Vibrational Analysis of Thermoelastic Beams on Dual-Parameter Foundations via the Fractional Three-Phase-Lag Approach

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The manuscript presented by the authors addresses an interesting problem, and I find it well-written and of sufficient value for those interested in similar topics.

The abstract seems adequate; it is well-presented, providing sufficient information about the problem, its possible solution, the approach taken, the mathematical tools used, and the expected results.
The introduction is presented in an organized and quite explicit manner, showing the objective of the study, explaining the approach, and discussing previous results from related problems that support this approach.
The mathematical development is presented coherently and clearly enough. This ensures that the results obtained are well understood, as is the rationale for using the mathematical concepts and developments. The figures presented are consistent with the numerical results and demonstrate the appropriateness of the methodology.
The discussion section is adequately developed and leads to the conclusions that the authors present in a concise and explicit manner.

Regardless of these comments, I suggest the authors conduct a final review of the manuscript, with the aim of avoiding any errors. Aside from the review I've already done, I believe it's necessary to ensure there are no issues.

Author Response

Reviewer 1

We thank the reviewer for their thorough-evaluation and valuable suggestions. This feedback has been vital in refining our work. Our detailed, point by point responses to each suggestion are provided below.

Comments: The manuscript presented by the authors addresses an interesting problem, and I find it well-written and of sufficient value for those interested in similar topics.

The abstract seems adequate; it is well-presented, providing sufficient information about the problem, its possible solution, the approach taken, the mathematical tools used, and the expected results.
The introduction is presented in an organized and quite explicit manner, showing the objective of the study, explaining the approach, and discussing previous results from related problems that support this approach.
The mathematical development is presented coherently and clearly enough. This ensures that the results obtained are well understood, as is the rationale for using the mathematical concepts and developments. The figures presented are consistent with the numerical results and demonstrate the appropriateness of the methodology.
The discussion section is adequately developed and leads to the conclusions that the authors present in a concise and explicit manner.

Regardless of these comments, I suggest the authors conduct a final review of the manuscript, with the aim of avoiding any errors. Aside from the review I've already done, I believe it's necessary to ensure there are no issues.

Response:

We sincerely appreciate the reviewer's constructive feedback and confirmation of the validity of our methodology and results. We are pleased to see that the mathematical developments and diagrams have clearly illustrated our findings. As suggested, we have thoroughly revised the manuscript to improve the language and ensure its complete accuracy. Thank you again for the valuable feedback.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

Dear Editor

 

I read the article with great interest.  

1. The manuscript addresses an important and timely problem and has the potential to make a meaningful contribution to the field.

2. The overall structure of the paper is clear, and the objectives are well defined.

3. The methodology adopted in the study is appropriate and logically developed.

4. The results are presented clearly and supported by suitable figures and analysis.

5. The discussion section is informative and links the findings with relevant existing literature.

6. However, the Introduction section requires extensive improvement to better position the novelty and significance of the work. In particular, the part related to analytical solutions should be expanded and strengthened. The authors are encouraged to consult relevant studies such as “A discussion on the Lie symmetry analysis, travelling wave solutions and conservation laws of new generalized stochastic potential-KdV equation” and “Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary” to enrich the analytical background.

7. For the fractional-order modeling, the manuscript should provide a solid and well-justified rationale explaining why fractional derivatives are employed, how they differ from classical derivatives, and why they are more impactful or realistic for the current problem.

8. The Introduction would also benefit from referencing recent analytical works on fractional operators, such as “Generalized Mittag-Leffler Kernel Form Solutions of Free Convection Heat and Mass Transfer Flow of Maxwell Fluid with Newtonian Heating: Prabhakar Fractional Derivative Approach”, to better highlight the mathematical relevance of the chosen fractional framework.

9. The authors should further strengthen the application aspect by discussing practical motivations and real-world relevance, for example by consulting studies like “Dynamics of fractional order delay model of coronavirus disease”, which demonstrate the effectiveness of fractional models in applied sciences.

10. With these improvements—particularly a more comprehensive, well-motivated Introduction and clearer justification of the analytical and fractional approaches—the manuscript would be significantly strengthened and suitable for publication after minor to moderate revision.

11. What is the rationale for selecting this specific fractional derivative, and how does it compare with other commonly used fractional operators? 
12. Are the physical assumptions and boundary/initial conditions clearly justified and realistic for the modeled problem?
13. Is the analytical solution method sufficiently general, or does it rely on restrictive assumptions that limit applicability? 
14. How is the validity of the analytical results verified (e.g., comparison with limiting classical cases or existing results)?
15. Are all mathematical steps in the solution procedure presented clearly enough to ensure reproducibility?
16. How sensitive are the results to variations in key fractional-order parameters?
17. Is numerical validation or graphical comparison used to support the analytical findings, and if not, why? 
18. Are dimensionless variables and scaling arguments properly introduced and justified? 
19. Does the methodology adequately address potential sources of error or limitations of the analytical approach? 
20. Can the proposed method be extended to more general geometries or boundary conditions, and what are its current limitations? 

