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.

中文翻译：

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用于在两个尺度上模拟动态问题的通用、隐式、大应变 FE$^2$ 框架

在本文中，我们本着 FE$^2$ 方法的精神，提出了一种用于动态加载的全耦合、两尺度均质化方法。该框架考虑了线性动量的平衡，包括微观尺度的惯性，以捕捉微观异质性引起的可能动态影响。有限应变公式适用于考虑几何非线性，从而能够研究例如塑性或纤维拉出，这可能与大变形有关。建立一致的运动学比例链接作为整个代表性体积元素的位移约束。导出一致的宏观材料切线模量，包括封闭形式的微惯性。这些可以通过所有微观有限元的循环轻松计算，只需应用现有的组装和求解程序。因此，使其适用于标准的有限元程序架构。提出了分层周期性材料的数值示例，并与直接数值模拟进行了比较，以证明所提出框架的能力。