Saint-Venant end effects for plane deformations of linear piezoelectric solids

The recent developments in smart structures technology have stimulated renewed interest in the fundamental theory and applications of linear piezoelectricity. In this paper, we investigate the decay of Saint-Venant end effects for plane deformations of a piezoelectric semi-infinite strip. First of all, we develop the theory of plane deformations for a general anisotropic linear piezoelectric solid. Just as in the mechanical case, not all linear homogeneous anisotropic piezoelectric cylindrical solids will sustain a non-trivial state of plane deformation. The governing system of four second-order partial differential equations for the two in-plane displacements and electric potential are overdetermined in general. Sufficient conditions on the elastic and piezoelectric constants are established that do allow for a state of plane deformation. The resulting traction boundary-value problem with prescribed surface charge is an oblique derivative boundary-value problem for a coupled elliptic system of three second-order partial differential equations. The special case of a piezoelectric material transversely isotropic about the poling axis is then considered. Thus the results are valid for the hexagonal crystal class 6mm. The geometry is then specialized to be a two-dimensional semi-infinite strip and the poling axis is the axis transverse to the longitudinal direction. We consider such a strip with sides traction-free, subject to zero surface charge and self-equilibrated conditions at the end and with tractions and surface charge assumed to decay to zero as the axial variable tends to infinity. A formulation of the problem in terms of an Airy-type stress function and an induction function is adopted. The governing partial differential equations are a coupled system of a fourth and third-order equation for these two functions. On seeking solutions that exponentially decay in the axial direction one obtains an eigenvalue problem for a coupled system of fourth and second-order ordinary differential equations. This problem is the piezoelectric analog of the well-known eigenvalue problem arising in the case of an anisotropic elastic strip. It is shown that the problem can be uncoupled to an eigenvalue problem for a single sixth-order ordinary differential equation with complex eigenvalues characterized as roots of transcendental equations governing symmetric and anti-symmetric deformations and electric fields. Assuming completeness of the eigenfunctions, the rate of decay of end effects is then given by the real part of the eigenvalue with smallest positive real part. Numerical results are given for PZT-5H, PZT-5, PZT-4 and Ceramic-B. It is shown that end effects for plane deformations of these piezoceramics penetrate further into the strip than their counterparts for purely elastic isotropic materials.     

End effects for plane deformations of an elastic anisotropic semi-infinite strip

In the linear theory of elasticity, Saint-Venant's principle is used to justify the neglect of edge effects when determining stresses in a body. For isotropic materials, the validity of this is well established. However for anisotropic and composite materials, experimental results have shown that edge effects may persist much farther into the material than for isotropic materials and as a result cannot be neglected. This paper further examines the effects of material anisotropy on the exponential decay rate for stresses in a semi-infinite elastic strip. A linearly elastic semi-infinite strip in a state of plane stress/strain subject to a self-equilibrated end load is considered first for a specially orthotropic material and then for the general anisotropic material. The problem is governed by a fourth-order elliptic partial differential equation with constant coefficients. In the former case, just a single dimensionless material parameter appears, while in the latter, only three dimensionless parameters are required. Energy methods are used to establish lower bounds on the actual stress decay rate. Both analytic and numerical estimates are obtained in terms of the elastic constants of the material and results are shown for several contemporary engineering materials. When compared with the exact stress decay rate computed numerically from the eigenvalues of a fourth-order ordinary differential equation, the results in some cases show a high degree of accuracy. In particular, for strongly orthotropic materials, an asymptotic estimate provides extremely accurate estimates for the decay rate. Results of the type obtained here have several important practical applications. For example, they provide physical insight into the mechanical testing of anisotropic and laminated composite structures (including the off-axis tension test), are useful in assessing the influence of fasteners, joints, etc. on the behavior of composite structures and allow for tailoring a material with specific properties to ensure that local stresses attenuate at a desired rate.     

End Effects in Anti-plane Shear for an Inhomogeneous Isotropic Linearly Elastic Semi-infinite Strip

The purpose of this research is to further investigate the effects of material inhomogeneity on the decay of Saint-Venant end effects in linear isotropic elasticity. This is carried out within the context of anti-plane shear deformations of an inhomogeneous isotropic elastic solid. The mathematical issues involve the effects of spatial inhomogeneity on the decay rates of solutions to Dirichlet or Neumann boundary-value problems for a second-order linear elliptic partial differential equation with variable coefficients on a semi-infinite strip. In previous work [1], the elastic coefficients were assumed to be smooth functions of the transverse coordinate so that the material was inhomogeneous in the lateral direction only. Here we develop a new technique, based on a change of variable, to study generally inhomogeneous isotropic materials. The governing partial differential equation is transformed to a Helmholtz equation with a variable coefficient, which facilitates analysis of the influence of material inhomogeneity on the diffusion of end effects. For certain classes of inhomogeneous materials, an explicit optimal decay estimate is established. The results of this paper are applicable to continuously inhomogeneous materials and, in particular, to functionally graded materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

