Piezoelectric study on circularly cylindrical layered media

In this paper, an analytical solution in series form for the problem of a circularly cylindrical layered piezoelectric composite consisting of N dissimilar layers is presented within the framework of linear piezoelectricity. Each layer of the composite is assumed to be transversely isotropic with respect to the longitudinal direction (x₃ direction), and the composite is subject to arbitrary electromechanical singularities infinitely extended in a direction perpendicular to the x₁–x₂ plane such that only in-plane electric fields and out-of-plane displacement are produced. The alternating technique in conjunction with the method of analytical continuation is applied to derive the general multilayered media solution in an explicit series form, whose convergence is guaranteed numerically. The distributions of the shear stress and electric field are found to be dependent on the material combinations and the magnitude and position of the electromechanical singularities. An exactly closed form solution is obtained and discussed graphically for a practical example.     

Solution of a mixed dynamic problem on the displacements given on the boundary of a half-plane lying on the surface of an isotropic homogeneous elastic half-space

We consider the problem on the motion of an isotropic elastic body occupying the half-space z ≥ 0 on whose boundary, along the half-plane x ≥ 0, the horizontal components of displacement are given, while the remaining part of the boundary is stress-free. We seek the solution by the method of integral Laplace transforms with respect to time t and Fourier transforms with respect to the coordinates x, y; the problem is reduced to a system of Wiener-Hopf equations, which can be solved by the methods of singular-integral equations and circulants. We invert the integral transforms and reduce the solution to the Smirnov-Sobolev form. We calculate the tangential stress intensity coefficients near the boundary z = 0, x = 0, |y| < ∞ of the half-plane. The circulant method for solving the Wiener-Hopf system was proposed in [1]. A static problem similar to that considered in the present paper was solved earlier. The Hilbert problem was reduced to a system of Fredholm integral equations in [2]. In the present paper, we solve the above problem by reducing the solution to quadratures and a quasiregular system of Fredholm integral equations. We give a numerical solution of the Fredholm equations and calculate the integrals for the tangential stress intensity coefficients.     

Contact problems for several transversely isotropic elastic layers on a smooth elastic half-space

The idea of generalized images, first used by the author for the case of crack problems, is applied here to solve a contact problem for n transversely isotropic elastic layers, with smooth interfaces, resting on a smooth elastic half-space, made of a different transversely isotropic material. A rigid punch of arbitrary shape is pressed against the top layer’s free surface. The governing integral equation is derived for the case of two layers; it is mathematically equivalent to that of an electrostatic problem of an infinite row of coaxial charged disks in the shape of the domain of contact. This result is then generalized for an arbitrary number of layers. As a comparison, the method of integral transforms is also used to solve the problem. The main difference of our integral transform approach with the existing ones is in separating of our half-space solution from the integral transform terms. It is shown that both methods lead to the same results, thus giving a new interpretation to the integral transform as a sum of an infinite series of generalized images.     

Asymptotic expansion of a class of integral transforms via Mellin transforms

An asymptotic expansion for large λ of functions I(λ) defined by definite integrals of the form $$I(\lambda ) = \mathop \smallint \limits_0^\infty h(\lambda t)f(t)dt$$ is obtained in the case where h(t)=O(exp(-βt ^p )) as t→∞ with β, ?>0. To obtain the expansion for such integral transforms, I(λ) is first represented as a contour integral involving M [h; z], the Mellin transform of the kernel h(t) evaluated at z, and M[f; 1-z], the Mellin transform of the function f(t) evaluated at 1-z. By assuming a rather general asymptotic expansion for f(t) near t=0, it is shown that M[f; 1-z] can be continued into the right-half plane as a meromorphic function with poles that can be located and classified. The desired asymptotic expansion of I is then obtained by systematically moving the contour in its integral representation to the right. Each term in the expansion arises as a residue contribution corresponding to a pole of M[f; 1-z]. It is then shown how the expansion, originally found for large positive λ, can be extended to complex λ. Finally several examples are considered which illustrate the scope of our expansion theorems.     

