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Technical Brief

# Torsional Buckling by Joining Prestrained and Unstrained Elastomeric Strips With Application as Bilinear Elastic SpringOPEN ACCESS

[+] Author and Article Information
Raudel Avila

Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208
e-mail: roavila@u.northwestern.edu

Yeguang Xue

Department of Mechanical Engineering,
Northwestern University,
Evanston, IL 60208
e-mail: yeguang-xue@u.northwestern.edu

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received July 7, 2017; final manuscript received July 19, 2017; published online August 3, 2017. Editor: Yonggang Huang.

J. Appl. Mech 84(10), 104502 (Aug 03, 2017) (5 pages) Paper No: JAM-17-1358; doi: 10.1115/1.4037347 History: Received July 07, 2017; Revised July 19, 2017

## Abstract

Controlled formation of complex three-dimensional (3D) geometries has always attracted wide interest especially in micro/nanoscale where traditional fabrication techniques fail to apply. Recent advances employed buckling as a promising complementary assembling technique and the method can be used for high-performance electronics materials, such as silicon. This paper describes a new buckling pattern generated by joining multiple prestrained and unstrained elastomeric strips. After releasing, periodic twisting of the system along the releasing direction is generated and bilinear force–displacement relationship is revealed from finite element analysis (FEA). The finding enriches the classes of geometries that can be achieved from structural buckling. Also, compared to other buckling phenomena, the lateral dimension of the system does not change during the buckling process, which makes the structure perfect for elastic spring elements that can be arranged closely to each other without interference.

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## Introduction

Complex three-dimensional (3D) structures, which are ubiquitous in nature, have attracted increasing interest to study their formation recently. Many complex 3D morphologies can be reproduced by simple rules. For example, the complex shapes of seed pods are related to a differential swelling operation [1], and the looped pattern of the gut is driven by different growth rates of tissues [2]. Huang et al. [3,4] revealed that simply by joining two straight elastomeric strips, one prestrained and the other unstrained, either helices or hemihelices structures form, depending on the cross section aspect ratio and prestrain. The buckling behaviors of slender structure elements (rods, plates, and shells) under compression play an important role in the process of 3D morphologies evolution [5].

Inspired by the complex 3D structures emerged from buckling, many fabrication techniques have been developed by harnessing the deterministic buckling, which are of particular interest in micro/nanoscale where traditional fabrication fails to apply. For example, the wrinkling patterns resulted from compression of a system consisting of stiff thin film bonded with compliant substrate [69] are widely utilized in providing electronic devices with stretchability [7] or defining superhydrophobic surfaces [10]. Moreover, controlled buckling of thin ribbons or membranes with prestrained elastomeric substrate provides an important method for the micro/nanofabrication of silicon and other inorganic semiconductor materials [1117]. In this note, we describe a new class of buckling patterns by joining multiple strips with different geometric cross-sectional configurations in order to achieve a different type of torsional buckling patterns, rather than the helices/hemihelices from joining one prestrained strip and one unstrained strip [3]. Compared to other buckling phenomena, the most important property here is that the lateral dimension does not change during the buckling process, which makes the structure more suitable for structural spring elements because they can be arranged closely without the interference.

## Results

A typical process involves using two types of elastomeric strips (Fig. 1(a)), the longer one (type A, length $L$) with rectangular cross section ($W×H$) and the shorter one (type B, length $L′$) with square cross section ($W′×W′$). In the first step, the shorter type B strip is stretched by the strain $χ=(L−L′)/L′$ such that both strips have the same length $L$ (Fig. 1(b)). The widths of two different types of strips, $W′$ and $W$, satisfy $W′=W1+χ$, such that $W′$ becomes the same as W after the stretch $χ$ due to incompressibility of the elastomers, i.e., the widths of two different types of strips match (both are $W$) after elongation of the shorter type B strip. In the second step, four longer type A strips are bonded to each side of the stretched shorter type B strip (Fig. 1(c)). Finally, releasing the two ends after bonding generates the buckling pattern, which is discussed later. During the release, one end fully is fixed and the other end is moved toward the fixed end while allowing rotation along the prestretched direction under quasi-static condition.

