On Love-type waves in a finitely deformed magnetoelastic layered half-space

In this paper, the propagation of Love-type waves in a homogeneously and finitely deformed layered half-space of an incompressible non-conducting magnetoelastic material in the presence of an initial uniform magnetic field is analyzed. The equations and boundary conditions governing linearized incremental motions superimposed on an underlying deformation and magnetic field for a magnetoelastic material are summarized and then specialized to a form appropriate for the study of Love-type waves in a layered half-space. The wave propagation problem is then analyzed for different directions of the initial magnetic field for two different magnetoelastic energy functions, which are generalizations of the standard neo-Hookean and Mooney?CRivlin elasticity models. The resulting wave speed characteristics in general depend significantly on the initial magnetic field as well as on the initial finite deformation, and the results are illustrated graphically for different combinations of these parameters. In the absence of a layer, shear horizontal surface waves do not exist in a purely elastic material, but the presence of a magnetic field normal to the sagittal plane makes such waves possible, these being analogous to Bleustein?CGulyaev waves in piezoelectric materials. Such waves are discussed briefly at the end of the paper.     

Elastic Wave Propagation in Soft Microstructured Composites Undergoing Finite Deformations

We consider the propagation of elastic waves in soft composite materials undergoing large deformations. The analysis is performed in terms of small amplitude motions superimposed on a deformed state. By consideration of 2D periodic laminates and 3D fiber composites, we find that an applied deformation influences the elastic waves through the change in the microstructure, and through the change in the local material properties. These effects can be significantly amplified by the deformation induced elastic instability phenomenon leading to microstructure transformations. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On complex variable method in finite elasticity

We highlight the alternative presentation of the Cauchy-Riemann conditions for the analyticity of a complex variable function and consider plane equilibrium problem for an elastic transversely isotropic layer, in finite deformation. We state the fundamental problems and consider traction boundary value problem, as an example of fundamental problem-one. A simple solution of “Lame's problem” for an infinite layer is obtained. The profile of the deformed contour is given; and this depends on the order of the term used in the power series specification for the complex potential and on the material constants of the medium.     

About embedded eigenvalues for a spectral problem arising in the study of elastic surface waves in a topographical waveguide

In this paper, we are interested with the spectral study of an operator given by an elastic topographical waveguide, a deformed half‐space, of which the cross‐section is a local perturbation of a homogeneous half‐plane. We look for guided waves propagating more rapidly than Rayleigh waves (which mathematically would correspond to embedded eigenvalues) and prove that there are no guided waves propagating more rapidly than S‐waves. Thanks to the boundary of the deformed half‐plane and some reduced equations, these eventual eigenmodes must locally vanish. Adapting to our case a unique continuation principle for the elasticity system, we conclude that these eigenmodes vanish everywhere. Copyright © 2002 John Wiley & Sons, Ltd.     

Longitudinal and Transverse Waves in Finite Elastic Strain. Hadamard and Green Materials

This work is concerned with the propagation of purely longitudinaland purely transverse waves in homogeneously deformed isotropicelastic materials. Two types of compressible material are alsodiscussed. A Hadamard material, so called by John in the hyperelasticcase, is one in which longitudinal waves may propagate in everydirection when the material is homogeneously deformed. A secondmaterial, called a "Green material" is introduced. In it twotransverse waves can propagate in every direction when the materialis homogeneously deformed. It is seen that a Mooney materialis the only isotropic incompressible elastic material in whichtwo transverse waves can propagate in every direction when itis homogeneously deformed, while the pressure stays constantthroughout the material. The propagation of finite amplitudewaves in these materials is discussed. Finally, it is shownthat the only motions which can be maintained in all homogeneouscompressible elastic Hadamard materials under the action ofsurface forces alone, are necessarily homogeneous and accelerationless.     

