# Modeling of uniaxial compression of fiber reinforcements using finite strains

Sébastien Comas-Cardona, Philippe Le Grognec, Christophe Binetruy, Patricia Krawczak

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Sébastien Comas-Cardona, Philippe Le Grognec, Christophe Binetruy, Patricia Krawczak. Modeling of uniaxial compression of fiber reinforcements using finite strains. 10th International Esaform Conference on Material Forming (ESAFORM 2007), Apr 2017, Zaragoza, Spain. pp.1064-1067, 10.1063/1.2729655. hal-01781208

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# Modeling of Uniaxial Compression of Fiber Reinforcements using Finite Strains

S. Comas-Cardona, P. Le Grognec, C. Binétruy and P. Krawczak

Polymers and Composites Technology & Mechanical Engineering Department
Ecole des Mines de Douai
941 rue Charles Bourseul, BP 10838
59508 Douai, France

**Abstract.** Liquid Composite Molding (LCM) processes are increasingly used to produce composite parts. Most of those processes involve compression of the fiber reinforcement and resin flow. In order to accurately model LCM processes, a good knowledge of fiber reinforcement behavior in compression is required. Several models have already been published, but none of them include permanent deformations. Also because of the large deformation involved in the processes, a finite strain formulation is proposed. Results are given for a glass twill-weave fabric.

**Keywords:** Fabrics, Unidirectional Compression, Finite Strain, Elastoplasticity

**PACS:** 81.05.Mh, Ni, Pj, Qk, 62.20 Fe, 81.05 Ni

## INTRODUCTION

An increasing number of thermosetting composite parts are produced using one of the Liquid Composite Molding (LCM) processes such as Resin Transfer Molding (RTM) or Vacuum Assisted Resin Transfer Molding (VARTM). For such molding, dry fiber reinforcements are placed within a mold or half mold, and then resin is either injected or infused through the fibers. Once the resin has cured, the part can be demolded.

During mold closing or vacuum application, the fiber reinforcements are compressed and their final thickness can be reduced by 50% with respect to their initial thickness. Those deformations are either forced (control of the thickness) or induced (control of the applied pressure). Moreover, those deformations have been shown to be partially permanent [1]. In order to maintain quality during manufacturing and control dimensional tolerances, a proper modeling of the fiber reinforcement in compression is required.

Previous work on the subject have shown the non-linear elasticity [2] and viscoelasticity [3] involved in the compression of such reinforcements. The measurement of permanent deformation has been shown to be measurable using load and unloading cycles [1]. However, no model or attempt to include permanent deformation during the compression of fiber reinforcements has been published.

Within this context, this study will aim at including permanent deformation using a non-linear elastoplastic law for uniaxial compression. Because of the large strains involved during manufacturing, a Lagrangian finite strain formulation is used. The parameters are determined from mechanical compression tests of dry and impregnated fiber reinforcement samples. Results are given for a glass twill-weave fabric.
## FINITE STRAIN FORMULATION

The following assumptions have been made when considering the compression of several plies of fiber reinforcements between parallel compression platens (FIG. 1):

* the in-plane (x, y) strains are negligible with respect to the one in the thickness (z) direction. The compression area of the samples is constant ($A_0$)

* the compression is homogeneous within the z direction

The deformation gradient tensor $F$ for a uniaxial compression in the z-direction can be expressed as follow:

$$F_{zz} = I_{xx} + \nabla u_{xx} = 1 + \alpha \quad (1)$$

where $I$ is the identity tensor, $u$ the displacement, $\alpha$ the relation shortening which is written: $\alpha = \Delta h/h_0$, where $h$ and $h_0$ are respectively the actual and initial sample thicknesses.

