Showing papers in "Journal of Elasticity in 1976"
TL;DR: A thermomechanical continuum theory involving a chemical reaction and mass transfer between two constituents is developed in this article as a model for bone remodeling, which describes an elastic material which adapts its structure to applied loading.
Abstract: A thermomechanical continuum theory involving a chemical reaction and mass transfer between two constituents is developed here as a model for bone remodeling. Bone remodeling is a collective term for the continual processes of growth, reinforcement and resorbtion which occur in living bone. The resulting theory describes an elastic material which adapts its structure to applied loading.
659 citations
TL;DR: In this paper, it was shown that a femur will not undergo bone remodeling as a result of a pure torsion about the long axis of the bone, but rather due to a pure curvature of the femur.
Abstract: The general theory of adaptive elastic materials proposed in [1] as a model for the physiological process of bone remodeling is specialized to the case of small strains in isothermal processes. Boundary value problems in the context of this specialized theory are specified and solved. It is shown that remodeling will not occur in a long bone such as a femur as a result of a pure torsion about the long axis of the bone. Criteria for biologically normal remodeling are established for steady stress and steady strain.
191 citations
TL;DR: By distributing the concentrated singularities such as a Kelvin doublet along the axis of symmetry, the authors describes the displacement field, in an elastic medium, for various modes of rotation and translation for a rigid prolate and oblate spheroid.
Abstract: By distributing the concentrated singularities such as a Kelvin doublet along the axis of symmetry we describe the displacement field, in an elastic medium, for various modes of rotation and translation for a rigid prolate and oblate spheroid. The limiting cases of a sphere, a slender body and a thin circular disk are also discussed. All the solutions are presented in a closed form.
96 citations
TL;DR: In this paper, a method due to Friedlander of accommodating disturbances of arbitrary form into the theory of surface waves in a semi-infinite isotropic elastic body is extended and shown to yield a simple closed form solution for the displacement field.
Abstract: A method due to Friedlander of accommodating disturbances of arbitrary form into the theory of surface waves in a semi-infinite isotropic elastic body is extended and shown to yield a simple closed form solution for the displacement field. An analogous treatment of interfacial waves of arbitrary form at a plane contact discontinuity separating different isotropic elastic materials is also given.
65 citations
TL;DR: In this article, a method of solving the Saint-Venant's problem for inhomogeneous and anisotropic elastic bodies is presented, where the authors use an iterative approach to solve the problem.
Abstract: A method of solving Saint-Venant's problem for inhomogeneous and anisotropic elastic bodies is presented.
48 citations
TL;DR: In this paper, two integral equation formulations for the determination of the vertical displacement and the bending moment around holes in an elastic plate are presented, each formulation consists of two equations, the first one an integral equation and an integro-differential equation and the second one two singular integral equations.
Abstract: Two integral equation formulations for the determination of the vertical displacement and the bending moment around holes in an elastic plate are presented. Each formulation consists of two equations, the first one an integral equation and an integro-differential equation and the second one two singular integral equations. The equations are solved using B-splines as approximations to the unknowns and the method is applied to the case of one elliptic hole in a twisted plate.
44 citations
TL;DR: The general theory of linear constraints in linear elasticity theory is outlined in this article, and a modified form of constraint theory is proposed for problems that are illposed in constraint theory although well-posed in the absence of any constraint.
Abstract: The general theory of linear constraints in linear elasticity theory is outlined. For problems that are ill-posed in constraint theory although well-posed in the absence of any constraint, a modified form of constraint theory is proposed.
41 citations
TL;DR: In this paper, it was shown that a simple tensile load produces a simple extension provided the empirical inequalities (Truesdell and Noll [1], eqn. 51.27) hold.
Abstract: It is shown that a simple tensile load produces a simple extension provided the empirical inequalities (Truesdell and Noll [1], eqn. 51.27) hold.
34 citations
TL;DR: In this paper, a solution for the relationship between load, displacement and inner contact radius for an axisymmetric, spherically concave, rigid punch, indenting an elastic half-space was obtained.
Abstract: A solution is obtained for the relationship between load, displacement and inner contact radius for an axisymmetric, spherically concave, rigid punch, indenting an elastic half-space. Analytic approximations are developed for the limiting cases in which the ratio of the inner and outer radii of the annular contact region is respectively small and close to unity. These approximations overlap well at intermediate values. The same method is applied to the conically concave punch and to a punch with a central hole.
31 citations
TL;DR: In this article, the authors studied the entropy inequality of elastic materials whose local state depends upon the first and second order gradients of the deformation, the temperature, its gradient, and the time rate of change of the temperature.