Influence of the nonlinear matrix material behaviour on the effective properties of textile-reinforced thermoplastics

For a consistent lightweight design the consideration of the nonlinear macroscopic material behaviour of composites, which is amongst others driven by damage and strain-rate effects on the mesoscale, is required. Therefore, a modelling approach using numerical homogenization techniques is applied to predict the effective nonlinear material behaviour of the composite based on the finite element simulation of a representative volume element (RVE). In this RVE suitable constitutive relations account for the material behaviour of each constituents. While the reinforcing glass fibres are assumed to remain linear elastic, a viscoplastic constitutive law is applied to represent the strain-rate dependent, inelastic deformation of the matrix material. In order to analyse the influence of the nonlinear matrix material behaviour on the global mechanical response of the composite, effective stress-strain-curves are computed for different load cases and compared to experimental observations. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

不可压缩弹性固体中的二维应力波分析

本文研究不可压缩弹性固体中的二维应力波.首先对一般的应变能函数给出了分析简单波和激波的基本方程,然后求出了波速和相应的本征向量,证明在一般情况下有两组简单波和两组激波,最后举了平面变形和反平面变形两个例子.在平面变形的情况下,平面激波的斜反射问题一般无解.     

Asymptotic behavior of nonlinear waves in elastic media with dispersion and dissipation

In the case of nonlinear elastic quasitransverse waves in composite media described by nonlinear hyperbolic equations, we study the nonuniqueness problem for solutions of a standard self-similar problem such as the problem of the decay of an arbitrary discontinuity. The system of equations is supplemented with terms describing dissipation and dispersion whose influence is manifested in small-scale processes. We construct solutions numerically and consider self-similar asymptotic approximations of the obtained solution of the equations with the initial data in the form of a “spreading” discontinuity for large times. We find the regularities for realizing various self-similar asymptotic approximations depending on the choice of the initial conditions including the dependence on the form of the functions determining the small-scale smoothing of the original discontinuity. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 147, No. 2, pp. 240–256, May, 2006.     

Instability of solitary waves in non-linear composite media

A problem on the dynamic instability of soliton solutions (solitary waves) of Hamilton's equations, describing plane waves in non-linear elastic composite media with or without anisotropy, is considered. In the anisotropic case, there are two two-parameter families of solitary waves: fast and slow and, when there is no anisotropy, there is one three-parameter family. A classification of the instability of solitary waves of the fast family in the anisotropic case and of representatives of families of solitary waves, the velocities of which lie outside of the range of stability when there is anisotropy and when there is no anisotropy, is given on the basis of a numerical solution of a Cauchy problem for the model equations. In this paper, the fundamental equations describing plane waves in non-linear, anisotropic, elastic composites are derived, the Hamilton form of the basic equations is presented, the symmetries in the anisotropic and isotropic cases are considered, the conserved quantities and the soliton solutions, that is, the solitary waves are presented, the nature of the instability of representatives of all three families is investigated, brief formulation of the results is presented and problems of the instability of the fast family in the anisotropic case and of representatives of the families, the velocities of which lie outside of the range of stability in the presence and absence of anisotropy (explosive instability), are discussed.     

Deformation of orthotropic composites with unidirectional ellipsoidal inclusions under matrix microdamages

In the present paper, a model of deformation of stochastic composites under microdamaging is developed for the case of orthotropic composite, when the microdamages are accumulated in the matrix. The composite is treated as an isotropic matrix strengthened by three-axial ellipsoidal inclusions with orthotropic symmetry of elastic properties. It is assumed that the loading process leads to accumulation of damages in the matrix. Fractured microvolumes are modeled by a system of randomly distributed quasispherical pores. The porosity balance equation and relations for determining the effective elastic moduli for the case of a composite with orthotropic components are taken as the basic relations. The fracture criterion is assumed to be given as the limit value of the intensity of average shear stresses occurring in the undamaged part of the material. Based on the analytical and numerical approach, an algorithm for the determination of nonlinear deformation properties of such a material is constructed. The nonlinearity of composite deformations is caused by the accumulation of microdamages in the matrix. Using the numerical solution, nonlinear stress-strain diagrams for the orthotropic composite in the case of biaxial extension are obtained. Published in Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 121–130, January–March, 2008.     