For a Lagrangian formulation, the constitutive law relates the material stress $\Sigma$ and the Green strain tensor $E$ which are:

$$\Sigma_{zz} = F_{zz}^{-1} \Pi_{xx} = \frac{-\lambda}{1 + \alpha} \qquad E_{zz} = \frac{1}{2} (F_{zz}^{T} F_{zz} - I_{zz}) = \alpha + \frac{\alpha^2}{2} \quad (2)$$

where $\lambda$ is the nominal stress applied on the fiber reinforcement sample and $\Pi$ the first Piola-Kirchhof stress.

<figure><img src="image_2.png" /><figcaption>FIGURE 1. Unidirectional compression geometry and displacement field.</figcaption></figure>

## NON-LINEAR ELASTIC-PLASTIC MODEL

*Loading.* During loading, in order to comply with the non-linearity observed in previous studies [2], the following constitutive law is chosen:

$$d\Sigma_{zz} = C(E_{zz})dE_{zz} \quad (3)$$

Note that $E_{zz}$ includes elastic and plastic deformation occurring during loading of the fiber reinforcements.
The elastic $C^e$ and hardening $H$ moduli can be expressed as $C^e = \frac{d\Sigma_{zz}}{dE_{zz}^e}$ and $H = \frac{d\Sigma_{zz}}{dE_{zz}^p}$

*Unloading.* A non-linear elastic behavior can be chosen during unloading of the fiber reinforcements. The constitutive equation is written as:

$$d\Sigma_{zz} = C^e(E_{zz}^e)dE_{zz} \quad (4)$$

## EXPERIMENTAL

Glass twill-weave reinforcement samples are submitted to compression at a speed of 0.5 mm/min. A sample, constituted of several dry plies, is inserted between the two platens. The stress $\Sigma_{zz}$ and strain $E_{zz}$ are calculated from the tensile test machine recordings (force $F$ and sample thickness $h$) and the sample geometry (compression area $A_0$).

$$\Sigma_{zz} = \frac{-\lambda}{1+\alpha} = \frac{-F}{A_0(1+\alpha)} \qquad E_{zz} = \alpha + \frac{\alpha^2}{2} \quad (5)$$

## RESULTS AND DISCUSSION

Different of compression tests can be carried out. First a regular compression (loading) test can be performed on the sample. Using that test combined with the proper methodology to measure plastic (permanent) deformation [4], one can extract the loading elastic tangent modulus using Eq. 3.

An other test for fabrics is based on compression loading followed by unloading. An example of the material's response is given in FIG. 2. Looking at the unloading steps in FIG. 2, considering it as non-linear elastic and using the plastic deformation involved, one can calculate the unloading elastic tangent modulus (FIG. 3). It is noteworthy that all unloading tangent moduli create a master curve. Finally, when comparing loading and unloading elastic tangent modulus curves, they both remarkably superimpose (FIG. 4).

<figure><img src="image_3.png" /><figcaption>FIGURE 2. Dry fiber reinforcement sample response to loading followed by unloading compressions.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>FIGURE 3. Elastic modulus during different unloading steps for the case of dry samples.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>FIGURE 4. Elastic modulus for loading and unloading steps for the case of dry samples.</figcaption></figure>

CONCLUSION

A non-linear elastic-plastic behavior has been proposed to model the uniaxial com-
pression of fiber reinforcements. That proposition is strengthened by the fact that both
loading and unloading elastic tangent moduli overlay. Further work is directed towards
the compression of thick samples, in which the assumption of homogenous through-
thickness deformation does not apply anymore.

REFERENCES

1. A. Somashekar, S. Bickerton, and D. Bhattacharyya, *Composites Part A* **37**, 858–867 (2006).

2. S. Toll, and J. Manson, *Proceedings of the FRC Conf., Newcastle upon Tyne, GB* pp. 25/1–25/10 (1994).

3. S. Bickerton, M. Buntain, and A. Somashekar, *Composites Part A* **34**, 431–444 (2003).

4. S. Comas-Cardona, P. Le Grognec, C. Binetruy, and P. Krawczak, *Composites Science and Technology*, *In press* (2007).