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.

许多材料的宏观行为是复杂的，并且其机制的最终结果是在各种不同的尺度上运行。跨尺度对物质行为的不完全了解是整体物质行为的认识不确定性的来源。但是，由于材料响应的复杂性以及积分计算的计算成本过高，因此很难评估这种不确定性。在本文中，我们利用物质响应的多尺度和层次性质来开发一种无需整体计算即可量化物质响应总体不确定性的方法。具体来说，我们将不确定性限制在每个尺度上，然后将部分不确定性组合在一起，从而为整体或整体不确定性提供一个界限。该范围提供了不确定性的保守估计。重要的是，这种方法不需要非常昂贵的积分计算。我们演示了多晶镁板的弹道冲击问题的框架。镁及其合金作为有前途的轻质结构和保护材料目前受到关注。最后，我们指出，该方法还可以用于按材料设计方法研究对较小规模的特定机制的整体响应的敏感性。镁及其合金作为有前途的轻质结构和保护材料目前受到关注。最后，我们指出，该方法还可以用于按材料设计方法研究对较小规模的特定机制的整体响应的敏感性。镁及其合金作为有前途的轻质结构和保护材料目前受到关注。最后，我们指出，该方法还可以用于按材料设计方法研究对较小规模的特定机制的整体响应的敏感性。