Auxetic vibration behaviours of periodic tetrahedral units with a shared edge

A very low-frequency mode supported within an auxetic structure is presented. We propose a constrained periodic framework with corner-to-corner and edge-to-edge sharing of tetrahedra and develop a kinematic model incorporating two types of linear springs to calculate the momentum term under infinitesimal transformations. The modal analysis shows that the microstructure with its two degrees of freedom has both low- and high-frequency modes under auxetic transformations. The low-frequency mode approaches zero frequency when the corresponding spring constant tends to zero. With regard to coupled eigenmodes, the stress–strain relationship of the uniaxial forced vibration covers a wide range. When excited, a very slow motion is clearly observed along with a structural expansion for almost zero values of the linear elastic modulus.

referees' reports. Please find the referees' comments along with any feedback from the Editors below my signature.
We invite you to respond to the comments and revise your manuscript. Below the referees' and Editors' comments (where applicable) we provide additional requirements. Final acceptance of your manuscript is dependent on these requirements being met. We provide guidance below to help you prepare your revision.
Please submit your revised manuscript and required files (see below) no later than 7 days from today's (ie 08-Sep-2021) date. Note: the ScholarOne system will 'lock' if submission of the revision is attempted 7 or more days after the deadline. If you do not think you will be able to meet this deadline please contact the editorial office immediately.
Please note article processing charges apply to papers accepted for publication in Royal Society Open Science (https://royalsocietypublishing.org/rsos/charges). Charges will also apply to papers transferred to the journal from other Royal Society Publishing journals, as well as papers submitted as part of our collaboration with the Royal Society of Chemistry (https://royalsocietypublishing.org/rsos/chemistry). Fee waivers are available but must be requested when you submit your revision (https://royalsocietypublishing.org/rsos/waivers).
Thank you for submitting your manuscript to Royal Society Open Science and we look forward to receiving your revision. If you have any questions at all, please do not hesitate to get in touch.

Reviewer: 2
Comments to the Author(s) The paper presents the auxetic vibration behaviour of periodic tetrahedral units with a shared edge. According to the reviewer's opinion, the paper is well-structured and clear. The topic is interesting and falls within the aim of the journal. In addition, the results are well-presented and could be helpful to further develop the same topic. Therefore, the paper can be accepted for publication in the current form.

===PREPARING YOUR MANUSCRIPT===
Your revised paper should include the changes requested by the referees and Editors of your manuscript. You should provide two versions of this manuscript and both versions must be provided in an editable format: one version identifying all the changes that have been made (for instance, in coloured highlight, in bold text, or tracked changes); a 'clean' version of the new manuscript that incorporates the changes made, but does not highlight them. This version will be used for typesetting.
Please ensure that any equations included in the paper are editable text and not embedded images.
Please ensure that you include an acknowledgements' section before your reference list/bibliography. This should acknowledge anyone who assisted with your work, but does not qualify as an author per the guidelines at https://royalsociety.org/journals/ethicspolicies/openness/.
While not essential, it will speed up the preparation of your manuscript proof if you format your references/bibliography in Vancouver style (please see https://royalsociety.org/journals/authors/author-guidelines/#formatting). You should include DOIs for as many of the references as possible.
If you have been asked to revise the written English in your submission as a condition of publication, you must do so, and you are expected to provide evidence that you have received language editing support. The journal would prefer that you use a professional language editing service and provide a certificate of editing, but a signed letter from a colleague who is a native speaker of English is acceptable. Note the journal has arranged a number of discounts for authors using professional language editing services (https://royalsociety.org/journals/authors/benefits/language-editing/).

===PREPARING YOUR REVISION IN SCHOLARONE===
To revise your manuscript, log into https://mc.manuscriptcentral.com/rsos and enter your Author Centre -this may be accessed by clicking on "Author" in the dark toolbar at the top of the page (just below the journal name). You will find your manuscript listed under "Manuscripts with Decisions". Under "Actions", click on "Create a Revision".
Attach your point-by-point response to referees and Editors at Step 1 'View and respond to decision letter'. This document should be uploaded in an editable file type (.doc or .docx are preferred). This is essential.
Please ensure that you include a summary of your paper at Step 2 'Type, Title, & Abstract'. This should be no more than 100 words to explain to a non-scientific audience the key findings of your research. This will be included in a weekly highlights email circulated by the Royal Society press office to national UK, international, and scientific news outlets to promote your work.

