Phase Field Modeling of Fracture in Plates and Shells

Abstract

The numerical modeling of failure mechanisms in plates and shells due to fracture based on sharp crack discontinuities is extremely demanding and suffers in situations with complex crack topologies. This drawback can be overcome by a diffusive crack modeling, which is based on the introduction of a crack phase field. In this paper, we extend ideas recently outlined in [1,2] towards the phase field modeling of fracture in dimension-reduced continua with application to Kirchhoff plates and shells. The introduction of history fields, containing the maximum reference energy obtained in history, provides a very transparent representation of the coupled balance equations and allows the construction of an extremely robust operator split technique. The performance of the proposed models is demonstrated by means of representative numerical examples. Approximation of Crack Topology by Phase FieldWe develop a crack line theory where the crack is considered as a line on the reference surface of the thin-walled structure. To this end, we introduce the time-dependent crack phase field d ∈ [0, 1] characterizing for d = 0 the unbroken and for d = 1 the fully broken state of the material. A regularized crack phase field d(x, t) is obtained from the minimization principleThe crack functional L in terms of the crack line density function γ(d, ∇d) governs the regularization by the length scale parameter l giving for l → 0 the sharp crack topology. Incremental Variational Principle of Diffusive Fracture in PlatesThe basic fields of the coupled problem are the deflection field w and the fracture phase field d. The global energy storage functional considering standard linear theory of elasticity for plates readswhere we introduced the reference energy of the Kirchhoff plate ψ 0,b in terms of the curvature κ(w)and K := EtThe degradation function g(d) = (1 − d) 2 describes the degradation of ψ 0,b with evolving damage. The rate of the energy functional is governed by the bending moment m := ∂ κ ψ, the energetic force f := −∂ d ψ, and the micro force χ :We propose an extended dissipation functional in terms of β dual to d, the Lagrange parameter λ, and the threshold function t c for a rate independent class of dissipation functions satisfying the non-reversible evolution of the crack phase fieldThe rate potential functional Π λ = E + D λ − P balances the internal power E + D λ with the power due to external loading P . Now, the governing equations are obtained from the argument of virtual power that we base on the variational statement {ẇ,ḋ, β, λ} = Arg{ staṫ w,ḋ,β,λ≥0Variation of the functional with respect to the four field variables gives the coupled Euler-Lagrange equations

Phase-field description of brittle fracture in plates and shells

“…We note that (21)- (22) are naturally nonlinear strain measures accounting for large deformations. In this paper, we restrict ourselves to small deformation problems and use linearized versions of (21)- (22), which are presented in detail in Appendix A. It is crucial to note that both ε m and κ induce the same type of strain components, and the total strain ε at any point in the shell is given by Figure 1 illustrates equation (23), showing that the split into membrane and bending strains simply represents a split of ε(θ 3 ) into a constant and a symmetrically linear part.…”

“…Approaches to combine shell models with a phase-field description of fracture using a single phase field were presented so far in [22,23]. In the following, we briefly review these approaches and make some observations, which motivate the new approach proposed in the next section.…”

“…The split of the strain energy into membrane and bending contributions, however, does not take this effect into account and might lead to fracture although the strains are purely compressive. The second problem of the approach in [22] is the fact that the bending energy is degraded entirely without any split into tension and compression terms. Considering a case of pure bending (typically a plate), such an approach effectively corresponds to an isotropic fracture model.…”

“…Ulmer et. al [22] presented an approach for brittle fracture in thin plates and shells, where shells are considered as a combination of a plate and a standard membrane. Accordingly, the elastic energy is divided into bending and membrane contributions.…”

“…We first briefly review the existing approaches [22,23] and observe that they do not permit a correct description of fracture in the typical stress states of plates and shells, i.e., pure bending or combined bending and membrane stresses. Based on these observations, we propose a new approach which leads to a correct description of the mechanical response in such situations.…”