Dynamic (Harmonic) Interfacial Stress Field in a Half-Plane Covered with a Prestretched Soft Layer

A half-plane covered with a prestretched layer is considered under the action of a periodic dynamic (harmonic) lineal load applied to the free surface of the layer. Within the framework of a piecewise homegeneous body model, with the use of equations of the three-dimensional linearized theory of elastic waves in initially stressed bodies, the problem of stress state is formulated. It is assumed that the materials of the layer and half-plane are linearly elastic, homogeneous, and isotropic, and a plane strain state is considered. The corresponding boundary-value problems are solved analyticaly by employing the exponential Fourier transformations. Numerical results are obtained in the case where the elastic modulus of the half-plane material is greater than that of the layer material. It is established that, because of softening of the layer material, the stresses on the interplane increase mainly in the vicinity of the acting force and this increase has a local character. Moreover, it is established that the prestretching of the cover layer decreases the absolute values of these stresses.     

Influence of initial stresses on fracture of composite materials containing interacting cracks

Considered in this study are the axially-symmetric problems of fracture of composite materials with interacting cracks, which are subjected to initial (residual) stresses acting along the cracks planes. An analytical approach within the framework of three-dimensional linearized mechanics of solids is used. Two geometric schemes of cracks location are studied: a circular crack is located parallel to the surface of a semi-infinite composite with initial stresses, and two parallel co-axial penny-shaped cracks are contained in an infinite composite material with initial stresses. The cracks are assumed to be under a normal or a radial shear load. Analysis involves reducing the problems to systems of second-kind Fredholm integral equations, where the solutions are identified with harmonic potential functions. Representations of the stress intensity factors near the cracks edges are obtained. These stress intensity factors are influenced by the initial stresses. The presence of the free boundary and the interaction between cracks has a significant effect on the stress intensity factors as well. The parameters of fracture for two types of composites (a laminar composite made of aluminum/boron/silicate glass with epoxy-maleic resin and a carbon/plastic composite with stochastic reinforcement by short ellipsoidal carbon fibers) are analyzed numerically. The dependence of the stress intensity factors on the initial stresses, physical-mechanical parameters of the composites, and the geometric parameters of the problem are investigated.     

Dispersion of lamb waves in a three-layer plate made from compressible materials with finite initial deformations

A three-layer plate made from compressible elastic materials with finite initial deformations caused by compressive forces applied to their faces is considered. By using the three-dimensional linearized theory of propagation of elastic waves in bodies with initial stresses, the propagation of Lamb waves in this plate is investigated.     

Generalized magneto-thermoelasticity with two temperature and initial stress under Green–Naghdi theory

Magneto-thermoelastic interactions in an initially stressed isotropic homogeneous elastic half-space with two temperature are studied using mathematical methods under the purview of the Green–Naghdi theory with type II and III. The medium is considered to be permeated by a uniform magnetic field. The normal mode analysis is used to obtain the exact expressions for the displacement components, force stresses, temperature and couple stresses distribution. The variations of the considered variables through the horizontal distance are illustrated graphically. Comparisons are made with the results between type II and III. Numerical work is also performed for a suitable material with the aim of illustrating the results.     

Axisymmetric contact problems for a prestressed incompressible elastic layer

Two axisymmetric problems of the indentation without friction of an elastic punch into the upper face of a layer when there is a uniform field of initial stresses in the layer are considered. The model of an isotropic incompressible non-linearly elastic material, specified by a Mooney potential, is used. Two cases are investigated: when the lower face of the layer is rigidly clamped after it is prestressed, and when the lower face of the layer is supported on a rigid base without friction after it is prestressed. It is assumed that the additional stresses due to the action of the punch on the layer are small compared with the initial stresses; this enables the problem of determining the additional stresses to be linearized. The problem is reduced to solving integral equations of the first kind with symmetrical irregular kernels relative to the pressure in the contact area. Approximate solutions of the integral equations are constructed by the method of orthogonal polynomials for large values of the parameter characterizing the relative layer thickness. The case of a punch with a plane base is considered as an example.     