Abstract: Elastic materials whose local state depends upon the first and second order gradients of the deformation, the temperature, its gradient and the time rate of change of the temperature are studied according to an inequality proposed by Green and Laws. It is shown that in such materials either thermal disturbances can propagate with finite speed in the linear theory, and the constitutive quantities do not depend upon the second order gradients of the deformation or the constitutive quantities may depend upon the second order gradients of the deformation and in the linear theory thermal disturbances do not propagate with finite speed. In the latter case the entropy inequality reduces to the Clausius-Duhem inequality.
24 citations
TL;DR: In this paper, the two-dimensional boundary value problems of elastic half-spaces were solved using Somigliana's method and the solutions were compared with those derived from dislocation considerations.
Abstract: In this paper Somigliana's method has been utilized to solve the two-dimensional boundary value problems of elastic half-spaces. We have considered traction and displacement problem, shear problem, contact and crack problems and the solutions, thus derived, have been compared with those derived from dislocation considerations.
TL;DR: In this article, self-similar problems of indentation of an elastic half-space by rigid cones or wedges are solved, assuming perfect adhesion, when the velocities of the indentation is large enough for the area of contact to spread faster than the speed of P-waves.
Abstract: Self-similar problems of indentation of an elastic half-space by rigid cones or wedges are solved, assuming perfect adhesion, when the velocity of indentation is large enough for the area of contact to spread faster than the speed of P-waves. In contrast to the earlier study of the wholly subsonic case [2], the present problems can be solved in closed form without approximation. It emerges, too, that the no slip condition would be satisfied for a range of values of a finite coefficient of friction, in contrast to the situation in [2], where any finite friction is bound to allow some slip. A variety of wave fronts exist in the present problems and all of their amplitudes are found explicitly and discussed.
TL;DR: In this paper, the problem of wave propagation within the framework of a complete linear theory of elastic plates is treated using the method of wave curves, and a complete classification of the various extensional and bending waves is obtained, along with the corresponding speeds of propagation.
Abstract: The problem of wave propagation within the framework of a complete linear theory of elastic plates is treated using the method of wave curves. A complete classification of the various extensional and bending waves is obtained, along with the corresponding speeds of propagation. These are shown to correspond to the phase velocities of harmonic waves for infinite wave number. The decay and coupling equations are found, and the problem of waves due to a punch applied to the plate surface is treated.
TL;DR: In this article, some basic equations recently derived by Clements are used to consider two contact problems in anisotropic thermoelasticity: the first problem concerns the determination of the thermal stress due to a heated load over a section of the boundary.
Abstract: Some basic equations recently derived by Clements are used to consider two contact problems in anisotropic thermoelasticity. The first problem concerns the determination of the thermal stress in an anisotropic half-space due to a heated load over a section of the boundary. The second problem concerns the indentation of a half-space by a heated rigid punch. In particular, indentation by a cylindrical punch is considered and some numerical results obtained.
TL;DR: For small deformations of isotropic incompressible hyperelastic materials which are superimposed upon the simultaneous inflation and extension of a cylindrical tube, new closed form solutions are derived without any restrictions on the strain-energy function as mentioned in this paper.
Abstract: For small deformations of isotropic incompressible hyperelastic materials which are superimposed upon the simultaneous inflation and extension of a cylindrical tube, new closed form solutions are derived without any restrictions on the strain-energy function. These solutions are used to derive the load-deflection relation for small radial deformations of pre-compressed long bonded cylindrical rubber bush mountings. They are also used to formulate the n=1 buckling criterion for long cylindrical tubes which are subjected to uniform external pressure.
TL;DR: For small axially symmetric deformations of isotropic incompressible hyperelastic materials which are superimposed upon the symmetrical expansion of a spherical shell, new closed form solutions are derived without any restrictions on the strain-energy function as discussed by the authors.
Abstract: For small axially symmetric deformations of isotropic incompressible hyperelastic materials which are super-imposed upon the symmetrical expansion of a spherical shell, new closed form solutions are derived without any restrictions on the strain-energy function. These solutions are used to derive the n=1 buckling criterion for thick-walled spherical shells which are subjected to uniform external pressure. They are also used to deduce an upper bound to the force deflection relation for small superimposed translational deflections of bonded pre-compressed spherical rubber bush mountings.
TL;DR: In this paper, a Green's function approach is used to formulate and obtain the stress field, under torsional loads in a radially finite solid cylinder with radially variable elastic modulus.
Abstract: A Green's function approach is used to formulate and obtain the stress field, under torsional loads in a radially finite solid cylinder with radially variable elastic modulus. With this approach a certain dual static-geometric analogy in the solution is readily proved and applied to generate the solution with stress boundary conditions from that with displacement boundary conditions and vice-versa. The problem is solved using both boundary conditions and for an exponentially varying shear modulus. In particular, under displacement boundary conditions, the stress field in the solid with a generalised Reissner-Sagoci boundary condition is easily deduced. With stress boundary conditions, the criteria for crack propagation in such elastic models are also obtained using the Griffith-Irwin condition of rupture.
