Improving constitutive equations in multiscale modelling by means of the sufficiency criterion using the example of nano wire contraction

Abstract

We demonstrate how two concepts from statistical modelling, namely the sufficiency criterion and the Rao-Blackwell-Kolmogorov theorem, can improve mechanical models. Applied to the contraction of relaxed nano wires, we start with a rough and simple equation. A set of sufficient parameters is then used to derive an improved equation that resembles the real solution in shape and quantity. The analytical solution and molecular simulations will finally highlight the functionality and the advantages of the scheme.Statistical and mechanical models share the same challenge of determining closing equations, e.g., estimators in statistics and constitutive equations in multiscale systems. In statistics, there is a powerful tool that improves a given estimator: the sufficiency criterion and its use in the Rao-Blackwell-Kolmogorov theorem. Interestingly, the improved estimator can be proven to be never worse and only requires a sufficient set of parameters [1,3]. The improvement scheme consists of three steps:

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