Influence of Distortion in the Homogenization of Fiber-Reinforced Composites

Abstract

We consider fiber-reinforced composites, where the fibers present some distortion in their longitudinal direction. The length scale of this distorsion is much larger than the spacing of the fibers. We derive asymptotic formulas for a conduction problem. Reiterated homogenization, i.e. homogenizing with respect to the smallest length scale, then with respect to the largest does not capture the effects of distortion. Instead, we use a representation formula to show how distortion influences the correction terms.

Multiscale asymptotic method of optimal control on the boundary for heat equations of composite materials

“…[1,4,20]). With the sizes of physical devices ever decreasing, the thermal analysis of these structures and the optimal control of the temperature field are becoming more important.…”

“…The second one is how to treat the multiscale asymptotic solution near the boundary ∂Ω. The new contributions obtained in this paper were the determination of the convergence rate for the approximate solutions by using a suitable cut-off function for problems (1)- (4).…”

“…[16, p. 20]). 3 ) are satisfied, then one can show that the optimal control u ε,k (x, t) of (1)- (4) is characterized through the unique solution {y ε,k , ψ ε,k , u ε,k } of the optimality system given by ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ d dt y ε,k u ε,k + A ε y ε,k u ε,k = f, (x, t) ∈ Ω × (0, T ), σ ε y ε,k = u ε,k (x, t), (x, t) ∈ ∂Ω × (0, T ),…”