Perturbation Formula for Phase Velocity of Rayleigh Waves in Prestressed Anisotropic Media

Herein we consider Rayleigh waves propagating along the free surface of a macroscopically homogeneous, anisotropic, prestressed half-space. We adopt the formulation of linear elasticity with initial stress and assume that the deviation of the prestressed anisotropic medium from a comparative ‘unperturbed’, unstressed and isotropic state, as formally caused by the initial stress and by the anisotropic part of the incremental elasticity tensor, be small. No assumption, however, is made on the material anisotropy of the incremental elasticity tensor. With the help of the Stroh formalism, we derive a first-order perturbation formula for the shift of phase velocity of Rayleigh waves from its comparative isotropic value. Our perturbation formula does not agree totally with that which was derived some years ago by Delsanto and Clark, and we provide another argument as further support for our version of the formula. According to our first-order formula, the anisotropy-induced velocity shifts of Rayleigh waves, taken in totality of all propagation directions on the free surface, carry information only on 13 elastic constants of the anisotropic part of the incremental elasticity tensor. The remaining eight elastic constants are those which would become zero if were monoclinic with the two-fold symmetry axis normal to the free surface of the material half-space in question.     

Plate-Like Smart Structures: Reduced Models and Numerical Simulations

In this paper asymptotic models describing the mechanical and electric equilibrium state of two types of smart structures are presented and justified. The first structure consists of an anisotropic elastic thin plate with two surface bonded anisotropic piezoelectric patches and the second one is an anisotropic elastic sandwich thin plate with an inserted anisotropic piezoelectric patch. The two unknowns of the corresponding asymptotic models, the mechanical displacements of the structures and the electric potentials of the patches, are partially decoupled. The former are the solution of modified Kirchhoff-Love plate models, while the latter can be derived as explicit functions of the mechanical displacements. Moreover, different formulas for the electric potential arise as a consequence of diverse electric boundary conditions. We report numerical simulations with these asymptotic models.     

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.     

Stress effect decay estimates for anisotropic material in a semi-infinite strip

IntroductionInthetheoryofplanedeformationsoflinearelastotatics,Saint-Venant'sprincipleplaysanimportantroleinbothoftheoryandpracticalapplicahonsandisoftenusedtojushfyapproximationthatneglectedgeeffects.ForhomogenousisotropicmaterialthevalidityofSaintVenant'sprincipleiswellestablished.However,forhomogenousanisotropicmaterial,experimentalresultshaveshownthatedgeeffectsmaypersistmuchfartherintotheinteriorofthebodythanforisotropicmaterialandasaresultcannotbeneglected.Asweknow,theelasticityproblem…     

Application of the method of matched asymptotic expansions to the calculation of the stationary thermal propagation of the front of an exothermic reaction in a condensed medium

In this paper we use the method of matched asymptotic expansions to establish a two-term formula for the speed of propagation of the front of an exothermic reaction in a condensed medium whose thermophysical characteristics depend on the concentration of the reacting matter and the temperature. As the parameter of the expansion we use the ratio of the activation temperature to the adiabatic combustion temperature. The results are applied to the case of the combustion of nonvolatile condensed systems. We compare the approximate formula obtained with the results of a numerical integration.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 5, pp. 106–112, September–October, 1972.     

Continuous Dependence on Base Data in an Elastic Prismatic Cylinder

This paper establishes the continuous dependence of the displacement on the base geometry and load of the decay behaviour in a prismatic semi-infinite cylinder occupied by an anisotropic homogeneous linear elastic material. The cylinder is in equilibrium under zero body-force, zero displacement on the lateral boundary and pointwise specified displacement on the base. A differential inequality is derived which on integration leads to an inequality in terms of the perturbation of the base geometry and differences in the base displacement. Continuous dependence is an immediate consequence. This revised version was published online in July 2006 with corrections to the Cover Date.     

