Curved creases redistribute global bending stiffness in corrugations: theory and experimentation

Abstract

Corrugations offer a convenient way to make thin, lightweight sheets into stiff structures. However, traditional, v-shaped corrugations made from straight creases result in highly anisotropic stiffness which leads to undesirable flexibility in some directions of loading. In this paper, we explore the bending stiffness of curved-crease corrugations with a planar midsurface—developable corrugations made by folding thin sheets about curves and without linerboard covers on the top or bottom. The curved-crease corrugations break symmetry in the pattern and can redistribute stiffness to resist bending deformations in multiple directions. To study these systems, we formulate a framework for predicting the bending stiffness of any planar-midsurface corrugation from its multiple geometric features at different scales. We use the framework to create two predictive methods that provide valuable insight into the global stiffness of corrugations without a detailed analysis. Results from these methods match well with experimental, three-point bending tests of five corrugation geometries made from polyester film. We found that corrugations with elliptical or parabolic curved-creases that intersect with one edge of the pattern are best at redistributing stiffness in multiple directions. While a straight-crease pattern has a stiffness of about 4 N/mm in one direction and about 0 N/mm in the other, a parabolic crease pattern has a stiffness of about 2.5 N/mm in both directions. These curved-crease corrugations can enable the self-assembly and fabrication of practical, stiff structures from simple, developable sheets.

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Acknowledgements

We thank Jack Riley (University of Michigan) for his preliminary experimental work on curved parabolas that verified their unique characteristics and fueled our curiosity for curved-crease corrugations, as a whole.

Funding

The authors thank the Office of Naval Research for their financial support (Grant No. N00014-18-1-2015). The first author thanks the National Science Foundation Graduate Research Fellowship Program for their support (Grant No. DGE 1256260).

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Correspondence to Evgueni T. Filipov.

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Appendix: Details on the crease pattern geometries

Appendix: Details on the crease pattern geometries

In this appendix, we give the specific functions used to generate the flat crease patterns shown in Fig. 3. We also present the values we used in the functions so that our results can be further examined or replicated. Note that for each of the five geometries, \(W = 100\) mm, \(H = 100\) mm, \(n_s = 5\), and \(i \in \{1, 2, 3, 4, 5\}\).

Straight-crease corrugation For the straight-crease corrugation, there is one input besides the constraining assumptions: \(a_i\). The function defining the \(i{{{\text {th}}}}\) curve of the crease pattern is,

$$\begin{aligned} y_i = a_i, \end{aligned}$$
(11)

where,

$$\begin{aligned} a_i = \dfrac{H}{n_s}i. \end{aligned}$$
(12)

The functions are defined in the domain \(-W/2 \le x \le W/2\). An example crease pattern used in the analyses and experimentation is shown in Fig. 3a with corresponding parameter values in Table 1.

Table 1 Geometric parameter values for example straight-crease pattern (see Fig. 3a)

Parabolic-point corrugation For the parabolic-point corrugation, there are two inputs besides the constraining assumptions: \(b_i\) and \(c_i\). The function defining the \(i{{{\text {th}}}}\) curve is,

$$\begin{aligned} y_i = b_i x_i^2 + c_i, \end{aligned}$$
(13)

where,

$$\begin{aligned} c_i = \dfrac{H}{n_s}i; \quad \quad b_i = -\dfrac{4c_i}{W^2} = -\dfrac{4H}{W^2n_s}i. \end{aligned}$$
(14)

The functions are defined in the domain \(-r_i \le x_i \le r_i\), where \(r_i = \sqrt{-c_i/b_i} = W/2\). An example crease pattern is shown in Fig. 3b with corresponding values in Table 2.

Table 2 Geometric parameter values for example parabolic-point pattern (see Fig. 3b)

Parabolic-edge corrugation For the parabolic-edge corrugation, two inputs additional inputs are required: \(b_i\) and \(c_i\). The function defining the \(i{{{\text {th}}}}\) curve is,

$$\begin{aligned} y_i = b_i x_i^2 + c_i, \end{aligned}$$
(15)

where,

$$\begin{aligned} b_i = -\dfrac{4H}{W^2}; \quad \quad c_i = \dfrac{H}{n_s}i. \end{aligned}$$
(16)

Each curve is defined in the domain \(-r_i \le x \le\, r_i\), where \(r_i = \sqrt{-c_i/b_i} = \sqrt{W^2 i/(4n_s)}\). An example crease pattern is shown in Fig. 3c with corresponding values in Table 3.

Table 3 Geometric parameter values for example parabolic-edge pattern (see Fig. 3c)

Elliptical-point corrugation For the elliptical-point corrugation, three inputs besides the constraining assumptions are required: \(d_i\), \(e_i\), and \(f_i\). The function defining the \(i{{{\text {th}}}}\) curve is,

$$\begin{aligned} y_i = e_i\sqrt{1-\left( \dfrac{x_i}{d_i}\right) ^2} + f_i, \end{aligned}$$
(17)

where,

$$\begin{aligned} d_i = \dfrac{W}{2}; \quad e_i = \dfrac{H}{n_s}i; \quad f_i = 0. \end{aligned}$$
(18)

Each curve is defined in the domain \(-r_i \le x_i \le r_i\), where \(r_i = d_i\sqrt{1-(-f_i/e_i)^2} = W/2\). An example crease pattern is shown in Fig. 3d with corresponding values in Table 4.

Table 4 Geometric parameter values for example elliptical-point pattern (see Fig. 3d)

Elliptical-edge corrugation For the elliptical-edge corrugation, three inputs are required besides the constraining assumptions: \(d_i\), \(e_i\), and \(f_i\). The function defining the \(i{{{\text {th}}}}\) curve is,

$$\begin{aligned} y_i = e_i\sqrt{1-\left( \dfrac{x_i}{d_i}\right) ^2}+ f_i, \end{aligned}$$
(19)

where,

$$\begin{aligned} d_i = \dfrac{W}{2}; \quad e_i = H; \quad f_i = H\left( \dfrac{i}{n_s}-1\right) . \end{aligned}$$
(20)

Each curve is defined in the domain \(-r_i \le x_i \le r_i\), where \(r_i = d_i\sqrt{1-(-f_i/e_i)^2} = W/2\sqrt{2i/n_s - (i/n_s)^2}\). An example crease pattern is shown in Fig. 3e with corresponding values in Table 5.

Table 5 Geometric parameter values for example elliptical-edge pattern (see Fig. 3e)

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Woodruff, S.R., Filipov, E.T. Curved creases redistribute global bending stiffness in corrugations: theory and experimentation. Meccanica (2020). https://doi.org/10.1007/s11012-020-01200-7

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Keywords

  • Corrugations
  • Curved-crease origami
  • Lightweight structures
  • Developable folding
  • Multi-directional stiffness