Ferroelastic dynamics and strain compatibility

We derive underdamped evolution equations for the order-parameter (OP) strains of a proper ferroelastic material undergoing a structural transition, using Lagrangian variations with Rayleigh dissipation, and a free energy as a polynomial expansion in the N = n + N_op symmetry-adapted strains. The N_op strain equations are structurally similar in form to the Lagrange-Rayleigh one-dimensional strain dynamics of Bales and Gooding (BG), with "strain accelerations" proportional to a Laplacian acting on a sum of the free-energy strain derivative and frictional strain force assuming geometric linearity. The tensorial St. Venant's elastic compatibility constraints that forbid defects, are used to determine the n non-order-parameter strains in terms of the OP strains, generating anisotropic and long-range OP contributions to the free energy, friction, and noise. The same OP equations are obtained by either varying the displacement vector components, or by varying the N strains subject to the N_c compatibility constraints. A Fokker-Planck equation, based on the BG dynamics in more than one dimension with noise terms, is set up. The BG dynamics corresponds to a set of nonidentical nonlinear (strain) oscillators labeled by wave vector k→, with competing short- and long-range couplings. The oscillators have different "strain-mass" densities ρ(k) ∼ 1/k² and dampings ∼ 1/ρ(k) ∼ k², so the lighter large-k oscillators equilibrate first, corresponding to earlier formation of smaller-scale oriented textures. This produces a sequential-scale scenario for post-quench nucleation, elastic patterning, and hierarchical growth. Neglecting inertial effects yields a late-time dynamics for identifying extremal free-energy states, that is, of the time-dependent Ginzburg-Landau form, with nonlocal, anisotropic Onsager coefficients that become constants for special parameter values. We consider in detail the two-dimensional (2D) unit-cell transitions from a triangular to a centered rectangular lattice (N_op = 2,n = 1,N_c= 1) and from a square to a rectangular lattice (N_op= 1,n = 2,N_c= 1) for which the OP compatibility kernel is retarded in time, or frequency dependent in Fourier space (in fact, acoustically resonant in ω/k). We present structural dynamics for all other 2D symmetry-allowed proper ferroelastic transitions: the procedure is also applicable to the 3D case. Simulations of the BG evolution equations confirm the inherent richness of the static and dynamic texturings, including strain oscillations, domain-wall propagation at near-sound speeds, grain-boundary motion, and nonlocal "elastic photocopying" of imposed local stress patterns.