# The Conformal Design of an Island-Bridge Structure on a Non-Developable Surface for Stretchable Electronics

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Conformal Criterion for Island

#### 2.1. Conformal Modelling for Island

_{island}, width of w

_{island}, and thickness of t

_{island}(${t}_{\mathrm{island}}\ll {w}_{\mathrm{island}}\le {l}_{\mathrm{island}}$) is mapped onto a part of the torus surface under the assumption that no tension exists in width direction [19], which produces a rectangle conformal zone of length l

_{island}. Let the coordinates x and y denote the distance from the island center along the length and width direction, respectively, as shown in Figure 1b. Figure A1 shows the situation when the width direction is deviated from the bending direction of curvature ${\kappa}_{2}$ with a deflection angle θ. The relationship between conformal strain energy on the island and deflection angle θ is shown in Figure A2. It can be seen from the result that the island has lowest strain energy when θ = 0, which means a most steady state. So, we adopt this state to perform the analysis.

#### 2.2. Adhesion Experiment for Island

_{island}, Poisson’s ratio ν

_{island}, yield strain, and work of adhesion γ. The operational processes and test results for tension test and peel test are listed in Appendix C. For the adhesive PVC sticker used in the experiment, Young’s modulus, Poisson’s ratio, yield strain, and work of adhesion $\gamma $ are ${E}_{\mathrm{island}}=1.29\mathrm{GPa}$, ${\nu}_{\mathrm{island}}=0.32$, ${\epsilon}_{\mathrm{critical}}=2\%$, and $\gamma =7.596\mathrm{N}/\mathrm{m}$, respectively. The non-dimensional parameter $\xi $ for this experiment is 0.147, which is less than the critical one, which corresponds to a weak adhesion condition. The theoretical non-dimensional critical conformal width given by the first equation in Equation (8) is 0.2868, which is quite close to the experimental one (${\kappa}_{2}{w}_{\mathrm{critical}}=0.3$). It is worth noting that with the further increase of the width of island after ${\kappa}_{2}{w}_{\mathrm{island}}>0.4$, the width of the conformal region will reduce (for ${\kappa}_{2}{w}_{\mathrm{island}}=0.5$, the conformal width is 0.26 and for ${\kappa}_{2}{w}_{\mathrm{island}}=0.6$, the conformal width is 0.24), which may come from the influence of the un-conformal region. As the width of island gets bigger, the un-conformal region gets bigger too, so the strain energy in the un-conformal region will be larger and larger. However, for the problem solving the critical width, this part of energy is not under consideration.

## 3. Mechanics of Stretchable Bridges

#### 3.1. Tensile Stiffness Design for Bridges

_{2}and spacing l

_{1}and has a rectangular cross section with width w

_{bridge}and thickness t

_{bridge}. The serpentine bridge made of single layer PI with Young’s modulus ${E}_{\mathrm{PI}}=2.5\mathrm{GPa}$ and Poisson’s ratio ${\nu}_{\mathrm{PI}}=0.34$ is analyzed to given the scaling laws of axis force, and its dependency on the geometric parameters mentioned above. The tensile stiffness can be solved by taking a derivative of the axis force with respect to axial displacement. The serpentine bridge is clamped at two ends and pulls from an axial direction (x direction in Figure 1c). Four-node shell elements are used to model the serpentine bridge, and high-quality meshes are adopted to guarantee the accuracy of those analyses. A two-step method is used for FEA simulations. Firstly, the buckling analysis is adopted to get buckling strain and buckling modes for the serpentine bridge. Then, using the buckling modes from step 1 as initial imperfection to continue a nonlinear static analysis, a small enough damping is added to the model to ensure the convergence of the analysis.

#### 3.2. Stretchability Demands for Bridges

_{island}and distance s between two islands is mapped onto a sphere with radius R. The total length and width of the array are ${l}_{tot}=n{w}_{island}+\left(n-1\right)s$ and ${w}_{tot}=m{w}_{island}+\left(m-1\right)s$, respectively.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Conformal Strain Energy with Angle of Deviation

**Figure A2.**The conformal strain energy per unit area in island with initial angle θ at different width-length ratios (

**a**) and different curvature ratios (

**b**).

## Appendix B. Relative Error between Approximate Solution and Exact Solution

**Figure A3.**The relative error between approximate solution and accurate solution with ${\kappa}_{2}{w}_{\mathrm{island}}$: (

