AI 中文总结
研究针对相场模型应用于工程尺度部件时计算成本高的问题，提出基于FE$^2$的均匀化框架，通过建立微观 - 宏观关系实现理论在双尺度方案中的应用，经实例验证可合理准确预测宏观平均场演化。
AI 中文摘要
相场模型已成为模拟材料中复杂微观结构演化的标准工具，但其应用于工程尺度部件时，常因解析精细尺度特征所需的高昂计算成本而受阻。为应对这一挑战，我们提出了一个相场理论的一致均匀化框架。通过根据古尔丁微力制定的微观均匀性的希尔 - 曼德尔型条件，为代表性体积单元（RVE）导出了一个适定的边值问题，建立了序参量及其梯度的严格微观 - 宏观关系。该理论在计算双尺度（FE$^2$）方案中实现，并通过直接数值模拟进行了验证。研究了两个不同的例子：一个最小的艾伦 - 卡恩模型和一个用于应力驱动马氏体相变的机械耦合模型。结果表明，所提出的框架能够以合理的精度可靠地预测宏观平均场的空间和时间演化。
英文摘要
Phase-field models have become a standard tool for simulating complex microstructure evolution in materials, but their application to engineering-scale components is often hindered by prohibitive computational costs arising from the need to resolve fine-scale features. To address this challenge, we propose a consistent homogenization framework for phase-field theory. By enforcing a Hill-Mandel-type condition of micro-homogeneity formulated in terms of Gurtin's microforces, a well-posed boundary value problem is derived for the representative volume element (RVE), establishing rigorous micro-macro relations for both the order parameter and its gradient. The theory is implemented within a computational two-scale (FE$^2$) scheme and validated against direct numerical simulations. Two distinct examples are investigated: a minimal Allen-Cahn model and a mechanically-coupled model for stress-driven martensitic phase transformation. The results demonstrate that the proposed framework can reliably predict the spatial and temporal evolution of the macroscopically averaged fields with reasonable accuracy.
