**5. Rheological Behavior and Modeling in the case of mono dimensional loading according to direction (3), perpendicular to the layers**

Test campaigns on real blocks and small size samples [2] are able to show that the viscoelastic behavior is nonlinear (Fig. 5). Moreover, it is the subject of several rheological complex phenomena (hardening, aging, accommodation...) (Fig. 7) [3].

Initial density is an important parameter which influences behavior. In fact, the more great it is the more the behavior improves. The same density changes during the solicitation resulting hardening.

At first glance, the complex reality of the material makes it difficult to apply principles of mechanics of continuous materials. This is made possible by the adoption of some simplifying assumptions.

The assumptions are:

348 Viscoelasticity – From Theory to Biological Applications

**Figure 16.** Slope A2 (direction 2) – stress curves at constant initial density

**Figure 17.** Initial residual stress (3) - initial density curve

First, we assume that there is no slippage or detachment between the elementary leaves and the volume used is large enough to be able to erase the influence of microscopic details. Then, it is considered that the material is continuous, homogeneous and orthotropic revolution.

Finally, the aging process can be ignored.

On the other hand, specific plastic deformation is negligible compared to the overall deformation of material due to the change in volume of air trapped into the cells, or by compressibility and evacuation (fig.10).

Because the material is assimilated to the alveolar one, the behavior is represented by the following model (fig. 18) [8]:

The material is the subject of superposition of tow behaviors:


In the case of alveolar material with open cells, the behavior of gas does not take place.

When the stress is applied quickly (case of variable stress or imposed deformation velocity), the material can be considered as alveolar with closed cells. But, when it is maintained constant for a long time (case of creep), the material can be considered as alveolar with open cells.

In conclusion, the alveolar material is characterized by an elastic limit σe.

For the material under study, the elastic limit was identified by simple compression tests with constant deformation velocity (fig. 19).

The variation of elastic limit (σe) of this material was identified experimentally and can be assimilated by a straight line whose equation is:

$$\begin{aligned} \varepsilon &= \varepsilon\_1 + A \log(\frac{t}{t\_0}) \\\\ \varepsilon &= \varepsilon\_1 + \varepsilon\_2 \\\\ \sigma &= E\_1 \varepsilon\_1 = \eta \varepsilon\_2 \\\\ \varepsilon^\bullet &= \frac{d\varepsilon}{dt} \quad \& \quad \sigma = \frac{d\sigma}{dt} \end{aligned}$$

$$\mathcal{E}^{\bullet} = \left(\frac{\mathbf{d}\_{\mathrm{p}} \cdot d\_{0}}{d\_{0}^{\mathrm{n}}}\right) \left(\frac{1}{P}\right) \sigma^{\bullet} + A \exp\left[\frac{\cdot \left(\boldsymbol{\varepsilon} \cdot \boldsymbol{\varepsilon}\_{0}\right)}{\mathbf{A}}\right]$$

$$\mathcal{E}\_0 := \left(\frac{\mathbf{d}\_p \cdot d\_0}{d\_0^n}\right) \left(\frac{1}{P}\right) \sigma^2$$

$$\mathbf{A} = \left(\frac{\mathbf{d}\_p \cdot d\_0}{d\_0^m}\right) \left(\frac{1}{P}\right) \sigma$$

Non Linear Viscoelastic Model Applied on Compressed Plastic Films for Light-Weight Embankment 353

p 0

*d d P*

*n p*

0 : instantaneous deformation

: unidirectional deformation velocity A,n,m : parameters to be identified

The comparison of theoretical and experimental results becomes possible by numerical resolution of the differential equation in its three forms. The concordance between reality and theory is good and the simplification gives us reasons of satisfaction.(fig.20, fig.21,

 

d - 1

d0 : initial density

The experimental results enabled us to identify all necessary parameters.

**Figure 20.** Non linear model, compression with constant deformation velocity

The hypothesis of non-aging behavior is adopted. The relation "stress-strain" in case of

 

*F*

**6. Three-dimensional model, constant stress** 

constant stress and small strain is represented by:

0

**5.4. Validation** 

fig.22, fig.23) [1],[3],[7].

