A mathematical theory is developed through generalized Sundman transformation to show the existence of classes of quadratic Liénard type equations which admit exact and explicit general trigonometric solutions but with amplitude-dependent frequency. The application of the theory to compute also exact and explicit general periodic solutions to nonlinear differential equations like inverted Painlevé-Gambier equations in terms of trigonometric or Jacobian elliptic functions is highlighted by some illustrative examples.

In this paper, a simple nonlinear Maxwell model consisting of a nonlinear spring connected in series with a nonlinear dashpot obeying a power-law with constant material parameters, for representing successfully the time-dependent properties of a variety of viscoelastic materials, is proposed. Numerical examples are performed to illustrate the sensitivity of the model to material parameters

The paper is designed to provide a theoretical mechanical model taking into account elastic, viscous and inertial nonlinearities simultaneously, for predicting the time dependent response of some materials under a constant loading. Numerical illustrations are carried out to investigate the effects of rheological coefficients action on the model.

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.

<abstract><p>In the attractive research field of nonlinear differential equations, there are a few studies devoted to finding exact and explicit harmonic and isochronous periodic solutions and limit cycles. In this contribution, we present some classes of polynomial mixed Lienard-type differential equations that can generate many equations with exact solutions. These classes of equations constitute counterexamples of the classical existence theorems.</p></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.

In this work, the classical mechanical Voigt model is modified and extended to finite deformations by using a rational elastic spring force function to describe accurately the nonlinear time-dependent deformation response of some viscoelastic materials. As theoretical results, a hyperlogistic-type function has been found as the deformation versus time relationship. This growth model appeared powerful to reproduce mathematically as shown by numerical works, any S-shaped experimental data. Compared with some previous models, the present one-dimensional formulation gives the advantage to assure or to control via an explicit material parameter, to speak, via the coefficient of inertia, the nonlinearity of the model. The proposed model demonstrated then the importance to consider in the material modeling the inertial coefficient.

In this paper, we present an exceptional Lienard equation consisting of a modified Van der Pol-Helmholtz oscillator equation. The equation, a frequency-dependent damping oscillator, does not satisfy the classical existence theorems but, nevertheless, has an isochronous centre at the origin. We exhibit the exact and explicit general harmonic and isochronous solutions by using the first integral approach. The numerical results match very well analytical solutions.Mathematics Subject Classi cation (2010). 34A05, 34A12, 34A34, 34C25, 34C60.