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

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

I have read the paper with interest and I would submit to your attention the following comments.

The main point concerns methodological aspects. Each model needs to be verified - i.e. the robustness of the mathematical function to describe the physical phenomenon - and validated by experimental verification. The number of samples used for the experiment is limited to one specimen for each stress condition. This procedure may be sufficient for verification, but not for validation.  You already mention in the text the role of density, which could have been assessed even at this stage, but other possible elements should be considered such as grain orientation and/or microfibril angle. In other words, with the experimental verification you've done, you can't assess how the material variability can influence the stress response that you've attributed to the different loading conditions. As it stands, you can't assess the effect of material variability on the a and b parameters. In order to be able to propose your model for predicting the creep behaviour of the two selected species, a larger validation is needed (larger number of specimens - possibly not from only one tree - for each stress condition). 

Without this validation, the paper should clearly explain that with this experiment the model is verified but not yet validated (starting from the title, which could be changed to - Verification of a new ......).

Minor comments concern: deeper explanation of the reason that led to the definition of the loading time that limited the assessment to primary and secondary creep (even this aspect should be more clearly reported in the paper aim and in the summary).

How the MOR of the two types was determined. Did you perform a mechanical test or did you use reference values?

In the methodological description it is not clear what was the RH% condition of the experimental tests and what was the accuracy of its control during loading. Perhaps a more precise explanation of the difference between creep and mechano-sorptive creep could help to understand the contribution of the present model.

Author Response

Without this validation, the paper should clearly explain that with this experiment the model is verified but not yet validated (starting from the title, which could be changed to - Verification of a new ......).

Agree.

The paper has been reformatted to clarify that this study does not directly validate the proposed model. A complete validation would require incorporating additional factors such as density, microfibril angle, and a larger number of tests at each stress level using samples from different trees. These aspects are identified as perspectives for future work. Accordingly, the title of the paper has been revised and it is now entitled: "Formulation and numerical verification of a new rheological model for the creep behavior of tropical wood species using variable-order fractional elements."

 

The number of samples used for the experiment is limited to one specimen for each stress condition. This procedure may be sufficient for verification, but not for validation. In other words, with the experimental verification you've done, you can't assess how the material variability can influence the stress response that you've attributed to the different loading conditions

Agree.

Since we were not able to measure the density of each specimen prior to testing, and considering other factors such as the number of samples per stress level, the fiber orientation, and the fact that the samples originated from the same tree trunk, the study focuses on a numerical verification of the proposed model against the collected experimental data rather than a full validation. The model thus aims to assess the influence of the stress level on the creep behavior, and not the influence of material variability. This point has been corrected (lines 13–15). Accordingly, the paper has been reformulated (lines 111–114 of commented manuscript) to reflect this clarification, and the limitations you highlighted have been included as perspectives for improving our models in future work (lines 444–448 of commented manuscript). Clarifications on the number of samples per stress level have been made (lines 151–152 of commented manuscript) and additional files provide the full raw data set.  

 

Minor comments concern: deeper explanation of the reason that led to the definition of the loading time that limited the assessment to primary and secondary creep (even this aspect should be more clearly reported in the paper aim and in the summary).

Agree.

The clarification that this study focuses on primary and secondary creep was provided in the abstract (line 1-2 of commented manuscript), as well as at the end of the introduction within the objectives, and in the conclusion (lines 417-419 of commented manuscript). The decision to focus on primary and secondary creep was based on technical constraints related to the experimental setup used during the test campaign. The creep stresses were therefore limited to one-third of the material’s modulus of rupture (MOR) and the tests were conducted under laboratory conditions with nearly constant temperature and relative humidity. This is why the study, for now, is restricted to the first two phases of creep. Investigating tertiary creep would have required more severe loading conditions, both mechanical and environmental.

How the MOR of the two types was determined. Did you perform a mechanical test or did you use reference values?

Regarding the MOR, the values used were not determined within the scope of this study, but were obtained from samples taken from the same tree and tested under the same laboratory conditions (see references 41 and 42 of the reviewed manuscript).

