The Effect of Micromechanics Models: 2D and 3D Numerical Modeling for Predicting the Mechanical Properties of PP/Alfa Short Fiber Composites

Round 1

Reviewer 1 Report

The paper is devoted to the analysis of the mechanical properties of a composite consisting of the polypropylene matrix and the natural short fibres. The manuscript reports a comparison of the results obtained by using well known micromechanical models and the “Projected fibre” method.  The Young modulus of fibre has been identified by a simple method of projecting the experimental value of Young modulus of composite on the relationship between composite’s Young modulus and fibre’s Young modulus determined by computer models. The identification was carried out on the basis of different homogenization approaches, very large discrepancies between results corresponding to different homogenization methods can be noticed. The topic is, undoubtedly, of interest to the readers of the Journal of Composites Science. Nonetheless, the manuscript has got a significant number of weaknesses. The manuscript in its present form does not contain all details which are essential to verify the obtained results. I recommend major revision according to the comments listed below. 

Point 1. The literature references included in the Introduction are incomplete and does not take into account other important research works in the field of micromechanics and homogenization. Moreover, references to works presenting methods of the inverse identification of the properties of the phases were not mentioned in the Introduction at all. 

Point 2. The Young modulus of composite measured experimentally is used as input data to the identification of the fibre’s properties. In order to make the identification reliable, the orientation distribution of fibres assumed in models should agree with the real distribution in the specimen. Therefore, the question is if the orientation distribution of fibres in the specimen was measured?

Point 3. Young modulus and the Poisson ratio of the matrix, as well as the Poisson ratio of fibre, have not been given.

Point 4. Identification has been focused on the Young modulus of fibre, however, the paper does not pay any attention to the Poisson ratio of fibre. The question is if the Poisson ratio was chosen arbitrarily? Why it is also not a variable which should be identified? I suggest conducting the identification by solving an optimization problem with two variables (Young modulus and Poisson ratio).

Point 5. How the random orientation (2D or 3D?) was considered by micromechanical models? - equations describing the mean-field methods (table 1) seems to not contain any orientation averaging. Please provide more details on this aspect.

Point 6. There is missing information related to the aspect ratio of fibre assumed in models.

Point 7. The “Projected fibre” approach has been described very superficially. There is no information related to the size of the models, number of fibres, discretization, treatment of orientation distribution. There are very large discrepancies between results provided by 2D and 3D models (especially in the identification of fibre’s young Modulus), what is the reason for such a big difference? 

Point 8. Are both 2D and 3D PF models assume 2D random orientation distribution?

Point 9. Table 1 - I suggest formatting the equations in table 1 as the equations with appropriate numbering (and consequently delete table 1). Parameter “n” from the equations is not explained. 

Point 10. The Authors once use the term “self-consistent” another time “self-coherent” – please unify the description

Author Response

Responses to Reviewer 1

The paper is devoted to the analysis of the mechanical properties of a composite consisting of the polypropylene matrix and the natural short fibres. The manuscript reports a comparison of the results obtained by using well known micromechanical models and the “Projected fibre” method.  The Young modulus of fibre has been identified by a simple method of projecting the experimental value of Young modulus of composite on the relationship between composite’s Young modulus and fibre’s Young modulus determined by computer models. The identification was carried out on the basis of different homogenization approaches, very large discrepancies between results corresponding to different homogenization methods can be noticed. The topic is, undoubtedly, of interest to the readers of the Journal of Composites Science. Nonetheless, the manuscript has got a significant number of weaknesses. The manuscript in its present form does not contain all details which are essential to verify the obtained results. I recommend major revision according to the comments listed below. 

Response: Thank you for your comments. We have gone through your comments carefully

and tried our best to address them one by one. We hope the manuscript has been improved accordingly

Point 1. The literature references included in the Introduction are incomplete and does not take into account other important research works in the field of micromechanics and homogenization. Moreover, references to works presenting methods of the inverse identification of the properties of the phases were not mentioned in the Introduction at all. 

Response:

As suggested by the reviewer, we have included more references on homogenization models and inverse identification to our article.  Please see changes in red in the revised manuscript.