At
Step 3 'File upload' you should include the following files: --Your revised manuscript in editable file format (.doc, .docx, or .tex preferred). You should upload two versions: 1) One version identifying all the changes that have been made (for instance, in coloured highlight, in bold text, or tracked changes); 2) A 'clean' version of the new manuscript that incorporates the changes made, but does not highlight them.
--An individual file of each figure (EPS or print-quality PDF preferred [either format should be produced directly from original creation package], or original software format).
--An editable file of all figure and table captions. Note: you may upload the figure, table, and caption files in a single Zip folder.
--If you are requesting a discretionary waiver for the article processing charge, the waiver form must be included at this step.
--If you are providing image files for potential cover images, please upload these at this step, and inform the editorial office you have done so. You must hold the copyright to any image provided.
--A copy of your point-by-point response to referees and Editors. This will expedite the preparation of your proof.

At
Step 6 'Details & comments', you should review and respond to the queries on the electronic submission form. In particular, we would ask that you do the following: --Ensure that your data access statement meets the requirements at https://royalsociety.org/journals/authors/author-guidelines/#data. You should ensure that you cite the dataset in your reference list. If you have deposited data etc in the Dryad repository, please only include the 'For publication' link at this stage. You should remove the 'For review' link.
--If you are requesting an article processing charge waiver, you must select the relevant waiver option (if requesting a discretionary waiver, the form should have been uploaded at Step 3 'File upload' above).
--If you have uploaded ESM files, please ensure you follow the guidance at https://royalsociety.org/journals/authors/author-guidelines/#supplementary-material to include a suitable title and informative caption. An example of appropriate titling and captioning may be found at https://figshare.com/articles/Table_S2_from_Is_there_a_trade-off_between_peak_performance_and_performance_breadth_across_temperatures_for_aerobic_sc ope_in_teleost_fishes_/3843624.

At
Step 7 'Review & submit', you must view the PDF proof of the manuscript before you will be able to submit the revision. Note: if any parts of the electronic submission form have not been completed, these will be noted by red message boxes.

See Appendix A.
Decision letter (RSOS-210768.R1) We hope you are keeping well at this difficult and unusual time. We continue to value your support of the journal in these challenging circumstances. If Royal Society Open Science can assist you at all, please don't hesitate to let us know at the email address below.
Dear Dr Tanaka, I am pleased to inform you that your manuscript entitled "Auxetic vibration behaviours of periodic tetrahedral units with a shared edge" is now accepted for publication in Royal Society Open Science.
Please ensure that you send to the editorial office an editable version of your accepted manuscript, and individual files for each figure and table included in your manuscript. You can send these in a zip folder if more convenient. Failure to provide these files may delay the processing of your proof. You may disregard this request if you have already provided these files to the editorial office.
If you have not already done so, please remember to make any data sets or code libraries 'live' prior to publication, and update any links as needed when you receive a proof to check -for instance, from a private 'for review' URL to a publicly accessible 'for publication' URL. It is good practice to also add data sets, code and other digital materials to your reference list.
You can expect to receive a proof of your article in the near future. Please contact the editorial office (openscience@royalsociety.org) and the production office (openscience_proofs@royalsociety.org) to let us know if you are likely to be away from e-mail contact --if you are going to be away, please nominate a co-author (if available) to manage the proofing process, and ensure they are copied into your email to the journal. Due to rapid publication and an extremely tight schedule, if comments are not received, your paper may experience a delay in publication.
Please see the Royal Society Publishing guidance on how you may share your accepted author manuscript at https://royalsociety.org/journals/ethics-policies/media-embargo/. After publication, some additional ways to effectively promote your article can also be found here https://royalsociety.org/blog/2020/07/promoting-your-latest-paper-and-tracking-yourresults/.
On behalf of the Editors of Royal Society Open Science, thank you for your support of the journal and we look forward to your continued contributions to Royal Society Open Science.

(i) Point-by-Point Responses to Reviewer Comments
Below, we provide point-by-point responses to the reviewers' comments. In the revised manuscript, all changes are marked in blue.

