Finite Elements of Multiscale Mixtures (FE2M) in Three Dimensions: Theory, Numerical Implementation, and Analyses

Abstract The remarkable macro-mechanics of many soft tissues and fluid-saturated, porous engineered materials derive from the complex micro-mechanics of their constituents interacting. To facilitate mechanistic understanding and improved analyses (e.g. of experimental results) of the multi-scale mechanics of such materials we aimed to establish a computational framework specific to heterogeneous, fluid-saturated, soft tissues and engineered materials. The FE2 method is a general approach in finite elements (FEs) to describe the behavior of heterogeneous materials that circumvents the macroscopic constitutive model, instead a micro-scale representative volume element (RVE) analyses uses the same FE formulation to describe the nonlinear material behaviors resulting from the microstructure. We combined the theory of mixtures and the FE2 method (i.e. finite elements of multiscale mixtures, the FE2M method) to solve 3-D, two-scale, non-linear, coupled, and time-dependent boundary value problems for poroelastic materials. We established this multi-scale framework within FEBio (University of Utah) to perform finite element discretizations of boundary value problems on both macro- and micro-scales and thus facilitate new studies of fluid-saturated, biphasic/poroelastic materials. We investigated the performance of our FE2M framework using numerical experiments, specifically 3-D simulations of square plates with regular circular holes and included two different boundary conditions. Specifically, we studied the error behaviors, convergence rates, and run times, as well as the influence of size effects, i.e. the difference in scale between the macroscale (global) models and the microscale (local) RVE models for biphasic materials. Our results show that there is a size effect but that, with careful application of the framework, we can make the error arbitrarily small. Our FE2M framework is ideally suited to bridge the joint, tissue, and intra-tissue scales inherent in the biomechanics of soft tissues, but have not yet been applied to multi-phase, heterogeneous and fibrous materials. We provide free, public access to our FE2M framework as a plugin for FEBio, including a short user manual and input files for simple test cases, at github.uconn.edu/imLab/FE2M Resources.