Narrow Results

We prove the existence and uniqueness theorems for a free boundary problem formulated as a mathematical model for metal matrix composites manufacturing. The process consists of the injection of a liquid metal through a rigid porous solid, accompanied by partial solidification and subsequent remelting. A technologically relevant qualitative property here investigated is a condition ensuring that solidification does not proceed to a point that the molten metal flow is stopped before the complete filling of the mould.

Asymptotic homogenization technique and two-scale convergence is used for analysis of macro-strength, fatigue and creep durability of composites with a periodic structure. The linear damage accumulation rule is employed in the phenomenological micro-durability conditions (for each component of the composite) under varying cyclic or creep loading. Both local and nonlocal strength and durability conditions are analyzed. The strong convergence of the strength and fatigue and creep damage measures as the structure period tends to zero is proved and their limiting values are estimated.

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.