In the methodological description it is not clear what was the RH% condition of the experimental tests and what was the accuracy of its control during loading. Perhaps a more precise explanation of the difference between creep and mechano-sorptive creep could help to understand the contribution of the present model.

Agree.

The experimental tests were conducted under laboratory conditions with an environment of 65% ± 5% relative humidity and 20 °C ± 2 °C, similarly to the pre-conditioning environment of the samples, as mentioned in the original manuscript (lines 122–124 of commented manuscript). The control of these environmental conditions was ensured by a digital thermo-hygrometer, and the room was kept closed during measurements to minimize fluctuations. Lines 129–133 of the commented manuscript provide these details and also specify that the model studied is suited for creep behavior, but not for mechano-sorptive creep.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

1. Summary:

The manuscript proposes a new rheological model using variable-order fractional calculus to describe the creep behavior of two tropical wood species, Sapelli and Wengé. The model incorporates stress-dependent viscosity and is benchmarked against constant-order fractional models. The main contribution is the improved fit and physical interpretability offered by the new model using fewer rheological elements. The work is ambitious, and the modeling framework is mathematically rigorous, with potentially valuable implications for materials science and timber engineering2.

General Comments:

• Scientific novelty and relevance

The application of variable-order fractional models to tropical wood species is novel and timely. However, the manuscript fails to make a compelling case for why this specific model structure (linear α(t), polynomial stress dependence of υ) is superior to others beyond data fitting.

• Hypothesis clarity and testability

The manuscript lacks a clear research hypothesis. It transitions directly into model formulation without a concise statement of what is being tested or compared. The notion that the new model provides "deeper insight" into the viscoelastic behavior is vague unless the physical interpretation of parameters (e.g., a, b, υ) is better grounded in materials science theory.

• Experimental robustness

Only four stress levels per wood species were tested, and no replicates are mentioned. There is no information about variability among samples or how many specimens were tested per condition. This is a significant omission. Were the results reproducible? No confidence intervals or error bars are shown in any figures.

1. Specific Comments:

• Line 115–125 (Materials and Setup): How many replicates were used for each stress level and species? Was density variability among specimens evaluated? Why was sapwood exclusively used?
• Line 220–243 (Results and fitting): The R² values are impressive (>0.99), but the residuals are not shown. Are there systematic deviations in the primary creep region? Provide residual plots to better support model adequacy.
• Tables 1 and 2 (Model parameters): Units are confusing. Why is E₀ in 10³ MPa for one species and 10⁴ MPa for the other? Are these magnitudes consistent with known elastic moduli for Sapelli and Wengé?
• Line 294–320 (Sensitivity analysis): Figures 10 and 11 are visually illustrative, but this analysis is qualitative. Please quantify the sensitivity (e.g., using partial derivatives, Sobol indices, or similar metrics).
• Figure 12 (υ vs. stress trend): The justification for the cubic polynomial stress dependence of υ is weak and purely empirical. Why not test other forms (e.g., exponential, logarithmic)? Also, polynomial extrapolation is risky without a physical basis.
• Line 345–350 (J_mod equation): Eq. (20) is presented without derivation. Since it introduces a nonlinear stress term into the denominator, it is essential to justify mathematically and physically how this affects linearity assumptions in the original model.
• Line 375–379 (Conclusions): The final claim of enabling prediction under “various loading histories” is overstated. The models were tested only under static loading. Dynamic, cyclic, or time-varying loadings were not studied. Figures 3 & 4: Graphs lack error bars or confidence intervals in visual form, even though statistical significance is discussed. These should be added.
• Figure 2: Quality is poor. The combination diagram should be redrawn in vector format with clear labels and scales.

Recommendation:

Major revisions are required before this manuscript can be considered for publication. The authors need to:

    Clarify sample sizes, replicate testing, and variability.

    Include uncertainty metrics and residual analysis.

    Strengthen the validation of the model beyond fitting (cross-validation, test datasets).

    Provide the raw data and fitting scripts in a public repository.

Author Response

Scientific novelty and relevance

The application of variable-order fractional models to tropical wood species is novel and timely. However, the manuscript fails to make a compelling case for why this specific model structure (linear α(t), polynomial stress dependence of υ) is superior to others beyond data fitting.

