# Formulation and Numerical Verification of a New Rheological Model for Creep Behavior of Tropical Wood Species Based on Modified Variable-Order Fractional Element

Loic Chrislin Nguedjio, Jeanne Sandrine, Mabekou Takam, Benoit Blaysat, Pierre Kisito Talla, Rostand Moutou Pitti

► To cite this version:

Loic Chrislin Nguedjio, Jeanne Sandrine, Mabekou Takam, Benoit Blaysat, Pierre Kisito Talla, et al. Formulation and Numerical Verification of a New Rheological Model for Creep Behavior of Tropical Wood Species Based on Modified Variable-Order Fractional Element. Forests, 2025, 16 (5), pp.824. 10.3390/f16050824. [hal-05069274](https://hal.science/hal-05069274v1)

HAL Id: hal-05069274

https://hal.science/hal-05069274v1

Submitted on 15 May 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

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

Distributed under a Creative Commons Attribution 4.0 International License
# Formulation and numerical verification of a new rheological model for creep behavior of tropical wood species based on modified variable-order fractional element

Loic Chrislin Nguedjio<sup>1,2* ID</sup>, Jeanne Sandrine Mabekou Takam<sup>2</sup>, Benoit Blaysat<sup>1,3 ID</sup>, Pierre Kisito Talla<sup>2</sup> and Rostand Moutou Pitti<sup>1,4 ID</sup>

**Abstract:** This paper aims to develop a rheological model with fewer parameters that accurately describes the primary and secondary creep behavior of wood materials. The models studied are grounded in Riemann-Liouville fractional calculus theory. A comparison was conducted between the constant-order fractional Zener model and the variable-order fractional Maxwell model, with four parameters each. Using experimental creep data from four-point bending tests on two tropical wood species, along with an optimization algorithm, the variable-order fractional model demonstrated greater effectiveness. The selected fractional derivative order, modelled as a linearly increasing function of time, helped to elucidate the internal mechanisms in the wood structure during creep tests. Analyzing the parameters of this order function enabled an interpretation of their physical meanings, showing a direct link to the material's mechanical properties. The Sobol indices have demonstrated that the slope of this function is the most influential factor in determining the model's behavior. Furthermore, to enhance descriptive performance, this model was adjusted by incorporating stress nonlinearity to account for the effects of the variation of constant loading level in wood. Consequently, this new formulation of rheological models, based on variable-order fractional derivatives, not only allows for a satisfactory simulation of the primary and secondary creep of wood, but also provides deeper insights into the mechanisms driving the viscoelastic behavior of this material.

**Keywords:** Tropical species; fractional calculus; viscoelastic behavior; variable order; Maxwell model; Zener model.

## 1. Introduction

Wood material is known for its complex and delayed behavior, which is difficult to understand due to its heterogeneous, anisotropic nature, and particularly its physical and mechanical properties that vary depending on the conditions it is subjected to[1–3]. These intrinsic properties that complicate the prediction of its mechanical behavior under different environmental and mechanical stresses limit its use. Among the various behaviors that wood exhibits, viscoelastic behaviors such as creep are observed, which generally occur when the wood material responds to a sustained load over time [4–7]. Understanding

<footnote n="1">¹ Université Clermont Auvergne, Clermont Auvergne INP, CNRS, Institut Pascal, Clermont-Ferrand, F-63000, France; loic.nguedjio@doctorant.uca.fr, benoit.blaysat@uca.fr, rostand.moutou_pitti@uca.fr</footnote>

<footnote n="2">² Mechanics and Modeling of Physical Systems Research Unit (UR-2MSP), Department of Physics, Faculty of Sciences, University of Dschang, Dschang, BP 067, Cameroon; loic.nguedjio@doctorant.uca.fr, sandrinengamga@yahoo.com, tpierrekisito@yahoo.com</footnote>

<footnote n="3">³ Institut Universitaire de France, Paris, France; benoit.blaysat@uca.fr</footnote>

<footnote n="4">⁴ CENAREST, IRT, Libreville, BP14070, Gabon; rostand.moutou_pitti@uca.fr</footnote>

<footnote>* Correspondence: loic.nguedjio@doctorant.uca.fr; Tel.: +33-07-51-54-76-00</footnote>
and modelling this behavior remains a major goal for materials science to achieve. In this regard, several researchers have modelled the viscoelastic behavior of wood using empirical models, classical rheological models, and fractional rheological models.

One of the most widely used empirical models in the literature is Findley's model. Zhao *et al.* [8] used this model to study the creep of wood plastic composites under tensile stress, and numerical simulations showed a good correlation between experimental data and the model. Zhu *et al.* [9] found similar results with this model while studying the creep of wood fiber-based polymer composites; the success of this model is also supported by several other authors [10–13]. Despite the good correlation between creep experimental data and Findley's empirical model, the major issue remains that the model only applies to very low stress levels. Classical rheological models emerged to address this gap. The classical rheological Burger model is used by Ma *et al.* [14] to study the creep properties of *Eucalyptus* wood, and by Lagaña *et al.* [15] to study the creep under bending of Spruce wood. The results obtained show that this four-parameter model can describe wood creep even at very high temperatures. The performance of this model is also recognized for simulations of wood material creep in various contexts, including tensile-compression and variable environments[16–19]. There are also modified versions of the classical rheological Burger model that are inspired by the original formulation. Shimazaki *et al.* [20] replaced the dashpots of the original model with non-Newtonian dashpots to describe the creep of wood under shear stress, achieving good correlations with experimental compression data on Japanese Hinoki cypress samples. Sometimes, classical models require the combination of a high number of elements to represent certain behaviors of wood, which leads to generalized classical rheological models. Vidal-Sallé *et al.* [21] propose a generalized Maxwell model to analyze the nonlinear response and mechano-sorptive creep of wood, with the model's performance demonstrated through finite element simulations. The generalized classical Kelvin-Voigt model is used by Dubois *et al.* [22] to describe wood behavior under various mechanical and environmental conditions. The proposed model performs well following a creep-recovery tensile test campaign and is validated by a series of finite element simulations. Kazemi *et al.* [23] demonstrate that a generalized 5-parameter Kelvin-Voigt model can effectively capture the biaxial deformations of steel specimens subjected to temperature variations, supporting the reliability of classical rheological models for representing viscoelastic behavior.

With this last category of rheological models, which require a considerable number of parameters, it becomes numerically challenging to determine all these parameters, coupled with the industrial costs of design. From this perspective, classical rheological models become obsolete for certain types of modelling. Additionally, classical rheological models are unable to accurately represent a significant part of the memory effect of wood under stress [24], a gap that seems to be filled by new approaches based on fractional calculus[25,26]. Thanks to their strong memory effect and their ability to reduce the number of parameters, fractional rheological models have become essential in viscoelasticity [27], as demonstrated by Bonfanti *et al.* [28]. Comparing the classical Kelvin-Voigt model with 8 parameters to the fractional Thomson model with 3 parameters, Nguedjio *et al.*[29] show significant numerical and rheological gains in modelling tropical *Sapelli* wood. Their study reveals that the fractional Thomson model describes the creep of this tropical wood species better than the classical model with up to 8 parameters, with deformation history also being well represented by the fractional model. Atchounga *et al.* [30] use the fractional Zener model to study constant bending creep of the tropical *Wengé* wood and show that the model fits well with experimental data. Foadieng *et al.* [31] opt for the fractional Maxwell model to describe the creep of tropical *Assamela* wood. By comparing experimental data obtained from four-point bending to this model, they achieve satisfactory correlations. Nguedjio et al.
[32] also studied the coupled creep-recovery viscoelastic behavior of Sapelli wood, showing that the fractional Maxwell model is best suited for simulating these behaviors. Indeed, their paper makes an effort to enhance the predictive performance of fractional rheological models, with the Maxwell model proving the least affected by the mechanical variability of this wood species. Long et al. [33] affirm that previous fractional models effectively capture viscoelastic behavior. After studying fractional models modified with exponential and logarithmic kernels, they conclude that these modified versions are generally less effective than the basic fractional models. However, despite the remarkable performance of these fractional models, which allow us to determine a fractional viscoelastic coefficient characterizing the wood's mechanical properties, we still lack information on how this mechanical properties evolve over time.

