Composite Material Elastic Effective Coefficients Optimization by Means of a Micromechanical Mechanical Model

Round 1

Reviewer 1 Report

This work is important in the field of composite materials within the framework of elastic media; therefore, their results are interesting and provide a theoretical basis, and engineering and computational aspects for the development of new models applied to the field of composite materials. I suggest to follow the review report.

Comments for author File: Comments.pdf

Author Response

We thank the reviewer for devoting his/her time to examine our paper and for providing constructive criticism and helpful suggestions. In preparing the revised manuscript, we took into account all comments, requests and suggestions, as explained next in detail. All the important changes in the text are highlighted by yellow color in the revised manuscript.

Modifications in Response to Comments of Reviewer

Answers are provided per comment.

The paper addresses the development of an efficient methodology to predict the effective elastic properties of periodic fiber reinforced composite materials with strongly bonded long fibers, based on the homogenization scheme and an optimization algorithm. In particular, the GMC (generalized method of cells) homogenization approach is combined with the covariance matrix adaptation evolution strategy (CMA-ES) to identify the effective elasticity tensor of periodic uniaxial fiber reinforced composite materials. The results show that the designed structure profiles have great influence on the effective properties of the present heterogeneous models.

This work is important in the field of composite materials within the framework of elastic media; therefore, their results are interesting and provide engineering and computational aspects for the development of new models applied to the field of composite materials.

The reviewer would like to make the following remarks:

1. The literature review of the introduction section is not sufficient in the framework of the micromechanical model applied to elastic heterogeneous media and applications. There are multiphase models based on the Mori-Tanaka method, generalized method of cells, generalized self-consistent scheme, asymptotic homogenization methods, representative cell method, FEM, and others, that address the study of the effective properties of elastic fiber-reinforced composites. Please, provide sufficient background and include all relevant references. Also, it should expose the main results obtained by previous works in this area, and it should be useful to point out the difference between this study and some other related works mentioned in the introduction.

Thank you for your comment. The introduction has been revised and appropriately highlighted.

The following references have been included

[4]      H. J. Bohm, A SHORT INTRODUCTION TO BASIC ASPECTS OF CONTINUUM MICROMECHANICS (ILSB-Arbeitsbericht 206). 2010.

[5]      J. D. Eshelby, “The determination of the elastic field of an ellipsoidal inclusion, and related problems,” Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, vol. 241, no. 1226, pp. 376–396, Aug. 1957, doi: 10.1098/rspa.1957.0133.

[7]      J. Pan and L. Bian, “A re-formulation of the Mori–Tanaka method for predicting material properties of fiber-reinforced polymers/composites,” Colloid and Polymer Science, vol. 297, no. 4, pp. 529–543, Apr. 2019, doi: 10.1007/s00396-019-04472-y.

[8]      M. Barral, G. Chatzigeorgiou, F. Meraghni, and R. Léon, “Homogenization using modified Mori-Tanaka and TFA framework for elastoplastic-viscoelastic-viscoplastic composites: Theory and numerical validation,” International Journal of Plasticity, vol. 127, p. 102632, Apr. 2020, doi: 10.1016/j.ijplas.2019.11.011.

[9]      S. Mercier, A. Molinari, S. Berbenni, and M. Berveiller, “Comparison of different homogenization approaches for elastic–viscoplastic materials,” Modelling and Simulation in Materials Science and Engineering, vol. 20, no. 2, p. 024004, Mar. 2012, doi: 10.1088/0965-0393/20/2/024004.

[10]    F. Desrumaux, F. Meraghni, and M. L. Benzeggagh, “Generalised Mori-Tanaka Scheme to Model Anisotropic Damage Using Numerical Eshelby Tensor,” Journal of Composite Materials, vol. 35, no. 7, pp. 603–624, Apr. 2001, doi: 10.1177/002199801772662091.

