Fiber suspension flow and homogenization of elastic properties under uncertainty

Abstract

To numerically investigate the stochastic influence of uncertain input parameters on fiber suspension flow and homogenization problems, point collocation and spectral projection polynomial chaos is applied on the following problems. (1) The Jeffery equation in homogenous shear flow under uncertain aspect ratio. (2) the Folgar-Tucker equation in homogenous shear flow under uncertain fiber interaction coefficient. (3) the Folgar-Tucker equation in plane channel flow under uncertain aspect ratio. For problem (1), compared to Monte Carlo simulation, polynomial chaos expansions provide poor results with implausible oscillations for the means of the second-order fiber orientation tensor. For problems (2) and (3) the results of both polynomial chaos methods are plausible and match each other closely. The asymptotic states of the second-order fiber orientation tensor and its distribution in problems (2) and (3) depend non-linearly on the respective uncertain input parameters. In problem (3) the initial distributions of the components of the second-order fiber orientation tensor collapse into a strongly skewed distribution with small range in areas where the uncertain fiber aspect ratio only weakly influences the orientation state. For problem (3) the effective Young’s modulus depending on the fiber orientation state is obtained through mean-field homogenization with the Mori-Tanaka method. The deterministic effective elastic properties depend more strongly on the fiber aspect ratio than the second-order fiber orientation tensor in the investigated range of inputs. Non-linear effects on the distributions of the effective Young’s modulus are less pronounced than for the second order-fiber orientation tensor. The strongest non-linear relation between the uncertain fiber aspect ratio and the effective Young’s modulus is found in areas where the fiber aspect ratio affected the 11-component of the second-order fiber orientation tensor the most.