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...

We explore the nonlinear variational modelling of two-dimensional (2D) crystal plasticity based on strain energies which are invariant under the full symmetry group of 2D lattices. We use a natural parameterization of strain space via the upper complex Poincaré half-plane. This transparently displays the constraints imposed by lattice symmetry on the energy landscape. Quasi-static energy minimization naturally induces bursty plastic flow and shape change in the crystal due to the underlying coordinated basin-hopping local strain activity. This is mediated by the nucleation, interaction, and annihilation of lattice defects occurring with no need for auxiliary hypotheses. Numerical simulations highlight the marked effect of symmetry on all these processes. The kinematical atlas induced by symmetry on strain space elucidates how the arrangement of the energy extremals and the possible bifurcations of the strain-jump paths affect the plastification mechanisms and defect-pattern complexity in the lattice.

By using modular functions on the upper complex half-plane, we study a class of strain energies for crystalline materials whose global invariance originates from the full symmetry group of the underlying lattice. This follows Ericksen’s suggestion which aimed at extending the Landau-type theories to encompass the behavior of crystals undergoing structural phase transformation, with twinning, microstructure formation, and possibly associated plasticity effects. Here we investigate such Ericksen-Landau strain energies for the modelling of reconstructive transformations, focusing on the prototypical case of the square-hexagonal phase change in 2D crystals. We study the bifurcation and valley-floor network of these potentials, and use one in the simulation of a quasi-static shearing test. We observe typical effects associated with the micro-mechanics of phase transformation in crystals, in particular, the bursty progress of the structural phase change, characterized by intermittent stress-relaxation through microstructure formation, mediated, in this reconstructive case, by defect nucleation and movement in the lattice.