A COMPUTATIONALLY EFFICIENT XFEM BASED STOCHASTIC MULTISCALE FRAMEWORK FOR THE ANALYSIS OF HETEROGENEOUS MATERIALS

Abstract

Composite materials have high strength to weight ratio and good mechanical and thermal properties. These materials are used in various applications under very harsh service environment. The reliability and safety of an engineering structure undergoing extreme service loads can be assessed through experiments. However, performing large number of experiments is expensive, time consuming and practically impossible. Numerical methods can be used as an alternative for predicting the failure of the structure under mechanical and thermal loads. In order to obtain a reliable numerical solution, the investigators must be certain about behavior and properties of the materials. The behavior of any material is greatly affected by its microstructural morphology. It has been noticed that the heterogeneous composite materials possess large scatter in their properties due to presence of reinforcement particles, pores, micro-cracks and various other defects. The analysis of these materials can be performed using analytical and numerical schemes. The development of analytical schemes for the modeling of heterogeneous material is limited to simple geometries. On the other hand, complex geometric features can be easily modelled through numerical techniques. Over the years, the primary focus of the research has been to incorporate actual microstructural morphology in the numerical models. The classical approach of the continuum mechanics is insufficient to include such microstructure features during the analysis. Direct numerical simulations using finite element method (FEM) requires huge computational power and memory to analyze heterogeneous materials. The major challenge with FEM and other mesh based methods in analyzing heterogeneous material is to generate conformal mesh. Extended finite element method (XFEM) resolves this issue by adding additional enrichment functions in the trial function approximation. In the past two decades, multiscale techniques have helped in properly analyzing the heterogeneous materials. During the implementation of multiscale technique, the solution variables at different scale are obtained and correlated. Generally, most of the multiscale framework focused on the prediction of elastic properties of the materials. The modeling of variation in the tensile properties of the materials have rarely been demonstrated due to large memory and high computational cost requirement. To overcome these issues, XFEM is integrated into stochastic multiscale framework for modelling heterogeneous materials. The present thesis mainly focuses on the development of a computationally efficient stochastic multiscale framework based on XFEM for predicting the variation in the elastic modulus and tensile strength of heterogeneous material. At first, single scale XFEM based stochastic multiscale framework is developed. In this approach, the XFEM is implemented at micro-scale and FEM is implemented at macro-scale. Abstract iii The properties of the heterogeneities are assumed to be known. To model microstructure morphology, stochastic techniques are used for modelling the effect of size, shape, clustering and volume fraction of the heterogeneities (pores and reinforcement particles). Adaptive hanging node mesh refinement technique is used for improving the efficiency. The convergence analysis is performed for adaptive XFEM, which includes RVE size estimation and mesh sensitivity analysis. The numerical results obtained using adaptive XFEM are found accurate and efficient as compared to FEM. The micro-scale analysis predicts that the variation in the elastic modulus and tensile strength is mainly due to the change in the volume fraction of heterogeneities. The macro-scale analyses are performed by stochastically distributing micro-scale properties over macro-domain. The analysis predicts a large scatter in the tensile strength and small scatter in the elastic modulus. Next, a two scale XFEM based stochastic multiscale framework is developed. In this scheme, XFEM is implemented at both micro and macro-scales. NBG-18 graphite is considered as an example for the study. The size of reinforcement particles in the NBG-18 is large as compared to pores. Therefore, pores are modelled at micro-scale and reinforcement particles are modelled at macro-scale. At micro-scale, the properties are evaluated for each pore diameter. The range of pore diameter is based on the experimental probability distribution function. At macro-scale, the reinforcement particles are modeled stochastically for a given range and volume fraction. The fracture plane orientation is assigned randomly for the elements associated with reinforcement particles. The micro-scale properties are assigned stochastically to all the remaining elements of the macro-domain. The numerical variation in the tensile strength property obtained from macro-scale analysis is found in good agreement with the experimental data. Finally, multi-split XFEM approach is developed to further enhance the computational efficiency of the analysis. The multi-split XFEM is capable of modeling multiple discontinuities within a single element. Therefore, mesh density of the domain becomes independent of the gap among the heterogeneities. The concept of level set correction is introduced to reduce the error encountered during mapping of level set values. The accuracy of the multi-split XFEM is examined through convergence analysis. The numerical analysis proved that multi-split XFEM is about 10% computationally more efficient than adaptive XFEM. The capability of XFEM based stochastic multiscale framework has been demonstrated by predicting the tensile strength probability of the heterogeneous materials. The elastic modulus values lie within the analytical bounds and tensile strength properties are found in good agreement with the experimental data. Adaptive hanging node mesh refinement technique greatly enhances the computational efficiency and accuracy of adaptive XFEM. The developed multi-split XFEM is computationally more efficient than adaptive XFEM.