Home | About | Journals | Submit | Contact Us | Français |

**|**HHS Author Manuscripts**|**PMC5042350

Formats

Article sections

- Abstract
- 1. Introduction
- 2. A finite deformation model of the serpentine interconnects
- 3. Results and discussion
- 4. Conclusions
- References

Authors

Related links

Int J Solids Struct. Author manuscript; available in PMC 2017 August 1.

Published in final edited form as:

Published online 2016 April 27. doi: 10.1016/j.ijsolstr.2016.04.030

PMCID: PMC5042350

NIHMSID: NIHMS783142

Zhichao Fan,^{1} Yihui Zhang,^{1,}^{*} Qiang Ma,^{1} Fan Zhang,^{1} Haoran Fu,^{1} Keh-Chih Hwang,^{1} and Yonggang Huang^{2,}^{*}

The publisher's final edited version of this article is available at Int J Solids Struct

Lithographically defined interconnects with filamentary, serpentine configurations have been widely used in various forms of stretchable electronic devices, owing to the ultra-high stretchability that can be achieved and the relative simple geometry that facilitates the design and fabrication. Theoretical models of serpentine interconnects developed previously for predicting the performance of stretchability were mainly based on the theory of infinitesimal deformation. This assumption, however, does not hold for the interconnects that undergo large levels of deformations before the structural failure. Here, an analytic model of serpentine interconnects is developed starting from the finite deformation theory of planar, curved beams. Finite element analyses (FEA) of the serpentine interconnects with a wide range of geometric parameters were performed to validate the developed model. Comparisons of the predicted stretchability to the estimations of linear models provide quantitative insights into the effect of finite deformation. Both the theoretical and numerical results indicate that a considerable overestimation (e.g., > 50% relatively) of the stretchability can be induced by the linear model for many representative shapes of serpentine interconnects. Furthermore, a simplified analytic solution of the stretchability is obtained by using an approximate model to characterize the nonlinear effect. The developed models can be used to facilitate the designs of serpentine interconnects in future applications.

During the last decade, stretchable electronics has been attracting increasing attention, owing to their combination of high performance and extreme deformability that cannot be achieved by traditional electronics (Lacour et al., 2006; Kim et al., 2008; Sekitani et al., 2008; Sekitani et al., 2009; Sekitani and Someya, 2010; Li et al., 2011; Kim et al., 2012; Li et al., 2013; Lin et al., 2015; Lu and Yang, 2015; Xu et al., 2015; Yao and Zhu, 2015). Promising applications enabled by such technologies have been reported, such as spherical-shaped digital cameras (Ko et al., 2008; Song et al., 2013), “epidermal” health/wellness monitors (Lanzara et al., 2010; Kim et al., 2011; Jung et al., 2014) and sensitive electronic skins (Someya et al., 2004; Wagner et al., 2004; Mannsfeld et al., 2010; Kim et al., 2014; Yang et al., 2015). A key challenge in the design of stretchable electronics involves the balance of ultra-stretchability with high-performance, due to the brittleness of traditional high-quality electronic materials (Sun et al., 2012). An efficient solution employs the island-bridge architecture (Kim et al., 2008; Ko et al., 2008; Kim et al., 2009; Lee et al., 2011), in which the functional elements sit on the islands and the electrical interconnects form the bridges. Because of the considerable difference in the stiffness, the islands keep almost undeformed upon stretching, while the bridges unravel to accommodate the applied strain (Lacour et al., 2006; Lee et al., 2011). Therefore, a variety of design strategies have been proposed for the electrical interconnects, including the curvy configurations formed through postbuckling of straight ribbons (Sun et al., 2006; Ko et al., 2008; Xu et al., 2011), the planar filamentary configurations in the serpentine or fractal patterns (Gonzalez et al., 2009; Kim et al., 2009; Xu et al., 2013; Fan et al., 2014; Huang et al., 2014; Xu et al., 2014; Zhang et al., 2014; Zhu et al., 2014), and the planar configurations involving Kirigami patterns (Filipov et al., 2015; Song et al., 2015; Shyu et al., 2015; Zhang et al., 2015b). Of these design strategies, the serpentine interconnects have been widely exploited in various functional systems (Kim et al., 2008; Kim et al., 2011; Yang et al., 2015; Zhang et al., 2015a), because of the efficiency in the stretchability and relative simple geometry that facilitates the design and fabrication. In practical applications, the functional device systems interconnected by serpentine traces are usually integrated with an elastomeric substrate, as illustrated in Fig. 1a. Upon stretching, the serpentine ribbons might buckle out of plane or simply deform in the initial plane, depending on the magnitude of thickness/width ratio (Li et al., 2005; Lanzara et al., 2010; Zhang et al., 2013b). For relatively large thickness/width ratios (e.g., > 1.0) that are demanded in applications (e.g., radio-frequency coils, photovoltaic modules) (Yoon et al., 2008; Xu et al., 2013) where low electrical resistances are essential, the serpentine interconnects are governed by in-plane bending deformations. Another class of wide, ultra-thin interconnects that undergo buckled, out-of-plane deformations is also widely used to accommodate large levels of stretching without inducing failure in the materials, because of the characteristic of the wet etching lithography process (Hsu et al., 2011). Due to the complex, spatial deformations during postbuckling, the analytical modeling of buckled serpentine interconnects is very challenging, which is out of the scope of the current study. For the non-buckled serpentine interconnects, theoretical models have been developed to analyze the deformation and stretchability (Zhang et al., 2013a; Widlund et al., 2014; Yang et al., 2014). These models offer insights into the relationships between the stretchability and the various geometric parameters, on the premise that the effect of nonlinear deformation can be neglected. This assumption, however, does not hold for the interconnects that undergo large levels of deformations before the structural failure, where the prediction based on the theory of infinitesimal deformation could lead to an evident overestimation (e.g., > 50% relatively) of the stretchability.