[11]    N. Charalambakis, G. Chatzigeorgiou, Y. Chemisky, and F. Meraghni, “Mathematical homogenization of inelastic dissipative materials: a survey and recent progress,” Continuum Mechanics and Thermodynamics, vol. 30, no. 1, pp. 1–51, Jan. 2018, doi: 10.1007/s00161-017-0587-5.

[12]    Y. BENVENISTE, “Revisiting the generalized self-consistent scheme in composites: Clarification of some aspects and a new formulation,” Journal of the Mechanics and Physics of Solids, vol. 56, no. 10, pp. 2984–3002, Oct. 2008, doi: 10.1016/j.jmps.2008.06.006.

[13]    Z. Hashin, “Thermoelastic properties of fiber composites with imperfect interface,” Mechanics of Materials, vol. 8, no. 4, pp. 333–348, Feb. 1990, doi: 10.1016/0167-6636(90)90051-G.

[14]    E. Hervé-Luanco and S. Joannès, “Multiscale modelling of transport phenomena for materials with n-layered embedded fibres. Part I: Analytical and numerical-based approaches,” International Journal of Solids and Structures, vol. 97–98, pp. 625–636, Oct. 2016, doi: 10.1016/j.ijsolstr.2016.05.015.

[15]    Y. Benveniste and G. W. Milton, “The effective medium and the average field approximations vis-à-vis the Hashin–Shtrikman bounds. II. The generalized self-consistent scheme in matrix-based composites,” Journal of the Mechanics and Physics of Solids, vol. 58, no. 7, pp. 1039–1056, Jul. 2010, doi: 10.1016/j.jmps.2010.04.013.

[16]    R. M. Christensen and K. H. Lo, “Solutions for effective shear properties in three phase sphere and cylinder models,” Journal of the Mechanics and Physics of Solids, vol. 27, no. 4, pp. 315–330, Aug. 1979, doi: 10.1016/0022-5096(79)90032-2.

[17]    Z. Sekkate, A. Aboutajeddine, and A. Seddouki, “Elastoplastic mean-field homogenization: recent advances review,” Mechanics of Advanced Materials and Structures, vol. 29, no. 3, pp. 449–474, Jan. 2022, doi: 10.1080/15376494.2020.1776431.

[18]    J. M. Ortolano, J. A. Hernandez, and J. A. Oliver, A Comparative Study on Homogenization Strategies for Multi-Scale Analysis of Materials, Monograph CIMNE., vol. 135. 2013.

[19]    J. Aboudi, “Micromechanical Analysis of Composites by the Method of Cells,” Applied Mechanics Reviews, vol. 42, no. 7, pp. 193–221, Jul. 1989, doi: 10.1115/1.3152428.

[20]    J. Wang, J. H. Andreasen, and B. L. Karihaloo, “The solution of an inhomogeneity in a finite plane region and its application to composite materials,” Composites Science and Technology, vol. 60, no. 1, pp. 75–82, Jan. 2000, doi: 10.1016/S0266-3538(99)00103-7.

[21]    A. S. Sangani and W. Lu, “Elastic coefficients of composites containing spherical inclusions in a periodic array,” Journal of the Mechanics and Physics of Solids, vol. 35, no. 1, pp. 1–21, 1987, doi: 10.1016/0022-5096(87)90024-X.

[22]    I. Cohen, “Simple algebraic approximations for the effective elastic moduli of cubic arrays of spheres,” Journal of the Mechanics and Physics of Solids, vol. 52, no. 9, pp. 2167–2183, Sep. 2004, doi: 10.1016/j.jmps.2004.02.008.

2. I recommend providing more details about of the GMC method, the statement of the main problem and boundary and interface conditions to make a more self-contained work.

Thank you for your comment. The paragraph  2.1 has been revised and appropriately highlighted.