Some qualitative effects in the exact solutions of the Lamé problem for large deformations

Qualitative effects in the solution of a number of radially symmetric and plane axisymmetric problems for bodies made of non-linearly elastic incompressible materials are analysed for large deformations. In the case of problems of the axisymmetric plane deformation of cylindrical bodies, the lack of uniqueness of the solution for a given follower load in the case of a Bartenev–Khazanovich material and the existence of a limiting load in the case of a Treloar (neo-Hookian) material have been studied in detail and the dependences of the limiting load on the ratio of the external and internal radii of a hollow cylinder in the undeformed state have been presented. A similar study has been carried out for constitutive relations of a special form that well describe the properties of rubber. For this material, the lack of uniqueness of the solution is revealed for fairly high loads. The axisymmetric problem of the plane stress state of a circular ring made of a Bartenev–Khazanovich material has been solved and a lack of uniqueness of the solution for a given follower load was discovered in the case when the dimensions of the ring are given in the undeformed state. Similar studies have been carried out for Chernykh and Treloar materials in the case of the problem of the radially symmetric deformation of a spherical shell. It was established that, in the case of a Chernykh material, the lack of uniqueness of the solution depends considerably on the constant characterizing the physical non-linearity. The limit case of the deformation of a spherical cavity in an infinitely extended body has been investigated. The effect of an unbounded increase in the boundary stresses is observed for finite external loads, that appears in the case of the problem of the plane axisymmetric deformation of a cylindrical cavity in an infinitely extended body made of a Bartenev–Khazanovich material and in the case of the problem of the radially symmetric deformation of an infinitely extended body made of a Chernykh material with a spherical cavity.     

Buckling and postcritical behaviour of the elastic infinite plate strip resting on linear elastic foundation

In this paper the von Kármán model for thin, elastic, infinite plate strip resting on a linear elastic foundation of Winkler type is studied. The infinite plate strip is simply-supported and subjected to evenly distributed compressive loads. The critical values of bifurcation parameters and buckling modes for given frequency of longitudinal waves are found on the basis of investigation of linearized problem. The mathematical nonlinear model is reduced to operator equation with Fredholm type operator of index 0 depending on parameters defined in corresponding Hölder spaces. The Lyapunov-Schmidt reduction and the Crandall-Rabinowitz bifurcation theorem (gradient case) are used to examine the postcritical behaviour of the plate. It is proved that there exists maximal frequency of longitudinal waves depending on the compressive load and the stiffness modulus of foundation.     

Propagation of axisymmetric waves in an initially twisted circular compound bimaterial cylinder with a soft inner and a stiff outer constituents

Within the scope of a piecewise homogeneous body model, with the use of the three-dimensional linearized theory of propagation of elastic waves in an initially stressed body, the problem on the propagation of axisymmetric waves in an initially twisted circular compound bimaterial cylinder is studied. A mathematical formulation of the problem is presented, and the corresponding solution method is proposed and employed. Numerical results are presented and discussed for the case where the material of the outer cylinder is stiffer than the inner solid one. In particular, it is established that, as a result of initial twisting of the compound cylinder, new axisymmetric wave modes appear in it.     

On the influence of initial stresses on the velocities of the stoneley waves

Wave propagation along a plane boundary separating compressible, previously deformed bodies with elastic potential of arbitrary form, is studied. The linearized theory of wave propagation in bodies with finite initial deformation is used. A case in which one of the bodies is a liquid, is studied. It is shown that in the case of the Murnaghan and harmonic type potentials the wave velocities depend linearly on the initial stresses. In contrast with the case of an unbounded isotropic body /1/, here the character of the dependence is not influenced by the choice of the form of the potential. In the absence of the initial stresses the relations obtained coincide with the results of /2/.     

