Crystal elasto-plasticity on the Poincaré half-plane

Abstract

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.

Ericksen-Landau Modular Strain Energies for Reconstructive Phase Transformations in 2D Crystals

“…This produces the inhomogeneous progress of the structural phase change, characterized by jagged stress relaxation via bursty microstructure development in the body, also mediated, in the reconstructive case, by defect nucleation and movement in the lattice. The present simulations also confirm the role, highlighted yet in [37], of the energy's valley floors as largely establishing the deformation pathways for a crystal's intermittent evolution under an external driving.…”

“…which are all the isometries of H given by the group of linear fractional transformations with integral entries, supplemented by the map z → −z (see [37] for details). Figure 1 represents geometrically this action, where the fundamental domain 2…”

“…where the smooth function σ d (J ) should be such that it guarantees [37]: (a) the full GLperiodicity ( 1) for ( 6), rather than the sole invariance under SL(2, Z) exhibited by J ; and, (b) ensure the existence of a positive-definite elastic tensor for any stable lattice configuration, thus avoiding the existence of equilibria with degenerate Hessian, as required by elasticity theory. Examples of such potentials, suitable for the elasto-plasticity of 2D crystals, and for their reconstructive transformations, are discussed explicitly hereafter.…”

“…In this spirit, [25,37] proposed an explicit class of smooth J -based strain energies with a unique ground state, up to full lattice symmetry, exploring the ensuing variational modelling of 2D crystal elasto-plasticity. We notice that smoothness, as obtained here and in [25,37], is the standard assumption for potentials in Landau theory and elasticity theory, while the reduced C 2 -smoothness of [6] would induce jump discontinuities in the third-and higher-order moduli [38], affecting the connected material properties.…”

Phase transformations, fracture, and other structural changes in inelastic materials

“…Their consideration is based on the approximation of the more complex 2D energy landscape ψ c xi with additional intermediate minima, which is obtained using molecular static or first principle simulations. An alternative approach to dislocations based on the energy landscape in the total strain space is presented in [8,15,124].…”