Synclastic Behavior of the Auxetic Core for Furniture Panels
Round 1
Reviewer 1 Report
Comments and Suggestions for AuthorsThe most significant breakthrough of this work lies in demonstrating that M-type honeycomb cores with positive Poisson's ratios can achieve synclastic curvature, defying the conventional wisdom that such surface formation is exclusively facilitated by auxetic (negative Poisson's ratio) structures. Overall, the work demonstrates significant novelty; however, the following critical issues must be addressed prior to publication:
- The Abstract section requires substantial refinement to effectively articulate: research significance, methodological approach, key findings, and primary objectives.
- The finite element implementation methodology lacks sufficient clarity. Could you elaborate on: mesh type selection (e.g., shell vs. solid elements), element sizing strategy, and contact interface treatment in critical regions?
- The finite element implementation methodology lacks sufficient clarity. Could you elaborate on: mesh type selection (e.g., shell vs. solid elements), element sizing strategy, and contact interface treatment in critical regions?
- Methodological consideration: The R-core specimens utilized machined WoodEpox@ composites, while H-core and M-core employed paper-based materials. Significant disparities in mechanical properties (e.g., stiffness, toughness) and manufacturing processes exist between these systems. Could such heterogeneity potentially confound comparative bending analyses? We inquire why identical material/processing protocols weren't adopted to isolate architectural variables.
Author Response
Dear Reviewer,
We sincerely appreciate your thoughtful and constructive review, as well as your recognition of the manuscript's main contribution, which demonstrates synclastic curvature with an M-type honeycomb, despite positive effective Poisson's ratios. Below, we address each of your comments in turn. All changes have been made in the text in red.
Comments 1: Abstract requires substantial refinement (significance, method, key findings, objectives)
Response 1: Thank you for the suggestion. We thoroughly revised the Abstract to state the research gap and significance, the objectives, the experimental and FE approaches, and the key quantitative findings.
The cores of honeycomb panels are usually made of hexagonal cells. Due to their structure, they create anticlastic surfaces that are difficult to use in furniture design. Synclastic surfaces in lightweight sandwich panels are typically associated with auxetic cores characterized by a negative Poisson's ratio. This study aimed to transform the hexagonal cell cores into cells with a negative or positive Poisson's ratio (NPR, PPR), enabling these cores to form synclastic surfaces. New core structures for synclastic furniture sandwich honeycomb panels were modeled numerically and experimentally. It has been demonstrated that reentrant cells with NPR create synclastic surfaces, and new shapes of core cells, created by transforming hexagonal cells with PPR, also enable the formation of synclastic surfaces. Cores' synclasticity was assessed in two orthogonal planes using physical models and Finite Element Analysis (FEA). A new and original discovery is the demonstration that not only auxetic but also modified hexagonal cells with Poisson's ratios of =0.545 and =0.512, respectively, exhibit excellent synclastic properties. The agreement between FEA and experiment was very high. The results show that not only NPR but also cell topology provides a practical route to the synclastic formation of cores without the use of auxetic materials.
Comments 2: The finite element implementation methodology lacks sufficient clarity. Could you elaborate on: mesh type selection (e.g., shell vs. solid elements), element sizing strategy, and contact interface treatment in critical regions?
Comments 3: The finite element implementation methodology lacks sufficient clarity. Could you elaborate on: mesh type selection (e.g., shell vs. solid elements), element sizing strategy, and contact interface treatment in critical regions?
Response 2,3: Thank you for pointing this out. We expanded §2.4 and made the modeling choices explicit. Comments 2 and 3 concern the same issue; we address them together. All changes have been made in the text in red.
2.4.1. Compression tests
Cores were meshed with 3D solid C3D8R elements (reduced integration, hourglass control) with a global target size of ≈0.25 mm. The number of nodes was, on average, 320000, and the number of elements was 190000.
2.4.2. Bending test
A nonlinear static analysis with large displacements (On) and small strains (Off) was performed (Newmark time integration, Newton–Raphson iterations, Direct Sparse solver, automatic stepping 0–1 s). The material followed Table 1 (plasticity with an input σ–ε curve). Geometric symmetry (¼ model) was exploited to reduce computational cost. The model reproduced the actual contact conditions of the core with a spherical support, represented by a hemisphere with a radius of 66 mm. Surface-to-surface contact was defined between the core and the dome, with a Coulomb friction coefficient of 0.2. A blended curvature-based mesh (High Quality, 16-point Jacobian check) was used, with a size ranging from 2 mm to 10 mm, resulting in 385,509 nodes and 195,432 elements. Additional runs with mesh sizes ranging from 0.1 mm to 5 mm, resulting in 1.14 × 10^6 nodes and 0.57 × 10^6 elements, did not significantly alter the deflection profiles or agreement metrics (see below). Solid (tetrahedral) elements were preferred over shells to capture through-thickness stiffness and contact compliance near the spherical support. The applied load was gravitational, corresponding to the core’s self-weight under an acceleration of −9.81 m/s² (Figure 8). The mechanical properties of the paper are summarized in Table 1.