Nonlinear deformation of Three-Component Composites

The model of nonlinear deformation of stochastic composites under microdamaging is developed for the case of threecomponent composite, when the microdamages are accumulated in the matrix. The composite is treated as isotropic matrix strengthened by two different types of spheroidal inclusions with transversally-isotropic symmetry of elastic properties. Fractured microvolumes are modeled by a system of randomly distributed quasispherical pores. The porosity balance equation and relations for determining the effective elastic modules for the case of transversally-isotropic components are taken as basic relations. The fracture criterion is assumed to be given as the limit value of the intensity of average shear stresses occurring in the undamaged part of the material. The algorithm for determination of nonlinear deformative properties of such a material is constructed. The nonlinear stress-strain diagrams for three-component concrete for the case of uniaxial tension are obtained. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Self-similar asymptotics of wave problems and the structures of non-classical discontinuities in non-linearly elastic media with dispersion and dissipation

Solutions of the non-linear hyperbolic equations describing quasi-transverse waves in composite elastic media are investigated within the framework of a previously proposed model, which takes into account small dissipative and dispersion processes. It is well known for this model that if a solution of the problem of the decay of an arbitrary discontinuity is constructed using Riemann waves and discontinuities having a structure, the solution turns out to be non-unique. In order to study the problem of non-uniqueness, solutions of non-self-similar problems are constructed numerically within the framework of the proposed model with initial data in the form of a “smooth” step. With time passing the solutions acquire a self-similar asymptotic form, corresponding to a certain solution of the problem of the decay of an arbitrary discontinuity. It is shown that, by changing the method of smoothing the step, one can construct any of the self-similar asymptotic forms, as was done previously in Ref. [Chugainova AP. The asymptotic behaviour of non-linear waves in elastic media with dispersion and dissipation. Teor Mat Fiz 2006;147(2):240–56] for media with terms of opposite sign, responsible for the non-linearity, although the set of admissible discontinuities and the structure of the solutions of the problems in these cases turn out to be different.     

Amplitude modulation of nonlinear waves in a thick elastic tube filled with an inviscid fluid

In the present work, employing the nonlinear equations of an incompressible, isotropic and elastic thick tube and the approximate equations of an incompressible inviscid fluid, and then utilizing the reductive perturbation technique the amplitude modulation of weakly nonlinear waves is examined. It is shown that the amplitude modulation of these waves is governed by a nonlinear Schrödinger(NLS) equation. The range of modulational instability of the monochromatic wave solution with the initial deformation, material and geometrical characteristics is discussed for some elastic materials.     

纤维复合材料中弹性波散射与动应力

基于弹性波动理论,对纤维增强复合材料结构中弹性波多重散射与动应力集中问题进行了分析研究,给出了介质各区域弹性波分析解的表达式.根据位移与应力在各区界面处的连续条件,确定了未知弹性波模式系数.采用Hankel函数的加法定理,将不同局部坐标系中散射波场的表达式变换到了同一个局部坐标系中,以给出弹性波模式系数和动应力集中因子的表达式.分析了多相纤维基体中两个散射体的间距、界层区材料性质以及界层区和纤维核区截面尺寸的变化,对各区界面动应力集中系数的影响.分析表明,两个散射体的间距、界层区材料性质和结构尺寸的变化对复合材料的力学特性具有显著影响.作为算例,给出了纤维增强复合材料结构中各区界面动应力集中系数的数值结果,并对其进行了分析讨论.     

带有摩擦的单边接触大变形问题的研究(Ⅰ)——增量变分方程

本文以非线性连续体几何场论为基本理论和方法,建立了拖带坐标下弹塑性大变形增量变分方程的更一般表示式.给出了二维、三维连续体接触边界变化率公式,得到了变边界接触大变形增量变分公式和速率型变分不等式,为有限元计算求解带有摩擦弹塑性大变形接触问题提供了理论基础.