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 $\boldsymbol{F}$ , while in the latter, this functional is approximated by a function of F $\boldsymbol{F}$ and its higher-order gradients ∇ 1 F , … , ∇ n F $\nabla^{1}\!\boldsymbol{F},\ldots ,\nabla^{n}\!\boldsymbol{F}$ . 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 $\boldsymbol{P}$ , the higher-order stress measures conjugate to ∇ 1 F , … , ∇ n F $\nabla^{1}\!\boldsymbol{F},\ldots ,\nabla^{n}\!\boldsymbol{F}$ , 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 $n=2$ ) case is discussed both theoretically and computationally in the last part of the work.

中文翻译：

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基于平衡统计热力学的非局部热弹性

这项工作的目的是基于原子考虑和平衡统计热力学 (EST) 为非简单材料的热弹性制定能量本构关系。特别是，考虑了（不受限制的）规范和（受限制的）准调和公式。在典型情况下，（空间）非局域性是由于原子受到连续变形的影响均匀且仿射地改变位置的假设的放宽所致。在准谐波情况下，对平均原子位置（即柯西-伯恩）的类似假设是放松的。考虑了两种类型的空间非局域性，即强和弱。在前一种情况下，原子位置（或平均位置）是变形梯度 F $\boldsymbol{F}$ 的函数，而在后一种情况下，该泛函近似于 F $\boldsymbol{F}$ 及其高阶梯度 ∇ 1 F , … , ∇ n F $\nabla^{1}\!\boldsymbol{F},\ldots ,\ nabla^{n}\!\boldsymbol{F}$ 。在此基础上，得到了热弹性自由能、熵、内能和应力的正则和拟调和非局域模型关系。此外，此类关系是针对热弹性材料特性（例如刚度）制定的。在工作的第二部分，（非极性）简单材料的连续热力学的基本关系被推广到高阶变形梯度（即弱非局部）连续，并应用于高能热弹性。相应的公式特别基于 (i) 能量平衡的欧几里得框架无差异和 (ii) 耗散原理。在标准情况下，(i) 的必要条件是线性动量平衡和（广义）基尔霍夫应力的对称性（即角动量平衡）。在 (ii) 的上下文中，自由能密度特别决定了第一 Piola-Kirchhoff 应力 P $\boldsymbol{P}$ ，高阶应力测量与 ∇ 1 F , … , ∇ n F $\nabla 共轭^{1}\!\boldsymbol{F},\ldots ,\nabla^{n}\!\boldsymbol{F}$ ，以及广义基尔霍夫应力。在相应的（弱非局部的）正则自由能上对现象学自由能进行建模，产生基于 EST 的熵、所有应力测量和热弹性材料特性的本构形式。或者，可以将前者的能量建模为后者的近似值。