# The viscoelastic paradox in a nonlinear Kelvin–Voigt type model of dynamic fracture

## TL;DR

This paper studies a nonlinear model of fracture in viscoelastic materials and shows that energy used to create cracks is not accounted for in the model's energy balance.

## Contribution

The paper introduces a nonlinear Kelvin–Voigt type model and proves the existence of solutions while highlighting the viscoelastic paradox.

## Key findings

- A solution exists for the viscoelastic dynamic system on a time-dependent cracked domain.
- The energy-dissipation balance excludes energy used to increase the crack.
- The nonlinear model exhibits the viscoelastic paradox, similar to the linear case.

## 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 discretization-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.