Three-dimensional Green's functions in anisotropic trimaterials

Three-dimensional elastostatic Green's functions in anisotropic trimaterials are derived, for the first time, by applying the generalized Stroh's formalism and Fourier transforms. The Green's functions are expressed as a series summation with the first term corresponding to the full-space solution and other terms to the image solutions due to the interfaces. The most remarkable feature of the present solution is that the image solutions can be expressed by a simple line integral over a finite interval [0,2π]. By partitioning the trimaterial Green's function into a full-space solution and a complementary part, the line integral involves only regular functions if the singularity is within one of the three materials, being treated analytically owning to the explicit expression of the full-space solution. When the singularity is on the interface, which occurs if the field and source points are both on the same interface, the involved singularity is handled with the interfacial Green's functions.A numerical example is presented for a trimaterial system made of two anisotropic half spaces bonded perfectly by an isotropic adhesive layer, showing clearly the effect of material layering on the Green's displacements and stresses. Furthermore, by comparing the present Green's solution to the direct (two-dimensional) 2D integral expression which is also derived in this paper, it is shown that, the computational time for the calculation of the Green's function can be substantially reduced using the present solution, instead of the direct 2D integral method.     

Static analysis of anisotropic functionally graded magneto-electro-elastic beams subjected to arbitrary loading

This paper considers the analytical and semi-analytical solutions for anisotropic functionally graded magneto-electro-elastic beams subjected to an arbitrary load, which can be expanded in terms of sinusoidal series. For the generalized plane stress problem, the stress function, electric displacement function and magnetic induction function are assumed to consist of two parts, respectively. One is a product of a trigonometric function of the longitudinal coordinate (x) and an undetermined function of the thickness coordinate (z), and the other a linear polynomial of x with unknown coefficients depending on z. The governing equations satisfied by these z-dependent functions are derived. The analytical expressions of stresses, electric displacements, magnetic induction, axial force, bending moment, shear force, average electric displacement, average magnetic induction, displacements, electric potential and magnetic potential are then deduced, with integral constants determinable from the boundary conditions. The analytical solution is derived for beam with material coefficients varying exponentially along the thickness, while the semi-analytical solution is sought by making use of the sub-layer approximation for beam with an arbitrary variation of material parameters along the thickness. The present analysis is applicable to beams with various boundary conditions at the two ends. Two numerical examples are presented for validation of the theory and illustration of the effects of certain parameters.     

Proving conditional attractivity of a closed set with a family of liapunov functions

Considering a closed set M of some x-space and a solution x(t), y(t) of a differential system x = X(x, y, t), y = Y(x, y, t), we give sufficient conditions in order that x(t) approaches M. We use several auxiliary functions and employ Salvadori's method of a one parameter family of Liapunov functions. An application is given to the two-body problem in the presence of some friction forces and when the reference frame is non-inertial.     

Piezoelastic study on singularities interacting with circular and straight interfaces

The piezoelastic investigation for a circular inclusion embedded in a sandwich has been carried out. Each medium of the composite is assumed to be transversely isotropic with hexagonal symmetry, which has an isotropic basal plane of x₁x₂-plane and a poling direction of x₃-axis. The electromechanical loadings considered in this paper include a point force and a point charge located in the middle layer of the sandwich. An efficient procedure is established by combining the analytical continuation method and alternating technique to derive the general forms of the piezoelastic fields in terms of the corresponding problem. Numerical results are provided for a number of particular examples to study the influence of material combinations, geometry, and loading condition on both the mechanical and electric response.     

On the transformation toughening of a crack along an interface between a shape memory alloy and an isotropic medium

In this study, a bilinear cohesive zone model is employed to describe the transformation toughening behavior of a slowly propagating crack along an interface between a shape memory alloy and a linear elastic or elasto-plastic isotropic material. Small scale transformation zones and plane strain conditions are assumed. The crack growth is numerically simulated within a finite element scheme and its transformation toughening is obtained by means of resistance curves. It is found that the choice of the cohesive strength t₀ and the stress intensity factor phase angle φ greatly influence the toughening behavior of the bimaterial. The presented methodology is generalized for the case of an interface crack between a fiber reinforced shape memory alloy composite and a linear elastic, isotropic material. The effect of the cohesive strength t₀, as well as the fiber volume fraction are examined.     