As an example, we use longer type A strips of the dimension $W=0.3 mm$, $H=1.2 mm$, and $L=100.0 mm$ and the shorter type B strip of the dimensions $W′=0.42 mm$ and $L′=50.4 mm$. All the strips have Young's moduli of $E=1 MPa$. After stretching the shorter type B strip by the strain $χ=98.4%$, both the length and width of the two strips match exactly. The dimensions of the shorter type B strip are chosen such that the uniaxial stress after prestretching is $σpre=1 MPa$, which is the same as its Young's modulus.

Figure 2(a) shows the deformed structure after the prestretch is fully released (from 100 mm to 91 mm), obtained by the finite element analysis (FEA). The axis of center strip remains straight after releasing because not only the side strips are symmetrically arranged but also the prestretched center strip avoids the most common Euler-type buckling mode for slender bar structures. The cross-sectional images at different positions along $x$-axis (Fig. 2(b)) suggest periodic twisting along the center axis, and the twisting angle $ϕ$ can be approximately described by

Display Formula

(1)$ϕ=ϕ0 cos(2πλx)$

where $ϕ0$ is the maximum twisting angle with respect to the unreleased position, and $λ$ is the wavelength. For the system described earlier, the average wavelength after full release is $λ≈8.7 mm$. Following the twisting of the center strip, a wavy form of out-of-plane deformation occurs for each side strip. The membrane energy of the side strips, resulting from the wavy form of out-of-plane deformation, decreases. In order to have stable buckling pattern, the aspect ratio of side strips $H/W$ must be large enough such that the introduced bending energy from wavy form of deformation is overwhelmed by the decrease of membrane energy, and the total energy of the system decreases. This is confirmed by FEA that no stable buckling pattern occurs when $H/W≤1$. Similar torsional buckling patterns can also be produced by changing the cross-sectional shape of the prestretched center strip from square to other regular polygons, such as hexagon or octagon, or by attaching more side strips (Fig. 2(c)).

Finite element analysis further reveals the mechanical behavior of the system by symmetrically attaching unstrained strips to a prestrained center strip. It displays a bilinear force–displacement relationship (Fig. 3) due to reduction of the tensile stiffness after buckling. Figure 3(a) shows the force–displacement curves of systems with exactly same dimensions after bonding as in Figs. 2(a) and 2(b) except that the prestress in the center strip takes different values, i.e., the same $W$, $H$, and $L$ but different $W′$ and $L′$ corresponding to the prestress. The tensile stiffness before and after buckling are $knonbuckle=0.0161 N/mm$ and $kbuckle=0.0079 N/mm$, respectively. Using the initial length of the unstrained strips ($L=100 mm$) as a reference, the displacement at the transition point is slightly below zero, which corresponds to the critical buckling strain. The prestress $σpre$ in the center strip controls the distance from the transition point to the fully released state (zero net force in the cross section). Furthermore, the stiffness ratio of nonbuckling to buckling states $knonbuckle/kbuckle$ can also be tuned by changing $H/W$. Figure 3(b) shows the force–displacement curves of systems for various side strip height $H$. The resulting properties are summarized in Table 1, which show that the ratio $knonbuckle/kbuckle$ increases with $H/W$. Therefore, the transition point and the tensile stiffness of buckling and nonbuckling state can be changed to achieve the desired bilinear force–displacement relationship.

Compared to many other buckled structures, the system by joining unstrained elastomeric strips with a prestrained center strip has an important advantage—though the length changes, the cross-sectional dimensions barely change in the transition from nonbuckling to buckling states. For Euler buckling of beams, the maximum lateral displacement is the buckle amplitude, which could be much larger than its original cross-sectional dimension. For helices/hemihelices, the cross-sectional dimension changes significantly after it is straightened. On the contrary, for the system discussed earlier, the center axis remains straight during torsional buckling, and twisting along the axis does not change its cross-sectional dimension. Such a property combined with its programmable bilinear mechanical behavior makes the structure perfect for elastic spring elements—multiple bilinear springs with different transition points can be combined in parallel to achieve a desired, continuous force–displacement curve [18,19], and the structures discussed can be arranged very closely without interference.

