezoelasticity by means of Lure operator method. As a special case, the governing differential equations for transversely isotropic elastic plates are obtained directly.     

Three-dimensional piezoelectric boundary element analysis of transversely isotropic half-space

In this paper, we present a boundary element method (BEM) solution technique for studying the three-dimensional transversely-isotropic piezoelectric half-space problems. The use of mixed alternative point force solutions for half and full-space problems presented are necessary to overcome the computation difficulties especially in the calculation of the derivatives with respect to z. Infinite boundary elements are introduced to model the surface of the half-space only when stresses at the internal points are required to be evaluated. The integration over the infinite boundary elements is bounded and some limitations of the infinite element construction are relaxed. Closed-form solutions for uniformly distributed mechanical and electrical loads acting on a circular area on the surface of half-space are derived. This theoretical work serves as a good verification tool for numerical computation. In this paper, the numerical and theoretical results show good agreement. Numerical analysis via the finite element method (FEM) is also carried out using the commercial solver ANSYS. These FEM results are used to verify against the accuracy of the BEM solution. Finally, numerical results for the case of Hertzian pressure applied to an imperfect half-space are presented. The effects of the coupled mechanical–electrical influences as well as the presence of voids are examined. This work was supported by NTU Academic Research Funds. The finite element simulation using the ANSYS code was conducted by Mr. Ji Ren. Also, the authors wish to acknowledge the journal editor and anonymous reviewers for their helpful suggestions and comments leading to improvement of the paper.     

A quasi-incompressible and quasi-inextensible element formulation for transversely isotropic materials

The contribution presents a new finite element formulation for quasi-inextensible and quasi-incompressible finite hyperelastic behavior of transeversely isotropic materials and addresses its computational aspects. The material formulation is presented in purely Eulerian setting and based on the additive decomposition of the free energy function into isotropic and anisotropic parts, where the former is further decomposed into isochoric and volumetric parts. For the quasi-incompressible response, the Q1P0 element formulation is outlined briefly, where the pressure-type Lagrange multiplier and its conjugate enter the variational formulation as an extended set of variables. Using the similar argumentation, an extended Hu-Washizu–type mixed variational potential is introduced, where the volume averaged fiber stretch and fiber stress are additional field variables. Within this context, the resulting Euler-Lagrange equations and the element formulation resulting from the extended variational principle are derived. The numerical implementation exploits the underlying variational structure, leading to a canonical symmetric structure. The efficiency of the proposed approached is demonstrated through representative boundary value problems. The superiority of the proposed element formulation over the standard Q1 and Q1P0 element formulation is studied through convergence analyses. The proposed finite element formulation is modular and exhibits very robust performance for fiber reinforced elastomers in the inextensibility limit.     

Three-dimensional vibration analysis of a transversely isotropic piezoelectric cylindrical panel

Summary. In this work, based on three-dimensional piezoelectric elasticity, an exact analysis of the free vibrations of a simply supported, homogeneous, transversely isotropic cylindrical panel is presented. Three displacement potential functions are introduced so that the equations of motion and Gauss equation are uncoupled and simplified. It is noticed that a purely transverse (SH) mode is independent of piezoelectric effects and the rest of the motion. The equations for free vibration problems are further reduced to four second-order ordinary differential equations, after expanding the displacement and electric potential functions with an orthogonal series. The dispersion relations for an electrically shorted and charge free simply supported cylindrical panel with stress free edges have been obtained and discussed. A modified Bessel function solution with complex arguments is directly used for complex eigenvalues. In order to clarify the developed method and to compare the results to the existing areas, numerical examples are presented and the computed functions are illustrated graphically.     

On simple constitutive restrictions for transversely isotropic nonlinearly elastic materials and isotropic magneto-sensitive elastomers

The aim of this paper is to extend the analysis of constitutive restrictions proposed for isotropic nonlinearly elastic materials to transversely isotropic elastic solids and isotropic magneto-sensitive elastomers. These two models are considered because their more general constitutive equations, which are given as strain-energy functions depending on certain invariants, show a similar formulation. The restrictions imposed on the constitutive relations are based on different physically admissible behaviors and given in terms of inequalities referred to simply as constitutive inequalities. The general considerations studied are used to illustrate the constitutive structure of some examples.     

Analysis of viscoelastic behaviour of transversely isotropic materials

In this paper a constitutive equation to describe the mechanical behaviour of materials, reinforced with unidirectional fibres, is presented. The material behaviour of both matrix and fibres may be viscoelastic. The constitutive equation is a linear relation between the second Piola–Kirchhoff stress tensor and the Green–Lagrange strain tensor. The effective relaxation functions in the constitutive equation are composed of component relaxation functions employing the structural model of Hashin and Rosen. A two-dimensional membrane element incorporating this constitutive equation is implemented in a finite element program. The results of several calculations are presented in order to demonstrate the possibilities of the numerical tool. One calculation concerns a square membrane with a circular hole in its centre. The effect of fibre orientation on deformation and stresses will be displayed for this structure as well as for another membrane structure.     

