In this paper we present a fully-coupled, two-scale homogenization method for dynamic loading in the spirit of FE\(^2\) methods. The framework considers the balance of linear momentum including inertia at the microscale to capture possible dynamic effects arising from micro heterogeneities. A finite-strain formulation is adapted to account for geometrical nonlinearities enabling the study of e.g. plasticity or fiber pullout, which may be associated with large deformations. A consistent kinematic scale link is established as displacement constraint on the whole representative volume element. The consistent macroscopic material tangent moduli are derived including micro inertia in closed form. These can easily be calculated with a loop over all microscopic finite elements, only applying existing assembly and solving procedures. Thus, making it suitable for standard finite element program architectures. Numerical examples of a layered periodic material are presented and compared to direct numerical simulations to demonstrate the capability of the proposed framework. In addition, a simulation of a split Hopkinson tension test showcases the applicability of the framework to engineering problems.

在本文中，我们本着FE \（^ 2 \）的精神，提出了一种用于动力载荷的全耦合，两尺度均质化方法方法。该框架考虑了包括微惯性在内的线性动量的平衡，以捕获由微观异质性引起的可能的动力效应。有限应变公式适用于解决几何非线性问题，从而可以研究可塑性或纤维拉拔现象，这些问题可能与大变形有关。建立一致的运动比例尺链接作为对整个代表性体积元素的位移约束。得出一致的宏观材料切线模量，包括闭合形式的微观惯性。只需应用现有的装配和求解程序，就可以轻松地在所有微观有限元上进行循环计算。因此，使其适合于标准的有限元程序体系结构。提出了层状周期性材料的数值示例，并将其与直接数值模拟进行比较，以证明所提出框架的功能。此外，模拟霍普金森拉伸试验的结果证明了该框架对工程问题的适用性。