This paper addresses the attainability of the Hashin-Shtrikman bounds for multiphase composites, including those of conductive materials and elastic materials. It presents a new derivation of these bounds that yield a necessary and sufficient condition for optimal microstructures. A key idea is a simple characterization of the gradient Young measures associated with optimal microstructures.

The paper presents the solution of spatial transient problem impact a moving heat flux source induced by laser radiation on surface half-space using superposition principle and method functions. hyperbolic equation thermal conductivity accounting for relaxation time is used to model heating process. It assumed that distributed symmetrically with respect center spot. combined numerical analytica...