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.
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 is strongly
-measurable, it is
-a.s. the point-wise limit of simple functions, [Citation15].
2 Since in our model the elastic constant is equal to one, the critical strain in (Equation13
(13)
(13) ) and the critical peak stress
are interchangeable.