On a class of conservation laws in linearized and finite elastostatics

The gauge theory of dislocations: conservation and balance laws

“…With the known Lie-point symmetries he constructed conserved currents and path-independent integrals and related them to torsion and bending problems of rods, plates and shells. Knowles and Sternberg [5] extended these results for the case of finite elasticity. Fletcher [6] continued further and calculated the variational symmetries and conserved integrals in linear elastodynamics.…”

Energy-momentum tensors in nonsimple elastic dielectrics

“…This approach requires considerably less work as compared to the techniques employed previously and is easily extendable to electromagnetic materials of grade N. We note that Noether's theorem ~as been used by Knowles and Sternberg [9] to derive conservation laws in linearized and finite elastostatics, by Golebiewska-Herrmann [10] to obtain a unified formulation leading to all conservation laws of continuum mechanics, by Pak and Herrmann [11] to obtain conservation laws and the material momentum tensor for an elastic dielectric, and by Maugin [12] to obtain pseudo-momentum and Eshelby's material tensor in electromagneto-mechanical framework. Ma,ugin [12] noted that the work can be extended to nonsimple hyperelastic solids but did not provide any results.…”

“…Noether's theorem states that if the action A is invariant under a set oftransformations of the coordinates and the fields, then there exist conserved currents :fa such that (9) where -.Jalif = .c~a lJf…”

Exact results with the J-integral applied to free-boundary flows

“…the resulting field equations need not be linear (the original applications in crack theory were principally to nonlinear elasticity). Also, this route via the Noether theorem, while not used in the discovery of the J-integral, has led to the discovery of a further path-independent integral L when there is invariance under infinitesimal rotation about a given axis and to yet another integral M when there is invariance relative to a self-similar change of scale (see Gunther 1962;Knowles & Sternberg 1972;Budiansky & Rice 1973;Rice 1985). For example, in the two-dimensional case of Laplacian fields, and a closed contour C that surrounds no singularities, these additional integrals are…”

“…It allows an easy determination of the asymptotic stress and strain in the close vicinity of a crack, for example. It is not restricted to the linear elasticity case where the field in the sample is biharmonic (Eshelby 1970;Knowles & Sternberg 1972), and can also take into account plasticity for non-growing cracks, within the approximation of 'deformation theory' (Hutchinson 1968;Rice & Rosengren 1968). In this paper we present some applications of this technique when the field in the flow is Laplacian.…”