## Conclusion

A new buckling pattern generated by joining multiple prestrained and unstrained elastomeric strips is studied in this note. Periodic twisting deformation occurs after the release of prestrain, and the side strips buckle into wavy shape to reduce the membrane energy resulting from the normal stress. The finding enriches the classes of 3D geometries that can be achieved through buckling approach. Compared to other buckling phenomena, the system discussed here has the advantage that the lateral dimension does not change during the buckling process, which makes the structure perfect for elastic spring elements that can be arranged closely to each other without the interference.

## Methods

###### Finite Element Analysis.

The finite element analysis was performed using the commercial FEA software abaqus/explicit [20] and 3D linear reduced integration elements (ABAQUS element type C3D8R) are used. The bonding between strips implemented by sharing nodes and a uniaxial stress field is assigned to the strips that are prestrained. Quasi-static conditions were ensured by keeping the kinetic energy negligible compared to the strain energy. In all the simulations, the accuracy of each mesh was ascertained through a mesh refinement study. Mooney–Rivlin hyperelastic model was used in the simulations, and the strain energy density $U$ takes the form of

$U=C10(I1¯−3)+C01(I2¯−3)+1D1(J−1)2$

where $C10$, $C01$, and $D1$ are the material constants, and $I1¯$, $I2¯$, and $J$ are the strain invariants defined as

$I1¯=J−2/3(λ12+λ22+λ32)I2¯=J−4/3(λ12λ22+λ22λ32+λ32λ12)J=λ1λ2λ3$

where $λi$ are the principal stretches. The compatibility with small deformation relates the material constants $C10$, $C01$, and $D1$, Young's modulus $E$, and Poisson's ratio $ν$ by $C10=C01/4=E/20(1+ν)$ and $D1=6(1−2ν)/E$. In all the calculations, Young's modulus of the elastomer is taken as $E=1 MPa$ and Poisson's ratio is $ν=0.49$.

## Acknowledgements

R.A. gratefully acknowledges the support from the NSF Grant No. 1121262, 3M Corporation, and Research Experience for Undergraduates (REU) program provided by Material Research Science and Engineering Center at Northwestern University. Y.X. gratefully acknowledges the support from the Ryan Fellowship and the Northwestern University International Institute for Nanotechnology. The authors thank the useful discussions with Jiang-hong Yuan from Southwest Jiaotong University, Chengdu, China.

## Funding Data

• National Science Foundation (Grant No. 1121262).

