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Homogeneous Dislocation Nucleation in Landau Theory of Crystal Plasticity

1.1. Introduction

Crystalline solids exhibit plasticity when macroscopic stresses exceed certain thresholds. It is well-known that plastic deformation of crystals is originated from the generation and motion of interacting dislocations, which evolve collectively in a complex energy landscape driven by the applied loading and long-range mutual interactions (Wilson 1954). Controlling crystal plasticity is needed in a variety of applications from metal hardening (Cottrell 2002) and fatigue failure (Irastorza-Landa et al. 2016) to nano-scale forming (Chen et al. 2010) and micro-pillar optimization (Pan et al. 2019; Zhang et al. 2017).

The current trend of manufacturing small-scale metallic crystalline materials calls for a deeper understanding of their mechanical behavior at micro- and nano-scales. At small scales, a smooth description of plastic flow breaks down, plastic response exhibits strong intermittency (Csikor et al. 2007; Devincre et al. 2008; Ispánovity et al. 2014), and mechanical properties of materials depend strongly on size, initial microstructure, quenched disorder, and prior deformation [ see, e. g. Zhang et al. (2016, 2017)]. These properties render inadequate phenomenological continuum plasticity theory that adopts the smooth description of crystal plasticity (Forest 1998; Franciosi and Zaoui 1991) although it has been very successful in reproducing the most important plasticity phenomenology such as yield, hardening, and shakedown (Lubliner 2008). This inadequacy led to the use or development of other approaches going beyond the phenomenological continuum theory.

Molecular dynamics simulations that rely minimally on phenomenology are widely used to study crystal plasticity; the main advantage is that it is being formulated without any auxiliary hypotheses beyond the choice of interatomic potential (Bulatov et al. 1998; Moretti et al. 2011; Zepeda-Ruiz et al. 2017). However, the major drawback in molecular theories is the limitation on the accessible time and length scales, typically 104-106 atoms (which is equivalent to a few nanometers) and a time span of a few nanoseconds (Baruffi et al. 2019). An intermediate discrete dislocation dynamics approach describes the dynamics of elastically interacting defects (dislocations) but it requires phenomenological rules describing short-range interactions, annihilation, and nucleation (Devincre et al. 2008; Queyreau et al. 2010; Shilkrot et al. 2004). A bridge between the discrete dislocation dynamics theory and molecular dynamics has recently emerged in the form of phase-field crystal theory, but such a detailed description still remains prohibitively expensive when one deals with a large number of dislocations (Elder et al. 2002; Salvalaglio et al. 2019; Skaugen et al. 2018); quasicontinuum numerical approaches that attempt to match phenomenological continuum theory with microscopic molecular dynamics at selective points face the same problem (Kochmann and Amelang 2016). Dynamics of many defects can be also described by an evolving continuum dislocation density; on the other hand, despite many interesting recent advances, a rigorous coarse-graining in a strongly interacting system of many dislocations still remains a major challenge (Acharya and Roy 2006; Chen et al. 2013; El-Azab 2000; Groma 2019; LeSar 2014; Sandfeld et al. 2011; Valdenaire et al. 2016; Xia and El-Azab 2015). A very powerful meso-scale approach is the phase-field method based on the Ginzburg-Landau theory and it has been successful in modeling dislocation dynamics by both employing the continuum microelasticity theory to describe the elastic interactions (Khachaturyan 1967) and incorporating the ?-surface into the crystalline energy to describe the core structures (Finel and Rodney 2000; Hunter et al. 2011; Rodney et al. 2003; Ruffini et al. 2017; Shen and Wang 2004; Wang et al. 2001; Zheng et al. 2018).

