Risk-averse estimates of effective properties in heterogeneous elasticity

In this work, we propose a theoretical framework for computing pessimistic and optimistic estimates of effective properties in the case of heterogeneous elastic materials with uncertain microscopic elastic properties. We rely on a risk-averse measure widely used in finance called the conditional-value at risk (CVaR). The CVaR computes the conditional expectation of events occurring above a given risk level, thereby characterizing the extreme tails of the probability distribution of a random variable. In the context of elastic materials, we propose to use the CVaR on the elastic free energy to compute an optimistic estimate of the global stiffness for some confidence level alpha. Similarly, we also use the CVaR on the complementary elastic energy to compute a pessimistic estimate of the global stiffness. The obtained CVaR estimates benefit from a convex optimization formulation. The resulting material behavior is still elastic but not necessarily linear anymore. We discuss approximate formulations recovering a linear elastic behavior. We apply the proposed formulations to the micromechanical estimates of effective elastic properties of random heterogeneous materials.