Landau-Type Theory of Planar Crystal Plasticity

Abstract

We show that nonlinear continuum elasticity can be effective in modeling plastic flows in crystals if it is viewed as Landau theory with an infinite number of equivalent energy wells whose configuration is dictated by the symmetry group GL (3, Z). Quasi-static loading can be then handled by athermal dynamics, while lattice based discretization can play the role of regularization. As a proof of principle we study in this Letter dislocation nucleation in a homogeneously sheared 2D crystal and show that the global tensorial invariance of the elastic energy foments the development of complexity in the configuration of collectively nucleating defects. A crucial role in this process is played by the unstable higher symmetry crystallographic phases, traditionally thought to be unrelated to plastic flow in lower symmetry lattices.Crystal plasticity is the simplest among yield phenomena in solids [1], and yet it has been compared in complexity to fluid turbulence [2,3]. The intrinsic irregularity of plastic flow in crystals [4] is due to short and long range interaction of crystal defects (dislocations) [5] dragged by the applied loading through a rugged energy landscape [6][7][8]. Fundamental understanding of plastic flow in crystals is crucial for improving hardening properties of materials [9], extending their fatigue life [10], controlling their forming at sub-micron scales [11] and building new materials [12].Macroscopic crystal plasticity relies on a phenomenological continuum description of plastic deformation in terms of a finite number of order parameters representing amplitudes of pre-designed mechanisms. These mechanisms are coupled elastically and operate according to friction type dynamics [13][14][15][16][17]. The alternative microscopic approaches, relying instead on molecular dynamics [18][19][20][21][22][23][24][25], can handle only macroscopically insignificant time and length scales [26]. An intermediate discrete dislocation dynamics approach focuses on long range interaction of few dislocations, while their short range interaction is still treated phenomenologically [27][28][29]. Collective dynamics of many dislocations can be also described by the dislocation density field, however, rigorous coarse-graining in such strongly interacting system still remains a major challenge [30][31][32][33][34][35][36][37].A highly successful computational bridge between microscopic and macroscopic approaches is provided by the quasi-continuum finite element method which uses adaptive meshing and employs ab initio approaches to guide the constitutive response at different mesh scales [38][39][40][41][42]. Its drawbacks, however, are spurious effects due to matching of FEM representations at different scales and a high computational cost of reconstructing the constitutive response at the smallest scales [43].In this Letter we propose a synthetic approach dea-Figure 1. Schematic representation of a lattice invariant shear and the associated energy barriers along the simple shear loading path ∇y = 1 + α(e1 ⊗ e ⊥ 1 ). Alt...

Crystal elasto-plasticity on the Poincaré half-plane

“…2(d)-(e)-(f) matches well the features of dislocations' elastic field both far from and in the proximity of the core region, see also Fig. 4 in Baggio et al (2019). Fig.…”

“…1 and 2(a)). (7) is represented by a γ-evolving cloud of points on the Dedekind tessellation of H. This distribution of local-strain density gives a very informative picture of the main properties of the plastification field, see also Balandraud et al (2015), Baggio et al (2019). The Supplementary Video V1 shows the entire simulation, reporting also the γ-dependent histograms for the local rotation angle θ and of det F.…”

“…(9), energies of this form can describe plasticity and phase transformations in 2D crystals, as well as their coupling. Our present aim is deriving a family of simplest potentials in (4)-(5)-(10) by using functions σ d (J) in (10) which have a single minimizer z 0 in D, see also Baggio et al (2019). A single minimizer, GL-related to z 0 , is therefore also on each tile in the Dedekind tessellation of Fig.…”

“…As in Folkins (1991), Parry (1998), Baggio et al (2019), here we use a natural parameterization of 2D strain space by means of the upper complex Poincaré half-plane H, one of the best known models of the hyperbolic plane (Kilford, 2008;Terras, 2013). On H the well-known Dedekind tessellation (Mumford et al, 2002;Ye et al, 2005) transparently displays the action of GL-symmetry on strain space, and thus on a crystal's energy landscape.…”

“…By design, in the present study of crystal plasticity we restrict ourselves to GL-energies with a minimum complexity of the associated topography. We thus consider a family of potentials with a single minimizer (periodically replicated through GL-periodicity) and a very simple explicit expression, see also Baggio et al (2019). This is motivated by the aim of obtaining plastic-flow results without possibly spurious effects due to the presence of other metastable lattice configurations.…”