Modeling the Nonlinear Rheological Behavior of Materials with a Hyper-Exponential Type Function

Abstract

<p>This paper describes a nonlinear rheological model consisting of a modified and extended classical Voigt model for predicting the time dependent deformation of a variety of viscoelastic materials exhibiting elastic, viscous and inertial nonlinearities simultaneously. The usefulness of the model is illustrated by numerical examples.</p

A Nonlinear Mechanical Model for Predicting the Dynamics Response of Materials under a Constant Loading

“…We can interpret each term present in Equation (9) following Monsia (2011f). The first term in Equation (9) is proportional to the basic inertial stress, the second to a nonlinear quadratic viscous stress, the third term to the linear viscous stress, the fourth term to the classical linear elastic stress and the last term to a quadratic nonlinear elastic stress.…”

“…Note that in this present one-dimensional model, the stresses and strain are scalar functions and the coefficients a, b and c are time independent material parameters. Thus, noting t  the total stress due to the external exciting force acting on the material, the superposition of elastic stress, viscous stress and inertial stress is given by (Monsia 2011f…”

A Theoretical Characterization of Time Dependent Materials by Using a Hyperlogistic-Type Model

“…The viscous and inertial stresses are derived as first and second time derivatives of a similar function to the pure elastic stress. Therefore, the superposition of these three stress components, for a nonlinear elastic spring force function ) (  of deformation  , which is a scalar function, leads to the fundamental equation (Monsia, 2011g;Monsia, 2012) …”

“…Recently again, Monsia (2011f) using again the Bauer's approach (1984), proposed a nonlinear rheological model based on a substitution of linear elastic and damping forces in Voigt model by nonlinear elastic and damping forces with inclusion of a body for providing a theoretical basis to empirical exponential or logistic formulas used by several authors for fitting the stress-strain experimental data of arteries. Very lately, Monsia (2011g) by application of the Bauer's theory (1984), developed successfully for a variety of materials a nonlinear rheological model that can be used to predict and describe their dynamical behavior. By using a hyperbolic law as elastic spring force function (Monsia, 2011g), the strain versus time variation has been described as a hyper-exponential type function that appeared useful to reproduce the typical nonlinear exponential deformation of some viscoelastic materials.…”

“…Very lately, Monsia (2011g) by application of the Bauer's theory (1984), developed successfully for a variety of materials a nonlinear rheological model that can be used to predict and describe their dynamical behavior. By using a hyperbolic law as elastic spring force function (Monsia, 2011g), the strain versus time variation has been described as a hyper-exponential type function that appeared useful to reproduce the typical nonlinear exponential deformation of some viscoelastic materials. By considering also the same hyperbolic law and applying the Bauer's theory, Monsia (2012) formulated for some materials, a rheological model to describe successfully their time dependent response under an applied constant load.…”