This monograph treats prognosis of thermoinelastic properties essential for design of polymeric composite materials reinforced by spherical radially inhomogenous inclusions (¿functionally graded composites¿). The key subjects are: 1. synthesis of composites based on model systems with regular and singular transition layers; 2. a development of software for finding effective viscoelastic characteristics; 3. analysis of viscoelastic behavior of impact resistant and vibrodamping polymeric systems with regular and anomalous behavior; 4. a prognosis method with wide variation of filler concentration and loading conditions and 5. damage analysis for linear and nonlinear viscoelastic polymeric composites compared with experimental data. For a simplified description of interaction of particles and thermal residual stresses, Eshelbian self-consistent effective field method is employed. Damage evolution is described by a modified Vakulenkös approach to endochronic thermodynamics. The creep of the underground train support caused by triangular history of compressive stress is treated. The book is intended for researchers and engineers in material science and mechanical engineering.