3. The authors should prove that the created RUC satisfies the two following conditions: (1) periodic boundaries, (2) intersection of fiber reinforced

Thank you for your comment. All the experimental data used in our research to validate our model were carried out from various researchers mostly on cylindrical test specimens and collected in the Word Wide Failure exercise. Composites microstructure is not explicitly described in the aforementioned exercise, therefore since the AS4/3501-6 material was in the form of prepregs we assumed periodic configuration. We added the following sentence

Line 169 “The AS4/3501-6 material comes in the form of prepreg, which allows the approximation of its microstructure as a periodic. Stress and strain fields in such periodic configurations are investigated by a periodically repeating unit cell (RUC).”

4. I would like to suggest an analysis of limit cases, which could give to the reader more and better reliability of the proposed model. A theoretical validation with other theoretical approach is recommendable.

Thank you for your comment. As we mentioned on our research work we focused on the elastic behaviour of the composite behaviour. Therefore an analysis of limit cases to study inelastic response or failure is beyond the scope of the current research work. We aim to extend our research work to these fields in the near future.

5. There are references that are not found, please check them, see for example, in lines 88 and 316. Also, please check the reference style.

Thank you for your comment. Missing references has been corrected.

6. Some tensor notations should be checked.

Thank you for your comment. Theoretical part has been revised and appropriately highlighted.

7. The quality of the images should be improved

Thank you for your comment. The images have been revised.

8. Finally, the paper is well organized, but it needs grammatical and spell-check

Thank you for your comment. Grammatical errors have been corrected.

The present contribution has novelty aspects. Therefore, this work requires minor revision before to be published in Applied Mechanics

Reviewer 2 Report

R8: In what sense is it efficient? It is a numerical method which requires a specific expertise, costs for software licence, therefore implies resources that are not forgranded. There are analytical methods which are faster and can be programmed in a Excel format which deliver the same when not more accurate results. Thus please specify the word "efficient"

R50: See previous comment. Moreover, a software license have to reimburst and an expertise on the specific software and numerical analysis is required. On the contrary an analytical solution is faster, cheaper and requires only  basic knowledge, see e.g. https://doi.org/10.5281/zenodo.3878156. Thus, please elaborate on the argument why to chose a numerical analysis.

R59-63: Sentence too long, has to be revised. A reference is required on the Covariance Adaptation matrix optimization

R63-68: Sentence too long, has to be revised.

R88-89: Reference not found

R135-136: Figure 2 is blury. Has to be revised.

R140: "the knowledge of" is overflown, could be deleted

R209: "is excited" is commonly used for dynamic loads. Here you are talking about quai-static loads. Therefore, consider revising.

R259: use the same units for strain in the text and the diagrams. Otherwise, although you mentioned it correct, it is a bit confusing. 

R275: I am not convinced that the discripancies between your model and the experimental data are caused due to the different fiber volume fraction or the varaibility of the carbon fiber stiffness. My assumption is that the test specimens were misaligned i.e. the fibers were not parallel to the load direction, especially the tensile ones along the fibers. This will definately influence the longitudinal, the transverse stiffness and the shear modulus and will help your model to converge. My assumption is moreover based on the following facts: a. with the rule of mixtures, for a 60% FVF laminate I am calculating 136GPa as you do and b. the carbon fiber quality is not variating that much within a material batch i.e. I do trust the fiber stiffness that you are reporting. c. the specimens must have been cut grom a pre-preg laminate configuration with few or no transverse fibers, well oriented along one UD direction.     

Therefore, recalculate the stiffness properties assuming a fiber off-axis misalignment. Otherwise, you are basing your model on wrong assumptions which are also influencing the final fitting.

R277-278: Correct the sentence shifting

R280: "is estimated", rather it is derived

R283-286: "We notice...of fibers", The longitudinal stiffness is rather dominated from the fiber volume fraction and the texture of the fabric (wooven, NCF, pultruded, etc) i.e. on the local waviness. It does not have to do with the packing arrangement. See the well established "Rule of mixtures". Therefore, consider revising.

