Showing papers in "Journal of Elasticity in 1975"
TL;DR: In this paper, a non-linear theory of elasticity is set up in the most convenient form (lagrangian coordinates and stress tensor), and the appropriate energy-momentum tensor is derived, and it is shown that the integral of its normal component over a closed surface gives the force (as the term is used in the theory of solids) on defects and inhomogeneities within the surface.
Abstract: The application to continuum mechanics of the general methods of the classical theory of fields is advocated and illustrated by the example of the static elastic field. The non-linear theory of elasticity is set up in the most convenient form (lagrangian coordinates and stress tensor). The appropriate energy-momentum tensor is derived, and it is shown that the integral of its normal component over a closed surface gives the force (as the term is used in the theory of solids) on defects and inhomogeneities within the surface. Other topics discussed are Gunther's and related integrals, symmetrization of the energy-momentum tensor, and the Eulerian formulation. Some further extensions, existing and potential, are indicated.
675 citations
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TL;DR: For elastic bars, the authors discuss some material instabilities for a barre élastique, and discuté quelques instabilités matérielles.
Abstract: For elastic bars, we discuss some material instabilities.RésuméPour une barre élastique nous discutons quelques instabilités matérielles.
540 citations
TL;DR: In this paper, it was shown that ellipticity prevails only if the principal stretches are suitably restricted and break down, in particular at a local state of uni-axial tension or compression of sufficiently severe intensity.
Abstract: This paper deals with the ellipticity of the equations of finite elastostatics for a compressible material that corresponds to a special choice of the strain-energy density and has received repeated attention in the literature. The possible failure of ellipticity of the appropriate system of displacement equations of equilibrium at solutions involving large deformations was suggested by certain difficulties encountered in an attempt to determine the deformations and stresses arising in such a material near the tip of a crack. It is shown here that ellipticity prevails only if the principal stretches are suitably restricted and breaks down, in particular, at a local state of uni-axial tension or compression of sufficiently severe intensity.
204 citations
TL;DR: In this paper, the indentation of an elastic half-space by an axisymmetric punch under a monotonically applied normal force is formulated as a mixed boundary value problem under the assumption of Coulomb friction with coefficient μ in the region of contact.
Abstract: The indentation of an elastic half-space by an axisymmetric punch under a monotonically applied normal force is formulated as a mixed boundary value problem under the assumption of Coulomb friction with coefficient μ in the region of contact. Within an inner circle the contact is adhesive, while in the surrounding annulus the surface moves inwards with increasing load. The slip boundary between the two regions depends on μ and the Poisson ratio v, and is found uniquely as an eigenvalue of a certain integral equation.
202 citations
TL;DR: In this paper, the displacement induced by angular dislocation in an elastic half space is investigated and the surface data are seen to exhibit a very simple dependence on the elastic constants, which can be used to construct the fields for any polygonal loop by superposition.
Abstract: The solution for an angular dislocation allows one to construct the fields for any polygonal loop by superposition. The paper presents the displacements induced by the angular dislocation in an elastic half space. In view of potential applications in geophysics, particular attention is paid to the elastic fields at the free surface. The surface data are seen to exhibit a very simple dependence on the elastic constants.
183 citations
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TL;DR: In this article, the Somigliana formula is used to reduce an arbitrary elastic crack problem to a system of three integral equations for the components of displacement discontinuity for the case of a penny-shaped crack situated in an infinite isotropic medium.
Abstract: The Somigliana formula is used to reduce an arbitrary elastic crack problem to a system of three integral equations for the components of displacement discontinuity. For the case of a penny shaped crack situated in an infinite isotropic medium with the crack faces subjected to arbitrary tractions, the integral equations are solved explicitly. In particular integral formulae are obtained for the stresses on the plane of the crack beyond the crack-tip, and hence for the stress intensity factors. The special case of uni-directional shear traction on the crack is examined.
86 citations
TL;DR: For equilibrium states of elastic materials some general formulae of conservation type have been established in recent papers by Knowles and Sternberg and by Green as mentioned in this paper, which arise naturally from the application of standard integral identities to an energy-momentum tensor first introduced into elastostatics by Eshelby.
