Sensitivity of a homogeneous and isotropic second-gradient continuum model for particle-based materials with respect to uncertainties

This work concerns the probabilistic analysis of particle-based materials. More precisely, this work is devoted to the stochastic modeling of the geometric and constitutive microscale parameters associated with particle-pair interactions of an existing model for particle-based materials. Such an issue is addressed with a probabilistic methodology that relies on the maximum entropy principle from information theory. After defining and improving the chosen second-gradient continuum model for particle-based materials, it is shown that for micro-homogeneous and micro-isotropic materials, the involved microscale parameters turn out to be statistically independent. More precisely, the particle-pair distance between two consecutive particles is a uniformly distributed random variable and the specific micro-scale stiffness parameters are Gamma-distributed random variables. A homogeneous and isotropic 2D concrete plate subjected to an axial load is considered for illustration purposes. A stochastic solver based on Monte Carlo numerical simulations and mixed finite element (FE) method are chosen. The FE discretization is applied to the weak formulation of the equivalent continuum model with random mechanical properties. On the contrary, the particle-pair distance and the micro-scale stiffness parameters, which are the parameters of the boundary value problem formulated for the equivalent continuum model, are modeled as random variables. Finally, uncertainties propagation is discussed and statistical fluctuations of the macro mechanical response are found to be significant.