To enhance the ability of fractional models to describe the viscoelastic behavior of composite materials, a new formulation of fractional calculus has been introduced, called variable order fractional calculus [34,35]. Previously described fractional models are based on constant-order fractional calculus, and the introduction of variable order is expected to better represent the behavior of materials that change significantly over time or under different loading conditions, including wood. Although the application of this new fractional modelling formulation is still poorly documented, especially for wood, other materials exhibiting complex viscoelastic behavior have already been studied using this method. Xiang et al. [36] used the variable-order Maxwell fractional model to study the creep of fibrous polymer composites. They postulated that the fractional order is a linear function of time and demonstrated that, in addition to its strong memory effect, this variable order reflects the changing mechanical properties of the composite during the creep phenomenon. This linear form of fractional order has been confirmed as they studied the relaxation of glassy polymers [37], comparing exponential and sinusoidal order functions to the linear form, with the latter yielding the best results. Yuxiao et al. [38] show that variable-order fractional models offer advantages over other models in describing changes over time in the complex behavior of various systems, a conclusion also supported by Zhou et al. [39], who studied soil creep, and Meng et al. [40], who investigated the viscoelastic behavior of polymer composites.

In this article, constant-order and variable-order fractional models are used to study the primary and secondary creep of tropical wood species. The variable order model was further adapted by incorporating a nonlinear element to account for the effect of stress level, resulting in a new variable-order fractional model suited for studying creep in tropical wood species. Following this introduction, we will present the experimental method as well as a brief detail of the calculation method for the creep function of the model. Then, the results of the numerical simulations will be presented and discussed, and we will conclude with a summary and future perspectives.

## 2. Materials and Methods

### 2.1. *Materials and Experimental setup*

Two species of wood from the tropical forests of Cameroon were selected for this study: Sapelli (Entandrophragma cylindricum) and Wengé (Millettia laurentii). The first species is a lightweight wood with a constant bending strength (MOR) of 102 MPa at 12% relative humidity and an average density of 0.67 kg per cubic meter [41], while the second species is stiffer, with an MOR of up to 144 MPa and average density of 0.78 kg per cubic meter [42]. The samples were taken from the same tree trunk of each species in the southern Cameroon forest. They were then cut into test specimens with standardized dimensions according to the French standard NF B 51-008 [43]: a consistent length of 360 mm and a square cross section of 20 mm x 20 mm, sequentially taken from the sapwood.
These samples were stored in the laboratory for three weeks under ambient conditions, approximately 20 °C ± 2 °C and 65% ± 5% relative humidity and the tests were performed in the same climatic conditions. The test room is kept closed during the experiments, as the models used for subsequent numerical simulations are intended for creep under constant mechanical and climatic conditions. At the end of conditioning period, the samples were weighed to ensure a uniform mass, confirming their readiness for the subsequent creep tests. Creep tests were performed using a four-point constant bending setup (see Fig.1). In accordance with the NF EN 384 [44] standard, each specimen was positioned on two simple supports and subjected to two equal loads, applied symmetrically at equal distances from the supports. After preparing the specimens to the specified dimensions, strain gauges were attached before positioning each specimen between the supports of the bending system (Fig.2). The strain gauges, provided by Measurements Group Inc., had a gauge factor of Kj = 2.05 ± 0.5%. Additionally, an ET616 extensometer from Deltalab was connected to each specimen to monitor the maximum strain in the central region with an accuracy of 1 micrometer per meter. While the deformation was recorded electronically, the maximum stress σ in the specimen's central region was determined using the pressure indicated by the gauge, according to the relationship described in Eq.(1):

$$ \sigma(t) = \frac{3dP(t)}{bh^2}, \qquad (1) $$

where P (N) represents the total applied load at time t, d (mm) the lever arm defined as the distance between outer and inner loading points, b (mm) the width of the specimen, and h (mm) the specimen's height [32].

<figure><img src="image_2.png" /><figcaption>Figure 1. (a) Experimental device, (b) Conceptual diagram [30].</figcaption></figure>

This study focuses on the first two phases of creep, namely primary creep and secondary creep. For this reason, the applied load levels were all below one-third of the rupture load of the wood species. At the end of the experimental campaign, five samples were tested at each of seven loading levels for Sapelli and two samples at each of six loading levels for Wengé. The loadings were controlled at 16.5 MPa (W1), 23.1 MPa (W2), 38.1 MPa (W3), and 43.0 MPa (W4) for Wengé, and at 7.4 MPa (S1), 20.6 MPa (S2), 25.0 MPa (S3), and 29.4 MPa (S4) for Sapelli.
<figure><img src="image_3.png" /><figcaption>Figure 2. Wood specimen : dimensions (mm) and gauge position.</figcaption></figure>

## 2.2. Description of models

### 2.2.1. Constant-order fractional rheological model

The constant-order fractional models developed in this study are obtained by replacing the dashpot of conventional models with a rheological element called a spring-pot. Specifically, the spring-pot serves as an asymptotic representation of an amalgamation of elastic (spring) and viscous (dashpot) elements interconnected both in series and in parallel, whose equivalent scheme is as indicated in the Fig.4 [45]. This component exhibits an intermediate behavior lying between elasticity and viscosity, and its governing law is expressed as [46]:

$$ \sigma(t) = \nu D^{\alpha} \varepsilon(t) \quad 0 < \alpha < 1, \qquad (2) $$

where $\nu$ represents the fractional viscosity coefficient, $\alpha$ the constant fractional order [46] and $D^\alpha$ the fractional derivative, as defined in the Riemann-Liouville sense following Eq.(3) [47]:

$$ (D_a^\alpha f)^{RL}(x) = \frac{d}{dx} \frac{1}{\Gamma(1-\alpha)} \int_a^x \frac{f(t)}{(x-t)^\alpha} dt, \quad \alpha \ni 0; 1[, \qquad (3) $$

where $\Gamma(*)$(Eq.(4)) is the Gamma function [48]:

$$ \Gamma(z) = \int_{0}^{+\infty} e^{-x} x^{z-1} dx, \quad z \ni \mathbb{R}_{+}^{*}. \qquad (4) $$

In a creep test, which the stress history $\sigma(t) = \sigma_0.H(t)$, the compliance of the spring-pot system is expressed by the following equation [49]:

$$ J(t) = \frac{t^{\alpha}}{\nu\Gamma(1+\alpha)}, \qquad (5) $$

and the creep function is deduced as follows:

$$ \varepsilon(t) = \sigma_0 \cdot J(t). \qquad (6) $$

The constant-order fractional Zener model is used in this study. It consists of three rheological elements: two springs following Hooke's law and one spring-pot that follows the law given in Eq.(2), with all these elements assembled according to the schematic shown in Fig.3:

