Numerical prediction of disorder effects in solid foams using a probabilistic homogenization scheme

• Numerical prediction of material uncertainties in polydisperse solid foams.
• Finite element scheme with effective deformations as enhanced degrees of freedom.
• Probabilistic homogenization.
• Discretization of the space of the random variables.
• Scheme applicable to effective stiffness and effective strength.

The present study is concerned with a numerical prediction of uncertainties in the macroscopic mechanical properties of microheterogeneous materials with uncertain microstructure. As a model material, solid foams are employed. The stochastic information on the uncertainty is gained in multiple numerical homogenization analyses of small-scale testing volume elements. The local relative density, the cell size distribution, the cell geometry and the spatial orientation of the testing volume elements are assumed to form the set of the relevant stochastic variables. Selected microstructural cases are analyzed for their macroscopic material response. Based on the probability distributions for the stochastic variables defining the microstructures of the testing volume elements, the probability distributions for the mesoscopic material properties are obtained. For the numerical homogenization of the testing volume elements, an enhanced finite element technique is employed, where the components of the macroscopic deformation gradient are introduced as generalized degrees of freedom. Assuming periodic boundary conditions, the global degrees of freedom interact with the conventional displacement degrees of freedom of the discretized microstructure via special boundary coupling elements. The mesoscopic stresses are obtained in a rather efficient manner as the generalized reaction forces for the global degrees of freedom.