Point 2. The Young modulus of composite measured experimentally is used as input data to the identification of the fibre’s properties. In order to make the identification reliable, the orientation distribution of fibres assumed in models should agree with the real distribution in the specimen. Therefore, the question is if the orientation distribution of fibres in the specimen was measured?

Response:

Thank you very much for pointing this out. We have made several observations with  Binocular microscope and Scanning electron microscope to estimate the dimensions of the fibres after the elaboration process. This also allows as to confirm their random dispersion inside the specimen. An example of these images was added to the article. (Please refer to the figure in the attached author response file.)

Point 3. Young modulus and the Poisson ratio of the matrix, as well as the Poisson ratio of fibre, have not been given.

Response:

We apologize for not mentioning these informations in our original manuscript. The Young modulus and the Poisson ratio of the matrix were added to the revised manuscript. However, we didn’t measure the Young modulus and the Poisson ratio of alfa fibres. We mentioned in the manuscript that we used the values obtained by khaldi et al. ( 28.43 ± 4.07GPa for the Young’s modulus and 0,34 for the Poisson ratio)

Point 4. Identification has been focused on the Young modulus of fibre, however, the paper does not pay any attention to the Poisson ratio of fibre. The question is if the Poisson ratio was chosen arbitrarily? Why it is also not a variable which should be identified? I suggest conducting the identification by solving an optimization problem with two variables (Young modulus and Poisson ratio).

Response:

We fully agree with your comment. It’s an interesting approach. Unfortunately, we didn’t measure the Poisson ratio experimental values for our composites. In collaboration with our partners in the university Southern Brittany, we project to use the nanoindentation technique to estimate the Poisson ratio of the composites and alfa fibres. This will be the topic of our next article.

Point 5. How the random orientation (2D or 3D?) was considered by micromechanical models? - equations describing the mean-field methods (table 1) seems to not contain any orientation averaging. Please provide more details on this aspect.

Response:

Thank you for pointing this out. In fact, in both numerical and analytical models, the orientations are discretized into N families. Each of these N families of reinforcements has a particular orientation and is therefore considered as N different phases. 

Point 6. There is missing information related to the aspect ratio of fibre assumed in models.

Response:

We apologize for not mentioning this information in the submitted manuscript. Indeed, the aspect ratio of the fibre was estimated over 40 measurements on the Binocular microscope and the Scanning electron microscope observations. We found an aspect ratio of about 13.33. (Please refer to the figure in the attached author response file.)

Point 7. The “Projected fibre” approach has been described very superficially. There is no information related to the size of the models, number of fibres, discretization, treatment of orientation distribution. There are very large discrepancies between results provided by 2D and 3D models (especially in the identification of fibre’s young Modulus), what is the reason for such a big difference? 

Response:

We agree with the reviewer’s assessment. Accordingly, we have added these details to the revised manuscript. See change in red.

For the 3D approach it will be interesting to reconsider the technics used to generate the random distribution of fibres in order to improve results computed of fibre Young’s modulus (Obtained using inversed approach). 

Point 8. Are both 2D and 3D PF models assume 2D random orientation distribution?

Response:

Yes, both 2D and 3D models assume 2D random orientation distribution for the fibres.

Point 9. Table 1 - I suggest formatting the equations in table 1 as the equations with appropriate numbering (and consequently delete table 1). Parameter “n” from the equations is not explained. 

Response:

Table 1 was Revised accordingly

To represent the random distribution of the fibres inside the matrix, we consider each direction as a phase. The parameter “n” represent the number of phases. We added this information to the revised manuscript. See changes in red.

Point 10. The Authors once use the term “self-consistent” another time “self-coherent” – please unify the description

Response:

Sorry for our inattention. The correct term is “self-consistent”. We went through the entire manuscript to correct this mistake.

Author Response File: Author Response.docx

Reviewer 2 Report

The research presented is of some interest in the context of recent sensitivity to environmental problems. However, some aspects need to be pointed out.

Line 125

“displacement filed u”

Probably, this is “field”.

Line 176

“Young’s modulus of this fiber is 28.43 ± 4.07GPa”

Is 28.43 correct? It seems rather strange that it is the same upper bound value as a few lines before:

“In the literature, bundles of alfa fiber are known to have Young's modulus between 172 18.2 and 28.43 GPa”.

