On a doubly nonlinear model for the evolution of damaging in viscoelastic materials

Abstract

We consider a model describing the evolution of damage in visco-elastic materials, where both the stiffness and the viscosity properties are assumed to degenerate as the damaging is complete. The equation of motion ruling the evolution of macroscopic displacement is hyperbolic. The evolution of the damage parameter is described by a doubly nonlinear parabolic variational inclusion, due to the presence of two maximal monotone graphs involving the phase parameter and its time derivative. Existence of a solution is proved in some subinterval of time in which the damage process is not complete. Uniqueness is established in the case when one of the two monotone graphs is assumed to be Lipschitz continuous.

Global existence of strong solutions to the one-dimensional full model for phase transitions in thermoviscoelastic materials

“…Hence, χ is forced to take values in [0,1], with the convention that χ = 0 when the body is completely damaged, and χ = 1 in the damage-free case. Hence, in [7], [8] the equation for macroscopic movements has a degenerating character related to the parameter χ, which is however different from the one in (1.3). For, in their case the coefficients of the elliptic operators in the stress-strain relation vanish only as χ ց 0, contrary to the twofold degeneracy of the equation (1.6 c), which we shall further comment later on.…”

“…The model is analyzed in [20] and in [21] pertains to nonlinear thermoviscoplasticity: in the one-dimensional (in space) case, the authors prove the global well-posedness of a PDE system, incorporating both hysteresis effects and modelling phase change, which however does not display a degenerating character. Degenerating phase parameters appear in models for damaging phenomena, see [6], [7], [8]. In this case, the phase variable χ is related to the local proportion of damaged material.…”

“…For, in their case the coefficients of the elliptic operators in the stress-strain relation vanish only as χ ց 0, contrary to the twofold degeneracy of the equation (1.6 c), which we shall further comment later on. Within this framework, in [7], [8] local in time well-posedness results are proved for the resulting PDE system.…”

A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains

“…Observe that such results rely on suitable smoothness assumptions on the domain Ω. The latter are also at the core of the analysis developed in the subsequent paper [BSS05], where the (doubly nonlinear) evolution equation for the damage parameter z (with q = 2) is coupled with a parabolic equation for u, in the context of linear viscoelasticity. Therein, the usage of the regularizing term A 2 z t is avoided.…”

Well-posedness and long-time behavior for a class of doubly nonlinear equations

“…Equations of the type of (1.1) stem in connection with phase change phenomena [14,17,23,32,46,47], gas flow through porous media [68], damage [15,16,30,31,54], and, in the specific case α(λr) = α(r) for all λ > 0 and r ∈ R (which is however not included in the present analysis), elastoplasticity [25,50,51,52], brittle fractures [26], ferroelectricity [56], and general rate-independent systems [29,48,49,53,55].…”