A variational approach to the quasistatic limit of viscous dynamic evolutions in finite dimension

Abstract

In this paper we study the vanishing inertia and viscosity limit of a second order system set in an Euclidean space, driven by a possibly nonconvex time-dependent potential satisfying very general assumptions. By means of a variational approach, we show that the solutions of the singularly perturbed problem converge to a curve of stationary points of the energy and characterize the behavior of the limit evolution at jump times. At those times, the left and right limits of the evolution are connected by a finite number of heteroclinic solutions to the unscaled equation.

The notions of Inertial Balanced Viscosity and Inertial Virtual Viscosity solution for rate-independent systems

“…Proof. By reasoning as in [30,Theorem 5.4] it is easy to see that the map t → E(t, u + (t)) − t 0 ∂ t E(r, u(r)) dr is nonincreasing; it essentially follows from the energy balance (4.1) by dropping the dissipated energy (i.e. the terms with R(•) and |•| V ) and controlling the kinetic energy in the limit ε → 0 by means of (iii) in Proposition 4.3.…”

“…Furthermore, the ratedependent nature of the cost forces one to consider minimization problems on an asymptotically infinite time horizon, and take the infimum over them. We also refer to [1,30] for a similar analysis with no rate-independent dissipation, namely considering R = 0, where an analogous notion of solution was developed. As it happened in [30], we prefer to consider a notion of solution which does not depend on the chosen representative of u in its Lebesgue class, which is done by considering left and right limits only in the energy balance.…”

On the Approximation of Quasistatic Evolutions for the Debonding of a Thin Film via Vanishing Inertia and Viscosity

“…Going further in the analysis without that assumptions requires a deep understanding of the measure µ introduced in Proposition 4.7. This kind of study has been developed in [26] in finite dimension, but a generalisation to our infinite dimensional setting seemed hard to us. The idea in [26] relies on the introduction of a suitable cost function which measures the energy gap of a limit solution after a jump in time, and hence characterises their counterpart of measure µ.…”

“…perfect plasticity and to [16], [17] for the undamped version of the debonding model we analyse in this work. The issue of quasistatic limit has also been studied in a finite-dimensional setting where, starting from the works [1], [28] and with the contribution of [22], an almost complete understanding on the topic has been reached in [26]. A common feature appearing both in finite both in infinite dimension is the validation of the quasistatic approximation only in presence of a damping term in the dynamic model.…”

Discrete approximation of dynamic phase-field fracture in visco-elastic materials

“…e.g., [29,47]. More recently, research focus in both engineering applications [8,51,49] and in applied analysis [12,13,14,32,44,46,50] is put on the investigation of dynamic fracture.…”