含正交排列夹杂和缺陷材料的等效弹性模量和损伤

研究含正交排列夹杂和缺陷材料的等效弹性模量和损伤,推导了以Eshelby－Mori－Tanaka方法求解多相各向异性复合材料等效弹性模量的简便计算公式,针对含三相正交椭球状夹杂的正交各向异性材料,得到了由细观参量（夹杂的形状、方位和体积分数）表示的等效弹性模量的解析表达式．在此基础上,提出了一个宏细观结合的正交各向异性损伤模型,从而建立了以细观量为参量的含损伤材料的应力应变关系．最后,对影响材料损伤的细观结构参数进行了分析．     

Indentation of a Punch with a Fine-Grained Base into an Elastic Foundation

The linear contact problem for a system of small punches located periodically on a part of the boundary of an elastic foundation is studied. An averaged contact problem is derived using the Marchenko–Khruslov averaging theory. An asymptotic formula is obtained for the translational capacity of a smooth punch with a fine-grained flat base.     

An approach for obtaining approximate formulas for the Rayleigh wave velocity

In this paper, we introduce an approach for finding analytical approximate formulas for the Rayleigh wave velocity for isotropic elastic solids and anisotropic elastic media as well. The approach is based on the least-square principle. To demonstrate its application, we applied it in order to obtain an explanation for Bergmann’s approximation, the earliest known approximation of the Rayleigh wave velocity for isotropic elastic solids, and used it to establish a new approximation. By employing this approach, the best approximate polynomials of the second order of the cubic power and the quartic power in the interval [0, 1] were found. By using the best approximate polynomial of the second order of the cubic power, we derived an approximate formula for the Rayleigh wave speed in isotropic elastic solids which is slightly better than the one given recently by Rahman and Michelitsch by employing Lanczos’s approximation. Also by using this second order polynomial, analytical approximate expressions for orthotropic, incompressible and compressible elastic solids were found. For incompressible case, it is shown that the approximation is comparable with Rahman and Michelitsch’s approximation, while for the compressible case, it is shown that our approximate formulas are more accurate than Mozhaev’s ones. Remarkably, by using the best approximate polynomials of the second order of the cubic power and the quartic power in the interval [0, 1], we derived an approximate formula of the Rayleigh wave velocity in incompressible monoclinic materials, where the explicit exact formulas of the Rayleigh wave velocity so far are not available.     

Complete Asymptotic Expansions of Solutions of the System of Elastostatics in the Presence of an Inclusion of Small Diameter and Detection of an Inclusion

We consider the system of elastostatics for an elastic medium consisting of an imperfection of small diameter, embedded in a homogeneous reference medium. The Lamé constants of the imperfection are different from those of the background medium. We establish a complete asymptotic formula for the displacement vector in terms of the reference Lamé constants, the location of the imperfection and its geometry. Our derivation is rigorous, and based on layer potential techniques. The asymptotic expansions in this paper are valid for an elastic imperfection with Lipschitz boundaries. In the course of derivation of the asymptotic formula, we introduce the concept of (generalized) elastic moment tensors (Pólya–Szegö tensor) and prove that the first order elastic moment tensor is symmetric and positive (negative)-definite. We also obtain estimation of its eigenvalue. We then apply these asymptotic formulas for the purpose of identifying with high precision the order of magnitude of the diameter of the elastic inclusion, its location, and its elastic moment tensors.     

Instabilities and loss of ellipticity in fiber-reinforced compressible non-linearly elastic solids under plane deformation

In a recent paper we examined the loss of ellipticity and its interpretation in terms of fiber kinking and other instability phenomena in respect of a fiber-reinforced incompressible elastic material. Here we provide a corresponding analysis for fiber-reinforced compressible elastic materials. The analysis concerns a material model which consists of an isotropic base material augmented by a reinforcement dependent on the fiber direction. The assessment of loss of ellipticity can be cast in terms of the eigenvalues of the acoustic tensors associated with the isotropic and anisotropic parts of the strain-energy function. For the anisotropic part, two different reinforcing models are examined and it is shown that, depending on the choice of model and whether the fiber is under compression or extension, loss of ellipticity may be associated with, in particular, a weak surface of discontinuity normal to or parallel to the deformed fiber direction or at an intermediate angle. Under compression the associated failure interpretations include fiber kinking and fiber splitting, while under extension fiber de-bonding and matrix failure are included.     

