Nonlinear elasticity in proper ferroelastics

Nonlinear elasticity in proper ferroelastics is investigated using the Landau theory of phase transitions. In proper ferroelastics, where certain combinations of strain components $e_s$ correspond to the order parameter, coefficients in the Landau free energy are found to coincide with special combinations of elastic constants of second and higher orders, so that both linear and nonlinear elasticity can be directly accounted for by the Landau theory. Accordingly, four categories of strain-induced ferroelastics are distinguished and their nonlinear elastic properties are established. The temperature variation of the second-, third-, and fourth-order elastic constants is also described. A measure of the nonlinearity coefficient $L_1$, expressing the nonlinear elastic energy stored at a ferroelastic transition, is defined and calculated for each of the preceding categories of transitions. Numerical models are discussed for illustrative examples of type-I (Te${\mathrm{O}}_2$), type-II (${\mathrm{V}}_3$Si), and type-III (LaNb${\mathrm{O}}_4$) ferroelastics. In "pseudoproper" ferroelastics, where spontaneous strain is a secondary order parameter, elastic properties are accounted for in the Landau free energy via elastic energies of second and higher orders. With the use of the specific case of La${\mathrm{P}}_5$ ${\mathrm{O}}_{14}$, it is shown that pseudoproper ferroelastics have distinctive elastic (linear and nonlinear) behavior, the magnitude of $L_1$ depending on the strength of the linear coupling between the order parameter and the spontaneous strain. Available examples of pseudoproper ferroelastics are briefly discussed.