Diffuse-interface polycrystal plasticity: Expressing grain boundaries as geometrically necessary dislocations

The standard way of modeling plasticity in polycrystals is by using the crystal plasticity model for single crystals in each grain, and imposing suitable traction and slip boundary conditions across grain boundaries. In this fashion, the system is modeled as a collection of boundary-value problems with matching boundary conditions. In this paper, we develop a diffuse-interface crystal plasticity model for polycrystalline materials that results in a single boundary-value problem with a single crystal as the reference configuration. Using a multiplicative decomposition of the deformation gradient into lattice and plastic parts, i.e. F(X,t) = F^L(X,t) F^P(X,t), an initial stress-free polycrystal is constructed by imposing F^L to be a piecewise constant rotation field R^0(X), and F^P = R^0(X)^T, thereby having F(X,0) = I, and zero elastic strain. This model serves as a precursor to higher order crystal plasticity models with grain boundary energy and evolution.Comment: 18 pages, 7 figure

The Green tensor of Mindlin's anisotropic first strain gradient elasticity

We derive the Green tensor of Mindlin's anisotropic first strain gradient elasticity. The Green tensor is valid for arbitrary anisotropic materials, with up to 21 elastic constants and 171 gradient elastic constants in the general case of triclinic media. In contrast to its classical counterpart, the Green tensor is non-singular at the origin, and it converges to the classical tensor a few characteristic lengths away from the origin. Therefore, the Green tensor of Mindlin's first strain gradient elasticity can be regarded as a physical regularization of the classical anisotropic Green tensor. The isotropic Green tensor and other special cases are recovered as particular instances of the general anisotropic result. The Green tensor is implemented numerically and applied to the Kelvin problem with elastic constants determined from interatomic potentials. Results are compared to molecular statics calculations carried out with the same potentials

Statistics of grain microstructure evolution under anisotropic grain boundary energies and mobilities using threshold-dynamics

This paper investigates the statistical behavior of two-dimensional grain microstructures during grain growth under anisotropic grain boundary characters. We employ the threshold-dynamics method, which allows for unparalleled computational speed, to simulate the full-field curvature motion of grain boundaries in a large polycrystal ensemble. Two sets of numerical experiments are performed to explore the effect of grain boundary anisotropy on the evolution of microstructure features. In the first experiment, we focus on abnormal grain growth and find that grain boundary anisotropy introduces a statistical preference for certain grain orientations. This leads to changes in the overall grain size distribution from the isotropic case. In the second experiment, we examine the texture development and growth of twin grain boundaries at different initial microstructures. We find that both phenomena are more pronounced when the initial microstructure has a dominant fraction of high-angle grain boundaries. Our results suggest effective grain boundary engineering strategies for improving material properties.Comment: 25pages, Figure

Results on the interaction between atomistic and continuum models

University of Minnesota Ph.D. dissertation. August 2014. Major: Aerospace Engineering and Mechanics. Advisor: Ellad B. Tadmor. 1 computer file (PDF); v, 108 pages, appendix A.In this thesis, we develop continuum notions for atomistic systems which play an important role in developing accurate constitutive relations for continuum models, and robust multiscale methods for studying systems with multiple length and time scales. We use a unified framework to study the Irving--Kirkwood and Murdoch--Hardy procedures used to obtain definitions for continuum fields in atomistic systems. We identify and investigate the following three problems. 1. Continuum fields derived for atomistic systems using the Irving--Kirkwood or the Murdoch--Hardy procedures correspond to a spatial description. Due to the absence of a deformation mapping field in atomistic simulations, it is uncommon to define atomistic fields in the reference configuration. We show that the Murdoch--Hardy procedure can be modified to obtain pointwise continuum fields in the reference configuration using the motion of particles as a surrogate for the deformation mapping. In particular, we obtain definitions for the first and second atomistic Piola--Kirchhoff stress tensors. An interesting feature of the atomistic first Piola--Kirchhoff stress tensor is the absence of a kinetic contribution, which in the atomistic Cauchy stress tensor accounts for thermal fluctuations. We show that this effect is also included in the atomistic first Piola--Kirchhoff stress tensor through the motion of the particles. 2. We investigate the non-uniqueness of the atomistic stress tensor stemming from the non-uniqueness of the potential energy representation. In particular, we show using rigidity theory that the distribution associated with the potential part of the atomistic stress tensor can be decomposed into an irrotational part that is independent of the potential energy representation, and a traction-free solenoidal part. Therefore, we have identified for the atomistic stress tensor a discrete analog of the continuum generalized Beltrami representation (a version of the vector Helmholtz decomposition for symmetric tensors). 3. We show that an ambiguity in the original Irving--Kirkwood procedure resulting due to the non-uniqueness of the energy decomposition between particles can be completely avoided through an alternate derivation for the energy balance. It is found that the expressions for the specific internal energy and the heat flux obtained through the alternate derivation are quite different from the original Irving--Kirkwood procedure and appear to be more physically reasonable. Next, we apply spatial averaging to the pointwise field to obtain the corresponding macroscopic quantities. These lead to expressions suitable for computation in molecular dynamics simulations

Bicrystallography-informed Frenkel-Kontorova model for interlayer dislocations in strained 2D heterostructures

In recent years, van der Waals (vdW) heterostructures and homostructures, which consist of stacks of two-dimensional (2D) materials, have risen to prominence due to their association with exotic quantum phenomena. Atomistic scale relaxation effects play an extremely important role in the electronic scale quantum physics of these systems. We investigate such structural relaxation effects in this work using atomistic and mesoscale models, within the context of twisted bilayer graphene -- a well-known heterostructure system that features moire patterns arising from the lattices of the two graphene layers. For small twist angles, atomic relaxation effects in this system are associated with the natural emergence of interface dislocations or strain solitons, which result from the cyclic nature of the generalized stacking fault energy (GSFE), that measures the interface energy based on the relative movement of the two layers. In this work, we first demonstrate using atomistic simulations that atomic reconstruction in bilayer graphene under a large twist also results from interface dislocations, although the Burgers vectors of such dislocations are considerably smaller than those observed in small-twist systems. To reveal the translational invariance of the heterointerface responsible for the formation of such dislocations, we derive the translational symmetry of the GSFE of a 2D heterostructure using the notions of coincident site lattices (CSLs) and displacement shift complete lattices (DSCLs). The workhorse for this exercise is a recently developed Smith normal form bicrystallography framework. Next, we construct a bicrystallography-informed and frame-invariant Frenkel-Kontorova model, which can predict the formation of strain solitons in arbitrary 2D heterostructures, and apply it to study a heterostrained, large-twist bilayer graphene system