Nonlinear Creep of Viscoelastic Organic Fibers under Tension

The nonlinearity of the creep of nylon fibers is justified based on the similarity of a set of isochronous creep curves, which also includes the instantaneous deformation curve. Nonlinear hereditary constitutive equations of creep are derived. The real values of the influence function are determined as the basic rheological characteristic of the material. The applicability of Boltzmann's, Abel's, Rzhanitsyn's, and Rabotnov's kernels is estimated quantitatively. The choice of an Abel-type kernel is justified. The calculated and experimental data are in satisfactory agreement for a loading duration of up to 1,000 hours and an order of magnitude change in the stresses __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 7, pp. 102–115, July 2005.     

Calculating creep strains in linear viscoelastic materials under nonstationary uniaxial loading

The creep strains in linear viscoelastic materials under nonstationary loading of various types (incremental loading, complete unloading, and cyclic loading) are determined. Boltzmann–Volterra hereditary theory with fractional exponential kernel is used. Nonstationary loads are specified by Heaviside functions. The calculated results are validated by experimentally determining nonstationary creep strains of glass-reinforced plastic, plastic laminate, polymer concrete, duralumin, and nylon     

The integral representation of fractionally exponential functions and their application to dynamic problems of linear visco-elasticity

Fractionally exponential functions are written in the integral form and distribution functions with an Abelian singularity are obtained for the corresponding relaxation and retardation spectra. A principle is stated, defining the dynamic problems for which weakly singular functions can be used as the kernels of the integral operators. A one-dimensional sound wave traveling in a semiinfinite visco-elastic medium is considered. The generalized exponential functions of fractional order, proposed by Yu. N. Rabotnov [1, 2] as the kernels of Boltzmann-Volterra integral relations, have found wide applications in theory of linear visco-elasticity. This is explained partly by the great mathematical flexibility of the F-operators when applying the Volterra principle to the solution of elastically hereditary problems and partly by the fact that almost all weakly singular kernels possessing an Abelian singularity are connected in some way or other with the F-functions. For example, the resolvent of the elementary weakly singular Abelian kernel is an F-function. The product of an exponential function with an Abelian kernel represents a particular case of the product of two F-functions with different fractional parameters, while the resolvent of such a kernel is the product of an exponential function with an F-function [3, 4]. Since the e-functions are defined by slowly convergent series, their various asymptotic forms [2, 5–8] are commonly used in practical calculations. The theory of F-functions can be developed further in the context of their integral representations, which enables a more exact physical interpretation to be given to their parameters and on occasion simplifies computational operations.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 11, No. 1, pp. 103–110, January–February, 1970.In conclusion, the author thanks Yu. N. Rabotnov for discussion.     

Application of fractional exponential hereditary kernels in the nonlinear theory of viscoelasticity

It is proved that fractional exponential hereditary kernels of nonlinear viscoelasticity can be used to evaluate creep strains and stress relaxation. A nonlinear theory of viscoelasticity with time-independent nonlinearity described as a nonlinear curve of instantaneous elastoplastic deformation is used. The calculated results are validated against experimental data on the viscoelastic deformation of laminated and unidirectional fibrous composites and their components under the conditions of constant stresses, complete unloading, incremental loading, pure torsion, and constant strains     

Calculating the Linear Creep Strains of Viscoelastic Fibers under Tension

The linearity domain for the viscoelastic properties of high-molecular organic fibers is determined. The linearity criteria are coincidence of experimental compliance curves and linearity of isochronic creep curves. Statistical criteria are used to establish linearity. The influence function in the constitutive equation of linear viscoelasticity is an Abel-type power kernel. The calculated and experimental creep strains are in good agreement both at the initial stage of deformation and after long-term loading__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 5, pp. 97–106, May 2005.     

Effective properties of a linear viscoelastic composite

The relaxation moduli of a composite are determined. The relaxation of its components is described by various few-parameter kernels: Mittag-Leffler functions of different orders and Rzhanitsyn kernel. It is assumed that the composite components are made of model materials with volume relaxation. The Laplace transform and fractional rational approximation are used to develop an algorithm for reducing the relaxation functions of the composite to one class (series of decreasing exponents or exponents of fractional order). The relaxation moduli of a unidirectionally reinforced composite are determined as an example     

A method for determining the parameters of the hereditary kernels in the nonlinear theory of viscoelasticity

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     

Dynamic stability of a viscoelastic plate with concentrated masses

The paper addresses the geometrically nonlinear problem of dynamic stability of a viscoelastic plate with concentrated masses. The Bubnov-Galerkin method based on polynomial approximation is used to reduce the problem to a system of nonlinear Volterra-type integro-differential equations with singular relaxation kernels. This system is solved by numerical method based on quadrature formulas. The critical loads are found and their dependence on the arrangement and number of concentrated masses is studied for a wide range of mechanical and geometrical parameters of the plate. The choice of a relaxation kernel for dynamic problems for viscoelastic thin-walled plate-like structures is justified. Results produced by different theories are compared __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 109–118, February 2008.     

STATISTIC MODELING OF THE CREEP BEHAVIOR OF METAL MATRIX COMPOSITES BASED ON FINITE ELEMENT ANALYSIS

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考虑老化的混凝土粘弹性分数导数模型

混凝土是一种具有分形结构的材料。采用分数微积分模型来研究具有分形结构材料的老化规律目前尚未见到。本文的目的是采用含分数阶导数的类标准线性体来模拟考虑老化的混凝土的蠕变和松弛规律。给出了分数导数与Abel核之间的关系。讨论了类标准线性体的蠕变柔量和松弛模量及其在考虑老化的混凝土中的应用。与传统的混凝土流变模型相比较表明，类标准线性体可以更好地同时拟合混凝土在不同龄期的蠕变和松弛曲线。而且其形式简单、统一，在计算过程中需要调整的参数很少。可以预见，类标准线性体在混凝土的结构设计和计算中将有着广泛的应用前景。