The aim of this paper was to verify, for the first time, the applicability of variable-order fractional models to the viscoelastic behavior of tropical wood species. Beyond simply fitting the experimental data, the initial results showed that this category of models could effectively describe the creep behavior of these wood species. Furthermore, within the scope of this study, it was observed that for short-term creep, a time-linear variable-order fractional function was able to capture the progressive softening of the polymer chains within the wood’s internal structure (lines 359–374 of commented manuscript), an insight that was not accessible with constant-order fractional models.

Hypothesis clarity and testability

The manuscript lacks a clear research hypothesis. It transitions directly into model formulation without a concise statement of what is being tested or compared. The notion that the new model provides "deeper insight" into the viscoelastic behavior is vague unless the physical interpretation of parameters (e.g., a, b, υ) is better grounded in materials science theory.

The idea behind developing a variable-order fractional model for tropical woods stems from the fact that classical rheological models (such as generalized Burger, Kelvin, and Maxwell models) require more parameters and exhibit a weak memory effect (lines 62–68 of commented manuscript). The literature shows that constant-order fractional models (already applied to tropical wood species) better capture the memory effect while reducing the number of parameters (lines 182–186 of commented manuscript). However, they do not provide information on how creep mechanisms evolve over time. The introduction of a variable-order fractional approach offers new insights into this aspect. Materials science has already clearly established the mechanical and physical interpretations of the parameters a and b for polymeric materials (references 36 and 37), but not yet for tropical woods in the context of variable-order fractional models one of the issues addressed in this paper.

Experimental robustness

Only four stress levels per wood species were tested, and no replicates are mentioned. There is no information about variability among samples or how many specimens were tested per condition. This is a significant omission. Were the results reproducible? No confidence intervals or error bars are shown in any figures.

Agree.

The experimental data were provided as supplementary files during the resubmission of the paper. Lines 151–153 of commented manuscript specify the number of specimens tested, and the four specimens discussed in this paper were selected to enable comparison with results already available for these wood species. For Sapelli wood, seven stress levels were experimentally tested with an average of five specimens per level; for Wengé wood, six stress levels were tested with two specimens per level. The experimental data showed good reproducibility. Variability in terms of differences in density and sample origin was not addressed in this paper and is planned for future research (lines 444–448 of commented manuscript). The figures 8 and 9 were redrawn to include error bars for greater clarity.

Specific Comments:

1. Line 115–125 (Materials and Setup): How many replicates were used for each stress level and species? Was density variability among specimens evaluated? Why was sapwood exclusively used?

For Sapelli wood, 5 to 6 specimens were used per stress level, and for Wengé wood, 2 specimens per stress level. The density of each specimen prior to testing could not be evaluated (this will be addressed in future work to include specimen variability in the model). We chose the sapwood because, for the available tree trunks, it was the most uniform and homogeneous part, which allowed for machining specimens of suitable dimensions for standardized testing.

2.Line 220–243 (Results and fitting): The R² values are impressive (>0.99), but the residuals are not shown. Are there systematic deviations in the primary creep region? Provide residual plots to better support model adequacy.

Done. The values of R² reflect the average accuracy of fit over the entire process. However, for primary creep, larger deviations in the fit can be observed. We also plotted the residual curves, which justify these adjustments (Figures A1 and A2).

3.Tables 1 and 2 (Model parameters): Units are confusing. Why is E₀ in 10³ MPa for one species and 10⁴ MPa for the other? Are these magnitudes consistent with known elastic moduli for Sapelli and Wengé?

The tropical wood Wengé is stiffer than the tropical wood Sapelli, which highlights the difference in the order of magnitude of their elastic moduli. These values are in good agreement with those reported for these species, as demonstrated by other studies (reference 30).

4.Line 294–320 (Sensitivity analysis): Figures 10 and 11 are visually illustrative, but this analysis is qualitative. Please quantify the sensitivity (e.g., using partial derivatives, Sobol indices, or similar metrics).

Done.