Regards 

Author Response

Reviewer 2

We would like to express our sincere gratitude to the Reviewer for their careful and comprehensive-evaluation of our article. We are particularly-encouraged by the Reviewer’s positive assessment of our mathematical-framework and the clarity of our presentation. Their thoughtful comments and constructive-suggestions have been instrumental in further enhancing the quality of this work. Following the Reviewer's recommendation, we have conducted a rigorous final review of the text to ensure complete technical and linguistic precision. Our point-by-point responses to the feedback are provided below.

I read the article with great interest.

• The manuscript addresses an important and timely problem and has the potential to make a meaningful contribution to the field.
• The overall structure of the paper is clear, and the objectives are well defined.
• The methodology adopted in the study is appropriate and logically developed.
• The results are presented clearly and supported by suitable figures and analysis.
• The discussion section is informative and links the findings with relevant existing literature.

Comment 1: However, the Introduction section requires extensive improvement to better position the novelty and significance of the work. In particular, the part related to analytical solutions should be expanded and strengthened. The authors are encouraged to consult relevant studies such as “A discussion on the Lie symmetry analysis, travelling wave solutions and conservation laws of new generalized stochastic potential-KdV equation” and “Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary” to enrich the analytical background.

Comment 2: For the fractional-order modeling, the manuscript should provide a solid and well-justified rationale explaining why fractional derivatives are employed, how they differ from classical derivatives, and why they are more impactful or realistic for the current problem.

Response1 and 2: We extend our sincere thanks to the reviewer for their constructive feedback. Based on these suggestions, the introduction has been significantly improved for greater clarity and ease of presentation. Furthermore, the discussion of analytical solutions has been broadened to encompass a wider range of relevant studies. Specifically, we have added a detailed analysis of “Analytical results on the unsteady rotational flow of fractional-order non-Newtonian fluids with shear stress on the boundary” and other related studies. Als, we have provided a detailed justification in the introduction for the use of fractional derivatives. Unlike conventional derivatives, which assume an immediate and local response. We thank the reviewer once again for his valuable suggestion which helped to enrich the quality of the manuscript.

Comment 3: The Introduction would also benefit from referencing recent analytical works on fractional operators, such as “Generalized Mittag-Leffler Kernel Form Solutions of Free Convection Heat and Mass Transfer Flow of Maxwell Fluid with Newtonian Heating: Prabhakar Fractional Derivative Approach”, to better highlight the mathematical relevance of the chosen fractional framework.

Response: We extend our sincere thanks to the reviewer for this constructive suggestion. We agree that referencing recent analytical developments in fractional operators strengthens the mathematical foundation of our model. Accordingly, we have updated the introduction to include a discussion of the diverse applications of fractional derivatives, with particular emphasis on the application of the Prabhakar fractional derivative and generalized Mitag-Leffler nuclei in heat transfer problems. This addition helps position our three-phase-lag fractional (FTPL) model within the broader framework of rigorous analytical research. Thank you again for the suggestion, which has enriched our work.

 

Comment 4: The authors should further strengthen the application aspect by discussing practical motivations and real-world relevance, for example by consulting studies like “Dynamics of fractional order delay model of coronavirus disease”, which demonstrate the effectiveness of fractional models in applied sciences.

Response: We thank the reviewer for this constructive suggestion. We agree that highlighting the practical implications of fractional order models enhances the influence of the research. Based on the reviewer's recommendation, we have included a discussion of the practical significance of these models, specifically referencing the proposed study on coronavirus dynamics. We emphasized how fractional derivatives, along with time delays, provide a more accurate memory-based framework for modeling biological transmission compared to classical integer order models. Thank you again for the constructive suggestion.  

Comment 5: With these improvements—particularly a more comprehensive, well-motivated Introduction and clearer justification of the analytical and fractional approaches—the manuscript would be significantly strengthened and suitable for publication after minor to moderate revision.

Response: We thank the reviewer for their positive assessment of our work and valuable suggestions. We agree that strengthening the motivation in the introduction and clarifying the methodological choices will improve the impact and readability of the research. We have revised the research accordingly. Thank you for enriching the discussion, which has greatly enhanced our study.

Comment 6: What is the rationale for selecting this specific fractional derivative, and how does it compare with other commonly used fractional operators? 

Response: We thank the reviewer for this valuable question about the mathematical basis of our model. The choice of fractional derivative used in this study is based on several key factors:

First, the effects of memory and heredity: Unlike conventional derivatives, the chosen fractional operator is non-local. It incorporates a convolutional kernel that takes into account the history of thermal and mechanical states. This is physically essential for micro-structures, where heat transfer is not instantaneous but rather influenced by previous temperature gradients.

Second, the kernel's versatility: We chose this operator because its kernel effectively bridges the gap between different transport systems. While other operators, such as Riemann-Liouville or Caputo derivatives, are commonly used, the fractional order approach incorporated into the three-phase-lag (TPL) model specifically addresses the phase lag behavior simultaneously. This provides a more balanced coupling than standard operators, which may only affect one variable.

Third, physical consistency at the microscopic level: Many commonly used fractional operators can lead to mathematical anomalies or non-zero constants for the derivative of a constant. The operator used here was chosen to ensure full compliance with the physical requirements of generalized thermoelastic model, such as the requirement of finite velocities for heat waves.