R302: "N23", rather v23

R316: MIssing reference

Figure 7: Why is the evolution of the Young's Modulus along the fibers not 100% linear? Are there any artifacts affecting the results? Experimentally and theoretically it is expected to be linear. 

Figure 7: Why are the matrix dominated properties change asynchronous in terms of the fiber volume fraction? Does this make sense? 

R318-321: You are founding all your research on the assumption that there is an effect of the fiber volume fraction.

1. Do you have any evidence of that e.g. what is the thickness of the referenced specimens and thus derive the expected FVF?

2. What if there is a fiber misalignment? Why did you exclude that? This would increase the transverse and shear stiffness while decreasing the longitudinal.

3. What is the texture of the tested fabrics? NCF, Prepregs, In mould laid rovings? Are there any transverse reinforcements on the ply level? This data will affect the layer stiffness. How do you cope with that? 

R321-323: In a numerical analysis the output is always dependent on the input. How do you evaluate/benchmark/validate your findings? How do they compare with experimental data? Please add a qualitative example from literature data in Figure 7 for the development of all mechanical properties, either experimental or analytical.

R384: The presented result have to be elaborated further. A Table is requiered comparing the predicted laminate mechanical properties compared with the experimental ones, e.g. the stiffness along the fibers is still 8% higher than the experimental. In terms of weight and therefore money for a stiffness driven design of a structure this is not the most optimal result. How do you quantify the nearly perfect match?  

Author Response

We thank the reviewer for devoting his/her time to examine our paper and for providing constructive criticism and helpful suggestions. In preparing the revised manuscript, we took into account all comments, requests and suggestions, as explained next in detail. All the important changes in the text are highlighted by yellow color in the revised manuscript.

Modifications in Response to Comments of Reviewer

Answers are provided per comment.

R8: In what sense is it efficient? It is a numerical method which requires a specific expertise, costs for software licence, therefore implies resources that are not forgranded. There are analytical methods which are faster and can be programmed in a Excel format which deliver the same when not more accurate results. Thus please specify the word "efficient"

Thank you for your comment. Efficiency of the applied numerical method lies on its robustness and versatility against analytical approaches. GMC is a free tool and therefore no paid software license is required, thus no resources are needed. Indeed, using such numerical method, a fundamental expertise is demanded, something that is also imperative in creating a script module in Excel (holding paid license) either.

Furthermore, this work presents a framework, that could be adopted and extended in the analysis of more complex and demanding composite structures using Finite Element Analysis. Thus, an analytical computation would be insufficient and cumbersome.

R50: See previous comment. Moreover, a software license have to reimburst and an expertise on the specific software and numerical analysis is required. On the contrary an analytical solution is faster, cheaper and requires only basic knowledge, see e.g. https://doi.org/10.5281/zenodo.3878156. Thus, please elaborate on the argument why to chose a numerical analysis.

Thank you for your comment. Truly, an analytical solution is faster and cheaper, but lacks the advantages of a numerical analysis as mentioned in our previous comment. Is it also noted that this work deals with the idea of coupling a numerical analysis to a optimization algorithm so as to produce high-fidelity models based on experimental data, that could then introduced in a numerical FEM analysis.

R59-63: Sentence too long, has to be revised. A reference is required on the Covariance Adaptation matrix optimization

Thank you for your comment. The sentence was revised and appropriately highlighted. The following references have been included. Changed have been highlighted in text.

[31]      Hansen, N., The CMA Evolution Strategy A Comparing Review. Towards a New Evolutionary Computation, 2006. 192(1): p. 75-102.

[34]      I. Zacharakis, D. Giagopoulos, A. Arailopoulos, O. Markogiannaki, Optimal Finite Element Modeling of Filament Wound CFRP Tubes Optimal Finite Element Modeling of Filament Wound CFRP Tubes, Engineering Structures. 253 (2022) 113808. https://doi.org/10.1016/j.engstruct.2021.113808.

