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

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, modeled 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 non-linearity to account for the effects of the variation in 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.

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 [1,2,3]. 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 [4,5,6,7]. Understanding and modeling this behavior remains a major goal for materials science to achieve. In this regard, several researchers have modeled 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 [10,11,12,13]. 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 [16,17,18,19]. 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 non-linear 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 five-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 modeling. 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 eight parameters to the fractional Thomson model with three parameters, Nguedjio et al. [29] show significant numerical and rheological gains in modeling tropical Entandrophragma cylindricum 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 eight 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 Millettia laurentii wood and show that the model fits well with experimental data. Emmanuel et al. [31] opt for the fractional Maxwell model to describe the creep of tropical Pericopsis elata 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 Entandrophragma cylindricum 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 modeling 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 non-linear 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 brief details 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: Entandrophragma cylindricum and 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 [43,44]: a consistent length of 360 mm and a square cross section of 20 mm × 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 was 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 Figure 1). In accordance with the NF EN 408 [45] 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 (Figure 2). The strain gauges, provided by Measurements Group Inc. (Raleigh, NC, USA), had a gauge factor of $Kj=2.05\%\pm 0.5\%$. Additionally, an ET616 extensometer from Deltalab (Rubí, Barcelona, Spain) was connected to each specimen to monitor the maximum strain in the central region with an accuracy of 1 μm per meter. While the deformation was recorded electronically, the maximum stress $\sigma$ in the specimen’s central region was determined using the pressure indicated by the gauge, according to the relationship described in Equation (1):

$$\sigma \left(t\right)={\displaystyle \frac{3dP\left(t\right)}{b{h}^2}},$$

(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].

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 Entandrophragma cylindricum (Sapele) and two samples at each of six loading levels for Millettia laurentii (Wenge). The loadings were controlled at 16.5 MPa (W1), 23.1 MPa (W2), 38.1 MPa (W3), and 43.0 MPa (W4) for Wenge and at 7.4 MPa (S1), 20.6 MPa (S2), 25.0 MPa (S3), and 29.4 MPa (S4) for Sapele.

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 Figure 3 [46]. This component exhibits an intermediate behavior lying between elasticity and viscosity, and its governing law is expressed as follows [47]:

$$\sigma \left(t\right)=\upsilon D^{\alpha }\epsilon \left(t\right)0<\alpha <1,$$

(2)

where $\upsilon$ represents the fractional viscosity coefficient; $\alpha$ the constant fractional order [47]; and $D^{\alpha }$ the fractional derivative, as defined in the Riemann–Liouville sense following Equation (3) [48]:

$${\left(D_a^{\alpha }f\right)}^{RL}\left(x\right)={\displaystyle \frac{d}{dx}}{\displaystyle \frac{1}{\mathsf{\Gamma }(1-\alpha )}}{\int }_a^x{\displaystyle \frac{f\left(t\right)}{{(x-t)}^{\alpha }}}dt,\alpha \ni ]0;1[,$$

(3)

where $\mathsf{\Gamma }(\ast )$ (Equation (4)) is the Gamma function [49]:

$$\mathsf{\Gamma }\left(z\right)={\int }_0^{+\infty }{e}^{-x}{x}^{z-1}dx,z\ni {\Re }_{+}^{*}.$$

(4)

In a creep test, which the stress history $\sigma \left(t\right)={\sigma }_0·H\left(t\right)$, the compliance of the spring–pot system is expressed by the following equation [50]:

$$J\left(t\right)={\displaystyle \frac{t^{\alpha }}{\upsilon \mathsf{\Gamma }(1+\alpha )}},$$

(5)

and the creep function is deduce as follows:

$$\epsilon \left(t\right)={\sigma }_0·J\left(t\right).$$

(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 Equation (2), with all these elements assembled according to the schematic shown in Figure 3.

The differential equation of this model is given by the following [26]:

$$E_0{D}^{\alpha }\epsilon \left(t\right)+b\epsilon \left(t\right)=D^{\alpha }\sigma \left(t\right)+a\sigma \left(t\right),$$

(7)

where $a={\displaystyle \frac{E_0+E_1}{\upsilon }}$ and $b={\displaystyle \frac{E_0{E}_1}{\upsilon }}$. Using the Laplace transform to solve this equation, we obtain the following compliance function [26,51]:

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

(8)

where $J_i=1/E_i,(i=0,1$) and $E_{\xi ,\beta }(*)$ defined by Equation (9) is the generalized Mittag–Leffler function [49]. The creep function is also given by Equation (6).

$$E_{\xi ,\beta }\left(z\right)=\sum _{k=0}^{+\infty }{\displaystyle \frac{z^k}{\mathsf{\Gamma }(\xi k+\beta )}},\xi ,\beta >0.$$

(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 [31,52]. 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 Equation (2) can be written as a function of the modulus of elasticity $E_0$ of the spring and the relaxation time $\tau ={\displaystyle \frac{\eta }{E_0}}$:

$$\upsilon ={\tau }^{\alpha }{E}_0,$$

(10)

where we can clearly see the following:

• If $\alpha =0$, $\upsilon =E_0$, the behavior of the material depends only on the characteristics of the material at the present time: this is the elastic behavior;
• If $\alpha =1$, $\upsilon =\tau E_0=\eta$, the behavior will depend on infinitely neighboring instants; this is the viscous behavior [53].

