论文标题
带有散装表面动力学的相位场的连续框架
A continuum framework for phase field with bulk-surface dynamics
论文作者
论文摘要
这种连续的力学理论旨在详细介绍Fischer，Maass和Dieterich [1]，Goldstein，Miranville和Schimperna [2]和Knopf，Lam，Liu＆Metzger提出的动态边界条件的潜在理性机制。作为副产品，我们概括了这些理论。这些类型的动态边界条件通过相位场的整体和表面部分微分方程之间的耦合来描述。我们在此连续框架内的出发点是在任意零件$ \ Mathcal {p} $上假定的虚拟幂的原理，其中边界$ \ partial \ partial \ mathcal {p} $可能会失去平滑度。也就是说，正常字段可能沿边缘$ \ partial^2 \ Mathcal {p} $不连续。但是，表征正常场不连续性的边缘被认为是光滑的。我们的结果可以总结如下。我们为散装耦合的虚拟力原理提供了广义版，以及部分自由能不平衡的广义版本。接下来，我们将表面和边缘微裂缝的显式形式以及散装和表面相位场的场方程。最后一组场方程在某种程度上类似于散装和表面的cahn--hilliard方程。最后，我们提供了一组合适的组成关系和热力学一致的边界条件。
This continuum mechanical theory aims at detailing the underlying rational mechanics of dynamic boundary conditions proposed by Fischer, Maass, & Dieterich [1], Goldstein, Miranville, & Schimperna [2], and Knopf, Lam, Liu & Metzger, [3]. As a byproduct, we generalize these theories. These types of dynamic boundary conditions are described by the coupling between the bulk and surface partial differential equations for phase fields. Our point of departure within this continuum framework is the principle of virtual powers postulated on an arbitrary part $\mathcal{P}$ where the boundary $\partial\mathcal{P}$ may lose smoothness. That is, the normal field may be discontinuous along an edge $\partial^2\mathcal{P}$. However, the edges characterizing the discontinuity of the normal field are considered smooth. Our results may be summarized as follows. We provide a generalized version of the principle of virtual powers for the bulk-surface coupling along with a generalized version of the partwise free-energy imbalance. Next, we derive the explicit form of the surface and edge microtractions along with the field equations for the bulk and surface phase fields. The final set of field equations somewhat resembles the Cahn--Hilliard equation for both the bulk and surface. Lastly, we provide a suitable set of constitutive relations and thermodynamically consistent boundary conditions.
