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Three practically useful reductions of phenomenological mesoscopic field dislocation mechanics (PMFDM) are explored to evaluate some of their capabilities and limitations. Lower-order gradient plasticity arises as one of these reduced frameworks, and a boundary condition appropriate for the modelling of constrained plastic flow in 3D is demonstrated through FEM simulation.To elaborate and fix ideas in a simplified context, we motivate the averaging procedure giving rise to PMFDM in a 1D context and show the possibility of obtaining a diffusion term in coarse evolution through a perturbative approach. Formally, the diffusion appears as a small, first-order correction to mean transport of excess dislocation density. Physically, it arises from a more refined modelling of the slip distortion rate and not as an additional energetic ‘back’ stress contribution. The approximate nature of the perturbative approach is assessed to highlight the importance of the transport term in the equation. We also comment on the emergence of stochasticity and memory effects in coarse response.

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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...

Size effects at initial yield (prior to stage II) of idealized micron-sized specimens are modeled within a continuum model of plasticity. Two different aspects are considered: specification of a density of dislocation sources that represent the emission of dislocation dipoles, and the presence of an initial, spatially inhomogeneous excess dislocation content. Discreteness of the source distribution appears to lead to a stochastic response in stress-strain curves, with the stochasticity diminishing as the number of sources increases. Variability in stress-strain response due to variations of source distribution is also shown. These size effects at initial yield are inferred to be due to physical length scales in dislocation mobility and the discrete description of sources that induce internal-stress-related effects, and not due to length-scale effects in the mean-field strain-hardening response (as represented through a constitutive equation).