Adjustable positive and negative hygrothermal expansion metamaterial inspired by the Maltese cross

A metamaterial that can manifest both positive and negative coefficients of moisture and thermal expansion is presented herein, based on inspiration from the Maltese cross. Each unit of the metamaterial consists of a pair of equal-armed crosses pin-joined at their junctions to permit rotation, but elastically restrained by a bimaterial spiral spring, and four pairs of hinge rods to translate the relative rotational motion of the pair of equal-armed crosses into translational motion of the connecting rods. The effective coefficients of moisture and thermal expansion models were developed for small and large changes in the hygrothermal conditions using infinitesimal (approximate) and finite (exact) motion analyses, respectively, with the former giving constant effective coefficients with respect to environmental changes. Results indicate that the approximate method underestimates the magnitude of both the effective expansion coefficients under cooling and drying but overestimates magnitudes of both coefficients during heating and moistening, and that the change in both expansion coefficients is more drastic during cooling and drying than during heating and moistening. In addition to providing another micro-lattice geometry for effecting expansion coefficients of either signs, this metamaterial exhibits auxetic property.


Recommendation?
Accept with minor revision (please list in comments)

Comments to the Author(s)
In this manuscript, the author proposed a metamaterial based on the geometry of the Maltese cross. The effective coefficients of moisture and thermal expansions can be engineered in this type of metamaterials, to have either positive or negative values. The analytical mechanics study unveils that finite motion analysis produces more accurate prediction (compared to infinitesimal motion analysis) of the effective coefficients of hygrothermal expansion for large changes in hygrothermal conditions. The methods and results are sound and consistent. The manuscript can be considered for publication in Royal Society Open Science following the author' addressing the following minor issues.
(1) Out of the 32 references in total, 14 reference papers were written by the author himself, and another 5 by Prof J.N. Grima. It may leave the readers an impression that the field of this manuscript is not very active. Please rework the references to be more inclusive and systematic.
(2) Why is the Maltese cross-inspired geometry necessary in the design of hygrothermal metamaterials? Is it better than other simpler geometries which may also induce negative (or positive) effective coefficient of hygrothermal expansion?
(3) Page 8 Line 48: The author mentioned the "conservation of the arc length of the bimaterial strip." Is it valid to omit the axial expansion of the bimaterial strip so that the total arc length stays unchanged? Please comment on the assumptions here and their validity. How small is the hygrothermal expansion/contraction of the truss members (i.e., the arms and the rods)? Is it valid to omit their contribution to the overall effective coefficient of hygrothermal expansion?
(4) What is the range of length scales for the metamaterial to perform as predicted? Does the author anticipate any available form of experimental validation of this proposed Maltese crossinspired design?
(5) Can this Maltese cross-inspired layout be extended to 3D space for construction of 3D materials with tunable coefficient of hygrothermal expansion? (6) Typos and other issues Page 3 Line 57: "In" should be used in "is such a manner that." Page 19 Line 41: "bimaterrial" Figure 4 left: the schematic of spiral does not look perfect. Figure 6: A', B' and K' are not displayed correctly.

Review form: Reviewer 2
Is the manuscript scientifically sound in its present form? Yes

Are the interpretations and conclusions justified by the results? No
Is the language acceptable? Yes

Introduction
The state of art presented by the Authors is medium long (32 references) but there are still lots of interesting papers about metamaterials or structures with negative properties. Structures or metamaterials should be discussed more precisely in the introduction state of the art review section. Metamaterials are artificially made materials that do not exist in nature. The term derives from the Greek word meta, meaning beyond. More specifically, metamaterials are composites that have the desired combination of properties that cannot be obtained by combining the properties of their constituents. The term was coined in 1999 by Rodger Walser The unfortunate fact is that no definition exists that would be universally applicable or universally accepted.
There are a few books about Metamaterials that should be mentioned in the introduction, eg. Maier 2017 and. Some papers were published in the special issue of journals e.g. Special issue: "Advances in Mechanical Metamaterials" in Materials, https://www.mdpi.com/journal/materials/special_issues/mechanic_mater Simple mechanical and thermodynamical models, which show auxetic behavior was found in the 80s of the 20th century (see papers of Gibson; Almgren, Kolpakov, Wojciechowski). Since 1987, when negative Poisson's ratio foams have been developed by Lakes, it is known that materials and structures showing the negative Poisson's ratio do exist in nature. The mechanical response of these materials can be drastically changed depending on the number of applied loads, and auxetic materials are expected to have unusual, possibly enhanced geometrical and mechanical characteristics such as synclastic curvature in bending, deformationdependent permeability, high shear stiffness, indentation resistance, and fracture toughness, and improved damping and sound absorption properties. Some papers of Gibson and co-authors should be considered as papers important to the manuscript (Gibson 1981. Lorna Gibson one of the first researchers published works about cellular materials with negative PR. Auxetic material and structures or composite with auxetic phase/layer show better damping properties than materials with positive Poisson's ratio . Biomaterial metamaterials can exhibit temperature dependency of mechanical properties e.g. thermoauxeticity (Jopek 2018) or anomalous deformation ).