(a) Schematic illustration of an island-bridge system with serpentine interconnects. (b) A clamped serpentine interconnect subjected to an axial stretching (*U*_{app}) at the two ends. (c) A planar, curved beam (*AB*) transformed into (*ab*) after in-plane deformation **...**

This paper aims to formulate an analytic model of serpentine interconnects in the framework of finite deformation, which provides a more accurate prediction of stretchability than the linear models (Zhang et al., 2013a; Widlund et al., 2014; Yang et al., 2014). To characterize the limit of elastic deformation for the serpentine interconnects made of metals, we introduce the elastic stretchability that is defined as the magnitude of applied strain needed to induce plastic yielding in the interconnects (Zhang et al., 2013b). For the serpentine interconnects made of relatively brittle materials (i.e., silicon, PMMA), we are usually interested in the total stretchability that is defined as the applied strain needed to induce fracture in the interconnects. In the current study, we consider suspended serpentine interconnects that can deform freely during unraveling, such that the constraint of substrate can be neglected. The developed model, as validated by finite element analyses (FEA), is exploited to analyze the nonlinear geometric effect on the stretchability for a wide range of geometric parameters and material systems. Furthermore, an analytic solution of the stretchability is obtained by using an approximate model to account for the nonlinear effect, which agrees well with the results of precise model and FEA. The results suggest that the neglect of such nonlinear effect can result in a considerable overestimation (e.g., > 50% relatively) of stretchability for relatively slender geometries, which might be detrimental in practical applications.

A typical serpentine interconnect consists of three straight wires (length *L* or *L*/2) and two arcs with identical radius (*R*) and arc angle (*α*), as illustrated in Fig. 1b. Three dimensionless parameters, the width/radius ratio = *w/R*, the arm length/radius ratio = *L/R* and the arc angle *α*, can be then adopted to characterize its shape. The width/radius ratio () of the non-buckled interconnect is usually much smaller than 0.5, such that it can be modeled as a curved, Euler-Bernoulli beam.

Consider a general planar, curved beam *AB* (width *w*) in the Cartesian coordinates (*X,Y*), which is subjected to forces or displacement loading at the ends, as shown in Fig. 1c. Let **E**_{1} and **E**_{2} denote the unit vectors along the tangential and normal directions of central axis, and *Θ* the angle by rotating the *X* axis anticlockwise to **E**_{1}. After deformation, the material point * P*(

$$\lambda =\frac{\mathrm{d}s}{\mathrm{d}S}=\frac{\sqrt{{\left(\mathrm{d}x\right)}^{2}+{\left(\mathrm{d}y\right)}^{2}}}{\mathrm{d}S},$$

(1)

where *S* and *s* is the arc-length coordinate in the undeformed and deformed configurations, respectively. The basic geometric equations include

$$\frac{\mathrm{d}x}{\mathrm{d}S}=\lambda \phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}\theta ,\phantom{\rule{1em}{0ex}}\frac{\mathrm{d}y}{\mathrm{d}S}=\lambda \phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}\phantom{\rule{thinmathspace}{0ex}}\theta ,$$

(2)

$$\Delta \widehat{\kappa}=-\frac{\mathrm{d}(\theta -\u03f4)}{\mathrm{d}S}=-\frac{\mathrm{d}\theta}{\mathrm{d}S}-K,$$

(3)

where $\Delta \widehat{\kappa}$ is the curvature change corresponding to the work conjugate of bending moment (Su et al., 2012), and *K* = −d*Θ*/d*S* is the initial curvature. The equilibrium equations can be written in terms of the internal forces (**t** = t_{1}**e**_{1} + *t*_{2}**e**_{2}) and moment (*m*) as

$$\frac{\mathrm{d}{t}_{1}}{\mathrm{d}S}+(K+\Delta \widehat{\kappa}){t}_{2}=0,\phantom{\rule{1em}{0ex}}\frac{\mathrm{d}{t}_{2}}{\mathrm{d}S}-(K+\Delta \widehat{\kappa}){t}_{1}=0,\phantom{\rule{1em}{0ex}}\frac{\mathrm{d}m}{\mathrm{d}S}-\lambda {t}_{2}=0,$$

