## Computational modeling of failure in composite laminates

### Frans van der Meer

There is no state of the art computational model that is good enough for predictive simulation of the complete failure process in laminates. Already on the single ply level controversy exists. Much work has been done in recent years in the development of continuum models, but these fail to predict the correct failure mechanism in cases where matrix cracking in off-axis plies is part of the global mechanism. The way forward is to model matrix cracks as true discontinuities in the displacement field, in which case the orientation of the cracks can easily be controlled. A mesh-independent representation with partition of unity based methods or, similarly, the phantom node method is to be preferred, because with these methods cracks can initiate and grow at arbitrary locations in the model, wherever the stress field gives rise to it.

This requires an initially rigid mixed mode cohesive law. Unfortunately, straightforward formulation of such a law leads to a singularity; the traction is not uniquely defined for zero crack opening and zero damage. Robustness of the simulation requires that this singularity is removed from the description.Two methods exist that do this. Firstly, it is possible to relate the cohesive traction not only to the displacement jump, but also to the stress in the surrounding material. Secondly, one can start from a cohesive law with a finite initial stiffness and then apply a shift to the origin to mimic initially rigid behavior. With both methods a cohesive law is derived that satisfies the Benzeggagh-Kenane relation for mixed mode fracture energy.

The constitutive model for the single ply is completed with a damage/plasticity law for shear nonlinearity and a continuum damage model for fiber failure. The ply model can be used as a building block for analysis of complex failure mechanisms in laminates.

For full-laminate analysis, an additional failure process is possible, namely delamination. The proposed laminate model consists of a single layer of elements for each unidirectional ply, interconnected with interface elements. These interface elements are equipped with a cohesive law for delamination. Notably, the interaction between the phantom node method in the plies and the interface elements is accurate without updating the interface elements, particularly when a nodal integration scheme is used. A complicating aspect of laminate analysis, is that matrix cracking may occur in a distributed fashion because of mutual constraint between the plies. It is possible to model many cracks with the phantom node method. However, the number of cracks must be limited with a minimum crack spacing parameter to keep the problem well posed and to allow for coalescence of cracks. Because laminate failure is a highly nonlinear process, a carefully designed solution algorithm is indispensable to bring simulations to a successful end. Sharp snapbacks may occur because the stiff fibers may unload suddenly when failure progresses through the laminate. In order to follow the equilibrium path through these snapbacks, an arclength method is needed. The dissipation-based arclength method is used for this purpose, because it is robust and generic. The formulation is extended for cases with both permanent deformations and damage. Adaptive time stepping is crucial and in some cases a modified Newton-Raphson scheme is to be preferred.

The discontinuities that are represented matrix cracks, which may be numerous, are inserted during the computation. This is handled after equilibrium has been found. After crack propagation, equilibrium is sought again before the next time step is entered, but multiple crack segments may be introduced at once. The framework is validated against experimental observations for several laminate cases. Subcritical damage consisting of delamination and matrix cracks is generally captured well. Different failure mechanisms can be described and the appropriate one is `chosen’ automatically because the different processes are incorporated realistically. The fiber failure model lacks a representation statistical strength distribution of the fibers and is therefore not predictive in brittle cases where the strength in fiber direction is a key parameter.

This work has resulted in a robust integrated framework for computational modeling of failure in composite laminates. The numerical results are objective with respect to discretization. Different failure processes and their interaction are represented such that the simulations not only provide insight in when the laminate fails, but also in how it fails.