A unilateral L2-gradient flow and its quasi-static limit in phase-field fracture by an alternate minimizing movement

1 Introduction

Phase-field approaches are widely used to simulate crack propagation in academical and industrial applications: even within the linear-elastic setting there are plenty of phase-field models, based on different choices of potentials and evolution laws, see e.g. [2, 6, 10, 11, 18, 23, 25, 26, 34, 35] or the recent review [3]. In the present work, we will study in detail a couple of evolutions generated by a phase-field energy of the form

where Ω⊂ℝ2 is a bounded Lipschitz domain, u~⁢(t)=u+g⁢(t) is the displacement fields with u∈H01⁢(Ω) (so that u~⁢(t)=g⁢(t) on ∂⁡Ω), W⁢(D⁢u~) is a linear elastic energy density, v∈H1⁢(Ω;[0,1]) is the phase-field variable, Gc>0 is toughness while η>0 is a regularization parameter.

Functionals like ℱ provide an elliptic regularization of free discontinuity functional: for instance, neglecting boundary conditions, for ε→0+ and 0<ηε=o⁢(ε) the Γ-limit of the functionals

is of the form

where J⁢(u) is the set of discontinuity points of the displacement field u and ℋ1 denotes Hausdorff measure. Roughly speaking, the set J⁢(u) represents the crack. Convergence has rigorously been proved in the framework of SBD2 and GSBD2 spaces respectively in [12] and [20] while in the scalar framework of the space GSBV2 it was proved in the well-known paper [5]. In our work, we will study an evolution in time, rather than a limit for ε→0+, hence we will work with the functional ℱ, omitting, for simplicity of notation, the dependence on the “internal length” ε.

Our starting point is a time-discrete evolution generated by an alternate unilateral minimizing movement. For τ>0 let tn=n⁢τ∈[0,T] for n∈ℕ be the time discretization. Then the incremental problem is the following: given (un-1,vn-1) (at time tn-1) the configuration (un,vn) at time tn is obtained by solving

Similar discrete schemes have been used in applications, e.g. in [23, 35]. Note that the updated configuration is determined by a single iteration in each variable, that each minimization problem is well posed (thanks to η>0) and that the constraint v≤vn-1 models irreversibility. This scheme takes, of course, full advantage of the separate quadratic structure of ℱ⁢(t,⋅,⋅). Our first result proves that (as τ→0) the time-discrete evolutions converge to a time-continuous evolution t↦(u⁢(t),v⁢(t)), where v⁢(⋅) is monotone non-increasing and satisfies for every t∈[0,T] the energy balance

where |∂v-⁡ℱ⁢(t,u,v)|L2=sup⁡{-∂v⁡ℱ⁢(t,u,v)⁢[ξ]:ξ≤0,∥ξ∥L2≤1} is the unilateral slope. Note that, by irreversibility, only negative variations are allowed; for this reason a minus sign is added to the notation |∂v⁡ℱ|L2 of the (unconstrained) slope. Equation (1.2) can be considered as De Giorgi’s integral characterization of gradient flows; indeed, for a.e. t∈[0,T] the time continuous limit solves also the following system of PDEs:

where 𝝈v⁢(u~)=(v2+η)⁢𝝈⁢(u~) denotes the phase-field stress. Technically, we will see that v∈W1,2⁢(0,T;L2⁢(Ω)) and that v⁢(t)⁢W⁢(D⁢u~⁢(t))+Gc⁢(v⁢(t)-1)-Gc⁢Δ⁢v⁢(t) is a finite Radon measure with positive part [⋅]+ in L2⁢(Ω). To better understand the variational structure behind this evolution problem, note that formally the partial derivatives of ℱ read

The positive part [⋅]+ appearing in (1.3) comes from the irreversibility constraint; more precisely, the term -[v⁢(t)⁢W⁢(D⁢u~⁢(t))+Gc⁢(v⁢(t)-1)-Gc⁢Δ⁢v⁢(t)]+ is (in a suitable sense) the “projection” of -∂v⁡ℱ⁢(t,u⁢(t),v⁢(t)) on the set of negative variations ξ, therefore the parabolic equation can be interpreted as a unilateral gradient flow, constrained by irreversibility.

In a larger perspective, an alternate minimizing scheme has been employed in different ways also in a dynamic visco-elastic setting [24], in another gradient flow setting [7] and in a quasi-static setting [21]. In general, alternate scheme are very useful in numerical simulation since they require only the minimization of convex (in our case, quadratic) functionals. On the other hand, different approaches can provide existence of solutions for (1.3) or similar problems. For instance: [22] obtains existence introducing a unilateral minimizing movement for a “reduced” (non-convex) energy, which in our setting would be of the form ℱ⁢(t,ut,v,u) for ut,v∈argmin⁡{ℱ⁢(t,u,v):u∈H01}, while [9] proves existence, for a similar problem, by a fixed point argument. In the applications, “Ginzburg–Landau” models are used in phase-field fracture for instance [1, 18, 23].