(4a-c)

In practical applications, the serpentine interconnects can be made of plastic metals (e.g., Au, Cu, Al, etc.) (Brosteaux et al., 2007; Gonzalez et al., 2008), or relatively brittle materials [e.g., silicon, polymethyl methacrylate (PMMA)] (Kim et al., 2008; Kim et al., 2014). Here, a linear, elastic constitutive relation is adopted to characterize the material behavior:

$${t}_{1}=EA(\lambda -1),\phantom{\rule{1em}{0ex}}m=EI\Delta \widehat{\kappa},$$

(5a,b)

where *EA* and *EI* are the tensile and bending stiffnesses, respectively. Such linear relation is valid before the plastic yielding (or fracture) occurs in the metals (or brittle materials).

According to the equilibrium equations, the internal forces (*t*_{1} and *t*_{2}) can be correlated to those (*t*_{1}* _{R}* and

$${t}_{1}=\mathrm{cos}(\theta -{\theta}_{R}){t}_{1R}+\mathrm{sin}(\theta -{\theta}_{R}){t}_{2R},\phantom{\rule{1em}{0ex}}{t}_{2}=\mathrm{cos}(\theta -{\theta}_{R}){t}_{2R}-\mathrm{sin}(\theta -{\theta}_{R}){t}_{1R}.$$

(6a,b)

where the subscript ‘*R*’ denotes the right end (*x* = *x _{R}* and

$$\frac{{\mathrm{d}}^{2}\theta}{\mathrm{d}{S}^{2}}=-\frac{1}{EI}\left(\frac{{t}_{1}}{EA}+1\right){t}_{2},$$

(7)

For a given bending moment (*m _{R}*) at the right end, (d

$$\mathrm{d}S=\pm \frac{\mathrm{d}\theta}{{F}_{1}\left(\theta \right)},$$

(8)

where

$${F}_{1}\left(\theta \right)=\sqrt{\frac{2}{EIEA}\left\{\begin{array}{c}\mathrm{sin}\left(\frac{\theta -{\theta}_{R}}{2}\right)\left[{t}_{1R}\phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}\left(\frac{\theta -{\theta}_{R}}{2}\right)-{t}_{2R}\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\left(\frac{\theta -{\theta}_{R}}{2}\right)\right]\hfill \\ \left[2EA+{t}_{1R}\left(1+\mathrm{cos}\left(\theta -{\theta}_{R}\right)\right)+{t}_{2R}\phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}\left(\theta -{\theta}_{R}\right)\right]\hfill \end{array}\right\}+{\left(\frac{{m}_{R}}{EI}+K\right)}^{2}},$$

(9)

and the symbol ‘ ± ’ is determined by the sign of d*θ*/d*S*. If there is one point [*S** (0, *L _{S}*)] within the deformed beam that satisfies (d

$$\mathrm{sign}\left({\phantom{\mid}\frac{\mathrm{d}\theta}{\mathrm{d}S}\mid}_{S\in ({S}^{\ast},{L}_{S}]}\right)=\mathrm{sign}\left({\phantom{\mid}\frac{\mathrm{d}\theta}{\mathrm{d}S}\mid}_{S={L}_{S}}\right)=-\mathrm{sign}\left(K+\frac{{m}_{R}}{EI}\right),$$

(10a)

$$\mathrm{sign}\left({\phantom{\mid}\frac{\mathrm{d}\theta}{\mathrm{d}S}\mid}_{S\in [0,{S}^{\ast})}\right)=-\mathrm{sign}\left({\phantom{\mid}\frac{\mathrm{d}\theta}{\mathrm{d}S}\mid}_{S={L}_{S}}\right)=\mathrm{sign}\left(K+\frac{{m}_{R}}{EI}\right).$$

(10b)

Substitution of *θ** into the integration of Eq. (8), with the aid of Eq. (10a), gives the following equation for solving *S**,

$${S}^{\ast}={L}_{S}+\mathrm{sign}\left(K+\frac{{m}_{R}}{EI}\right){\int}_{{\theta}^{\ast}}^{{\theta}_{R}}\frac{\mathrm{d}\theta}{{F}_{1}\left(\theta \right)}.$$

(11)

If the solution of *S** is out of the domain [0, *L _{S}*], then d

Integration of Eq. (8) from the left to the right end gives an equation for determination of the angle *θ _{L}* at the left end, i.e.,

$${L}_{S}={\int}_{{\theta}_{L}}^{{\theta}_{R}}\mathrm{sign}\left(\frac{\mathrm{d}\theta}{\mathrm{d}S}\right)\frac{\mathrm{d}\theta}{{F}_{1}\left(\theta \right)},$$

