Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics

Negri, Matteo

doi:10.1051/cocv/2014004

Cited by 35 publications

(26 citation statements)

References 34 publications

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“…Our first and main goal is to prove the convergence of discrete solutions and to derive a precise characterization of the limit evolution; our second task is the study of its mechanical properties. By our analysis we obtained the following results: the time-continuous evolution is characterized "mathematically" in terms of a (non-degenerate, parametrized) BV -evolution [17,19,23] while "mechanically" it satisfies a phase-field version of Griffith's criterion, at least in the regime of stable propagation.…”

Section: Introductionmentioning

confidence: 96%

Convergence of alternate minimization schemes for phase-field fracture and damage

Knees

Negri

2017

Math. Models Methods Appl. Sci.

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We consider time-discrete evolutions for a phase-field model (for fracture and damage) obtained by alternate minimization schemes. First, we characterize their time-continuous limit in terms of parametrized [Formula: see text]-evolutions, introducing a suitable family of “intrinsic energy norms”. Further, we show that the limit evolution satisfies Griffith’s criterion, for a phase-field energy release, and that the irreversibility constraint is thermodynamically consistent.

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Section: Introductionmentioning

confidence: 96%

Convergence of alternate minimization schemes for phase-field fracture and damage

Knees

Negri

2017

Math. Models Methods Appl. Sci.

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“…Part I. The proof follows closely that of [32,Theorem 4.4]. If s < S, then s ∈ [ , S ε n ) for ε n ≪ ; thus (5.13), with λ = , provides…”

Section: Rescaled Parametrized Gradient Flowsmentioning

confidence: 69%

“…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].…”

Section: Introductionmentioning

confidence: 71%

A unilateral L2L^{2}-gradient flow and its quasi-static limit in phase-field fracture by an alternate minimizing movement

Negri

2017

Advances in Calculus of Variations

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We consider an evolution in phase-field fracture which combines, in a system of PDEs, an irreversible gradient-flow for the phase-field variable with the equilibrium equation for the displacement field. We introduce a discretization in time and define a discrete solution by means of a 1-step alternate minimization scheme, with a quadratic {L^{2}}-penalty in the phase-field variable (i.e. an alternate minimizing movement). First, we prove that discrete solutions converge to a solution of our system of PDEs. Then we show that the vanishing viscosity limit is a quasi-static (parametrized) BV-evolution. All these solutions are described both in terms of energy balance and, equivalently, by PDEs within the natural framework of {W^{1,2}(0,T;L^{2})}.

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“…However, it is worth mentioning that our approach, although derived independently, resembles similar methods which have appeared recently in the literature. In [32], a related scheme has been for instance investigated in order to obtain a general existence result in a nonconvex but smooth setting. The author also takes into account viscous dissipation effects, and provides a constructive time rescaling, where the evolutions have a continuous dependence on time.…”

Section: 2mentioning

confidence: 99%

Linearly constrained evolutions of critical points and an application to cohesive fractures

Artina

Cagnetti

Fornasier

et al. 2017

Math. Models Methods Appl. Sci.

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We introduce a novel constructive approach to define time evolution of critical points of an energy functional. Our procedure, which is different from other more established approaches based on viscosity approximations in infinite dimension, is prone to efficient and consistent numerical implementations, and allows for an existence proof under very general assumptions. We consider in particular rather nonsmooth and nonconvex energy functionals, provided the domain of the energy is finite dimensional. Nevertheless, in the infinite dimensional case study of a cohesive fracture model, we prove a consistency theorem of a discrete-to-continuum limit. We show that a quasistatic evolution can be indeed recovered as a limit of evolutions of critical points of finite dimensional discretizations of the energy, constructed according to our scheme. To illustrate the results, we provide several numerical experiments both in one and two dimensions. These agree with the crack initiation criterion, which states that a fracture appears only when the stress overcomes a certain threshold, depending on the material.Moreover, we say that a measurable function v : [0, T ] → E is an approximable quasistatic evolution with initial condition v 0 and constraint f , if for every t ∈ [0, T ] there exists a sequence δ k → 0 + and a sequence (v δ k ) k∈N of discrete quasistatic evolutions with time step δ k , initial condition v 0 , and constraint f , such thatWe are now ready to state the main results of the paper (see Theorem 2.15 and Theorem 2.16).Dissipative systems. (0)). Under suitable assumptions on J, ψ, A, and f (see Theorem 2.15) we prove that thereis an approximable quasistatic evolution with initial condition v 0 and constraint f ;(C) Energy inequality: the function s → q(s),ḟ (s) F belongs to L 1 (0, T ) andis defined by (2.9) and ·, · F denotes the duality product in F.Any function v ∈ BV ([0, T ]; E) satisfying (A), (B), and (C) is a rate independent evolution. We also remark that, as a consequence of the local stability (B), the energy inequality becomes an equality in all the nontrivial intervals where the solution v(t) is absolutely continuous (see Theorem 2.18, where also the nondissipative case is treated). In addition, in such intervals the doubly nonlinear inclusion A * q(t) ∈ ∂J(v(t)) + ∂ψ(v(t)) is satisfied. It is however a well-known fact that, due to nonconvexity, our solutions can in general develop time discontinuities, where additional dissipation appears (see [27]). Non dissipative systems. When there is no dissipation we can still prove an existence result, although the evolution obtained is in general expected to be less regular in time. This is due to the fact that the absence of dissipation causes loss of compactness, since simple estimates of the total variation of the approximating solutions are now missing. This is undoubtedly a point of great interest in the analysis of the degenerate case (see also [2] for an abstract approach in the unconstrained setting). To compensate the loss of compactness, we need ...

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Quasi-static rate-independent evolutions: characterization, existence, approximation and application to fracture mechanics

Cited by 35 publications

References 34 publications

Convergence of alternate minimization schemes for phase-field fracture and damage

Convergence of alternate minimization schemes for phase-field fracture and damage

A unilateral L2L^{2}-gradient flow and its quasi-static limit in phase-field fracture by an alternate minimizing movement

Linearly constrained evolutions of critical points and an application to cohesive fractures

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