Figures 10 and 11 illustrate the sensitivity effects of creep to certain key parameters, notably the fractional viscosity coefficient in our case. This sensitivity analysis helps bridge the gap between the mathematical formulation, graphical variations, and the behaviors observed during laboratory tests, a typical approach commonly used in materials science. Additionally, we also performed a quantitative analysis of this sensitivity by calculating Sobol indices (Figure 12 and Table 4).

5.Figure 12 (υ vs. stress trend): The justification for the cubic polynomial stress dependence of υ is weak and purely empirical. Why not test other forms (e.g., exponential, logarithmic)? Also, polynomial extrapolation is risky without a physical basis.

The choice of the polynomial form specifically for this paper is justified by the observed nonlinearity, which is supported by other studies already conducted in the field of wood behavior modeling (see reference 20, for example). Exponential and logarithmic forms were tested, but they were unsuccessful in capturing the observed nonlinearity. Nevertheless, we provided a polynomial form that closely matches the observed nonlinearity, knowing that it can be approximated by either an exponential or logarithmic form through a Taylor expansion.

6.Line 345–350 (J_mod equation): Eq. (20) is presented without derivation. Since it introduces a nonlinear stress term into the denominator, it is essential to justify mathematically and physically how this affects linearity assumptions in the original model.

The initial model (Equation 17) assumes linearity with respect to stress. However, from both a physical and mechanical standpoint, the observed material behavior exhibits non-linearities, as stress influences both deformation and strain rate. This highlights 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 is crucial to faithfully represent the creep behavior (lines 386–395 of commented manuscript).

7.Line 375–379 (Conclusions): The final claim of enabling prediction under “various loading histories” is overstated. The models were tested only under static loading. Dynamic, cyclic, or time-varying loadings were not studied. Figures 3 & 4: Graphs lack error bars or confidence intervals in visual form, even though statistical significance is discussed. These should be added.

Done.

The statement was moderated by specifying that the model would predict behavior under static loading, provided that other factors, such as sampling variability, are included. Error bars were also included to provide more precision regarding the fitting curves (Figures 8 and 9).

8.Figure 2: Quality is poor. The combination diagram should be redrawn in vector format with clear labels and scales.

Done. The combined graphs were redrawn and saved in vector format to improve quality (see Figures 6 and 7).

Recommendation:

Clarify sample sizes, replicate testing, and variability.

Done. Lines 126-129 and 151-153.

Include uncertainty metrics and residual analysis.

Done. The figures were redrawn to include error bars, and the residual curves were also provided and discussed (Figures 8, 9, A1, and A2).

Provide the raw data and fitting scripts in a public repository.

Done.

Supplementary documents containing the raw data and the fitting code were included during the resubmission.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

1. Discuss the numerical scheme Figure 5 in detail
2. Provide technical insight about Figures 6, 7and 8
3. Technically discuss how the formation of the graph as shown in Figure 9 is achieved.
4. Give a reason for the variation of S and W values as shown in Figure 10.
5. Explain how to achieve the sensitivity of creep, as illustrated in Figure 11.
6. why is the Entandrophragma cylindricum stress variation slighter than the Millettia
   Laurentii?
7. Discuss in detail about Table 4 and 5.
8. State the work's novelty in the conclusion portion. 

Comments on the Quality of English Language

English could be improved for better understanding 

Author Response

1.Discuss the numerical scheme Figure 5 in detail

Agree. Lines 235-240, commented manuscript.

The numerical scheme was discussed, highlighting the various steps taken during the simulations to provide more details on the advantages and disadvantages of this method. Lines 235 to 240 of the commented manuscript were added for this purpose. As described by the algorithmic diagram in Figure 5, after entering the experimental strains (εi), the corresponding times (ti​) at which these strains were measured, the regulation parameter (γ), and the tolerance (τ), the creep function or objective function is defined. The model parameters to be determined are then initialized based on the mechanical order of magnitude associated with each parameter. If these initial values are poorly chosen, the solution will diverge, as the least squares problem is solved using the Cholesky method. However, the Levenberg-Marquardt method has the advantage of dynamically regulating the values of γ to find a good balance between speed and robustness. The optimal parameters depend on the tolerance set, and thus on the number of iterations; the higher the number of iterations, the more reliable the solution, though with the risk of overfitting the data. Conversely, the fewer the iterations, the larger the residuals, so a reasonable minimum and maximum tolerance must be set.