(12)

With *θ _{L}* solved, the internal forces (

$$\left\{\begin{array}{c}\hfill {x}_{L}\hfill \\ \hfill {y}_{L}\hfill \end{array}\right\}=\left\{\begin{array}{c}\hfill {x}_{R}\hfill \\ \hfill {y}_{R}\hfill \end{array}\right\}-{\int}_{{\theta}_{L}}^{{\theta}_{R}}\frac{\mathrm{sign}\left(\frac{\mathrm{d}\theta}{\mathrm{d}S}\right)}{{F}_{1}\left(\theta \right)}\left[\frac{\mathrm{cos}(\theta -{\theta}_{R}){t}_{1R}+\mathrm{sin}(\theta -{\theta}_{R}){t}_{2R}}{EA}+1\right]\left\{\begin{array}{c}\hfill \mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}\theta \hfill \\ \hfill \mathrm{sin}\phantom{\rule{thinmathspace}{0ex}}\theta \hfill \end{array}\right\}\mathrm{d}\theta .$$

(13)

So far, we have established a set of equations to determine all of the displacement and force components (*t*_{1}* _{L}*,

The above model of a general beam with a constant initial curvature is then utilized to analyze the finite deformation of a serpentine interconnect. For the island-bridge design under an external stretching, the serpentine interconnect is typically subject to the displacement-type boundary conditions, as shown in Fig. 1b. The effective applied strain (*ε _{app}*) of the serpentine interconnect is defined as

$${\epsilon}_{app}=\frac{{U}_{app}}{4R\phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}(\alpha \u22152)+2L\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}(\alpha \u22152)},$$

(14)

where *U _{app}* is the applied tensile displacement. Due to the large stiffness of islands, the clamped boundaries (i.e., vanishing rotational angle and transverse displacement component) can be adopted to model the serpentine interconnect at the two ends. Considering the anti-symmetric geometry with respect to the center point, we only analyze a half of the interconnect as illustrated in Fig. 2, where the center point is pinned, corresponding to a zero moment. This model consists of two straight segments (denoted by ‘

$$\begin{array}{cc}\hfill {x}_{IIR}=& \left(\frac{L}{4}+{x}_{IL}\right)\mathrm{cos}\frac{\alpha}{2}-({y}_{IL}-R)\mathrm{sin}\frac{\alpha}{2},{y}_{IIR}=\left(\frac{L}{4}+{x}_{IL}\right)\mathrm{sin}\frac{\alpha}{2}+({y}_{IL}-R)\mathrm{cos}\frac{\alpha}{2}\hfill \\ \hfill {\theta}_{IIR}=& {\theta}_{IL}+\frac{\alpha}{2},{t}_{1IIR}={t}_{1IL},{t}_{2IIR}={t}_{2IL},{m}_{IIR}={m}_{IL},\hfill \end{array}$$

(15a)

$$\begin{array}{cc}\hfill {x}_{IIIR}=& \frac{L}{4}+{x}_{IIL}\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\frac{\alpha}{2}-{y}_{IIL}\phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}\frac{\alpha}{2},\phantom{\rule{1em}{0ex}}{y}_{IIIR}=R+{x}_{IIL}\phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}\frac{\alpha}{2}+{y}_{IIL}\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\frac{\alpha}{2},\hfill \\ \hfill {\theta}_{IIIR}=& {\theta}_{IIL}+\frac{\alpha}{2},\phantom{\rule{1em}{0ex}}{t}_{1IIIR}={t}_{1IIL},\phantom{\rule{1em}{0ex}}{t}_{2IIIR}={t}_{2IIL},\phantom{\rule{1em}{0ex}}{m}_{IIIR}={m}_{IIL}.\hfill \end{array}$$

(15b)

The boundary conditions can be written as

$${x}_{IR}=\frac{L}{4}+\frac{1}{2}{U}_{app}\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\frac{\alpha}{2},\phantom{\rule{1em}{0ex}}{y}_{IR}=-\frac{1}{2}{U}_{app}\phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}\left(\frac{\alpha}{2}\right),\phantom{\rule{1em}{0ex}}{\theta}_{IR}=0,\phantom{\rule{1em}{0ex}}{x}_{IIL}=-\frac{L}{4},\phantom{\rule{1em}{0ex}}{y}_{IIIL}=0,\phantom{\rule{1em}{0ex}}{m}_{IIIL}=0.$$

(16)

By using the equilibrium equation of the interconnect in the deformed configuration and the boundary condition *m _{IIIL}* = 0, the bending moment (

$${m}_{IR}=\left({t}_{1IR}\phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}\frac{\alpha}{2}+{t}_{2IR}\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\frac{\alpha}{2}\right)\left(2R\phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}\frac{\alpha}{2}+L\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\frac{\alpha}{2}+\frac{{U}_{app}}{2}\right).$$

