Diffusive-length-scale adjustable phase field fracture model for large/small structures

In phase field fracture (PFF) method, the sharp crack is approximated by a phase field crack zone whose size is characterized by a diffusive length scale. Recently, the diffusive length scale is usually regarded as a constant material parameter determined by the fracture toughness, material strength, and young's modulus. As a result, the application of the PFF method poses challenges when dealing with structures whose sizes are much too large or small compared to the constant diffusive length scale. In details, for a large-scale structure, a significant computational burden is inevitable due to the limitation imposed by the constant diffusive crack length scale on the element size (i.e. the element size is generally at least smaller than half of the diffusive crack length scale to achieve the sufficient precision for the phase field process zone). For a small-scale structure, the crack patterns tend to be unclear and unrealistic due to the excessively large diffusive crack zone. To address these limitations, we propose a novel PFF method to make the relation between the diffusive length scale and the material parameters adjustable via modifying the energetic degradation functions. It is found that with the increase of the diffusive crack length scale, a transition from quasi-brittleness to brittleness is observed in the constitutive relationship curves, which coincides with the classical size effect on structural strength. Further, the Bažant's size effect in classical fracture mechanics can be reproduced by the present PFF method through scaling the size of a geometrically similar structure, i.e. a large-scale structure exhibits toughness-dominated fracture while a small-scale structure behavior strength-dominated fracture. The present PFF method efficiently addresses the mismatch of diffusive length scales for various material constituents by examining phase field crack growth in particle-reinforced composite plates. By adjusting the diffusive crack length scale based on structure size, it avoids high computational costs from large structures and unrealistic crack patterns from small ones. Moreover, it outperforms traditional PFF methods using a constant diffusive length scale in handling composite materials.