Abstract: For equilibrium states of elastic materials some general formulae of conservation type have been established in recent papers by Knowles and Sternberg and by Green. It is shown that these results arise naturally from the application of standard integral identities to an energy-momentum tensor first introduced into elastostatics by Eshelby. A duality is exhibited between the energy-momentum tensor and the Cauchy stress which leads directly to inverse deformation relations for elastic solids due originally to Shield.
80 citations
TL;DR: In this article, the axially symmetric problem for a penny-shaped cut is solved in the case the displacements are prescribed on its upper surface and stresses on its lower surface.
Abstract: The axially symmetric problem for a penny-shaped cut is solved in the case the displacements are prescribed on its upper surface and stresses on its lower surface. The solution is achieved by using integral transforms along with certain representations that reduce the general problem to the solution of uncoupled Hilbert problems. For polynomial loadings the Hilbert problems can be solved exactly by methods given by Muskhelishvili. As in the related plane problem two singularities at the edge of the disc are noted in the solution.
39 citations
TL;DR: In this paper, a succinct analysis of interfacial discontinuities in anisotropic elastic solids is presented, combined with known results on the ellipsoidal inclusion problem to provide some general formulae for the determination of stress (and strain) concentration factors.
Abstract: The paper contains a succinct analysis of interfacial discontinuities in anisotropic elastic solids. The results are combined with known results on the ellipsoidal inclusion problem to provide some general formulae for the determination of stress (and strain) concentration factors. Some explicit results are given for cavities in an infinite matrix under arbitrary uniform loading at infinity.
37 citations
TL;DR: In this article, the transformation relations between the relevant variables (and hence also the response functions) of the traditional formulation and one which can be obtained relative to one of the major surfaces are derived.
Abstract: The equations of motion in terms of resultants and the constitutive equations in the theory of shells and plates are ordinarily derived from the three-dimensional equations relative to an interior surface, often taken to be the middle surface in the reference configuration of the shell-like body. This usual formulation is not, in general, applicable to contact problems of shells for which one of the major surfaces, e.g., the upper surface, is specified as the reference surface. The present paper is concerned with the transformation relations between the relevant variables (and hence also the response functions) of the traditional formulation and one which can be obtained relative to one of the major surfaces. Our derivations, carried out both from the three-dimensional equations and by a direct approach, utilize only the conservation laws and the field equations without recourse to constitutive equations. Hence, the results are applicable to any shell-like medium and their validity is not limited to elastic shells alone.
20 citations
TL;DR: In this paper, the propagation of small-amplitude second-sound waves is discussed and the results compared with those predicted by the theory of Lord and Shulman [1].
Abstract: Standard techniques of continuum mechanics are used to describe the flow of certain microscopic excitations which are believed to give rise to the second-sound phenomena in some materials at low temperatures. Appropriate balance laws are formulated, and constitutive equations for an elastic solid are postulated. The propagation of small-amplitude second-sound waves is discussed and the results compared with those predicted by the theory of Lord and Shulman [1].
TL;DR: In this paper, a two-dimensional continuum model with couple stress for a gridwork-reinforced composite is developed based on the calculation of equivalent potential and kinetic energies stored in the representative medium.
Abstract: A two-dimensional continuum model with couple stress for a gridwork-reinforced composite is developed. The derivation is based on the calculation of equivalent potential and kinetic energies stored in the representative medium. Hamilton's principle is used to derive the equations of motion and the boundary conditions. Deformation variables are defined and the constitutive relations are subsequently derived. The in-plane transverse vibration problem is investigated as an evaluation example for which both the continuum approach and the discrete element method are employed to compute the frequencies. Numerical results show that solutions according to both methods agree reasonably well.
TL;DR: In this paper, it was shown that the limiting values of the integrands are essentially determined by the curvature of the surface of the elastic body and by the gradient of the solution of the integral equation.
Abstract: The integrals in certain singular integral equations of the theory of elasticity are defined in the sense of the Cauchy principal value. The existence of the Cauchy principal value has been proved for the plane problem by numerous authors (see Muskhelishvili [1], [2]) and for the three-dimensional problem by Kupradze and co-workers [3]. The knowledge of the limiting values of the integrands at the test point is essential for the numerical treatment. In this paper it is shown that the limiting values of the integrands are essentially determined by the curvature of the surface of the elastic body and by the gradient of the solution of the integral equation. A special regard is payed to test points at which the curvature and the gradient are discontinuous.