Saint-Venant Decay Rates for an Isotropic Inhomogeneous Linearly Elastic Solid in Anti-Plane Shear

The purpose of this research is to investigate the effects of material inhomogeneity on the decay of Saint-Venant end effects in linear isotropic elasticity. This question is addressed within the context of anti-plane shear deformations of an inhomogeneous isotropic elastic solid. The mathematical issues involve the effects of spatial inhomogeneity on the decay rates of solutions to Dirichlet or Neumann boundary-value problems for a second-order linear elliptic partial differential equation with variable coefficients on a semi-infinite strip. The elastic coefficients are assumed to be smooth functions of the transverse coordinate. The estimated rate of exponential decay with distance from the loaded end (a lower bound for the exact rate of decay) is characterized in terms of the smallest positive eigenvalue of a Sturm–Liouville problem with variable coefficients. Analytic lower bounds for this eigenvalue are used to obtain the desired estimated decay rates. Numerical techniques are also employed to assess the accuracy of the analytic results. A related eigenvalue optimization question is discussed and its implications for the issue of material tailoring is addressed. The results of this paper are applicable to continuously inhomogeneous materials and, in particular, to functionally graded materials. This revised version was published online in August 2006 with corrections to the Cover Date.     

The method of geometrical optics for differential equations of the fourth order as applied to low-frequency plasma oscillations

The theory of osculations of a spatially inhomogeneous plasma [1] draws substantially on the theory of geometrical optics as applied to differential equations of the second order. The theory of asymptotic solutions for equations of the second order has now been thoroughly developed [2]. The quasi-classical quantization rules determining the spectrum of eigenvalues of such equations are written in the form of the well-known Bohr -Sommerfeld integrals [3]. However, in analyzing the spectrum of oscillations of an inhomogeneous plasma it is insufficient in many cases to confine oneself to equations of the second order. For example, in an inhomogeneous magnetoactive plasma, even when the thermal motion of the particles is neglected, the field equations, generally speaking, reduce to a differential equation of the fourth order. Equations of the fourth order also arise in investigating the stability of the hydrodynamical flow of a viscous fluid [4].Certain special forms of fourth-order equations were studied in [4–6]. The authors of [6] obtained a quasi-classical quantization rule for equations of the fourth order with a small parameter associated with the leading derivative. The present paper investigates the general fourth-order equation with real coefficients. Asymptotic solutions of such an equation are obtained with an accuracy to terms of the first order in the approximation of geometrical optics, and quasi-classical quantization rules are established for various concrete cases. Using the theory thus developed, a new spectrum of oscillations is determined, characteristic only for an inhomogeneous plasma in a magnetic field.In conclusion, the authors express their gratitude to V. P. Silin who suggested the idea of matching the quasi-classical solutions, and also to Yu. N. Dnestrovskii and D. P. Kostomarova for discussing the paper and offering valuable criticism.     

Spatial Decay Estimates for a Coupled System of Second-Order Quasilinear Partial Differential Equations Arising in Thermoelastic Finite Anti-plane Shear

The spatial decay behavior of solutions of a coupled system of second-order quasilinear partial differential equations, in divergence form, defined on a two-dimensional semi-infinite strip, is investigated. Such equations arise in the theory of anti-plane shear deformations for isotropic nonlinearly thermoelastic solids. Differential inequality techniques are employed to obtain exponential decay estimates. The results are illustrated by several examples. The results are relevant to Saint-Venant principles for nonlinear thermoelasticity as well as to theorems of Phragmen-Lindelof type. This revised version was published online in August 2006 with corrections to the Cover Date.     

Formulation of Problems in the General Kirchhoff—Love Theory of Inhomogeneous Anisotropic Plates

In this paper we study the procedure of reducing the three-dimensional problem of elasticity theory for a thin inhomogeneous anisotropic plate to a two-dimensional problem in the median plane. The plate is in equilibrium under the action of volume and surface forces of general form. À notion of internal force factors is introduced. The equations for force factors (the equilibrium equations in the median plane) are obtained from the thickness-averaged three-dimensional equations of elasticity theory. In order to establish the relation between the internal force factors and the characteristics of the deformed middle surface, we use some prior assumptions on the distribution of displacements along the thickness of the plate. To arrange these assumptions in order, the displacements of plate points are expanded into Taylor series in the transverse coordinate with consideration of the physical hypotheses on the deformation of a material fiber being originally perpendicular to the median plane. The well-known Kirchhoff—Love hypothesis is considered in detail. À closed system of equations for the theory of inhomogeneous anisotropic plates is obtained on the basis of the Kirchhoff—Love hypothesis. The boundary conditions are formulated from the Lagrange variational principle.     

