Dynamic interaction of parallel cracks in elastic bodies

We study time dependences of the stress intensity factors in an infinite isotropic elastic body with parallel cracks whose faces are subjected to impact loads. The original problem is reduced to the solution of a system of boundary integral equations with a parameter corresponding to the wave number in the decomposition of the dynamic process into the set of monochromatic components. The numerical reconstruction of the dependences of unknown quantities on time is carried out for two circular cracks loaded by tensile forces in the form of the Heaviside function. The presence of stress intensity factors of all three types is a distinctive feature of the considered problem. Ya. S. Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lvov, Ukraine. Translated from Problemy Prochnosti, No. 3, pp. 94–102, May–June, 1998.     

On free oscillations of elastic incompressible bodies in finite shear

Large amplitude free oscillations of thick-walled elastic, incompressible bodies are studied resulting from three types of shearing deformations. The material strain-energy function is expressed in a general form of finite series expansion so that the governing equations of motion in each case reduce to a system of non-linear second order partial differential equations. The degree of non-linearity is shown to be dependent on the number of terms retained in the power series. Approximate results for the non-linear systems are gained by a regular perturbation scheme. Illustrative problems using a strain-energy function for some rubber-like materials are also included.     

Stress concentration effects in microstretch elastic bodies

This paper is concerned with the plane strain problem of the equilibrium theory of microstretch elastic bodies. First, we study the problem of stress concentration in the neighbourhood of a circular hole located in a plane subjected to the action of constant loads at a great distance from the hole. Then, the problem of a rigid inclusion in an infinite body is studied.     

Cracks in gradient elastic bodies with surface energy

In the present paper the effect of higher order gradients on the structure of line-crack tips is determined. In particular we introduce in the constitutive equations of the linear deformation of an elastic solid a volumetric energy term, which includes the contribution of the strain gradient, and a surface energy gradient dependent term and then determine the effect of these terms on the structure of the mode-III crack tip and the associated stress and strain fields. By making use of the solution in terms of Fourier transform of the equation of elastic equilibrium we solve the half-plane boundary value problems of: (a) specified tractions, and (b) prescribed displacements, along the crack surface, respectively.Presented at Fourth Greek National Congress on Mechanics, 26–29 June 1995, held at Xanthi, Greece.     

A simple finite element‐based framework for the analysis of elastic pseudo‐rigid bodies

This article advocates a general procedure for the numerical investigation of pseudo‐rigid bodies. The equations of motion for pseudo‐rigid bodies are shown to be mathematically equivalent to those corresponding to certain constant‐strain finite element approximations for general deformable continua. A straightforward algorithmic implementation is achieved in a classical finite element framework. Also, a penalty formulation is suggested for modelling contact between pseudo‐rigid bodies. Representative planar simulations using a non‐linear elastic model demonstrate the predictive capacity of the pseudo‐rigid theory, as well as the robustness of the proposed computational procedure. Copyright © 1999 John Wiley & Sons, Ltd.     

Pulse propagation in finite elastic inhomogeneous media

The transfer matrix of a finite elastic bar is derived and the reflection and transmission functions are obtained. The matrix formalism is combined with Fourier component decomposition and applied to simulate acoustic pulse propagations in elastic inhomogeneous periodic and finite media. The numerical results are compared with experimental data to conclude about the validity of this method.     

Some experiments on the elastic stability of some highly elastic bodies

Uniaxial compression experiments on solid circular bars, rectangular bars, and thick-walled circular tubes made of rubber materials are described, and the data are compared with known theoretical results derived for neo-Hookean materials. It is found in all cases that the general nature of certain global analytical estimates are in good qualitative agreement with existence of a transition slenderness ratio that separates buckling characteristic of long bars from axisymmetrie bulging characteristic of short bars. The data agrees qualitatively also with other more precise analytical results for circular and rectangular bars, but only as regards Euler type buckling. Specific analytical results for axisymmetrie deformation of circular tubes are found to be without experimental foundation.     

Thermodynamic influences on shock waves in inhomogeneous elastic bodies

Summary In this paper, the differential equation governing the behavior of shock waves propagating in inhomogeneous thermoelastic bodies is derived. The implications of this equation on the local behavior of the amplitudes are examined. It is shown that the behavior depends on the relative magnitudes of a number , called the critical jump in strain gradient, and the jump in strain gradient across the shocks.

