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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.

To better understanding mechanical responses of materials subjected to deformations or forces, theoretical models are required. These models provide important analytical tools for predicting and simulating material functions. When a material is subjected to deformations, the resulting stress is related to the strain by a mathematical relationship known as the constitutive law. This constitutive equation can, thus, provide usefulness in determining rheological properties of materials. Since viscoelastic materials exhibit both combined viscous and elastic material behaviors, their constitutive equation must mathematically relate stress, strain and their time derivatives. For this purpose, these constitutive equations are often determined from combinations of springs and dashpots arranged in series and/or parallel [

In this study, a simple nonlinear Maxwell model (

In this part we describe the theoretical rheological model and derive the governing differential equation including the nonlinear restoring force and damping effects. Most viscoelastic materials are highly influenced by the nonlinear elastic and viscous damping terms so that, their rheological material properties are nonlinear timedependent. For this, a best description of these materials must proceed from the use of nonlinear theories. To build our proposed viscoelastic model, we start from the classical linear Maxwell model [

where and are the strains of the nonlinear spring and the nonlinear dashpot, respectively. η is the viscosity module, and E is the elasticity module. The dot denotes the time derivative and, m and n are nonlinearity parameters. By differentiations with respect to time and making appropriate substitutions, one can deduce from Equation (1) the constitutive differential equation

Equation (2) represents mathematically in the single differential form the relation between the total stress induced in the material under a strain history . This equation is a first-order nonlinear ordinary differential equation in for a given strain history when the numbers m and n are identically different from the number one.

If , and denote the mass, length and time dimension, respectively, the dimension of the stress varies as . The strain is a dimensionless quantity. Therefore, in Equation (2) the coefficient E possesses the same dimension with the stress, that of η varies as

.

We derive in this section the hyperlogistic-type solution allowing the description of the time-dependent stress induced in the material studied. For this, we consider that the material under consideration is subjected to a linear strain-path control, that is to say

where is the rate of application of the strain.

Thus, Equation (2) becomes

In order to solve Equation (4) we proceed to the following change of variable

or

Differentiating Equation (5) with respect to time yields

Substituting these relationships (Equation (6) and (7)) into Equation (4), the resulting equation becomes

Equation (8) is also a first-order ordinary differential equation in y, which can be solved analytically with the suitable boundary conditions of the mechanical problem considered in hyper-exponential or hyperlogistic-type function for special values of the exponent mn.

In this particular case where , Equation (8) becomes a simple linear first-order ordinary differential equation

which can be easily solved analytically using the initial condition

,

Thus, we can obtain as solution

From the Equation (6) we may deduce taking into account the Equation (10) the stress versus time as

Equation (11) gives the time variation of the stress in the viscoelastic material studied. It models the timedependent stress as a hyper-exponential function showing that the initial stress is different from zero. Moreover, Equation (11) predicts a stress that asymptotically approaches a maximum value with increasing time.

In this particular case where , Equation (8) becomes a first-order Riccati nonlinear ordinary differential equation [

which can be easily solved analytically using the initial condition

,

Therefore, we can obtain as solution

(13)

We can deduce from Equation (6) taking into consideration the above Equation (13) the stress versus time

(14)

Equation (14) describes the time variation of the stress in the viscoelastic material studied as a hyperlogistic-type function, which is powerful to reproduce any S-shaped curve [9,10].

In this part some numerical examples concerning the time-dependent stress are presented to illustrate the ability of the model to reproduce the mechanical response of the viscoelastic material studied. The dependence of the stress versus time curve on the material parameters is also discussed.

In

The stress curves at various values of the elasticity module E for the material under study are shown in

We observe from

or

with

and the slope reduces to zero. The red color corresponds to , the blue to , and the green to . The other parameters are , , , .

, , , , . It can be observed from

In

affects the maximum value of the stress. The graph shows that an increasing η, increases the maximum stress and increases also the time needed to attain the maximum stress. The red color corresponds to , the blue to , and the green to . The other parameters are , , , .

It can be observed from

The stress curves at various values of the nonlinearity parameter n for the material under consideration are shown in

We observe from

The previous numerical examples show that theoretical models are important tools for the prediction and simulation of viscoelastic behavior of materials. In this work, a simple nonlinear viscoelastic model is presented. The present model has been developed following two working hypothesis. The first postulates that the material properties can be divided into a nonlinear pure elastic component obeying a power-law and acting in series with a nonlinear damping element obeying a power-law and capturing the time-dependent deviation from the equilibrium state. The second hypothesis assumes that the material is subjected to a linear strain-path control. Under these restrictions, the model predicted the timedependent stress induced in the material as a hyperlogistic-type function, which is able to reproduce any S-shaped curve as shown by numerical examples. These predicted results by the proposed model are in very agreement with those published in the literature. The proposed model is an extension of the classical Maxwell model to large deformations by means of two parameters m and n. It appeared evident that for and , the present model reduces to the well-known Maxwell model. Consequently, these parameters assure the role of nonlinearity coefficients.

A complete characterization of viscoelastic materials is very difficult to perform, due to the fact that the mechanical response of these materials is time-dependent and history-dependent, and moreover, their stress-strain curve is nonlinear. Following this viewpoint, nonlinear theoretical models are necessary to better predict and understand the time-dependent behavior of materials. For this purpose, a nonlinear generalized Maxwell model has been developed. The model allowed, according to the obtained results, describing mathematically and accurately the nonlinear time-dependent stress in some viscoelastic materials, as a hyperlogistic-type function, that is powerful to represent any sigmoid curve. The present model, in particular, is shown to be very sensitive to the magnitude of the strain rate. Altogether, more experimental results and practical tests are needed to further validate the feasibility of this model.