Third-Order Padé Thermoelastic Constants of Solid Rocks

Abstract

Classical third-order thermoelastic constants are generally formulated by the theory of small-amplitude acoustic waves in cubic crystals during heat treatments. Investigating higher-order thermoelastic constants for higher temperature is a challenging task because of more undetermined constants involved. However, even at low temperatures, these Taylor-type thermoelastic constants encounter divergence in characterizing the temperature-dependent velocity changes of elastic waves in solid rocks as a complete polycrystal compound of different mineral lithologies. Therefore, we propose third-order Padé-type thermoelastic constants derived by the approximation of Padé rational function to the total strain energy. The Padé thermoelastic constants are characteristics of a reasonable theoretical prediction for acoustic velocities of solid rocks even at high temperature. The results demonstrate that the third-order Padé thermoelasticity can characterize thermally induced velocity changes more accurately than the conventional third-order Taylor thermoelasticity, and have the same accuracy for the corresponding higher-order thermoelastic model. The Padé approximation could be considered a more versatile model for describing thermal velocity changes for polycrystals and solid rocks. The physics of the Padé coefficients is relevant to the thermal expansion mismatch and thermally induced deformation of microcracks. Introduction Temperature significantly changes the mechanical and physical properties of rocks, which becomes importance in many fields of earth sciences and geological engineering. Temperature-induced variations in elastic properties generally present strong nonlinearity even at low temperature because of the differential thermal expansion of multimineral rocks. The classical theory of thermoelasticity is formulated on account of the Taylor power series of the Helmholtz free energy functions (Dillon, 1962). The resulting second- and third-order thermoelastic constants have been widely used for crystals, but with certain insufficiencies in representing the temperature-dependent velocity changes of elastic waves for rocks like a completely polycrystal compound of varying mineral lithologies. The investigation of higher-order thermoelastic constants could be useful for understanding the nature of nonlinear behavior in heating rocks, but involves more undetermined constants and becomes a challenging task. As a more effective alternative, this article addresses a nonlinear thermoelasticity for solid rocks relied on the Padé approximation of Helmholtz free energy functions.

Third‐order elasticity of transversely isotropic field shales

“…Incorporating finite strain would only result in a marginal correction to the linear strain and would not contribute to the understanding of the overburden dynamic trends of this study. It can be mentioned that finite strains can be accounted for in a static TOE model such as Wang and Schmitt (2021) suggested for isotropic materials. However, this will complicate the calculation of the static model, and consequently also the dynamic TOE model in our case, because a strain implies a change of moduli, which in turn changes the strain.…”

“…The TOE theory enables prediction of how velocities depend on strain. The expansion of the elastic strain energy of a medium around a reference state of zero strain, truncating at the third‐order terms, takes the common form (Brugger, 1964; Fuck & Tsvankin, 2009; Lubarda, 1997; Rasolofosaon, 1998; Shapiro, 2017; Sinha, 1982; Thurston, 1974; Wang & Schmitt, 2021) $$$$\begin{equation}W = \frac{1}{2}{C_{ijkl}}{\varepsilon _{ij}}{\varepsilon _{kl}} + \frac{1}{6}{C_{ijklmn}}{\varepsilon _{ij}}{\varepsilon _{kl}}{\varepsilon _{mn}},\end{equation}$$$$where $${C_{ijkl}}$$ and $${C_{ijklmn}}$$ are coefficients of the second‐order elastic (SOE) and TOE stiffnesses, respectively, $${\varepsilon _{ij}}$$ are the strain components, and i , j , k , l , m and n are dummy indices where summation over repeated indices in each term is implied. Note that the strains in the experiments and simulations herein are of the order 10 −3 or less, that is infinitesimal.…”