Phase Field Approximation of Dynamic Brittle Fracture

Abstract

The numerical assessment of fracture has gained importance in fields like the safety analysis of technical structures or the hydraulic fracturing process. The modelling technique discussed in this work is the phase field method which introduces an additional scalar field. The smooth phase field distinguishes broken from undamaged material and thus describes cracks in a continuum. The model consists of two coupled partial differential equations -the equation of motion including the constitutive behaviour of the material and a phase field evolution equation. The crack growth follows implicitly from the solution of this system of PDEs. The numerical solution with finite elements can be accelerated with an algorithm that performs computationally extensive tasks on a graphic processing unit (GPU). A numerical example illustrates the capability of the model to reproduce realistic features of dynamic brittle fracture.In this work a linear elastic body Ω with Lamé parameters λ, µ and mass density ρ is considered. The external boundary is denoted by ∂Ω = ∂Ω u ∪∂Ω t . Furthermore, the body may contain internal discontinuities, i.e. cracks, Γ, see Fig. 1. The displacement of a point x is denoted as u (x, t) where the field u (x, t) satisfies Dirichlet boundary conditions u (x, t) = u * (x, t) on ∂Ω u and Neumann boundary conditions σn = t * on ∂Ω t . Herein, σ is the stress tensor. The discontinuities Γ are approximated by a phase field s (x, t) which varies smoothly between s = 1 in undamaged material and s = 0 in the broken material (see Fig. 2). By means of s the fracture energy is approximated bywhere G c is the fracture resistance and is a length scale characterizing the crack width. In the limit → 0 the phase field ∂Ω u Ω Γ ∂Ω t t * n Fig. 1: Body with internal discontinuities (sharp cracks) Γ. s(x, t) ∂Ω t ∂Ω u Ω n t * Fig. 2: Approximation of internal discontinuities by a phase field s(x, t).approximation of the fracture energy (1) is exact, see [1]. The phase field s is connected to the elastic energy density ψ e to model the loss of stiffness in the broken material. In order to achieve a more realistic crack behaviour in compression, the compressive part of ψ e is not affected by the crack field. As in [2] the elastic energy density readswith tr − (ε) = min {0, tr(ε)}, tr + (ε) = max {0, tr(ε)}, the n-dimensional bulk modulus K n = λ + 2 n µ and the deviatoric part e of the linearized strain tensor ε = 1 2 ∇u + (∇u) T . The Cauchy stress tensor is given by σ = ∂ψ e ∂ε .

Effects of spatial heterogeneity and material anisotropy on the fracture pattern and macroscopic effective toughness of Mancos Shale in Brazilian tests

“…In this work, a phase field fracture model based on the generalized Griffith's theory is adopted to simulate the crack initiation and propagation for brittle materials [e.g., Kuhn and Müller , ; Borden et al ., ; Hofacker and Miehe , ; Verhoosel and Borst , ; Schlüter et al ., ; Heister et al ., ]. Since the phase field fracture model employs an implicit function instead of strong discontinuities to represent cracks, it does not require complicated numerical treatments such as generation of enrichment functions [ Dolbow et al ., ], dynamic insertion of cohesive elements [ Zhang and Paulino , ], or modification of the mesh [ Mota et al ., ].…”

Validation simulations for the variational approach to fracture

“…Over the past decade, this approach was applied to many areas including elastic fracture [18,17,16], thermoelastic fracture [43,21], thin-film fracture [42,41,45], thin-shells [6], electro-mechanical fracture [1,2], or dynamic fracture [40,20,15,57] to name a few. A major difficulty associated with the variational approach to fracture is the reliance on global energy minimization, which can sometimes result in unrealistic crack paths by making equally admissible "near" and "far" points in configuration space.…”

The concept of representative crack elements for phase‐field fracture: Anisotropic elasticity and thermo‐elasticity

“…Thus, describing these features is still under development for the phase‐field method. May et al, Strobl and Seelig, Schlüter, and Steinke and Kaliske have shown, that the well‐known volumetric‐deviatoric split (V‐D) and spectral split approaches with tension/compression decomposition lead to misleading predictions for the force transfer through the crack. They have proposed and developed a model for the crack kinematics in case of isotropic, linear elastic material.…”