Phase‐Field Simulation of Crack Propagation at Adhesive Interfaces in Brittle Materials

When it comes to failure of a heterogeneous material, the adhesive interfaces linking the different constituents can crucially influence its mechanical behavior. In this contribution, a diffuse description of interfaces in the context of the phase‐field approach to brittle fracture is outlined. Two‐ and three‐dimensional simulations of crack phenomena are presented and serve for validation of the model against LEFM.

1 Phase-field modeling of heterogeneous materials 1

.1 Diffuse modeling approach for adhesive interfaces
Starting point for the phase-field modeling of brittle fracture is the total pseudo energy functional in which the MIEHE split [1] of the strain energy into a fraction ψ el + attributed to the tensile part of the deformation and a compression part ψ el − is applied. Over a finite width governed by the crack length scale parameter c , the phase-field variable d bridges between the intact (d = 0) or fully broken (d = 1) material state.
Following the previous work of HANSEN-DÖRR et al. [2], adhesive interfaces are incorporated defining space-dependent functions for the fracture toughness G c (x). For this purpose, the interface is described in a diffuse manner in analogy to the regularization of the crack over the length scale c . In addition to the fracture toughness G i c , it is assigned a finite characteristic width i . Accordingly, the fracture toughness function depicting the interface can be interpreted as static phase-field. In principle, a HEAVISIDE-like jump between the fracture toughness values G i c and G b c assigned to the interface and the bulk material, respectively, can bee assumed, see Fig. 1 (a). However, a jump of the fracture toughness values negatively influenced the convergence of numerical solvers. Therefore, a smooth, GAUSSIAN-like transition is prescribed for the simulation of crack phenomena. For the case of heterogeneous elastic properties in the bulk material, a hyperbolic tangent-like approximation of the jump of the elastic constants at the interface midline is assumed as an approximation.

Bulk material influence on effective interface properties
For adhesive interfaces which are incorporated in the phase-field model as described above, an influence of the fracture toughness of the bulk material G b c on the dissipation due to an interfacial crack arises from the diffuse description of both the crack and the interface. For a one-dimensional setting and the HEAVISIDE-like interface description, this effect can be described analytically considering the dissipation per cross-section which corresponds to the effective fracture toughness of the interface. When evaluating the functional (2) for a heterogeneous fracture toughness, the alteration of the crack phase-field profile with respect to the homogeneous case has to be taken into account, cf. Fig. 1 (a). In addition to the analytical investigation, a numerical study on this lengthscale interaction effect has been conducted for both the HEAVISIDE-like and the GAUSSIAN-like fracture toughness functions. In this study, a crack propagating along an interface in a two-dimensional setting has been considered and the concept of configurational forces [3] has been employed. The results of this study are depicted in Fig. 1 (b) together with the analytical results from (2) for the HEAVISIDE-like interface description and in Fig. 1 (c) for the GAUSSIAN-like fracture toughness function, respectively. Based on the numerical results, a compensation procedure is established: An artificially lowered value of the interfacial fracture toughness is defined such that the actual fracture toughness G i,act c recovers the physical value, cf. [2,4] for details.

Simulation of crack phenomena
In order to demonstrate the predictive capability of the present model, two-and three-dimensional simulations of crack phenomena have been conducted. The setup and the numerical results are shown in Fig. 2. Good qualitative agreement between the simulations and analytical predictions from LEFM [5] is obtained. = 8, and homogeneous elastic constants, crack branching into the interface is simulated (•d > 0.8) in accordance with LEFM [5] which is followed by kinking of one of the tips out into the bulk. (c) -Elastic heterogeinity (E2/E1 = 1/3 , 1 , 3 ) has a significant impact on the crack path. For a constant ratio G b c /G i,act c , interfacial failure becomes the more energetically favorable the stiffer the material beyond the interface is. Furthermore, approaching the interface, the crack is deviated towards the interface midline for E2/E1 < 0 and vice versa which coincides with LEFM predictions [5].