Bifurcation of cavitation solutions for incompressible transversely isotropic hyper-elastic materials

In this paper, the bifurcation problem of void formation and growth in a solid circular cylinder, composed of an incompressible, transversely isotropic hyper-elastic material, under a uniform radial tensile boundary dead load and an axial stretch is examined. At first, the deformation of the cylinder, containing an undetermined parameter-the void radius, is given by using the condition of incompressibility of the material. Then the exact analytic formulas to determine the critical load and the bifurcation values for the parameter are obtained by solving the differential equation for the deformation function. Thus, an analytic solution for bifurcation problems in incompressible anisotropic hyper-elastic materials is obtained. The solution depends on the degree of anisotropy of the material. It shows that the bifurcation may occur locally to the right or to the left, depending on the degree of anisotropy, and the condition for the bifurcation to the right or to the left is discussed. The stress distributions subsequent to the cavitation are given and the jumping and concentration of stresses are discussed. The stability of solutions is discussed through comparison of the associated potential energies. The bifurcation to the left is a `snap cavitation'. The growth of a pre-existing void in the cylinder is also observed. The results for a similar problem in three dimensions were obtained by Polignone and Horgan.     

Explicit formulations of tangent stiffness tensors for isotropic materials

A compact explicit expression for the tangent stiffness tensor is presented. Throughout the analysis, the formulation holds for general isotropic elastic materials and does not require solving eigenvector problems. On the theoretical side, a very simple solution of a tensor equation is obtained. Then the expressions for the derivatives of general symmetric isotropic tensor functions of a symmetric tensor are developed. On the computational side, particular attention is given to the consideration of the special case, Green elastic materials, in which the strain energy does not admit a closed‐form expression in terms of principal invariants. Finally, a simple formulation of the tangent stiffness tensor for Ogden material model is supplied. Copyright © 2006 John Wiley & Sons, Ltd.     

A local thermal non-equilibrium model for transversely isotropic porothermoelastic materials

This work presents a local thermal non-equilibrium (LTNE) poroelasticity model for transversely isotropic porous media and applies the model to the problem of a cylindrical hole in an infinite porous medium subjected to convective cooling on the boundary of the cavity. The LTNE thermo-poroelasticity equations are solved using Laplace transform, and numerical examples are presented to examine the effects of LTNE and material anisotropy on the pore pressure and thermal stresses around the cavity. The results show that the thermal pore pressure and the magnitudes of thermal stresses increase significantly with an increase in the thermal expansion coefficient in the transverse direction. However, the elastic modulus anisotropy alone has only a marginal effect on the thermal pore pressure and stresses. The results also confirm that the LTNE effects become more pronounced when the convective heat transfer boundary conditions with moderate Biot numbers are considered.     

Three-dimensional extended displacement discontinuity method for vertical cracks in transversely isotropic piezoelectric media

The conventional displacement discontinuity method is extended to study a vertical crack under electrically impermeable condition, running parallel to the poling direction and normal to the plane of isotropy in three-dimensional transversely isotropic piezoelectric media. The extended Green's functions specifically for extended point displacement discontinuities are derived based on the Green's functions of extended point forces and the Somigliana identity. The hyper-singular displacement discontinuity boundary integral equations are also derived. The asymptotical behavior near the crack tips along the crack front is studied and the ordinary 1/2 singularity is obtained at the tips. The extended field intensity factors are expressed in terms of the extended displacement discontinuity on crack faces. Numerical results on the extended field intensity factors for a vertical square crack are presented using the proposed extended displacement discontinuity method.     

Orientation-dependent piezoelectric Eshelby S-tensor for a lamellar structure in a transversely isotropic medium

Summary.  This paper provides explicit expressions for the orientation-dependent piezoelectric Eshelby S-tensor with a lamellar structure in a transversely isotropic medium. Many piezoelectric or ferroelectric materials prefer to form a lamellar structure during a change of nano-scale microstructure and domain switch; such a lamellar morphology is energetically more favorable than other shapes in a diffusionless process. According to the above observation, this paper seeks to derive the Eshelby tensor in a coupled field with the consideration of lamellar structure. In general, Eshelby's tensor can be obtained through a Green's function technique, but it is usually given in an integral form. Due to the geometric simplicity of a lamellar structure, a simple explicit form of piezoelectric S-tensor can be obtained directly from the interfacial continuous conditions. In a transversely isotropic medium, the angle between the global symmetric axis and one of the local parallel to the surface of the lamellar inclusion is the key geometrical factor in the construction of the S-tensor. Two special cases with 90° and 180° orientation that bear significant relevance to domain switch in ferroelectric ceramics are discussed in detail. Received September 2, 2002; revised December 3, 2002 Published online: May 8, 2003 This work was supported by the National Science Foundation, Surface Engineering and Materials Design Program, under grant CMS-0093808.     

