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

Abstract

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 ...

Viscous Corrections of the Time Incremental Minimization Scheme and Visco-Energetic Solutions to Rate-Independent Evolution Problems

“…More degenerate situations when α = 0 can also be considered, both from the continuous (see [1]) and the discrete point of view (see [3], who considers a different dependence with respect to time, given by a time-dependent linear constraint): the main difficulty here relies on the loss of time-compactness, since simple estimates of the total variation of the approximating curves are missing.…”

“…Besides applications to various models where the Energetic approach have been successfully employed (we plan to apply the present theory to elastoplasticty and crack propagation models, following the approaches of [24,45,34,10]), further important developments have to be better understood: one of the most interesting one concerns the case of a viscosity term δ which is not "controlled" by the distance d. This situation may occur when d = 0, causing severe compactness issues (this is the most difficult case, see [1,3]), or when the evolution involves a coupling between quasi-static and viscous laws [11], leading to a problematic formulation of the energy balance.…”

Optimal Regularity and Structure of the Free Boundary for Minimizers in Cohesive Zone Models

“…In recent years, a variational formulation of fracture evolution has been proposed by Francfort and Marigo [18], and later developed by Dal Maso and Toader [14], and Dal Maso, Francfort, and Toader [12,13] (see also [19] and the references therein, for a variational theory of rate independent processes). Such evolution is based on the idea that at any given time the configuration of the elastic body is an absolute minimiser of the energy functional (see also [1,8,15], and [11] in the context of plasticity where, more in general, critical points of the energy are allowed).…”

“…In this paper we study optimal regularity and free boundary for minimizers of an energy functional arising in cohesive zone models for fracture mechanics. Such models describe the situation in which the energy density of the fracture depends on the distance between the lips of the crack (see for instance [1,8,9,10,16]). We consider the energy functional associated to an elastic body occupying the open stripe R n × (−A, A), with n ≥ 2 and A > 0.…”

Convergence analysis of time-discretisation schemes for rate-independent systems

“…In Section 3.1, we show the convergence of discrete solutions under the assumption η N → ∞ for N → ∞ and obtain again BV-solutions in the limit. Observe that (1.6)-(1.7) is a modified version of an algorithm studied in [ACFS17]: instead of the term R(v − z k,i−1 ) the authors in [ACFS17] use the term R(v − z k ) in (1.6) and they study the convergence of the scheme for fixed η > 0 and N → ∞. For the version (1.6) we show that the sequence (z k,i ) i∈N itself converges to a critical point (i.e.…”