Effective heat conductivity of composites with stochastic microstructure using asymptotic homogenization

This contribution presents a comprehensive methodology aimed at determining the effective heat conductivity of composites with stochastic microstructure by analyzing micro Computerized Tomography (μCT) images. We revisit asymptotic homogenization for multiscale analysis of transient heat problems and derive systems of partial differential equations (PDEs) governing the homogenized problem and two new cell problems, which are solved numerically using the Finite Element (FE) method to identify the effective thermal conductivity. The methodology does not require enforcing Dirichlet Boundary Conditions (BCs) on the interfaces, making it well-suited for analyzing stochastic microstructures with irregular interfaces. Following image preprocessing and segmentation to identify the pores (void) and the solid matrix, the workflow transforms the segmented image into a periodic computational domain suitable for the upscaling procedure to identify the effective thermal conductivity tensor. We employ the open-source computing platform FEniCSx, along with its Multi-Point Constraints (MPC), to solve the computational problems and enforce the periodic boundary condition (PBC), eliminating the need for one-by-one mapping of inlet and outlet computational nodes. To validate the methodology, we apply it to model a bi-laminated composite and compare the obtained results with analytical values. This is followed by statistical descriptions of μCT images of several samples, together with a comprehensive representativity analysis using multiple RVE realizations approach. We find the results of statistical descriptions useful to guide us in selecting suitable RVE sizes.