This video is my presentation on the journal article “Intrinsic Nonlinear Elasticity: An Exterior Calculus Formulation” at the 17th U.S. National Congress on Computational Mechanics (USNCCM17), in Albuquerque, New Mexico.
USNCCM is the premier U.S. venue for showcasing the latest research in the broad field of computational mechanics, bringing together top researchers and practitioners in academia, government, and industry from around the world.
Rashad, R., Brugnoli, A., Califano, F. et al. Intrinsic Nonlinear Elasticity: An Exterior Calculus Formulation. Journal of Nonlinear Science 33, 84 (2023).
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.