Modeling of Hereditary Materials Relaxation by Abel Kernel

Abstract

The present work is devoted to mathematical modeling of the process of the hereditary materials relaxation. Nonlinear integral equation of a hereditary type is proposed. The Abel kernel with two unknown parameters is adopted as the kernel of the integral equation: α∈(0,1),δ>0. Two new characteristics were introduced: 1) experimental rheological parameter of relaxation; 2) calculated (model) rheological parameter of relaxation. Using the least squares method, expressions are obtained to determine unknown parameters of the Abel kernel. A mathematical expression is given to approximate the process of the hereditary materials relaxation. Using examples of rheonomic materials different in structure (polyurethane matrix, propellant, polyoxymethylene, fiberglass), it is shown that the proposed methods allow to determine Abel kernel parameters with a high accuracy and to model the process of relaxation of rheonomic materials different in structure during a long period of time: from 102 to 1.8·106 seconds (500 hours).

Unsteady-State Creep of an Asphalt Concrete

“…Equation ( 5) contains three unknown parameters: ε 0 , α and δ. The works [36,50,51] propose to consider α as the known parameter, and the parameters ε 0 and δ are calculated according to the expressions:…”

“…In other words, the material "flows" under the impact of load like a viscous liquid on this site. The works [37][38][39]51] show experimentally that a creep curve of an asphalt concrete is approximated on the unsteadystate creep site with a high accuracy by a straight line, and therefore, it can be characterized by only one parameter-the steady-state creep rate . ε I I .…”