Under fatigue-loading, short-fiber reinforced thermoplastic materials typically show a progressive degradation of the stiffness tensor. The stiffness degradation prior to failure is of primary interest from an engineering perspective, as it determines when fatigue cracks nucleate. Efficient modeling of this fatigue stage allows the engineer to monitor the fatigue-process prior to failure and design criteria which ensure a safe application of the component under investigation.

We propose a multiscale model for the stiffness degradation in thermoplastic materials based on resolving the fiber microstructure. For a start, we propose a specific fatigue-damage model for the matrix, and the degradation of the thermoplastic composite arises from a rigorous homogenization procedure. The fatigue-damage model for the matrix is rather special, as its convex nature precludes localization, permits a well-defined upscaling, and is thus well-adapted to model the phase of stable stiffness degradation under fatigue loading. We demonstrate the capabilities of the full-field model by comparing the predictions on fully resolved fiber microstructures to experimental data.

Furthermore, we introduce an associated model-order reduction strategy to enable component-scale simulations of the local stiffness degradation under fatigue loading. With model-order reduction in mind and upon implicit discretization in time, we transform the minimization of the incremental potential into an equivalent mixed formulation, which combines two rather attractive features. More precisely, upon order reduction, this mixed formulation permits precomputing all necessary quantities in advance, yet, retains its well-posedness in the process. We study the characteristics of the model-order reduction technique, and demonstrate its capabilities on component scale. Compared to similar approaches, the proposed model leads to improvements in runtime by more than an order of magnitude.

在疲劳载荷下，短纤维增强热塑性材料通常表现出刚度张量的逐渐退化。故障之前的刚度退化是从工程的角度看主要感兴趣的，因为它决定何时疲劳裂纹成核。该疲劳阶段的有效建模使工程师能够在失效和设计标准之前监控疲劳过程，从而确保所研究组件的安全应用。

我们提出了一种基于解析纤维微观结构的热塑性材料刚度退化的多尺度模型。首先，我们为基体提出了一个特定的疲劳损伤模型，热塑性复合材料的降解源于严格的均质化程序。矩阵的疲劳损伤模型相当特殊，因为它的凸面性质排除了局部化，允许定义明确的放大，因此非常适合模拟疲劳载荷下稳定刚度退化的阶段。我们通过将完全解析的纤维微结构的预测与实验数据进行比较来证明全场模型的能力。

此外，我们引入了相关的模型阶数减少策略，以实现疲劳载荷下局部刚度退化的组件级模拟。考虑到模型阶数减少和隐式离散化，我们将增量潜力的最小化转换为等效的混合公式，它结合了两个相当有吸引力的特征。更准确地说，在订单减少时，这种混合公式允许预先计算所有必要的数量，但在此过程中仍保持其适定性。我们研究了模型降阶技术的特性，并展示了其在组件规模上的能力。与类似的方法相比，所提出的模型使运行时间提高了一个数量级以上。