An explicit phase field modeling of brittle plates with nonlocal operator method under certain loading condition

Abstract

Due to its complexities and challenges, fracture modeling still faces many pressing issues despite increasing research efforts. One common method is the finite element method, which is commonly utilized in this field. However, FEM has some disadvantages when it comes to predicting failures, because it relies on continuum mechanics, which is a mathematical model that is not valid when there are cracks or other discontinuities in the displacement field. To address this problem, a new approach developed, called peridynamics, which is a different way of formulating the equations of continuum mechanics that is better suited to predicting failure in structures. PD can overcome the limitation of FEM, which requires describing and tracking of the crack path during the growth. While peridynamics can be used to a broad range of materials and mechanical problems, they are most often used to study for brittle fracture. On the other hand, PD can suffer from hourglass modes, which causes instability and it requires a constant horizon which needs significant computational costs. The nonlocal operator method (NOM) is a method for developing nonlocal forms using the concept of dual-support, and it also allows for the creation of implicit formulations of nonlocal theory. It is notable for its ability to be used with both variational and weighted residual methods, and it is a generalized version of the DH-PD approach that extends the idea of nonlocality. In addition to this, NOM allows for the variation of the energy functional in nonlocal theories. Traditional integral equations are defined in a single integral domain, while NOM and other dual-support approaches are based on the use of two integral domains. In the nonlocal operator method (NOM), partial derivatives are calculated using nonlocal versions of gradient, curl and divergence operators. These operators are used to approximate the local operators in the limit as the internal length scale approaches zero. The nonlocal operator method does not need the use of shape functions like FEM. Instead, it directly obtains discrete equations through the use of nonlocal operators, which greatly simplifies the numerical implementation. The phase field method is a numerical technique used to model brittle fracture in materials. It is based on the use of a phase field parameter, which changes continuously as the structure evolves, to simulate the evolution of cracks. One of the main advantages of this approach is that it can simulate multiple cracks, regardless of their number or shape, in a continuous manner without the need to explicitly track their positions. In this thesis, a nonlocal operator method combined with an explicit phase field method was used to model the propagation of quasi-static fractures and compare the computational efficiency of this approach with numerical models based on implicit methods from the literature. The strong form of the governing equations was derived based on the energy form of the phase field model, and both the mechanical field and phase field were updated using an explicit time integration. Numerical benchmark problems, including an L-shaped panel, a three-point bending test, and a notched plate with holes, were analyzed and the results were found to be in good agreement with previous work. To develop the computational performance of the explicit model, a hybrid implicit/explicit model was also proposed. Additionally, a local damping technique was used to decrease the ratio of kinetic energy to internal energy in the explicit phase field model and apply mass scaling, which saved computational time in the cases studied.