General remark
As it was mentioned by the Author "the Maltese cross metamaterial can deform with negative Poisson's ratio characteristics when a force is applied parallel to one of the on-axis directions". It will be very valuable to paper to present some basic characteristics of PR of structure.
Bimaterial spiral spring complicates the possibility of manufacturing the proposed metamaterial structure. Did Author consider a more simple mechanism than a spiral?
Other minor remarks P4. L47-49. Is it possible to construct "Maltese cross"-like structures with angles fi and theta which are different from 45 deg and 22.5 deg, respectively? How the properties of these structures will be different?
P17. L3-6. Please add references with common biomaterial strip analyzed in the paper.
P17. T1. Please add to table 1 values of PR and CME for given materials.
P19. L41-44. There are two assumptions: equal Young's modulus and equal CME (see Eq. 5.2), so there is not possible to make general conclusions from results. Some additional references to consider for extension and improvement of the introduction section Review form: Reviewer 3 Is the manuscript scientifically sound in its present form? No

Are the interpretations and conclusions justified by the results? Yes
Is the language acceptable? Yes

Do you have any ethical concerns with this paper? No
Have you any concerns about statistical analyses in this paper? No

Recommendation?
Major revision is needed (please make suggestions in comments)

Comments to the Author(s)
The manuscript is essentially interesting and shows how one can obtain anomalous behaviour from what is being termed as the 'Maltese Cross' motif as a result of changes in temperature or change in moisture content.
There are a number of aspects in the paper which should be improved. These include: (1) Assumptions for the model are not clearly stated. For example, the derivation assumes that the ligaments do not change length nor do they bend. The latter assumption is easy to justify if the hinges are friction-less and offer no resistance, otherwise even this assumption needs to be stated, justified, and discussed. (2) The mathematical model hinges on the validity of the equations 2.3 and 2.4. Are these derived by the author, or quoted from others? If derived by author, can a derivation be presented, and ideally some form of validation? (3) The main figures in the manuscript, in my opinion, are those in Figure 2 and 3. I would recommend some additional colours to fully distinguish the different component units in the system. This will highlight that the 'cross' is a single unit, but the short ligaments (which are coloured in the same red tone) are not. There are also some errors in the colours in Figure 3 (a grey ligament should be red).
Decision letter (RSOS-210593.R0) 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 Professor Lim
On behalf of the Editors, we are pleased to inform you that your Manuscript RSOS-210593 "Adjustable Positive and Negative Hygrothermal Expansion Metamaterial Inspired by the Maltese Cross" has been accepted for publication in Royal Society Open Science subject to minor revision in accordance with the 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 28-Jun-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.
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Reviewer comments to Author: Reviewer: 1 Comments to the Author(s) In this manuscript, the author proposed a metamaterial based on the geometry of the Maltese cross. The effective coefficients of moisture and thermal expansions can be engineered in this type of metamaterials, to have either positive or negative values. The analytical mechanics study unveils that finite motion analysis produces more accurate prediction (compared to infinitesimal motion analysis) of the effective coefficients of hygrothermal expansion for large changes in hygrothermal conditions. The methods and results are sound and consistent. The manuscript can be considered for publication in Royal Society Open Science following the author' addressing the following minor issues.
(1) Out of the 32 references in total, 14 reference papers were written by the author himself, and another 5 by Prof J.N. Grima. It may leave the readers an impression that the field of this manuscript is not very active. Please rework the references to be more inclusive and systematic.
(2) Why is the Maltese cross-inspired geometry necessary in the design of hygrothermal metamaterials? Is it better than other simpler geometries which may also induce negative (or positive) effective coefficient of hygrothermal expansion?
(3) Page 8 Line 48: The author mentioned the "conservation of the arc length of the bimaterial strip." Is it valid to omit the axial expansion of the bimaterial strip so that the total arc length stays unchanged? Please comment on the assumptions here and their validity. How small is the hygrothermal expansion/contraction of the truss members (i.e., the arms and the rods)? Is it valid to omit their contribution to the overall effective coefficient of hygrothermal expansion?
(4) What is the range of length scales for the metamaterial to perform as predicted? Does the author anticipate any available form of experimental validation of this proposed Maltese crossinspired design?
(5) Can this Maltese cross-inspired layout be extended to 3D space for construction of 3D materials with tunable coefficient of hygrothermal expansion? (6) Typos and other issues Page 3 Line 57: "In" should be used in "is such a manner that." Page 19 Line 41: "bimaterrial" Figure 4 left: the schematic of spiral does not look perfect.

