In this paper, we derive a nonlinear strain gradient theory of thermoelastic materials with microtemperatures taking into account micro-inertia effects as well. The elastic behaviour is assumed to be consistent with Mindlin’s Form II gradient elasticity theory, while the thermal behaviour is based on the entropy balance of type III postulated by Green and Naghdi for both temperature and microtemperatures. The work is motivated by increasing use of materials having microstructure at both mechanical and thermal levels. The equations of the linear theory are also obtained. Then, we use the semigroup theory to prove the well-posedness of the obtained problem. Because of the coupling between high-order derivatives and microtemperatures, the obtained equations do not have exponential decay. A frictional damping for the elastic component, whose form depends on the micro-inertia, is shown to lead to exponential stability for the type III model.

中文翻译：

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Mindlin 的 II 型梯度热弹性与 III 型微观温度的指数稳定性

在本文中，我们推导出了具有微温度的热弹性材料的非线性应变梯度理论，同时考虑了微惯性效应。假设弹性行为与 Mindlin 的 II 型梯度弹性理论一致，而热行为则基于 Green 和 Naghdi 对温度和微温假设的 III 型熵平衡。这项工作的动机是增加使用具有机械和热水平微观结构的材料。还获得了线性理论的方程。然后，我们使用半群理论来证明所得到问题的适定性。由于高阶导数和微观温度之间的耦合，得到的方程没有指数衰减。弹性组件的摩擦阻尼，