(17)

Then the boundary condition *m _{IIIL}* = 0 can be replaced by Eq. (17). For a prescribed displacement (

$${\epsilon}_{max-nonlinear}=\stackrel{\u2012}{w}{F}_{2}\left(\stackrel{\u2012}{L},\alpha ,{\epsilon}_{app}\right),$$

(18)

where the membrane strain is neglected, *F*_{2}(*α,,ε _{app}*) is a function that can be determined numerically using the above model, and the subscript ‘

Schematic illustration of the mechanics model for the serpentine interconnect: (a) half of the serpentine interconnect with the left end pinned and the right end clamped and stretched out by *U*_{app}/2. Free-body diagrams of the three segments, (b) part **...**

To provide a clear understanding of the finite deformation effect on the stretchability, the analytic models (Zhang et al., 2013a; Widlund et al., 2014; Yang et al., 2014) of infinitesimal deformation are revisited for comparison. By neglecting the strain energies due to membrane and shear deformations, the Castigliano's energy method (Widlund et al., 2014) can be adopted to obtain the relationships between the constraint forces and the applied strain. According to this linear model, the peak value (*ε _{max-linear}*) of maximum principal strain might occur at the interconnect end (

$${\epsilon}_{max-linear}=\mathrm{max}\left({\epsilon}_{max1-linear},{\epsilon}_{max2-linear}\right),$$

(19)

where

$$\begin{array}{cc}\hfill {\epsilon}_{max1-linear}=& 3\stackrel{\u2012}{w}{\epsilon}_{app}[(2+\stackrel{\u2012}{L}\phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}\phantom{\rule{thinmathspace}{0ex}}\alpha -2\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}\alpha ){g}_{1}(\alpha ,\stackrel{\u2012}{L})-(\stackrel{\u2012}{L}+2\phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}\phantom{\rule{thinmathspace}{0ex}}\alpha +\stackrel{\u2012}{L}\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}\alpha ){f}_{1}(\alpha ,\stackrel{\u2012}{L})]\hfill \\ \hfill =& \stackrel{\u2012}{w}{f}_{2}(\alpha ,\stackrel{\u2012}{L}){\epsilon}_{app}\hfill \end{array}$$

(20a)

is the maximum principal strain at the interconnect end, and

$$\begin{array}{cc}\hfill {\epsilon}_{max2-linear}=& 3\stackrel{\u2012}{w}{\epsilon}_{app}[(\stackrel{\u2012}{L}+2\phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}\phantom{\rule{thinmathspace}{0ex}}{\theta}_{peak}){f}_{1}(\alpha ,\stackrel{\u2012}{L})-2(1-\mathrm{cos},{\theta}_{peak}){g}_{1}(\alpha ,\stackrel{\u2012}{L})]\hfill \\ \hfill =& \stackrel{\u2012}{w}{g}_{2}(\alpha ,\stackrel{\u2012}{L}){\epsilon}_{app}\hfill \end{array}$$

(20b)

is the peak strain in the arc segment. The expressions of *f*_{1}(*α, *) and *g*_{1}(*α, *) are given in Appendix, and *θ _{peak}* = arctan [

Calculations based on the nonlinear FEA were carried out to validate the developed analytic model. Euler-Bernoulli beam elements with relatively large thickness/width ratios (e.g., > 1.5) were adopted to model the serpentine interconnects, in which the out-of-plane deformations were suppressed. Refined meshes were used to ensure the computational accuracy. Figure 3 presents the results of peak strain (*ε _{max}*) and the associated deformed configurations for two representative serpentine interconnects, under different levels of stretching (

Analytic and FEA results of the maximum strain and the associated evolutions of deformed configurations for two serpentine interconnects under different levels of applied strain *ε*_{app} The geometric parameters are ( = 0.04, **...**

To characterize the total stretchability (*ε _{total–stretchability}*) of serpentine interconnects made of relatively brittle materials (i.e., silicon, PMMA), we introduce a simple fracture criterion based on the maximum principal strain, and takes the corresponding threshold (

$${\delta}_{elastic-stretchability}=\frac{{\epsilon}_{elastic-stretchability}^{linear}-{\epsilon}_{elastic-stretchability}^{nonlinear}}{{\epsilon}_{elastic-stretchability}^{nonlinear}},\phantom{\rule{1em}{0ex}}{\delta}_{total-stretchability}=\frac{{\epsilon}_{total-stretchability}^{linear}-{\epsilon}_{total-stretchability}^{nonlinear}}{{\epsilon}_{total-stretchability}^{nonlinear}}.$$

(21)

Obviously, the relative error of elastic stretchability (~ 0.16 in Fig. 3a and ~ 0.036 in Fig. 3b) is much smaller than that of total stretchability (~ 0.89 in Fig. 3a and ~ 0.88 in Fig. 3b).

