**4. Experimental studies**

338 Viscoelasticity – From Theory to Biological Applications

**Figure 1.** Photo of a block

not airtight.

**3. Physical reality of material** 

apply the principles of the Continum Mechanics

into the cells, by compressibility or and by fleeing.

As the volume and the density are fixed, we put the necessary quantity of plastic in a parallelepiped shape press. After being compressed under high pressure (about 7 bar), and

After the manufacturing process which is at unidirectional pressing, the material is in the form of stacked almost identical layers. A certain volume of air is trapped inside the plastics sheets and forms more or less closed cells. This action gives lightness and makes the material alveolar.

In reality, the material is anisotropic and discontinuous, but macroscopically, it may be considered an orthotropic, continuous material with a revolution case whose axis coincides with the direction of pressing during manufacturing. (fig.2). In this case, we will be able to

On the other hand, the internal structure of the material is formed by a network of cells which are filled with air and have irregular geometric shapes. The flexible Walls of cells are

The global deformation of loaded material is the superposition of the proper deformation of the plastic sheets and the deformation due to the changing of the volume of air including

legated the resulting block stayed in shape after it is released. [1].(Fig.1)

The experimental studies of the behavior of the material began by compression and creep tests on full size blocks and completed by several kinds of tests on reduced size samples.

In the purpose to obtain a reduced size sample, tow apparatus were conceived and built. The first one enabled us to make cylindrical samples and the second one is conceived to produce cubic samples.

The dimensions of real blocks or samples are carefully chosen in order to neutralize the microscopic details and to be able to apply the Continum Mechanics principles.

In addition, two experimental machines are designed and constructed specifically for this type of material.

A first apparatus is intended to perform simple compression tests (Fig. 3). The load applied by a bar, was transmitted by a rigid plate that moves while remaining horizontal. This system that prevents the differential deformation is in the form of three rod lubricated to facilitate sliding.[6].

**Figure 3.** Apparatus for simple compression test

Secondly, a biaxial apparatus is developed (Fig. 4). In a direction, the applied load using the weight is transmitted through a rigid plate. In the other direction, the stress is measured by a dynamometer attached to another plate of the same nature. Its movement has approximately the same deformation of the ring. Other strains are measured by conventional means (comparator). The guidance system consists of four lubricated rigid rods.

Non Linear Viscoelastic Model Applied on Compressed Plastic Films for Light-Weight Embankment 341

After testing and verifying the proper functioning of both apparatus, wide open test

For all kind of test, real blocks and reduced samples of the same or different densities are tested according to a path of applied stresses step by step. This approach helps us to

Simple compression tests along the axis of revolution of the material are carried out on real size blocks as on samples of small size with different densities (0.4, 0.5 or 0.6). The stress is applied through stages. Each strain is maintained long enough to highlight the phenomenon

**Figure 5.** Deformation- time curve in logarithmic scale, under constant stress, initial density 0.4, loading


understand more the behavior of this discontinuous and anisotropic material.

**4.1. Testing** 

campaigns are made.


The carried out tests are :

of creep (Fig. 5, 6, 7) [2].

by steps. Real blocks size.

**4.2. Simple compression test (creep)** 

The choice of biaxial testing was justified by the inability to perform simple compression tests, at the same time in the directions (1) and (2). In fact, the material is in the form of a stack of sheets clamped by metal wires. The stress in the direction, parallel to the layers causes an increase in tension in the wires and the behavior of material will be managed by their rigidity and their number. From a mechanical point of view, the sample will be charged on certain places on time (in contact with the wire) and the deformation will not be uniform on both open sides [6].

**Figure 4.** Biaxial apparatus

#### **4.1. Testing**

340 Viscoelasticity – From Theory to Biological Applications

uniform on both open sides [6].

**Figure 4.** Biaxial apparatus

rods.

Secondly, a biaxial apparatus is developed (Fig. 4). In a direction, the applied load using the weight is transmitted through a rigid plate. In the other direction, the stress is measured by a dynamometer attached to another plate of the same nature. Its movement has approximately the same deformation of the ring. Other strains are measured by conventional means (comparator). The guidance system consists of four lubricated rigid

The choice of biaxial testing was justified by the inability to perform simple compression tests, at the same time in the directions (1) and (2). In fact, the material is in the form of a stack of sheets clamped by metal wires. The stress in the direction, parallel to the layers causes an increase in tension in the wires and the behavior of material will be managed by their rigidity and their number. From a mechanical point of view, the sample will be charged on certain places on time (in contact with the wire) and the deformation will not be After testing and verifying the proper functioning of both apparatus, wide open test campaigns are made.

The carried out tests are :


For all kind of test, real blocks and reduced samples of the same or different densities are tested according to a path of applied stresses step by step. This approach helps us to understand more the behavior of this discontinuous and anisotropic material.

#### **4.2. Simple compression test (creep)**

Simple compression tests along the axis of revolution of the material are carried out on real size blocks as on samples of small size with different densities (0.4, 0.5 or 0.6). The stress is applied through stages. Each strain is maintained long enough to highlight the phenomenon of creep (Fig. 5, 6, 7) [2].

**Figure 5.** Deformation- time curve in logarithmic scale, under constant stress, initial density 0.4, loading by steps. Real blocks size.

$$
\varepsilon\_3 = \varepsilon\_{0,3} + A\_3 \log(\frac{t}{t\_0})
$$

$$
\varepsilon\_2 = \varepsilon\_{0,2} + A\_2 \log(\frac{t}{t\_0})
$$

Non Linear Viscoelastic Model Applied on Compressed Plastic Films for Light-Weight Embankment 347

**Figure 14.** Instantaneous deformation (direction 2) – stress curves at constant initial density

**Figure 15.** Instantaneous deformation (direction 2) – initial density curves at constant stress

**Figure 12.** Stress (σ3) – Stress (σ2) for different densities

**Figure 13.** Deformation- time curve in logarithmic scale, under constant stress according to direction 2, initial density 0.32

**Figure 12.** Stress (σ3) – Stress (σ2) for different densities

initial density 0.32

**Figure 13.** Deformation- time curve in logarithmic scale, under constant stress according to direction 2,

**Figure 14.** Instantaneous deformation (direction 2) – stress curves at constant initial density

**Figure 15.** Instantaneous deformation (direction 2) – initial density curves at constant stress

Non Linear Viscoelastic Model Applied on Compressed Plastic Films for Light-Weight Embankment 349

**5. Rheological Behavior and Modeling in the case of mono dimensional** 

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

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

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

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

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

Because the material is assimilated to the alveolar one, the behavior is represented by the

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

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

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

For the material under study, the elastic limit was identified by simple compression tests

The variation of elastic limit (σe) of this material was identified experimentally and can be

**loading according to direction (3), perpendicular to the layers** 

complex phenomena (hardening, aging, accommodation...) (Fig. 7) [3].

resulting hardening.

simplifying assumptions.

Finally, the aging process can be ignored.

compressibility and evacuation (fig.10).


with constant deformation velocity (fig. 19).

assimilated by a straight line whose equation is:


The material is the subject of superposition of tow behaviors:

following model (fig. 18) [8]:

The assumptions are:

revolution.

cells.

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

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