Well-posedness of a random coefficient damage mechanics model*

Abstract

We study a one-dimensional damage mechanics model in the presence of random materials properties. The model is formulated as a quasilinear partial differential equation of visco-elastic dynamics with a random field coefficient. We prove that in a transformed coordinate system the problem is well-posed as an abstract evolution equation in Banach spaces, and on the probability space it has a strongly measurable and Bochner integrable solution. We also establish the existence of weak solutions in the underlying physical coordinate system. We present numerical examples that demonstrate propagation of uncertainty in the stress–strain relation based on properties of the random damage field.

Keywords:

• Viscoelasticity
• damage mechanics
• random coefficient differential equation
• mild solutions

AMS CLASSIFICATIONS:

• 35K90
• 35Q74
• 35R60
• 74E35

Acknowledgements

G. S. would also like to thank D. M. Ambrose (Drexel University) for several helpful conversations during this project.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 Recall that if $X(\omega )$ is strongly $\mathbb{P}$-measurable, it is $\mathbb{P}$-a.s. the point-wise limit of simple functions, [15].

2 Since in our model the elastic constant is equal to one, the critical strain ${ϵ}_{\star }$ in (13(13) $\tilde{\mathcal{D}}(ϵ,x;\omega )=\{\begin{array}{ll}{\displaystyle 0}& ϵ\le {ϵ}_{\star }(x;\omega ),\\ {\displaystyle \frac{{ϵ}_{\star }(x;\omega )+\mathrm{\Delta }ϵ(x;\omega )}{\mathrm{\Delta }ϵ(x;\omega )}(1-\frac{{ϵ}_{\star }(x;\omega )}{ϵ})}& {ϵ}_{\star }(x;\omega )<ϵ\le {ϵ}_{\star }(x;\omega )+\mathrm{\Delta }ϵ(x;\omega ),\\ {\displaystyle 1}& ϵ>{ϵ}_{\star }(x;\omega )+\mathrm{\Delta }ϵ(x;\omega ),\end{array}$(13) ) and the critical peak stress ${\sigma }_{\star }$ are interchangeable.

Additional information

Funding

This work was supported by the U.S. Army Research Office Award Army Research Laboratory W911NF-19-1-0243. G. S. completed his contribution to this work under the support of U.S. National Science Foundation Grant DMS-1818726.