The Kirchhoff–Helmholtz integral for anisotropic elastic media

The Kirchhoff–Helmholtz integral is a powerful tool to model the scattered wavefield from a smooth interface in acoustic or isotropic elastic media due to a given incident wavefield and observation points sufficiently far away from the interface. This integral makes use of the Kirchhoff approximation of the unknown scattered wavefield and its normal derivative at the interface in terms of the corresponding quantities of the known incident field. An attractive property of the Kirchhoff–Helmholtz integral is that its asymptotic evaluation recovers the zero-order ray theory approximation of the reflected wavefield at all observation points where that theory is valid. Here, we extend the Kirchhoff–Helmholtz modeling integral to general anisotropic elastic media. It uses the natural extension of the Kirchhoff approximation of the scattered wavefield and its normal derivative for those media. The anisotropic Kirchhoff–Helmholtz integral also asymptotically provides the zero-order ray theory approximation of the reflected response from the interface. In connection with the asymptotic evaluation of the Kirchhoff–Helmholtz integral, we also derive an extension to anisotropic media of a useful decomposition formula of the geometrical spreading of a primary reflection ray.     

A thick anisotropic plate element in the framework of an absolute nodal coordinate formulation

In this research, the incorporation of material anisotropy is proposed for the large-deformation analyses of highly flexible dynamical systems. The anisotropic effects are studied in terms of a generalized elastic forces (GEFs) derivation for a continuum-based, thick, and fully parameterized absolute nodal coordinate formulation plate element, of which the membrane and bending deformation effects are coupled. The GEFs are first derived for a fully anisotropic, linearly elastic material, characterized by 21 independent material parameters. Using the same approach, the GEFs are obtained for an orthotropic material, characterized by nine material parameters. Furthermore, the analysis is extended to the case of nonlinear elasticity; the GEFs are introduced for a nonlinear Cauchy-elastic material, characterized by four in-plane orthotropic material parameters. Numerical simulations are performed to validate the theory for statics and dynamics and to observe the anisotropic responses in terms of displacements, stresses, and strains. The presented formulations are suitable for studying the nonlinear dynamical behavior of advanced elastic materials of an arbitrary degree of anisotropy.     

考虑材料各向异性的熔丝制造PLA点阵结构弹性各向同性设计

增材制造技术的兴起激发了国内外学者对结构创新设计的热情. 然而, 增材制造材料的各向异性为结构力学性能的预测与设计带来了一定的困难. 为了准确预测熔丝制造聚乳酸(PLA)材料和点阵结构的弹性性能, 并实现点阵结构的弹性各向同性设计, 首先, 本文采用正交各向异性弹性模型来描述PLA材料的弹性行为, 通过实验和计算得到了正交各向异性模型需要的9个独立的弹性常数. 然后, 设计了一种力学性能可调的二维组合桁架点阵结构, 基于代表体元法, 在不考虑材料各向异性的情况下推导出了其平面内等效弹性性能的解析表达式及弹性各向同性条件. 最后, 根据PLA材料的各向异性调整点阵结构内部杆件的弹性模量和厚度, 并基于代表体元法重新推导出了点阵结构平面内等效弹性性能的解析表达式及其弹性各向同性条件. 研究结果表明, 正交各向异性弹性模型适用于描述熔丝制造PLA材料的弹性行为, 基于该模型能够准确预测PLA材料在任意方向上的弹性模量. 在预测与设计熔丝制造点阵结构的力学性能时需要充分考虑材料的各向异性. 在考虑材料的各向异性之后, 基于代表体元法调整点阵结构的几何尺寸, 能够实现部分点阵结构的弹性各向同性设计.      

Topological derivative for an anisotropic and heterogeneous heat diffusion problem

The topological derivative measures the sensitivity of a given shape functional with respect to an infinitesimal singular domain perturbation. According to the literature, the topological derivative has been fully developed for a wide range of physical phenomenon modeled by partial differential equations, considering homogeneous and isotropic constitutive behavior. In fact, only a few works dealing with heterogeneous and anisotropic material behavior can be found in the literature, and, in general, the derived formulas are given in an abstract form. In this work, we derive the topological derivative in its closed form for the total potential energy associated to an anisotropic and heterogeneous heat diffusion problem, when a small circular inclusion of the same nature of the bulk phase is introduced at an arbitrary point of the domain. In addition, we provide a full mathematical justification for the derived formula and develop precise estimates for the remainders of the topological asymptotic expansion. Finally, the influence of the heterogeneity and anisotropy are shown through some numerical examples of heat conductor topology optimization.     