The differential equation of this model is given by [50]:

$$ E_0 D^\alpha \varepsilon(t) + b\varepsilon(t) = D^\alpha \sigma(t) + a\sigma(t), \qquad (7) $$
<figure><img src="image_4.png" /><figcaption>Figure 3. Zener fractional model.</figcaption></figure>

where $a = \frac{E_0 + E_1}{v}$ and $b = \frac{E_0 E_1}{v}$. Using the Laplace transform to solve this equation, we obtain the following compliance function [50,51]:

$$J(t) = J_0 + J_1 \left[ 1 - E_{\alpha,1} \left( -\frac{E_1}{v} t^{\alpha} \right) \right], \quad (8)$$

where $J_i = 1/E_i$, ($i = 0, 1$) and $E_{\xi,\beta}(\cdot)$ defined by Eq.(9) is the generalized Mittag-Leffler function [48]. The creep function is also given by Eq.(6).

$$E_{\xi,\beta}(z) = \sum_{k=0}^{+\infty} \frac{z^k}{\Gamma(\xi k + \beta)}, \qquad \xi, \beta > 0. \quad (9)$$

### 2.2.2. Variable-order fractional rheological model

Existing research shows that constant-order fractional rheological models (already applied to tropical wood species) enhance the consolidation of the memory effect by reducing the number of parameters [31,59]. However, they do not provide insight into the evolution of creep mechanisms over time, highlighting the need for a variable-order fractional model. The fractional viscosity coefficient of Eq.(2) can be written as a function of the modulus of elasticity $E_0$ of the spring and the relaxation time $\tau = \frac{\eta}{E_0}$:

$$v = \tau^{\alpha} E_0, \quad (10)$$

where we can clearly see that:

* if $\alpha = 0, v = E_0$: the behavior of the material depends only on the characteristics of the material at the present time: this is elastic behavior;

* if $\alpha = 1, v = \tau E_0 = \eta$: the behavior will depend on infinitely neighbouring instants, this is the viscous behavior [52].

We also define the fractional derivative in the Riemann-Liouville sense with variable order $\alpha(t)$ according to Lorenzo and Hartley [53] as a convolution product:

$$(D_a^{-\alpha(t)} f)^{RL}(x) = \int_a^x \frac{f(t) (x-t)^{\alpha(t)-1}}{\Gamma(\alpha(t))} dt, \quad \alpha(t) \ni 0; 1[.] \quad (11)$$

In this equation, the order of the derivative varies with time, which implies not only an enhanced memory effect but also the history of the fractional order [53]. By treating the order of the derivative as a time-dependent variable and using the relation from Eq.(10), the behavior law of the spring-pot from Eq.(2) takes the following form [54,55]:

$$\sigma(t) = E_0 \tau^{\alpha(t)} D^{\alpha(t)} \varepsilon(t) \quad 0 < \alpha < 1. \quad (12)$$
Eq.(12) can be written as follows by finding an initial expression for the function $\varepsilon(t)$ [36]:

$$ \varepsilon(t) = \frac{1}{E_0} \tau^{-\alpha(t)} D^{-\alpha(t)} \sigma(t) \quad 0 < \alpha < 1. \qquad (13) $$

By substituting the convolution product (Eq.(11)) into Eq.(13) and performing integration by parts, we obtain the expression of the compliance function in the following form [14]:

$$ J(t) = J_0 \frac{\left(\frac{t}{\tau}\right)^{\alpha(t)}}{\Gamma(1 + \alpha(t))}. \qquad (14) $$

The variable-order fractional model chosen for this study is the fractional Maxwell model, which consists of a spring with an elasticity modulus $E_0$ and a spring-pot connected in series, where the spring-pot follows the behavior described by Eq.(12). Fig.4 illustrates the arrangement of these two rheological elements.

<figure><img src="image_5.png" /><figcaption>Figure 4. Variable-order fractional Maxwell model.</figcaption></figure>

The compliance function of this model is derived from the principle of superposition of the sum of the compliance of the spring and that of Eq.(14), as shown in Eq.(15):

$$ J(t) = J_0 \left[ 1 + \frac{\left(\frac{t}{\tau}\right)^{\alpha(t)}}{\Gamma(1 + \alpha(t))} \right]. \qquad (15) $$

This same solution was obtained by Giusti et al. [56] by starting from the constitutive differential equation of the variable-order fractional Maxwell model.

The function $\alpha(t)$ can take the form of either an exponential or even a sinusoidal function [57], but for the purposes of this study, the fractional order takes the form of a linear function of time, as shown in Eq.(16), because it represents the continuous change in the material during creep testing:

$$ \alpha(t) = at + b, \qquad (16) $$

$a$ is the rate of change of the fractional order and represents the slope of the line representing the order function, while $b$ is the initial state of the order and represents the y-intercept of this function. Ultimately, the compliance function of the variable-order Maxwell fractional model is obtained by substituting the order $\alpha(t)$ from Eq.(16) into the form given in Eq.(15) and we obtain the following form:

$$ J(t) = J_0 \left[ 1 + \frac{\left(\frac{t}{\tau}\right)^{at+b}}{\Gamma(1 + at + b)} \right], \qquad (17) $$

and the creep function is given by Eq.(6).
## 2.3. Method for determining creep parameters

In this methodology, the optimal parameters of the models need to be determined. The parameter $E_0$ (MPa) of the variable-order Maxwell fractional model is obtained from the initial instantaneous deformation using the following formula:

$$E_0 = \frac{\sigma}{\varepsilon_0}, \qquad (18)$$

where $\varepsilon_0$ is the initial instantaneous deformation. The other parameters $v$ (MPa.min$^\alpha$), $a$ and $b$ are determined using the Levenberg-Marquardt optimization algorithm according to the numerical scheme described in Fig. 5.

<figure><img src="image_6.png" /><figcaption>Figure 5. Numerical scheme implementation for computing optimal parameters.</figcaption></figure>

This algorithm simultaneously combines gradient descent and Gauss-Newton methods, which makes it particularly robust in the sense that convergence is fast with few
iterations [58]. The experimental data are compared with the model's creep function to predict the strain at each point in time, and the values of the model parameters are determined by iteratively reducing the sum of the squares of the errors between the function and the measured data points through a sequence of parameter value updates. The advantage of this algorithm is that it allows optimized parameter values to be obtained with good accuracy compared with other algorithms, notably gradient descent and Gauss-Newton taken separately [58].

As outlined in the algorithmic scheme in Fig. 5, in addition to the input data, the model parameters to be determined require an initialization that considers the mechanical order of magnitude associated with each parameter. If these initial values are poorly selected, the solution may diverge, as the least squares problem is solved using the Cholesky method. However, the Levenberg-Marquardt method has the advantage of dynamically adjusting the values of damping parameter to strike a balance between speed and robustness.

### 3. Results and Discussion

Figs. 6 and 7 obtained in accordance with the protocol described in the previous section present the experimental two stages creep data of Sapelli and Wengé, respectively. For each of the two wood species, the creep tests were conducted over a duration of 10 hours (600 minutes). In the following paragraphs, we present the results of the numerical simulations according to the principle of Fig. 5 which allow us to validate the proposed models.