Finally, comparison with other operators: Although the Caputo derivative is preferred for its ability to handle classical initial conditions, and the Prabhakar derivative is known for its generalized Mittag-Leffler nucleus, the framework we have chosen is specifically optimized for the analysis of damping and thermoelastic vibrations. It offers a more direct physical explanation of the relaxation-time compared to the more abstract fractional nuclei used in purely mathematical contexts.

Comment 7: Are the physical assumptions and boundary/initial conditions clearly justified and realistic for the modeled problem?

Response: We appreciate the reviewer's focus on the model's foundations. We have ensured that all physical assumptions and boundary/initial conditions are rigorously justified to reflect the real-world constraints of the problem. As detailed in the revised manuscript, our approach is based on the following:

• Physical Assumptions: This research employs a unified analytical framework to study transient thermal-elastic vibration in microbeams. The underlying assumption is the application of a partial three-phase-lag (FTPL)-model. This model is specifically justified as a means of accurately describing the interrelated, scale-dependent thermal-elastic responses by incorporating complex thermal relaxation effects. This model is more realistic for microscale problems because it accounts for the memory effects and non-local behaviors inherent in advanced materials, which conventional models often fail to capture.
• Boundary and Initial Conditions: The model considers the micro-beam based on a two-parameter- model. This model is justified as a realistic representation for engineering applications because it considers both compressive stiffness and shear interactions, providing a more consistent approach to describing complex stress-strain properties compared to simpler models.
• Limiter and Initial Conditions. Realistic Modeling: The article explicitly states that these conditions allow for a more accurate characterization of unconventional heat transfer and complex thermoelastic interactions inherent in microstructures. By incorporating foundation stiffness and thermal inertia effects as intrinsically related factors, the model provides critical design parameters directly applicable to the dynamic-stability of resonators in real-world micro/nanoelectronics systems.

Comment 8: Is the analytical solution method sufficiently general, or does it rely on restrictive assumptions that limit applicability? 

Response: We appreciate the reviewer's inquiry regarding the scope and generality of our analytical approach. We assure you that the solution-method is sufficiently broad and was strategically chosen to provide a unified-framework for characterizing complex thermoelastic behaviors that conventional models fail to capture. Rather than relying on restrictive assumptions, this methodology offers a more flexible and realistic approach to modeling advanced materials. By incorporating fractional order parameters and two parameter basis interactions, the proposed method extends the application of thermoelastic analysis to materials and structures exhibiting non-classical, scale-dependent, and memory-dependent phenomena. Consequently, the model provides robust and general predictive power, suitable for a wide range of high-resolution engineering-applications. Thank you for your interest in the quality of our article.

Comment 9: How is the validity of the analytical results verified (e.g., comparison with limiting classical cases or existing results)?

Response: The authors appreciate the reviewer's inquiry regarding the validation of our analytical-results. We agree that rigorous validation is essential to ensure the integrity of scientific-research. The analytical results of this study were validated using a systematic comparative framework. This process involved two main approaches: first, a direct comparison of our results with established and available results in the scientific literature; and second, the reduction of our generalized fractional-equations to classical boundary states to ensure consistency with conventional models. These comprehensive comparisons demonstrate the robustness of our mathematical-framework and its improved predictive accuracy for thermoelastic phenomena at the microscopic level. Thank you for raising this important point, which is fundamental to the quality of scientific research.

Comment 10: Are all mathematical steps in the solution procedure presented clearly enough to ensure reproducibility?

Response: We extend our sincere thanks to the reviewer for their keen interest in the clarity of our methodology and their inquiry regarding the reproducibility of the mathematical-steps.

We confirm that the mathematical procedures are presented in sufficient detail to ensure the reproducibility of our results by other researchers. The research demonstrates a precise and coherent logical sequence, progressing from the fundamental-physical laws and governing partial differential equations to the final analytical formulas. Through the clear application of the Laplace-transform, the boundary conditions of the Pasternak-foundation, and the numerical inversion coefficients, we have maintained a high level of transparency essential for scientific verification and reproducibility.

Comment 11: How sensitive are the results to variations in key fractional-order parameters?

Response: We appreciate your inquiry regarding the sensitivity of the results to changes in the principal fractional order coefficients. We assure you that this manuscript provides a detailed quantitative and qualitative assessment of the sensitivity of the results to changes in the fractional-order coefficient . Based on the findings presented, the sensitivity analysis can be summarized as follows:

• Impact on Heat Distribution: The temperature distribution is strongly influenced by the fractional order parameters. The study shows that is a crucial indicator for controlling the heat wave propagation velocity. Changes in this coefficient lead to variations in the heat peak and the depth of heat-penetration within the microbeam.
• Coupled Mechanical Response: Due to the interaction of the thermal and mechanical domains, displacement, bending moment, and deflection all exhibit strong dependence on the value of . The research demonstrates that a precise understanding and careful selection of the fractional order coefficient are essential for accurate stress-strain analysis in the structures of MEMS.
• Increased Degrees of Freedom: The research highlights that introducing the fractional-order parameterssignificantly improves our ability to model complex thermal-elastic behavior by providing an additional degree of freedom. This allows the model to detect subtle-damping effects and extreme stress distributions that conventional models typically underestimate.
• Predictive Accuracy: Sensitivity analysis confirms that the fractional three-phase-lag (FTPL) model offers superior predictive power, particularly in characterizing thermal wave scattering. Small variations in allow the model to switch between different-transport modes, making it highly sensitive and adaptable to the properties of different materials at the microscopic level.

