Angela Mihai - Stochastic hyperelastic models

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For natural and industrial elastic materials, uncertainties in the mechanical responses arise from the inherent micro-structural inhomogeneity, sample-to-sample intrinsic variability, and observational data, which are sparse, inferred from indirect measurements, and polluted by noise. For these materials, deterministic approaches, which are based on average data values, can greatly underestimate or overestimate their properties, and stochastic representations accounting also for data dispersion are needed.

In this talk, I will present an explicit strategy for constructing stochastic hyperelastic models described by a strain-energy function where the parameters are random variables characterised by probability density functions. These models are able to propagate uncertainties from input data to output quantities of interest. In particular, for a stochastic hyperelastic body with a simple geometry, I will show analytically that, by contrast to the deterministic elastic problem where a single critical value strictly separates the cases where an instability can or cannot occur, for the stochastic problem, there is a probabilistic interval where the stable and unstable states always compete in the sense that both have a quantifiable chance to be found. More complex but still tractable problems can be treated in a similar manner.