The effect of finite deformation depends highly on various geometric parameters (*α, , *), as evidenced by the results of *δ _{elastic–stretchability}* and

As mentioned in Section 2, the maximum strain might occur at two different locations (i.e., the arc segments or the two ends). According to the results of nonlinear theory and FEA, the nonlinear effect on the maximum strain of the arc segment is nearly negligible for *ε _{max}* below 2%, with two examples shown in Fig. 5a,d. This might be attributed to the shift of maximum strain point within the arc, as the applied strain increases. Thereby, the maximum strain of the arc segment is approximately the same as the result [

$${\epsilon}_{max1-nonlinear}={\epsilon}_{max1-linear}+\stackrel{\u2012}{w}G(\stackrel{\u2012}{L},\alpha ){\epsilon}_{app}^{2},$$

(22)

where *ε*_{max1-linear} is the prediction based on the linear theory, as given by Eq. (20a); *G*(*, α*) is a function depending on the length/radius ratio and arc angle, and can be determined from the calculations using the precise nonlinear model in Section 2. Typical values of *G*(*, α*) are given in Table 1 for a wide range of and *α*. The maximum strain of the entire interconnect can be then written as

$${\epsilon}_{max-nonlinear}=\mathrm{max}({\epsilon}_{max1-nonlinear},{\epsilon}_{max2-linear}),$$

(23)

This approximate model offers a good accuracy (e.g., < 10% relatively) of predictions for most practical applications (e.g., *ε _{max}*/ <1/2). Based on this approximate model, both the elastic and total stretchabilities can be obtained as

$${\epsilon}_{stretchability}=\mathrm{min}\left(\frac{{\epsilon}_{th}}{\stackrel{\u2012}{w}{g}_{2}(\stackrel{\u2012}{L},\alpha )},\frac{-{f}_{2}(\stackrel{\u2012}{L},\alpha )+\sqrt{{\left({f}_{2}(\stackrel{\u2012}{L},\alpha )\right)}^{2}+4{\epsilon}_{th}G(\stackrel{\u2012}{L},\alpha )\u2215\stackrel{\u2012}{w}}}{2G(\stackrel{\u2012}{L},\alpha )}\right),$$

(24)

where the threshold (*ε _{th}*) of material failure corresponds to the fracture strain (2%) for the total stretchability of brittle materials, and the yield strain (0.3%) for the elastic stretchability of metals.

The maximum strain (*ε*_{max}) at the interconnect end and that of the arc segment versus the applied strain (*ε*_{app}) of two representative serpentine interconnects, with geometric parameters of ( = 0.04, **...**

Figure 5a,d shows the maximum strains (*ε _{max}*

All of the above analyses are focused on the single-period serpentine interconnects that can be joined with islands in practical applications. In addition to this type of construction, the multiple-period filamentary serpentine microstructures are also widely used in stretchable electronics (Yang et al., 2014). For such multiple-period serpentine geometries, the finite-deformation model developed in this paper can be also extended to analyze their stretchabilities. Consider a long serpentine microstructure with (*n*) periodic cells, we still take half of microstructure to simplify the analyses, considering the anti-symmetric geometry with respect to the mid-point of the entire microstructure. This model can be then divided into (*n*) arc segments and (*n*+1) straight segments. For each segment, the finite deformation model [Eqs. (1) ~ (13)] can be still used to determine the components (*t*_{1}, *t*_{2}, *m*, *θ*, *x*, *y*) of one end from those at the other end. The continuity and boundary conditions of this multiple-period model are similar to Eqs. (15) and (16). Then, we can adopt the same numerical algorithm to solve the above equations and continuity/boundary conditions. For example, a wide range of trial solutions for the inner forces at the right-end boundary can be sought to calculate the two coordinates at the left-end boundary, and the calculations terminate until the two coordinates satisfy the displacement boundary conditions to a sufficient accuracy. Substituting the solutions of inner forces and displacements/angles into the finite deformation model then gives all of the relevant information during stretching.

This work presents a finite deformation model of the non-buckled serpentine interconnects under large levels of stretching. Both the predicted maximum strain and deformed configurations agree well with the corresponding FEA results. The comparison of predicted stretchabilities to the counterparts of linear model elucidates quantitatively the effect of nonlinear finite deformation. Systematic analyses suggest that the linear model can result in a considerable overestimation (e.g., > 50% relatively) of total stretchability for many representative shapes of serpentine interconnects. Differently, the linear prediction of the elastic stretchability is overall acceptable for most metallic serpentine shapes with moderately wide ribbons (e.g., >0.1). Furthermore, an approximate model is proposed to characterize the nonlinear effect of finite deformation, which provides good predictions of stretchabilities over a wide range of geometric parameters. These results can serve as design guidelines in the optimization of serpentine interconnects for achieving highly stretchable functional devices.