Comment 12: Is numerical validation or graphical comparison used to support the analytical findings, and if not, why? 

Response: We appreciate your inquiry regarding the use of numerical verification or graphical comparison to support the analytical results. We confirm that both methods were extensively employed in this study to support and validate the analytical findings. This study relies on a robust combination of numerical-calculations and graphical-comparison to validate the analytical framework and demonstrate its practical applicability in the design of microelectromechanical systems (MEMS).

Comment 13: Are dimensionless variables and scaling arguments properly introduced and justified? 

Response: Thank you for your inquiry on this point. The dimensionless variables and measurement arguments have already been thoroughly presented and justified within this study. The rationale for the measurement approach can be summarized as follows:

• Unification of equations: To ensure the generality of the results and their independence from the material's dimensions or units, the governing equations were transformed using a consistent set of dimensionless-parameters. This allows for a more accurate comparison between different beam geometries at the microscale.
• Scale-based justification: The measurement arguments are specifically designed for the microscale, where the surface-to-volume ratio and thermal relaxation times become the dominant factors. This is justified by the need to observe scale-dependent phenomena, which characterize micro-electromechanical systems and are absent in macroscale models.
• Simplification of complex couplings: The introduction of dimensionless variables simplifies the handling of the complex-coupling between the fractional three-phase-lag (FTPL) model and Pasternak-foundation parameters. By standardizing variables such as temperature, displacement, and time, the physical influence of the fractional-order parametersand the stiffness of the foundation can be clearly isolated and defined during parametric analysis.
• Compatibility with numerical inversion: The dimensionless form is essential for the stability and accuracy of the Laplace-transform and its numerical inversion, ensuring an accurate representation of dynamic stability and diffusion-characteristics in numerical results.

Comment 14: Does the methodology adequately address potential sources of error or limitations of the analytical approach? 

Response: We appreciate the reviewer's attention to these methodological considerations. Indeed, this framework addresses potential sources of error by replacing restrictive classical assumptions with more sophisticated analytical tools. Specifically, the fractional three-phase-lag (FTPL) model resolves the infinite heat diffusion velocity paradox, while Pasternak-foundation provides a more realistic calculation of shear interactions to ensure the reliability of the foundation. Furthermore, we maintain scale accuracy by using non-local elasticity to correct for size effects. The entire analytical procedure is validated by comparing the results with classical boundary cases, ensuring a robust and mathematically consistent solution for micro-scale applications.

Comment 15: Can the proposed method be extended to more general geometries or boundary conditions, and what are its current limitations? 

Response: We appreciate the reviewer's interest in the scope of the proposed methodology. Indeed, this methodology is designed as a unified analytical framework that can be extended to more general geometries and diverse boundary-conditions, although some computational considerations are involved. Therefore, this methodology provides a solid and generalizable foundation for elastic thermal analysis. Its underlying logic which combines non-local elasticity with fractional-order thermal conductivity is intentionally broad enough to allow for future-applications in more complex structural-designs.

 

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This study introduces a unified analytical framework for investigating the transient thermoelastic vibration of a micro-scale-beam resting on a dual-parameter foundations. For this purpose, the authors apply the fractional three-phase lag (FTPL) generalized thermoelastic-model to accurately characterize the scale-dependent, coupled thermal and elastic responses. In general, the work is valuable. The analysis results can be helpful for the design and optimization of the micromachine structures. Therefore, the following revision suggestions should be considered in the revision stage.

1. In abstract, at line 26, the authors claim “foundation stiffness and thermal inertial effects are intrinsically linked”, however, the reviewer cannot find this link from this study.
2. Although in abstract, the author stated that “The calculated results are systematically compared with established classical theories to validate the model's robustness …”, the analytical solution obtained in this study has not been validated.
3. Now that the J1, J2 and J3 after Equation (24) are the same as those after Equation (20), they don't need to present it again. J1, J2 and J3 can be listed and numbered separately as an equation to facilitate later mention. Similar improvement may be applied for J4 after Equation (28).
4. The title for Section 5 is inappropriate, not only is there a problem with varying font sizes, but there are also issues with the wording.
5. At lines 289-292, the geometric and physical parameters of the micro-scale-beam, from [17] may be given in the form of table, thus the interested readers can see the authors' taken values at a glance.
6. The authors need to re-read carefully the entire text to avoid any typos, for example, at lines 89, 133, 134, 237 and so on.
7. The authors should check every value carefully presented in tables to avoid errors. For example, in Table 1, when x is 2, the FLS model gives the value of 0.000754267, which is obviously wrong since it is greater than the value of 0.000327042 when x is equal to 1.9.
8. In Table 2, the style of the data should be consistent, so that the interested readers can see how it changes at a glance, rather than mixed. Similar issues may be found in last line in Table 3, the first and last lines in Table 4, and so on.
9. For all figures, the scale size of the axis is too small to see clearly, in addition, the number style should be uniform, and the quality of the figures also needs to be further improved.
10. The conclusion section should list the main findings of this study, rather than the purpose for which these findings helped to achieve the engineering design. Therefore, the wording and style of the conclusion section need to be further adjusted.