<figure><img src="image_7.png" /><figcaption>Figure 6. Creep of *Entandrophragma cylindricum*.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 7. Creep of Millettia Laurentii.</figcaption></figure>

3.1. Prediction of creep using the variable-order Maxwell fractional model

The curves are obtained by fitting the variable-order Maxwell fractional model, as given in Eq. (17), to the experimental data using the Levenberg–Marquardt algorithm. The results of the fittings are presented in Figs. 8 and 9.

Direct observations of these curves show that the proposed variable-order model satisfactorily simulates the experimental creep data for both wood species. The primary creep, which lasts about 100 minutes (10 in the Log(t)), appears to present the most discrepancies between the experimental data and the model, as some experimental points are not captured by the model during this phase. The secondary creep, which extends beyond 100 minutes, is significantly better captured by our model across all tested specimens. Generally, we attribute these discrepancies to potential fluctuations in environmental conditions during the tests, especially the humidity level, which can significantly affect wood behavior. Other reasons for these discrepancies may be related to the experimental data acquisition equipment. The model's effectiveness is numerically assessed using the coefficient of determination *R*² calculated by the algorithm, which indicates the accuracy of the results proposed by the model. The experimental data are fitted by the variable-order Maxwell fractional model with an average accuracy of 0.99 for all tested specimens of the two selected wood species, demonstrating the satisfactory effectiveness of this model. An analysis of the residual curves further supports these findings. The residuals, representing the differences between the experimental data and the model predictions, are generally centered around zero (See Figs. A1 and A2), indicating that the model effectively captures the overall behavior without significant unmodeled systematic trends. Nevertheless, a systematic deviation is observed during the primary creep phase, as revealed by the residual analysis, highlighting that this phase is less accurately described by the model.
<figure><img src="image_9.png" /><figcaption>**Figure 8.** Creep prediction of *Entandrophragma cylindricum* using variable order factional Maxwell model: (a) S1; (b) S2; (c) S3; (d) S4.</figcaption></figure>

The optimal parameters calculated are presented in Tables (1) and (2) for **Sapelli** and **Wengé**, respectively. It is observed that the parameter *E*₀ is practically unaffected by changes in the stress level, indicating that it can be considered a material characteristic, specifically the rigidity. The parameters a and b are also minimally affected by the stress level and can be directly related to the intrinsic behavior of the material. In contrast, the parameter *v* appears to be more influenced by the stress level, especially for **Wengé**. Indeed, the anisotropy and heterogeneity of the wood material, which induce variability in the specimens, might be responsible for these slight discrepancies in the model parameters. Since the tested specimens have difficulty maintaining the same density due to these intrinsic properties, the modification of the internal structure of each specimen is unlikely to occur in exactly the same manner. Additionally, potential fluctuations in environmental conditions such as humidity and temperature during the experimental tests could also contribute to the observed differences. Similar discrepancies have been observed by Atchounga *et al.* [30] in the creep parameters of **Wengé** wood using the constant-order Zener fractional model, by Nguedjio *et al.* [29,32] in the creep parameters of **Sapelli** wood using four constant-order fractional models (Maxwell, Zener, Thomson, and Burger), by Songsong *et al.* [59] in the creep parameters of bamboo scrimber using both classical and constant-order fractional Maxwell models, by Moucka *et al.* [60] in fractional viscoelastic modelling of porcine skin using generalized Poynting-Thomson model, and in several other works [61–63]. The possible causes of these discrepancies are also related to the delayed
intrinsic behavior of the material and the sensitivity of the material's characteristics to environmental conditions.

<figure><img src="image_10.png" /><figcaption>**Figure 9.** Creep prediction of *Millettia Laurentii* using variable order fractional Maxwell model: (a) W1; (b) W2; (c) W3; (d) W4.</figcaption></figure>

<table><caption>**Table 1.** Parameters of creep variable order fractional Maxwell model (*Entandrophragma cylindricum*).</caption><thead><tr><th>σ<sub>0</sub> (MPa)</th><th>E<sub>0</sub> (10<sup>3</sup>MPa)</th><th>ν(10<sup>7</sup>MPa.min<sup>α</sup>)</th><th>a (10<sup>-5</sup>)</th><th>b (10<sup>-1</sup>)</th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td>7.4</td><td>5.73</td><td>2.43</td><td>3.79</td><td>3.26</td><td>0.9876</td></tr><tr><td>14.7</td><td>5.07</td><td>4.30</td><td>3.85</td><td>3.09</td><td>0.9925</td></tr><tr><td>20.6</td><td>5.25</td><td>3.32</td><td>3.21</td><td>2.98</td><td>0.9894</td></tr><tr><td>29.4</td><td>5.32</td><td>3.45</td><td>6.18</td><td>4.02</td><td>0.9928</td></tr></tbody></table>

<table><caption>**Table 2.** Parameters of creep variable order fractional Maxwell model (*Millettia Laurentii*).</caption><thead><tr><th>σ<sub>0</sub> (MPa)</th><th>E<sub>0</sub> (10<sup>4</sup>MPa)</th><th>ν(10<sup>6</sup>MPa.min<sup>α</sup>)</th><th>a (10<sup>-5</sup>)</th><th>b (10<sup>-1</sup>)</th><th>R<sup>2</sup></th></tr></thead><tbody><tr><td>16.5</td><td>1.578</td><td>2.14</td><td>5.71</td><td>2.79</td><td>0.9916</td></tr><tr><td>23.1</td><td>1.055</td><td>0.11</td><td>3.45</td><td>3.02</td><td>0.9952</td></tr><tr><td>38.1</td><td>0.807</td><td>6.13</td><td>3.03</td><td>3.03</td><td>0.9892</td></tr><tr><td>43.0</td><td>1.028</td><td>6.56</td><td>2.80</td><td>3.23</td><td>0.9837</td></tr></tbody></table>
## 3.2. Comparison between the constant-order fractional model and the variable-order fractional model

The advantage of rheological models is that they make it possible to predict the behavior of the material over a long period of time based on relatively short experiments, other tests that could be carried out in the laboratory would require much longer experiments to achieve the same goal. A rheological model would be considered suitable for a material if it describes with the highest possible accuracy most of the effects related to the material's behavior and has a reduced number of parameters to determine and rheological elements, in order to minimize simulation time and industrial costs. Atchounga et al. [30] used the constant-order Zener fractional model for the creep of tropical Wengé wood, while Nguedjio et al. [29] used the same model for the creep of Sapelli, the two wood species studied in this work using the variable-order Maxwell fractional model. The results of the comparisons between these two fractional models are presented in Table (3).

<table><caption>**Table 3.** Comparison between the constant-order fractional Zener model and the variable-order fractional Maxwell model.</caption><thead><tr><th>Models</th><th>Fractional Zener constant order [29,30]</th><th>Fractional Maxwell variable order</th></tr></thead><tbody><tr><td>Number of parameters to be determined numerically</td><td>4</td><td>4</td></tr><tr><td>Number of rheological elements</td><td>3</td><td>2</td></tr><tr><td>Precision R<sup>2</sup> (Sapelli)</td><td>[0.9600-0.9718]</td><td>[0.9876-0.9928]</td></tr><tr><td>Precision R<sup>2</sup> (Wengé)</td><td>[0.9570-0.9860]</td><td>[0.9837-0.9952]</td></tr></tbody></table>

The analysis of the values in this comparative table allows us to choose the most appropriate model for studying the creep of the two wood species. Numerically, both models require the determination of four parameters using the same algorithm, namely the Levenberg-Marquardt algorithm. However, the constant-order Zener model requires the assembly of three rheological elements, whereas the variable-order Maxwell model requires only two; thus, the variable-order model is preferable from this perspective but is not a consistent selection criterion. When examining the precision of the results, it is observed that the variable-order model provides results with greater accuracy compared to the constant-order model across all tested specimens. Furthermore, the variable order offers an opportunity to understand how the material's mechanical properties change over time and gives a physical meaning to the fractional order, which is not the case for the constant order model. In the following paragraph, we will explore the precise relationship between this variable order and the viscoelastic deformation mechanism. Considering all these results, it is reasonable to choose the variable order model for describing the creep of Sapelli and Wengé tropical wood.