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Thermodynamische Einflüsse auf Stoßwellen in inhomogenen elastischen Körpern

Zusammenfassung In dieser Arbeit wird die das Verhalten von in inhomogenen thermoelastischen Körpern sich ausbreitenden Stoßwellen charakterisierende Differentialgleichung abgeleitet. Die Folgerungen auf das lokale Verhalten der Amplituden wird untersucht. Es wird gezeigt, daß das Verhalten von der relativen Größe einer Zahl , genannt kritischer Sprung des Verzerrungsgradienten, und dem Sprung im Verzerrungsgradienten über den Stoß abhängt.

     

On finite deformations of elastic membranes

The established theory of the Finite Deformations of Elastic Membranes is based on the assumption that the considered membrane is deformed under the action of edge tractions and pressures applied on its two faces. It is further tacitly assumed that the pressures are small. Though the assumption of small pressure-difference is reasonable, the pressure themselves need not be small. Neither the surface tractions need be normal to the corresponding surfaces. In this paper the equations of Finite Elasticity are used for the derivation of a generalized Membrane Theory. No assumptions are made about the surface tractions. A small non-dimensional parameter is introduced in the three-dimensional equations, where 2h is the variable thickness of the membrane and L a characteristic length. The zero and first-order equations are obtained. From the zero-order equations it follows that the pressures applied on the faces of the membrane are equal to the zero-order and the shear tractions equal and opposite to the same order. From the first-order equations it is deduced that the pressure difference is small. It is also proved that for zero shear tractions and small pressures the equations derived here reduce to the usual equations of the non-linear Membrane Theory.     

A finite element analysis of some boundary value problems for a new type of constitutive relation for elastic bodies

Recently, there has been interest in the study of a new class of constitutive relation, wherein thelinearized strain tensor is assumed to be a function of the stresses. In this communication, some boundaryvalue problems are solved using the finite element method and the solid material being described by such aconstitutive relation, where the stresses can be arbitrarily ‘large’, but strains remain small. Three problems areanalyzed, namely the traction of a plate with hyperbolic boundaries, a plate with a point load, and the tractionof a plate with an elliptic hole. The results for the stresses and strains are compared with the predictions thatare obtained by using the constitutive equation of the classical linearized theory of elasticity.     

A 2.5D finite/infinite element approach for modelling visco‐elastic bodies subjected to moving loads

The objective of this study is to propose a 2.5D finite/infinite element procedure for dealing with the ground vibrations induced by moving loads. Besides the two in‐plane degrees of freedom (DOFs) per node conventionally used for plane strain elements, an extra DOF is introduced to account for the out‐of‐plane wave transmission. The profile of the half‐space is divided into a near field and a semi‐infinite far field. The near field containing loads and irregular structures is simulated by the finite elements, while the far field covering the soils extending to infinity by the infinite elements with due account taken of the radiation effects for moving loads. Enhanced by the automated mesh expansion procedure proposed previously by the writers, the far field impedances for all the lower frequencies are generated repetitively from the mesh created for the highest frequency considered. Finally, the accuracy of the proposed method is verified through comparison with a number of analytical solutions. Copyright © 2001 John Wiley & Sons, Ltd.     

Perturbations about a finite elastic inflation

An exact general analytic solution for a class of boundary value problems involving perturbations about a finite inflation of a slab containing a circular hole or inclusion is obtained. The equilibrium equations for the perturbed state are derived in terms of a general strain-energy function and solved exactly for Mooney-Rivlin materials. The method is not, however, restricted to this particular class of materials. Applications are made to the case where a perturbational uniaxial tension is acting at sections far from the cavity or inclusion and to the case where a small shear is applied at the edge of the hole. The deformation, the stress field and stress concentration around the hole are investigated in detail and computational results are presented graphically.     