0 0 0 0 0 0 P 0 n a d b m a d b , , , : parameters to be identify by experience P : atmpspheric pressure d : proper density of plastic d : initial density *aba b* 

#### **5.2. Nonlinear model with hardening**

Until now, the initial density is taken as constant. To be able to take into account the hardening phenomenon, we will introduce the density as a function according to the strain. Necessary analysis and calculation are done. The new version of the model is presented a differential equation as the following [1]:

$$\begin{aligned} \boldsymbol{\sigma}^{\bullet} &= \left( \frac{\mathbf{d}\_{\mathrm{p}} \cdot \boldsymbol{\alpha}}{\boldsymbol{\alpha}^{\mathrm{a}\boldsymbol{\alpha} + \mathbf{b}}} \right) \bigg( \frac{1}{P} \right) \boldsymbol{\sigma}^{\bullet} + \left( \frac{\mathbf{d}\_{\mathrm{p}} \cdot \boldsymbol{\alpha}}{\boldsymbol{\alpha}^{\mathrm{a}\_{1}\boldsymbol{\alpha} + \mathbf{b}\_{1}}} \right) \exp\left[ \frac{\cdot \left( \boldsymbol{\varepsilon} \cdot \boldsymbol{\varepsilon}\_{0} \right)}{\mathbf{A}} \right] \quad & \boldsymbol{\varepsilon}\_{0} = \left( \frac{\mathbf{d}\_{\mathrm{p}} \cdot \boldsymbol{\alpha}}{\boldsymbol{\alpha}^{\mathrm{a}\boldsymbol{\alpha} + \mathbf{b}}} \right) \bigg( \frac{1}{P} \right) \boldsymbol{\sigma}^{\mathrm{a}} \\\\ & \mathbf{A} = \left( \frac{\mathbf{d}\_{\mathrm{p}} \cdot \boldsymbol{\alpha}}{\boldsymbol{\alpha}^{\mathrm{a}\_{1}\boldsymbol{\alpha} + \mathbf{b}\_{1}}} \right) \bigg( \frac{1}{P} \right) \boldsymbol{\sigma} \quad ; \quad \boldsymbol{\alpha} = \frac{\mathbf{d}\_{0}}{1 \cdot \boldsymbol{\alpha}} \end{aligned}$$
  $d\boldsymbol{p} : \text{proper density of plastic}$ 

0 d : initial density

#### **5.3. Linearized model with no hardening**

The simplification of the nonlinear model is carried out in two ways [7]:

Adoption of linear forms of the elastic and the delayed deformations according to the stress.

Adoption of linear forms of the elastic and a constant delayed one according to the stress. In another word, the linear viscoelasticity is applied.

In any case, the initial density is taken as a parameter and the model is presented as the following [5]:

$$
\varepsilon^{\bullet} = \left(\frac{\mathbf{d\_p} \cdot d\_0}{d\_p^{\bullet}}\right) \left(\frac{1}{P}\right) \sigma^{\bullet} + A \exp\left[\frac{\cdot \left(\varepsilon \cdot \varepsilon\_0\right)}{\mathbf{A}}\right],
$$

$$
\varepsilon\_0 = \left(\frac{\mathbf{d}\_p \cdot d\_0}{d\_p^n}\right) \left(\frac{1}{P}\right) \sigma
$$

0 : instantaneous deformation : unidirectional deformation velocity A,n,m : parameters to be identified d0 : initial density 

The experimental results enabled us to identify all necessary parameters.

#### **5.4. Validation**

352 Viscoelasticity – From Theory to Biological Applications

0 0 0 0

d : initial density

a b a b

0

**5.3. Linearized model with no hardening** 

another word, the linear viscoelasticity is applied.

*dp*

 

P : atmpspheric pressure d : proper density of plastic

0 0

*aba b*

n a d b m a d b

 

P 0

**5.2. Nonlinear model with hardening** 

differential equation as the following [1]:

stress.

following [5]:

p 0 0 d - 1 <sup>A</sup>*<sup>m</sup> d d P*

, , , : parameters to be identify by experience

Until now, the initial density is taken as constant. To be able to take into account the hardening phenomenon, we will introduce the density as a function according to the strain. Necessary analysis and calculation are done. The new version of the model is presented a

> 1 1 p p 0

 

d - 1 d - - - exp

 *P* A

1 1

a b

d : initial density

The simplification of the nonlinear model is carried out in two ways [7]:

*p*

*d*

*d P*

 

p 0 a b d - 1 *<sup>P</sup>* 

 

 

p 0

d - 1 <sup>d</sup> A ; 1 -

Adoption of linear forms of the elastic and the delayed deformations according to the

Adoption of linear forms of the elastic and a constant delayed one according to the stress. In

In any case, the initial density is taken as a parameter and the model is presented as the

 

p 0 <sup>0</sup> d - 1 - - exp *<sup>n</sup>* <sup>A</sup>

*A*

 

*P*

: proper density of plastic

The comparison of theoretical and experimental results becomes possible by numerical resolution of the differential equation in its three forms. The concordance between reality and theory is good and the simplification gives us reasons of satisfaction.(fig.20, fig.21, fig.22, fig.23) [1],[3],[7].