Comments 4: Methodological consideration: The R-core specimens utilized machined WoodEpox@ composites, while H-core and M-core employed paper-based materials. Significant disparities in mechanical properties (e.g., stiffness, toughness) and manufacturing processes exist between these systems. Could such heterogeneity potentially confound comparative bending analyses? We inquire why identical material/processing protocols weren't adopted to isolate architectural variables.
Response 4: We appreciate this critical point. We clarified the scope and limitations to avoid cross-material confounds.
The quantitative bending comparison is limited to H and M cores manufactured from the same paper to isolate topological effects. As the Reviewer said, significant disparities in mechanical properties and manufacturing processes exist between R, H, M systems. This heterogeneity definitely confounds comparative bending analyses. For these reasons, the R core (WoodEpox®) is included as an auxetic baseline and was not used for cross-material bending benchmarks, because, as shown in the work [16], R-type structures guarantee synclastic surfaces. The authors have added an appropriate explanation marked in red.
Respectfully,
The Authors
Author Response File: 
Author Response.docx
Reviewer 2 Report
Comments and Suggestions for AuthorsThis manuscript systematically investigates the synelastic behavior of honeycomb core materials and proposes a method for achieving synclastic surfaces through cell shape modification, demonstrating significant innovation and application potential in the fields of furniture and composite materials. The following issues remain to be addressed:
1.While the literature review adequately covers honeycomb structures, auxetic materials, and their applications in fields such as aerospace, support specifically related to the furniture industry is somewhat limited. It is recommended to incorporate more relevant studies.
2.The number of experimental samples for each cell type is not clearly stated. It is suggested that sample size information be provided to enhance statistical reliability.
3.Although the synelastic behavior of M-type cells is proposed, the underlying mechanical mechanism—particularly how the removal of cell walls influences overall deformation behavior—is not thoroughly explained. Additional theoretical analysis from a structural mechanics perspective is recommended.
4.The paper-based material (Testliner-2) mentioned in the study is sensitive to humidity and temperature, yet the influence of these environmental factors on synelastic behavior has not been evaluated. Supplementary experiments or discussion on this aspect are advised.
5.The language is generally fluent, but some sentences are excessively long. It is suggested that these be simplified to improve readability, particularly in the Methods section.
6.There is a lack of practical application examples of furniture panels: although the furniture industry is mentioned, no specific case studies or prototype demonstrations of furniture component forming are provided. Inclusion of such examples is recommended.
7.Future research directions could be further expanded. It is recommended that the Conclusion section include suggestions for research on multi-material systems, multi-scale structures, or behavior under dynamic loading.
Author Response
Dear Reviewer,
We sincerely appreciate your positive assessment and thoughtful suggestions. Below, we address each point in turn and indicate the corresponding revisions to the manuscript. All changes have been made in the text in blue.
Comments 1: While the literature review adequately covers honeycomb structures, auxetic materials, and their applications in fields such as aerospace, support specifically related to the furniture industry is somewhat limited. It is recommended to incorporate more relevant studies.
Response 1: Thank you very much for your kind suggestion. We expanded the Introduction to better reflect furniture and interior applications of paper-core sandwich panels (lightweight, indentation resistance, tight radii; RH/T considerations) and to state the gap our study addresses explicitly.
In furniture and interior applications, paper-honeycomb sandwich panels are valued for their low mass, indentation resistance, and the ability to achieve tight radii and curved skins while maintaining adequate stiffness. Moreover, the air relative humidity and temperature sensitivity of ligno-cellulosic cores motivate climate-controlled testing and design allowances [7][8][9][10].
Smardzewski, J.; Kramski, D. Modelling Stiffness of Furniture Manufactured from Honeycomb Panels Depending on Changing Climate Conditions. Thin-Walled Structures 2019, 137, doi:10.1016/j.tws.2019.01.019.
Peliński, K.; Smardzewski, J.; Narojczyk, J. Stiffness of Synclastic Wood-Based Auxetic Sandwich Panels. physica status solidi (b) 2020, 257, 1900749, doi:https://doi.org/10.1002/pssb.201900749.
Smardzewski, J.; Tokarczyk, M. Lightweight Honeycomb Furniture Panels with Discreetly Located Strengthening Blocks. Compos Struct 2024, 331, 117927, doi:10.1016/J.COMPSTRUCT.2024.117927.
Mrówczyński, D.; Gajewski, T.; Cornaggia, A.; Garbowski, T. Impact of Temperature and Humidity on Key Mechanical Properties of Corrugated Board. Applied Sciences 2024, 14, doi:10.3390/app142412012.
Comments 2: The number of experimental samples for each cell type is not clearly stated. It is suggested that sample size information be provided to enhance statistical reliability.
Response 2: Thank you for the suggestion.
For the physical tests of R [20], H, and M-type cores, only one specimen of each type was used. This decision was made for two reasons. To ensure consistency with the numerical experiment, which was also conducted on a single model of each core type. The authors' experience shows that the large size of a core composed of repeating cells favorably influences the homogenization of the entire structure, eliminating the effect of geometric imperfections on its mechanical properties. Appropriate text was added in blue.