Author Response

Reviewer 3

We are grateful to the Reviewer for their thorough and supportive-review. We appreciate the acknowledgment that our methodology and results are well presented and coherent. To address the Reviewer's final-suggestion, we have meticulously re-examined the entire article, polished the language and verified all data to ensure the work is error free. Below, we present our specific-responses to each of the Reviewer's comments.

This study introduces a unified analytical framework for investigating the transient thermoelastic vibration of a micro-scale-beam resting on a dual-parameter foundations. For this purpose, the authors apply the fractional three-phase lag (FTPL) generalized thermoelastic-model to accurately characterize the scale-dependent, coupled thermal and elastic responses. In general, the work is valuable. The analysis results can be helpful for the design and optimization of the micromachine structures. Therefore, the following revision suggestions should be considered in the revision stage.

Comment 1: In abstract, at line 26, the authors claim “foundation stiffness and thermal inertial effects are intrinsically linked”, however, the reviewer cannot find this link from this study.

Response: We thank the reviewer for their constructive feedback, which contributes to improving the quality of our work. The term "intrinsic link" refers to the interrelationship within our mathematical framework, where the foundation stiffness and thermal inertia effects together determine the dynamic-response of the threshold. This is illustrated by:

• The governing equation (Eq. 8): The partial differential equation integrates the foundation parametersand thermal components into a single unified equation, demonstrating their mutual influence on mechanical stability.
• Thermal wave propagation: In the fractional three-phase-lag(FTPL) model, the thermal wave propagation is significantly influenced by the two-parameter foundation.
• The dimensional coefficients (Eqs. 27-29 in the revised manuscript): The coefficients from to  explicitly combine the foundation stiffness and thermal constants, mathematically linking the thermal moment to the bending stiffness.
• Parametric correlation: The numerical-results show that the underlying differences directly alter the temperature-distribution and mechanical responses, confirming that thermal inertia cannot be fully determined without considering the mechanical constraints.

Thank you again for raising this discussion, which improves the quality of our work.

Comment 2: Although in abstract, the author stated that “The calculated results are systematically compared with established classical theories to validate the model's robustness …”, the analytical solution obtained in this study has not been validated.

Response:  We thank the reviewer for his valuable feedback on the validation of our analytical solution. We recognize the importance of ensuring the robustness of the model and have provided evidence of its validity throughout the study, for example:

• Comparison with Established Theories: In Section 6, we demonstrate the possibility of simplifying our unified analytical framework to align with several well-known thermoelasticity models. By setting certain parameters to zero we show how our solution reduces to the classical coupled fractional thermoelasticity(FCTE) model and the fractional Lord and Schulman (FLS) model.
• Conformity with Previous References: The geometric and physical parameters used in our numerical results were directly adopted from previous references as [17]. Our results, as shown in Figures 1–12 and Tables 1–12, were systematically compared with these and other models to confirm the model's accuracy and originality.
• Visual and Numerical Verification: Section 7.1 and its accompanying Figures 1–4 and Tables 1–4 provide a direct graphical and numerical comparison between our fractional three-phase-lag (FTPL) model and existing models such as (FCTE) and (FLS). For example, Table 1 lists temperature values ​​across the different models, showing how they converge or diverge under specific conditions.
• Systematic Verification: The analytical solution was derived using the Laplace transform technique and verified by Riemann sum approximation, a standard procedure used in similar validated studies. We believe that these comparisons and reductions to special cases constitute strong validation of the proposed analytical solution.

We will emphasize this more clearly in the revised version of the study. We reiterate our thanks to you for raising the pivotal discussion that contributed to improving the quality of the article.

Comment 3: Now that the  and  after Equation (24) are the same as those after Equation (20), they don't need to present it again.  and   can be listed and numbered separately as an equation to facilitate later mention. Similar improvement may be applied for  after Equation (28).

Response:  We thank the reviewer for their constructive feedback on the presentation of equations and the definitions of coefficients  and . We agree that standardizing these definitions will improve the readability of the paper and facilitate future citations. Based on your suggestions, the values ​​for  and  have been listed and numbered as independent equations upon their first mention. All subsequent mentions have been updated to refer this newly numbered equation to avoid repetition. , which follows equation (28), was not mentioned in subsequent equations and therefore does not require additional numbering. We believe these improvements will contribute to a smoother and more professional presentation of our analytical framework. Thank you again for your valuable feedback.

Comment 4: The title for Section 5 is inappropriate, not only is there a problem with varying font sizes, but there are also issues with the wording.

