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Finsler-Geometric Continuum Mechanics

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Technical Report,01 Oct 2015,01 May 2016

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US Army Research Laboratory Aberdeen Proving Ground United States

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Concepts from Finsler differential geometry are applied toward a theory of deformable continua with microstructure. The general model accounts for finite strains, nonlinear elasticity, and various kinds of structural defects in a solid body. The general kinematic structure of the theory includes macroscopic and microscopic displacement fields i.e., a multiscale theory whereby the latter are represented mathematically by the director vector of pseudo-Finsler space, not necessarily of unit magnitude. Variational methods are applied to derive Euler-Lagrange equations for static equilibrium and Neumann boundary conditions. The theory is specialized in turn to physical problems of tensile fracture, shear localization, and cavitation in solid bodies. The pseudo-Finsler approach is demonstrated to be more general than classical approaches and can reproduce phase field solutions when certain simplifying assumptions are imposed. Upon invoking a conformal or Weyl-type transformation of the fundamental tensor, analytical and numerical solutions of representative example problems offer new physical insight into coupling of microscopic dilatation with fracture or slip.

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