Reviewer #1:
The paper investigates a periodic framework structure made of articulated regular tetrahedra. The framework is remarkable for several reasons: although over-constrained, it has two degrees of freedom for periodic deformations, the initial configuration has the eight tetrahedra of a unit cell positioned as a stellated octahedron and a particular auxetic deformation is singled out (unimode transformation

Response:
We are grateful for your supportive comments. The two typos pointed out by the reviewer were corrected in the revised manuscript.

Reviewer #2:
The paper presents the auxetic vibration behaviour of periodic tetrahedral units with a shared edge. According to the reviewer's opinion, the paper is well-structured and clear. The topic is interesting and falls within the aim of the journal. In addition, the results are well-presented and could be helpful to further develop the same topic. Therefore, the paper can be accepted for publication in the current form.

Response:
We are grateful for your supportive comments.

(ii) General Modifications Made by the Authors
After submission, we realized that some statements and mathematical descriptions should be improved to make this manuscript more readable. These modifications explained below did not change the structure, essential discussion or conclusions of the manuscript. In the revised manuscript, all changes are marked in red. We have also amended several symbolic errors and subtle expressions, which are not described below as they are of minor importance.

[Figures 7, 8, and 9]
In the original manuscript, we used the angular frequency variable with a unit of [kg −1/2 ·m −1 ]. = D 2 Three figures (Figs. 7-9) have either a vertical or horizontal axis corresponding to and therefore are not uniquely determined by the diagrams. To regain uniqueness, we replaced with in each of the 5 figures (see below) and revised the manuscript accordingly. The eigenvector in Equation (4.8) represents a frequency mode of the structure and therefore should not have a unit in general. In the original manuscript, however, we expressed that 2 = ( -2 ) #(4.8) with units for the moment of inertia. In the revised manuscript, we have amended this non-dimensional description with the following; 2 = ( 1 -2 ( ) ) .#(4.8) In consequence of this change in Eq. (4.8), we have also recalculated the term in the last paragraph of Section 4(a) in page 11, and substituted the following result: D 1 = -2 12 and D 3 = 13 2 12 [Equation (4.14)] We made an error in defining the stress in Eq. (4.14). In the revised manuscript, we redefine the stress * 1 as . Then, the RHS of Eq. (4.14) and of (4.15) are revised as instead of . The * 1 ≡ /(4 2 2 ) * 1 /2 * 1 definition is validated later in the same section, which is why we have added the following statement after the definition.
"The definition is validated later through our investigation of the elastic modulus." The modification of the stress equation yields the correct elastic modulus in Eq. Thus, all the relevant terms of are replaced with in the main text and Figure 9. To justify this D /(2 ) D / redefinition, we have added in page 13 in the revised manuscript, the following statement: "…, whereas the latter instance yields a force-displacement relationship * 1 (4 2 ) = 2 D (2 * 1 ) = 2 D (2 D 3 ) for the unimode transformation. The elastic relationship per unit cell validates the definition of . " * 1

[The 3rd paragraph of the introduction]
In the revised manuscript, to avoid misleading the readers, we have amended the statement referring to Ref.
"Recently, computations have predicted several types of polycrystal materials having negative values for the directional and/or the homogeneous Poisson ratios that arise through tetrahedral rotations [27]."

[Conclusions]
The structure we proposed in this study is locked physically in the higher vibration mode. However, our additional investigations after submission revealed that this vibration-contact problem can be avoided by considering the configuration of the unclosed structure, initially rotated by . During modelling, the 0 > 0 structure holds similar vibrational characteristics with those of the original closed structure (see Fig. S1 below). , obtained in multi-0 > 0 body dynamics simulations. Within , high or low frequency modes 0°< 0 < 2 -1 2 ≈ 49.6°h ave no interferences through self-contact of components. The inset shows an example with . 0 = 20°I n analysing the mechanism, the plots were generated during numerical simulations using multi-body dynamics software (Adams ® , MSC Software Corp.). Note that the numerical points for , indicated by 0 = 0 open circles, agree with theoretical predictions (dashed curves). Following these latest findings, we have revised the manuscript adding a statement in the 4th paragraph of Conclusions; "The self-contact problem can be resolved by an unclosed structure with applied in Eq. (2.12) that 0 > 0 holds similar vibrational characteristics (high and low frequency modes) in the allowable range of ."

[Data Accessibility]
We have modified the data accessibility statement and citation as follows: "