**a**) for strain and (

**b**) for conformal strain energy.

## Appendix C. Material Parameter Test

**Figure A4.**Experimental apparatus and experimental data for material parameter test: (

**a**) an universal mechanical tester (INSTRON 5944); (

**b**) a home-made peel platform with an angle-adjustable jig, in which the X or Z-motion of the translation stage is able to be driven by two independent linear/electric actuators, and its Y-motion depends on a manually single axis table; (

**c**) stress-strain curve for PVC sticker in tension test; and (

**d**) peel force for PVC sticker in peel test with peel angle $\beta ={135}^{\circ}$ and peel rate ${v}_{\mathrm{peel}}=1\mathrm{mm}/\mathrm{s}$.

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**Figure 1.**(

**a**) An island-bridge structure array on a non-developable surface; (

**b**) theory model of island on a torus surface under control by two principal curvatures, κ

_{1}and κ

_{2}; (

**c**) schematic of geometric parameters for a serpentine bridge with m unit cells; (

**d**) a numbered island-bridge structure array with m rows and n columns of islands.

**Figure 2.**Strain and strain energy in the island during conformal contact: (

**a**) maximum strain in island with non-dimensional width κ

_{2}w

_{island}at κ

_{2}t

_{island}= 10

^{−6}; (

**b**) the ratio of stretching strain energy to bending strain energy with non-dimensional parameter η.

**Figure 3.**The non-dimensional critical conformal width ${\kappa}_{2}{w}_{\mathrm{critical}}$ with $\xi $ for ${\epsilon}_{\mathrm{critical}}=1\%$ and ${\nu}_{\mathrm{island}}=0.32$. Two regions, weak adhesion and strong adhesion, are divided by ${\xi}_{\mathrm{critical}}=0.56$.

**Figure 4.**The conformal behaviors between sphere and PVC islands with different width: (

**a**) ${\kappa}_{2}{w}_{\mathrm{critical}}=0.2$, (

**b**) ${\kappa}_{2}{w}_{\mathrm{critical}}=0.3$, (

**c**) ${\kappa}_{2}{w}_{\mathrm{critical}}=0.4$, (

**d**) ${\kappa}_{2}{w}_{\mathrm{critical}}=0.5$, (

**e**) ${\kappa}_{2}{w}_{\mathrm{critical}}=0.6$, and (

**f**) enlarge view of wrinkle in (

**c**).

**Figure 5.**The axial force of a serpentine interconnect under stretching, obtained from the finite element analysis with different parameters: (

**a**) applying strain, (

**b**) wave numbers of bridge, (

**c**) thickness of bridge, (

**d**) width of bridge.

**Figure 6.**The max principal strain in the bridge versus the applied strain, and the corresponding deformation configurations in xy and yz viewport: (

**a**) ε

_{appl}= 21%, (

**b**) ε

_{appl}= 22%, (

**c**) ε

_{appl}= 40%, (

**d**) ε

_{appl}= 60%, (

**e**) ε

_{appl}= 80%, and (

**f**) ε

_{appl}= 100%.

**Figure 7.**Demands for stretchability of the bridges given by geometric method: location-dependent property of demands for stretchability in the array (

**a**) and at the first row (

**b**) for the horizontal bridges; maximum demand for stretchability of the device with the number of islands (

**c**) and area coverage (

**d**).

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**MDPI and ACS Style**

Xiao, L.; Zhu, C.; Xiong, W.; Huang, Y.; Yin, Z.
The Conformal Design of an Island-Bridge Structure on a Non-Developable Surface for Stretchable Electronics. *Micromachines* **2018**, *9*, 392.
https://doi.org/10.3390/mi9080392

**AMA Style**

Xiao L, Zhu C, Xiong W, Huang Y, Yin Z.
The Conformal Design of an Island-Bridge Structure on a Non-Developable Surface for Stretchable Electronics. *Micromachines*. 2018; 9(8):392.
https://doi.org/10.3390/mi9080392

**Chicago/Turabian Style**

Xiao, Lin, Chen Zhu, Wennan Xiong, YongAn Huang, and Zhouping Yin.
2018. "The Conformal Design of an Island-Bridge Structure on a Non-Developable Surface for Stretchable Electronics" *Micromachines* 9, no. 8: 392.
https://doi.org/10.3390/mi9080392