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The problem of structural design of polymeric and composite viscoelastic materials is currently ofgreat interest. The development of new methods of calculation of the stress–strain state of viscoelastic solidsis also a current mathematical problem, because when solving boundary value problems one needs to considerthe full history of exposure to loads and temperature on the structure. The article seeks to build an iterativealgorithm for calculating the stress–strain state of viscoelastic structures, enabling a complete separation of timeand space variables, thereby making it possible to determine the stresses and displacements at any time withoutregard to the loading history. It presents a modified theoretical basis of the iterative algorithm and providesanalytical solutions of variational problems based on which the measure of the rate of convergence of theiterative process is determined. It also presents the conditions for the separation of space and time variables.The formulation of the iterative algorithm, convergence rate estimates, numerical computation results, andcomparisons with exact solutions are provided in the tension plate problem example