The viscoelastic paradox in a nonlinear Kelvin-Voigt type model of dynamic fracture
Maicol Caponi, Alessandro Carbotti, Francesco Sapio
TL;DR
We address the problem of dynamic fracture in viscoelastic media described by an implicit nonlinear constitutive law on prescribed crack paths. We develop a time-discretization scheme with regularized monotone operators, prove discrete energy estimates, and use compactness and Browder–Minty theory to obtain a weak solution in which the constitutive law is satisfied. The main contributions are the existence (and uniqueness under monotonicity) results for a nonlinear viscoelastic system on moving cracks and a derived energy-dissipation balance that reveals a viscoelastic paradox in the nonlinear Kelvin–Voigt model. This work extends dynamic fracture theory to implicit nonlinear viscoelasticity and connects with phase-field perspectives, highlighting how viscous effects can inhibit crack growth under Griffith-type balances.
Abstract
In this paper we consider a dynamic model of fracture for viscoelastic materials, in which the constitutive relation, involving the Cauchy stress and the strain tensors, is given in an implicit nonlinear form. We prove the existence of a solution to the associated viscoelastic dynamic system on a prescribed time-dependent cracked domain via a discretisation-in-time argument. Moreover, we show that such a solution satisfies an energy-dissipation balance in which the energy used to increase the crack does not appear. As a consequence, in analogy to the linear case this nonlinear model exhibits the so-called viscoelastic paradox.