Analysis of Staggered Evolutions for Nonlinear Energies in Phase Field Fracture

Abstract

We consider a class of separately convex phase field energies employed in fracture mechanics, featuring non-interpenetration and a general softening behavior. We analyze the time-discrete evolutions generated by a staggered minimization scheme, where the fracture irreversibility is modeled by a monotonicity constraint on the phase field variable. We characterize the time-continuous limits of the discrete solutions in terms of balanced viscosity evolutions, parametrized by their arc-length with respect to the L 2 -norm (for the phase field) and the H 1 -norm (for the displacement field). By a careful study of the energy balance we deduce that time-continuous evolutions may still exhibit an alternate behavior in discontinuity times.

Irreversibility and alternate minimization in phase field fracture: a viscosity approach

“…The theoretical investigation of the relationship between alternate minimization schemes for phase field models of fracture and rate-independent processes has instead only recently started with the works [2,3,6,33] (see also [31,38] for the application of alternating algorithms in different physical frameworks). Here, we describe the algorithm adopted in [33].…”

“…We mention that a discrete version of [33] in a space-discrete (finite element) setting has been studied in [2] together with the limit of the solutions in a discrete to continuum sense, i.e., as the mesh becomes finer and finer. The results of [33] have been further generalized in [6] to not separately quadratic energy functionals. An example of such energies, that will be considered also in this paper, is inspired by the phase field model introduced in [10,20].…”

“…Here we only mention that in [20] it has been proven that, in dimension n = 2, F ε Γ-converges to G in (1.2) for u ∈ SBD 2 (Ω; R 2 ) satisfying the non-interpenetration constraint [u] • ν u ≥ 0, where ν u is the approximate unit normal to J u and [u] stands for the amplitude of the jump of u across J u . Despite the sound mathematical results obtained in [2,6,33], one of the drawback of the alternating scheme (1.3)-(1.4) lies in the irreversibility condition z ≤ z k i,j−1 , which forces the phase field variable to be non-increasing along the whole algorithm. This requirement, indeed, could lead to an accumulation of numerical error, making the fracture simulation very inaccurate.…”

Γ-convergence for high order phase field fracture: Continuum and isogeometric formulations

“…In the discrete setting, we show that for η ε = o(ε) and h = o(ε) (the element size) the Γ-limit of F ε,h is again the above Griffith's functional F in GSBD 2 . Comparing with the continuum setting, the discrete Γ-limsup inequality requires to take into account the fact that "interpolation" in the space of tensor product B-splines does not preserve L ∞ -bounds; as a consequence the projection v h of the continuum phase-field profile v, which is a natural candidate for the recovery sequence, may not take value in [0,1]. This technical issue is by-passed using an ad hoc local modification of v h , at the price of introducing an additional approximation error, vanishing in the limit for ε → 0.…”

“…For phase field fracture, this is usually obtained by (time discrete) incremental problems, based on alternate minimization, or staggered, schemes [11]. A characterization of the time-continuous evolution (in the limit as the time step vanishes) has been proved for first order phase-field functionals in [29] (for the dynamic case) and in [34,27,1] (for the quasi-static case). Finally, we remark that algorithms based on second order functionals proved to be numerically very efficient; indeed, alternate minimization schemes converge to an equilibrium configuration faster than first order problems (see e.g., [9, Figure 10 and Tables 4, 5] and similarly [14, 2 Setting and statement of the Γ-convergence results…”