Geometric Structure-Preserving Discretization Schemes for Nonlinear Elasticity
Final performance rept. 1 Jul 2012-31 May 2015
GEORGIA TECH RESEARCH CORP ATLANTA
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We introduced a smooth complex for nonlinear elasticity that can be considered as the tensorial analogue of the standard grad-curl-div complex. This mathematical structure simultaneously describes the kinematics and the kinetics of large deformations. The relation between this complex and the de Rham complex allows one to readily derive the necessary and sufficient conditions for the compatibility of displacement gradient and the existence of stress functions on non-contractible bodies. The main application of the nonlinear elasticity complex is in developing mixed finite element methods for large deformations, which will be pursued in a future project. To this end, the smooth complex should be extended to also include less smooth tensors. We introduced this extension by using the so-called partly Sobolev spaces. The result is a Hilbert complex involving second-order tensors on flat compact manifolds with boundary. We then used the general framework of Hilbert complexes to write Hodge-type and Helmholtz-type orthogonal decompositions for second-order tensors. As some applications of these decompositions in continuum mechanics, one can study the strain compatibility equations of nonlinear elasticity in the presence of Dirichlet boundary conditions.
- Theoretical Mathematics