Finite element approximation of the fields of bulk and interfacial line defects

Abstract

A generalized disclination (g.disclination) theory [AF15] has been recently introduced that goes beyond treating standard translational and rotational Volterra defects in a continuously distributed defects approach; it is capable of treating the kinematics and dynamics of terminating lines of elastic strain and rotation discontinuities. In this work, a numerical method is developed to solve for the stress and distortion fields of g.disclination systems. Problems of small and finite deformation theory are considered. The fields of a single disclination, a single dislocation treated as a disclination dipole, a tilt grain boundary, a misfitting grain boundary with disconnections, a through twin boundary, a terminating twin boundary, a through grain boundary, a star disclination/penta-twin, a disclination loop (with twist and wedge segments), and a plate, a lenticular, and a needle inclusion are approximated. It is demonstrated that while the far-field topological identity of a dislocation of appropriate strength and a disclination-dipole plus a slip dislocation comprising a disconnection are the same, the latter microstructure is energetically favorable. This underscores the complementary importance of all of topology, geometry, and energetics in understanding defect mechanics. It is established that finite element approximations of fields of interfacial and bulk line defects can be achieved in a systematic and routine manner, thus contributing to the study of intricate defect microstructures in the scientific understanding and predictive design of materials. Our work also represents one systematic way of studying the interaction of (g.)disclinations and dislocations as topological defects, a subject of considerable subtlety and conceptual importance [Mer79, AMK17]. energies for finite bodies. In [AF12, AF15], a continuum model is introduced for the g.disclination static equilibrium as well as dynamic behaviors, where the singularities are well-handled. The Weingarten theorem for g.disclinations established in [AF15] is characterized further in [ZA16], with the derivation of explicit formula for important topological properties of canonical g.disclination configurations. Relationships between the representations of the dislocation, disclination, and the g.disclination from the Weingarten point of view and in g.disclination theory are established therein. Concrete connections are also established between g.disclinations as mathematical objects and the physical ideas of interfacial and bulk line defects like defected grain and phase boundaries, dislocations, and disclinations. The papers [AF12, AF15, ZA16] explain the theoretical and physical basis for the results obtained in the present work.This paper focuses on the applications of the g.disclination model through computation. The goal is to show that the g.disclination model is capable of solving various material-defect problems, within both the small and finite deformation settings. Finite element schemes to solve for the stress and energy density fields of g.disc...

Fast Fourier transform-based micromechanics of interfacial line defects in crystalline materials

“…As a last remark, it is here highlighted that the numerical method is very fast for complex anisotropic and heterogeneous elasticity as the phase boundary "terrace" in anisotropic bi-materials. Even though the Fourier-based approach is developped in small deformation, the numerical results are consistent with the FEM results reported by Zhang et al [2018] who used a more general framework in a finite deformation setting. Furthermore, the method is an alternative numerical method to analytical methods based on the sextic equation and Stroh formalism for anisotropic elasticity which was more developed for the dislocation theory [Stroh, 1958;Barnett and Lothe, 1974].…”

“…Some arrays of disconnections can be also used to model martensitic interfaces (habit planes) or hetero-interfaces forming a terrace structure observed with TEM (transmission electron microscopy) [Pond et al, 2003[Pond et al, , 2007Wang et al, 2011]. Numerical calculations with the finite element method (FEM) were provided by Zhang et al [2018] for such complex defects in small and finite deformation settings considering generalized disclinations [Acharya and Fressengeas, 2012].…”

Continuum mechanics of moving defects in growing bodies

“…Physical considerations related to predicting stress fields of terminating twin boundaries and the stress-free, compatible, elastic, twinning shear distortions of through-twin boundaries [ZAP18] motivate the introduction of the following Stokes-Helmholtz (SH) decompositions:…”

On the relevance of generalized disclinations in defect mechanics

“…These equations are solved along with balance of linear and angular momentum involving Cauchy stresses and couple-stresses (with constitutive assumptions) to obtain g.disclination and dislocation stress and couple stress fields. In the companion paper [ZAP16] we solve these equations along with div T = 0 with T representing the Cauchy stress as a function of W , and we ignore couple stresses for simplicity.…”