**Figure 20.** Non linear model, compression with constant deformation velocity

#### **6. Three-dimensional model, constant stress**

The hypothesis of non-aging behavior is adopted. The relation "stress-strain" in case of constant stress and small strain is represented by:

$$
\stackrel{\rightarrow}{\sigma}\_{\sigma} = F \stackrel{\rightarrow}{\otimes}\_{\mathcal{E}}
$$

Non Linear Viscoelastic Model Applied on Compressed Plastic Films for Light-Weight Embankment 355

F is a matrix composed by f1, f12, f3, f13, f4, five functions to be identified. They are a function of time, stress and initial density. The role of the stress perpendicular to the layers (σ3) is paramount. It marks the nonlinearity of the behavior. For the other stresses, the linearization

> 11 1 12 13 11 22 12 1 13 22 33 13 13 3 33 12 1 12 12 23 4 23 13 4 13

*ff f f ff ff f*

The identification of the functions is carried out from experimental results:

biaxial testing.

f4 is equal to <sup>ଵ</sup>

(3)....

ீଵଷ

0 00 0 0 0 00

*f f*

000 0 0 000 0 0

f1 and f13 identified by performing tests of simple compression along the axis of revolution.

f2 and f12 are determined by simple compression tests according to the two other axes or by

identifiable by a distortion test. Without this test, it can be approached by assuming that the material is isotropic respectively with the characteristics specified in the directions (1) and

0 m

*p <sup>B</sup> d d*

> 13 3 3 f *f*

Pa <sup>t</sup> <sup>1</sup> 1 1 ln Pa <sup>t</sup> <sup>d</sup>

After necessary calculation, the characteristic functions take the following forms [6]:

*<sup>d</sup> B A*

33

p

Pa

d

3

*f*

0 00 0 00

*f*

where G13 is the shear modulus around the directions (1) and (2). It is

0 33 33 0 0

33

Pa

*f*

 

 

 

 

 

 

of the behavior is adopted with a quite acceptable accuracy [6].

**Figure 21.** Non linear model, creep tests by stages

**Figure 22.** Non linear hardening model, settlement of embankment.

**Figure 23.** Linearized model. Measured and calculated deformations for an initial density: 0.493, creep tests by stages.

F is a matrix composed by f1, f12, f3, f13, f4, five functions to be identified. They are a function of time, stress and initial density. The role of the stress perpendicular to the layers (σ3) is paramount. It marks the nonlinearity of the behavior. For the other stresses, the linearization of the behavior is adopted with a quite acceptable accuracy [6].

354 Viscoelasticity – From Theory to Biological Applications

**Figure 21.** Non linear model, creep tests by stages

**Figure 22.** Non linear hardening model, settlement of embankment.

tests by stages.

**Figure 23.** Linearized model. Measured and calculated deformations for an initial density: 0.493, creep

The identification of the functions is carried out from experimental results:

f1 and f13 identified by performing tests of simple compression along the axis of revolution.

f2 and f12 are determined by simple compression tests according to the two other axes or by biaxial testing.

f4 is equal to <sup>ଵ</sup> ீଵଷ where G13 is the shear modulus around the directions (1) and (2). It is identifiable by a distortion test. Without this test, it can be approached by assuming that the material is isotropic respectively with the characteristics specified in the directions (1) and (3)....

