TY - UNPB
T1 - Intrinsic nonlinear elasticity
T2 - An exterior calculus formulation
AU - Rashad, Ramy
AU - Brugnoli, Andrea
AU - Califano, Federico
AU - Luesink, Erwin
AU - Stramigioli, Stefano
PY - 2023/3/10
Y1 - 2023/3/10
N2 - 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 principle 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. Second, the de Rham complex structure relating the spaces of bundle-valued forms to each other.
AB - 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 principle 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. Second, the de Rham complex structure relating the spaces of bundle-valued forms to each other.
KW - math-ph
KW - math.DG
KW - math.DS
KW - math.MP
U2 - 10.48550/arXiv.2303.06082
DO - 10.48550/arXiv.2303.06082
M3 - Preprint
BT - Intrinsic nonlinear elasticity
PB - ArXiv.org
ER -