## References

Armon, S. , Efrati, E. , Kupferman, R. , and Sharon, E. , 2011, “ Geometry and Mechanics in the Opening of Chiral Seed Pods,” Science, 333(6050), pp. 1726–1730. [PubMed]
Savin, T. , Kurpios, N. A. , Shyer, A. E. , Florescu, P. , Liang, H. Y. , Mahadevan, L. , and Tabin, C. J. , 2011, “ On the Growth and Form of the Gut,” Nature, 476(7358), pp. 57–62. [PubMed]
Huang, J. S. , Liu, J. , Kroll, B. , Bertoldi, K. , and Clarke, D. R. , 2012, “ Spontaneous and Deterministic Three-Dimensional Curling of Pre-Strained Elastomeric Bi-Strips,” Soft Matter, 8(23), pp. 6291–6300.
Liu, J. , Huang, J. S. , Su, T. X. , Bertoldi, K. , and Clarke, D. R. , 2014, “ Structural Transition From Helices to Hemihelices,” PLoS One, 9(4), p. e0139525.
Reis, P. M. , 2015, “ A Perspective on the Revival of Structural (In) Stability With Novel Opportunities for Function: From Buckliphobia to Buckliphilia,” ASME J. Appl. Mech., 82(11), p. 111001.
Huang, Z. Y. , Hong, W. , and Suo, Z. , 2005, “ Nonlinear Analyses of Wrinkles in a Film Bonded to a Compliant Substrate,” J. Mech. Phys. Solids, 53(9), pp. 2101–2118.
Jiang, H. Q. , Khang, D. Y. , Song, J. Z. , Sun, Y. G. , Huang, Y. G. , and Rogers, J. A. , 2007, “ Finite Deformation Mechanics in Buckled Thin Films on Compliant Supports,” Proc. Natl. Acad. Sci. U.S.A., 104(40), pp. 15607–15612. [PubMed]
Raayai-Ardakani, S. , Yague, J. L. , Gleason, K. K. , and Boyce, M. C. , 2016, “ Mechanics of Graded Wrinkling,” ASME J. Appl. Mech., 83(12), p. 121011.
Liu, Y. L. , Li, M. J. , Liu, J. R. , and Chen, X. , 2017, “ Mechanism of Surface Wrinkle Modulation for a Stiff Film on Compliant Substrate,” ASME J. Appl. Mech., 84(5), p. 051011.
Lee, W. K. , Jung, W. B. , Nagel, S. R. , and Odom, T. W. , 2016, “ Stretchable Superhydrophobicity From Monolithic, Three-Dimensional Hierarchical Wrinkles,” Nano Lett., 16(6), pp. 3774–3779. [PubMed]
Xu, S. , Yan, Z. , Jang, K. I. , Huang, W. , Fu, H. R. , Kim, J. , Wei, Z. , Flavin, M. , McCracken, J. , Wang, R. , Badea, A. , Liu, Y. , Xiao, D. Q. , Zhou, G. Y. , Lee, J. , Chung, H. U. , Cheng, H. Y. , Ren, W. , Banks, A. , Li, X. L. , Paik, U. , Nuzzo, R. G. , Huang, Y. G. , Zhang, Y. H. , and Rogers, J. A. , 2015, “ Assembly of Micro/Nanomaterials Into Complex, Three-Dimensional Architectures by Compressive Buckling,” Science, 347(6218), pp. 154–159. [PubMed]
Zhang, Y. H. , Yan, Z. , Nan, K. W. , Xiao, D. Q. , Liu, Y. H. , Luan, H. W. , Fu, H. R. , Wang, X. Z. , Yang, Q. L. , Wang, J. C. , Ren, W. , Si, H. Z. , Liu, F. , Yang, L. H. , Li, H. J. , Wang, J. T. , Guo, X. L. , Luo, H. Y. , Wang, L. , Huang, Y. G. , and Rogers, J. A. , 2015, “ A Mechanically Driven Form of Kirigami as a Route to 3D Mesostructures in Micro/Nanomembranes,” Proc. Natl. Acad. Sci. U.S.A., 112(38), pp. 11757–11764. [PubMed]
Hanna, B. H. , Magleby, S. P. , Lang, R. J. , and Howell, L. L. , 2015, “ Force-Deflection Modeling for Generalized Origami Waterbomb-Base Mechanisms,” ASME J. Appl. Mech., 82(8), p. 081001.
Yan, Z. , Zhang, F. , Wang, J. C. , Liu, F. , Guo, X. L. , Nan, K. W. , Lin, Q. , Gao, M. Y. , Xiao, D. Q. , Shi, Y. , Qiu, Y. T. , Luan, H. W. , Kim, J. H. , Wang, Y. Q. , Luo, H. Y. , Han, M. D. , Huang, Y. G. , Zhang, Y. H. , and Rogers, J. A. , 2016, “ Controlled Mechanical Buckling for Origami-Inspired Construction of 3D Microstructures in Advanced Materials,” Adv. Funct. Mater., 26(16), pp. 2629–2639. [PubMed]
Chen, Y. L. , Zhu, Y. , Chen, X. , and Liu, Y. L. , 2016, “ Mechanism of the Transition From In-Plane Buckling to Helical Buckling for a Stiff Nanowire on an Elastomeric Substrate,” ASME J. Appl. Mech., 83(4), p. 041011.
Wang, A. , Avila, R. , and Ma, Y. J. , 2017, “ Mechanics Design for Buckling of Thin Ribbons on an Elastomeric Substrate Without Material Failure,” ASME J. Appl. Mech., 84(9), p. 094501.
Wang, F. , Guo, X. G. , Xu, J. X. , Zhang, Y. H. , and Chen, C. Q. , 2017, “ Patterning Curved Three-Dimensional Structures With Programmable Kirigami Designs,” ASME J. Appl. Mech., 84(6), p. 061007.
Ma, Y. J. , Feng, X. , Rogers, J. A. , Huang, Y. G. , and Zhang, Y. H. , 2017, “ Design and Application of ‘J-Shaped’ Stress-Strain Behavior in Stretchable Electronics: A Review,” Lab Chip, 17(10), pp. 1689–1704. [PubMed]
Ma, Y. J. , Jang, K. I. , Wang, L. , Jung, H. N. , Kwak, J. W. , Xue, Y. G. , Chen, H. , Yang, Y. Y. , Shi, D. W. , Feng, X. , Rogers, J. A. , and Huang, Y. G. , 2016, “ Design of Strain-Limiting Substrate Materials for Stretchable and Flexible Electronics,” Adv. Funct. Mater., 26(29), pp. 5345–5351.
Dassault Systèmes, 2015, “ ABAQUS 2016 Analysis User's Manual,” Dassault Systèmes, Providence, RI.
View article in PDF format.