Saint-Venant End Effects in Anti-plane Shear for Classes of Linear Piezoelectric Materials

This paper is concerned with further investigation of the effect of mechanical/electrical coupling on the decay of Saint-Venant end effects in linear piezoelectricity. Saint-Venant's principle and related results for elasticity theory have received considerable attention in the literature but relatively little is known about analogous issues in piezoelectricity. The current rapidly developing smart structures technology provides motivation for the investigation of such problems. The decay of Saint-Venant end effects is investigated in the context of anti-plane shear deformations for linear homogeneous piezoelectric solids. For a rather general class of anisotropic piezoelectric materials, the governing partial differential equations of equilibrium are a coupled system of second-order partial differential equations for the mechanical displacement u and electric potential ?. The traction boundary-value problem with prescribed surface charge can be formulated as an oblique derivative boundary-value problem for this elliptic system. Energy-decay estimates using differential inequality methods are used to study the axial decay of solutions on a semi-infinite strip subjected to non-zero boundary conditions only at the near end. This analysis is carried out for a rather general class of materials (the tetragonal ${\bar 4}$ crystal class). The boundary-value problem involves a full coupling of mechanical and electrical effects. There are four independent material constants appearing in the problem. An explicit estimated decay rate (a lower bound for the actual decay rate) is obtained in terms of two dimensionless piezoelectric parameters d ₀,r, the first of which provides a measure of the degree of piezoelectric coupling. The estimated decay rate is shown to be monotone decreasing with increasing values of the coupling parameter d ₀. In the limit as d ₀→0, we recover the exact decay rate for the purely mechanical case. Thus, for the tetragonal ${\bar 4}$ class of materials, piezoelectric end effects are predicted to penetrate further into the strip than their elastic counterparts, confirming recent results obtained in other contexts in linear piezoelectricity.     

Some extensions of the initial value problem for second order evolution equations

The initial problem for second order linear evolution equation systems is discussed by using the contraction semigroup theory. A kind of initial value problem for second order is also discussed with variable coefficients for evolution equations by using the analytical semigroup theory, and is unified with the solutions of the initial value problem for this class of equations and those of first order temporally inhomogeneous evolution equations. This is an important class of equations in mathematical mechanics.     

Cauchy Problem and Exponential Stability for the Inhomogeneous Landau Equation

This work deals with the inhomogeneous Landau equation on the torus in the cases of hard, Maxwellian and moderately soft potentials. We first investigate the linearized equation and we prove exponential decay estimates for the associated semigroup. We then turn to the nonlinear equation and we use the linearized semigroup decay in order to construct solutions in a close-to-equilibrium setting. Finally, we prove an exponential stability for such a solution, with a rate as close as we want to the optimal rate given by the semigroup decay.     

Finite-amplitude acoustic pulses in a waveguide layer with longitudinal inhomogeneity

This paper considers the propagation of a weakly nonlinear acoustic pulse in a slightly curved waveguide layer which is strongly inhomogeneous in the transverse direction and weakly inhomogeneous in the longitudinal direction. The basic system of hydrodynamic equations reduces to a nonlinear wave equation, whose coefficients are determined using the equations of state of the medium. It is established that as the adiabatic exponent passes through the value γ = 3/2, the nature of the pulse propagation changes: for large values of γ, the medium is focusing, and for smaller values, it is defocusing. It is shown that the pulse propagation process is characterized by three scales: the high-frequency filling is modulated by the envelope, whose evolution, in turn, is determined by the moderate-rate evolution of the envelope phase and slow amplitude variation. A generalized nonlinear Schrödinger equation with the coefficients dependent on the longitudinal coordinate is derived for the pulse envelope. An explicit soliton solution of this equation is constructed for some types of longitudinal inhomogeneity.     

The plane self-similar anisotropic and angularly inhomogeneous wedge under power law tractions and the asymptotic analysis of the stress field

In this study the generally anisotropic and angularly inhomogeneous wedge, under power law tractions of order n of the radial coordinate r at its external faces is considered. At first, using variable separable relations in the equilibrium equations, the strain–stress relations and the strain compatibility equation, a differential system of equations is constructed and investigated. Decoupling this system, an ordinary differential equation is derived and the stress and displacement fields may be determined. The proposed procedure is also applied to the elastostatic problem of an isotropic and angularly inhomogeneous wedge. In the sequel William's asymptotic analysis in the case of angular inhomogeneity is examined. Finally, applications for the case of an angularly inhomogeneous wedge-shape dam and for the asymptotic procedure in an isotropic wedge with angularly varying shear modulus, are made.     