## 3.3. Analysis of the sensitivity of parameters *a* and *b* on the creep mechanism

To determine the benefit of adopting a variable fractional order rather than a constant order, we performed a sensitivity analysis of the parameters *a* and *b* related to the order function. This analysis aims to provide a possible physical interpretation of the parameters *a* and *b* related to the material's behavior under creep stress. This analysis is based on the parameter control method, which involves varying one parameter while keeping all other parameters constant [29,64].

First, in Fig. 10, we plotted the evolution of the fractional order as a function of time. Compared to classical rheological models [32], variable order fractional models can reflect the softening and macro molecular movement of a material from an ideal solid (order 0) to a fluid (order 1), representing viscoelastic behavior such as creep. As shown by the
curves in Fig.10, the fractional order increases linearly as the creep time progresses. This indicates a continuous softening over time of the polymer chains within the wood's internal structure as deformation evolves, a result also observed by Xiang *et al.* [36] in their study on the creep of natural fiber-based composites. Furthermore, the fractional order values are higher for **Sapelli** (S) and lower for **Wenge** (W) because the second species is softer and more rigid than the first. As a result, the polymer bonds in **Wenge** are more difficult to break, leading to lower fractional order values and consequently, a smaller viscosity coefficient under identical stress conditions. These curves also show that the order function is significantly affected by the parameters a and b, suggesting that these two parameters are directly related to the material's characteristics.

<figure><img src="image_11.png" /><figcaption>**Figure 10.** Evolution of the fractional order as a function of time at different stress levels: (a) *Entandrophragma cylindricum*; (b) *Millettia Laurentii*.</figcaption></figure>

By fixing the values of *E*₀, *σ*, *v* and *b* and varying the values of *a* in the function of Eq.(17), we obtained the curves in Fig.13 representing the influence of the parameter *a* on the creep of **Sapelli** and **Wenge**. The values of *a* are chosen such that the order function *α*(*t*) always between 0 and 1. It is observed that when the values of the parameter *a* decrease, there is an increase in deformations as well as an increase in the rate of creep. In other words, a low value of the parameter *a* within an appropriate range indicates a rapid rate of change in the variable order as well as the material's properties, which can lead to lower resistance to creep deformation. Therefore, the parameter *a* can be considered an indicator of the rate of change in mechanical properties during creep. Additionally, a higher value of the parameter *a* (e.g. on the order of 10⁻⁴) would result in a fractional order greater than 1, and in this case, the behavior would no longer be viscoelastic.

The analysis of the influence of the parameter *b* on creep has been similarly obtained using the parameter control method by fixing the values of *E*₀, *σ*, *v* and *a* and varying the values of *b* in the function of Eq.(17). Given the expression of the order function in this study, *α*(*t*) = *at* + *b*, the parameter *b* represents the fractional order at the beginning of creep. Therefore, it can be said that the parameter *b* represents the initial state of the material's properties at the start of creep.

A quantitative sensitivity analysis of the parameters *a* and *b* on the creep behavior was performed using the Maxwell variable order fractional model for the two wood species studied in this paper. The results, based on the calculation of Sobol indices (first-order index, second-order index, and total index), are presented in Fig.12 and Table 4. The Sobol index method quantifies the relative importance of the model parameters in influencing the behavior described by the model. The results clearly indicate that the parameter *a* has the most significant influence on the model's behavior, as evidenced by all the calculated Sobol
indices. From a mechanical perspective, *a* represents the rate at which internal mechanisms within the wood's structure operate during a creep test. This dominant effect of *a* is consistent with the understanding that creep deformations in wood originate from the internal structure and depend primarily on the rate at which the polymer bonds break. The parameter *b* has less influence on the behavior of the model, and the interaction between the two parameters is weak (as shown in Fig. 12 with the almost zero S2 index). This is because the model is predominantly influenced by the parameter *a*, indicating that the internal mechanisms occurring within the wood's structure play a much more significant role than the initial test conditions.

<figure><img src="image_12.png" /><figcaption>**Figure 11.** The sensitivity of creep strain to parameter *a*: (a) *Entandrophragma cylindricum*; (b) *Milletia Laurentii*.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Figure 12.** Quantitative analysis of the sensitivity of parameters *a* and *b* on the Maxwell variable-order fractional model.</figcaption></figure>
<table><caption>**Table 4.** Values of sobol index.</caption><thead><tr><th>Sobol Index</th><th>First Order (S1)</th><th>Total (ST)</th><th>Second Order (S2)</th></tr></thead><tbody><tr><td>a</td><td>0.99</td><td>0.99</td><td>8.077 × 10<sup>-6</sup></td></tr><tr><td>b</td><td>4.29 × 10<sup>-6</sup></td><td>2.77 × 10<sup>-6</sup></td><td>8.077 × 10<sup>-6</sup></td></tr></tbody></table>

**4. Proposal of modified model that include Spring-pot with stress-dependent changes**

The results indicate that while the Maxwell variable-order fractional model effectively describes creep mechanisms, it remains sensitive to stress variations (particularly the viscosity coefficient $v$), which affects its predictive performance under different loading conditions. This section aims to propose a modification to the model by introducing nonlinearity to reduce its sensitivity to stress variations. The drawback of this approach lies in the significant increase in the number of model parameters when adopting such an evolution law. To align with the ideal of maintaining good model performance with fewer parameters, the literature generally favors limiting this type of evolution to a maximum of two parameters, ensuring that only one additional parameter is introduced compared to the initial model [20]. The initial model (Eq.(17)) assumes linearity with respect to stress. However, from a physical and mechanical standpoint, the observed material behavior exhibits non linearities, as stress influences both deformation and strain rate. This reflects the need for a more sophisticated relationship between stress and strain. Consequently, we introduced a non-linear term to better capture the internal mechanisms of the material, such as the rupture of polymer bonds in wood, which inherently follow non-linear dynamics. The modified model provides a more accurate description of the observed behavior without invalidating the assumptions of the original model: under low stress conditions, a Taylor series expansion shows that the model reverts to linearity, while at higher stress levels, incorporating non linearity becomes essential to faithfully represent the creep behavior. We propose an evolution of the parameter $v$ in the following polynomial form:

$$v(\sigma) = k(\sigma + m)^n, \qquad (19)$$

$k$, $m$ and $n$ are constants. $k$ is homogeneous to a time while $m$ is homogeneous to a stress. $n$ is chosen according to the evolution trend of $v$ as a function of the stress $\sigma$, in this paper, we have fixed $n$ to 3 because we observe a polynomial trend of order 3 as shown in Fig.13. This leads to a spring-pot where the fractional viscosity is a nonlinear function of stress. The compliance of the modified fractional Maxwell model with variable order is then expressed as follows based on the same developments which led to the formulation of Eq.(17):

$$J_{\text{mod}}(t) = J_0 \left[ 1 + \frac{(t)^{at+b}}{k(\sigma + m)^n \Gamma(1 + at + b)} \right]. \qquad (20)$$