Response: We extend our sincere thanks and appreciation to the reviewer for their valuable feedback on Section 5. We acknowledge some issues with the title and inconsistent font sizes. The section title has been revised, and the font size and style have been corrected to align with the titles of the other sections in the research. We have also conducted a thorough review of the research to ensure consistent formatting across all sections. Thank you again for your valuable feedback.

Comment 5: At lines 289-292, the geometric and physical parameters of the micro-scale-beam, from [17] may be given in the form of table, thus the interested readers can see the authors' taken values at a glance.

Response: We thank the reviewer for his valuable recommendation. We agree that presenting the engineering and physical parameters in a table will improve the manuscript's clarity for interested readers. Based on the values ​​derived from study [17] and detailed in the text, a table has been included in the revised manuscript. Sincerely,

Comment 6: The authors need to re-read carefully the entire text to avoid any typos, for example, at lines 89, 133, 134, 237 and soon.

Response: Thank you for carefully reading the manuscript and pointing out these typographical errors. We have thoroughly reviewed the entire text, paying particular attention to the lines you noted. We have also conducted an additional comprehensive spell-check and manual review of the document to ensure that all these errors have been corrected. We believe these changes have significantly improved the quality of the manuscript.

Comment 7: The authors should check every value carefully presented in tables to avoid errors. For example, in Table 1, when x is 2, the FLS model gives the value of 0.000754267, which is obviously wrong since it is greater than the value of 0.000327042 when x is equal to 1.9.

Response: We thank the reviewer for checking the numerical data in the tables. We appreciate you bringing to our attention the discrepancy in Table 1 regarding the temperature values. Upon investigation, we determined this to be a typographical error during the final formatting of the table. The correct value should reflect the decreasing temperature distribution along the x-axis, consistent with the physical behavior of the model shown in Figure 1. We carefully reviewed the software used in the Results section and then correctly inserted all tables (1–12) to ensure the consistency of the numerical data across all models (FCTE), (FLS), (FTPL), and (TPL). Thank you again for your interest and commitment to the quality of this article.

Comment 8: In Table 2, the style of the data should be consistent, so that the interested readers can see how it changes at a glance, rather than mixed. Similar issues may be found in last line in Table 3, the first and last lines in Table 4, and so on.

Response: We extend our sincere thanks to the reviewer for their valuable feedback on the formatting of the data in our tables. We agree that consistency is essential for clear comparisons. We have reviewed Table 2, Table 3, and Table 4 and noted the inconsistencies in the accuracy of decimal numbers and scientific notation mentioned in your comment. To address this issue, all numerical data in these tables has been standardized to a fixed number of decimal places (e.g., six decimal places) to facilitate comparisons between rows. Furthermore, in cases where values ​​are very small (such as the last rows of the tables), we have used a consistent format for scientific notation instead of mixing standard and exponential formulas. These stylistic improvements have also been implemented in the Numerical Results section to ensure consistency. Thank you again for your constructive feedback.

Comment 9: For all figures, the scale size of the axis is too small to see clearly, in addition, the number style should be uniform, and the quality of the figures also needs to be further improved.

Response: We appreciate your constructive feedback on the visual presentation of our results. We agree that clear and consistent figures are essential for effectively communicating our findings. In response to your feedback, we have improved the font size of the axis labels and numerical scales for all Figures (1- 12) to ensure clarity in the final version. We have also standardized the formatting of numbers across all axes for a uniform and professional appearance. Thank you for your valuable comments, which have significantly contributed to improving the quality of our manuscript.

Comment 10: The conclusion section should list the main findings of this study, rather than the purpose for which these findings helped to achieve the engineering design. Therefore, the wording and style of the conclusion section need to be further adjusted.

Response: We appreciate the reviewer's valuable feedback on the conclusion section. We agree that a strong conclusion should focus on the scientific findings of the study. Therefore, we have revised the conclusion to highlight the main results, and you will find it in the revised version of the paper. Thank you very much for all the comments that contributed to improving the overall quality of the research.

Author Response File: Author Response.pdf

Reviewer 4 Report

Comments and Suggestions for Authors

In this study, the authors employed the fractional three-phase lag model to investigate a generalized thermoelastic beams supported by two-parameter elastic foundation. Extensive comparative analysis and parameter research are conducted and some important findings are obtained. Totally speaking, the work is interesting, while at the same time this paper still need to be further improved, according to the following comments.

