A dynamic model for viscoelastic materials with prescribed growing cracks

Abstract

In this paper, we prove the existence of solutions for a class of viscoelastic dynamic systems on time-dependent cracked domains, with possibly degenerate viscosity coefficients. Under stronger regularity assumptions we also show a uniqueness result. Finally, we exhibit an example where the energy-dissipation balance is not satisfied, showing there is an additional dissipation due to the crack growth.

A dynamic model for viscoelasticity in domains with time-dependent cracks

“…In this section, we illustrate a second method to find solutions to the viscoelastic dynamic system (3.16)-(3.20) according to Definition 3.3. This method is based on a minimizing movement approach deriving from the theory of gradient flows, and it is a classical tool used to prove the existence of solutions in the context of fractures, see, e.g., [4,7,9]. Be means of this method, we are also able to provide an energy-dissipation inequality satisfied by the solution, and consequently, by means of this inequality, we prove that such a solution satisfies the initial conditions (3.20) in a stronger sense than the one stated in (3.22).…”

“…4 and 5 we deal with the existence of a solution to the viscoelastic dynamic model; in particular in Sect. 4, we provide a solution by means of a generalization of Lax-Milgram's theorem by Lions. After that, in Sect.…”

An existence result for the fractional Kelvin–Voigt’s model on time-dependent cracked domains

“…In the classical theory of linear viscoelasticity, the constitutive stress-strain relation of the so-called Kelvin-Voigt's model is given by σ (t) = Ceu(t) + Beu(t) in \ t , t ∈ (0, T ), (1.2) where C and B are two positive tensors acting on the space of symmetric matrices, and ev denotes the symmetric part of the gradient of a function v (which is defined as ev := 1 2 (∇v + ∇v T )). The local model associated with (1.2) has already been widely studied and we can find several existence results in the literature; we refer to [2,3,6,7,17,24] for existence and uniqueness results in the pure elastodynamics case (B = 0) and in the classic Kelvin-Voigt's one.…”

Nonlinear dynamic fracture problems with polynomial and strain-limiting constitutive relations

“…Another deficiency with the model (1.9), (1.10) is the presence of the viscous term u t . When taking the limit in the approximation parameter, the viscoelastic paradox can occur [13], that is, the only possible solution of the corresponding sharpinterface problem is constant in time. However, in such dynamic fracture models, our interest lies in non-stationary solutions.…”