Hilbert complexes of nonlinear elasticity

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• 作者：Arzhang Angoshtari ; Arash Yavari
• 关键词：Nonlinear elasticity ; Hilbert complex ; Orthogonal decompositions
• 刊名：Zeitschrift f¨¹r angewandte Mathematik und Physik
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：67
• 期：6
• 全文大小：923 KB
• 刊物主题：Theoretical and Applied Mechanics; Mathematical Methods in Physics;
• 出版者：Springer Basel
• ISSN：1420-9039
• 卷排序：67

文摘

We introduce some Hilbert complexes involving second-order tensors on flat compact manifolds with boundary that describe the kinematics and the kinetics of motion in nonlinear elasticity. We then use 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 nonlinear elasticity, we study the strain compatibility equations of linear and nonlinear elasticity in the presence of Dirichlet boundary conditions and the existence of stress functions on non-contractible bodies. As an application of these Hilbert complexes in computational mechanics, we briefly discuss the derivation of a new class of mixed finite element methods for nonlinear elasticity.