Unlike most of the existing literature, where either a single physical process of environmental degradation at multiple scales is assumed or multiple physical processes on a single scale are considered, MDS-MP postulates governing equations of the coupled mechanical, thermal and moisture diffusion processes at the microscale of interest and systematically derives the macroscale equations by means of reduced order homogenization of multiple physical processes. In the following example, temperature dependent diffusivities are assumed to follow Arrhenius law and the mechanical constitutive equations are assumed to be governed by the viscoplasticity theory based on overstress (VBO model). The boundary sorption on the exposed boundaries is assumed to be governed by Henry’s equation.
The above diffusion-reaction model has been studied for oxidation thickness predictions of PMR-15 composite at 288°C, 316°C and 343°C. Figure 1 (bottom right) depict oxidation thickness as a function of aging time and temperature. It can be seen that the simulation results and in excellent agreement with the experiment. For validation, we consider the experimental data for unidirectional G30-500/PMR-15 composites aged in air at 288 °C. The anisotropic oxidative response was observed where the materials degraded preferentially at the specimen surface perpendicular to the fiber. The oxidation growth in the transverse direction has been observed to be the same to the oxidation growth in neat resin, whereas the growth in axial direction is much higher. Figure 1 (bottom left) shows that oxidation substantially increases in both the axial (X) and transverse (Y) directions with aging time and the dominance of the axial oxidation degradation as compared to the transverse degradation in composites due to higher effective diffusivity.
For the second multiscale-multiphysics example, consider a beam with top half made of electroactive material and lower half made of material with no electromechanical coupling (Figure 2). The electroactive material is a periodic composite with unit cell consisting of matrix material and horizontal electroactive fiber. Both materials have the same mechanical properties (linear isotropic material with Young modulus and Poisson ratio ), with fiber additionally having electromechanical coupling – electrostriction. Figure 2 shows also the comparison of direct numerical simulation (DNS) simulation with MDS-MP for different values of a. The MDS-MP simulations are performed for two different meshes with 96 elements and with 192 elements to study the convergence of MDS-MP solution to DNS solution. For a=20, the error between DNS and MDS-MP was 4.8% for 96 element mesh and 1.3% for 192 element mesh. For a=40, the error was 4.9% and 1.4%, respectively. For a=100, error was 5%.