Elastic Energy-momentum Tensor Research Articles

Generating elastic solutions of the Einstein field equations from the Schwarzschild vacuum solution

The problem of generating solutions of the Einstein field equations with an elastic energy–momentum tensor from the Schwarzschild vacuum solution by means of conformal transformations is analyzed. Applying the formulation of relativistic elasticity, suitable conformal factors are obtained for static and non-static elastic spacetime configurations and particular solutions are presented. This work shows that the technique used here permits generating new elastic matter solutions from a vacuum spacetime.

The pointwise Eshelby force on the interface between a transformed inclusion and its surrounding matrix

Eshelby showed that the pointwise force F on and normal to the interface between a transformed inclusion and its surrounding matrix is the jump in the normal component of the elastic energy-momentum tensor across the interface. Gavazza later showed, using an entirely different approach, that this thermodynamic driving force F has a much simpler form involving only the average of the stress tensors at adjacent points on opposite sides of the interface and the “transformation strain” tensor. The equivalence of and connection between the two formulae was apparently first shown by Eshelby in a personal letter to Gavazza (attached as an appendix to this paper), although the brevity of the letter makes following Eshelby’s proof a little difficult. Here we expand Eshelby’s hitherto unpublished proof of the equivalence of the two expressions in what we believe is a clearer fashion.

Relation between Eshelby stress and Eshelby fourth-order tensor within an ellipsoidal inclusion

The pioneering work by John D. Eshelby in the 1950s and the 1960s on the theory of materials with defects has opened the doors to what today we call configurational mechanics or, in his honour, Eshelbian mechanics. Two of the main results that Eshelby obtained in this field are the use of the elastic energy-momentum tensor to calculate the net force on a defect and the study of materials with inclusions from the geometrical point of view. In Continuum Mechanics, the energy-momentum tensor is now commonly referred to as the Eshelby stress and is the physical quantity that captures the presence of singularities, such as point defects, inclusions, dislocations. In the study of materials with inclusions, Eshelby established a method for the calculation of the strain and stress fields, which entails a fourth-order tensor that relates the strain in the inclusion to the virtual strain (transformation strain or eigenstrain) defining the geometrical misfit between inclusion and matrix. Surprisingly, perhaps, the scientific communities in these two streams of research seem to have had little or no interaction, i.e. virtually all those researchers that have worked in terms of the Eshelby stress have never used the Eshelby fourth-order tensor, and vice versa. To the best of our knowledge, there exists no explicit mathematical relation between the two objects. Therefore, the objective of this paper is to study the relationship between the Eshelby stress and the Eshelby fourth-order tensor within an ellipsoidal inclusion, in the infinitesimal theory of elasticity. Of the three cases that shall be analysed, the first two are commonly referred to as “homogeneous inclusion” and “inhomogeneous inclusion” in the literature, while we refer to the latter as to “general inclusion”, since it describes both the other two as particular cases.

Uniqueness of Equilibrium Solutions in Second-Order Gradient Nonlinear Elasticity

We consider a three-dimensional elastic body whose material response function depends not only on the gradient of the deformation, but also on its second gradient. Using the elastic energy-momentum tensor as derived by Eshelby [2] we generalize a well-known uniqueness result of Knops and Stuart [8] for a Dirichlet boundary value problem associated with this response function.

The energy-momentum tensor and material uniformity in finite elasticity

Eshelby's elastic energy-momentum tensor is shown to satisfy a differential identity which, in the general case of a uniform elastic body with inhomogeneities, is expressible in terms of the torsion of the material connection.

The elastic energy-momentum tensor in special relativity

We consider the standard nonrelativistic theory of a continuous, elastic medium with finite deformations, according to which the elastic energy is a function only of the state of strain, and the elastic stress tensor is proportional to the strain gradient of the elastic energy in appropriate coordinates. We derive a special relativistic, energy-momentum tensor, which yields the standard class of theories in the nonrelativistic limit, from the requirement that it depend only on the state of deformation (including the minimal dependence on velocity consistent with covariance), plus conservation laws. The result agrees with an earlier theory proposed by B. DeWitt ( in “Gravitation: An Introduction to Current Research” (L. Witten, Ed.), pp. 305–318, Wiley, New York, 1962), who generalized the nonrelativistic Lagrangian to general relativity. The elastic momentum density turns out to be of order v c 2 , and therefore absent in the nonrelativistic theory.

The elastic energy-momentum tensor in special relativity: David N. Williams. Randall Laboratory of Physics, The University of Michigan, Ann Arbor, Michigan 48109-1120

The elastic energy-momentum tensor in special relativity: David N. Williams. Randall Laboratory of Physics, The University of Michigan, Ann Arbor, Michigan 48109-1120

On the screw dislocation in an inhomogeneous elastic medium: the case of continuously varying elastic moduli

The elastic fields associated with a screw dislocation at (x *, 0) in an infinite inhomogeneous elastic medium of an arbitrarily varying shear modulus u(x) are obtained by Fourier analysis for the case in which u(x) and its derivatives are everywhere continuous and bounded. Using the elastic energy-momentum tensor, the self glide force on the dislocation is calculated. The resulting expression for self-force contains a term and (b24r)(du/dx)x* In 1ke, where k=(1u(x*))(du/dx)x* and e is the dislocation core radius. Thus k may be viewed as a reciprocal characteristic length which is introduced by the inhomogeneity. Also included is an extension of the results to include the effects of a free surface at x = 0.