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Abstract

It is important to consider the microstructure of a material when studying the macroscopic mechanical properties. Although special equipments have been used for micromechanics study through experimental tests, it is limited by instruments and reproducibility. In contrast, numerical methods have advantages on modelling of different scales, both in space and time. The efficiency, economy, reproducibility and flexibility make numerical modelling very convenient to perform micromechanics study. Current numerical methods used in micromechanics can be grouped into continuous methods, discontinuous methods and coupled methods. Continuous methods are good at the stress analysis in the pre-failure stage while the discontinuous methods are good at the motion analysis in the post-failure stage. Combining continuous and discontinuous methods, coupled methods are developed to deal with problems with different features, e.g., multiscale analysis. The numerical manifold method (NMM) is one of the coupled methods which can provide a uniform framework for continuous-discontinuous analysis. In NMM, the mathematical interpolation and the physical integration are separated. The distinct cover system makes NMM very suitable for large deformation analysis considering both continuous and discontinuous behaviors. However, NMM is still a polyhedron-based numerical method. The geometrical operations (topology and contact detection) are the obstacle especially in 3D. Inspired by micro-based numerical methods, the particle manifold method (PMM) is developed as an extension of NMM. PMM uses a mathematical cover system to describe the motion and deformation of a particle-based physical domain. By introducing the concept of particle into NMM, PMM takes the advantages of easy topological and contact operations with particles. Furthermore, the particle representation is much more suitable for micromechanics study. In this thesis, the methodology, formulations and implementation of the method are presented. After establishment of theoretical basement and validation of numerical implementation, four aspects of PMM are further developed. The first aspect of PMM is the particle contact model and its applications. As an important discontinuous feature, the contact problem in PMM is explicitly described by the particle-particle (P-P) model. The P-P model is very simple to implement through the concept of the penalty method. Different constitutive laws can be introduced using the penalty number, which is the only artificial parameter. The P-P model also simplifies the geometrical operation and the description of rough surface. With the P-P model, PMM is used to model P-wave transmission across rock fractures with different properties. Through the comparisons with theoretical solutions, PMM is validated for the modelling of wave transmission across both continuous and discontinuous rock fractures. The second aspect of PMM is the micro failure model and its applications. The failure and fracturing analysis of PMM is performed through particle models which can be grouped into continuous models and discontinuous models. The Particle-Link (P-L) model is developed based on continuous particles. Commonly adopted macro failure criteria (e.g., Mohr-Coulomb criterion) can be easily implemented for the P-L model. The other Particle-Contact (P-C) model is developed based on discontinuous particles. Corresponding to the particle failure models, the technique of topological detection between polyhedron and particle ensures that degrees of freedom (DOFs) are properly assigned when the model geometry changes. With these micro particle models, brittle fracturing problems under both static and dynamic loadings are simulated by PMM. It is proven that PMM is fully capable for the micromechanics study. Another aspect of PMM is the code design and parallel computing. During the whole calculation procedure in PMM, solving the global system equation is the key part and consumes most of the computational time. Therefore, an accurate and efficient solver is very important to PMM. Therefore, a sparse and block preconditioned conjugate gradient method (SBPCG) is developed. To achieve high performance, a modern GPU parallel algorithm using CUDA (NVIDIA's parallel computing architecture) is developed for SBPCG. The testing results show that GPU parallelism is a good option for the particle-based method which is data-intensive. The last aspect of PMM is the multiscale algorithm. Due to the same mathematical and mechanical framework adopted by both NMM and PMM, the multiscale particle manifold method (MPMM) is achieved without much difficulty. The proposed multiscale model consists of a unified mathematical cover system, a macro material zone represented by polygons and a micro material zone represented by particles. The numerical demonstrations show that MPMM (1) performs well in accuracy, (2) is good at presenting material’s micro structure and (3) is suitable for discontinuous analysis. With the proposed MPMM, the dynamic Brazilian disc test is simulated and the effect of loading rate on material strength is investigated. A novel micro view is proposed to explain the true effect of loading rate. It is believed that the micro intrinsic strength of the material exits and it does not change with the loading rate. In summary, the particle manifold method (PMM) is developed for multiscale continuous-discontinuous analysis in this thesis. The micro concept is successfully introduced into the continuous-discontinuous framework of NMM. The distinct particle representation combined with topological detection provides flexibility to implement micro failure model and micro contact model. PMM also has the potential to achieve high performance through modern GPU parallelism. The multiscale algorithm of PMM is straightforward. In future, micromechanics study of PMM and 3D implementation are the main extending directions.

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