# Study of nonlinear creep behavior of Entandrophragma Cylindricum wood through Zener fractional rheological model

Loic Chrislin Nguedjio, Jeanne Sandrine Takam Mabekou, Rostand Moutou

Pitti, Benoît Blaysat, Pierre Kisito Talla

<figure><img src="image_1.png" /></figure>

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Loic Chrislin Nguedjio, Jeanne Sandrine Takam Mabekou, Rostand Moutou Pitti, Benoît Blaysat, Pierre Kisito Talla. Study of nonlinear creep behavior of Entandrophragma Cylindricum wood through Zener fractional rheological model. Annual Cobference of Society of experimental Mechanics, Jun 2024, Portland (Maine), United States. hal-04801516

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Submitted on 25 Nov 2024

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Study of nonlinear creep behavior of *Entandrophragma Cylindricum* wood through
Zener fractional rheological model

Loïc Chrislin NGUEDJIO¹, Sandrine NGAMGA MABEKOU¹, Rostand MOUTOU PITTI²,³ Benoit BLAYSAT², Pierre Kisito TALLA¹

¹Unité de Recherche de Mécanique et de Modélisation des Systèmes des Physiques (UR2MSP) ;
Département de Physiques, Faculté des Sciences, Université de Dschang, BP 67, Dschang, Cameroun

²Université Clermont Auvergne, Clermont Auvergne INP, CNRS, Institut Pascal, F-63000, Clermont-
Ferrand, France

³CENAREST, IRT, BP14070, Libreville, Gabon

ABSTRACT

Structures made of wood material face daily up to the creep problem. This recurring phenomenon affects the strength and rigidity of wood in service, leading to short-term and long-term deformation and sometimes the structure failure. To model and predict this phenomenon, classic rheological models are used, but their general creep formulation does not always guarantee an adequate description of the material's strain. In this context, the aim of the research work presented in this article is to study the nonlinear creep behavior of *Entandrophragma cylindricum* wood using the Zener rheological model based on fractional calculus. To achieve this objective, we developed the model's creep function and then used it to simulate the experimental data using a Levenberg-Marquardt optimization algorithm. The rheological parameters were also determined. The results of the numerical simulations show that the 3 parameter's Zener model fits the creep of *Entandrophragma cylindricum* wood with satisfactory correlations and prove that this model can be used in engineering as a model for predicting the deformations of this wood material, with an average reliability of 97%.

**Keywords:** Wood, creep, *Entandrophragma cylindricum*, fractional calculus, Zener model

INTRODUCTION

In recent years, the demand for wood materials in construction industry has grown considerably. Thanks to its low density, the wood material offers a better strength-weight ratio and its cellular structure makes it light, thus reducing the overall mass of the structure while keeping it very solid. Wood material has excellent characteristics which make it a non-disputable choice. According to a study carried out by the Passive House institute in 2015, 55% of buildings are certified to be built in wood material [1], which demonstrates the great potential of this material by giving it the place of best candidate against its competitors due to its mechanical, thermal and ecological qualities. Once in service in a structure, wood material is subject to a set of constraints (environmental constraints, external and internal loads, etc.) which push it to exhibit faulty behaviors such as creep [2], which causes damage, progressive ruin and breakage of the structural element. It is becoming urgent to find ways and means to control this harmful behavior in order to guarantee the safety of wooden structures. Rheological modeling is considered today as a solution that provides satisfactory results [3,4]. In addition, fractional rheological models have the advantage of well reconstructing the time history of deformations [5] during modeling while requiring fewer parameters compared to their classical counterparts [6]. The objective of this work is to model and predict the deformations of *Entandrophragma cylindricum* wood (Sapelli) used in structures. For this reason the fractional Zener model is proposed.
## BACKGROUND

The fractional Zener model (Figure 1) proposed in this study consists of a spring connected in series with the basic fractional Kelvin-Voigt model. Under constant stress, the creep function obtained for this model is given by the following Equation 1:

$$ \varepsilon(t) = \sigma \cdot a \cdot E_{\alpha,1}(-ab \cdot t^{\alpha}) + \sigma \cdot c \cdot t^{\alpha} \cdot E_{\alpha,\alpha+1}(-ab \cdot t^{\alpha}), \quad (1) $$

where $\frac{1}{E_0}$, $b = \frac{E_0 E_1}{v}$ and $c = \frac{E_0 + E_1}{v}$. $E_0$ and $E_1$ are the elastic moduli of the material, v is the fractional viscosity coefficient and $\alpha$ is the fractional derivative order [7,8].

