The shear wave diffraction on tunnel cavities in an elastic layer and a half-layer

Summary  The antiplane steady dynamic problem of the theory of elasticity for an isotropic layer and a half-layer weakened by tunnel cavities of arbitrary cross section is investigated. A radiating monochromatic shear wave (SH-wave) is considered as loading. Using the constructed Green's functions for a layer and a half- layer, the corresponding boundary problems are reduced to the Fredholm's integral equations of the second kind. The obtained algorithm is realized numerically by the quadrature method. Results specifying the influence of the openings configuration, their number and relative position, types of edge conditions and inertia effects on tangential stress concentration are given. Received 13 July 1999; accepted for publication 20 July 2000     

A crack of arbitrary shape inside an infinite transversely isotropic cylinder under arbitrary load

Summary  In the first part of the article an infinite circular cylinder is considered, made of transversely isotropic elastic material and weakened by a plane crack perpendicular to its axis O z. The crack is opened by an arbitrary normal stress. The second part is devoted to the same crack loaded by an arbitrary tangential stress. The complete solution in both cases is presented as a sum of the solution of a similar problem of a crack in an infinite space and an integral transform term, the parameters of which are determined from a set of linear algebraic equations derived from the boundary conditions. Governing integral equations with respect to the yet unknown crack displacement discontinuities are obtained. In the case of a circular crack, these equations can be inverted and solved by the method of consecutive interations. Received 30 November 2000; accepted for publication 3 May 2001     

Moving load response in micropolar thermoelastic medium without energy dissipation possessing cubic symmetry

The steady state response of a micropolar thermoelastic medium without energy dissipation possessing cubic symmetry due to a moving load has been studied. Fourier transform has been employed and the transform has been inverted by using a numerical inversion technique. The components of displacement, stress, microrotation and temperature distribution in the physical domain are obtained numerically. The results of normal displacement, normal force stress, tangential couple stress and temperature distribution have been compared for micropolar cubic crystal and micropolar isotropic solid. The numerical results are illustrated graphically for a particular material. Some special cases have also been deduced.     

Time harmonic sources at micropolar thermoelastic medium possessing cubic symmetry with one relaxation time

The response of a micropolar thermoelastic medium possessing cubic symmetry with one relaxation time due to time harmonic sources has been investigated. Fourier transform has been employed and the transform has been inverted by using a numerical inversion technique. The components of displacement, stress, microrotation and temperature distribution in the physical domain are obtained numerically. The results of normal displacement, normal force stress, tangential couple stress and temperature distribution have been compared for micropolar cubic crystal and isotropic micropolar solid. The numerical results are illustrated graphically for a particular material. Some special cases have also been deduced.     

Deformation in micropolar cubic crystal due to various sources

The response of a micropolar cubic crystal due to various sources has been studied. The eigenvalue approach using Laplace and Fourier transforms has been employed to solve the problem. The integral transforms have been inverted by using a numerical technique to obtain the displacement, microrotation and stress components in the physical domain. The results of normal displacement, normal force stress and tangential couple stress have been compared for micropolar cubic crystal and micropolar isotropic solid and illustrated graphically.     

Stress state of isotropic layer with crack

The stress state of an elastic isotropic layer with a finite through crack is considered. At the boundary planes of the layer, the normal component of the displacement vector and the tangential stress are zero. The crack surface is subject to normal forces that vary arbitrarily. On the basis of three-dimensional elasticity theory, a method of solving the problem is proposed. Numerical results characterizing the behavior of the normal-stress intensity coefficient are obtained. Translated from Prikladnaya Mekhanika, Vol. 33, No. 1, pp. 43–51, January, 1997.     

General treatment of mode III interfacial crack problems in piezoelectric materials

Summary  The anti-plane problem of N collinear interfacial cracks between dissimilar transversely isotropic piezoelectric media, which are subjected to piecewise uniform out-of-plane mechanical loading combined with in-plane electric loading at infinity, and also a line loading at an arbitrary point, is addressed by using the complex function method. In comparison with other relevant works, the present study has two features: one is that the analysis is based on the permeable crack model, i.e. the cracks are considered as permeable thin slits, and, thus, both the normal component of electric displacement and the tangential component of electric field are assumed to be continuous across these slits. The other feature is that explicit closed-form solutions are given not only in piezoelectric media, but also inside cracks when the media are subjected to the most general loading. It is shown that the singularities of electric displacement and electric field in the media are always dependent on that of stress for the general case of loading, and all the singularities of field variables are independent of the applied uniform electric loads at infinity. For the interfacial cracks the electric field is square-root singular at the crack tips and shows jumps across the interface, while the normal component of the electric field is linearly variable inside the crack, but the tangential component is square-root singular. However, for a homogeneous medium with collinear cracks, the electric field is always nonsingular in the medium while the electric displacement exhibits square-root singularity. Moreover, in this case, the electric field inside any crack is equal to a constant when uniform loads are applied at infinity. Received 22 November 1999; accepted for publication 20 July 2000     

