Phase‐Field Simulations of Cracks under Dynamic Loading

Phase‐field fracture simulations have been established to simulate crack propagation in fracture mechanics. This contribution sets the focus on different driving forces for crack growth and on the simulation of waves propagating along the fractured interfaces.


Introduction
Phase-field models became very popular recently to predict the location of crack initialization and the crack path itself, cf. [7][8][9]. The main idea is to characterize the state of the material by an additional parameter z(x, t) : Ω 0 × T → [0, 1] within the reference domain Ω 0 and the time interval T = [0, T ], T ∈ R + . In particular, the intact material is characterized by z = 0 and the fully broken state by z = 1. The total energy E tot consists of the free Helmholtz-energy density Ψ(u, z) and of surface energy contributions where the latter term has to be regularized because of the moving boundaries Γ 0 (t): The choice of the crack density functional γ is not unique; in this work we apply a second order phase-field approach of the form: γ(z, ∇z) = 1 2lc z 2 + lc 2 ∇z · ∇z. The length-scale parameter l c is a measure for the width of the diffuse interface zone and acts two-fold, as a mesh dependent and as a material dependent parameter. Detailed investigations about its influence can be found in the literature, cf. [1,7,8].
The energy minimization problem of (1) corresponds to the coupling of the balance of linear momentum and the evolution equation of the phase-field z, where Y summarizes the generalized thermodynamic forces Y = Y e + Y f = Y e − δ z γ with the crack driving force Y e and τ is a numerically determined retardation time.

Phase-field model
Commonly the crack driving force is deduced from an energy optimization, Y e = δ z Ψ. Since crack growth requires a state of tension and does not grow under compression with same energy this asymmetry has to be incorporated in the model. This applies for both linear and finite elasticity. Exemplarily, we outline the common decomposition in linear elasticity which is based on the decomposition of the principal strains such that the strain energy density results in with ǫ + = 3 a=1 ǫ a + n a ⊗ n a and • ± = 1 2 (• ± | • |), • ∈ {ǫ a , tr ǫ} being the positive part of the eigen strains, cf. [8]. In finite elasticity we choose an additive decomposition of the invariants as proposed in [4] for which the strain energy density remains polyconvex. Please note that these decompositions are modeling assumptions and can be varied in different ways. In any case the assumption of decomposition results in different crack driving forces and also influences the crack propagation.
This situation motivated us to investigate the choice of crack driving force in more detail. In addition to variational crack driving forces also ad-hoc assumptions for Y e motivated by established failure criteria are introduced. Exemplarily, we focus here on the maximum principal stress criterion which can be traced back to the Rankine failure criterion. The driving force is 2 of 2 Section 7: Coupled problems

Wave propagation and Brazilian test
For purpose of illustration we focus on two aspects of our simulations and study the wave propagation and a dynamic Brazilian test in compression. The corresponding experiments are performed in a Split-Hopkinson Pressure Bar setup. The wave propagation in the bars is simulated by the linear wave equation ρü − ∇ · (C : ǫ) = f . This second order equation is reformulated in a first order Friedrich system which is then solved by a discontinuous Galerkin method for hyperbolic conservation laws in space and the midpoint rule in time. This discrete system conserves energy; the corresponding unidirectional wave propagation through the bar is depicted in Fig. 1. The Brazilian test acts as a classical test to identify the tensile resistance of brittle material. A cylindrical specimen is compressed from two opposite sides until the tension within the specimen results in failure. The maximum stress occurs in the middle of the specimen and is given by Frocht [3] as with the specimen diameter D and the length L. The experimental setup is depicted in the left picture of Fig. 2 for a specimen of concrete (E = 50 000 N/mm 2 , ν = 0.2). A detailed investigation can be found in [2]. The final results based on the phase-field method are demonstrated in the right plots of Fig. 2. The crack initialization and also the crack path itself can be observed in the numerical simulation and coincide nicely with the experiments, cf. [6]. The choice of the crack driving force is important because the standard variational ansatz does not lead to physically meaningful results here, cf. [2].