R63-68: Sentence too long, has to be revised.

Thank you for your comment. The sentence was revised and appropriately highlighted.

R88-89: Reference not found

Thank you for your comment. Missing reference has been corrected.

R135-136: Figure 2 is blury. Has to be revised.

Thank you for your comment. Quality of Figure 2 has been enhanced.

R140: "the knowledge of" is overflown, could be deleted

Thank you for your comment. Phrase was deleted.

R209: "is excited" is commonly used for dynamic loads. Here you are talking about quai-static loads. Therefore, consider revising.

Thank you for your comment. The sentence was revised and appropriately highlighted.

R259: use the same units for strain in the text and the diagrams. Otherwise, although you mentioned it correct, it is a bit confusing.

Thank you for your comment. Units for strains were matched to the ones on the diagrams and appropriately highlighted.

R275: I am not convinced that the discripancies between your model and the experimental data are caused due to the different fiber volume fraction or the varaibility of the carbon fiber stiffness. My assumption is that the test specimens were misaligned i.e. the fibers were not parallel to the load direction, especially the tensile ones along the fibers. This will definately influence the longitudinal, the transverse stiffness and the shear modulus and will help your model to converge. My assumption is moreover based on the following facts: a. with the rule of mixtures, for a 60% FVF laminate I am calculating 136GPa as you do and b. the carbon fiber quality is not variating that much within a material batch i.e. I do trust the fiber stiffness that you are reporting. c. the specimens must have been cut grom a pre-preg laminate configuration with few or no transverse fibers, well oriented along one UD direction.

Thank you for your comment. Indeed, this is a very correct and important observation.

Therefore, recalculate the stiffness properties assuming a fiber off-axis misalignment. Otherwise, you are basing your model on wrong assumptions which are also influencing the final fitting.

Thank you for your comment. Following your assumption on the off-axis fiber misalignment, recalculating of the properties in order to produce a more reliable model, lies beyond the main scope of this work. This is because, the main goal of this work is to present a novel framework where coupling of a robust micromechanics method implemented through GMC software to a state-of-the-art optimization algorithm CMA-ES, could be utilized in order to finely tune the properties of a composite laminate component, that could be further confidently introduced in finite element analysis, through used defined material properties. The reliability of the experimental data used is not examined, experiments are not questioned and thus are regarded as credible. Since we have not conducted any experiments, no comments but only assumptions could be made that lay within an uncertainty area. The continuation of this work would be based exactly on this, that is conducting experiments, where all data are available, and assumptions could be made on a solid ground.

R277-278: Correct the sentence shifting

Thank you for your comment. Sentence shifting was corrected.

R280: "is estimated", rather it is derived

Thank you for your comment. “estimated” has been replaced by “derived”. Correction has been appropriately highlighted.

R283-286: "We notice...of fibers", The longitudinal stiffness is rather dominated from the fiber volume fraction and the texture of the fabric (wooven, NCF, pultruded, etc) i.e. on the local waviness. It does not have to do with the packing arrangement. See the well established "Rule of mixtures". Therefore, consider revising.

Thank you for your comment. We agree with the corrections and revised. Correction has been appropriately highlighted.

R302: "N23", rather v23

Thank you for your comment. Correction has been highlighted in text.

R316: MIssing reference

Thank you for your comment. Missing reference has been filled in.

Figure 7: Why is the evolution of the Young's Modulus along the fibers not 100% linear? Are there any artifacts affecting the results? Experimentally and theoretically it is expected to be linear.

Figure 7: Why are the matrix dominated properties change asynchronous in terms of the fiber volume fraction? Does this make sense?

Thank you for your comment. You are right obviously the behavior is linear. The error was in the scale of the Figure. Corrected.

R318-321: You are founding all your research on the assumption that there is an effect of the fiber volume fraction.