Stress analysis of functionally graded rotating discs: analytical and numerical solutions

This study deals with stress analysis of annular rotating discs made of functionally graded materials(FGMs).Elasticity modulus and density of the discs are assumed to vary radially according to a power law function,but the material is of constant Poisson’s ratio.A gradient parameter n is chosen between 0 and 1.0.When n = 0,the disc becomes a homogeneous isotropic material.Tangential and radial stress distributions and displacements on the disc are investigated for various gradient parameters n by means of the diverse elasticity modulus and density by using analytical and numerical solutions.Finally,a homogenous tangential stress distribution and the lowest radial stresses along the radius of a rotating disc are approximately obtained for the gradient parameter n = 1.0 compared with the homogeneous,isotropic case n = 0.This means that a disc made of FGMs has the capability of higher angular rotations compared with the homogeneous isotropic disc.     

Shape optimization of two equal holes in an infinite elastic plate

Abstract

The optimal design of the stress state in elastic plate structures with openings is a problem of great significance in engineering practice. Achieving proper shape of hole can reduce stress concentration around the boundaries remarkably. The optimal shape of a single hole in an infinite plate under uniform stresses has been obtained by complex variable method based on different optimal criteria. The complex variable method is particularly suitable for the hole shape optimization in infinite plate, in which the continuous hole boundary can be represented by the mapping function. It can also be used to solve the shape optimization problems of two or more holes. However, because of the difficulty of finding the mapping function for multi connected domain, the holes are mapped onto slits or separately mapped onto a circle. In this article, the two symmetrical and identical holes are mapped onto an annulus simultaneously by the newly found mapping function, which has a general form. The maximum tangential stress around the boundaries is minimized to achieve the optimal hole shape. And the coefficients of mapping function which describe the boundary are calculated by differential-evolution algorithm.     

The thermoelastic equilibrium of a transversally isotropic medium with an elliptic crack under symmetric loading

The stress intensity factors (SIFs) are evaluated for flat elliptical cracks located in a transversally isotropic material (cracks are assumed perpendicular to the transtropy axis) under an arbitrary load and symmetric temperature. The SIFs for an elliptical crack in a transversally isotropic medium are determined using the formulas (derived by the author in his previous studies) of transition from an isotropic to transversally isotropic material and the relative problem for an isotropic medium. It is proved that these formulas can be employed for an arbitrary homogeneous transversally isotropic material (no matter whether the roots of some characteristic equation of the material are real or complex) with an arbitrary flat crack or a system of coplanar flat cracks, including elliptical ones, under an arbitrary load and symmetric temperature. A transversally isotropic material with two coplanar elliptical cracks is considered as an illustrative example. The dependences of the SIFs on the parameters of cracks and their arrangement at a decreasing temperature are presented. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 4, pp. 96–105, April, 2000.     

Deformation due to time harmonic sources in micropolar thermoelastic medium possessing cubic symmetry with two relaxation times

The response of a micropolar thermoelastic medium possessing cubic symmetry with two relaxation times due to time harmonic sources is investigated. Fourier transform is employed and the transform is inverted by using a numerical inversion technique. The components of displacement, stress, microrotation and temperature distribution in the physical domain are obtained numerically. The results of normal displacement, normal force stress, tangential couple stress and temperature distribution are compared for micropolar cubic crystal and micropolar isotropic solid. The numerical results are illustrated graphically for a particular material. Some special cases are also deduced.     

Elastic half-space with several domains subjected to tangential displacements

Summary The mixed boundary value problem for a transversely isotropic elastic half-space is considered for the case when uniform tangential displacements are prescribed over several domains of arbitrary shape, and the rest of the half-space boundary is stress free. The problem can be interpreted either as that of two elastic half-spaces interconnected by several regions of general shape and subjected to remote shear loading, or as a contact problem of several flexible punches, connected to the half-space, with different tangential displacements prescribed. A general theorem is established which relates the resulting tangential forces, acting on each domain, with their generalized displacements through a system of linear algebraic equations. The theorem is applied to the case of arbitrarily located elliptical domains subjected to uniform tangential displacements. Several specific examples are considered.

