Computational homogenisation of phase-field fracture

In this manuscript, the computational homogenisation of phase-field fractures is addressed. To this end, a variationally consistent two-scale phase-field fracture framework is developed, which formulates the coupled momentum balance and phase-field evolution equations at the macro-scale as well as at the Representative Volume Element (RVE) scale. The phase-field variable represent fractures at the RVE scale, however, at the macro-scale, it is treated as an auxiliary variable. The latter interpretation follows from the homogenisation of the phase-field through volume or a surface-average. For either homogenisation choices, the set of macro-scale and sub-scale equations, and the pertinent macro-homogeneity satisfying boundary conditions are established. As a special case, the concept of selective homogenisation is introduced, where the phase-field is chosen to live only in the RVE domain, thereby eliminating the macro-scale phase-field evolution equation. Numerical experiments demonstrate the local macro-scale material behaviour of the selective homogenisation based two-scale phase-field fracture model, while its non-selective counterpart yields a non-local macro-scale material behaviour

Computational homogenisation and solution strategies for phase-field fracture

The computational modelling of fracture not only provides a deep insight into the underlying mechanisms that trigger a fracture but also offers information on the post-fracture behaviour (e.g., residual strength) of engineering materials and structures. In this context, the phase-field model for fracture is a popular approach, due to its ability to operate on fixed meshes without the need for explicit tracking of the fracture path, and the straight-forward handling of complex fracture topology. Nevertheless, the model does have its set of computational challenges viz., non-convexity of the energy functional, variational inequality due to fracture irreversibility, and the need for extremely fine meshes to resolve the fracture zone. In the first part of this thesis, two novel methods are proposed to tackle the fracture irreversibility, (i) a micromorphic approach that results in local irreversibile evolution of the phase-field, and (ii) a slack variable approach that replaces the fracture irreversibility inequality constraint with an equivalent equality constraint. Benchmark problems are solved using a monolithic Newton-Raphson solution technique to demonstrate the efficiency of both methods.The second aspect addressed in this thesis concerns multi-scale problems. In such problems, features such as the micro-cracks are extremely small (several orders of magnitude) compared to the structure itself. Resolving these features may result in a prohibitively computationally expensive problem. In order to address this issue, a computational homogenisation framework for the phase-field fracture is developed. The framework allows the computational of macro (engineering)-scale quantities using different homogenising (averaging) approaches over a microstructure. It is demonstrated that, based on the choice of the homogenisation approaches, local and non-local macro-scale material behaviour is obtained