On finite amplitude waves in heterogeneous elastic solids

The perturbation technique of the ‘stretching of the coordinates’ is used to obtain first and second order perturbation solutions of finite amplitude plane waves which propagate into an elastic half-space whose material property varies in the direction of the propagation. The interaction between nonlinearity and heterogeneity is discussed, and the results are illustrated by means of two examples: the longitudinal waves propagating in an elastic half-space with harmonic heterogeneity; the shear wave in a half-space whose property varies as A[1 + ?(X/L)ⁿ], where A, ? ? 1, L, and n are constants, and X denotes the initial particle position measured normal to the plane boundary.     

Crack interaction in an infinite elastic strip

The problem considers an arbitrary number of colinear and unequal size Griffith cracks opened by a non-uniform internal pressure in an infinite elastic strip. The cracks are located halfway between and parallel to the surfaces of the 2-dimensional medium. By appropriate integral transformations the mixed boundary value problem is reduced to singular integral equations. The stress intensity factors, crack openings and crack energies are then determined for many different cases.     

Topological derivatives for fundamental frequencies of elastic bodies

In this article a new method for topological optimization of fundamental frequencies of elastic bodies, which could be considered as an improvement on the bubble method, is introduced. The method is based on generalized topological derivatives. For a body with different types of inclusion the vector genus is introduced. The dimension of the genus is the number of different elastic properties of the inclusions being introduced. The disturbances of stress and strain fields in an elastic matrix due to a newly inserted elastic inhomogeneity are given explicitly in terms of the stresses and strains in the initial body. The iterative positioning of inclusions is carried out by determination of the preferable position of the new inhomogeneity at the extreme points of the characteristic function. The characteristic function was derived using Eshelby's method. The expressions for optimal ratios of the semi-axes of the ellipse and angular orientation of newly inserted infinitesimally small inclusions of elliptical form are derived in closed analytical form.     

Modelling cracks in finite bodies by distributed dislocation dipoles

The method of continuous distribution of dislocations is extended here to model cracks in finite geometries. The cracks themselves are still modelled by distributed dislocations, whereas the finite boundaries are represented by a continuous distribution of dislocation dipoles. The use of dislocation dipoles, instead of dislocations, provides a unified formulation to treat both simple and arbitrary boundaries in a numerical solution. The method gives a set of singular integral equations with Cauchy kernels, which can be readily solved using Gauss–Chebyshev quadratures for finite bodies of simple shapes. When applied to arbitrary geometries, the continuous distribution of infinitesimal dislocation dipoles is approximated by a discrete distribution of finite dislocation dipoles. Both the stress intensity factor and the T -stress are evaluated for some well-known crack problems, in an attempt to assess the performance of the methods and to provide some new engineering data.     

On complex potentials in the theory of microstretch elastic bodies

The paper is concerned with the plane strain problem in the equilibrium theory of microstretch elastic solids. We show that the complex variable technique of the classical theory of elasticity can be extended to the theory of microstretch elastic bodies. The method is used to study the effect of the stress concentration around a circular hole.     

Some results on finite amplitude elastic waves

Summary In this paper some special types of finite amplitude wave motions are considered, for which kinematical non-linearities do not arise in the equations of motion of an elastic solid. Consequently, only constitutive non-linearities occur and, for special classes of materials, solutions may be read off from corresponding solutions in the linear theory. These include SH-waves¹ and Love waves in layered or inhomogeneous media. Finite amplitude transverse circularly-polarized harmonic progressive waves are shown to propagate in any compressible or incompressible isotropic elastic material. Some effects of homogeneous pre-stress are also investigated.

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Zusammenfassung Es werden Spezialfälle von Wellenbewegungen mit endlicher Amplitude betrachtet, für die keine kinematischen Nichtlinearitäten in den Bewegungsgleichungen des elastischen Festkörpers auftreten. Es kommen also nur Material-Nichtlinearitäten vor. Für gewisse Klassen von Materialien können die Lösungen aus den entsprechenden der linearen Theorie gewonnen werden. Hierher gehören horizontal polarisierte Scherwellen und Lovesche Wellen in geschichteten oder inhomogenen Medien. Es wird gezeigt, daß sich harmonische, zirkularpolarisierte Transversalwellen in jedem kompressiblen oder inkompressiblen elastischen Material fortpflanzen. Einige Effekte homogener Vorspannung werden ebenfalls untersucht.