1. At present, what is the current state of numerical simulation for this problem? Did the authors consider using numerical simulation to verify the analytical results they obtained?
2. If there are experimental data for the same problem, the author should strive to use the experimental results to verify the validity of the obtained analytical solution.
3. The authors need to provide a simple diagram of the mechanical model of the beam, in which the geometric dimensions, the cross section type, the establishment of the coordinate system, the material properties, the elastic foundation, and the temperature change mode, of the beam, can all be clearly seen at a glance.
4. The beta 1-8 at lines 248-250 should be listed as a separate equation, like the previous J1-J3 and J4
5. The boundary condition of fixed ends presented in Equation (30) is problematic. Since for the fixed boundary conditions of beams, the deflection w and its first order derivative to x (that is, rotation angle) are zero at two ends. For the simply-supported end boundary conditions, the second order derivative of w to x (this is, the bending moment) is zero. Please check this boundary condition of beam ends.
6. Are Figures 1-4 and Tables 1-4 in one-to-one correspondence? If yes, why the data from the figures is not consistent to data from the tables. If no, what is their transformation relationship? For example, in Figure 1, the changing range of x is 0-2, while in Table 1, this range is changed as 1-2. Similar issues may be found between Figures 5-8 and Tables 5-8, as well as between Figures 9-12 and Tables 9-12.
7. The physical quantities A and rho in Equation (6) were not introduced when they first appeared in this paper, although they represent area and density, which are well-known concepts. Please review the entire text for similar issues.
8. One of the characteristics of this work is that the solutions to the problems are presented in an analytical manner. Given that there are many equations and derivations in this paper, the author should carefully check them to avoid any errors.
9. At lines 299 and 477, the authors refer to Section 8, which is obviously wrong, please check them.
10. Please kindly give all authors’ names in References. In addition, the style of References should be strictly follow the style from the template for this journal.

Author Response

Reviewer 4

We are grateful to the Reviewer for their thorough and supportive review. We appreciate his praise for our methodology and results, noting their quality and coherence. In response to his recent suggestion, we have meticulously re-examined the entire manuscript, refined the language and proofread all data to ensure the work is error-free. Below, we provide our detailed responses to each of the reviewer's comments.

In this study, the authors employed the fractional three-phase lag model to investigate a generalized thermoelastic beams supported by two-parameter elastic foundation. Extensive comparative analysis and parameter research are conducted and some important findings are obtained. Totally speaking, the work is interesting, while at the same time this paper still need to be further improved, according to the following comments.

Comment 1: At present, what is the current state of numerical simulation for this problem? Did the authors consider using numerical simulation to verify the analytical results they obtained?

Response: We thank the reviewer for their valuable and constructive comments on the numerical simulation and verification of our results.

Regarding the current state of numerical-simulations of this problem, the field is undergoing a shift from classical models based on Euler-Bernoulli and Fourier equations which fail to capture effects at small scales—to more sophisticated frameworks. Modern simulations now prioritize generalized thermoelasticity models (such as the CTE, LS, and TPL models) to account for finite heat diffusion velocities and thermal relaxation. Furthermore, the integration of fractional calculus (specifically, the FTPL model) has become the preferred method for accurately depicting complex memory effects and thermoelastic behavior in advanced materials. These models are often combined with two-parameter basic models, such as the Pasternak-foundation model, to improve the simulation of shear and compression interactions that are overlooked by simpler models.

To validate our analytical results, we systematically compared them with established classical theories, such as the (FCTE) and (FLS) models. We used geometric and physical parameters from previously documented studies and applied inverse numerical techniques, such as the Riemann-sum approximation, to transform the analytical Laplace field data into verifiable physical distributions. This transition from traditional Euler-Bernoulli and Fourier models to these advanced frameworks-along with the use of inverse Laplace-methods to transform mathematical derivations into concrete numerical proofs ensures the consistency of our results with existing scientific data and provides a more realistic analysis of shear-compression interactions-compared to Winkler's foundation approach.

Thank you again for your constructive comments.

Comment 2: If there are experimental data for the same problem, the author should strive to use the experimental results to verify the validity of the obtained analytical solution.

Response: We thank the reviewer for their valuable suggestion regarding the use of experimental data for verification. Currently, direct experimental data for the state of a micro-beam, based on the two-parameter-Pasternak model and subjected to the fractional three-phase-lag (FTPL) elastic convection, are limited in the published literature. This is primarily due to the difficulty of isolating and measuring the effects of thermal relaxation and high-frequency memory at the micro-scale in a laboratory setting.

In the absence of direct experimental results, we validated our analytical solution by verifying the boundary-state and simplifying our model to conform to classical theories. We found that the results perfectly match these established reference parameters. We also compared our results with data published in peer-reviewed scientific journals, using physical and engineering parameters from documented previous studies, and compared our numerical results with theirs to ensure consistency. Finally, we performed a mathematical verification using the Riemann-sum approximation to invert the numerical Laplace-transform, a method known for its high accuracy and stability in verifying complex analytical transformations.

We acknowledge that experimental verification is the gold standard, and we intend to continue using or incorporating this data into our future work as advanced measurement techniques for fractional thermal elasticity become more readily available.

Thank you again for your valuable suggestion, which will undoubtedly open new avenues for our future research.

Comment 3: The authors need to provide a simple diagram of the mechanical model of the beam, in which the geometric dimensions, the cross-section type, the establishment of the coordinate system, the material properties, the elastic foundation, and the temperature change mode, of the beam, can all be clearly seen at a glance.

Response: We appreciate the reviewer's constructive suggestion, and in response, we have included a comprehensive diagram illustrating the mechanical model in the revised manuscript. This diagram clearly demonstrates the key elements of our study. Thank you for this constructive suggestion, which greatly enhances the clarity and credibility of the manuscript.

Comment 4: The beta 1-8 at lines 248-250 should be listed as a separate equation, like the previous  and .

