An efficient method for determining the deformation function of a composite is discussed. The method is based on a fractional exponential representation of the deformation functions of the composite components. The viscoelastic solution is obtained using the Volterra principle. The deformation function is represented as a function of a base operator. Thus, the problem is solved by approximating the deformation function by a continued fraction and applying the method of operator continued fractions. A computational procedure is detailed and illustrated using data on longitudinal relaxation of polymethylmethacrylate. As an example, the deformation of a polymethylmethacrylate-based fibrous composite with viscoelastic properties is analyzed Introduction. Extensive use of polymers and polymer-based composites entails intensive development of the theory of viscoelasticity needed to solve practical deformation problems for composites and boundary-value problems for materials with time-dependent stress-strain state. Typical properties of these materials are relaxation (decreasing stresses at constant strains) and creep (increasing strains at constant stresses). To describe the deformation of viscoelastic materials, Boltzmann proposed a theory of hereditary viscoelasticity based on the superposition principle. According to this theory, the equation relating the strains and stresses in a linear nonaging viscoelastic material under uniaxial loading and isothermal conditions reads

In this paper the Laplace transform method is combined with Padé approximations to solve linear viscoelastic problems. This approach allows to avoid the usual difficulties of original function determination. An algorithm is given to find solution with arbitrary precision. As an example the solution for problem of viscoelastic orthotropic half-plane stress state under concentrated normal force is given.

An efficient method is proposed to determine the deformation function of a viscoelastic material from experimental data. The deformation function is assumed to be an integral operator with Rabotnov's fractional-exponential kernel or a sum of such kernels. This representation enables effective use of the method of operator continued fractions. To illustrate the method, deformation data for polymethylmethacrylate are used. The viscoelastic characteristics of a composite based on this material are obtained using the method of operator continued fractions Introduction. There are different methods to represent creep data using various functions of time. Such functions are needed to solve viscoelastic problems to obtain many characteristics of the material under investigation. Experimental creep data can be represented: (a) as a table if strains ε i = ε(t i ) at t i , i = 0, 1, …, n, are known (ε max is the maximum strain corresponding to the time t max ) and (b) graphically if strain ε = ε(t) at each t is known. This representation implies either a spline running through experimental points or already obtained standard creep curves.Many hereditary-elasticity problems are solved using the following analytic time-dependence of strain: