Intrinsic Nonlinear Elasticity: An Exterior Calculus Formulation

In this paper, we formulate the theory of nonlinear elasticity in a geometrically intrinsic manner using exterior calculus and bundle-valued differential forms. We represent kinematics variables, such as velocity and rate of strain, as intensive vector-valued forms, while kinetics variables, such as stress and momentum, as extensive covector-valued pseudo-forms. We treat the spatial, material and convective representations of the motion and show how to geometrically convert from one representation to the other. Furthermore, we show the equivalence of our exterior calculus formulation to standard formulations in the literature based on tensor calculus. In addition, we highlight two types of structures underlying the theory: first, the principal bundle structure relating the space of embeddings to the space of Riemannian metrics on the body and how the latter represents an intrinsic space of deformations and second, the de Rham complex structure relating the spaces of bundle-valued forms to each other.

Automated summary

In this paper, the theory of nonlinear elasticity is formulated using exterior calculus and bundle-valued differential forms in a geometrically intrinsic manner. The kinematics variables are represented as intensive vector-valued forms, while the kinetics variables are represented as extensive covector-valued pseudo-forms. The spatial, material and convective representations of the motion are discussed, and the geometric conversion between different representations is shown. The equivalence of the exterior calculus formulation to standard formulations based on tensor calculus is demonstrated. The underlying structures of the theory, including the principal bundle structure and the de Rham complex structure, are highlighted.