A Variational Framework for n-th Order Invariant Continuum Mechanics
Variational methods have found many applications in computational continuum mechanics as they suggest numerical algorithms for the solutions of concrete problems. However, in the modern axiomatic foundations of continuum mechanics as reflected in the works of Truesdell, Noll and others (e.g., [1,2]) the variational approach is nonexistent. The arguments against the application of variational principles in the formulation of the theory of continuum mechanics are given in . These arguments may be summarized as follows. (1) Regarding the claim that variational principles are invariant it is argued that “now…the principles of tensor analysis offer us a simpler and direct method…for obtaining invariant statements. (2) “The boundary conditions emerging from a variational principle depend upon what boundary integrals… are included in the statement of the principle and the selection of the boundary integrals is not always dictated by the physical idea which the variational principle is assumed to express.” (3) Regarding the claim that variational principles are statements about the system as a whole and the elegance that variational principles sometimes exhibit, it is argued that unlike the “beautiful variational statements for systems of mass points” the variational principles in continuum mechanics are usually “awkward or unnatural”. (4) “No variational principle has ever been shown to yield Cauchy’s fundamental theorem in its basic sense as asserting that existence of the stress vector implies the existence of the stress tensor”.
KeywordsVector Bundle Variational Principle Virtual Work Cotangent Bundle Boundary Integral
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