Determining the parameters of the fractional exponential heredity kernels of linear viscoelastic materials

Abstract

The parameters of the fractional exponential creep and relaxation kernels of linear viscoelastic materials are determined. Methods that approximate the kernel by using the Mittag-Leffler function, the Laplace-Carson transform, and direct approximation of the creep function by the original equation are analyzed. The parameters of fractional exponential kernels are determined for aramid fibers, parapolyamide fibers, glass-reinforced plastic, and polymer concrete. It is shown that the kernel parameters calculated through the direct approximation of the creep function provide the best agreement between theory and experiment. The methods are experimentally validated for constant-stress and variable-stress loading in the modes of additional loading and complete unloading Introduction. Most methods for solving problems of linear viscoelasticity (Laplace transforms, operational methods) are based on various analytic representations of the creep and relaxation kernels [13]. These kernels should meet some requirements such as accurate description of the relationship among stresses, strains, and time, possibility for the extrapolation of experimental data, and availability for practical calculations. Another important requirement is the possibility of using a simple effective method for the mathematical processing of original experimental data to determine the kernel parameters.There are many analytic expressions for the creep and relaxation kernels. The structure of the kernels and methods to determine their parameters are detailed in [3, 7-9, 13, 15]. Rabotnov's fractional exponential function [13-15] seems to be the most promising kernel as highly flexible in describing the viscoelastic properties of real materials. In [5,20], it is proposed to use a sum of fractional exponential functions as a heredity kernel.A fractional exponential function is well understood, tabulated, found to be related to the Mittag-Leffler function and other special functions, and proved to satisfy a multiplication theorem similar to that for Abel operators [14-16, 18, 21, 23, 25]. However, it remains difficult to determine the parameters of fractional exponential kernels because the series in the analytic expression of the kernel is poorly converging.It is proposed to approximate the fractional exponential kernel by the Mittag-Leffler function [18], to use the Laplace-Carson transform [6], and to approximate an experimental bounded creep function [2]. The Mittag-Leffler function will be used to determine the creep kernel parameters of thermoset polymers [1], the approximation of a bounded creep function will be used to determine the creep kernel parameters of polyethylene of medium hardness [2,7], and the Laplace-Carson transform will be used to determine the creep kernel parameters of polyethylene and some glass-reinforced plastics [4,6,12,17].The present paper generalizes the direct approximation of an experimental creep function to unlimited test duration, validates all the methods for organic fibers, glass-reinforced plastic, and polymer co...

Determination of the Creep Parameters of Linear Viscoelastic Materials

“…Possibly, in order to reduce the length of material in published works containing an approximation of creep curves of various materials, as a rule, information on creep strain is given graphically, for example, in works [9,10,[12][13][14][18][19][20].…”

“…Some methods for determining the parameters of Rabotnov's creep kernel are known: using Mittag-Leffler's function [11,12], using Laplace-Carson's transform [9,[12][13][14], and a method of direct approximation [9,[12][13][14][15].…”

Calculating creep strains in linear viscoelastic materials under nonstationary uniaxial loading

“…As such materials, we use FM 10001 nylon fibers [26], laminated plastic under tension [20], ÒÑ8/3-250 glass-reinforced plastic [14] stretched along the weft (j = 90°) and at an angle (j = 45°) to the axis of symmetry, and contact-molded glass-reinforced plastic [5] 2) and its resolvent (1.3) contain three parameters a, b, and lto be determined in creep and relaxation tests. We determine these parameters from creep tests at constant stresses using the values of the creep function J t k j ( )obtained as in [3,21]. In this case, formula (1.4) becomes…”

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

“…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 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].…”