Y.Z. acknowledges support from the National Science Foundation of China (Grant No. 11502129) and the Thousand Young Talents Program of China. Y.H. acknowledges the support from NSF (CMMI-1300846 and CMMI-1400169) and the NIH (grant #R01EB019337). K.C.H. acknowledges the support from the National Basic Research Program of China (Grant No. 2015CB351900).

The expressions of *f*_{1}(*α*, ) and *g*_{1}(*α*, ) in Eq. (20) are given by

$${f}_{1}(\stackrel{\u2012}{L},\alpha )=\frac{8\left(\stackrel{\u2012}{L}\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\frac{\alpha}{2}+2\phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}\frac{\alpha}{2}\right)\left\{\begin{array}{c}6[(5-5\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}\alpha ){\stackrel{\u2012}{L}}^{2}+4\alpha \stackrel{\u2012}{L}+12\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}\alpha -12]\mathrm{cos}\frac{\alpha}{2}+\hfill \\ [(5+5\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}\alpha ){\stackrel{\u2012}{L}}^{3}+(24-72\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}\alpha )\stackrel{\u2012}{L}+72\alpha ]\mathrm{sin}\frac{\alpha}{2}\hfill \end{array}\right\}}{h(\stackrel{\u2012}{L},\alpha )},$$

(A.1)

$${g}_{1}(\stackrel{\u2012}{L},\alpha )=\frac{4\left(\stackrel{\u2012}{L}\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\frac{\alpha}{2}+2\phantom{\rule{thinmathspace}{0ex}}\mathrm{sin}\frac{\alpha}{2}\right)\left\{\begin{array}{c}3(9{\stackrel{\u2012}{L}}^{3}+8\alpha {\stackrel{\u2012}{L}}^{2}+24\stackrel{\u2012}{L}+16\alpha )\mathrm{cos}\frac{\alpha}{2}+\stackrel{\u2012}{L}(5{\stackrel{\u2012}{L}}^{2}-72)\mathrm{cos}\frac{3\alpha}{2}\hfill \\ +12[(7+5\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}\alpha ){\stackrel{\u2012}{L}}^{2}+4\alpha \stackrel{\u2012}{L}+4-12\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}\alpha ]\mathrm{sin}\frac{\alpha}{2}\hfill \end{array}\right\}}{h(\stackrel{\u2012}{L},\alpha )},$$

(A.2)

where

$$h(\stackrel{\u2012}{L},\alpha )=7({\stackrel{\u2012}{L}}^{4}+60{\stackrel{\u2012}{L}}^{2}+288){\stackrel{\u2012}{L}}^{2}+24({\stackrel{\u2012}{L}}^{4}+44{\stackrel{\u2012}{L}}^{2}+288)\stackrel{\u2012}{L}\alpha +576({\stackrel{\u2012}{L}}^{2}+6){\alpha}^{2}-5184+48[6({\stackrel{\u2012}{L}}^{2}-16)\stackrel{\u2012}{L}\alpha -(5{\stackrel{\u2012}{L}}^{2}+144){\stackrel{\u2012}{L}}^{2}+192]\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}\alpha +24[(3{\stackrel{\u2012}{L}}^{4}+44{\stackrel{\u2012}{L}}^{2}-576)\stackrel{\u2012}{L}+24(3{\stackrel{\u2012}{L}}^{2}-8)\alpha ]\mathrm{sin}\phantom{\rule{thinmathspace}{0ex}}\alpha -[(7{\stackrel{\u2012}{L}}^{4}+180{\stackrel{\u2012}{L}}^{2}-4896){\stackrel{\u2012}{L}}^{2}+24({\stackrel{\u2012}{L}}^{4}-40{\stackrel{\u2012}{L}}^{2}+96)\stackrel{\u2012}{L}\alpha +4032]\mathrm{cos}\left(2\alpha \right)-12[(3{\stackrel{\u2012}{L}}^{4}+140{\stackrel{\u2012}{L}}^{2}-576)\stackrel{\u2012}{L}+4(5{\stackrel{\u2012}{L}}^{4}-42{\stackrel{\u2012}{L}}^{2}+24)\alpha ]\mathrm{sin}\left(2\alpha \right).$$

(A.3)

