The phase-field modeling of fracture evolution in ductile materials with application to paperboard mechanics

Paperboard is one of the many layers composing the layered packaging material employed in the food industry for preservation of liquid products. Its purpose is to provide the mechanical properties to the final package in terms of strength and stiffness. The yearly production of this kind of packages amounts at several hundred billions units. This number motivates the intense research activity aimed at characterizing and improving the packaging material to reduce waste. Despite experimental tests may provide useful information, practical applications require a mathematical modeling based on a profound understanding of the material response. The mathematical description of paperboard response has been extensively explored in the literature in the elastoplastic regime preceding the onset of damage and fracture. The main objective of the current work is to extend the state-of-the-art elastoplastic modeling of paperboard material to include the development of damage and subsequent crack propagation. In Computational Mechanics, the modeling of fracture evolution introduces the fundamental challenge of dealing with discontinuous displacements. In the last two decades, the phase-field approach has overcome this problem changing the description of the crack topology. Indeed, the sharp geometry is substituted with a smooth regularization governed by the order parameter called extit{phase-field}. In the present work, a computationally efficient and explicit algorithm for the rigorous enforcement of the irreversibility constraint in the phase-field modeling of brittle fracture is presented. The proposed approach relies on the alternate minimization of the total energy functional. The phase-field evolution turns out to be governed by a {complementarity boundary-value problem}, where the complementarity stems from the irreversibility, while the boundary-value problem stems from the presence of the gradient term in the phase-field functional. A solution strategy based on the Projected Successive Over-Relaxation (PSOR) method for constrained optimization, where an iterative explicit scheme is used for the solution of symmetric linear complementarity problems, is presented. A variational formulation of ductile fracture, based on a phase-field modeling of crack propagation, is then proposed for isotropic materials both in small and large deformations. The formulation is based on an effective stress approach, combined with an AT1 phase-field model. Starting from established variational statements of finite-step elastoplasticity for generalized standard materials, a mixed variational statement is consistently derived, incorporating in a rigorous way a variational finite-step update for both the elastoplastic and the phase-field dissipations. The complex interaction between ductile and brittle dissipation mechanisms is modeled by assuming a plasticity driven crack propagation model. A non-variational function of the equivalent plastic strain is then introduced to modulate the phase-field dissipation based on the developed plastic strains. In the context of small strains, a gradient-extended plasticity framework has been proposed to prevent the pathological mesh-dependence due to the combination of the softening response and the continuing plastic deformation induced by the effective stress approach. Particular care has been devoted to the formulation of a consistent Newton-Raphson scheme for the gradient-extended model in the case of Mises plasticity, with a global return mapping and relative tangent matrix, supplemented by a line-search scheme, for fixed phase field. The resulting algorithm has proved to be very robust and computationally effective. To approach the phase-field modeling of fracture in paperboard the proposed ductile fracture formulation has been extended to orthotropic materials, being ductility and orthotropy the fundamental features of the paperboard mechanical response. The resulting orthotropic, small strain phase field model for ductile fracture, based on a state-of-the-art elastoplastic in-plane model, has been applied to the simulation of failure experimental tests on paperboard strips, with excellent results in terms of accuracy and scale independence. As a first step towards the inclusion in the model of the large strain out-of-plane paperboard behavior, a large strain isotropic elastoplastic model for ductile phase-field fracture has been proposed, based on a variational update of the large strain finite-step elastoplastic phase-field problem. The gradient extension of the model and its application to orthotropic paperboard are then left for a future development.