## References

Armon, S. , Efrati, E. , Kupferman, R. , and Sharon, E. , 2011, “ Geometry and Mechanics in the Opening of Chiral Seed Pods,” Science, 333(6050), pp. 1726–1730. [PubMed]
Savin, T. , Kurpios, N. A. , Shyer, A. E. , Florescu, P. , Liang, H. Y. , Mahadevan, L. , and Tabin, C. J. , 2011, “ On the Growth and Form of the Gut,” Nature, 476(7358), pp. 57–62. [PubMed]
Huang, J. S. , Liu, J. , Kroll, B. , Bertoldi, K. , and Clarke, D. R. , 2012, “ Spontaneous and Deterministic Three-Dimensional Curling of Pre-Strained Elastomeric Bi-Strips,” Soft Matter, 8(23), pp. 6291–6300.
Liu, J. , Huang, J. S. , Su, T. X. , Bertoldi, K. , and Clarke, D. R. , 2014, “ Structural Transition From Helices to Hemihelices,” PLoS One, 9(4), p. e0139525.
Reis, P. M. , 2015, “ A Perspective on the Revival of Structural (In) Stability With Novel Opportunities for Function: From Buckliphobia to Buckliphilia,” ASME J. Appl. Mech., 82(11), p. 111001.
Huang, Z. Y. , Hong, W. , and Suo, Z. , 2005, “ Nonlinear Analyses of Wrinkles in a Film Bonded to a Compliant Substrate,” J. Mech. Phys. Solids, 53(9), pp. 2101–2118.
Jiang, H. Q. , Khang, D. Y. , Song, J. Z. , Sun, Y. G. , Huang, Y. G. , and Rogers, J. A. , 2007, “ Finite Deformation Mechanics in Buckled Thin Films on Compliant Supports,” Proc. Natl. Acad. Sci. U.S.A., 104(40), pp. 15607–15612. [PubMed]
Raayai-Ardakani, S. , Yague, J. L. , Gleason, K. K. , and Boyce, M. C. , 2016, “ Mechanics of Graded Wrinkling,” ASME J. Appl. Mech., 83(12), p. 121011.
Liu, Y. L. , Li, M. J. , Liu, J. R. , and Chen, X. , 2017, “ Mechanism of Surface Wrinkle Modulation for a Stiff Film on Compliant Substrate,” ASME J. Appl. Mech., 84(5), p. 051011.
Lee, W. K. , Jung, W. B. , Nagel, S. R. , and Odom, T. W. , 2016, “ Stretchable Superhydrophobicity From Monolithic, Three-Dimensional Hierarchical Wrinkles,” Nano Lett., 16(6), pp. 3774–3779. [PubMed]
Xu, S. , Yan, Z. , Jang, K. I. , Huang, W. , Fu, H. R. , Kim, J. , Wei, Z. , Flavin, M. , McCracken, J. , Wang, R. , Badea, A. , Liu, Y. , Xiao, D. Q. , Zhou, G. Y. , Lee, J. , Chung, H. U. , Cheng, H. Y. , Ren, W. , Banks, A. , Li, X. L. , Paik, U. , Nuzzo, R. G. , Huang, Y. G. , Zhang, Y. H. , and Rogers, J. A. , 2015, “ Assembly of Micro/Nanomaterials Into Complex, Three-Dimensional Architectures by Compressive Buckling,” Science, 347(6218), pp. 154–159. [PubMed]
Zhang, Y. H. , Yan, Z. , Nan, K. W. , Xiao, D. Q. , Liu, Y. H. , Luan, H. W. , Fu, H. R. , Wang, X. Z. , Yang, Q. L. , Wang, J. C. , Ren, W. , Si, H. Z. , Liu, F. , Yang, L. H. , Li, H. J. , Wang, J. T. , Guo, X. L. , Luo, H. Y. , Wang, L. , Huang, Y. G. , and Rogers, J. A. , 2015, “ A Mechanically Driven Form of Kirigami as a Route to 3D Mesostructures in Micro/Nanomembranes,” Proc. Natl. Acad. Sci. U.S.A., 112(38), pp. 11757–11764. [PubMed]
Hanna, B. H. , Magleby, S. P. , Lang, R. J. , and Howell, L. L. , 2015, “ Force-Deflection Modeling for Generalized Origami Waterbomb-Base Mechanisms,” ASME J. Appl. Mech., 82(8), p. 081001.
