Determining the Parameters of the Hereditary Kernels of Nonlinear Viscoelastic Materials in Tension

Abstract

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):