Finite Elements of Multiscale Mixtures (FE2M)

Developing a 3-D mutiscale framework combining the Mixture Theory and the FE2-method (i.e. the FE2M method) in order to solve two-scale, non-linear, coupled and time dependent boundary value problems (BVPs) for poro-hyper-elastic, fluid-saturated porous media. Figure 12 shows the schematic representation of the motions of solid and fluid particles in the context of the theory of porous media. To solve the partial differential equations governing the macroscopice BVP, we used the finite element method twice: we evaluated the macroscopic material tangent and measures first P-K stress, Green Lagrange strain rate, right Cauchy-Green tensor, Jacobian, fluid volume fraction, and filtration velocity respectively at each Gauss integration point by a volume averaged solution of the underlying microscopic Representative Volume Element (RVE), as shown in Fig. 13. We considered 3-D unit cube for macrostructure, as shown in Fig. 14. Figure 15 shows the macroscopic results for both single and two-scale models for displacement fields. We also demonstrate the micro (RVE) response for elements 10, 410, 470, and 890.

Past Research

My research focused on developing numerical methods for important class of engineering applications such as microstructure evolution, contact mechanics, and frictional crack.

Polycrystalline Solidification with Phase Field Model

Materials engineering problems related to polycrystalline solids often require the prediction of grain growth and the stress analysis of polycrystalline materials. The novelty and intellectual contribution of this research originates from applying recently developed strong form–based collocation method to polycrystalline materials to assess the feasibility of the method in polycrystalline solidification analysis with a diffusive interface approach. We considered the effect of rigid inclusions on polycrystalline growth. For this analysis, the computational domain was discretized with 29,241 collocation points. The inclusions were modeled with the static phase field order parameters. In other words, the phase field variables φi corresponding to the inclusions were set to 1 (i.e., fully solidified solids) during the phase field model analysis without updating them. Figure 1 illustrates the evolution of the polycrystalline structure with six rigid inclusions at different time steps. Figure 1(i) shows the final steady-state polycrystalline structure.

Polycrystalline Stress Analysis

Performing stress analysis with the material interfaces, i.e., grain boundaries. We considered the polycrystalline structure consisting of 30 grains with 6 inclusions; the boundary conditions applied to the computation are shown in Fig. 2. In this example, two different cases, i.e., Case 1 and Case 2, were considered to further investigate the effect of the relative rigidity of inclusions on the deformation of the polycrystalline materials. To this end, the same order of the Young’s modulus for both grains and inclusions were considered for Case 1. However, for Case 2, the Young’s modulus of the inclusions was one order higher than that of the grains in order to study the aforementioned effect of the harder inclusions in a polycrystalline structure. The contour plots of displacement and von Mises stress considering the same order (i.e., Case 1) of material properties for grains and inclusions is shown in Fig. 3.

Contact Mechanics

This study is the first attempt to model frictional contact using a meshfree collocation approach on a strong form. Frictional contact constraints are included in the strong form force balance equation as part of Neumann boundary conditions. The equation is then directly discretized with the precomputed derivative operators obtained from moving least-squares approximation using Taylor expansion of the displacement field through point-wise computations at collocation points. A half cylinder is pressed by a distributed pressure resulting in a normal contact force at the point of the contact. One of main advantages of the proposed collocation method is the easy adaptive refinement of the collocation points. This advantage is applied to this problem since high accuracy is required on the contact surface. To examine the mesh dependence of the method, in Fig. 4, three types of non-uniformly distributed collocation points we used are provided: (a) coarse over entire domain, (b) fine over entire domain, and (c) refined in the vicinity of contact region. Plots of the normal contact traction and the stress in the y-direction are presented in Fig. 5. In Fig. 5(a), we compare the normal contact traction from three collocation points in Fig. 4 along with the analytical solution. While the traction obtained from Fig. 4(c) is almost consistent with the analytical solution, the results from Fig. 4(a) and (b) are significantly deviated. In Fig. 5(b), high stress concentration near the contact region is observed as expected.

Frictional Crack

Strong form meshfree collocation method is proposed to model friction in crack surface. First, the proposed method is extended to solve multi-body frictional contact problems. Then, the near-tip field is modeled based on explicitly imposing the traction-free boundary condition on the crack surfaces and using visibility criterion to deal with the displacement discontinuity due to a crack. We consider a cracked body as shown in Fig. 6 in the framework of the point collocation method based on the strong formulation. We examine frictional sliding across an internal crack surface as shown in Fig. 7. The comparison of horizontal displacement contours obtained using the collocation method and the finite element method (FEM) using ABAQUS is plotted in Fig. 8. The contour plots are in excellent agreement with those obtained by FEM, and confirm that the proposed method gives consistent solutions for the frictional crack problem.

Nanocomposites Materials

In this study, we used the stochastic analysis and the computational homogenization method to analyse the effect of thickness and stiffness of the interphase region on the overall elastic properties of the clay/epoxy nanocomposites. Figure 9 shows a detailed view of the mesh around the clays. The mesh has a single element through the thickness of interphase zone as well as clay thickness. Table 1 summarizes the results of the stochastic analysis. The table shows the mean values and standard deviation of the Young’s modulus for 0.5%; 1%; 2%; 3% and 4% clay/epoxy nanocomposites. As it is expected, the Young’s modulus of the clay/epoxy nanocomposites increases with increasing in the clay contents. To validate the results of stochastic analysis with experimental results, Fig. 10 plots the results of Table 1 as well as the experimental results. Figure 11 shows the scatter plot of Young’s modulus of clay/epoxy nanocomposite versus the interphase thickness and the Young’s modulus of interphase region for 3% clay/epoxy nanocomposite. The x-axis is used to show the scale of two main parameters, thickness (hi) and stiffness of interphase layer (Ei) as an independent variable and y-axis measure the Young’s modulus of clay/epoxy nanocomposites as dependent variable.