High accuracy wave simulation – Revised derivation,numerical analysis and testing of a nearly analytic integration discrete method for solving acoustic wave equation

The nearly analytic integration discrete (NAID) method for solving the two-dimensional acoustic wave equation has been fully mathematically revised, analyzed and tested. The NAID method is an alternative numerical modeling method for generating synthetic seismograms. The acoustic wave equation is first transformed into a system of first-order ordinary differential equations (ODEs) with respect to time variable t, and then directly integrated at a small time interval of [t_n, t_n+1] to obtain semi-discrete ordinary differential equations. The integral kernel is expanded into a truncated Taylor series, to which the integration operator is explicitly applied. The high-order temporal derivatives involved in the integral kernel are replaced by high-order spatial derivatives, which then are approximately calculated as a weighted linear combination of the displacement, the particle-velocity, and their spatial gradients. In this article, we investigate the theoretical properties of the revised NAID method, including the discrete error and the stability criteria. Numerical results for constant and layered velocity models show that, comparing to the Lax–Wendroff correction (LWC) scheme and the staggered-grid finite difference method, the NAID method can effectively suppress the numerical dispersion and source-noises caused by the discretization of the acoustic wave equation with too-coarse spatial grids or when models have strong velocity contrasts between adjacent layers. The proposed NAID method has been applied in computing the acoustic wavefields for two heterogeneous models – the corner edge model and the Marmousi model. Promising numerical results illustrate that the NAID method provides an encouraging tool for large-scale and complex wave simulation and inversion problems based on the acoustic equation.     

A new constitutive equation for the long-term creep of polymers based on physical aging

Based on the observation that during long-term creep the viscosity of polymers will continue to increase due to physical aging, a new constitutive equation is derived to describe the long-term creep behavior of polymers that are chrono-rheologically simple. The theory is developed using the concept of effective time for such materials whose long-term creep compliances with various aging times are characterized by a horizontal shift on the log(t)-scale. The derivation makes use of the basic mathematical structure for such a horizontal shift, with a result that is both sufficient and necessary. A linear viscosity function is found to be required for such a material, and the corresponding shift rate for both the long-term creep and the short-time creep is found to increase with aging time t_e, reaching an asymptotic value of unity. This theory improves Struik's (1978) classic theory for the special class of chrono-rheologically simple materials, in that, when the aging time is sufficiently long, both theories are identical, but when it is short, the present one can account for the transition to the asymptotic state. The developed effective-time theory is then extended to a polymer–matrix composite to predict the effect of physical aging on the long-term creep of a fiber-reinforced composite material.     

Quadrature formulas of maximum algebraic accuracy for cauchy type integrals with complex exponents of the Jacobi weight function

The present paper presents a Gauss type quadrature formula for a Cauchy type integral whose density is the product of a Hölder function by the weight function (1 ? x) ^α (1 + x) ^β (Re α, Reβ > ?1) of orthogonal Jacobi polynomials. It is shown that at the roots of the function of the second kind corresponding to the Jacobi polynomial P _n ^(α,β) (x), the quadrature formula with n nodes gives the exact value of a Cauchy type integral for an arbitrary polynomial of order k ≤ 2n. This formula was tested when solving several contact and mixed problems of the theory of elasticity.     

Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load qrk

This paper considers the bending of transversely isotropic circular plates with elastic compliance coefficients being arbitrary functions of the thickness coordinate, subject to a transverse load in the form of qr^k (k is zero or a finite even number). The differential equations satisfied by stress functions for the particular problem are derived. An elaborate analysis procedure is then presented to derive these stress functions, from which the analytical expressions for the axial force, bending moment and displacements are obtained through integration. The method is then applied to the problem of transversely isotropic functionally graded circular plate subject to a uniform load, illustrating the procedure to determine the integral constants from the boundary conditions. Analytical elasticity solutions are presented for simply-supported and clamped plates, and, when degenerated, they coincide with the available solutions for an isotropic homogenous plate. Two numerical examples are finally presented to show the effect of material inhomogeneity on the elastic field in FGM plates.     

Explicit expression of derivatives of elastic Green’s functions for general anisotropic materials

The analytical expressions of Green’s function and their derivatives for three-dimensional anisotropic materials are presented here. By following the Fourier integral solutions developed by Barnett [Phys. Stat. Sol. (b) 49 (1972) 741], we characterize the contour integral formulations for the derivatives into three types of integrals H, M, and N. With Cauchy’s residues theorem and the roots of a sextic equation from Stroh eigenrelation, these integrals can be solved explicitly in terms of the Stroh eigenvalues P_i (i=1,2,3) on the oblique plane whose normal is the position vector. The results of Green’s functions and stress distributions for a transversely isotropic material are discussed in this paper.     