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Der elastische halbraum mit einigen randbereichen vorgegebener tangentialverschiebung

Übersicht Behandelt wird das gemischte Randwertproblem für einen transversal-isotropn olastischen Halbraum, wenn eine gleichmäßaige Tangentialverschiebung einiger Oberflächenbereiche beliebiger Gestalt vorgeschrieben und der Rest des Randes spannungsfrei ist. Die Aufgabenstellung kann auf zweierlei Weise interpretiert werden: Als zwei elastische Halbräume, die längs einiger beliebig gestalteter Randbereiche miteinander verbunden sind, oder als das Kontaktproblem des Halbraumes mit einigen flexiblen Stempeln, für die verschiedene Tangentialverschiebungen vorgegeben sind. Aufgestellt wird ein allgemeines Theorem, welches die resultierenden Tangentialkräfte auf allen Bereichen mit ihren Verschiebungen durch ein System linearer algebraischer Gleichungen verknüpft. Angewandt wird das Theorem auf elliptische Bereiche mit beliebiger Position und gleichförmiger Tangentialverschiebung. Einige spezielle Beispiele werden vorgestellt.

     

A damage constitutive model for sliding friction coupled to wear

Several new constitutive models are formulated for the planar interface of a soft body sliding on a rigid soil, describing stick-slip phenomena due to friction, and wear due to abrasion. Attention is focused on damage at the interface, by neglecting any interaction with damage of body and any propagation of damage inside the body. Models are formulated in the general framework of the Thermodynamics of the irreversible processes and account for suitable defined internal variables of phenomenological type, namely gap, isotropic friction hardening and wear. The main feature of the new presented models is that the formulation of the wear process at the interface is obtained in the contest of Damage Mechanics, and it is based on the formal analogy between abrasion of a soft body and ductile damage of an elastic-plastic material. By following this approach, a scalar wear field, an effective stress and appropriate state and dissipation potentials are defined, able to describe a tangential isotropic wear process due to stick-slip and to hardening mechanism. Both cases of linear and nonlinear friction hardening are formulated; moreover, wearable and no-wearable bodies are considered. Numerical results relevant to one-dimensional problems are illustrated for monotonic, forward-backward and cyclic displacement time-histories, showing evolution for stress, gap and wear. Results furnished by different models are compared and discussed.     

Penny-shaped and half-plane cracks in a transversely isotropic piezoelectric solid under arbitrary loading

Summary The problem of a penny-shaped crack in a transversely isotropic piezoelectric material loaded by both normal and tangential tractions and by electric charges is analyzed. Closed-form solutions are obtained for the full electroelastic fields as well as for the stress and electric displacement intensity factors. Solutions are also obtained for the (non-trivial) limiting case of a half-plane crack. The results are illustrated on the example of piezoceramics PZT-6B. Received 12 July 1999; accepted for publication 20 July 1999     

Creep damage and life analysis of anisotropic materials

Summary  This paper provides a short survey of some recent advances in the mathematical modelling of materials behaviour under creep conditions. The tertiary creep phase is accompanied by the formation of microscopic cracks on the grain boundaries in such a way so that damage accumulation occurs. The paper is divided into three parts. Firstly, the damage state in a uniaxial tension specimen is discussed and the time to rupture calculated. The second part is concerned with the creep behaviour of materials in multiaxial stress. Because of its microscopic nature, damage generally has an anisotropic character even if the material was originally isotropic. The fissure's orientation and length cause anisotropic macroscopic behaviour. Therefore, damage in an isotropic or anisotropic material, which is in a state of multiaxial stress, can only be described in a tensorial form. Thus, tensorial constitutive and evolution equations have been developed. Some examples for practical use are discussed. Finally, some own experiments are mentioned which have been carried out in order to validate the mathematical modelling. Received 16 July 1999; accepted for publication 8 March 2000     

Asymptotic Analysis of Finite Deformation in a Nonlinear Transversely Isotropic Incompressible Hyperelastic Half-space Subjected to a Tensile Point Load