We also define the fractional derivative in the Riemann–Liouville sense with variable-order $\alpha \left(t\right)$ according to Lorenzo and Hartley [54] as a convolution product:

$${\left(D_a^{-\alpha \left(t\right)}f\right)}^{RL}\left(x\right)={\int }_a^x{\displaystyle \frac{f\left(t\right)·{(x-t)}^{\alpha \left(t\right)-1}}{\mathsf{\Gamma }\left(\alpha \right(t\left)\right)}}dt,\alpha \left(t\right)\ni ]0;1[.$$

(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 [54]. By treating the order of the derivative as a time-dependent variable and using the relation from Equation (10), the behavior law of the spring–pot from Equation (2) takes the following form [55,56]:

$$\sigma \left(t\right)=E_0{\tau }^{\alpha \left(t\right)}{D}^{\alpha \left(t\right)}\epsilon \left(t\right)0<\alpha <1.$$

(12)

Equation (12) can be written as follows by finding an initial expression for the function $\epsilon \left(t\right)$ [36]:

$$\epsilon \left(t\right)={\displaystyle \frac{1}{E_0}}{\tau }^{-\alpha \left(t\right)}{D}^{-\alpha \left(t\right)}\sigma \left(t\right)0<\alpha <1.$$

(13)

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

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

(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 Equation (12). Figure 4 illustrates the arrangement of these two rheological elements.

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 Equation (14), as shown in Equation (15):

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

(15)

This same solution was obtained by Giusti et al. [57] by starting from the constitutive differential equation of the variable-order fractional Maxwell model.

The function $\alpha \left(t\right)$ can take the form of either an exponential or even a sinusoidal function [58], but for the purposes of this study, the fractional order takes the form of a linear function of time, as shown in Equation (16), because it represents the continuous change in the material during creep testing:

$$\alpha \left(t\right)=at+b,$$

(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 $\alpha \left(t\right)$ from Equation (16) into the form given in Equation (15), and we obtain the following form:

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

(17)

and the creep function is given by Equation (6).

2.3. Method for Determining Creep Parameters

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

$$E_0={\displaystyle \frac{\sigma }{{\epsilon }_0}},$$

(18)

where ${\epsilon }_0$ is the initial instantaneous deformation. The other parameters $\upsilon (MPa.mi{n}^{\alpha })$, a, and b are determined using the Levenberg–Marquardt optimization algorithm according to the numerical scheme described in Figure 5.

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 [59]. 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 [59].

As outlined in the algorithmic scheme in Figure 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

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

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 Equation (17), to the experimental data using the Levenberg–Marquardt algorithm. The results of the fittings are presented in Figure 8 and Figure 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 min (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 min, 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 performance of the verification 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 distributed in a non-homogeneous manner around zero (See Figure A1 and Figure A2). Nevertheless, a systematic deviation is observed, as revealed by the residual analysis, highlighting that the model reliably captures the creep behavior of tropical wood but still requires further investigation to achieve full validation.

The optimal parameters calculated are presented in

Table 1. Parameters of creep variable-order fractional Maxwell model (Entandrophragma cylindricum).

${\mathit{\sigma }}_0$ (MPa)	${\mathit{E}}_0$ (${10}^3$ MPa)	$\mathit{\upsilon }({10}^7$$\mathbf{MPa}·{\mathbf{min}}^{\mathit{\alpha }})$	a (${10}^{-5}$)	b (${10}^{-1}$)	${\mathit{R}}^2$
7.4	5.73	2.43	3.79	3.26	0.9876
14.7	5.07	4.30	3.85	3.09	0.9925
20.6	5.25	3.32	3.21	2.98	0.9894
29.4	5.32	3.45	6.18	4.02	0.9928

and

Table 2. Parameters of creep variable-order fractional Maxwell model (Millettia laurentii).

${\mathit{\sigma }}_0$ (MPa)	${\mathit{E}}_0$ (104 MPa)	$\mathit{\upsilon }$ (106 MPa·min$\mathit{\alpha }$)	a (${10}^{-5}$)	b (${10}^{-1}$)	${\mathit{R}}^2$
16.5	1.578	2.14	5.71	2.79	0.9916
23.1	1.055	0.11	3.45	3.02	0.9952
38.1	0.807	6.13	3.03	3.03	0.9892
43.0	1.028	6.56	2.80	3.23	0.9837

for Sapele and Wenge, 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 $\upsilon$ appears to be more influenced by the stress level, especially for Wenge. 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, 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 Wenge wood using the constant-order Zener fractional model, by Nguedjio et al. [29,32] in the creep parameters of Sapele wood using four constant-order fractional models (Maxwell, Zener, Thomson, and Burger), by Songsong et al. [52] in the creep parameters of bamboo scrimber using both classical and constant-order fractional Maxwell models, by Moucka et al. [60] in fractional viscoelastic modeling of porcine skin using the generalized Poynting–Thomson model, and in several other works [61,62,63]. 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.

