Gradient-enhanced crystal plasticity coupled with phase-field fracture modeling

Abstract

This study addresses ductile fracture of single grains in metals by modeling of the formation and propagation of transgranular cracks. A proposed model integrates gradient extended hardening, phase-field modeling for fracture, and crystal plasticity. It is presented in a thermodynamical framework in large deformation kinematics and accounts for damage irreversibility. A micromorphic approach for variationally and thermodynamically consistent damage irreversibility is adopted. The main objective of this work is to analyze the capability of the proposed model to predict transgranular crack propagation. Further, the micromorphic approach for damage irreversibility is evaluated in the context of the presented ductile phase-field model. This is done by analyzing the impact of gradient-enhanced hardening considering micro-free and micro-hard boundary conditions, studying the effect of the micromorphic regularization parameter, evaluating the performance of the model in ratcheting loading and and testing its capability to predict three-dimensional crack propagation. In order to solve the fully coupled global and local equation systems, a staggered solution scheme that extends to the local level is presented.

Figures (7)

• Figure 1: Staggered iteration scheme for the global system solver. The global and local variables are grouped in two groups extending the staggered solution scheme to the local equation system. The staggered algorithm solves for one set of variables at a time while freezing the other set. The iteration between the sets is stopped once the global residuals fulfill the all convergence criteria presented in Table \ref{['tab:global_tolerances']} at the same time.
• Figure 2: Meshes for the 2D (left) and 3D (right) numerical examples. The cross-sectional geometry of the 3D-example is the same as for the 2D-example and geometrical measures are given in mm. Both examples employ unstructured meshes with a background element size of $4\,\mathrm{mm}$ and mesh refinements at the sides and in the center of the web, where the sample is expected to break. The refinements in the 2D-mesh consist of elements with an average size of $0.5\,\mathrm{mm}$, resulting in $19\,390$ triangular elements in total. In the 3D-mesh, center and sides employ elements of $1\,\mathrm{mm}$ average size and two positions are additionally refined with elements of $0.5\,\mathrm{mm}$ on average for imposing initial material inhomogenities. The 3D mesh consists of $123\,183$ tetrahedral elements in total.
• Figure 3: Reaction force response for the I-shaped beam exposed to different boundary conditions, as well as without gradient hardening ($l^\mathrm{g}=0.0\,\text{mm}$). All scenarios reach the softening regime. The micro-hard restriction of slip transfer on the boundaries leads to a stiffer response in the hardening regime. The last point of the curves corresponds to the last converged time step of the respective simulations.
• Figure 4: Degradation $g_\mathrm{e}$ at the final step of the three base scenarios. The well known diamond shape crack pattern (compare e.g. DeLorenzis2016) is recovered without gradient hardening. The behavior under micro-free boundary conditions is similar, but shows some smoothing of the crack shape. Under micro-hard boundary conditions plastic strains cannot develop on the boundary, thereby preventing the development of damage on the boundary.
• Figure 5: Effect of the micromorphic penalty parameter $\beta$: The top row shows results for a low penalty effect, the bottom row shows results for a sufficient penalty effect. If the penalty parameter is chosen too low a decoupling between the global and the local phase field occurs, resulting in a lack of regularization of the local phase field $\varphi$. In this case the local phase field $\varphi$ is almost exclusively driven by the the accumulated plastic strain $\epsilon^\mathrm{p}$, which leads to a high level of localization in the local phase field. The global phase field $d$ instead experiences a lack of localization, as the coupling to the local phase field $\varphi$ decreases.