Predicting the Dynamic Behavior of Materials with a Nonlinear Modified Voigt Model

Abstract

In this paper, a one-dimensional nonlinear modified and extended Voigt model with constant material parameters is formulated to represent mathematically the time deformation behavior of a variety of viscoelastic materials. A binomial law is used as a nonlinear elastic force function. Numerical illustrations performed show that the hyperlogistic-type solution obtained is very useful to reproduce any S-shaped experimental curve.

Simulating Nonlinear Oscillations of Viscoelastically Damped Mechanical Systems

“…Very recently, the Bauer's rheological-dynamical theory was formulated in a simple mathematical expression that may be described by a single second order evolution equation within the framework of continuum mechanics for investigating the dynamics of viscoelastic material systems [12,13]. This formulation has successfully been applied in several papers to model creep deformation [13], creep relaxation [6] and deformation restoration process under stress relaxation conditions [14,15] of a variety of viscoelastic solid bodies.…”

“…Therefore, an infinite number of functions φ may be designed. In doing so, various types of mathematical expressions for the function φ have been proposed in recent papers [6,[12][13][14][15]22]. In general, as performed by Bauer [7], the stiffness nonlinearity function φ(u) may be expanded in a power series as: ... ... ) (…”

“…Hence, the problem of integrability of ( 13) in terms of elementary standard functions is unnecessary to be considered again in this paper. Some equations similar to (13) in terms of elementary functions have recently been solved on the basis of mathematical transformations leading to a Riccati equation or a damped linear harmonic equation [12][13][14][15]22]. In the present work, (13) will be turned in the damped linear harmonic oscillatory equation.…”

“…where ) (t y designates the general solution of ( 14) A mathematical analysis of (14) indicates that the nature of (15) solution depends on the relative magnitudes of the damping factor λ and the stiffness module, to say, the natural frequency ω ο . In other words, the nature of solution depends on the roots of characteristic equation: 14), that can be real or complex numbers.…”

“…Viscoelastic systems have the specific ability to experience large deformations even for moderate force levels. Hence, the linear theory becomes quite unable to explain the dynamics of viscoelastic systems experiencing finite deformations [6,7,[12][13][14][15] . Geometrical, damping and material nonlinearities are fundamental causes for the viscoelastic behavior of systems to be nonlinear [5].…”