Based on Figure 4, a value greater than 40 GPa would be expected.

Lines 199-201

“This study was conducted with an alfa fiber Young’s modulus of 28.43 GPa and was limited to the fiber content of 40% by mass.”

“was limited” sounds as if you have performed the calculation only for the fiber content of 40% by mass. In contrast, Figure 5 shows that 40% is the upper bound value of the range of values examined. Please, rewrite in order to make the sentence clearer.

Lines 204-205

“From Figure 3, it can be seen how far apart the results of the models of Voigt, Reuss, Neerfeild-Hill, and Self coherent are from the experimental measurements.”

Is it actually Figure 3 or Figure 5?

The whole sentence is rather unclear and in need of revision.

Lines 215-225

Why did not you mention the Diluted model here? It is the best after Projected Fiber 2D and seems to provide even better (or comparable) results than Projected Fiber 2D for a fiber content of 40% by mass.

Lines 223-225 should actually be rewritten based on comparison with the Diluted model results.

Conclusions

Here, too, the results of the Diluted model are ignored. This is not honest from a scientific point of view. Authors are warmly invited to take them into account and comment on them appropriately.

Author Response

Responses to Reviewer 2

The research presented is of some interest in the context of recent sensitivity to environmental problems. However, some aspects need to be pointed out.

Response: Thank you for your comments. We have gone through your comments carefully

and tried our best to address them one by one. We hope the manuscript has been improved

accordingly.

Line 125

“displacement filed u” Probably, this is “field”.

Response:

Sorry for our inattention. We made the correction in the text. See changes in red.

Line 176

“Young’s modulus of this fiber is 28.43 ± 4.07GPa”

Is 28.43 correct? It seems rather strange that it is the same upper bound value as a few lines before:“In the literature, bundles of alfa fiber are known to have Young's modulus between 172 18.2 and 28.43 GPa”.Based on Figure 4, a value greater than 40 GPa would be expected.

Response:

We would like to thank the Reviewer for having raised this important point. Indeed, we have mentioned in a review article we published on alfa fibre fibre [El-Abbassi  et al.*]  that its Young’s modulus is between 18 and 58 GPa. However, in the studies where they found a high Young’s modulus, they have tested the single alfa fibre extracted through a complicated mechanical and chemical process. In our case, we used a simple alkaline treatment to extract bundles of alfa fibres with a diameter around 90 um. The works that had used a similar extraction process as us had their Young’s modulus between 18.2 and 28.43 GPa. Thereby, we choose to work with 28.43GPa. This information was added to the revised manuscript. See changes in red.

* El-Abbassi FE, Assarar M, Ayad R, Bourmaud A, Baley C. A review on alfa fiber (Stipa tenacissima L.): From the plant architecture to the reinforcement of polymer composites. Composites Part A: Applied Science and Manufacturing. 2020;128:105677

Lines 199-201

“This study was conducted with an alfa fiber Young’s modulus of 28.43 GPa and was limited to the fiber content of 40% by mass.”

“was limited” sounds as if you have performed the calculation only for the fiber content of 40% by mass. In contrast, Figure 5 shows that 40% is the upper bound value of the range of values examined. Please, rewrite in order to make the sentence clearer.

Response:

Thank you for pointing this out. We revised the sentence as follows: « Young’s modulus of composites reinforced with different percentages of alfa fiber from 0% to 40% by weight with an increment of 10% were calculated by using the homogenization models »

Lines 204-205

“From Figure 3, it can be seen how far apart the results of the models of Voigt, Reuss, Neerfeild-Hill, and Self coherent are from the experimental measurements.”

Is it actually Figure 3 or Figure 5?

The whole sentence is rather unclear and in need of revision.

Response:

 Thanks for your kind reminders. We revised the sentence as follows:

From Figure 5, it can be seen that the models of Voigt, Reuss, Neerfeild-Hill, and Self consistent didn’t give a good prediction of the composites PP/Alfa Young’s modulus. For example, the self-consistent, Voigt and Neerfeild have overestimated the Young’s modulus of PP/Alfa 30% by 34%, 140%, 54%, respectively. While the Reuss model have underestimated its Young’s modulus by 32% compared to the experimental results.