1. Do you have any evidence of that e.g. what is the thickness of the referenced specimens and thus derive the expected FVF?

Thank you for your comment. As mentioned previously the reliability of the experimental data used is not examined. So, there is no information about the thickness of the referenced specimens.

2. What if there is a fiber misalignment? Why did you exclude that? This would increase the transverse and shear stiffness while decreasing the longitudinal.

Thank you for your comment. As mentioned previously, this is a very important assumption. However, something like that would be thoroughly examined and addressed in case of self-conducted experiments. In this work took for granted the credibility of published experimental results and presented this numerical framework.

3. What is the texture of the tested fabrics? NCF, Prepregs, In mould laid rovings? Are there any transverse reinforcements on the ply level? This data will affect the layer stiffness. How do you cope with that?

Thank you for your comment. Please refer to the previous answer.

R321-323: In a numerical analysis the output is always dependent on the input. How do you evaluate/benchmark/validate your findings? How do they compare with experimental data? Please add a qualitative example from literature data in Figure 7 for the development of all mechanical properties, either experimental or analytical.

Thank you for your comment. We applied a two-step validation for our model.

The first validation of the predicted data was the comparison between predicted values and experimental data for constant fiber volume fracture Vf=0.60. As you can notice in Table.5, Table.6 and Table.7 the deviation between experimental and predicted data was bellow 20% for some effective properties and especially in the case of REC120 for E11, E22, v12 the predicted values were very close to the experimental (<15%). We believe that especially in the case of composite materials were the experimental data show a wide range of scatter, these deviations are acceptable and verify our approach. These trends can be found also in Mueller et al 1994.

In the case of influence of volume fraction, we validated qualitatively the results by comparing the curves of the predicted values of effective transverse Young’s modulus E22 and axial shear modulus G12 with those derived by the Method of Cells (MOC) for a glass-A/epoxy-A system, a graphite-A/epoxy-B system and a Modmor/epoxy-C system ["Micromechanics of Composite Materials,", Aboudi J., Elsevier 2013]. Aboudi also noticed a very good agreement for stiffnesses, whereas only reasonably good comparison was achieved for the transverse Poisson’s ratio. Similar results are presented in the same resource for effective properties predictions for a glass/epoxy composite (Tsai and Hahn, 1980) as a function of fiber volume fraction Vf for three different RUCs by using the Gerenalized Method of Cell (GMC) as a modeling scheme.

We revised the document and highlighted the new lines appropriately.

R384: The presented result have to be elaborated further. A Table is requiered comparing the predicted laminate mechanical properties compared with the experimental ones, e.g. the stiffness along the fibers is still 8% higher than the experimental. In terms of weight and therefore money for a stiffness driven design of a structure this is not the most optimal result. How do you quantify the nearly perfect match?

Thank you for your comment. A new Table 11 has been included comparing the predicted mechanical properties to the experimental ones. As far as the stiffness along the fibers  is concerned, a difference of 8.65% increase is predicted by the updated model. However, this prediction is only the tuned parameter, derived by the CMA-ES algorithm, aiming to minimize the discrepancies between experimental stress data and the equivalent GMC numerically computed. Thus, CMA-ES is used to reconcile discrepancies and achieve a high-fidelity analytical model that can confidently reflect the experimental measurements. The whole process does not aim to optimize a model in terms of weight and therefore cost of production. Concluding, the objective of the presented framework is not the optimization of the design of a structure, but the tuning of an analytical model using experimental information.

The nearly perfect match is numerically quantified by the overall minimum value computed by eq. 9 resulted in 9.1E-4 and visually confirmed by Figure 8.

Addition in text has been appropriately highlighted.

