The issue of evaluating strength and service life of critical engineering objects, whose exploitation properties are characterized by multi-parametric nonstationary thermal-mechanical effects, is discussed. Basic degradation mechanisms of structural materials (metals and alloys) characteristic of such objects are considered. The main criteria for mathematical models of fatigue damage accumulation are formulated. Based on the developed version of defining equations of damaged medium mechanics, the effect of the angle of inclination of the cooling channels on thermal-cyclic life of the models of flue tubes of combustion chambers gas turbine engines is numerically analyzed. Special attention is paid to studying the processes of complex thermal-cyclic deformation and fatigue damage accumulation, accompanied by rotation of the main sites of stress tensors, complete and plastic deformations, as well as the processes of fatigue damage accumulation. Comparison of the obtained numerical results with experimental data shows that the developed model of damaged medium mechanics adequately and accurately enough for engineering calculations quantitatively describes experimental data and can be effectively used for evaluating thermal-cyclic life of materials and structures under multiaxial non-proportional modes of thermal-cyclic loading. Keywords: thermal-cyclic life, damaged medium mechanics, complex deformation, models of flue tubes, cooling channel, inclination, macroscopic crack, strength, fracture.