TL;DR: In this article, a method is developed for the determination of the stresses in an elastic sheet with one or more unloaded holes, which is particularly well suited for finding the hoop stresses since it is based on a system of integral equations in which the hoop stress itself is one of the unknowns.
Abstract: A method is developed for the determination of the stresses in an elastic sheet with one or more unloaded holes. The method is particularly well suited for finding the hoop stresses since it is based on a system of integral equations in which the hoop stress itself is one of the unknowns. The derivation and the numerical solution of the equations is described. A number of applications of the method is also presented.
TL;DR: In this article, an ideal fiber-reinforced material is defined to be inextensible in one or more directions through each point of the material and a model for predicting the deformation and stresses produced in real materials which are very much stronger in resisting extension in those directions than in any other possible mode of deformation.
Abstract: An ideal fibre-reinforced material is defined to be inextensible in one or more directions through each point of the material. It has been found to be a good model for predicting the deformation and stresses produced in real materials which are very much stronger in resisting extension in those directions than in any other possible mode of deformation.
TL;DR: The duality between stress and deformation fields for plane deformations of a compressible isotropic hyperelastic material established by J. M. Hill [1] is generalized to deformations in this article.
Abstract: The duality between stress and deformation fields for plane deformations of a compressible isotropic hyperelastic material established by J. M. Hill [1]is generalized to deformations of a homogeneous elastic material without the restrictions of isotropy and hyperelasticity. At the same time a clarification of Hill's results is achieved.
TL;DR: In this paper, a closed chain of eight dual and reciprocal states of an elastic body was constructed, roughly speaking by interchanging the roles of stress and deformation, and simple formulae relating the strain-energy functions for all eight states can be written down.
Abstract: For certain equilibrium states of an elastic body dual states (in the sense of Hill [1] and Ogden [2]) can be constructed, roughly speaking by interchanging the roles of stress and deformation. Furthermore, for each of the original and dual states, a reciprocal state (in the sense of Adkins [3] and Shield [4]) can be found by interchanging initial and final coordinates. Although the resulting reciprocal states are not dually related, a closed chain of eight dual and reciprocal states can be constructed. In the case of plane strain simple formulae relating the strain-energy functions for all eight states can be written down.
TL;DR: In this paper, it was shown that any new solution cannot have more than one of the proper vectors of the deformation tensorc determining a vector field of constant non-zero abnormality.
Abstract: One is concerned with the problem of determining the static deformations which can be produced in every isotropic, homogeneous, incompressible elastic body by the action of surface tractions alone. It is shown that any new solution cannot have more than one of the proper vectors of the deformation tensorc determining a vector field of constant non-zero abnormality.
TL;DR: In this paper, the authors investigate the nonexistence and growth of weak solutions of nonlinear elastodynamics equations under various hypotheses on the data and on the form of the strain energy function.
Abstract: This paper investigates questions of nonexistence and growth of weak solutions of a system of equations of nonlinear elastodynamics under various hypotheses on the data and on the form of the strain energy function.
TL;DR: In this article, the contact behavior of a slender die indenting an elastic half-space was studied and it was shown that the problem of determining the pressure on the elastic halfspace may be reduced to a single variable integral equation, whose solution is commonly represented by an asymptotic series in a small parameter.
Abstract: The paper deals with the contact behaviour of a slender die indenting an elastic half-space. It is shown that the problem of determining the pressure on the elastic half-space may be reduced (with an error exponentially small relative to the elongation) to a single-variable integral equation, whose solution is commonly represented by an asymptotic series in a small parameter.
TL;DR: In this article, the authors used the matched asymptotic expansions to find an approximate expression for the applied couple as a function of the angle of rotation of the rigid inclusion, where the outer radius of the annulus is very large compared to the inner radius.
Abstract: A thin annular plate contains a rigid, circular, central inclusion. The plate is subjected to a large axisymmetric radial load at its outer edge, where it is also restrained against transverse displacement and rotation. A couple applied to the rigid inclusion causes it to rotate about its diameter out of the plane of the plate. We use the method of matched asymptotic expansions to find an approximate expression for the applied couple as a function of the angle of rotation of the rigid inclusion. If the outer radius of the annulus is very large compared to the inner radius, then the couple required to rotate a truly rigid inclusion is 25% higher than the couple required to rotate an inclusion whose membrane strain stiffness is the same as that of the plate (cf. ref. [3]) through the same small angle.