Info
This manuscript draft presents an analysis of a metamaterial inspired by the Maltese cross. The effective coefficients of moisture and thermal expansion models for this structure were developed for small and large changes in the hygrothermal conditions using approximate and exact motion analyses, respectively.

Recommendation
The manuscript can be recommended for publication in this journal after the Authors consider the minor remarks presented below.

Introduction
The state of art presented by the Authors is medium long (32 references) but there are still lots of interesting papers about metamaterials or structures with negative properties. Structures or metamaterials should be discussed more precisely in the introduction state of the art review section. Metamaterials are artificially made materials that do not exist in nature. The term derives from the Greek word meta, meaning beyond. More specifically, metamaterials are composites that have the desired combination of properties that cannot be obtained by combining the properties of their constituents. The term was coined in 1999 by Rodger Walser The unfortunate fact is that no definition exists that would be universally applicable or universally accepted.
There are a few books about Metamaterials that should be mentioned in the introduction, eg. Maier 2017 and Capolino 2009. Some papers were published in the special issue of journals e.g. Special issue: "Advances in Mechanical Metamaterials" in Materials, https://www.mdpi.com/journal/materials/special_issues/mechanic_mater Simple mechanical and thermodynamical models, which show auxetic behavior was found in the 80s of the 20th century (see papers of Gibson; Almgren, Kolpakov, Wojciechowski). Since 1987, when negative Poisson's ratio foams have been developed by Lakes, it is known that materials and structures showing the negative Poisson's ratio do exist in nature. The mechanical response of these materials can be drastically changed depending on the number of applied loads, and auxetic materials are expected to have unusual, possibly enhanced geometrical and mechanical characteristics such as synclastic curvature in bending, deformationdependent permeability, high shear stiffness, indentation resistance, and fracture toughness, and improved damping and sound absorption properties. Some papers of Gibson and co-authors should be considered as papers important to the manuscript (Gibson 1981. Lorna Gibson one of the first researchers published works about cellular materials with negative PR. Auxetic material and structures or composite with auxetic phase/layer show better damping properties than materials with positive Poisson's ratio ). Biomaterial metamaterials can exhibit temperature dependency of mechanical properties e.g. thermoauxeticity (Jopek 2018) or anomalous deformation ).

