Reproducing kernel finite volume methods for dynamic brittle fracture
Open Access
- Author:
- Yang, Saili
- Graduate Program:
- Civil Engineering (PHD)
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- October 08, 2021
- Committee Members:
- Pinlei Chen, Major Field Member
Charles Bakis, Outside Field Member
Christian Peco, Outside Unit Member
Mike Hillman, Chair, Minor Member & Dissertation Advisor
Patrick Fox, Program Head/Chair - Keywords:
- Reproducing kernel finite volume methods
Meshfree methods
Dynamic fracture
Cohesive zone methods
Phase-filed modelling - Abstract:
- Fracture simulation in solid mechanics still faces many challenges. Various numerical methods have been developed in this area. Some of the popular methods include virtual crack closure, extended methods, damage mechanics based methods, phase-field methods and cohesive zone methods. In practice, these methods encounter one or more than one of the following difficulties: (1) inability initiate a crack; (2) instability associated with material softening; (3) complex crack tracking processes with crack propagation; (4) complicated fracture parameters without clear physical interpretations. Even though the meshfree methods can avoid any mesh related issues, they also possess other issues such like imposing essential boundary conditions and numerical integration. In this work, a conforming reproducing kernel finite volume method (RKFM) is derived based on a global weak form. In this method, the essential boundary conditions can be directly enforced with collocation. The variational consistency conditions (for Galerkin exactness) have been examined, with associated numerical patch tests and convergence rate tests performed. It is found that the method can converge optimally with low-order quadrature, in contrast to conventional Galerkin meshfree methods. The comparison of using conforming and nonconforming cells is also made, where it is found the conforming condition is essential for convergence. In addition, the method has been extended to elastodynamics, where a one dimensional wave problem is used as a benchmark with good agreement with the analytical solution. The cohesive zone model (CZM), in which the fracture parameters possess clear physical meanings, is then implemented with RKFM. The cohesive traction in CZM can be treated as natural boundary conditions applied on the cracked cell surface. The crack separation is explicitly defined as a displacement jump using the reproducing kernel approximation. A cell conforming kernel is proposed under this framework for expediency. This approach is distinctly different from other CZM based methods where cohesive elements are inserted dynamically. The classical branching problem is tested for verification of this method in capturing the ability to capture dynamic branching, and provide results that are insensitive to the resolution of the discretization. This method has been also used to simulate a single edge notched specimen test for validation, which shows great consistency with the experimental results. A phase-field approach is further developed under the RKPM framework, where the hyperbolic version is considered for efficient explicit dynamics. In this process, a regularized strain energy is proposed as the driving force for phase-field updates. The dynamic crack branching problem is also tested with this method, where the method is shown to be effective, and also provides solutions insensitive to refinement. A discontinuous RKFM formulation is also given to enhance the stability when using a cell-conforming kernel. In this formulation, a discontinuous RK approximation is proposed. Different numerical traces are tested to alleviate the discontinuity across cells. An averaged numerical trace is shown to be effective in stabilizing the simulation with a nonuniform discretization. The implementation of CZM is not influenced in this discontinuous formulation. The discontinuous RK approximation and the CZM implementation are also applied in reproducing kernel particle methods (RKPM). In this investigation, quasi-linear RK approximation is found to be inconsistent with a singular kernel when using a direct nodal integration. A discontinuous stabilized conforming nodal integration method is given to deal with this issue. This method can also be combined with CZM and shows effectiveness in predicting the crack branching problem. Both the RKFM-CZM and RKFM-phase-field methods are applied to simulate the fracture process in a high performance concrete, where the numerical results are compared with the experimental results with good agreement obtained.