Reviewer 3 Report

The reviewed article (manuscript Applied Mechanics - 774584, titled: Composite material elastic effective coefficients optimization by means of a micromechanical mechanical model) presents interesting results in the field of the identify the effective elasticity tensor Cij of CFRP materials. In my opinion the topic is interesting, and the presented results are useful for the researchers in the related field. The methodology presented in the manuscript is based on a combination of the GMC micromechanical model and the state-of-the-art Covariance Matrix Adaptation optimization algorithm (CMA-ES). The results presented in this work confirmed the effectiveness of the proposed GMC - CMA-ES optimization scheme. I appreciate the effort that the authors have put in performing this study. I have no objections against the work by essence, but I have some comments concerning the improvement of article. It would improve the quality of the paper if the authors were willing to incorporate the following changes:

Older references are used in the article.

Line 88: make correction  in the sentence ... Error! Reference source  not found..

Line 165, line 175, Table 1 : make correction V_f , write the designation f (fraction) as a subscript

Line 246: make correction in Figure 4 - add a label (a) and (b)

Line 316: make correction  in the sentence ... Error! Reference source  not found..

Author Response

We thank the reviewer for devoting his/her time to examine our paper and for providing constructive criticism and helpful suggestions. In preparing the revised manuscript, we took into account all comments, requests and suggestions, as explained next in detail. All the important changes in the text are highlighted by yellow color in the revised manuscript.

Modifications in Response to Comments of Reviewer

Answers are provided per comment.

The reviewed article (manuscript Applied Mechanics - 774584, titled: Composite material elastic effective coefficients optimization by means of a micromechanical mechanical model) presents interesting results in the field of the identify the effective elasticity tensor Cij of CFRP materials. In my opinion the topic is interesting, and the presented results are useful for the researchers in the related field. The methodology presented in the manuscript is based on a combination of the GMC micromechanical model and the state-of-the-art Covariance Matrix Adaptation optimization algorithm (CMA-ES). The results presented in this work confirmed the effectiveness of the proposed GMC - CMA-ES optimization scheme. I appreciate the effort that the authors have put in performing this study. I have no objections against the work by essence, but I have some comments concerning the improvement of article. It would improve the quality of the paper if the authors were willing to incorporate the following changes:

Thank you for your comment.

Older references are used in the article.

Thank you for your comment. The following references have been added.

[4]   H. J. Bohm, A SHORT INTRODUCTION TO BASIC ASPECTS OF CONTINUUM MICROMECHANICS (ILSB-Arbeitsbericht 206). 2010.

[5]   J. D. Eshelby, “The determination of the elastic field of an ellipsoidal inclusion, and related problems,” Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, vol. 241, no. 1226, pp. 376–396, Aug. 1957, doi: 10.1098/rspa.1957.0133.

[7]   J. Pan and L. Bian, “A re-formulation of the Mori–Tanaka method for predicting material properties of fiber-reinforced polymers/composites,” Colloid and Polymer Science, vol. 297, no. 4, pp. 529–543, Apr. 2019, doi: 10.1007/s00396-019-04472-y.

[8]   M. Barral, G. Chatzigeorgiou, F. Meraghni, and R. Léon, “Homogenization using modified Mori-Tanaka and TFA framework for elastoplastic-viscoelastic-viscoplastic composites: Theory and numerical validation,” International Journal of Plasticity, vol. 127, p. 102632, Apr. 2020, doi: 10.1016/j.ijplas.2019.11.011.

[9]   S. Mercier, A. Molinari, S. Berbenni, and M. Berveiller, “Comparison of different homogenization approaches for elastic–viscoplastic materials,” Modelling and Simulation in Materials Science and Engineering, vol. 20, no. 2, p. 024004, Mar. 2012, doi: 10.1088/0965-0393/20/2/024004.

[10] F. Desrumaux, F. Meraghni, and M. L. Benzeggagh, “Generalised Mori-Tanaka Scheme to Model Anisotropic Damage Using Numerical Eshelby Tensor,” Journal of Composite Materials, vol. 35, no. 7, pp. 603–624, Apr. 2001, doi: 10.1177/002199801772662091.

