  The macroscopic behavior of many materials is complex and the end result of
mechanisms that operate across a broad range of disparate scales. An imperfect
knowledge of material behavior across scales is a source of epistemic
uncertainty of the overall material behavior. However, assessing this
uncertainty is difficult due to the complex nature of material response and the
prohibitive computational cost of integral calculations. In this paper, we
exploit the multiscale and hierarchical nature of material response to develop
an approach to quantify the overall uncertainty of material response without
the need for integral calculations. Specifically, we bound the uncertainty at
each scale and then combine the partial uncertainties in a way that provides a
bound on the overall or integral uncertainty. The bound provides a conservative
estimate on the uncertainty. Importantly, this approach does not require
integral calculations that are prohibitively expensive. We demonstrate the
framework on the problem of ballistic impact of a polycrystalline magnesium
plate. Magnesium and its alloys are of current interest as promising
light-weight structural and protective materials. Finally, we remark that the
approach can also be used to study the sensitivity of the overall response to
particular mechanisms at lower scales in a materials-by-design approach.