On singularities associated with the curvilinear anisotropic elastic symmetries

The objective of this paper is to describe a different approach to modeling the material symmetry associated with singularities that can occur in curvilinear anisotropic elastic symmetries. In this analysis, the intrinsic non-linearity of a cylindrically anisotropic problem is demonstrated. We prove that a simple homogenization process applied to a representative volume element containing the cylindrical anisotropic singularity removes the singularity. This geometric and interpretive approach is an aid to better modeling of material symmetry associated with these singularities.     

High frequency scattering due to a pair of time-harmonic antiplane forces on the faces of a finite interface crack between dissimilar anisotropic materials

The high-frequency elastodynamic problem involving the excitation of an interface crack of finite width lying between two dissimilar anisotropic elastic half-planes has been analyzed. The crack surface is excited by a pair of time-harmonic antiplane line sources situated at the middle of the cracked surface. The problem has first been reduced to one with the interface crack lying between two dissimilar isotropic elastic half-planes by a transformation of relevant co-ordinates and parameters. The problem has then been formulated as an extended Wiener–Hopf equation (cf. Noble, 1958) and the asymptotic solution for high-frequency has been derived. The expression for the stress intensity factor at the crack tips has been derived and the numerical results for different pairs of materials have been presented graphically.     

On the one-dimensional modeling of camber bending deformations in active anisotropic slender structures

The paper presents a one-dimensional model for anisotropic active slender structures that captures arbitrary cross-sectional deformations. The 1-D geometrically-nonlinear static problem is derived by an asymptotic reduction process from the equations of 3-D electroelasticity. In addition to the conventional (bending–extension–shear–twist) beam strain measures, it includes a Ritz approximation to account for arbitrary deformation shapes of the finite-size cross-sections. As a particular case, closed-form analytical expressions are derived for the linear static equilibrium of a composite thin strip with surface-mounted piezoelectric actuators. This solution is based on a boundary-layer approximation to the static equilibrium equations in regions where Saint-Venant’s principle for elastic bodies cannot be applied and includes camber bending deformations to account for the local bimoments induced by the distributed actuation in a finite-width strip.     

Stroh formulation for a generally constrained and pre-stressed elastic material

The Stroh formalism is essentially a spatial Hamiltonian formulation and has been recognized to be a powerful tool for solving elasticity problems involving generally anisotropic elastic materials for which conventional methods developed for isotropic materials become intractable. In this paper we develop the Stroh/Hamiltonian formulation for a generally constrained and prestressed elastic material. We derive the corresponding integral representation for the surface-impedance tensor and explain how it can be used, together with a matrix Riccati equation, to calculate the surface-wave speed. The proposed algorithm can deal with any form of constraint, pre-stress, and direction of wave propagation. As an illustration, previously known results are reproduced for surface waves in a pre-stressed incompressible elastic material and an unstressed inextensible fibre-reinforced composite, and an additional example is included analyzing the effects of pre-stress upon surface waves in an inextensible material.     

Three-dimensional elastic displacements induced by a dislocation of polygonal shape in anisotropic elastic crystals

Dislocations and the elastic fields they induce in anisotropic elastic crystals are basic for understanding and modeling the mechanical properties of crystalline solids. Unlike previous solutions that provide the strain and/or stress fields induced by dislocation loops, in this paper, we develop, for the first time, an approach to solve the more fundamental problem—the anisotropic elastic dislocation displacement field. By applying the point-force Green’s function for a three-dimensional anisotropic elastic material, the elastic displacement induced by a dislocation of polygonal shape is derived in terms of a simple line integral. It is shown that the singularities in the integrand of this integral are all removable. The proposed expression is applied to calculate the elastic displacements of dislocations of two different fundamental shapes, i.e. triangular and hexagonal. The results show that the displacement jump across the dislocation loop surface exactly equals the assigned Burgers vector, demonstrating that the proposed approach is accurate. The dislocation-induced displacement contours are also presented, which could be used as benchmarks for future numerical studies.