Lines 215-225

Why did not you mention the Diluted model here? It is the best after Projected Fiber 2D and seems to provide even better (or comparable) results than Projected Fiber 2D for a fiber content of 40% by mass.

Lines 223-225 should actually be rewritten based on comparison with the Diluted model results.

Response:

We understand your concern and have added to the revised manuscript a comparison between the Diluted model results and the experiments results. See changes in red.

Conclusions

Here, too, the results of the Diluted model are ignored. This is not honest from a scientific point of view. Authors are warmly invited to take them into account and comment on them appropriately.

Response:

We fully agree with your comment. We added in the conclusion that the Diluted model is more accurate than the Mori-Tanaka and the self-consistent methods. See changes in red.

Author Response File: Author Response.docx

Round 2

Reviewer 1 Report

The authors have improved the manuscript and responded to most of my comments and questions. Nonetheless, in my opinion, several points listed below still must be extended and clarified:

Point 1. In the response, the Authors stated that the random dispersion inside the specimen was confirmed by microscopic observations. The question is what was the measure of the randomness? – in this case, for instance, the second-order orientation tensor may be provided (this quantity is typically used for describing the orientation state of injection moulded composites). This issue is of paramount importance because the description of the orientation state may strongly influence the result of the inverse identification.

Point 2. In the response, the Authors claimed that N families of fibres were introduced to deal with the random orientation in micromechanical models. My question is how many of these families were applied. It will be also interesting to provide stiffness tensors obtained after homogenization involving discrete fibres families (for instance for one case of the volume fraction and material properties of the phases).

Point 3. The description of the FP approach has been extended in the revised version, nonetheless, it still does not cover details related to the numerical models (both 2D and 3D). The following information must be provided: the size of models (number of DOF), number of fibres (please add illustrations that presents dispersion of fibres used in 2D and 3D simulations), applied boundary conditions and how the homogenization is realised (an equation will be useful). It could be interesting to provide the effective stiffness tensors corresponding to the composite obtained by using 2D and 3D models (for instance for one case of the volume fraction and material properties of the phases).

Point 4. Conclusions- “…the Diluted model is more accurate than the Mori-Tanaka and self-consistent method” – this conclusion must be rephrased. In general, The Diluted model is less accurate than Mori-Tanaka and self-consistent models. Thus, instead, it will be better to state that usage of the Diluted model lead to the identification of the fibre’s Young modulus closer to the experimental value than M-T and S-C models (the Diluted model is not itself more accurate).

Point 5. I still encourage the Authors to provide additional literature references. In the revised version four additional references were added, however adding further ones is essential to provide enough wide background.

Author Response

The authors have improved the manuscript and responded to most of my comments and questions. Nonetheless, in my opinion, several points listed below still must be extended and clarified:

Thank you very much for agreeing with us to the intention of this manuscript. We have read your comments carefully and tried our best to address them one by one. We hope that the manuscript has been improved towards your standards after this revision.

Point 1. In the response, the Authors stated that the random dispersion inside the specimen was confirmed by microscopic observations. The question is what was the measure of the randomness? – in this case, for instance, the second-order orientation tensor may be provided (this quantity is typically used for describing the orientation state of injection moulded composites). This issue is of paramount importance because the description of the orientation state may strongly influence the result of the inverse identification.

Response:

We fully agree with your comment. The orientation state of the fibres has a great influence on the results of the inverse identification. In our case, the fibres length does not reach 15mm which can explain that we didn’t notice any preferred orientation that was induced by the extrusion and injection process. Indeed, the random distribution was confirmed by many microscopic observations on the fracture facies of our PP/Alfa composites (10%, 20%, 30%, and 40 wt%).  No other measurable experiments were performed. In order to take into account the reviewer's remark about the importance of observation issue, we will propose in a future work to use a tomography device that we have just acquired in our laboratory. The digitised measurement of the x,y coordinates of the ends for the 2D model and x,y,z for the 3D model, as well as the length and average diameter of the fibres, will allow an accurate construction of the finite element mesh of the random fibres, and consequently a reliable reverse identification.

