A statistical geometry approach to length scales in phase field modelling of fracture and strength of porous microstructures

Abstract

In phase field methods for fracture, versatility is acquired at the cost of the addition of a new parameter, a length scale parameter. The length scale parameter affects notch sensitivity, which in cellular materials is typically related to lengths of the material microstructure. Here, the relation between this length scale parameter and observable microstructural lengths of a cellular material is investigated numerically, specifically lengths derived using statistical geometry of random Voronoi tessellations. It is found that the fracture load of a homogeneous continuum model (i.e. a macroscopic model) coincides with that of a microstructured model if the length scale parameter is chosen to be the same in both models, while approximate macroscale stiffness and energy release rate are obtained by scaling the properties of the microstructured model with powers of the relative density. The correlation between the micro- and macroscale models is best when the length parameter is chosen as approximately two to three times the average cell size of the microstructure, depending on the relative density – which is also equal to approximately eight times a critical defect length of the Voronoi tessellation, regardless of relative density – as the microstructured material then behaves more like a continuum. If the length scale parameter needs to be smaller than twice the cell size or five times the critical length, the crack path is sensitive to features in the microstructure, and continuum modelling of the porous material cannot be advised.

Topics & Concepts