In the present work, the evolution of damage in periodic composite materials is investigated through a novel finite element-based multiscale computational approach. The proposed methodology is developed by means of the original combination of asymptotic homogenization with the phase field approach to nonlocal damage. This last is applied at the macroscale level on the equivalent homogeneous continuum, whose constitutive properties are obtained in closed form via a two-scale asymptotic homogenization scheme. The formulation considers different assumptions on the evolution of damage at the microscale (e.g., damage in the matrix and not in the inclusion/fiber), as well as the role played by the microstructural reinforcement, i.e. its volumetric content and shape. Numerical results show that the proposed formulation leads to an apparent tensile strength and a post-peak branch of unnotched and notched specimens dependent not only on the internal length scale of the phase field approach, as for homogeneous materials, but also on microstructural features. Down-scaling relations provide the full reconstruction of the microscopic fields at any point of the macroscopic model, as a simple post-processing operation.

中文翻译：

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周期性微结构材料损伤传播的相场方法

在目前的工作中，通过一种新的基于有限元的多尺度计算方法研究了周期性复合材料中损伤的演变。所提出的方法是通过渐近均匀化与非局部损伤的相场方法的原始组合开发的。后者在宏观尺度上应用于等效均匀连续体，其本构特性通过两尺度渐近均匀化方案以封闭形式获得。该公式考虑了对微观尺度损伤演变的不同假设（例如，基体中的损伤而不是夹杂物/纤维中的损伤），以及微结构增强材料所起的作用，即其体积含量和形状。数值结果表明，所提出的配方导致明显的拉伸强度和无缺口和缺口试样的峰后分支，不仅取决于相场方法的内部长度尺度，对于均质材料，还取决于微观结构特征。作为简单的后处理操作，按比例缩小关系在宏观模型的任何点提供微观场的完整重建。