Abstract
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 (RVE11The term ‘RVE’ is used interchangeably with sub-scale domain or microstructure, in this manuscript.) 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.
Highlights
An in-depth understanding of fracture processes in materials is essential for the prediction of fracture-induced failure in engineering structures
A novel two-scale phase-field fracture framework is proposed for computational homogenisation of fractures in complex microstructures
The framework has been developed using the Variationally Consistent Homogenisation technique (Larsson et al, 2010b), and it allows the use of several homogenisation measures
Summary
An in-depth understanding of fracture (initiation and propagation) processes in materials is essential for the prediction of fracture-induced failure in engineering structures. More elegant ways in the form of errororiented mesh refinement (Burke et al, 2010; Wick, 2016), refinement based on the phase-field reaching a certain threshold (Heister et al, 2015) and local increase of the tensile energy (Klinsmann et al, 2015), and multi-level hp refinement using the finite cell method (Nagaraja et al, 2019) exists in the phase-field fracture literature Despite these advancements, the development of robust and computationally efficient solution and meshing techniques are still topics of active research. The two-scale phase-field fracture framework is generic in the sense that it allows different choices pertaining to computational homogenisation of the microstructural quantities This aspect is explored at length in this manuscript, with (i.) volume and surface-average based homogenisation measures, and (ii.) selective homogenisation in the context of the phase-field variable.
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