Nonlinear elasticity of pre-stressed single crystals: resolving an old mess
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A general nonlinear theory for the elasticity of pre-stressed single crystals is presented. Various types of elastic moduli are defined, their importance is determined and relationships between them are presented. In particular, B moduli are present in the relationship between the Jaumann objective time derivative of the Cauchy stress and deformation rate and are broadly used in computational algorithms in various finite-element codes. Possible applications to simplified linear solutions for complex nonlinear elasticity problems are outlined and illustrated for a superdislocation. The effect of finite rotations is fully taken into account and analyzed. Different types of the bulk and shear moduli under different constraints are defined and connected to the effective properties of polycrystalline aggregates. Expressions for elastic energy and stress-strain relationships for small distortions with respect to pre-stressed configuration are derived in detail. Under hydrostatic initial load, general consistency conditions for elastic moduli and compliances are derived that follow from the existence of the generalized tensorial equation of state under hydrostatic loading obtained from single or polycrystal. It is shown that B moduli can be found from the expression for the Gibbs energy. However, higher order elastic constants defined from the Gibbs energy do not have any meaning since they do not directly participate in any of known equations, like stress-strain relationships and wave propagation equation. Deviatoric projection of B can also be found from the expression for the elastic energy for isochoric small strain increments and the missing components of B can be found from the consistency conditions. Numerous inconsistencies and errors in known works are analyzed.
This is a pre-print of the article Levitas, Valery I. "Nonlinear elasticity of pre-stressed single crystals: resolving an old mess." arXiv preprint arXiv:2105.10806 (2021). Posted with permission.