The objective of this new model is to ensure that creep description remains as independent as possible from stress level, allowing its application to any constant loading history, while preserving the memory effect introduced by the fractional spring pot kernel. Tables (5) and (6) show that, in addition to the stiffness $E_0$, which is largely insensitive to stress levels, the new coefficients $k$ and $m$ (used to compute viscosity $v$ via the nonlinear formulation in Eq. (19)) also remain unaffected by stress variations. As observed, the two new parameters remain consistent across all stress levels, supporting the interpretation that fractional viscosity can be regarded as an intrinsic material property. From this perspective, the fractional model gains credibility, as it allows for nearly identical parameter values independently of the applied stress level, provided the temperature and humidity remain
constant. This achieves a balance between the improved predictive performance of the model and the slight increase in the number of parameters, which has risen from four to five.

<figure><img src="image_14.png" /><figcaption>**Figure 13.** Evolution of viscosity *v* as a function of stress : (a) *Entandrophragma cylindricum*; (b) *Millettia Laurentii* (blue : data from Tables (1) and (2), red : fitting model from Eq.(19)).</figcaption></figure>

<table><caption>**Table 5.** Parameters of modified creep variable order fractional Maxwell model (*Entandrophragma cylindricum*).</caption><thead><tr><th>σ<sub>0</sub> (MPa)</th><th>E<sub>0</sub> (10<sup>3</sup> MPa)</th><th>k(10<sup>-4</sup>min<sup>a</sup>)</th><th>m(10<sup>7</sup> MPa)</th><th>a (10<sup>-5</sup>)</th><th>b (10<sup>-1</sup>)</th></tr></thead><tbody><tr><td>7.4</td><td>5.73</td><td>2.00</td><td>34.96</td><td>3.79</td><td>3.26</td></tr><tr><td>14.7</td><td>5.07</td><td>2.00</td><td>34.96</td><td>3.85</td><td>3.09</td></tr><tr><td>20.6</td><td>5.25</td><td>2.00</td><td>34.96</td><td>3.21</td><td>2.98</td></tr><tr><td>29.4</td><td>5.32</td><td>2.00</td><td>34.96</td><td>6.18</td><td>4.02</td></tr></tbody></table>

<table><caption>**Table 6.** Parameters of modified creep variable order fractional Maxwell model (*Millettia Laurentii*).</caption><thead><tr><th>σ<sub>0</sub> (MPa)</th><th>E<sub>0</sub> (10<sup>4</sup> MPa)</th><th>k(10<sup>-3</sup>min<sup>a</sup>)</th><th>m(10<sup>6</sup> MPa)</th><th>a (10<sup>-5</sup>)</th><th>b (10<sup>-1</sup>)</th></tr></thead><tbody><tr><td>16.5</td><td>1.578</td><td>18.00</td><td>32.13</td><td>5.71</td><td>2.79</td></tr><tr><td>23.1</td><td>1.055</td><td>18.00</td><td>32.13</td><td>3.45</td><td>3.02</td></tr><tr><td>38.1</td><td>0.807</td><td>18.00</td><td>32.13</td><td>3.03</td><td>3.03</td></tr><tr><td>43.0</td><td>1.028</td><td>18.00</td><td>32.13</td><td>2.80</td><td>3.23</td></tr></tbody></table>

## 5. Conclusions

This paper proposes an initial formulation and verification of a new variable-order fractional model aimed at capturing the primary and secondary creep behavior of tropical wood species, specifically **Sapelli** and **Wengé**. Two formulations of the Riemann-Liouville fractional derivative were employed to develop the rheological models: the constant-order derivative and the variable-order derivative. Fractional rheological models governed by the Spring-pot rheological element are capable of capturing the experimental creep data of wood and are better suited for this type of study due to their hereditary nature and their ability to use only a reduced number of parameters. Moreover, by comparing the constant-order fractional Zener model and the variable-order fractional Maxwell model, it was found that for both **Sapelli** and **Wengé** wood, the variable-order model performs better than the constant-order model. Specifically, the constant-order Zener model provides results with accuracy between 0.96 and 0.97, while the variable-order Maxwell model provides results with accuracy exceeding 0.99. This conclusion reinforces the fact that the Spring-pot

415

416

417

418

419

420

421

422

423

424

425

426

427

428
is a rheological element with strong memory characteristics expressed by the order of
the fractional derivative in its constitutive law. Furthermore, the linear time function of
order allowed for a better understanding of the role of the fractional derivative order in
rheological models of this type. Additionally, the sensitivity analysis of deformation to
the parameters a and b, representing respectively the slope and the intercept of the order
function, determined the physical significance of these parameters. Parameter a indicates
the rate of change of the material's mechanical properties during creep, and parameter
b indicates the initial state of these properties at the beginning of creep. Moreover, the
parameter a governs the model and the mechanisms responsible for creep, as highlighted
by the Sobol index, which has a value greater than 0.99 for this parameter. Finally, the paper
introduces a new formulation of the Maxwell variable-order fractional model, enabling
the consideration of deformations under various constant loading histories and offering a
more comprehensive description of the two first stages of creep behavior in tropical timber.
Future work will focus on modifying these fractional models by incorporating the effects
of wood sample variability such as density and grain angle through additional testing on
specimens sourced from a wider range of origins, to obtain more comprehensive models of
the real and overall behavior of wood material.

**Author Contributions:** **L.C. Nguedjio:** Conceptualization, Data curation, Funding acquisition, Investigation, Resources, Software, Writing – original draft, Writing – review & editing. **J.S. Mabekou Takam:** Data curation, Funding acquisition, Methodology, Supervision. **B. Blaysat:** Project administration, Supervision, Validation. **P.K. Talla:** Data curation, Investigation, Methodology, Project administration, Resources, Supervision, Validation, Writing – original draft, Writing – review & editing. **R. Moutou Pitti:** Funding acquisition, Project administration, Resources, Supervision, Validation, Visualization, Writing – original draft, Writing – review & editing.

**Funding:** "This research received no external funding"

**Data Availability Statement:** Data is available upon request.

**Acknowledgments:** The authors would like to express their gratitude to the French government through the Eiffel Excellence Program and the National Research Agency through the DIAMWOOD ANR PRCE N° 23-CE22-0006-03 project for the support provided for this work.

**Conflicts of Interest:** The authors declare that they have no competing interests.
# Appendix A.

## Appendix A.1. Residuals curves

<figure><img src="image_15.png" /><figcaption>Figure A1. Residuals curves of *Entandrophragma cylindricum* fitting : (a) S1; (b) S2; (c) S3; (d) S4.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Figure A2. Residuals curves of *Milletta Laurentii* fitting : (a) W1; (b) W2; (c) W3; (d) W4.</figcaption></figure>

## References

1. Huang, C.; Chui, Y.; Gong, M.; Chana, F.: Mechanical behaviour of wood compressed in radial direction: Part II. Influence of temperature and moisture content. *Journal of Bio and Biop 2020*, 5, 266–275.
2. Gao, S.; Tao, X.; Wang, X.; Wang, L.: Theoretical modeling of the effects of temperature and moisture content on the acoustic velocity of Pinus resinosa wood. *Jour of Fores Res*, **2018**, *29*, 541–548.