After necessary calculation, the characteristic functions take the following forms [6]:

$$f\_3 = \frac{1 - \frac{\mathbf{d}\_0 \left(\frac{\sigma\_{33}}{\mathbf{Pa}} + B\right)}{\mathbf{d}\_p \left(\frac{\sigma\_{33}}{\mathbf{Pa}} + B\right) - A} + \frac{d\_p - d\_0}{d\_0} \left[ \left(\frac{\sigma\_{33}}{\mathbf{Pa}} + 1\right)^{\mathbf{m}} - 1 \right] \left[ \ln \left(\frac{\mathbf{t}}{\mathbf{t}\_0}\right) \right]}{\frac{\sigma\_{33}}{\mathbf{Pa}}}$$

13 3 3 f *f*

$$\begin{aligned} f\_1 &= \left(d\_p - d\_0\right) \left[\frac{1}{d\_0^k} + \frac{1}{d\_0^l} \ln\left(\frac{t}{t\_0}\right)\right] + \\ a\_3^2 f\_3. \quad f\_{12} &= \alpha\_1 \left(d\_p - d\_0\right) \left[\frac{1}{d\_0^k} + \frac{1}{d\_0^l} \ln\left(\frac{t}{t\_0}\right)\right] + \\ a\_3^2 f\_3. f\_4 &= \frac{f\_3 \left(1 - \alpha\_3\right) + K \left(1 - \alpha\_3\right)}{2}; \\ \mathbf{K} &= \left(d\_p - d\_0\right) \left[\frac{1}{d\_0^k} + \frac{1}{d\_0^l} \ln\left(\frac{t}{t\_0}\right)\right] \end{aligned}$$

Non Linear Viscoelastic Model Applied on Compressed Plastic Films for Light-Weight Embankment 357

The development of mono axial and biaxial testing, the design and implementation of an appropriate apparatus helped us to perform many tests. Therefore, the characteristic functions of the material were determined satisfactorily. This allowed us to identify the

In a particular field of stress up to 200 kPa, the simplified model gives good satisfaction. The results are acceptable and the difference between theoretical and experimental ones does not overstep the bounds of uncertainty both in physical measurement and caracteristics

However, in several cases of application, the model in its many forms helps us to conceive and design structures by using this technique. Indeed, many projects were conceived and

*Lebanese University, University Institute of Technology (Saïda, Lebanon), Civil Engineering* 

[1] El Ghoche H., Boudissa M., Coulet C. - " Validation d'un Modèle Viscoélastique Appliqué à des Chantiers de Remblais Allégés En Blocs de Matières Plastiques Compressibles." ANNALES de l'Institut Technique du Bâtiment et des Travaux Publics.

[2] El Ghoche H., Coulet C., Gielly J. - "Modèle Viscoélastique Non Linéaire Ecrouissable pour des Blocs Compressibles de Matières Plastiques. Huitièmes Rencontres du Centre Jacques Cartier : Elasticité, Viscoélasticité et Contrôle Optimal. Aspects Théoriques et

[3] El Ghoche H..- " Modèle Viscoélastique Non Linéaire Ecrouissable Pour des Blocs Compressible De Matières Plastiques". Première Conférence Syrio-Libanaise de

[5] El Ghoche H, J. Alhajjar, Kh. Mochaorab. - "Géotechnique et Environnement : Substitution des Sols". Deuxième Partie Conseil National de la Recherche Scientifique-

[6] El Ghoche H - "Conception and Construction of Biaxial Apparatus for Identification of axial Orthotropic Material". Advanced Materials Research Vol. 324 (2011), pp 368-371.

Numériques. Université Claude Bernard. Lyon, France - 6 - 8 Décembre 1995

l'Ingénierie (CSLI-1). 13-14 octobre 1999. Université de Damas. Damas, Syrie. [4] El Ghoche H. - "Modélisation d'un matériau Alvéolaire. Application en Génie Civil." , Troisième Colloque Franco-Libanais Sur Les Sciences des Matériaux (CSM3), 16 - 18

essential parameters to the theoretical models developed.

Revue Technique et Scientifique. Septembre 1995.

Mai 2002 – Beyrouth Liban.

Liban, Juin 2004. 166 pages.

variability due to the material anisotropy.

realized in France (fig.24).

**Author details** 

Hayssam El Ghoche

*Department, Liban* 

**8. References** 

**7. Conclusion** 

Given the approximation of 4 *f* , we can neglect <sup>1</sup> and <sup>3</sup> by comparison to (1) and 4 *f* becomes:

$$f\_4 = \frac{\left(f\_3 + K\right)}{2},$$

t: time, t0: initial time, dp: density of plastic d0: initial density A, B, k, l: parameters depending on the initial density

**Figure 24.** One of realized projects in France
