The Theory Of Spatial Transformation For Crack Problems And Its Numerical Applications

The materials in the field of civil engineering often have natural internal defects, For example, the microcracks exist in concrete, and rock masses contain a number of multiscale flaws such as joints, cracks and faults, which are induced by the initial stage of formation of rock masses and successive tectonic motion processes. These internal defects may become of the key factor of the engineering accident. Therefore, the internal defects gain a lot of attention in many large-scale projects. The extended finite element method(XFEM) is a new numerical method, which has developed in the last 15 years. This method is applied to simulate the problems of discontinuity based on the partition of unity method. Two types of enrichment functions(Heaviside enrichment function and tip enrichment function) are introduced into XFEM. XFEM can simulate the discontinuity of the displacement field across the crack surface and the singularity at the crack tips without remeshing. This method overcome the limitations of the finite element method to deal with the problems of discontinuity, in which the finite element method have to spend much time due to the need of updating the mesh to match the geometry of the discontinuity. In this thesis, based on the XFEM, a new numerical method is proposed to analyze the non-linear problems of the propgation and coalescence of the cracks. This new numerical method called the theory of spatial transformation. The main work is summarized as follows:(1) An experimental study of rock-like materials containing multiple flaws under uniaxial compression is conducted. Firstly, according to the complete stress-strain curves of specimens, the effects of flaw angle and non-overlapping length on the mechanical properties of rock-like specimens are studied. The influences of the geometry of the pre-existing flaws on the mechanical properties of rock-like specimens are revealed. Then, during deformation, the real-time crack propagation and coalescence processes are captured by the high-speed digital video camera system HG-100 K. The effects of the flaw angle and non-overlapping length on the cracks initiation and cracks coalescence are investigated. The evolution laws of cracks affected by the flaw angles and non-overlapping length are revealed.(2) The theory of spatial transformation is proposed based on the XFEM to simulate the nonlinear problems in incremental form and the problems of multiple cracks. In this method, each node has 12 n DOFs in the domain containing n cracks. The simulation of the growth and propagation of cracks is conducted in theory of spatial transformation. The generalized boundary condition and its matrix are determined. The modified matrices of stiffness, mass and damping are also highly sparse. The proposed method equips high computational efficiency.(3) The theory of spatial transformation is applied to simulate the crack growth and coalescence under dynamic loads. Solution scheme of the dynamic problems in the framework of the theory of spatial transformation is established. The implicit Newmark time integration scheme is introduced to solve the time integration. It can be applied to deal with the problems with long dynamic response, and its solution is unconditionally stable. Moreover, three classical cases are investigated using the theory of spatial transformation to verify the correctness of the proposed method.(4) The theory of spatial transformation for geometrically nonlinear problems under total Lagrangian formulation is proposed. The total Lagrangian formulation is applied to analyze the geometrically nonlinear problems, especially for large deformation. During deformation, the Jacobian matrix is introduced to convert the second Piola-Kirchhoff stress tensor into the Cauchy stress tensor, which do not rely on material deformation history.(5) The theory of spatial transformation for frictional contact on crack slip is proposed. The modified Coulomb’s frictional model is introduced as the frictional constitutive model. The penalty method is used to incorporate the contact constraints into the theory. The thresholding method and image processing technique is introduced to deal with the pictures of real-time process of the specimens. The numerical results are in good agreement with the experimental results. Moreover, frictional contact on crack slip under dynamic loads in the framework of spatial transformation is established.(6) The theory of spatial transformation for crack branching is proposed. The formula of energy of boundary shift is deduced. As the model of boundary shift can be abstracted into the model of boundary cracking, the energy of boundary cracking is obtained. The energy release rates of the crack branching in the four quadrants are determined based on the energy of boundary cracking. Further, the energy release rate of the crack branching is introduced into the theory of spatial transformation to judge whether the crack tips will branch. If the crack tips branch, the criteria of crack branching are applied to determine the direction of the branching cracks, otherwise the criteria of maximum circumferential stress are applied to determine the direction of the cracks.