TL;DR: In this paper, a uniqueness theorem for the solution of a nonlinear initial-boundary value problem for a time dependent form of von Karman's equations is proven by the method of energy integrals.
Abstract: A uniqueness theorem for the solution of a nonlinear initial-boundary value problem for a time dependent form of von Karman's equations is proven. These equations are two coupled nonlinear fourth order partial differential equations which describe the bending of an elastic plate. The result is proved by the method of energy integrals.
TL;DR: In this paper, the stability under overall axial compression of a finitely inflated cylindrical membrane composed of highly elastic material is investigated and the critical loads for inflated tubes with closed ends and with either simply-supported or fixed ends are determined in terms of the material properties of the membrane.
Abstract: The stability under overall axial compression of a finitely inflated cylindrical membrane composed of highly elastic material is investigated. The critical loads for inflated tubes with closed ends and with either simply-supported or fixed ends are determined in terms of the material properties of the membrane. For long tubes the results are compared with the Euler formulae for the buckling load for struts in compression. An “equivalent Young's modulus” is derived, and it is shown that the critical loads can be obtained from the Euler formulae by using the dimensions of the inflated state and the equivalent Young's modulus.
TL;DR: In this paper, an iterative method is given and used to determine stress intensity factors for a star-like arrangement of cracks where each crack is at a distance s from the centre of the star.
Abstract: An iterative method is given and used to determine stress intensity factors for a starlike arrangement of cracks where each crack is at a distance s from the centre of the star. The problem is distinguished by the fact that in the limit e = 0 it is not possible to satisfy all the conditions of the problem. This suggests a singular perturbation procedure, however a formal iterative method is developed here from which the error at each step can be clearly seen. The method is not restricted to this problem, a rather different formulation of a similar method has been applied in [1] to determine stress intensity factors for cracks approaching or just passing through an interface. In fact the idea of the method should apply to any problem for which the limit solution e = 0 gives a different stress-singularity to that found in the e = 0 case.
TL;DR: In this paper, the authors used the ideal theory of fiber-reinforced or strong anisotropic materials to evaluate the stress distribution in an elastic component which contains a rigid inclusion and found that the stress and displacement fields in many two-dimensional inclusion problems are easily obtained.
Abstract: It is often difficult to evaluate the stress distribution in an anisotropic elastic component which contains a rigid inclusion. By using the ideal theory of fibre-reinforced or strong anisotropic materials it is found that the stress and displacement fields in many two-dimensional inclusion problems are easily obtained. These fields are found to depend upon the overall dimensions of the inclusion and are almost independent of its shape.
TL;DR: In this paper, the problem of plane steady vibration of an elastic wedge subject to harmonic normal and shearing tractions on its faces is reduced to a system of singular integral equations by the superposition of two half-plane solutions.
Abstract: The problem of plane steady vibration of an elastic wedge of arbitrary angle (less than 180 degrees) subject to harmonic normal and shearing tractions on its faces is reduced to a system of singular integral equations by the superposition of two half-plane solutions The integral equations have kernels with Cauchy singularities of a non-translation type, except for the 90 degree wedge The locations of these singularity lines are shown graphically as a function of wedge angle
TL;DR: In this paper, strong superadditivity properties of the stress function in the elastic torsion problem for multiply connected regions were proved. And the results were extended to associated Green's functions and other domain functions.
Abstract: In this note we prove strong superadditivity properties of the stress function in the elastic torsion problem for multiply connected regions. The results are extended to associated Green's functions and other domain functions.
TL;DR: In this article, the authors obtained upper bounds for the torsional rigidity of an isotropic right cylinder whose ends are restrained against warping in terms of the Saint-Venant rigidity, the polar moment of inertia and the lowest free membrane eigenvalue of the cross-section, the length of the cylinder and the elastic constants.
Abstract: We obtain upper bounds for the torsional rigidity of an isotropic right cylinder whose ends are restrained against warping in terms of the Saint-Venant torsional rigidity, the polar moment of inertia and the lowest free membrane eigenvalue of the cross-section, the length of the cylinder and the elastic constants They may be used to show that the two torsional rigidities tend to coincidence as the cylinder becomes infinitely long Various other implications of the bounds are also discussed