Spatial problems of stability theory of stratified,compressible, composite materials

We consider spatial, non-axially-symmetric problems of stability theory of stratified, compressible, composite materials for biaxial uniform compression with the use of three-dimensional, linearized stability theory of deformed bodies. It is shown in general form for arbitrary models of compressible elastic and elastoplastic bodies that the characteristic equation corresponding to spatial non-axially-symmetric problems reduces to the characteristic equation corresponding to planar problems of single-axis compression. In this context all numerical results obtained earlier for the planar problems mentioned are also equally valid for spatial, non-axially-symmetric problems with the corresponding notations for the wave formation parameters.Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 19, pp. 24–32, 1988.     

Torsional wave propagation in a pre-stressed circular cylinder embedded in a pre-stressed elastic medium

Within the framework of the piecewise homogeneous body model with utilization of the three-dimensional linearized theory of elastic waves in initially stressed bodies, the mathematical modeling of the torsional wave propagation in the initially stressed infinite body containing an initially stressed circular solid cylinder (case 1) and circular hollow cylinder (case 2) are proposed. In these cases, it has been assumed that in the constituents of the considered systems there exist only the normal homogeneous tensional or compressional initial stress acting along the cylinder, i.e. in the direction of wave propagation. In the case where the mentioned initial stresses are not present, the proposed mathematical modeling coincides with that proposed and investigated by other authors within the classical linear theory of elastic waves. The mechanical properties of the cylinder and surrounding infinite medium have been described by the Murnaghan potential. The numerical results related to the torsional wave dispersion and the influence of the mentioned initial stresses on this dispersion are presented and discussed.     

Moving Dugdale model

This study proposes a moving Dugdale model for modes I, II and III, presents a fully dynamic analysis of the problem with the help of complex function theory and gives an exact analytical solution. From this solution, the stress and displacement fields and the dynamic crack opening displacement for mode I, II and III are determined. Based on this, the author proposes a criterion to describe dynamic behaviour of a moving defect in solids.

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Zusammenfassung Diese Arbeit schlägt ein bewegliches Dugdale Modell für Mode I, II und III vor. Sie gibt eine vollständige dynamische Analyse des Problems mit der Methode der komplexen Funktionstheorie und liefert die exakte analytische Lösung. Aus dieser Lösung werden die Spannungs- und Verschiebungsfelder und damit die dynamischen Rißöffnungsverschiebungen für Mode I, II und III dieses Problems bestimmt. Auf dieser Grundlage schlägt der Autor ein Kriterium zur Beschreibung des dynamischen Verhaltens eines bewegten Defekts in einem Festkörper vor.

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This work is supported by the Research Fellowship of Alexander von Humboldt Foundation, Federal Republic of Germany.     

A Yoffe-type moving crack in one-dimensional hexagonal piezoelectric quasicrystals

A Yoffe-type moving crack in one-dimensional hexagonal piezoelectric quasicrystals is considered. The Fourier transform technique is used to solve a moving crack problem under the action of antiplane shear and inplane electric field. Full elastic stresses of phonon and phason fields and electric fields are derived for a crack running with constant speed in the periodic plane. Obtained results show that the coupled elastic fields inside piezoelectric quasicrystals depend on the speed of crack propagation, and exhibit the usual square-root singularity at the moving crack tip. Electric field and phason stresses do not have singularity and electric displacement and phonon stresses have the inverse square-root singularity at the crack tip for a permeable crack. The field intensity factors and energy release rates are obtained in closed form. The crack velocity does not affect the field intensity factors, but alters the dynamic energy release rate. Bifurcation angle of a moving crack in a 1D hexagonal piezoelectric quasicrystal is evaluated from the viewpoint of energy balance. Obtained results are helpful to better understanding crack advance in piezoelectric quasicrystals.     