3. Chiniforush, A.; Valipour, H.; Akbarnezhad, A.: Water vapor diffusivity of engineered wood: Effect of temperature and moisture content. *Const and Buil Mat*, **2019**, *224*, 1040–1055.

4. Tajvidi, M.; Falk, R. H.; Hermanson, J. C.: Analysis of cantilever-beam bending stress relaxation properties of thin wood composites. *Bioresources*, **2005**, *97*, 1995–2004.

5. Hunt, J. F.; Zhang, H.; Huang, Y.: Time-temperature superposition principle applied to a kenaf-fiber/high-density polyethylene composite. *Jour of Appl Pol Sci*, **2015**, *10*, 3131–3145.

6. Asyraf, M. R. M.; Ishak, M. R.; Sapuan, S. M.; Yidris, N.; Ilyas, R. A.: Woods and composites cantilever beam: A comprehensive review of experimental and numerical creep methodologies. *Jour of Mat Res and Tech*, **2020**, *9*, 6759–6776.

7. Chowdhury, M. A.; Alam, M. M.; Rahman, M. M.; Islam, M. A.: Model-based study of creep and recovery of a glassy polymer. In *Adv in Pol Tech*, **2022**.

8. Zhao, L.; Wei, Y.; Zhang, G. W.; Xi, F.: Short-term creep properties and creep model of wood-plastic composites. *Pol Comp*, **2022**, *43*, 924–933.

9. Zhu, Y.; Liu, P. Y.; Jiang, Z. H.: The creep behavior of wood-polymer composites. *Ad Mat Res*, **2013**, *815*, 632–638.

10. Georgiopoulos, P.; Kontou, E.; Christopoulos, A.: Short-term creep behavior of a biodegradable polymer reinforced with wood-fibers. *Comp Part B: Eng*, **2015**, *80*, 134–144.

11. Bouafif, H.; Koubaa, A.; Perré, P.; Cloutier, A.: Creep behaviour of HDPE/wood particle composites. *Int Jour of Micro and Mat Prop*, **2013**, *8*, 225–238.

12. Asyraf, M. R. M.; Ishak, M. R.; Sapuan, S. M.; Yidris, N.: Comparison of constant and long-term creep behaviors between balau wood and glass fiber reinforced polymer composite for cross-arm application. *Fib and Pol*, **2021**, *22*, 793–803.

13. Xi, F.; Zhao, L.; Wei, Y.; Yi, J.; Zhao, K.: Effect of temperature on the bending and creep properties of wood plastic composites. *Pol Comp*, **2023**, *44*, 4612–4622.

14. Ma, Y. R.; Li, X. J.; Deng, B.; Luo, Y. S.: Basic study on creep properties of eucalyptus wood. *Mad Cien tec*, **2014**, *911*, 232–237.

15. Lagana, M. K.; Babiak.: Creep parameters of spruce wood in high temperature environment. *Mad Cien tec*, **2008**, *10*, 19–24.

16. Dong, C.; Zhang, S.; Wang, J.; Chui, Y.H.: constant bending creep properties of furfurylated poplar wood. *Const and Buil Mat*, **2021**, *269*, 121–308.

17. Pot, G.; Toussaint, E.; Coutand, C.; Le Cam, JB: Experimental study of the viscoelastic properties of green poplar wood during maturation. *Jour of Mat Sci*, **2013**, *48*, 6065–6073.

18. Yildirim, N.; Shaler, S.; West, W.; Gajic, E.; Edgar, R.: The usability of Burger body model on determination of oriented strand boards’ creep behavior. *Ad Comp Let*, **2020**, *29*, 263–336.

19. Saadallah, Y.; Flilissa, S.; Hamadouche, B.: Viscoelastic creep in bending of olive wood (Olea Europaea L.). *Jour of the Ind Aca of Wood Sci*, **2024**, *21*, 58–64.

20. Shimazaki, K.; Ando, K.: Analysis of shear creep properties of wood via modified Burger models and off-axis compression test method. *Wood Sci and Tech*, **2024**, 1–18.

21. Vidal-Sallé, E.; Chassagne, P.: Constitutive equations for orthotropic nonlinear viscoelastic behaviour using a generalized Maxwell model: Application to wood material. *Mech of Time-Dep Mat*, **2007**, *11*, 127–142.

22. Dubois, F.; Husson, JM.; Sauvat, N.; Nicaise Manfoumbi: Modeling of the viscoelastic mechanosensitive behavior in wood. *Mech of Time-Dep Mat*, **2012**, *16*, 439–460.

23. Kazemi, A ; Baghani, M ; Shahsavari, H ; Abrinia, K.: A viscoelastic-viscoplastic constitutive model for high-temperature response of an advanced steel verified by biaxial measurement experiments. *Eur Jour of Mech-A/Sol*, **2023**, *97*, 104–821.

24. Songsong, S.: Study on the creep behavior of recombinant bamboo based on a modified generalized Kelvin model. *Wood Sci and Tech*, **2022**, *56*, 589–601.

25. Wang, P.; Cai, W.; Zhang, Y.; Wang, Z.: A fractional rheological model for loading-dependent rheological behavior of polymers. *Mech of Time-Dep Mat*, **2023**, 1–12.

26. Yang, X.-J.; Gao, F.; Srivastava, H.M.: New rheological models within local fractional derivative. *Rom Rep in Phys*, **2007**, *69*, 1–13.
27. Su, Xianglong; Xu, Wenxiang; Chen, Wen; Yang, Haixia: Fractional creep; relaxation models of viscoelastic materials via a non-Newtonian time-varying viscosity: physical interpretation. *Mech of Mat*, **2020**, *140*, 103–222.

28. Bonfanti, A.; Kaplan, J. L.; Charras, G.; Kabla, A.: Fractional viscoelastic models for power-law materials. *Soft Matter*, **2020**, *16*, 6002–6020.

29. Nguedjio, L. C.; Mabekou Takam, J. S.; Moutou Pitti, R.; Blaysat, B.; Zemtchou, F.; Mezatio, A. K.; Talla, P. K.: Modeling the nonlinear creep behavior of Entandrophragma cylindricum wood by a fractional derivative model. *Mech of Time-Dep Mat*, **2024**, *28*, 303–319.

30. Atchounga, P. K.; Njankouo, J. M.; Foadieng, E.; Talla, P. K.: Investigation of nonlinear creep behaviour of Millettia laurentii wood through Zener fractional rheological model. *Int Jour of Mat Sci*, **2021**, *2*, 01–07.

31. Emmanuel, F.; Christian Martial, F. W.; Ulrich Gael, A. T.; Kisito, T. P.; Medard, F.: Mechanical behavior of Pericopsis elata relative to age during growth. *Ad in Mat Sci and Eng*, **2021**, 1–14.

32. Nguedjio, L. C.; Mabekou Takam, J. S.; Moutou Pitti, R.; Blaysat, B.; Sauvat, N.; Gril, J.; Zemtchou, F.; Talla, P. K.: Analyzing creep-recovery behavior of tropical Entandrophragma cylindricum wood: Traditional and fractional modeling methods. *Int Jour of Sol and Struc*, **2025**, *306*, 113-122.

