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 [1] in which we replace the linear elastic spring with a nonlinear elastic spring (with stiffness E) obeying a power-law and also the linear dashpot with a nonlinear dashpot (with viscosity η) obeying a power-law as shown in Figure 1. Thus, the mechanical properties of the considered material are divided into two parts: a nonlinear elastic element which captures the nonlinear pure elastic behavior of the material at equilibrium, acting in series with a nonlinear damping element capturing the time dependent history response of the material. From the mathematical point of view, the nonlinear stiffness and the nonlinear damping terms are included in a model in order to loss the linearity in the differential constitutive equation that represents the dynamic properties of the mechanical system studied. Due to the fact that the elements are in series the total stress   and the total strain  can be written as

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.

Figure 2 exhibits the typical time-dependent stress curve with an increasing until a peak asymptotical value, obtained from Equation (11) with the value of coefficients, , , , . It can be seen from Figure 2 that the model is capable to represent mathematically and accurately the typical exponential stiffening of some viscoelastic materials, for example, soft living tissues and soils, as shown in [3] and “in press” [4-8]. The model predicts a mechanical response in which the slope, after reaching its maximum value at the inflexion point, declines gradually with increase time until the failure point at which the slope reduces to zero.

Figure 3(a), (b), (c), (d) and (e) shows the effect of material parameters on the time-stress response. The effects of these parameters are studied by varying one coefficient while keeping the other four constant. Figure 3(a) illustrates how the rate of application of the strain affects the maximum value of the stress. The graph shows that an increasing, increases the maximum stress and the slope, and has no significant effect on the time required to reach the maximum stress and on the initial value of the stress. The red color corresponds to , the blue to , and the green to . The other parameters are , , , .

In Figure 3(b) is shown the dependence of the stress on the viscosity coefficient η. An increase , increases the peak stress and the time needed to reach it. The slope increases also. But an increasing η, has no important effect on the initial value of the stress. The red color corresponds to , the blue to , and the green to . The other parameters are , , , .

The stress curves at various values of the elasticity module E for the material under study are shown in Figure 3(c). An increasing E, greatly and fast increases the value of the stress on the time period considered. The slope increases with increase E. The red color corresponds to , the blue to , and the green to . The other parameters are , , , .

Figure 3(d) shows the sensitivity of the stress-time curve to the nonlinearity parameter n. An increasing nonlinearity parameter n, has a high effect on the stress value in the time period considered. Indeed, an increasing n, significantly and fast increases the value of the stress and also the initial value of the stress increases with increase n. The red color corresponds to , the blue to , and the green to . The other parameters are , , , .

We observe from Figure 3(e) that an increasing initial value, has a significant effect on the stress value in the time period considered. In fact, an increase, increases significantly and fast the stress value and also the initial value of the stress. An increasing, reduces the slope. The stress becomes constant, that is to say,

or

with

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

Figure 4 exhibits the typical time-dependent stress curve with an increasing until a peak asymptotical value, obtained from Equation (14) with the value of coefficients

, , , , . It can be observed from Figure 4 that the model is able to represent accurately the typical time-dependent stress curve of a variety of viscoelastic materials, as mentioned above, for example, soft living tissues and soils, as shown in [3] and “in press” [4-8]. The stress versus time curve is nonlinear, with a nonlinear beginning initial portion, and illustrates then the sigmoid mechanical behavior of the viscoelastic material considered. The plotting showing a nonlinear sigmoid behavior indicates, consequently, the material stiffening followed by softening. The model predicts a mechanical response in which the slope, after reaching its maximum value at the inflexion point, declines gradually with increase time until the failure point at which the slope reduces to zero.

In Figure 5(a) is shown the dependence of the stress versus time curve on the strain rate. The graph indicates that the peak stress increases with increasing . The slope increases also with increase the rate of application of the strain. But, an increasing, has no important effect on the time required to attain the peak stress. The red color corresponds to , the blue to , and the green to . The other parameters are , , , .

Figure 5(b) illustrates how the viscosity coefficient η

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 Figure 5(c) the dependence of the stress on the elasticity module E. An increase E, has a great influence on the stress value. Indeed, it increases fast in the early periods of time. Then, this influence decreases as time tends to infinity, and the curves tend towards the same asymptotic value of the stress. The slope increases with increase E. The red color corresponds to , the blue to , and the green to . The other parameters are , , , .

The stress curves at various values of the nonlinearity parameter n for the material under consideration are shown in Figure 5(d). An increasing n, has a high effect on the stress value. In fact, the stress increases fast in the early periods of time. Then, this effect decreases as time tends to infinity, and the curves tend towards the same asymptotic value of the stress. The nonlinearity of the initial portion of curves becomes less important with increasing n. The red color corresponds to , the blue to , and the green to . The other parameters are , , , .

We observe from Figure 5(e) that an increasing initial value , has an important effect on the stress value. Indeed, the stress increases fast in the early periods of time. Then, this influence decreases as time tends to infinity, and the curves tend towards the same asymptotic value of the stress. Moreover, an increasing , increases the initial value of the stress. The red color corresponds to , the blue to , and the green to . The other parameters are , , , .

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.