General remark
As it was mentioned by the Author "the Maltese cross metamaterial can deform with negative Poisson's ratio characteristics when a force is applied parallel to one of the on-axis directions". It will be very valuable to paper to present some basic characteristics of PR of structure.
Bimaterial spiral spring complicates the possibility of manufacturing the proposed metamaterial structure. Did Author consider a more simple mechanism than a spiral?
Other minor remarks P4. L47-49. Is it possible to construct "Maltese cross"-like structures with angles fi and theta which are different from 45 deg and 22.5 deg, respectively? How the properties of these structures will be different?
P17. L3-6. Please add references with common biomaterial strip analyzed in the paper.
P17. T1. Please add to table 1 values of PR and CME for given materials.
P19. L41-44. There are two assumptions: equal Young's modulus and equal CME (see Eq. 5.2), so there is not possible to make general conclusions from results. The manuscript is essentially interesting and shows how one can obtain anomalous behaviour from what is being termed as the 'Maltese Cross' motif as a result of changes in temperature or change in moisture content.
There are a number of aspects in the paper which should be improved. These include: (1) Assumptions for the model are not clearly stated. For example, the derivation assumes that the ligaments do not change length nor do they bend. The latter assumption is easy to justify if the hinges are friction-less and offer no resistance, otherwise even this assumption needs to be stated, justified, and discussed.
(2) The mathematical model hinges on the validity of the equations 2.3 and 2.4. Are these derived by the author, or quoted from others? If derived by author, can a derivation be presented, and ideally some form of validation?
(3) The main figures in the manuscript, in my opinion, are those in Figure 2 and 3. I would recommend some additional colours to fully distinguish the different component units in the system. This will highlight that the 'cross' is a single unit, but the short ligaments (which are coloured in the same red tone) are not. There are also some errors in the colours in Figure 3 (a grey ligament should be red).
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Decision letter (RSOS-210593.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 Professor Lim, I am pleased to inform you that your manuscript entitled "Adjustable Positive and Negative Hygrothermal Expansion Metamaterial Inspired by the Maltese Cross" is now accepted for publication in Royal Society Open Science.
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Associate Editor Comments to Author (Dr Adil Al-Mayah):
The reviewers have made some excellent suggestions to improve the quality of the paper and its impact. Cited references need to be modified to include more related work related. Also, assumptions must be clearly stated. Figures modifications are highly recommended for a clear presentation.

Amendment
Thank you for your advice. I have increased the references from 32 to more than 50. Assumptions made have been clearly stated, such as the following in the Concept section.
Justification for the assumption appears as the last paragraph of the Results and Discussion section, as shown below.

The figures have been modified.
∆ > 0 or temperature increases ∆ > 0, the bimaterial spring becomes more curved. It is assumed that the crosses, hinge rods and connecting rods are rigid so that they do not elongate, shorten, bend or twist, and that the hinges are frictionless so as to permit free rotation at the pin joints. Based on the manner at which the bimaterial spring is attached as Having plotted the results of effective CTE and CME, it can now be justified why the CTE and CME of the crosses, hinge rods and connecting rods (collectively known as "linkages") can be neglected. With reference to table 1, the CTE of metals is generally in the order of 10 −6 −1 (to the order of 10 −5 −1 ) while perusal to figure 8 indicates that the calculated effective CTE is generally in the order of 10 −3 −1 (to the order of 10 −2 −1 ). Likewise, if we were to select a polymer for the linkages, then their CME would be in the order of 10 −3 [55] but reference to figure 10 reveals that the calculated effective CME is in the order of 10 0 . In both the CTE and CME cases, if the linkage material is selected from one of the materials of the bimaterial strip, the expansion coefficients are about 3 orders lower than those of the developed effective expansion coefficients, thereby justifying the assumption of zero expansion coefficients for the linkages in the analysis.

Reviewer: 1 Comments to the Author(s)
In this manuscript, the author proposed a metamaterial based on the geometry of the Maltese cross. The effective coefficients of moisture and thermal expansions can be engineered in this type of metamaterials, to have either positive or negative values. The analytical mechanics study unveils that finite motion analysis produces more accurate prediction (compared to infinitesimal motion analysis) of the effective coefficients of hygrothermal expansion for large changes in hygrothermal conditions. The methods and results are sound and consistent. The manuscript can be considered for publication in Royal Society Open Science following the author' addressing the following minor issues.

Comment
(1) Out of the 32 references in total, 14 reference papers were written by the author himself, and another 5 by Prof J.N. Grima. It may leave the readers an impression that the field of this manuscript is not very active. Please rework the references to be more inclusive and systematic.

Amendment
The number of references has almost doubled from 32 to 57, notably with the inclusion of works from various pioneers as well as more recent literature, as shown below:

Comment
(2) Why is the Maltese cross-inspired geometry necessary in the design of hygrothermal metamaterials? Is it better than other simpler geometries which may also induce negative (or positive) effective coefficient of hygrothermal expansion?

