Defects in nonlinear elastic crystals: differential geometry, finite kinematics, and second‐order analytical solutions

Abstract

A differential geometric description of crystals with continuous distributions of lattice defects and undergoing potentially large deformations is presented. This description is specialized to describe discrete defects, i.e., singular defect distributions. Three isolated defects are considered in detail: the screw dislocation, the wedge disclination, and the point defect. New analytical solutions are obtained for elastic fields of these defects in isotropic solids of finite extent, whereby terms up to second order in strain, involving elastic constants up to third order, are retained in the stress components. The strain measure used in the nonlinear elastic potential – a symmetric function, expressed in material coordinates, of the inverse deformation gradient – differs from that used in previous solutions for crystal defects, and is thought to provide a more realistic depiction of mechanics of large deformation than previous theory involving third‐order Lagrangian elastic constants and the Green strain tensor. For the screw dislocation and wedge disclination, effects of core pressure and/or possible contraction along the defect line are considered, and radial displacement contributions arise that are absent in the linear elastic solution, affecting dilatation. Stress components are shown to differ from those of linear elastic solutions near defect cores. Volume change from point defects is strongly affected by elastic nonlinearity.

Reformulation of Nonlinear Anisotropic Crystal Elastoplasticity for Impact Physics

“…New application of logarithmic strain-based theory (e-based theory) to shock compression of metals is presented here. Predictions are thermoelastic and strictly applicable only for very small volumes, such as in atomic simulations 14 or in the immediate vicinity of pinned defect cores, 15 wherein plastic deformation does not occur. Solutions to the planar thermoelastic shock problem in anisotropic crystals were derived fully for Lagrangian and Eulerian theory in Clayton 3 and for logarithmic theory in Clayton.…”

“…Results have been transitioned via publications. [3][4][5]12,13,15 A plan is underway to implement the model into multiscale simulations of armor and munitions at the US Army Research Laboratory. Specifically, developments from this Director's Research Initiative (DRI) project are expected to offer substantial improvements over prior analytical and computational studies of the finite strain response of metals, [18][19][20][21][22][23][24][25][26] ceramics, [27][28][29][30][31][32][33] concrete and geologic materials, 34,35 and energetic molecular crystals.…”

Shock compression of metal single crystals modeled via Finsler-geometric continuum theory

“…The order parameter is linked simultaneously to inelastic shear slip on six primary pyramidal systems [8,26] and volume change associated with nonlinear elastic dislocation core fields [27,28,29]. Other slip and twinning systems are not activated for this crystal orientation and loading mode.…”

Analysis of nonlinear elastic aspects of precursor attenuation in shock-compressed metallic crystals

“…No attempt is made here to explicitly compute precursor decay in a fully nonlinear setting for dislocation dynamics. Nonlinear elastic dislocation solutions are available only for special geometries and static problems [38], and standard superposition required for consideration of effects of multiple dislocations cannot be used in an exact nonlinear setting.…”