含断裂与流体流动的热力学背景下的Stefan问题
The Stefan problem in a thermomechanical context with fracture and fluid flow
AI总结：
本研究将经典仅考虑传热的Stefan问题拓展至热力学范畴，结合欧拉描述、粘弹性流变学、相场模型与L¹理论，严格证明了含断裂、流体流动及相变动力学效应的不完全熔化问题弱解的存在性。
摘要：
经典Stefan问题仅涉及固液相变过程中的传热，本文将其拓展至考虑力学效应的范畴。研究采用大位移下的欧拉描述，结合对流导数与Zaremba-Jaumann共旋时间导数，通过（客观）速率下的加性Green-Naghdi分解进行线性化。具体而言，液相为粘弹性流体，固相的蠕变与断裂则采用Jeffreys粘弹性流变学模型，结合相场模型与微（即“半”）可压缩材料概念展开研究。针对允许固液相变过程中存在动力学过热/过冷效应的松弛型Stefan问题，采用热方程的L¹理论。通过时间离散近似方法，严格证明了不完全熔化情形下弱解的存在性。
英文摘要：
The classical Stefan problem, concerning mere heat-transfer during solid-liquid phase transition, is here enhanced towards mechanical effects. The Eulerian description at large displacements is used with convective and Zaremba-Jaumann corotational time derivatives, linearized by using the additive Green-Naghdi's decomposition in (objective) rates. In particular, the liquid phase is a viscoelastic fluid while creep and rupture of the solid phase is considered in the Jeffreys viscoelastic rheology exploiting the phase-field model and a concept of slightly (so-called "semi") compressible materials. The $L^1$-theory for the heat equation is adopted for the Stefan problem relaxed by allowing for kinetic superheating/supercooling effects during the solid-liquid phase transition. A rigorous proof of existence of weak solutions is provided for an incomplete melting, employinga time-discretisation approximation.
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