2.Provide technical insight about Figures 6, 7and 8

Done.

Figures 6 and 7 present the deformations as a function versus time. Indeed, in accordance with the methodology and experimental protocol described in subsection 2.1 (lines 125 to 144 of the commented manuscript), the acquisition system allows for obtaining the deformation values at each selected time point. Once we have these data, we simply plot them, as shown in Figures 6 and 7.

3.Technically discuss how the formation of the graph as shown in Figure 9 is achieved.

Done.

Figures 8 and 9 are generated by fitting the experimental data using the model given by Equation (17), following the algorithm previously described. When we input the experimental data (deformations and times) and the model function (Equation (17)), the algorithm calculates the corresponding optimal parameters to make the model as accurate as possible relative to the experimental data, and we plot both curves on the same graph. This is explained in lines 249 to 251 of the commented manuscript.

4.Give a reason for the variation of S and W values as shown in Figure 10.

Done.

Figure 10 shows the evolution of the fractional order as a function of time. According to Equation (16), this fractional order is a linear and increasing function of time because, under constant loading, the studied wood species deform progressively, and thus, at their internal structures, the polymer bonds soften (lines 332–334 of commented manuscript). Furthermore, the fractional order values are higher for Sapelli (S) and lower for Wengé (W), as the latter is softer and stiffer than the former. Therefore, the polymer bonds are more difficult to break in Wengé, leading to lower fractional order values and consequently a smaller viscosity coefficient, assuming both species are conditioned in the same environment (lines 335–339 of the commented manuscript).

5.Explain how to achieve the sensitivity of creep, as illustrated in Figure 11.

Done.

Creep sensitivity is obtained using a technique we have called the control method. This involves fixing all parameters in an expression except for the one whose sensitivity we intend to study (lines 342–343 of commented manuscript). For Figure 11, the sensitivity to creep is performed using Equation (17), which gives the creep function, and we chose to analyze this for the parameters a and b of the fractional order because the fractional viscosity provides insight into the deformation mechanism of our wood species. Additional analysis elements have been provided (lines 359–374 of commented manuscript).

6.why is the Entandrophragma cylindricum stress variation slighter than the Millettia Laurentii?

The creep stresses of Sapelli (Entandrophragma cylindricum) are lower than those of Wengé (Millettia Laurentii). For this paper, only primary and secondary creep are studied. Therefore, in accordance with the experimental setup available during the tests, the applied stresses should be below one-third of the longitudinal bending rupture stress (MOR). For Sapelli, this gives a stress below 34 MPa, and for Wengé, below 48 MPa.

7.Discuss in detail about Table 4 and 5.

Done.

Additional elements have been added to further discuss the results of these tables (lines 408–413 of commented manuscript). Tables 4 and 5 are replicas of Tables 1 and 2, with the addition of two new coefficients from the modified model. The two new parameters remain constant regardless of the stress level. From this perspective, the fractional model is more credible, as it allows for nearly identical parameters regardless of the stress level, provided the environment remains at constant temperature and humidity.

8.State the work's novelty in the conclusion portion. 

Done. The section highlighting the novelty of the work (lines 417–419 of commented manuscript) has been added to the conclusion, in addition to the other new elements found in lines 433–441 of this part of the paper.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

Comments and Suggestions for Authors

Dear Authors,

Thank you for considering my suggestions and making changes accordingly.

Please have the patience to accept two further comments:

When you write in the conclusion "by incorporating the effects of wood sample variability, such as density and grain angle, through additional testing on samples from a wider range of origins to obtain more comprehensive models of the real and overall behaviour of wood materials", you are writing something different from my point of view.

You are writing something different from my point of view, the weakness of your model at this stage is that you don't know how it fits with the range of existence that the natural variability of the material can have on the creep behaviour. It's not about the effectiveness of the model, it's about the meaning of the parameter parameterisation (does the current 'a' value fit any piece of wood of that species?). This is the difference between verification and validation.

