Transient multi-scale analysis with micro-inertia effects using Direct FE^2 method

This paper presents an extension of Direct FE^2 method for the study of dynamic problems in heterogeneous materials. The proposed method can be formulated based on either the Hill–Mandel principle or the extended Hill–Mandel principle, the latter of which enforces the energy contributed by the internal force and the inertial force consistent at two scales. Unlike the traditional FE^2 method, it is not necessary to conduct two levels of finite element simulations linked by extensive information interchange. Instead, we reformulate the macroscopic variational statement with the microscopic contributions, leading to only a single coupled boundary value problem. The classical microscopic boundary condition used in the traditional FE^2 method can be employed but it is implemented through kinematical constraints between the macro nodes and the micro nodes in the Direct FE^2 method. The proposed method is illustrated by two numerical examples including a fiber-reinforced composite and an acoustic metamaterial. The results are verified by direct numerical simulations and it is shown that micro-inertia effects are not important in modeling low-velocity impact behavior of the composite but they are essential in capturing wave attenuation performance of locally resonant acoustic metamaterials.