Response: We extend our sincere thanks to the reviewer for their constructive suggestion, which significantly improved the clarity and credibility of the article. Based on your recommendation, we have modified the research format to display coefficients from  to  in a separate equation. This modification ensures consistency and clarity, enhancing the readability of the mathematical equations and the overall flow of the text. We appreciate your attention to detail, as this adjustment contributes to a more professional and organized-presentation of the governing equations. 

 

Comment 5: The boundary condition of fixed ends presented in Equation (30) is problematic. Since for the fixed boundary conditions of beams, the deflection w and its first order derivative to x (that is, rotation angle) are zero at two ends. For the simply-supported end boundary conditions, the second order derivative of w to x (this is, the bending moment) is zero. Please check this boundary condition of beam ends.

Response: We are very grateful for this constructive suggestion, which significantly improves the clarity and reliability of the manuscript. Regarding your comment on Eq. (30), we have carefully reviewed the boundary conditions. You are correct that for fixed limbs, both the deflection  and the angle of rotation should be zero. In contrast, for simply supported limbs, both the deflection $w$ and the bending moment  should be zero. Upon review, we found a typographical error in the description of Eq. (30). The mathematical formulation was intended to represent the terms of simply supported limits. We have corrected the text to accurately reflect that the model uses simply supported limits, ensuring complete consistency between the mathematical results and the physical interpretation. Thank you for pointing out this error; it ensures the technical accuracy of our work. Sincerely.

Comment 6: Are Figures 1-4 and Tables 1-4 in one-to-one correspondence? If yes, why the data from the figures is not consistent to data from the tables. If no, what is their transformation relationship? For example, in Figure 1, the changing range of x is 0-2, while in Table 1, this range is changed as 1-2. Similar issues may be found between Figures 5-8 and Tables 5-8, as well as between Figures 9-12 and Tables 9-12.

Response: We extend our sincere thanks to the reviewer for this constructive suggestion, which has greatly enhanced the clarity and reliability of the manuscript. Regarding the comment on the data ranges, we acknowledge typographical errors in the section. We have conducted a thorough review of all figures (1-12) and tables (1-12) to ensure complete consistency. The discrepancies mentioned, such as the  range being (0-2) in Figure 1 and (1-2) in Table 1, are due to manual input errors during the formatting-process. These errors have been corrected to ensure that the data ranges in the tables perfectly match the intervals shown in the corresponding figures. We thank you for your patience and attention to detail, which has helped us improve the technical-accuracy of our presentation.

Comment 7: The physical quantities A and rho in Equation (6) were not introduced when they first appeared in this paper, although they represent area and density, which are well-known concepts. Please review the entire text for similar issues.

Response: We extend our sincere thanks to the reviewer for their constructive suggestion, which greatly enhanced the clarity and credibility of the research. We apologize for the unintentional error in defining these variables. In the revised version, we ensured that the cross-sectional area and density were accurately defined when they first appeared in Equation (6). Furthermore, based on your recommendation, we conducted a thorough review of the entire text to identify and define any other parameters or symbols that may have been introduced without prior explanation. Sincerely,

Comment 8: One of the characteristics of this work is that the solutions to the problems are presented in an analytical manner. Given that there are many equations and derivations in this paper, the author should carefully check them to avoid any errors.

Response: We extend our sincere thanks to the reviewer for their constructive suggestion, which greatly enhanced the clarity and credibility of the research.

We fully agree that the accuracy of the analytical-derivations is fundamental to the integrity of this work. In response to your recommendation, we conducted a thorough, line-by-line review of all the equations and mathematical derivations presented in the study. Specifically, we verified the consistency of the equations, the Laplace domain to ensure there were no omitted or erroneous terms in the calculations, and the correctness of the final solutions.

We are confident that this comprehensive review has eliminated potential errors and ensured the mathematical-accuracy of the solutions presented. Thank you very much for your constructive suggestions.

Comment 9: At lines 299 and 477, the authors refer to Section 8, which is obviously wrong, please check them.

Response: We extend our sincere thanks to the reviewer for their constructive suggestion, which contributed to improving the quality of our research. We apologize for the unintentional error in the internal references. "Section 8" was incorrectly referenced on lines 299 and 477 due to page renumbering during the review process. We have carefully reviewed the research and corrected these references to the correct section (Section 7: Numerical results). We have also ensured that all references to sections, figures, and tables in the research are accurate and consistent. Sincerely,

 

Comment 10: Please kindly give all authors’ names in References. In addition, the style of References should be strictly follow the style from the template for this journal.

Response: We extend our sincere thanks to the reviewer for their constructive suggestion regarding the references, which has significantly improved the manuscript's clarity and credibility. We have thoroughly reviewed the references section to ensure its complete compliance with the journal's requirements, particularly the complete author lists and adherence to the required formatting. We have also conducted a final review of the bibliography to guarantee the accuracy of each reference and its conformity to the required style. Sincerely,

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

The revised version is well improved and I have no further suggestions 

Reviewer 3 Report

Comments and Suggestions for Authors

The revised version may be accepted for publication.

Reviewer 4 Report

Comments and Suggestions for Authors

The paper may be accepted in current form.