In this work, we use a recently developed complimentary mesoscopic approach (Baggio et al. 2019) that provides a nonlinear elasticity perspective on crystal plasticity and can be viewed as a far-reaching generalization of the Frenkel-Kontorova theory (Frenkel and Kontorova 1939). The approach is meso-scale in the sense it deals with macroscopic quantities such as stresses and strains, and at the same time, it accounts properly for the exact symmetry of the underlying crystal structure. The aim is bridging fully atomistic descriptions and macroscopic theory based on continuum mechanics. The approach exploits the global invariance of the energy in the space of metric tensors compatible with geometrically nonlinear kinematics of crystal lattices. It takes the form of Landau theory, in which geometrically nonlinear metric tensor measuring the local deformation is the order parameter, with an infinite number of equivalent energy wells whose position is governed by the infinite symmetry group and corresponds to lattice-invariant shears. Therefore, plastic deformation can be described by an escape from the reference well when the crystal is loaded and dislocations appear as domain boundaries. This approach can be traced back to a few classic papers by J. L. Ericksen (1977, 1980, 1983) and follows subsequent development from the work of Conti and Zanzotto (2004). Similar approaches with periodic energies based on geometrically linear kinematics have also been used to study many aspects of crystal plasticity including, but not limited, to the description of dislocation cores and dislocation nucleation and intermittent nature of plastic flows (see Bonilla et al. 2007; Carpio and Bonilla 2003, 2005; Geslin et al. 2014; Kovalev et al. 1993; Landau 1994; Lomdahl and Srolovitz 1986; Minami and Onuki 2007; Onuki 2003; Plans et al. 2007; Salman and Truskinovsky 2011, 2012; Srolovitz and Lomdahl 1986).

In this work, we use the model to study dislocation nucleation in a homogeneously sheared 2D square crystal. We consider athermal dynamics that reduces to parametric minimization of our elastic energy function with infinite periodicity. Our results suggest that the crystal does not necessarily follow the imposed deformation path such that a remarkable collective dislocation nucleation scenario takes place. The motivation for this particular study is due to the fact that, at small scales, it is experimentally possible to manufacture crystals with very low heterogeneity sources such as grain boundaries, precipitates, voids, cracks, and so on. Similarly, in nanocrystalline materials or ceramics with very fine grains, classical nucleation sources, e.g. Frank-Read, are not effective and nucleation occurs not only heterogeneously at pre-existent grain boundaries, but also homogeneously in grain interiors (Gutkin and Ovid'ko 2008). These peculiarities make relevant to develop a detailed mathematical modeling for a better understanding of homogeneous dislocation nucleation that remains a challenge in materials science. Most of the previous works rely on molecular theories. For instance, molecular dynamics simulations have been used to investigate nucleation in an initially defect-free crystal during nano-indentation (Miller and Rodney 2008; Zhu et al. 2004; Zimmerman et al. 2001) or also during compression/tension simulations to explain the resulting asymmetrical material behavior through non-Schmid effects (Tschopp et al. 2007). Simple shear molecular dynamics simulations have also been performed to examine the dependence of plastic flow on crystal orientation in single-crystal nickel (Horstemeyer et al. 2002). These studies aimed to develop nucleation criteria that can be used in higher scale models such as meso-scale discrete dislocation dynamics or continuum crystal plasticity models. On a more fundamental level, the minimum shear deformation required to induce plastic deformation (although its measurement does not seem feasible in experiments) has been studied using ab initio calculations (Ogata et al. 2002) to investigate the higher shear strength of aluminum than that of copper.

1.2. The model

We consider the deformation of a continuum body y = y(x), where y is the current configuration and x is the reference state. Due to Euclidean invariance, the strain-energy density of an elastic solid must depend on deformation gradient F = y through Cauchy-Green tensor C = FTF. It is straightforward to show that the strain-energy f(C) density possesses rotational invariance for F = Q ? O(2) [see Bhattacharya (1993), Salman (2009), Finel et al. (2010), and Salman et al. (2019)]. Second, in order to account for all deformations that map a Bravais lattice into itself, we must require that the strain energy density must also satisfy

where m belongs to in GL(2, Z) that denotes the group of 2 × 2 invertible matrices with entries in Z and det m = ±1 (Pitteri and Zanzotto 2003). This is the suitable group since two sets of lattice vectors and ea generate the same lattice. if and only if they are related as (Conti and Zanzotto 2004; Pitteri and Zanzotto 2003):

In the presence of such symmetry, the space of metric tensors C partitions into...