Nonlinear elasticity complex and a finite element diagram chase

02/05/2023

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by   Kaibo Hu, et al.

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In this paper, we present a nonlinear version of the linear elasticity (Calabi, Kröner, Riemannian deformation) complex which encodes isometric embedding, metric, curvature and the Bianchi identity. We reformulate the rigidity theorem and a fundamental theorem of Riemannian geometry as the exactness of this complex. Then we generalize an algebraic approach for constructing finite elements for the Bernstein-Gelfand-Gelfand (BGG) complexes. In particular, we discuss the reduction of degrees of freedom with injective connecting maps in the BGG diagrams. We derive a strain complex in two space dimensions with a diagram chase.