Yan, Z. , Zhang, F. , Wang, J. C. , Liu, F. , Guo, X. L. , Nan, K. W. , Lin, Q. , Gao, M. Y. , Xiao, D. Q. , Shi, Y. , Qiu, Y. T. , Luan, H. W. , Kim, J. H. , Wang, Y. Q. , Luo, H. Y. , Han, M. D. , Huang, Y. G. , Zhang, Y. H. , and Rogers, J. A. , 2016, “ Controlled Mechanical Buckling for Origami-Inspired Construction of 3D Microstructures in Advanced Materials,” Adv. Funct. Mater., 26(16), pp. 2629–2639. [PubMed]
Chen, Y. L. , Zhu, Y. , Chen, X. , and Liu, Y. L. , 2016, “ Mechanism of the Transition From In-Plane Buckling to Helical Buckling for a Stiff Nanowire on an Elastomeric Substrate,” ASME J. Appl. Mech., 83(4), p. 041011.
Wang, A. , Avila, R. , and Ma, Y. J. , 2017, “ Mechanics Design for Buckling of Thin Ribbons on an Elastomeric Substrate Without Material Failure,” ASME J. Appl. Mech., 84(9), p. 094501.
Wang, F. , Guo, X. G. , Xu, J. X. , Zhang, Y. H. , and Chen, C. Q. , 2017, “ Patterning Curved Three-Dimensional Structures With Programmable Kirigami Designs,” ASME J. Appl. Mech., 84(6), p. 061007.
Ma, Y. J. , Feng, X. , Rogers, J. A. , Huang, Y. G. , and Zhang, Y. H. , 2017, “ Design and Application of ‘J-Shaped’ Stress-Strain Behavior in Stretchable Electronics: A Review,” Lab Chip, 17(10), pp. 1689–1704. [PubMed]
Ma, Y. J. , Jang, K. I. , Wang, L. , Jung, H. N. , Kwak, J. W. , Xue, Y. G. , Chen, H. , Yang, Y. Y. , Shi, D. W. , Feng, X. , Rogers, J. A. , and Huang, Y. G. , 2016, “ Design of Strain-Limiting Substrate Materials for Stretchable and Flexible Electronics,” Adv. Funct. Mater., 26(29), pp. 5345–5351.
Dassault Systèmes, 2015, “ ABAQUS 2016 Analysis User's Manual,” Dassault Systèmes, Providence, RI.

## Figures

Fig. 1

Process to form the buckling pattern. (a) Prepare two different types of strips: type A strips with rectangular cross section (W×H) and length L and type B strip with square cross section (W′×W′) and length L′. (b) Elongate type B strip to the same length as type A strip and their width exactly match after elongation. (c) Attach type A strips to each side of the center type B strip and release the system after bonding to form the buckling pattern.

Fig. 2

(a) Buckling pattern predicted from FEA by releasing from the original length 100 mm to 91 mm. (b) A snapshot of the details of periodic buckling pattern. Twisting occurs along the stretching direction and the twisting angle ranges between −ϕ0 and +ϕ0. (c) Buckling pattern generated by replacing center strip of square-shaped cross section to hexagon-shaped cross section. The left side is the cross section of bonded system.

Fig. 3

Bilinear force–displacement relationship of the system by symmetrically attaching unstrained strips to a prestrained center strip. (a) The distance from the transition point to the fully released state (zero cross-sectional net force) is controlled by the prestress σpre in the center strip and (b) the tensile stiffness ratio of nonbuckling to buckling states knonbuckle/kbuckle can be tuned by changing H/W.

## Tables

Table 1 Stiffness in the nonbuckling and buckling states and their ratio for different H/W

## Errata

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