In the second part of the paper we consider the vanishing viscosity limit of (1.3). More precisely, we start with the system

where ε>0 is a “mobility parameter” or “viscosity”. Our goal is the characterization of the quasi-static limit, obtained as ε→0. Since in the limit we expect discontinuous evolutions and since the limit is rate independent, we first parametrize the evolutions by an arc-length parameter in L2. In this way, we get a Lipschitz map s↦(tε⁢(s),wε⁢(s),zε⁢(s)), where wε⁢(s)=uε∘tε⁢(s) and zε⁢(s)=vε∘tε⁢(s). Passing to the limit, as ε→0, we get a map s↦(t⁢(s),w⁢(s),z⁢(s)) which satisfies w⁢(s)∈argmin⁡{ℱ⁢(t⁢(s),w,z⁢(s)):w=g∘t⁢(s)⁢ on ⁢∂⁡Ω} together with the energy balance

It is noteworthy that this balance together with the Lipschitz continuity of parametrized solutions imply the main properties of the quasi-static parametrized solution s↦(t⁢(s),w⁢(s),z⁢(s)). Indeed, if t′⁢(s)>0 (i.e. in continuity points), we have equilibrium, i.e.

if t⁢(s) is constant in (s♭,s♯) (i.e. in discontinuity points) we have the following re-parametrization of (1.4)

where λ⁢(s)=∥[z⁢(s)⁢W⁢(D⁢w~⁢(s))+Gc⁢(z⁢(s)-1)-Gc⁢Δ⁢z⁢(s)]+∥L2. Hence the vanishing viscosity limit is labelled “parametrized BV-evolution” [28, 32].

It is interesting to compare, at least qualitatively, the quasi-static limit obtained here with the one obtained in [21]. The latter is based on the alternate minimization scheme of [11], which reads, in the one iteration version,

Note that in this scheme there is no additional viscosity. As a matter of fact, using the separate quadratic structure of the energy ℱ⁢(t,⋅,⋅), the above minimization problems, in u and v, are recast in [21] as the minimization of linear energies with suitable (state depending) dissipation-norms. Such dissipation-norms are called “intrinsic”, as opposed to “artificial” dissipations appearing in the vanishing viscosity approach, like the L2-norm in (1.1). Roughly speaking, the quasi-static limit of [21] is a parametrized BV-evolution for the energy ℱ with respect to these intrinsic dissipation-norms; for the detailed characterization together with several fine properties we refer to [21].

Last, but not least, let us provide some technical considerations about our results. First of all, the proofs of (1.2) and (1.3) rely essentially on the separate convexity of the energy ℱ⁢(t,⋅,⋅), the lower semi-continuity of the unilateral slope and a sort of upper gradient inequality, based on a measure theoretic argument employed in [15]. All these ingredients, put together, allow us to work with evolutions of class W1,2⁢(0,T;L2), which seems to be the natural weak setting for (1.2) and, in perspective, for more complex systems, e.g. [1], and higher-dimensional problems. We remark that our proof of existence for the unilateral gradient flow does not rely on the chain rule in W1,2⁢(0,T;H1) (see Lemma 7). However, in order to study the quasi-static limit, and prove (1.4), it is necessary to have a uniform bound on the length of the curves s↦(tε⁢(s),wε⁢(s),zε⁢(s)). This is a delicate technical point, which is obtained by means of a discrete Gronwall argument, cf. [22, 33], and which gives, as a by-product, a uniform bound in W1,2⁢(0,T;H1). This bound is used, together with the chain rule, in the proof (1.5). Finally, we remark that our analysis works for domains Ω contained in ℝ2 since it relies on Sobolev embeddings, see for instance Lemma 6. For similar technical reasons, in the N-dimensional setting gradient flows of this type have been studied, e.g. [7] and [22], by modifying the energy ℱ with some dimension depending variants; for instance ∫Ω|v-1|2+|∇⁡v|2⁢d⁢x=∥v-1∥H12 is replaced by ∥v-1∥W1,pp (for p>N) or by ∥v-1∥L22+|∇⁡v|H122 (for N=3).

2 Setting, energy and its derivatives

First of all we collect the set of assumptions used through the work.

Next, we provide the properties of energy and derivatives which will be used in the sequel.

If the displacement field u is sufficiently regular (and this is the case for our evolutions), variations of energy take a simple form; more precisely, if u∈W1,p⁢(Ω,ℝ2) for some p>2, then by Lemma 6, the energy ℱ⁢(t,u,⋅) is Gateaux differentiable with

In the evolution, irreversibility is modeled by monotonicity of v. For this reason, both the gradient flow and the BV-evolution will be defined in terms of the following unilateral L2-slope of ℱ⁢(t,u,⋅): if u∈W1,p⁢(Ω,ℝ2) for some p>2, let

For future convenience denote Ξ={ξ∈H1(Ω),ξ≤0,∥ξ∥L2≤1}, so that the unilateral slope is equivalently given by

By the regularity in time of g the partial time derivative takes the form

In particular, for v∈𝒱 by continuity and coercivity of the elastic energy we have

Finally, the energy ℱ⁢(t,⋅,v) is Fréchet differentiable in H01⁢(Ω,ℝ2), with respect to the natural norm of H01⁢(Ω,ℝ2), and for every ζ∈H01⁢(Ω;ℝ2) we have

where 〈⋅,⋅〉 denotes the duality between H01⁢(Ω;ℝ2) and H-1⁢(Ω;ℝ2).

3 Gradient flows