A method of determining the parameters of the hereditary kernels in a viscoelastic model with time-independent nonlinearity is tested. The parameters are determined by fitting the discrete values of the kernels that are obtained considering the similarity of isochronous creep curves and instantaneous deformation curve. The discrete values of the kernels in the zone of singularity are found using weight functions. The Abel function, a combination of power and exponential functions, and fractional-exponential functions are used as hereditary kernels. The method is tested by analyzing the creep, creep recovery, and stress relaxation in laminated composites, polymeric binders, and fiber reinforcements under uniaxial tension Introduction. The main difficulties faced in determining the parameters of the hereditary kernels of nonlinear viscoelastic materials are associated with the selection of the most adequate nonlinear viscoelastic model and with the inadequate accuracy of measurements in short-term tests in which dynamic effects are manifested. The most effective approaches to and methods of resolving such challenges are reviewed in [6,9,10,15,22,25].In [2], a method was proposed for determining the parameters of the hereditary kernels in a Rabotnov-type nonlinear theory of viscoelasticity with time-independent nonlinearity. The method is based on the unified isochronous deformation curve that reflects the similarity of isochronous creep curves to the instantaneous deformation curve. The nonlinearity of the constitutive equations is determined by the nonlinearity of the instantaneous deformation curve. We will use the creep kernel rather than the relaxation kernel because creep tests are easier to conduct than relaxation tests. The parameters of creep kernels are determined by fitting their discrete values obtained by differentiating the average similarity function. The discrete values of the kernels in the zone of singularity occurring in short-term tests are obtained using weight functions.Here the method of determining the parameters of the hereditary kernels of nonlinear viscoelastic materials outlined in [2] is tested by calculating creep strains at constant stresses, creep recovery under complete unloading, and stress relaxation.1. Problem Formulation. Subject of Analysis. We will test the method of determining the parameters of the hereditary kernels of nonlinear viscoelastic materials in the hereditary theory of viscoelasticity with time-independent nonlinearity. We start with the following one-dimensional constitutive equations [3,5,15,23]:

The analytical and experimental procedures of the method for determining the parameters of the hereditary kernels in the modified Rabotnov's nonlinear viscoelastic model are outlined. The method is based on the similarity of the isochronous creep curves to the instantaneous-deformation curve. The parameters of the kernels are determined by fitting the discrete values of the kernels that are found by differentiating the average similarity function. The discrete values of the kernels in the domain of singularity are calculated using weight functions Keywords: nonlinear viscoelasticity, parameters of hereditary kernel, creep rate, stress relaxation rate, similarity function, weight functionIntroduction. Selection of hereditary kernels and reliable determination of theirs resolvents and parameters constitute one of the basic problems in the hereditary theory of viscoelasticity. This problem is, as a rule, solved by seeking creep and relaxation kernels that characterize the mechanical properties of viscoelastic materials and appear in the constitutive equations relating stresses, strains, and time.There are various analytical interpretations of the creep and relaxation kernels and methods for determining their parameters [6,7,9]. The chief difficulties that arise in selecting kernels and methods to determine their parameters are due to the fact that the creep and relaxation rates tend to infinity as t ® 0 and asymptotically decrease as t ® ¥. Moreover, the hereditary kernels must be analytically invertible so that the parameters of one kernel can be used to find the values of another kernel.The quasistatic mechanical properties of materials in the linear theory of viscoelasticity can be quite accurately described in terms of creep and relaxation functions. These functions are set up using the data of constant-stress creep tests or constant-strain relaxation tests. A general approach, logical procedures, and their experimental and numerical implementation needed to determine mechanical characteristics in the linear theory of viscoelasticity are detailed in [6,7,9,12].The methods for determining mechanical characteristics in the nonlinear theory of viscoelasticity are more complicated and less justified than in the linear theory. This is primarily because the nonlinear theory uses several variations of constitutive equations and, respectively, several methods for determining viscoelastic characteristics.The most general approach to determining the mechanical characteristics of nonlinear viscoelastic materials uses the Volterra-Frechet multiple integral representation [7]. However, the great number and multidimensional nature of the hereditary kernels that enter into the multiple integral expressions, especially for high-order terms, considerably complicate the determination of the kernel parameters. To overcome these difficulties, simplified variations of the general nonlinear constitutive equation were proposed and numerical and experimental procedures for determining the necessary viscoelastic characteristics were dev...