Amendment
The Maltese cross is not the only way, but its linkage mechanism-as proposed in the paperpermits the rotation of the hinge rods  to a greater extent than the rotation of the cross as shown in figure 7. This enhances the displacement of the connecting rods, which helps to increase the expansion coefficients. A simpler geometry has been attempted recently [49], but its magnitudes of the CTE and CME are lower by 3 orders in comparison to the current work. This is now added as the second last paragraph of the Results and Discussion section, as furnished below.:

Comment
(3) Page 8 Line 48: The author mentioned the "conservation of the arc length of the bimaterial strip." Is it valid to omit the axial expansion of the bimaterial strip so that the total arc length stays unchanged? Please comment on the assumptions here and their validity. How small is the hygrothermal expansion/contraction of the truss members (i.e., the arms and the rods)? Is it valid to omit their contribution to the overall effective coefficient of hygrothermal expansion?

Amendment
It has previously been shown in the appendix A1 of reference [45] that by using commonly available bimaterial strips, the axial expansion and contraction is negligible, and therefore the omission of axial length change is a valid assumption. For example, if one were to use the copper-steel bimetallic strip and change the temperature by 100K, the magnitude of the percentage error for assuming constant arc length is only 0.138%. This piece of information has been incorporated to the paragraph after equation (2.2) as shown below: 56. Wu H, Cheng Y, Liu W, He R, Zhou M, Wu S, Song X, Chen Y. 2016 Effect of the particle size and the debinding process on the density of alumina ceramics fabricated by 3D printing based on stereolithography. Ceram. Int. 42(15 In comparison to a related but simpler metamaterial geometry consisting of counter-rotating crosses without the hinge rods and spiral springs [49], the magnitudes of the current effective CTE and CME are greater by 3 orders. This is attributed to the greater extent of hinge rod rotation in comparison to the cross rotation, which effectively accentuates the hygrothermal strain and, thereby, enhances the coefficients of expansion. The hygrothermal expansion from the crosses, hinge rods and connecting rods can be neglected because their expansion coefficient is 3 orders lower than the calculated expansion coefficients which assumes zero expansion of the cross and the rods. For example, if the crosses and rods were to be selected from any one of the metals listed in table 1 (these are the metals used for the bimaterial strips), then their CTE would be from the order of 10 −6 −1 (to the order of 10 −5 −1 ). With reference to figure 8, the calculated effective CTE is from the order of 10 −3 −1 (to the order of 10 −2 −1 ). Likewise, if we were to select a polymer for the crosses and the rods, then their CME would be in the order of 10 −3 [55]. Perusal to figure 10 reveals that the calculated effective CME is in the order of 10 0 . This clarification is now added as the last paragraph of the Results and Discussion section as follows:

Comment
(4) What is the range of length scales for the metamaterial to perform as predicted? Does the author anticipate any available form of experimental validation of this proposed Maltese cross-inspired design?

Amendment
If the metamaterial is built by 3D printing using typical nanoparticle size of order 10 1 , the size of the metamaterial's repetitive unit can go down to the order of 10 2 . This calculation is now furnished in the 3 rd last paragraph of the Results and Discussion section It has previously been shown, in appendix A1 of reference [45], that the change to the arc length is marginal such that the error introduced for assuming constancy in arc length is negligible. For example, in the case of the commonly used copper-steel bimaterial strip, the error for assuming constant arc length is in the order of 10 −3 [45]. Suppose each equal-armed Having plotted the results of effective CTE and CME, it can now be justified why the CTE and CME of the crosses, hinge rods and connecting rods (collectively known as "linkages") can be neglected. With reference to table 1, the CTE of metals is generally in the order of I would anticipate some form of experimental validation in a few years' time.

Comment
(5) Can this Maltese cross-inspired layout be extended to 3D space for construction of 3D materials with tunable coefficient of hygrothermal expansion?

Amendment
It is possible to extend the Maltese cross metamaterial to a 3D version with tunable coefficient of hygrothermal expansion, so I am including this statement as a suggestion for future work in the Conclusion section.
(6) Typos and other issues Page 3 Line 57: "In" should be used in "is such a manner that." Page 19 Line 41: "bimaterrial" Figure 4 left: the schematic of spiral does not look perfect. Figure 6: A', B' and K' are not displayed correctly.

