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 ${N=n+N}_{\mathrm{op}}$ symmetry-adapted strains. The ${N}_{\mathrm{op}}$ 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 $\stackrel{\ensuremath{\rightarrow}}{k},$ with competing short- and long-range couplings. The oscillators have different ``strain-mass'' densities $\ensuremath{\rho}(k)\ensuremath{\sim}{1/k}^{2}$ and dampings $\ensuremath{\sim}1/\ensuremath{\rho}(k)\ensuremath{\sim}{k}^{2},$ 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 ${(N}_{\mathrm{op}}{=2,n=1,N}_{c}=1)$ and from a square to a rectangular lattice ${(N}_{\mathrm{op}}{=1,n=2,N}_{c}=1)$ for which the OP compatibility kernel is retarded in time, or frequency dependent in Fourier space (in fact, acoustically resonant in $\ensuremath{\omega}/k).$ 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.