Periodic or Bounded Solutions of Carathéodory Systems of Ordinary Differential Equations

In this paper we extend the guiding function approach to show that there are periodic or bounded solutions for first order systems of ordinary differential equations of the form x′=f(t,x), a.e. t∈[a,b], where f satisfies the Carathéodory conditions. Our results generalize recent ones of Mawhin and Ward.     

Dyadic Green's functions for cylindrical waveguides with moving media

The dyadic Green's function for cylindrical waveguides of circular or rectangular cross section with a moving, isotropic, homogeneous medium is developed using the method of eigenfunction expansion. The orthogonality properties of the vector mode functions are discussed. In contrast to waveguides with a stationary medium, it is seen that the normalization factor in the case of the E mode introduces a pole in the integral representation for the Green's function which must be excluded from the integration contour.     

The inhomogeneous strain distributions within finite cylinders under the axial point load tests and the effects on the valence band of Si1−xGex alloy

An exact solution for inhomogeneous strain and stress distributions within a finite cubic isotropic cylinder of Si_1?xGe_x alloy under the axial Point Load Strength Test (PLST) is analytically derived. Lekhnitskii’s stress function is used to uncouple the equations of equilibrium, and a new expression for the stress function is proposed so that all of the governing equations and boundary conditions are satisfied exactly. The solution for isotropic cylinders under the axial PLST is covered as a special case. Numerical results show that the strain and stress distributions in the central region within half height and radius are relatively homogeneous, but strain and stress concentrations are usually developed near the point loads. The largest tensile strain and stress are always induced along the line joining the point loads, which gives theoretical explanation why most of the cylindrical specimens are split apart along the line joining the point loads under the axial PLST. In addition, by using envelope-function method, the effect of strain on the valence-band structure of Si_1?xGe_x alloy is analyzed. It is found that strain changes the band quantum gap and the shape of constant energy surfaces of the heavy-hole and the light-hole bands of Si_1?xGe_x alloy.     

Wave Propagation Parallel to the Layers in Elastic or Viscoelastic Layered Composites

Transient waves propagating parallel to the layers in a linear elastic or viscoelastic layered composite are studied. A step load in time is applied at the boundary x = 0 and the head-of-the-pulse asymptotic solution is obtained for large x and large time t. For viscoelastic composites the interaction between the dissipation and the dispersion is controlled by a parameter γ that contains the material mismatch of the layers and the distance: propagated by the waves. As the distance increases, so does γ, and the oscillatory response diminishes. For elastic composites, we show how the oscillatory response depends on the mismatch of the material properties and the thicknesses of the layers. We show that there are composites other than the one with zero mismatch for which the oscillatory response is almost nonexistent.     

Determination of boundary layers in the plane problem for three-layer strips. Part 2

In the first part of this paper, we considered the exact statement of the plane elasticity problem in displacements for strips made of various materials (problem A, an isotropic material; problem B, an orthotropic material with 2G ₁₂ < √E ₁ E ₂; problem C, an orthotropic material with 2G ₁₂ > √E ₁ E ₂). Further, we stated and solved the boundary layer problem (the problem on a solution decaying away from the boundary) for a sandwich strip of regular structure consisting of isotropic layers (problem AA). In the present paper, we use the solution of the plane problem to consider the problem for sandwich strips of regular structure with isotropic face layers and orthotropic filler (problem AB).     

A piezoelectric material strip with a crack perpendicular to its boundary surfaces

A cracked piezoelectric material strip under combining mechanical and electrical loads is considered. The crack is vertical to the top and bottom edges of the strip. The edges of the strip are parallel to the x-axis and perpendicular to the z-axis. When a piezoelectric ceramic is poled, it exhibits transversely isotropic behavior. Among many possible poled axis orientations, a particular orientation when the poling direction lies parallel to x-axis is examined in this paper. Both impermeable crack and permeable crack assumptions are considered. Numerical results are included for three kinds of fracture mechanics specimens, namely an edge-cracked strip, a double edge-cracked strip, and a center-cracked strip, subjected to uniform tensions and uniform electric displacement loads simultaneously, at the far ends. In addition, an edge-cracked strip under pure bending and uniform electric displacement loads at the far ends is also investigated in this paper.