Comments 3: Although the synelastic behavior of M-type cells is proposed, the underlying mechanical mechanism—particularly how the removal of cell walls influences overall deformation behavior—is not thoroughly explained. Additional theoretical analysis from a structural mechanics perspective is recommended.
Response 3: We appreciate your important suggestion. It is of particular importance to the authors as it forms part of ongoing research on this type of structure. Their goal is to propose a parametric mathematical model of M-type cells and cores that describes Poisson's ratios and elastic moduli as a function of changes in the main geometric and material parameters. Since this task also requires numerical verification using parametric models, we decided to present it in a subsequent paper. The aim of this paper was to demonstrate an innovation involving the creation of synclastic surfaces by a simple modification of the hexagonal cell. Therefore, the Reviewer's suggestion will be developed in the next full article.
Comments 4: The paper-based material (Testliner-2) mentioned in the study is sensitive to humidity and temperature, yet the influence of these environmental factors on synelastic behavior has not been evaluated. Supplementary experiments or discussion on this aspect are advised.
Response 4: The paper-based material used in the study is sensitive to relative humidity RH and temperature T. For these reasons, the authors clarified that all registrations were performed in a climate-controlled chamber and that cores were conditioned to maintain their geometry. We also added a brief note on limitations related to RH/T influence. All improvements in text are highlighted in blue.
All registrations were performed in a climate chamber under constant conditions T=25C, RH=45%. Cores were seasoned in a frame and cut immediately before testing to preserve cell geometry (Fig.5). During this procedure, no influence of T and RH changes on the measured core deflections was observed.
Comments 5: The language is generally fluent, but some sentences are excessively long. It is suggested that these be simplified to improve readability, particularly in the Methods section.
Response 5: We revised Materials & Methods to improve readability (shorter sentences, explicit lists). In particular, the FEA description was rewritten for clarity.
2.4.2. Bending test
A detailed description of the numerical modeling of cores with re-entrant cells is presented in [20]. Therefore, it was decided to present only the method of modeling cores with H and M types of cells, because the calculations of these structures were performed using SolidWorks software (Dassault Systèmes SolidWorks Corporation, Waltham, MA, USA). The model dimensions were set to 1140 mm × 635 mm for the M-type core and 1180 mm × 585 mm for the H-type core. A nonlinear static analysis with large displacements (On) and small strains (Off) was performed (Newmark time integration, Newton–Raphson iterations, Direct Sparse solver, automatic stepping 0–1 s). The material followed Table 1 (plasticity with an input σ–ε curve). Geometric symmetry (¼ model) was exploited to reduce computational cost. The model reproduced the actual contact conditions of the core with a spherical support, represented by a hemisphere with a radius of 66 mm. Surface-to-surface contact was defined between the core and the dome, with a Coulomb friction coefficient of 0.2. A blended curvature-based mesh (High Quality, 16-point Jacobian check) was used, with a size ranging from 2 mm to 10 mm, resulting in 385,509 nodes and 195,432 elements. Additional runs with mesh sizes of 0.1 mm to 5 mm, resulting in 1.14 × 10^6 nodes and 0.57 × 10^6 elements, did not materially change the deflection profiles or agreement metrics (see below). Solid (tetrahedral) elements were preferred over shells to capture through-thickness stiffness and contact compliance near the spherical support. The applied load was gravitational, corresponding to the core’s self-weight under an acceleration of −9.81 m/s² (Figure 8). The mechanical properties of the paper are summarized in Table 1.
Comments 6: There is a lack of practical application examples of furniture panels: although the furniture industry is mentioned, no specific case studies or prototype demonstrations of furniture component forming are provided. Inclusion of such examples is recommended.
Response 6: We have added a concise application note that connects our findings to current design and engineering needs in the furniture and transportation industries. All additions in text are highlighted in blue.
In furniture design, engineers continue to seek organic shapes with soft radii and smooth transitions. Historically, Baroque and Art Nouveau cabinetmakers, like contemporary furniture manufacturers, created such surfaces in presses using high pressure and temperature. Thanks to M-type cells, it will be possible to produce spherical doors, spherical cabinet sides and fronts, lampshades, and similar products without the need for high temperatures and high pressure.
Comments 7: Future research directions could be further expanded. It is recommended that the Conclusion section include suggestions for research on multi-material systems, multi-scale structures, or behavior under dynamic loading.
Response 7: Thank you for your valuable suggestion.
Further research will focus on modeling parametric multi-material systems and multi-scale structures composed of M-type cells. Modeling should include variable loads under variable climatic conditions, taking into account reinforcement systems.
All additions in text are highlighted in blue.
Respectfully,
The Authors
Author Response File: Author Response.docx
Round 2
Reviewer 2 Report
Comments and Suggestions for AuthorsThe manuscript has been revised according to the reviewers' comments and is now recommended for acceptance.