Amendment
These have been corrected.
The length scale depends on the manufacturing technique. It has been reported that aluminum oxide nanoparticles of size 50 [56] and more recently iron oxide nanoparticles of size 15 [57] have been successfully employed in 3D printing. Suppose a typical nanoparticle size of 30 is used for 3D printing, the spiral spring thickness ℎ 1 + ℎ 2 can be of thickness 600 while the pin joint diameter and the rod width can be 3 and 6 , respectively. If the lengths of the hinge rod and the cross arm be averaged to an order greater than their width, then a choice of ℎ = 43.3 and = 80 would satisfy the first of equation (3.3). If we let 1 = 2 = , as was adopted for plotting figures 8 (left) and figure 10, then the size of a repetitive unit is 2 1 0 = 2 2 0 = 247 . In other words, a metamaterial unit of length scale in the order of 10 2 can be achieved by 3D printing using nanoparticle size in the order of 10 1 .
metamaterial is recommended for future work. As with many other 2D metamaterials that have been extended to 3D, it is herein recommended that the Maltese cross metamaterial be extended to its 3D version for future work.

Introduction
The state of art presented by the Authors is medium long (32 references) but there are still lots of interesting papers about metamaterials or structures with negative properties. Structures or metamaterials should be discussed more precisely in the introduction state of the art review section. Metamaterials are artificially made materials that do not exist in nature. The term derives from the Greek word meta, meaning beyond. More specifically, metamaterials are composites that have the desired combination of properties that cannot be obtained by combining the properties of their constituents. The term was coined in 1999 by Rodger Walser The unfortunate fact is that no definition exists that would be universally applicable or universally accepted.
There are a few books about Metamaterials that should be mentioned in the introduction, eg. Maier 2017 and Capolino 2009. Some papers were published in the special issue of journals e.g. Special issue: "Advances in Mechanical Metamaterials" in Materials, https://www.mdpi.com/journal/materials/special_issues/mechanic_mater Simple mechanical and thermodynamical models, which show auxetic behavior was found in the 80s of the 20th century (see papers of Gibson; Almgren, Kolpakov, Wojciechowski).
Since 1987, when negative Poisson's ratio foams have been developed by Lakes, it is known that materials and structures showing the negative Poisson's ratio do exist in nature. The mechanical response of these materials can be drastically changed depending on the number of applied loads, and auxetic materials are expected to have unusual, possibly enhanced geometrical and mechanical characteristics such as synclastic curvature in bending, deformation-dependent permeability, high shear stiffness, indentation resistance, and fracture toughness, and improved damping and sound absorption properties. Some papers of Gibson and co-authors should be considered as papers important to the manuscript (Gibson 1981. Lorna Gibson one of the first researchers published works about cellular materials with negative PR. Auxetic material and structures or composite with auxetic phase/layer show better damping properties than materials with positive Poisson's ratio ). Biomaterial metamaterials can exhibit temperature dependency of mechanical properties e.g. thermoauxeticity (Jopek 2018) or anomalous deformation ).

Amendment
The author thanks the reviewer for suggesting a good historical background on metamaterials and auxetic systems, as well as the more recent literature. The Introduction section has been expanded as shown below: Metamaterials are materials whose micro-lattices are tailor-made so that effective characteristics are predominantly regulated by their microstructural lay-out instead of those by the base materials. Metamaterials are artificially-made materials that do not exist in nature. The term derives from the Greek word meta, meaning beyond. More specifically, metamaterials are composites that have the desired combination of properties that cannot be obtained by combining the properties of their constituents. The term was coined by Rodger Walser. The reader is referred to books that deal with electromagnetic metamaterials and their applications [6,7], elastic, acoustic and seismic metamaterials [8], and negative mechanical metamaterials [9], as well as special issues pertaining to metamaterials [10,11], to name a few. Gibson [12], Gibson et al. [13] and Gibson and Ashby [14] are one of the earliest to publish work on cellular materials with auxetic (negative Poisson's ratio) behaviour. Other pioneering works include those by Almgren [15], Kolpakov [16] and Wojciechowski [17,18], based on mechanical and thermodynamical models. Since 1987, when negative Poisson's ratio foams have been developed by Lakes [19], it is known that materials and structures showing the negative Poisson's ratio do exist in nature. The mechanical response of these materials can be drastically changed depending on the number of applied loads, and auxetic materials are expected to have unusual, possibly enhanced geometrical and mechanical characteristics such as synclastic curvature in bending, deformation-dependent permeability, high shear stiffness, indentation resistance, and fracture toughness, and improved damping and sound absorption properties.
Strek et al. [20] studied the contact problem of a composite plate covered with an auxetic layer. Auxetic material and structures or composite with auxetic phase/layer show better damping properties than materials with positive Poisson's ratio [21]. Some metamaterials have been shown to exhibit temperature dependency of mechanical properties, such as thermoauxeticity [22] or anomalous deformation [23,24]. Owing to the General remark As it was mentioned by the Author "the Maltese cross metamaterial can deform with negative Poisson's ratio characteristics when a force is applied parallel to one of the on-axis directions". It will be very valuable to paper to present some basic characteristics of PR of structure.