Green's function solutions for semi-infinite transversely isotropic materials

The Green's function problem of a semi-infinite transversely isotropic medium with the plane boundary parallel to the plane of isotropy is solved by using the potential function method. The Green's function solutions are expressed in terms of harmonic and bi-harmonic functions which are obtained by the separation of variables method. Closed form solutions for point forces applied in the interior of the medium are obtained. The present solution reduces to Sveklo's results when the point force is normal to the plane of isotropy and $\text{√(C}{}_{11}\text{C}{}_{33}\text{)-C}{}_{13}\text{-2C}{}_{44}\text{≠ 0}$. The Green's function solutions of Michell, Lekhnitzki and Hu, which deal with point forces applied at the free surface of a half-plane and $\text{√(C}{}_{11}\text{C}{}_{33}\text{)-C}{}_{13}\text{-2C}{}_{44}\text{≠ 0}$, can also be reproduced from the present approach. Furthermore, the present solution can be reduced to the results of Mindlin for semi-infinite isotropic materials by suitable substitutions of elastic constants.     

A note on penny-shaped cracks in transversely isotropic materials

This note contains some further discussion of the problem of a penny-shaped crack in a transversely isotropic solid. For uniform applied stress at infinity, the problem is solved using Eshelby's method. Particular attention is given to the interaction energy and the crack opening displacements. The results are given in a form which is convenient in the study of cracked solids.     

Boundary integral equation method for conductive cracks in two and three-dimensional transversely isotropic piezoelectric media

Using the fundamental solutions and the Somigliana identity of piezoelectric medium, the boundary integral equations are obtained for a conductive planar crack of arbitrary shape in three-dimensional transversely isotropic piezoelectric medium. The singular behaviors near the crack edge are studied by boundary integral equation approach, and the intensity factors are derived in terms of the displacement discontinuity and the electric displacement boundary value sum near the crack edge on crack faces. The boundary integral equations for two dimensional crack problems are deduced as a special case of infinite strip planar crack. Based on the analogy of the obtained boundary integral equations and those for cracks in conventional isotropic elastic material and for contact problem of half-space under the action of a rigid punch, an analysis method is proposed. As an example, the solution to conductive Griffith crack is derived.     

Unique real‐variable expressions of displacement and traction fundamental solutions covering all transversely isotropic elastic materials for 3D BEM

A general, efficient and robust boundary element method (BEM) formulation for the numerical solution of three‐dimensional linear elastic problems in transversely isotropic solids is developed in the present work. The BEM formulation is based on the closed‐form real‐variable expressions of the fundamental solution in displacements U_ik and in tractions T_ik, originated by a unit point force, valid for any combination of material properties and for any orientation of the radius vector between the source and field points. A compact expression of this kind for U_ik was introduced by Ting and Lee (Q. J. Mech. Appl. Math. 1997; 50 :407–426) in terms of the Stroh eigenvalues on the oblique plane normal to the radius vector. Working from this expression of U_ik, and after a revision of their final formula, a new approach (based on the application of the rotational symmetry of the material) for deducing the derivative kernel U_{ik, j} and the corresponding stress kernel Σ_ijk and traction kernel T_ik has been developed in the present work. These expressions of U_ik, U_{ik, j}, Σ_ijk and T_ik do not suffer from the difficulties of some previous expressions, obtained by other authors in different ways, with complex‐valued functions appearing for some combinations of material parameters and/or with division by zero for the radius vector at the rotational‐symmetry axis. The expressions of U_ik, U_{ik, j}, Σ_ijk and T_ik have been presented in a form suitable for an efficient computational implementation. The correctness of these expressions and of their implementation in a three‐dimensional collocational BEM code has been tested numerically by solving problems with known analytical solutions for different classes of transversely isotropic materials. Copyright © 2007 John Wiley & Sons, Ltd.     

The Bauschinger effect in transversely isotropic materials with translational strain-hardening

The dependence of the Bauschinger effect measure on the plastic strain value in tension in the direction of the axis of symmetry and in the orthogonal direction is investigated within the framework of the model of a transversely isotropic material with translational strain-hardening. A numerical example of determining the Bauschinger effect measure in martensitic 28Kh3SNMVFA steel is presented. The anisotropy of the yield strength in the region of small plastic strains is shown to be accompanied by a significant anisotropy of the Bauschinger effect measure.     

Interaction of antiplane cracks with elastic waves in transversely isotropic materials

Summary The interaction of plane time-harmonic SH-waves with micro-cracks in transversely isotropic materials is investigated. Elastic wave scattering by a single micro-crack is first analyzed. The scattered displacement is expressed as a Fourier integral containing the crack opening displacement. By using this representation formula and by invoking the traction-free boundary condition on the faces of the crack, a boundary integral equation for the unknown crack opening displacement is obtained. Expanding the crack opening displacement into a series of Chebyshev polynomials and adopting a Galerkin method, the boundary integral equation is converted into an infinite system of inear algebraic equations for the expansion coefficients which is solved numerically. Numerical results are presented for the elastodynamic stress intensity factors, the scattered far-field and the scattering cross section of a single crack. Then, propagation of plane time-harmonic SH-waves in a transversely isotropicmaterial permeated by a random and dilute distribution of micro-cracks is investigated. The effects of the micro-crack density on the attenuation coefficient and the phase velocity are analyzed by appealing to a simple energy consideration and by using Kramers-Kronig relations.