The formulation of linearized boundary integral equations of the anisotropic theory of elasticity and their application in geometrical inverse problems

An elastic bounded anisotropic solid with an elastic inclusion is considered. An oscillating source acts on part of the boundary of the solid and excites oscillations in it. Zero displacements are specified on the other part of the solid and zero forces on the remaining part. A variation in the shape of the surface of the solid and of the inclusion of continuous curvature is introduced and the problem of the theory of elasticity with respect to this variation is linearized. An algorithm for constructing integral representations for such linearized problems is described. The limiting properties of the linearized operators are investigated and special boundary integral equations of the anisotropic theory of elasticity are formulated, which relate the variations of the boundary strain and stress fields with the variations in the shape of the boundary surface. Examples are given of applications of these equations in geometrical inverse problems in which it is required to establish the unknown part of the body boundary or the shape of an elastic inclusion on the basis of information on the wave field on the part of the body surface accessible for observation.     

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/.     

On the three-dimensional stability loss problems of elements of structures of viscoelastic composite materials

Up to now, numerous problems of the stability loss for elements of structures made from composite materials have been investigated in the framework of the three-dimensional linearized theory of stability (TDLTS). It follows from the analysis of these investigations that the TDLTS was mainly applied to the design of elements of structures made from time-independent materials. For the solution of these problems for viscoelastic materials in the framework of the TDLTS, the dynamic investigation method and the critical deformation method are recommended in many references. However, it is known that a very reliable and frequently used approach for viscoelastic materials is the approach based on the study of the growth of insignificant initial imperfections in elements of structures with time. Taking into account the above-mentioned, an approach based on the growth of the initial imperfection for the investigation of the stability loss problems of elements of structures made from viscoelastic composite materials in the framework of TDLTS is proposed in the present paper. The composite material is modeled as an anisotropic, viscoelastic solid with averaged mechanical properties and all investigations are made on the strip simply supported at the ends.Yildiz Technical University, Dept. Math. Eng., 80750, Besiktas-Yildiz, Istanbul, Turkey. Published in Mekhanika Kompozitnykh Materialov, Vol. 34, No. 6, pp. 761–770, November–December, 1998.     

On the solution of creep theory problems for ageing bodies with growing slits and cavities : PMM, vol. 42, no. 6, 1978, pp. 1099–1106

There is considered a problem of creep theory for ageing homogeneous linearly deformed bodies with growing slits and cavities. Only the stresses or displacements are given on the moving sections of the contour. It is assumed that the Poisson's ratio is constant. Explicit representations are obtained from the stresses, strains, and displacements of the creep theory problem in terms of the stresses, strains, and displacements of elastically instantaneous problems. In particular, it follows from these representations that for the problem under consideration the Volterra principle is invalid in the general case. The results obtained extend the known theorems of Arutiunian which are valid for a domain with fixed boundaries [1–4]. The main results of the paper without the extension to the case of developing cavities were announced in [5].     

Forced Vibration of a Prestretched Two-Layer Slab on a Rigid Foundation

Within the framework of a piecewise homogeneous body model, with the use of the three-dimensional linearized theory of elastic waves in initially stressed bodies, the forced vibration of a prestretched two-layer slab resting on a rigid foundation is studied. To the upper plane of the slab, a harmonic point force is applied. It is assumed that the layer materials are incompressible, and their elastic properties are characterized by Treloar’s potential. Numerical results are presented for the case where the material stiffness of the lower layer is greater than that of the upper one. The influence of prestretching the layers on the frequency dependences of the normal stresses operating on the interface between the layers and between the slab and the rigid foundation are analyzed.__________Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 41, No. 3, pp. 335–350, May–June, 2005.     

两个各向同性半平面焊接的界面共线裂纹问题

复合材料焊接的界面裂纹问题是工程中碰到的实际问题,这方面已有不少有意义的研究工作（如献[3]-[6]等等）。本讨论两个各同性半平面焊接的界面共线裂纹问题,利用复变方法和解析函数边值问题题的基本理论,给出了弹性体应力分布封闭形式的解。     

Deformation of a composite elastic plane weakened by a periodic system of the arbitrarily loaded slits

The paper deals with the problems of periodic system of cuts distributed along the boundary of a bond connecting two elastic half-planes and acted upon by nonperiodic loads. In one problem it is assumed that the cuts are open, with normal and tangential stresses applied to their edges, while in another problem the edges touch each other and are loaded by tangential stresses. The method of solution is based on the simultaneous use of the discrete Fourier transformation and the theory of boundary value problems for automorphous analytic functions. The solutions are otained in quadratures. Other classes of problems to which the proposed methods can be applied, are described.

