Landau-Type Theory of Planar Crystal Plasticity

We show that nonlinear continuum elasticity can be effective in modeling plastic flows in crystals if it is viewed as a Landau theory with an infinite number of equivalent energy wells whose configuration is dictated by the symmetry group $\mathrm{GL}(2,\mathbb{Z})$. Quasistatic 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 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, typically thought to be unrelated to plastic flow.

Figure 1

Schematic representation of a lattice-invariant shear and the associated energy barriers along the simple shear loading path $\mathbf{\nabla }\mathbf{y}=\mathbb{1}+\alpha {\mathbf{e}}_1\otimes {\mathbf{e}}_1^{\perp }$. Alternating minimal-periodicity domains are marked in gray and white.

Figure 2

Structure of the $\mathrm{GL}(2,\mathbb{Z})$ periodicity domains in the space of metric tensors. (a) Partition of the complex half plane (Dedekind tessellation). (b) Equivalent partition of the section $\det \mathbf{C}=1$ of the space $\mathbf{C}$. Points ${\mathbf{S}}_{\mathbf{i}}$ represent the same square lattice, and points ${\mathbf{T}}_{\mathbf{i}}$ the same triangular lattice.

Figure 3

Energy landscapes corresponding to potential (1) with (a) $\beta =-1/4$ and (b) $\beta =4$. Colors indicate the energy level: blue, low; red, high. Black lines delimit the zones of linear stability for the homogeneous states. Green lines correspond to the simple shear loading paths discussed in the text.

Figure 4

Single edge dislocation in a square lattice linking the wells ${\mathbf{S}}_{\mathbf{1}}$ and ${\mathbf{S}}_{\mathbf{2}}$; its Burgers vector is horizontal, with the length equal to the side of the square unit cell. (a) Finite element nodes with coloring indicating the level of Cauchy stress ${\sigma }_{xy}$. (Inset) Image of this defect in the configurational space. (b) Stress profile along the glide plane. (Inset) The far field asymptotics. System size, $1000\times 1000$.

Figure 5

Collective dislocation nucleation. (a),(b) Square lattice. (c),(d) Triangular lattice. We show two representations of the same phenomenon. (a),(c) In the configurational space, where green lines are simple shear paths imposed by the loading device and blue dots indicate the metric tensor distribution among the elements. (b),(d) In the physical space. Colors indicate the level of the nodal Cauchy stresses ${\sigma }_{xy}$. System size, $200\times 200$.

Figure 6

Level sets of the energy density on the surface $\det \mathbf{C}=1$ around the point ${\mathbf{T}}_{\mathbf{1}}$. We use the parametrization $C_{11}=1/y$, $C_{22}=(x^2+y^2)/y$, $C_{12}=x/y$. (a) Klein-invariant-based potential. (b) Polynomial potential (1) with $\beta =-1/4$.