TL;DR: In this article, it was shown that a general penny-shaped crack situated in a basal plane of a hexagonal crystal can be studied in a straightforward way using the corresponding results appropriate to an isotropic medium.
Abstract: It is shown that a general penny-shaped crack situated in a basal plane of a hexagonal crystal can be studied in a straightforward way using the corresponding results appropriate to an isotropic medium. The stress intensity factors are derived, and discussed for the case in which the crack is subjected to a unidirectional shear traction.
TL;DR: In this article, a plane elastostatic contact problem for an infinite elastic wedge of arbitrary angle was solved using the Mellin transform formulation and the mixed boundary value problem was reduced to a singular integral equation with the contact stress as the unknown function.
Abstract: The paper deals with the plane elastostatic contact problem for an infinite elastic wedge of arbitrary angle. The medium is loaded through a frictionless rigid wedge of a given symmetric profile. Using the Mellin transform formulation the mixed boundary value problem is reduced to a singular integral equation with the contact stress as the unknown function. With the application of the results to the fracture of the medium in mind, the main emphasis in the study has been on the investigation of the singular nature of the stress state around the apex of the wedge and on the determination of the contact pressure.
TL;DR: In this paper, the priciple of complementary energy is given for the von Karman nonlinear plate theory and necessary conditions are three linear and static equilibrium equations in the interior and static boundary conditions on that part of the boundary surface, where forces are prescribed.
Abstract: In this paper, the priciple of complementary energy is given for the von Karman nonlinear plate theory. Thenecessary conditions are three linear and static equilibrium equations in the interior and static boundary conditions on that part of the boundary surface, where forces are prescribed. The stationary value of the complementary energy functional leads to the stress-displacement relations and the geometric boundary conditions.
TL;DR: In this paper, upper and lower bounds for the shear stress were derived for the strength of materials formula in the limit of vanishing thickness, based on the theory of flexure.
Abstract: Upper and lower bounds are derived for the shear stress as it is determined by Saint-Venant's theory of flexure, and used to establish the asymptotic character of the classical Strength of Materials formula in the limit of vanishing thickness.
TL;DR: In this paper, the problem of load-absorption of a pair of identical parallel elastic filaments in a fiber-reinforced composite material is investigated, where the filaments are assumed to have uniform circular cross-sections, are taken to be semi-infinite, and are supposed to be continuously bonded to an all-around infinite matrix of distinct elastic properties.
Abstract: This investigation is concerned with the interaction—as far as load-absorption is concerned—of a pair of identical parallel elastic filaments in a fiber-reinforced composite material. The filaments are assumed to have uniform circular cross-sections, are taken to be semi-infinite, and are supposed to be continuously bonded to an all-around infinite matrix of distinct elastic properties. At infinity the matrix is subjected to uniaxial tension parallel to the filaments. Two separate but related problems are treated. In the first both filaments extend to infinity in the same direction and their terminal cross-sections are coplanar. In the second problem the filaments extend to infinity in opposite directions and their terminal cross-sections need no longer be coplanar, the two filaments being permitted to overlap partly. An approximate scheme based in part on three-dimensional linear elasticity and developed originally by Muki and Sternberg is employed in the analysis. The problems are ultimately reduced to Fredholm integral equations which characterize the distribution of the axial filament force. The integral equations are analyzed asymptotically and numerically. Results are presented which show the variation of filament force with position and the effect on this variation of various relevant geometrical and material parameters. One result is of particular interest. In the second problem, involving the overlapping filaments, for certain cases the filament force exceeds its far-field asymptotic value for a portion of the filament length. Stated another way, this means that the filament is loaded in excess of that which one would calculate by equating axial strains.
TL;DR: In this article, the authors derived compatibility relations that must be satisfied behind the curve of a curved shock wave in elastic fluids, which arise because the shock strengths are not uniform over the shock surfaces.
Abstract: In addition to examining the infuence of thermodynamic effects on the evolutionary behavior of curved shock waves in elastic fluids, we derive certain compatibility relations which must be satisfied behind the shocks. These relations arise because the shock strengths are not uniform over the shock surfaces. We also present an alternate method of derivation of the well-known result concerning the vorticity jump.
TL;DR: In this paper, the contact problem for an elastic wedge of arbitrary angle is considered and an effective numerical solution of the integral equation is described and the results of three examples are presented.