The second comment concerns the sentence you added on line 268 "

indicating that the model effectively captures the overall behaviour without significant unmodeled systematic trends".

 I find this statement very questionable, if not incorrect. The distribution of your residual is not homogeneously distributed around the 0 line, but it clearly shows a consistent trend (it is one of the cases that shows very well how a very high value of R2 does not necessarily mean the capacity of the model to describe the physical phenomenon https://online.stat.psu.edu/stat462/node/120/). This doesn't allow us to say that there is no systematic trend - because it's simply not true - but above all it is an indication of a clear time-dependent effect that is shown in terms of trend by all samples, independently of species or loading conditions. It seems that if it resists a careful re-examination of the test condition, it is evidence of a material behaviour in the time domain, which consideration could help quite a lot in your future model's improvement.

Author Response

You are writing something different from my point of view, the weakness of your model at this stage is that you don't know how it fits with the range of existence that the natural variability of the material can have on the creep behaviour. It's not about the effectiveness of the model, it's about the meaning of the parameter parameterisation (does the current 'a' value fit any piece of wood of that species?). This is the difference between verification and validation.

Thank you for this important clarification regarding the natural variability of the material and its influence on the creep behavior of wood. As you rightly pointed out in your initial comments, the validation of our model is not yet ensured at this stage. Indeed, we have not yet systematically demonstrated to what extent the identified parameters particularly the parameter ‘a’ remain valid across the full range of natural variability of the material. This is precisely why we proposed a nonlinear formulation of the model, with the aim of reducing the sensitivity of the viscoelastic parameters to stress levels, as observed in this work. This approach led to results that we consider promising, as illustrated by the comparative analysis of Tables 1 and 3 (parameters E₀ and ʋ (Table 1) versus k and m (Table 3)). However, these same tables also show that the values of parameter ‘a’ vary with the applied stress, which represents a limitation in terms of validation. The fluctuation rate calculated around the average values of parameters ‘a’ and ‘b’ highlights this variability in the sense you described that is, the ability of the model’s parameterization to remain meaningful across creep behavior of the wood material. We believe that this variability largely stems from structural differences between the tested samples (e.g., density, grain angle, porosity, etc.). For this reason, we plan to expand our experimental campaign to include a wider range of samples with varying origins and structural characteristics, in order to identify more comprehensively the material variability factors affecting creep behavior and better account for them in future model development.

The conclusion of the paper was improved to incorporate this perspective and to further supported the results obtained (lines 445-450 of commented manuscript).

I find this statement very questionable, if not incorrect. The distribution of your residual is not homogeneously distributed around the 0 line, but it clearly shows a consistent trend (it is one of the cases that shows very well how a very high value of R² does not necessarily mean the capacity of the model to describe the physical phenomenon https://online.stat.psu.edu/stat462/node/120/). This doesn't allow us to say that there is no systematic trend - because it's simply not true - but above all it is an indication of a clear time-dependent effect that is shown in terms of trend by all samples, independently of species or loading conditions. It seems that if it resists a careful re-examination of the test condition, it is evidence of a material behaviour in the time domain, which consideration could help quite a lot in your future model's improvement.

Agree.

In light of your comment and the additional explanations provided by your online resource, the statement added in line 268 was indeed overstated and has been revised accordingly (lines 266-273 of commented manuscript). That sentence was originally intended to refer to the secondary creep phase, which appears to be better captured by the model, with a more homogeneous residual distribution than in the primary phase, where a clear systematic trend is observed as initially mentioned in this version of the manuscript. We sincerely thank you for this additional comment, which will certainly help us improve our model through further residual analyses based on more rigorous testing protocols.

Reviewer 2 Report

Comments and Suggestions for Authors

The authors have taken due account of the comments and recommendations in the original version of the article. 

Author Response

We sincerely thank the reviewer for acknowledging the efforts we made to address the comments and recommendations from the original version of the article. We appreciate the constructive feedback which has significantly contributed to improving the clarity and scientific rigor of our manuscript.