非线性粘性固体的热弹动力学
Thermo-elastodynamics of nonlinearly viscous solids
AI总结：
本文研究满足标架无差异原理的 Kelvin-Voigt 非线性粘性固体的热弹动力学，通过交错极小化移动格式与双曲变分方法相结合，并借助四阶 $p$-Laplacian 正则性理论，建立了动力学弱解的存在性。
摘要：
本文研究 Kelvin-Voigt 流变学中非线性粘性固体的热弹动力学，其中弹性应力张量和粘性应力张量均满足标架无差异原理。该系统包含一个在非简单材料框架下计入惯性的力平衡方程，以及一个由变形构型中的 Fourier 定律控制的热传导方程。我们将准静态热粘弹性的交错极小化移动格式与双曲型偏微分方程的变分方法相结合，主要结果是在动力学情形下建立了弱解的存在性。这一结果首先通过引入额外的耗散高阶正则化来实现。随后，通过过渡到满足总能量平衡的热传导方程的较弱形式，可以去除该正则化。后一种描述依赖于四阶 $p$-Laplacian 的正则性理论，该理论导出了变形量的正则性估计，超越了由能量界可获得的标准估计。除了对证明至关重要之外，这些额外的正则性性质可能具有独立的研究价值，并且在非线性粘弹性框架中——包括静态或准静态情形——似乎是新的。
英文摘要：
In this paper, we study the thermo-elastodynamics of nonlinearly viscous solids in the Kelvin-Voigt rheology where both the elastic and the viscous stress tensors comply with the frame-indifference principle. The system features a force balance including inertia in the frame of nonsimple materials and a heat-transfer equation which is governed by the Fourier law in the deformed configuration. Combining a staggered minimizing movement scheme for quasi-static thermoviscoelasticity with a variational approach to hyperbolic PDEs, our main result consists in establishing the existence of weak solutions in the dynamic case. This is first achieved by including an additional higher-order regularization for the dissipation. Afterwards, this regularization can be removed by passing to a weaker formulation of the heat-transfer equation which complies with a total energy balance. The latter description hinges on regularity theory for the fourth order $p$-Laplacian which induces regularity estimates of the deformation beyond the standard estimates available from energy bounds. Besides being crucial for the proof, these extra regularity properties might be of independent interest and seem to be new in the setting of nonlinear viscoelasticity, also in the static or quasi-static case.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