Amendment
The author thanks the reviewer for suggesting to explore the auxetic properties. A paper on mechanical properties of this metamaterial has just been submitted to a journal for consideration. As such, a brief remark is made in the Conclusion section as follows:

Comment
Bimaterial spiral spring complicates the possibility of manufacturing the proposed metamaterial structure. Did Author consider a more simple mechanism than a spiral?

Amendment
Yes, the author has considered a simpler mechanism consisting of a pair of counter-rotating crosses without the spiral spring, and the work was recently reported in reference [49], but the magnitudes of its CTE and CME are lower by 3 orders in comparison to those developed in this manuscript. This is now updated as the second last paragraph of the Results and Discussion section, as furnished below: Other minor remarks Comment P4. L47-49. Is it possible to construct "Maltese cross"-like structures with angles fi and theta which are different from 45 deg and 22.5 deg, respectively? How the properties of these structures will be different?
From figures 2 or 3, it can be inferred that in the absence of environmental change, the Maltese cross metamaterial can deform with negative Poisson's ratio characteristics when a force is applied parallel to one of the on-axes directions, with perfect auxeticity = −1 being achieved if the connecting rods aligned along both axes are of the same length. As such, an investigation on the effective Young's modulus and Poisson's ratio of the Maltese cross metamaterial is recommended for future work.
In comparison to a related but simpler metamaterial geometry consisting of counter-rotating crosses without the hinge rods and spiral springs [49], the magnitudes of the current effective CTE and CME are greater by 3 orders. This is attributed to the greater extent of hinge rod rotation in comparison to the cross rotation, which effectively accentuates the hygrothermal strain and, thereby, enhances the coefficients of expansion.

Amendment
Yes, it is possible to construct "Maltese cross"-like structures with angles fi and theta which are different from 45 deg and 22.5 deg, respectively. This is now briefly discussed in the 4 th last paragraph of the Result and Discussion section:

Comment
P17. L3-6. Please add references with common bimaterial strip analyzed in the paper.
The following has been amended just before table 1.

Comment
P17. T1. Please add to table 1 values of PR and CME for given materials.

Amendment
The PR of the metals have been added. However, their CME is undefined because the metals do not absorb moisture.
We have so far considered the original state being 0 = 22.5° and  0 = 45° in complying to the Maltese cross geometry. As one can expect, the overall properties can be adjusted if the angles were to be altered, even if we impose a similar condition 0°< 0 <  0 < 90°. For example, if we let  0 be large (i.e. close to 90°), a very small change of would bring about a much larger , thereby giving a much large infinitesimal expansion coefficient, but the range of deformation is limited, i.e. the deformation is arrested when  reaches 90°.

Reviewer: 3 Comments to the Author(s)
The manuscript is essentially interesting and shows how one can obtain anomalous behaviour from what is being termed as the 'Maltese Cross' motif as a result of changes in temperature or change in moisture content.
There are a number of aspects in the paper which should be improved. These include:

Comment
(1) Assumptions for the model are not clearly stated. For example, the derivation assumes that the ligaments do not change length nor do they bend. The latter assumption is easy to justify if the hinges are friction-less and offer no resistance, otherwise even this assumption needs to be stated, justified, and discussed.

Amendment
The assumptions have been explicitly incorporated into the first paragraph of the Concept section as shown below:

Comment
(2) The mathematical model hinges on the validity of the equations 2.3 and 2.4. Are these derived by the author, or quoted from others? If derived by author, can a derivation be presented, and ideally some form of validation?