Non-local Thermoelasticity Based on Equilibrium Statistical Thermodynamics

• Published in: Journal of elasticity Vol. 139; no. 1; pp. 37 - 59
• Main Authors: Po, Giacomo, Admal, Nikhil Chandra, Svendsen, Bob
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.04.2020
• Subjects: Automotive Engineering; Classical Mechanics; Physics; Physics and Astronomy; Statistical thermodynamics; Spatial non-locality; Atomistic thermoelasticity; 74A15; Higher-order deformation gradient thermoelasticity; 74A25; Quasi-harmonic approximation; Canonical ensemble; 74A20; 74A30
• ISSN: 0374-3535; 1573-2681
• DOI: 10.1007/s10659-019-09745-9

The purpose of this work is the formulation of energetic constitutive relations for thermoelasticity of non-simple materials based on atomistic considerations and equilibrium statistical thermodynamics (EST). In particular, both (unrestricted) canonical, and (restricted) quasi-harmonic, formulations are considered. In the canonical case, (spatial) non-locality results from relaxation of the assumption that atoms subject to continuum deformation change position uniformly and affinely. In the quasi-harmonic case, the analogous assumption on mean atomic position (i.e., Cauchy-Born) is relaxed. Two types of spatial non-locality, i.e., strong and weak, are considered. In the former case, atomic position (or mean position) is a functional of the deformation gradient F , while in the latter, this functional is approximated by a function of F and its higher-order gradients ∇ 1 F , … , ∇ n F . On this basis, canonical and quasi-harmonic non-local model relations are obtained for the thermoelastic free energy, entropy, internal energy, and stress. In addition, such relations are formulated for thermoelastic material properties (e.g., stiffness). In the second part of the work, basic relations from the continuum thermodynamics of (non-polar) simple materials are generalized to higher-order deformation gradient (i.e., weakly non-local) continua and applied to energetic thermoelasticity. The corresponding formulation is based in particular on (i) Euclidean frame-indifference of the energy balance and (ii) the dissipation principle. As in the standard case, necessary for (i) is linear momentum balance and the symmetry of the (generalized) Kirchhoff stress (i.e., angular momentum balance). In the context of (ii), the free energy density determines in particular the first Piola-Kirchhoff stress P , the higher-order stress measures conjugate to ∇ 1 F , … , ∇ n F , as well as the generalized Kirchhoff stress. Modeling the phenomenological free energy on the corresponding (weakly non-local) canonical free energy yields EST-based constitutive forms for the entropy, all stress measures, and thermoelastic material properties. Alternatively, one can model the former energy as an approximation to the latter. An example of this for the second-order ( n = 2 ) case is discussed both theoretically and computationally in the last part of the work.