Abstract: In this paper the contact problem for an elastic wedge of arbitrary angle is considered. It is assumed that the external load is applied to the medium through a rigid wedge and the coefficient of friction between the loading wedge and the elastic solid is constant. The problem is reduced to a singular integral equation of the second kind with the contact pressure as the unknown function. An effective numerical solution of the integral equation is described and the results of three examples are presented. The comparison of these results with those obtained from the frictionless wedge problem indicates that generally friction has the tendency of reducing the peak values of the stress intensity factors calculated at the wedge apex and at the end points of the contact area.
TL;DR: In this article, it was shown that for a bounded Hookean body, it is impossible for all three principal stresses (o-1, o-2, °3) to have relative minima (or minima) at an interior point P of a body unless all the extremes are improper, since otherwise the sum o-1 +a2+o-3 would have a proper relative maximum or minimum at P.
Abstract: A casual survey of particular solutions in elasticity theory suggests that, when a bounded Hookean body is loaded only by boundary forces that are in equilibrium, the absolute maximum tensile stress occurs at a boundary point. Using potential theory, G. Polya [1] investigated this question in 1930. He demonstrated that extremes of the cubical dilatation and the rotation necessarily occur on the boundary, and he proved several other theorems, but his conclusions concerning extreme principal stresses appear to be inferential. This question is discussed in this note. Normal stresses are compared algebraically; e.g., a tensile stress is considered to be greater than a compression stress. A stress will be said to have a proper (improper) relative maximum at a point P if, throughout a deleted neighborhood of P, its value is less than (less than or equal to) its value at P. Similarly, proper and improper relative minima are defined. It is not possible for all three principal stresses (o-1, o-2, °3) to have relative maxima (or minima) at an interior point P of a body unless all the extremes are improper, since otherwise the sum o-1 +a2+o-3 would have a proper relative maximum (or minimum) at P. However, this is impossible, since o-1 + a2 + a3 is a harmonic function, and it is known [2] that a harmonic function cannot have a proper relative extreme at an interior point of the region of regularity.
TL;DR: In this paper, the uniaxial theory is employed and two separate techniques are used for extracting information about the stress field for impact problems in inhomogeneous conical shells.
Abstract: Here we study transient elastic wave propagation in inhomogeneous conical shells. The uniaxial theory is employed and two separate techniques are used for extracting information about the stress field for impact problems. Firstly the formal Karal-Keller method is used enabling us to directly determine asymptotic wavefront expansions for the stress field. A transform technique, based on Eason's [1], is then used to obtain conditions on the physical parameters of the medium which give solutions to the governing equation in terms of Bessel functions and some particular problems are discussed. Previously unknown simple closed-form solutions are obtained. Our results have application to the propagation of waves in filamentary cones of constant wall thickness or in homogeneous cones having an axial temperature gradient.
TL;DR: In this paper, the authors derived fundamental matrix solutions of an infinite elastic dielectric plane subjected to a concentrated body force, electric force and charge density from the singular solutions of harmonic, biharmonic and Helmholtz equations.
Abstract: Representations of Galerkin type are obtained for the displacement vector, polarization vector and the potential fields in the static plane theory of elastic dielectrics using the method of associated matrices. Fundamental matrix solutions of an infinite elastic dielectric plane subjected to a concentrated body force, electric force and charge density are derived from the singular solutions of harmonic, biharmonic and Helmholtz equations. Using boundary operatorsY, Z, M, the fundamental matrix solutions, and Betti's formulae, a matrix Λ(x, y) is constructed and an integral representation for (u1,u2,P1,P2, ϕ) is obtained. Discontinuity theorems are stated for the double layer potential andQ operator of the single layer potential. By means of these theorems, the solutions of interior and exterior boundary value problems are reduced to the solution of a system of five singular integral equations. The index of one of the systems is shown to be zero and it is concluded that Fredholm theorems and its alternatives hold.
TL;DR: In this paper, the effect of fiber reinforcement on the dynamic universality of bending and shearing of a rectangular block, straightening and shearning of a sector of a circular tube, inflation, eversion, extension, bending, shearing, and azimuthal shearing were investigated.
Abstract: We consider the effect of a fiber-reinforcement on the dynamic universality of the following families of motions: bending and shearing of a rectangular block; straightening and shearing of a sector of a circular tube; inflation, eversion, extension, bending and shearing of a sector of a circular tube; inflation, extension, bending and azimuthal shearing of a sector of a circular tube
TL;DR: In this paper, a conical punch presses into an elastic layer resting on a rigid foundation, and numerical results for the radius of the circle of contact, the force required, and the displacement of the center of the punch are presented.
Abstract: A conical punch presses into an elastic layer resting on a rigid foundation. Precision numerical results for the radius of the circle of contact, the force required, and the displacement of the center of the punch are presented.