<figure><img src="image_2.png" /><figcaption>Figure 1: Zener's three-parameter fractional rheological model.</figcaption></figure>

The model is validated after carrying out a series of creep tests on specimens of the material. In this work, two cylindrical specimens (20mm x 20mm x 360mm) of the same *Entandrophragma cylindricum* tree trunk were tested in static four-point bending. In an environment with climatic conditions of 24°C temperature and 65% relative humidity, this test consists in applying a stress to a specimen, keeping it constant and measuring deformations over time (Figure 2) [9,10]. The result is a set of experimental points representing the material's creep data (Figure 3 and 4).

<figure><img src="image_3.png" /><figcaption>Figure 2: Test piece loaded in four-point bending</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3: Test piece loaded in four-point bending: specimens 1.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4: Test piece loaded in four-point bending: specimens 2.</figcaption></figure>

Under the two stress levels mentioned above, it can be seen from the curves in (Figure 3 and 4) that nonlinear creep in Sapelli occurs in three stages:

- Firstly, instantaneous elastic deformation of the material immediately after loading;

- Secondly, primary creep, which lasts for around 100 minutes and is characterized by a decrease in strain rate;

- And finally, secondary creep, which lasts until time t = 600 minutes, marking the end of the experiment. This is the longest stage and is characterized by deformation at almost constant speed.

It should be noted that the third creep phase (tertiary creep) is not represented here, as the stress levels used are still low compared with the material's failure stress.

The preceding experimental data are then driven by the model of Equation 1 using the Levenberg-Marquardt optimization algorithm. At the end of the simulations, the optimal rheological parameters of the model E, α and v are determined and the prediction curves superimposed on the experimental curves.

## ANALYSIS

The following figure 5 and 6 shows us the adjustments obtained.
<figure><img src="image_6.png" /><figcaption>Figure 5: Experimental creep curves: specimens 1.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Figure 6: Experimental creep curves: specimens 2.</figcaption></figure>

Figure 5 and 6 shows that the proposed model fits well with the first two stages (primary and secondary creep) of Sapelli wood. Nevertheless, we can observe experimental points that were not covered by the model, which may be due to the difficulty of keeping environmental conditions constant throughout the test. All in all, Zener's fractional model can be used to predict Sapelli creep with an average reliability rate of 97%.

In addition, the values of the optimal parameters determined by the algorithm are presented in (Table 1), where we can confirm the flexibility of Sapelli wood, as deformations increase with stress.

<table><caption>Table 1: Rheological parameters of the fractional Zener model.</caption><thead><tr><td>σ(MPa)</td><td>E<sub>0</sub>(10<sup>4</sup>MPa)</td><td>E<sub>1</sub>(10<sup>5</sup>MPa)</td><td>ν(10<sup>9</sup>MPa.min<sup>α</sup>)</td><td>α</td></tr></thead><tbody><tr><td>43,9</td><td>2,85</td><td>2,79</td><td>1,049</td><td>0,451</td></tr><tr><td>22,7</td><td>1,48</td><td>1,47</td><td>0,5636</td><td>0,446</td></tr></tbody></table>
It can be seen from this table that the higher the stress level, the greater the initial deformation and virtually all the values of deformation in the quasi-reversible zone, thus characterizing the softness of the sample. On the other hand, increasing the fractional order $\alpha$ proportionally decreases the strain values and this is due to the fact that when $\alpha$ is large, the fractional viscosity coefficient is also large, which reduces slippage at joints within the material.

## CONCLUSION

In this work, the non-linear creep of *Entandrophragma cylindricum* (Sapelli) wood was studied using an approach based on the theory of fractional calculus. The study showed that the proposed three-parameter Zener fractional model fits the creep of Sapelli with satisfactory correlations. This can therefore be used in engineering as a model for predicting the deformations of this wood material, with an average reliability of 97%. Future research will focus on studying this behavior at higher stress levels, taking into account fluctuations in environmental conditions. Tests will need to be conducted over a longer period to study the tertiary phase of creep and model behavior at the point of failure.

## ACKNOWLEDGEMENTS

The authors would like to thank the Mechanics and Modeling of Physical Systems Research Unit at the University of Dschang for providing the bending facilities.

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