# Determination of Nonlinear Creep Parameters for Hereditary Materials
## 1 Introduction  
Many natural (soils, rocks, wood, natural asphalt) and artificial (metals, their alloys, polymers, concretes, composites) materials, depending on temperature and load level, to one extent or another show their viscoelastic properties. Currently there are sufficiently developed viscoelastic theory and methods [1; 2; 3; 4; 5; 6] which allow the determination and description of viscoelastic properties of the materials. One can distinguish linear and nonlinear viscoelasticity [4; 5; 6; 7] in viscoelastic theory, as well as in elastic theory [8; 9]. One can say that currently the theory and methods of linear viscoelasticity have been developed sufficiently. In spite of the fact that as far back as 1913 [PERSON] [10] proposed to describe nonlinear viscoelasticity by binary integral equation, to date, nonlinear viscoelasticity theory and methods are on the stage of development and currently calculations for strength, stability, and longevity in many branches of engineering activity are made without consideration or with poor consideration of viscoelastic properties for the materials and elements of structures.  
One simple but efficient means for consideration of deformation nonlinearity for hereditary materials was proposed in 1948 by [PERSON] [11]. It was based on the similarity of isochronous creep curves of the materials. Meanwhile, the process of creep strain is described by the equation of Boltzmann-Volterra for linear viscoelastic materials, but strain in the left part of the equation has been replaced by the so-called \"curve of instantaneous deformation\", which is determined experimentally. [PERSON], as the author of the method, considers that the curve of instantaneous strain is a kind of ideal imaginary curve, which is impossible to obtain in reality, as in real conditions the strain rate is always a finite value [12; 13]. It can be obtained from isochronous creep curves at finite values of time, considering them similar.  
It is known that the selection of kernel of the integral equation and determination of its parameters is one of the most responsible actions in the description of mechanical behavior of viscoelastic materials. As it has been said in our previous work [14], the most universal is creep kernel in the form of the fractional exponential function of [PERSON] [4; 5; 6; 11]. The fractional exponential function is well studied and a special table [15] has been developed for simplification of calculations.  
Some methods for determining the parameters of [PERSON]'s fractional exponential kernel are known: direct approximation [16; 17; 18; 20; 19], the use of Laplace-Carson's transformation [16; 17; 18; 21; 22], interpolating by Lagrange polynomial [23], and the use of Mittag-Leffler's function [16; 24].  
This work proposes an effective algorithm for description of nonlinear deformation of hereditary materials based on [PERSON]'s method of isochronous creep curves, and also the methods have been improved for determination of creep kernel parameters, developed in our previous work [14].  
The term \"hereditary\" was introduced by [PERSON] and means delay of elastic strain in time [13]. [PERSON]. [PERSON] explains that the materials are considered \"hereditary\", described by creep kernel, depending on the difference of an argument: \(K(t-\tau)\)[25].