Finsler-Geometric Continuum Mechanics and the Micromechanics of Fracture in Crystals

Report No. ARL-RP-0593
Authors: JD Clayton
Date/Pages: March 2017; 56 pages
Abstract: A continuum theory for the mechanical response of solid bodies subjected to potentially finite deformation is further developed and applied to solve several new problems in the context of the micromechanics of crystalline solids. The theory invokes concepts from Finsler differential geometry, and it provides a diffuse interface description of fracture surfaces. The director or internal state vector is associated with an order parameter describing degradation of the solid. Here, the deformation gradient between pseudo-Finsler reference and spatial configuration spaces is decomposed into a product of two terms, neither necessarily integrable to a vector field. The first is the recoverable elastic deformation, the second is the residual deformation attributed to changes in free volume in failure zones. The latter is restricted to spherical or isotropic symmetry; resulting Euler–Lagrange equations for mechanical and state variable equilibrium are derived. Metric tensors and volume elements depend on the internal state via a conformal transformation, i.e., Weyl scaling. This version of the theory is first applied to tensile fracture of magnesium. Analytical solutions demonstrate the models capability to predict ductile versus brittle fracture depending on incorporation of Weyl scaling, with results aligned with molecular dynamics (MD) simulations. The second application is shear fracture in boron carbide: solutions depict weakening and tensile pressure in conjunction with structural collapse in shear transformation zones, as suggested by experiments, quantum mechanics, and/or MD simulations.
Distribution: Approved for public release
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Last Update / Reviewed: March 1, 2017