The Boussinesq problem, that is, determining the deformation in a hyperelastic half-space due to a point force normal to the boundary, is an important problem of engineering, geomechanics, and other fields to which elasticity theory is often applied. While linear solutions produce useful Greens functions, they also predict infinite displacements and other physically inconsistent results nearby and at the point of application of the load where the most critical and interesting material behavior occurs. To illuminate the deformation due to such a load in the region of interest, asymptotic analysis of the nonlinear Boussinesq problem has been considered in the context of isotropic hyperelasticity. Studies considering transversely isotropic materials have also been broadly used in the linear theory, but have not been treated within the nonlinear framework. In this paper we extend the nonlinearly elastic isotropic analysis to transverse isotropy, producing a more general theory which also better encompasses applications involving layered media. The governing equations for nonlinearly elastic, transversely isotropic solids are derived, conservation laws of elastostatics are invoked, asymptotic forms of the deformation solutions are hypothesized, and the differential equations governing deformation near the point load are determined. The analysis also develops sequences of simple tests to determine if a transversely isotropic material can possibly sustain a finite deflection under the point load. The results are applied to a variety of transversely isotropic materials, and the effects of the anisotropy considered is demonstrated by comparison of the resulting deformation with similar asymptotic solutions in the isotropic theory. Mathematics Subject Classifications (2000) 74B20, 74E10, 74G10, 74G15, 74G70.     

Micromechanical analysis of friction anisotropy in rough elastic contacts

Computational contact homogenization approach is applied to study friction anisotropy resulting from asperity interaction in elastic contacts. Contact of rough surfaces with anisotropic roughness is considered with asperity contact at the micro scale being governed by the isotropic Coulomb friction model. Application of a micro-to-macro scale transition scheme yields a macroscopic friction model with orientation- and pressure-dependent macroscopic friction coefficient. The macroscopic slip rule is found to exhibit a weak non-associativity in the tangential plane, although the slip rule at the microscale is associated in the tangential plane. Counterintuitive effects are observed for compressible materials, in particular, for auxetic materials.     

Short-term microdamage of a physically nonlinear fibrous material under simultaneous normal and tangential loads

The structural theory of short-term damage is generalized to the case where the undamaged isotropic matrix of a fibrous composite with transversely isotropic reinforcement deforms nonlinearly under loads that induce a combined stress state, microdamages occurring in the matrix alone. The basis for this generalization is the stochastic elasticity equations for a fibrous composite with porous matrix whose skeleton deforms nonlinearly. The Huber-Mises failure criterion is used to describe the damage of microvolumes in the matrix. The damaged microvolume balance equation is derived for the physically nonlinear material of the matrix based on the properties of the distribution function for the statistically homogeneous random field of ultimate microstrength. Together with the macrostress-macrostrain relationship, they constitute a closed-form system of equations. This system describes the coupled processes of physically nonlinear deformation and microdamage. Algorithms for calculating the dependences of macrostresses and microdamages on macrostrains are proposed. Stress-strain curves for a composite with a linearly hardened matrix under simultaneous normal and tangential loads are plotted. The effect of the volume fraction of reinforcement and tangential load on the curves is examined __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 3, pp. 48–59, March 2007.     

Surface effects in the deformation of an anisotropic elastic material with nano-sized elliptical hole

We examine the effect of surface energy on an anisotropic elastic material weakened by an elliptical hole. A closed-form, full-field solution is derived using the standard Stroh formalism. In particular, explicit expressions for the hoop stress, normal, in-plane tangential and out-of-plane displacement components along the edge of the hole are obtained. These expressions clearly demonstrate the effect of elastic anisotropy of the bulk material on the corresponding field variables. When the material becomes isotropic, the hoop stress agrees with the well-known result in the literature while both the in-plane tangential and out-of-plane displacements vanish and the normal displacement is constant along the entire boundary of the elliptical hole.     

Stress analysis of borehole subjected to fluid injection in transversely isotropic poroelastic medium

This paper proposes an analytical solution, using the combined Laplace–Fourier integral transform technique, for a borehole drilled in transversely isotropic porous medium and subjected to a fluid discharge over a finite length of its surface. Especially, the coupled boundary condition between the total radial stress and injection-induced pore pressure at the borehole surface is addressed in a rigorous fashion, which leads essentially to a set of dual integral equations that can be solved through standard numerical procedure. The study focuses on the calculation of stress fields around the borehole with particular attention given to the time-dependent effective tangential stress and pore pressure distributions. Numerical solutions are presented for verification with those recently derived for the limiting case of an isotropic medium and, more importantly, to investigate the influences of material anisotropy on the stress responses of the borehole and porous medium.