Surface waves in an exponentially graded, general anisotropic elastic material under the influence of gravity

In a recent paper Destrade [1] studied surface waves in an exponentially graded orthotropic elastic material. He showed that the quartic equation for the Stroh eigenvalue p is, after properly modified, a quadratic equation in p² with real coefficients. He also showed that the displacement and the stress decay at different rates with the depth x₂ of the half-space. Vinh and Seriani [2] considered the same problem and added the influence of gravity on surface waves. In this paper we generalize the problem to exponentially graded general anisotropic elastic materials. We prove that the coefficients of the sextic equation for p remain real and that the different decay rates for the displacement and the stress hold also for general anisotropic materials. A surface wave exists in the graded material under the influence of gravity if a surface wave can propagate in the homogeneous material without the influence of gravity in which the material parameters are taken at the surface of the graded half-space. As the wave number k → ∞, the surface wave speed approaches the surface wave speed for the homogeneous material. A new matrix differential equation for surface waves in an arbitrarily graded anisotropic elastic material under the influence of gravity is presented. Finally we discuss the existence of one-component surface waves in the exponentially graded anisotropic elastic material with or without the influence of gravity.     

On modeling the aggregate response of laminated thermoelastic media

Sufficient conditions are given that determine when solutions to general boundary value problems for inhomogeneous anisotropic laminated thermoelastic media are approximated, in the average, by solution of constant coefficient equations. These appropriate constant coefficients are computed and shown to be independent of the shape of the body and the loading on the body.     

Equivalent Diffusion in a Thin Film Flow with a Non-One-Dimensional Velocity Field and Anisotropic Diffusion Tensor

For describing the mass transfer processes in channels, Taylor's dispersion theory is widely used. This theory makes it possible, with asymptotic rigor, to replace the complete diffusion (heat conduction) equation with a convective term that depends on the coordinate transverse to the flow by an effective diffusion (dispersion) equation with constant coefficients, averaged over the channel cross-section. In numerous subsequent studies, Taylor's theory was generalized to include more complex situations, and novel algorithms for constructing the dispersion equations were proposed. For thin film flows a theory similar to Taylor's leads to a matrix of dispersion coefficients.In this study, Taylor's theory is extended to film flows with a non-one-dimensional velocity field and anisotropic diffusion tensor. These characteristics also depend to a considerable extent on the spatial coordinates and time. The dispersion equations obtained can be simplified in regions in which the effective diffusion coefficient tensor changes sharply.     

Saint-Venant decay rates for an anisotropic and non-homogeneous mixture of elastic solids in anti-plane shear

The purpose of this research is to investigate the influence of material inhomogeneity and anisotropy on the decay of Saint-Venant end effects in anti-plane shear deformations of linear mixtures of elastic solids. The spatial decay of solutions of a boundary value problem with variable coefficients on a semi-infinite strip is investigated. The results may be interpreted in terms of a Saint-Venant principle for anti-plane shear deformations of linear anisotropic mixtures of elastic solids. As our first results have a very general point of view, we study some examples in detail.     

Elastodynamic analysis of inhomogeneous anisotropic bodies

The integral equation method is presented for elastodynamic problems of inhomogeneous anisotropic bodies. Since fundamental solutions are not available for general inhomogeneous anisotropic media, we employ the fundamental solution for homogeneous elastostatics. The terms induced by material inhomogeneity and inertia force are regarded as body forces in elastostatics, and evaluated in the form of volume integrals. The scattering problems of elastic waves by inhomogeneous anisotropic inclusions are investigated for some test cases. Numerical results show the significant effects of inhomogeneity and anisotropy of materials on wave propagations.     

The anisotropic and angularly inhomogeneous elastic wedge under a monomial load distribution

In this study, the generally anisotropic and angularly inhomogeneous wedge under a monomial type of distributed loading of order n of, the radial coordinate r at its external faces is considered. At first, using variable separable relations in the equilibrium equations, the strain–stress relations and the strain compatibility equation, a differential system of equations, is constructed. Decoupling this system, an ordinary differential equation is derived and the stress and displacement fields may be determined. The proposed procedure is also applied to the elastostatic problem of an isotropic and angularly inhomogeneous wedge. The special cases of loading of order n=−1 and n=−2, where the self-similarity approach is not valid, are examined and the stress and displacements fields are derived. Applications are presented for the cases of an angularly inhomogeneous wedge and in the case of a bi-material isotropic wedge.     

双相各向异性介质弹性波场有限差分正演模拟

从双相各向异性介质模型出发,以Boit理论为基础,推导了斜方晶系各向异性介质-阶弹性波动方程,引入固、流体密度比和孔隙几何参数,将Biot方程系数简化为测量简单、物理意义明确的物理量,采用交错网格技术建立了各向异性孔隙介质波动方程的高精度差分格式,并首次对这类差分格式的频散特性和稳定性作了详细分析讨论,解决了计算稳定性和边界反射问题,与解析解的对比以及理论模型的数值模拟都表明,该方法不仅大大降低了计算量,提高了正演速度,并且具有良好的稳定性和精确性。