A new model is derived for the mechanical behavior of linear viscoelastic materials subjected to physical aging. According to the model, a viscoelastic medium is treated as a system of elastic springs (links between polymeric molecules) which replace each other. Two types of links are distinguished : links arisen at the instant of quenching (type I), and links emerging in the quenched material at a constant temperature (type ii). The mechanical behavior of an aging medium is determined by three material functions which characterize (i) the collapse of links of type I, (ii) the breakage of links of type II, and (iii) the rate of emergence of new links of type LI. We derive integral equations for these functions and find their solutions using data of the standard relaxation tests. To verify the model, we calculate the material response in the creep tests and compare results of numerical simulation with experimental data for an epoxy adhesive. The results obtained demonstrate fair agreement between experimental observations and their prediction. By using the model developed, we analyze numerically the behavior of a viscoelastic medium under time-varying loads. We study elongation of a specimen with a constant rate of strain, its recovery after creep tests, and steady shear oscillations of a layer. In the latter case, numerical results are compared with experimental data for polypropylene samples.