Point 2. In the response, the Authors claimed that N families of fibers were introduced to deal with the random orientation in micromechanical models. My question is how many of these families were applied. It will be also interesting to provide stiffness tensors obtained after homogenization involving discrete fibers families (for instance for one case of the volume fraction and material properties of the phases).

Response:

It is an important point. Actually, each direction is a family. So, in total, we have 91 families. Therefore, the summation over the n heterogeneities becomes a summation over the n families of orientations. As the orientation is completely random, all the fibre volume fractions will be identical and equal to the ratio of the total volume fraction to the number of families. Then, we went from the fixed frame of reference Oxyz to the frame of reference linked to the fibre Ox'y'z ' by using the three Euler angles. To the question of providing stiffness tensors got after homogenizations procedures, we think that response to this question will perhaps require one or two additional pages of development (we will exceed the limit fixed by the editor). This also takes us away from the paper's aim, which consists in confronting two modeling methodologies for material properties of biocomposites: micromechanical (with different models) and numerical using special finite elements (our present "Projected Fiber's " approach ). We will plan to develop the homogenization aspects and declination into stiffness tensors in a future issue.

Point 3. The description of the FP approach has been extended in the revised version, nonetheless, it still does not cover details related to the numerical models (both 2D and 3D). The following information must be provided: the size of models (number of DOF), number of fibers (please add illustrations that presents dispersion of fibers used in 2D and 3D simulations), applied boundary conditions and how the homogenization is realised (an equation will be useful). It could be interesting to provide the effective stiffness tensors corresponding to the composite obtained by using 2D and 3D models (for instance for one case of the volume fraction and material properties of the phases).

Response:

Thank you for pointing this out. In fact, we didn’t go through all these details because the 2D PF approach has been detailed in two articles co-signed by Rezak AYAD and Hocine KEBIR, who are co-authors of present article. The 1st paper (Kebir and Ayad, 2014) is reference number 2 of this paper. The 2nd article (Tiar & al, 2016) has been added to the list of references (number 4). The 3D approach will be detailed, theoretically and numerically, in a future article (in progress). Let's remind once again that we are limited by the number of pages for this article

Point 4. Conclusions- “…the Diluted model is more accurate than the Mori-Tanaka and self-consistent method” – this conclusion must be rephrased. In general, The Diluted model is less accurate than Mori-Tanaka and self-consistent models. Thus, instead, it will be better to state that usage of the Diluted model lead to the identification of the fiber’s Young modulus closer to the experimental value than M-T and S-C models (the Diluted model is not itself more accurate).

Response:

This is a fair point. The conclusion has been changed accordingly. See changes in blue.

Point 5. I still encourage the Authors to provide additional literature references. In the revised version four additional references were added, however adding further ones is essential to provide enough wide background.

Response:

This is a fair point. We added more references on the works presenting methods of the inverse identification of the properties of the phases. See changes in blue.

Author Response File: Author Response.docx

Reviewer 2 Report

The authors have adequately addressed the comments of the reviewer. Just one last comment as far the conclusions is concerned:

Lines 280-281:

“Note that analytical methods overestimate the effective elasticity modulus of studied composites”

Also specify here (as already done in lines 246-247) that this is not true for the Reuss model: note that analytical methods overestimate the effective elasticity modulus of studied composites, apart from the Reuss model.

Author Response

The authors have adequately addressed the comments of the reviewer. Just one last comment as far the conclusions is concerned:

Thank you very much for agreeing with us to the intention of this manuscript. We have read your comments carefully and tried our best to address them one by one. We hope that the manuscript has been improved towards your standards after this revision.

Lines 280-281:

“Note that analytical methods overestimate the effective elasticity modulus of studied composites”

Also specify here (as already done in lines 246-247) that this is not true for the Reuss model: note that analytical methods overestimate the effective elasticity modulus of studied composites, apart from the Reuss model.

Response:

This is a fair point. The conclusion has been changed accordingly. See changes in blue.

