Extending the Thick Level Set Approach: Plasticity, Parallel Computing and Cohesive Cracks

Developing accurate and robust numerical approaches that are capable of modeling fracture in solids has been a challenging undertaking in the computational mechanics community for decades. Models based on a continuous formulation or on a discontinuous one have been proposed by numerous authors, expanding upon abilities and disadvantages of these approaches. However, models attempting to bridge these two approaches have been less often encountered in the literature.

Over the last ten years, a new approach for modeling fracture in solids has been developed, coined the Thick Level Set (TLS) method, in which the damage evolution is linked to the movement of a damage front described with the level set method. This model offers an automatic transition from damage to fracture and deals with merging and branching cracks as well as crack initiation in a easy and robust manner. Furthermore, the TLS in its new (second) version, coined the TLSV2, is able to model explicitly the displacement discontinuity at the position of a crack.

These TLS features are very beneficial for the modeling of cusp crack patterns in resin-rich regions of fiber reinforced polymer composites under mode II loading. In this process, plasticity might occur prior to fracture, which begins with a series of inclined cracks that eventually merge to form what is at higher scale of observation understood as a single crack. When the crack reaches one of the boundaries of these resin-rich regions, the localized deformation in these parts is a sliding one, which is expected to be traction-free.

This in situ process has been reported as one of the reasons for the differences in terms of fracture energy between mode I and mode II crack growth since forming cusps requires the emergence of more crack surface than forming a single straight crack. Therefore, in order to simulate this fracture process under realistic boundary conditions, a model based on the TLS can be embedded in a ‘macroscopic’ setup, such as three-point bend end-notched flexure. A monolithic scheme with extreme refinement in a zone of interest is the most straightforward approach; however, for a specimen with realistic dimensions, this can be computationally unfeasible due to the computational resources needed to solve the large systems of equations involved in such problem. An ability to comprehensively model this microscopic process could help to achieve a better understanding of the mechanism behind the observed dependence of the fracture energy on the mode of fracture, which may in turn improve macroscale simulations.

This work focuses on extending the TLS method in order to profit from its full capabilities to deal with simulations of failure in solids under quasi-static loading conditions. For this purpose, several original numerical and theoretical components are proposed for reaching qualitative agreement with experimental observations of cusp formation in polymer matrix. In this context, the primary application of this thesis relies on the experimental observations at the microscopic level of such process. However, it is worth mentioning that the numerical tools developed in this thesis are not limited to the problem of cusps; in fact, they can be either used or easily extended to simulate other problems, for instance crack growth through the microstructure of cementitious materials with different aggregates.

First, the TLS is combined with plasticity in order to deal with ductile fracture since polymers may behave plastically prior to failure, particularly when loaded in shear. To accommodate for plasticity, several changes to the TLS framework are introduced. A strength-based criterion for initiation of damage based on the ultimate yield surface of such plasticity model is proposed. A mapping operator for transferring plastic history is included if the integration scheme in a finite element changes due to the evolution of the level set field. Furthermore, a new loading scheme is devised in order to take into account permanent strain.

Next, a generalized framework for the TLSV2 is introduced. The TLSV2 couples continuous and discontinuous approaches within a single framework, where the continuum part allows for handling crack initiation, branching and merging, whereas the discontinuous part brings the capability to handle discrete cracks with large crack opening or sliding without heavily distorted elements, as well as the possibility to model stiffness recovery upon contact.

Two major issues with the TLSV2 method that have not been dealt with since its inception are addressed in this thesis, and solutions are proposed. Firstly, the method depends on identifying the location of the skeleton curve of the level set field, on which the discontinuity in the displacement field is evaluated. The problem of locating the skeleton curve can be a complicated task, even more so because topological events may emerge as the analysis progresses, such as crack branching. The skeleton curve is determined through a combination of ball-shrinking and graph-based algorithms and then mapped onto the finite element mesh. Secondly, the cohesive forces and displacement discontinuity of the TLSV2 are modeled using the phantom node method. Furthermore, a new approach to compute the non-local crack driving force is introduced, and model calibration is discussed. The degree of stiffness recovery under compression that is still needed for the continuum part is investigated.

The TLS can be a computationally demanding approach. Therefore, a domain decomposition strategy is introduced in order to obtain a parallel implementation of the TLS method. To handle the numerical components specific to the TLS analysis steps involving level set update, equilibrium solution, and damage front advance, a parallel strategy is introduced for each of them. The most demanding task in terms of computational cost, i.e., solving the linearized system of equations from the equilibrium problem, is performed with a parallel iterative method profiting from the adopted domain decomposition method. A communication strategy is provided to deal with enriched nodes and new nodes necessary for the phantom node method belonging to shared regions of subdomains. Collective communication strategies are also proposed to deal with operations related to the level set update, damage front advance, and skeleton curve.

Numerical experiments demonstrate the accuracy and efficiency of the proposed framework in handling simulations of failure analysis with complex crack patterns in a sequential and parallel context.