Calculating creep strains in linear viscoelastic materials under nonstationary uniaxial loading

Abstract

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 Keywords: linear viscoelastic material, creep process, incremental loading, complete unloading, cyclic loading, stress relaxation, fractional exponential hereditary kernel Introduction. The basic task of one-dimensional creep theory is to derive, using some time operators, equations that relate strains and stresses as scalars [12]. It is assumed that a set of experimental creep curves at constant stresses has all necessary data to identify the parameters of any phenomenological creep theory. Such theories are usually validated by comparing theoretical and experimental relaxation and creep curves for nonstationary loading. Problems of relaxation and nonstationary creep are of practical interest as describing the real service conditions of many materials and structural members.To describe the relationship among strains, stresses, and time, widespread use is made of hereditary creep theories that take the loading history into account [7,11,12,[22][23][24]. The chief difficulties faced when using a hereditary theory are to choose a kernel for the integral equation, to find the associated resolvent, and to determine the parameters of the kernel. These tasks are easier to perform in the case of linear viscoelastic materials for which the relationship among strains, stresses, and time is uniquely described by a linear Volterra equation. Power and exponential functions and their aggregates are often used as resolvent kernels in linear problems. Some results of experimental validation of a linear hereditary theory with power and exponential kernels in the case of variable loading are presented in [1,2,8,9,25]. The agreement between calculation and experiment was achieved by increasing the number of kernel parameters (to eight) and conducting additional creep-recovery tests.Using the fractional exponential kernel proposed in [13] seems more promising. This kernel contains two unknown parameters, quire accurately describes creep curves of real materials at constant stresses, and is successfully used to solve boundary-value problems of linear hereditary creep. We will test the fractional exponential kernel by analyzing the relaxation of creep stresses and strains of linear viscoelastic polymeric, composite, and metallic materials under variable loading.1. Problem Formulation. Subject of Inquiry. In the one-dimensional case, the strain-stress relationship in the linear hereditary theory of viscoelasticity is described by dual Volterra equations of the second kind [7,11,12]:

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

“…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]: …”

“…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]: where the first equation describes creep, and the second equation describes relaxation; e( ) t and e t ( ) are the total strains consisting of elastic strain e e and creep strain e c at times t and t; s( ) t and s t ( ) are the stresses at times t and t; j 0 ( ) × is the function associated with the instantaneous deformation curve; K t ( ) -t is the creep kernel; R t ( ) -t is the relaxation kernel; l is the rheological parameter ( ) l > 0 ; t is the time of observation; t is the time preceding the time of observation.…”

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

“…1) at which the tangent modulus E ( ) * e is expressed as follows [1]: Table 1 collects the values of the elastic modulus E, the stress s * calculated by (1.6) and (1.7), and the parameters a b , , l of the fractional exponential hereditary kernels (1.2) calculated by the method from [2]. The parameters a b , , l were determined using experimental creep curves available in the literature (see [2,4,10] for the primary references).…”