The equations of elastostatics in a Riemannian manifold

Abstract

To begin with, we identify the equations of elastostatics in a Riemannian manifold, which generalize those of classical elasticity in the three-dimensional Euclidean space. Our approach relies on the principle of least energy, which asserts that the deformation of the elastic body arising in response to given loads minimizes over a specific set of admissible deformations the total energy of the elastic body, defined as the difference between the strain energy and the potential of the loads. Assuming that the strain energy is a function of the metric tensor field induced by the deformation, we first derive the principle of virtual work and the associated nonlinear boundary value problem of nonlinear elasticity from the expression of the total energy of the elastic body. We then show that this boundary value problem possesses a solution if the loads are sufficiently small (in a sense we specify). RésuméDans un premier temps, nous identifions leséquations de l'élastostatique dans une variété riemannienne, qui généralisent celles de la théorie classique de l'élasticité dans l'espace euclidien tridimensionnel. Notre approche repose sur le principe de moindre action, qui affirme que la déformation du corpsélastique sous l'action des forces externes minimise sur l'ensemble des déformations admissibles l'énergie totale du corpsélastique, définie comme la différence entre l'énergie de déformation et le potentiel des forces externes. Sous l'hypothèse que l'énérgie de déformation est une fonction du champ de tenseurs métriques induit par la déformation, nous dérivons dans un premier temps le principe des travaux virtuels et le problème aux limites associéà partir de l'expression de l'énergie totale du corpsélastique. Nous démontrons ensuite que ce problème aux limites admet une solution si les forces externes sont suffisamment petites (en un sens que nous précisons).

An Intrinsic Geometric Formulation of Hyper-Elasticity, Pressure Potential and Non-Holonomic Constraints

“…Here, we explain how to formulate Lagrangians (potentials) for surface forces (boundary conditions) and illustrate the proposed general method through the simple example of dead loads. The potential obtained is close to the one derived in [25], but with the notable difference that our formulation of the displacement uses the affine structure of space. The more complicated problem of prescribed pressure is discussed in section 6, both on the body and on the reference configuration.…”

“…These configurations are described using mappings from some abstract manifold with boundary, the body B, into the ambient (usually Euclidean) space E [62,61]. As mentioned by Truesdell and Noll [41,42] and later by several authors [64,49,28,10,6,25,60], the body does not have, from the pure differential geometry point of view, to be embedded in space and identified with some reference configuration. This point was emphasized by Noll [42,43], who called a formulation of continuum mechanics on the body B as intrinsic 1 and by Rougée [49,50] who refers to it as an intrinsic Lagrangian framework, whereas a formulation on a reference configuration is denominated as a standard Lagrangian approach.…”

“…Formulating the problem in an intrinsic manner, meaning on the body, was initiated by Rougée [51,52], who defined an elasticity law as a vector field on the manifold of Riemannian metrics on the body. However, the formulation of the boundary conditions on B rather than on a reference configuration in ambient space seems to remain an open problem in the general case [54,25,20].…”

Static Elasticity in a Riemannian Manifold

“…This paper is adapted from Grubic, LeFloch, and Mardare [17]. The definitions and notations used, but not defined here, can be found in Section 2.…”

Differential Geometry and Continuum Mechanics

“…This chapter is adapted from Grubic, LeFloch, and Mardare [GLM14]. The definitions and notations used, but not defined in this Introduction, can be found in Sect.…”