33. Long, Jianmin; Xiao, Rui; Chen, Wen.: Fractional viscoelastic models with non-singular kernels. *Mech of Mat*, **2018**, *127*, 55–64.

34. Fang, Z. W.; Sun, H. W.; Wang, H.: A fast method for variable-order Caputo fractional derivative with applications to time-fractional diffusion equations. *Comp Math with Appl*, **2020**, *80*, 1443–1458.

35. Sun, H.; Chang, A.; Zhang, Y.; Chen, W.: A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods; applications. *Frac Cal; Appl Anal*, **2019**, *22*, 27–59.

36. Xiang, G.; Yin, D.; Meng, R.; Lu, S.: Creep model for natural fiber polymer composites (NFPCs) based on variable order fractional derivatives: simulation; parameter study. *Jour of Appl Pol Sci 2020*, *24*, 48–96.

37. Xiang, G.; Yin, D.; Meng, R.; Cao, C.: Predictive model for stress relaxation behavior of glassy polymers based on variable-order fractional calculus. *Pol for Ad Tech*, **2021**, *32*, 703–713.

38. Yuxiao, K.; Shuhua, M.; Yonghong, Z.: Variable order fractional grey model; its application. *Appl Math Mod*, **2021**, *97*, 619–635.

39. Zhou, F. X.; Wang, L. Y.; Liu, Z. Y.; Zhao, W. C.: A viscoelastic-viscoplastic mechanical model of time-dependent materials based on variable-order fractional derivative. *Mech of Time-Dep Mat*, **2022**, *26*, 699–717.

40. Meng, R.; Yin, D.; Drapaca, C. S.: A variable order fractional constitutive model of the viscoelastic behavior of polymers. *Int Jour of Non-Lin Mech*, **2019**, *113*, 171–177.

41. CIRAD, Tropix, 2024: Physical and Mechanical Properties of tropical sapelli wood.

42. Njankouo, J.M.; Atchounga, P.; Foadieng, E.; Talla, P. K.: Investigation of physical, mechanical properties and long-term creep behavior of wengé wood (ww). *Appl Eng*, **2020**, *4*, 27-34.

43. Talla Fotsing, G.B.; Foadieng, E.; Fogué, M.; Talla, P. K.: Triaxial variation of the modulus of elasticity in the thermo-elastic range of six tropical wood species. *Wood Mat sci Eng*, **2023**, *18*, 120–129.

44. AFNOR, British Standard, BS EN 408:1995: Timber Structures-Structural Timber; Glued Laminated Timber-Determination of Some Physical; Mechanical Properties.

45. Krasnobrizha, A. (2015). Modeling hysteresis mechanisms of woven composites using a collaborative elasto-plastic damageable model with fractional derivatives, Ph.D. thesis, Ecole Centrale de Nantes.

46. Anatoli Krasnobrizha; Piotr Rozycki; Pierre Cosson; Laurent Gornet.: Modeling hysteresis mechanisms of woven composites using a collaborative elasto-plastic damageable fractional derivative model. *Mat & Tech 2016*, *104*, 407.

47. M. Hamza: Existence; uniqueness of the solution of an infinite-time impulsive fractional differential equation in Banach space, Université Badji Mokhtar, 2015.

48. Ali Khalouta: Solving linear; nonlinear partial differential equations using analytical approaches: extension to cases of fractional order PDEs, Ferhat Abbas Setif 1 University, 2019.
49. A. Stankiewicz.: Fractional Maxwell model of viscoelastic biological materials. In *BIO Web of Conferences* **2018**, 10, 20–32.

574

575

50. Yang, Xiao-Jun ; Gao, FENG ; Srivastava, Hari Mohan.: New rheological models within local fractional derivative. *Rom. Rep. Phys* **2017**, 69, 113.

576

577

51. Xu, Yingjun ; Cheng, Menghong ; Huang, Ruike ; Yu, Jianqi.: The General Kelvin Model ; Poynting Model Based on the General Fractional Calculus. In *IOP Conference Series: Earth ; Environmental Science* **2019**, 022–151.

578

579

580

52. Di Paola, Mario ; Alotta, Giuseppe ; Burlon, Antonio ; Failla, Giuseppe.: A novel approach to nonlinear variable-order fractional viscoelasticity. *Phi Trans Roy Soc A: Mathe, Phys ; Eng Sci* **2020**, 378, 2019–0296.

581

582

583

53. Di Paola, Mario ; Alotta, Giuseppe ; Burlon, Antonio ; Failla, Giuseppe.: Nonlinear Dynamics. *Non-lin Dyn* **2002**, 29, 57.

584

585

54. Kawada, Y. ; Yajima, T. ; Nagahama, H.: Fractional-order derivative ; time-dependent viscoelastic behaviour of rocks ; minerals. *Acta Geoph* **2013**, 61, 1690–1702.

586

587

55. Meng, R. ; Yin, D. ; Zhou, C. ; Wu, H.: Fractional description of time-dependent mechanical property evolution in materials with strain softening behavior. *Appl Math Mod* **2016**, 40, 398–406.

588

589

56. Giusti, A. ; Colombaro, I. ; Garra, R. ; Garrappa, R. ; Mentrelli, A.: On variable-order fractional linear viscoelasticity. *Fract Cal ; Appl Anal* **2024**, 1–15.

590

591

57. Sun, H. G. ; Chen, W. ; Wei, H. ; Chen, Y. Q. ; Mentrelli, A.: A comparative study of constant-order ; variable-order fractional models in characterizing memory property of systems. *The Eur Phys Jour Special Top* **2011**, 193, 185–192.

592

593

594

58. Gavin, H. P.: The Levenberg-Marquardt Algorithm for Nonlinear Least Squares Curve-Fitting Problems. Duke University, Department of Civil ; Environmental Engineering, **2019**.

595

596

597

598

599

59. S. Songsong ; W. Maosong: Evaluation of the applicability of different viscoelasticity constitutive models in bamboo scrimber short-term tensile creep property research. *Sci ; Eng of Comp Mat*, **2021**, 28, 363–371.

600

601

60. Moučka, Robert ; Sedlačík, Michal ; Pátíková, Zuzana.: Fractional viscoelastic models of porcine skin ; its gelatin-based surrogates. *Mech of Mat*, **2023**, 177, 104–559.

602

603

604

605

606

61. Zhang, X; Cen, J; Zhang, Y; Han, S; Gu, K; Yu, Y; Hou, J.: Comparative Study of Numerical Simulation on Short-Term Creep Behavior of Steam-Pretreated White Oak (Quercus alba L.) Wood. *Forest*, **2024**, 15, 21–66.

607

608

62. Xu, S; Cao, Y; Cao, X; Yang, P; Liu, X; Tang, R; Yan, Y; Wu, Q.: The Observation of Creep Strain Distribution in Laminated Veneer Lumber Subjected to Different Loading Regimes. *Forest*, **2024**, 15, 179.

609

610

611

612

613

614

63. Xing, D; Wang, X; Wang, S.: Temperature-Dependent Creep Behavior and Quasi-constant Mechanical Properties of Heat-Treated Wood. *Forest*, **2021**, 12, 968.

615

616

617

618

64. Zhou, HW ; Wang, CP ; Mishnaevsky, L Jr ; Duan, ZQ ; Ding, JY.: A fractional derivative approach to full creep regions in salt rock. *Mech of Time-Dep Mat*, **2013**, 17, 413–425.

619

620

621

622

623

624

625

626

627

628

629