Generally speaking, in the case of irregular loads, the solution is usually based on the theory of representation of the symmetry groups /1,2/, and in the case of certain types of symmetry, particularly the translational, on the discrete Fourier transforms /3– 6/. However the objects of transformation may be different in one and the same problem, and their choice affects significantly the solvability of the boundary value problem for the transformed quantities in the cell of periods. Below two problems of the theory of cracks are solved in quadratures to illustrate the effective simultaneous use of the discrete Fourier transformation and the Muskhelishvili method.     

Forced vibration of an initially stressed thick rectangular plate made of an orthotropic material with a cylindrical hole

The forced vibration of an initially statically stressed rectangular plate made of an orthotropic material is studied. The plate is simply supported along all its edges and contains an internal across-the-width cylindrical hole of rectangular cross section with rounded corners. The initial stresses are created by uniformly distributed normal forces applied to opposite end faces of the plate. Because of the hole, these stresses are not uniform in the plate and significantly affect the stress field caused by additional time-harmonic dynamical forces acting on the upper face of the plate. Hence, for solving the boundary-value problem considered, the superposition principle is unsuitable. Therefore, our investigations are carried out within the framework of the three-dimensional linearized theory of elastic waves in initially stressed bodies. The corresponding boundary- value problems on determining the initial and additional, dynamical stress states are solved by using the three-dimensional finite-element method. Numerical results on stress concentrations around the cylindrical hole and the fundamental frequencies, and on the influence of the initial stresses on the frequencies are presented.     

A linearized theory of thermoviscoelasticity

A generalized linearized theory of thermoviscoelasticity, including the effect of heat formation, is presented. The linearized equations of motion, of state, and for the energy are given together with the linearized boundary conditions for large initial deformations. Attention is drawn to the fact that the equations which have been derived can be used for the solution of problems concerning the stability of viscoelastic bodies, the propagation of waves in viscoelastic materials which are subjected to deformation, and problems concerning the stress-deformed state of viscoelastic elements. The problem of the propagation of plane waves in viscoelastic materials which are subjected to deformation is considered as an example.Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Mekhanika Polimerov, No. 2, pp. 214–221, March–April, 1972.     

Local near-surface buckling of a system consisting of an elastic (viscoelastic) substrate,a viscoelastic (elastic) bond layer,and an elastic (viscoelastic) covering layer

Within the framework of a piecewise homogenous body model and with the use of a three-dimensional linearized theory of stability (TLTS), the local near-surface buckling of a material system consisting of a viscoelastic (elastic) half-plane, an elastic (viscoelastic) bond layer, and a viscoelastic (elastic) covering layer is investigated. A plane-strain state is considered, and it is assumed that the near-surface buckling instability is caused by the evolution of a local initial curving (imperfection) of the elastic layer with time or with an external compressive force at fixed instants of time. The equations of TLTS are obtained from the three-dimensional geometrically nonlinear equations of the theory of viscoelasticity by using the boundary-form perturbation technique. A method for solving the problems considered by employing the Laplace and Fourier transformations is developed. It is supposed that the aforementioned elastic layer has an insignificant initial local imperfection, and the stability is lost if this imperfection starts to grow infinitely. Numerical results on the critical compressive force and the critical time are presented. The influence of rheological parameters of the viscoelastic materials on the critical time is investigated. The viscoelasticity of the materials is described by the Rabotnov fractional-exponential operator. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 43, No. 6, pp. 771–788, November–December, 2007.     

复合材料焊接线出现裂缝的平面弹性基本问题

本文用复变方法讨论了复合材料任意形状焊接线上出现若干条裂缝时的平面弹性第一和第二基本问题，把寻求复应力函数的问题分别归结为求解某种正则型奇异积分方程和正则型奇异积分方程组，并证明了其解存在且唯一。