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 Wenge wood, while Nguedjio et al. [29] used the same model for the creep of Sapele, 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. Comparison between the constant-order fractional Zener model and the variable-order fractional Maxwell model.

Models	Fractional Zener Constant Order [29,30]	Factional Maxwell Variable Order
Number of parameters to be determined numerically	4	4
Number of rheological elements	3	2
Precision $R^2$ (Sapele)	[0.9600–0.9718]	[0.9876–0.9928]
Precision $R^2$ (Wenge)	[0.9570–0.9860]	[0.9837–0.9952]

.

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 Sapele and Wenge 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 $\mathit{a}$ and $\mathit{b}$ related to the order function. This analysis aims to provide a possible physical interpretation of the parameters $\mathit{a}$ and $\mathit{b}$ 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 Figure 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 Figure 10, the fractional order increases linearly as the creep time progresses. This indicates 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 Sapele (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.

By fixing the values of $E_0$, σ, υ, and $\mathit{b}$ and varying the values of $\mathit{a}$ in the function of Equation (17), we obtained the curves in Figure 11 representing the influence of the parameter $\mathit{a}$ on the creep of Sapele and Wenge. The values of a are chosen such that the order function $\alpha \left(t\right)$ is always between 0 and 1. It is observed that when the values of the parameter $\mathit{a}$ 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 $\mathit{a}$ 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 $\mathit{a}$ can be considered an indicator of the rate of change in mechanical properties during creep. Additionally, a higher value of the parameter $\mathit{a}$ (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 $\mathit{b}$ on creep has been similarly obtained using the parameter control method by fixing the values of $E_0$, σ, υ, and $\mathit{a}$ and varying the values of $\mathit{b}$ in the function of Equation (17). Given the expression of the order function in this study, $\alpha \left(t\right)=\mathit{a}t+\mathit{b}$, the parameter $\mathit{b}$ represents the fractional order at the beginning of creep. Therefore, it can be said that the parameter $\mathit{b}$ 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 Figure 12 and

Table 4. Values of Sobol index.

Sobol Index	First Order (S1)	Total (ST)	Second Order (S2)
a	0.99	0.99	$8.077\times {10}^{-6}$
b	$4.29\times {10}^{-6}$	$2.77\times {10}^{-6}$	$8.077\times {10}^{-6}$

. 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 Figure 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.

4. Proposal of Modified Model That Includes 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 $\upsilon$), which affects its predictive performance under different loading conditions. This section aims to propose a modification to the model by introducing non-linearity 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 (Equation (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 $\upsilon$ in the following polynomial form:

$$\upsilon \left(\sigma \right)=k{(\sigma +m)}^n,$$

(19)

where 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 $\upsilon$ 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 Figure 13. This leads to a spring–pot where the fractional viscosity is a non-linear 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 Equation (17):

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

(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.

Table 5. Parameters of modified creep variable-order fractional Maxwell model (Entandrophragma cylindricum).

${\mathit{\sigma }}_0$ (MPa)	${\mathit{E}}_0$ (${10}^3$ MPa)	$\mathit{k}\left({10}^{-4}{\mathbf{min}}^{\mathit{\alpha }}\right)$	m (${10}^7$ MPa)	a$\left({10}^{-5}\right)$	b$\left({10}^{-1}\right)$
7.4	5.73	2.00	34.96	3.79	3.26
14.7	5.07	2.00	34.96	3.85	3.09
20.6	5.25	2.00	34.96	3.21	2.98
29.4	5.32	2.00	34.96	6.18	4.02

and

Table 6. Parameters of modified creep variable-order fractional Maxwell model (Millettia laurentii).

${\mathit{\sigma }}_0$ (MPa)	${\mathit{E}}_0$ (${10}^4$ MPa)	$\mathit{k}\left({10}^{-3}{\mathbf{min}}^{\mathit{\alpha }}\right)$	m (${10}^6$ MPa)	a$\left({10}^{-5}\right)$	b$\left({10}^{-1}\right)$
16.5	1.578	18.00	32.13	5.71	2.79
23.1	1.055	18.00	32.13	3.45	3.02
38.1	0.807	18.00	32.13	3.03	3.03
43.0	1.028	18.00	32.13	2.80	3.23

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 $\upsilon$ via the non-linear formulation in Equation (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.

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 Entandrophragma cylindricum and Millettia laurentii. 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 Entandrophragma cylindricum and Millettia laurentii 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 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 $\mathit{a}$ indicates the rate of change of the material’s mechanical properties during creep, and parameter $\mathit{b}$ 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, this 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. Furthermore, despite this modification, although the parameters $E_0$, k, and m can be associated with the behavior of the studied wood species, the parameters a and b still exhibit notable variability around their values, indicating the need for further investigation before they can be considered validated for modeling the creep behavior of tropical wood. Future work will focus on refine these models by addressing the observed variability in parameters a and b, while 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.