Phase-field modeling of ductile fracture across grain boundaries in polycrystals

Abstract

In this study, we address damage initiation and micro-crack formation in ductile failure of polycrystalline metals. We show how our recently published thermodynamic framework for ductile phase-field fracture of single crystals can be extended to polycyrstalline structures. A key feature of this framework is that is accounts for size effects by adopting gradient-enhanced (crystal) plasticity. Gradient-enhanced plasticity requires the definition of boundary conditions representing the plastic slip transmission resistance of the boundaries. In this work, we propose a novel type of micro-flexible boundary condition for gradient-plasticity, which couples the slip transmission resistance with the phase-field damage such that the resistance locally changes during the fracturing process. The formulation permits to maintain the effect of grain boundaries as obstacles for plastic slip during plastification, while also accounting for weakening of their resistance during the softening phase. In numerical experiments, the new damage-dependent boundary condition is compared to classical micro-free and micro-hard boundary conditions in polycrystals and it is demonstrated that it indeed produces a response that transitions from micro-hard to micro-free as the material fails. We show that the formulation maintains resistance to slip transmission during hardening, but can generate micro-cracks across grain boundaries during the fracture process. We further show examples of how the model can be used to simulate void coalescence and three-dimensional crack fronts in polycrystals.

Figures (8)

• Figure 1: Two- (left) and three-dimensional (right) grain structures employed for the numerical experiments. The two-dimensional structure consists of 10 grains and is used with and without the voids in the middle grain. Without voids, the middle grain is meshed in the same manner as the other grains. The three-dimensional grain structure consists of four grains. All set-ups are loaded by simple shear along the horizontal axis.
• Figure 2: Degradation and plastic strain response for micro-free boundary conditions on the inner grain boundaries. Both fully developed cracks cross grain boundaries. Localization of plastic strain initially occurs in the grain marked with arrow 1, close to the grain boundary. The bottom crack (arrow 2) develops later in the simulation. Its crack growth direction is impacted by the stress concentration caused by the grain boundary intersection marked with arrow 3. The micro-free boundary conditions are reflected by the accumulated plastic strain contour lines that are perpendicular to the grain boundaries, especially in the highly plastified regions (e.g. arrow 4). Micro-hard boundary conditions are prescribed on the outer boundaries of the grain structure for numerical stability.
• Figure 3: Degradation and plastic strain response for micro-hard boundary conditions on the inner grain boundaries. The crack developing within the upper grains is interrupted on the grain boundary marked by arrow 1. Localization of plastic strain initially occurs inside the grain right of that grain boundary. Plastic strain cannot develop on the grain boundaries. Micro-hard boundary conditions are prescribed on the outer boundaries of the grain structure.
• Figure 4: Degradation and plastic strain response for micro-flexible boundary conditions on the inner grain boundaries. The damage initiation side of the upper crack is indicated by arrow 1. The crack intiates on the right side of the grain boundary and then propagates to the left grain. The micro-flexible boundary conditions initially behave nearly micro-hard and upon damage development locally transition to micro-free behavior. The state of the micro-boundary conditions at the end of the simulation is color-coded on a white to black color scale in the figure. Micro-hard boundary conditions are prescribed on the outer boundaries of the grain structure for numerical stability.
• Figure 5: Volume averaged in-plane shear component of the second Piola-Kirchhoff stress tensor in the upper two grains, comparing different inner micro-boundary conditions. Micro-hard boundary conditions lead to a stiffer response than micro-free boundary conditions. As desired, micro-flexible boundary conditions initially behave similarly to micro-hard boundary conditions and then approach the behavior of micro-free boundary conditions during the softening period.