[11] N. Charalambakis, G. Chatzigeorgiou, Y. Chemisky, and F. Meraghni, “Mathematical homogenization of inelastic dissipative materials: a survey and recent progress,” Continuum Mechanics and Thermodynamics, vol. 30, no. 1, pp. 1–51, Jan. 2018, doi: 10.1007/s00161-017-0587-5.

[12] Y. BENVENISTE, “Revisiting the generalized self-consistent scheme in composites: Clarification of some aspects and a new formulation,” Journal of the Mechanics and Physics of Solids, vol. 56, no. 10, pp. 2984–3002, Oct. 2008, doi: 10.1016/j.jmps.2008.06.006.

[13] Z. Hashin, “Thermoelastic properties of fiber composites with imperfect interface,” Mechanics of Materials, vol. 8, no. 4, pp. 333–348, Feb. 1990, doi: 10.1016/0167-6636(90)90051-G.

[14] E. Hervé-Luanco and S. Joannès, “Multiscale modelling of transport phenomena for materials with n-layered embedded fibres. Part I: Analytical and numerical-based approaches,” International Journal of Solids and Structures, vol. 97–98, pp. 625–636, Oct. 2016, doi: 10.1016/j.ijsolstr.2016.05.015.

[15] Y. Benveniste and G. W. Milton, “The effective medium and the average field approximations vis-à-vis the Hashin–Shtrikman bounds. II. The generalized self-consistent scheme in matrix-based composites,” Journal of the Mechanics and Physics of Solids, vol. 58, no. 7, pp. 1039–1056, Jul. 2010, doi: 10.1016/j.jmps.2010.04.013.

[16] R. M. Christensen and K. H. Lo, “Solutions for effective shear properties in three phase sphere and cylinder models,” Journal of the Mechanics and Physics of Solids, vol. 27, no. 4, pp. 315–330, Aug. 1979, doi: 10.1016/0022-5096(79)90032-2.

[17] Z. Sekkate, A. Aboutajeddine, and A. Seddouki, “Elastoplastic mean-field homogenization: recent advances review,” Mechanics of Advanced Materials and Structures, vol. 29, no. 3, pp. 449–474, Jan. 2022, doi: 10.1080/15376494.2020.1776431.

[18] J. M. Ortolano, J. A. Hernandez, and J. A. Oliver, A Comparative Study on Homogenization Strategies for Multi-Scale Analysis of Materials, Monograph CIMNE., vol. 135. 2013.

[19] J. Aboudi, “Micromechanical Analysis of Composites by the Method of Cells,” Applied Mechanics Reviews, vol. 42, no. 7, pp. 193–221, Jul. 1989, doi: 10.1115/1.3152428.

[20] J. Wang, J. H. Andreasen, and B. L. Karihaloo, “The solution of an inhomogeneity in a finite plane region and its application to composite materials,” Composites Science and Technology, vol. 60, no. 1, pp. 75–82, Jan. 2000, doi: 10.1016/S0266-3538(99)00103-7.

[21] A. S. Sangani and W. Lu, “Elastic coefficients of composites containing spherical inclusions in a periodic array,” Journal of the Mechanics and Physics of Solids, vol. 35, no. 1, pp. 1–21, 1987, doi: 10.1016/0022-5096(87)90024-X.

[22] I. Cohen, “Simple algebraic approximations for the effective elastic moduli of cubic arrays of spheres,” Journal of the Mechanics and Physics of Solids, vol. 52, no. 9, pp. 2167–2183, Sep. 2004, doi: 10.1016/j.jmps.2004.02.008.

Line 88: make correction  in the sentence ... Error! Reference source  not found..

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Line 165, line 175, Table 1 : make correction Vf , write the designation f (fraction) as a subscript

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Line 246: make correction in Figure 4 - add a label (a) and (b)

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Round 2

Reviewer 2 Report

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