Author Response File: Author Response.docx

Round 3

Reviewer 1 Report

The Authors responded to most of my questions convincingly. However, I still encourage to improve the current version of the manuscript by considering the following points:

Point 1. In response, the Authors stated that it is inconvenient to present the details related to the numerical models which I have requested i.a. due to a limited number of pages. Therefore, I propose to remove fig. 3 as well as the associated description which can be found in the previous Author’s work (lines 155-166 can be found in the paper [2]). Instead please provide the information which I have asked for in the previous report related to the current models (2D and 3D): the size of models (number of DOF), number of fibres (please add illustrations that presents dispersion of fibres used in 2D and 3D simulations), applied boundary conditions and how the homogenization is realised. I believe that presenting this data will make the manuscript more clear and interesting for readers. Eventually, to keep more space, fig. 4 can be combined with fig. 5 into one figure.

Point 2. In my opinion, it is important to add in the Conclusions the information related to the limitations of the current study which may also point out the possible directions of future research. I suggest adding to the Conclusions comments related for example to the uncertainty of the orientation distribution (it was not precisely measured as the Authors stated in response to the previous report) and arbitrary selection of the fibre’s Poisson ratio (which was not being identified).

Author Response

Reviewer 1

The Authors responded to most of my questions convincingly. However, I still encourage to improve the current version of the manuscript by considering the following points:

Point 1. In response, the Authors stated that it is inconvenient to present the details related to the numerical models which I have requested i.a. due to a limited number of pages. Therefore, I propose to remove fig. 3 as well as the associated description which can be found in the previous Author’s work (lines 155-166 can be found in the paper [2]). Instead please provide the information which I have asked for in the previous report related to the current models (2D and 3D): the size of models (number of DOF), number of fibres (please add illustrations that presents dispersion of fibres used in 2D and 3D simulations), applied boundary conditions and how the homogenization is realised. I believe that presenting this data will make the manuscript more clear and interesting for readers. Eventually, to keep more space, fig. 4 can be combined with fig. 5 into one figure.

Response:

As suggested by the reviewer, we have removed fig. 3 and the lines 155-166 and added:

In the projected fiber approach, the fiber is modeled with a linear 2-node element (1 Degree of freedom / Node) merged inside a 3-node Constant Strain Triangle (2 DOF/ Node) for the 2D approach, while for the 3 D projected fiber approach, it is merged inside a 4-node tetrahedron element nodes (3 DOF/ Node). Then, instead of using Eshelbi tensors or Euler angles to get the global composite rigidity, the Projected Fiber approach consider the nodal degrees of freedom vector of the fiber as a projection on that of the resin element [2]. The random aspect of the short fibers is represented by considering 91 families. Therefore, the summation over the n heterogeneities becomes a summation over the n families of orientations. As the orientation is completely random, all the fibers volume fractions will be identical and equal to the ratio of the total volume fraction to the number of families. Both 2D and 3D approach, use the same principle for the Projection and random distribution of the fibers. The stiffness of the composite is written:

Where C is the stiffness tensor (m for matrix, f for fiber, ph for famillie, c for composite), A_f, and l_f are respectively the cross-section area and the length of the fibre element. For a family of fibers oriented at an angle α, the expression of the orientation tensor   is as follows :

In order to compare the results obtained by the FP approach to the experimental values of the tensile test, we have chosen the boundary conditions of the R.V.E so as to simulate a tensile test. To this purpose, we applied on one of the ends of the R.V.E a loading along the axis (ox) and on the opposite side a simple support in order to prevent any translation along the x axis and the y axis. A mobile support is also applied on the lower side parallel to the x axis (Figure 4) to prevent movements along y. (Please refer to the figure in the attached author response file.)

Point 2. In my opinion, it is important to add in the Conclusions the information related to the limitations of the current study which may also point out the possible directions of future research. I suggest adding to the Conclusions comments related for example to the uncertainty of the orientation distribution (it was not precisely measured as the Authors stated in response to the previous report) and arbitrary selection of the fibre’s Poisson ratio (which was not being identified).

Response:

We agree that these are limitations to the current study. So, as suggested, We have added to the conclusion:

As the orientation state of the fibers has a great influence on the results of the inverse identification, we project to use a tomography device to identify exactly the volume fraction of each orientation. Furthermore, inverse identification based on two variables (Young’s modulus and Poisson ratio) will make the results more accurate and represents work in progress. This approach will be reported in a future work.

Author Response File: Author Response.docx