**Publisher's Disclaimer: **This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

- Brosteaux D, Axisa F, Gonzalez M, Vanfleteren J. Design and fabrication of elastic interconnections for stretchable electronic circuits. IEEE Electron Device Lett. 2007;28:552–554.
- Fan JA, Yeo WH, Su YW, Hattori Y, Lee W, Jung SY, Zhang YH, Liu ZJ, Cheng HY, Falgout L, Bajema M, Coleman T, Gregoire D, Larsen RJ, Huang YG, Rogers JA. Fractal design concepts for stretchable electronics. Nat. Commun. 2014;5:3266. [PubMed]
- Filipov ET, Tomohiro T, Paulino GH. Origami tubes assembled into stiff, yet reconfigurable structures and metamaterials. Proc. Nat. Acad. Sci. USA. 2015;112:12321–12326. [PubMed]
- Gleskova H, Wagner S, Suo ZG. Failure resistance of amorphous silicon transistors under extreme in-plane strain. Appl. Phys. Lett. 1999;75:3011–3013.
- Gonzalez M, Axisa F, Bossuyt F, Hsu Y-Y, Vandevelde B, Vanfleteren J. Design and performance of metal conductors for stretchable electronic circuits. Circuit World. 2009;35:22–29.
- Gonzalez M, Axisa F, BuIcke MV, Brosteaux D, Vandevelde B, Vanfleteren J. Design of metal interconnects for stretchable electronic circuits. Microelectron. Reliab. 2008;48:825–832.
- Hsu YY, Gonzalez M, Bossuyt F, Axisa F, Vanfleteren J, De Wolf I. The effects of encapsulation on deformation behavior and failure mechanisms of stretchable interconnects. Thin Solid Films. 2011;519:2225–2234.
- Huang Y, Wang Y, Xiao L, Liu H, Dong W, Yin Z. Microfluidic serpentine antennas with designed mechanical tunability. Lab on a Chip. 2014;14:4205–4212. [PubMed]
- Jung S, Kim JH, Kim J, Choi S, Lee J, Park I, Hyeon T, Kim DH. Reverse-micelle-induced porous pressure-sensitive rubber for wearable human-machine interfaces. Adv. Mater. 2014;26:4825–4830. [PubMed]
- Kim D-H, Ghaffari R, Lu NS, Rogers JA. Flexible and stretchable electronics for biointegrated devices. Annu. Rev. Biomed. Eng. 2012;14:113–128. [PubMed]
- Kim D-H, Liu ZJ, Kim Y-S, Wu J, Song JZ, Kim H-S, Huang YG, Hwang K-C, Zhang YW, Rogers JA. Optimized structural designs for stretchable silicon integrated circuits. Small. 2009;5:2841–2847. [PubMed]
- Kim DH, Lu NS, Ma R, Kim YS, Kim RH, Wang SD, Wu J, Won SM, Tao H, Islam A, Yu KJ, Kim TI, Chowdhury R, Ying M, Xu LZ, Li M, Chung HJ, Keum H, McCormick M, Liu P, Zhang YW, Omenetto FG, Huang YG, Coleman T, Rogers JA. Epidermal electronics. Science. 2011;333:838–843. [PubMed]
- Kim DH, Song JZ, Choi WM, Kim HS, Kim RH, Liu ZJ, Huang YY, Hwang KC, Zhang YW, Rogers JA. Materials and noncoplanar mesh designs for integrated circuits with linear elastic responses to extreme mechanical deformations. Proc. Nat. Acad. Sci. USA. 2008;105:18675–18680. [PubMed]
- Kim J, Lee M, Shim HJ, Ghaffari R, Cho HR, Son D, Jung YH, Soh M, Choi C, Jung S, Chu K, Jeon D, Lee ST, Kim JH, Choi SH, Hyeon T, Kim DH. Stretchable silicon nanoribbon electronics for skin prosthesis. Nat. Commun. 2014;5:5747. [PubMed]
- Ko HC, Stoykovich MP, Song JZ, Malyarchuk V, Choi WM, Yu C-J, Geddes JB, III, Xiao J, Wang SD, Huang YG, Rogers JA. A hemispherical electronic eye camera based on compressible silicon optoelectronics. Nature. 2008;454:748–753. [PubMed]
- Lacour SP, Wagner S, Narayan RJ, Li T, Suo ZG. Stiff subcircuit islands of diamondlike carbon for stretchable electronics. J. Appl. Phys. 2006;100:014913.
- Lanzara G, Salowitz N, Guo ZQ, Chang F-K. A spider-web-like highly expandable sensor network for multifunctional materials. Adv. Mater. 2010;22:4643–4648. [PubMed]
- Lee J, Wu J, Shi MX, Yoon J, Park S-I, Li M, Liu ZJ, Huang YG, Rogers JA. Stretchable GaAs photovoltaics with designs that enable high areal coverage. Adv. Mater. 2011;23:986–991. [PubMed]
- Li T, Suo ZG, Lacour SP, Wagner S. Compliant thin film patterns of stiff materials as platforms for stretchable electronics. J. Mater. Res. 2005;20:3274–3277.
- Li Y, Shi X, Song J, Lü C, Kim T.-i., McCall JG, Bruchas MR, Rogers JA, Huang Y. Thermal analysis of injectable, cellular-scale optoelectronics with pulsed power. Proc. R. Soc. A. 2013;469:20130398. [PMC free article] [PubMed]
- Li YH, Fang B, Zhang JH, Song JZ. Surface effects on the wrinkling of piezoelectric films on compliant substrates. J. Appl. Phys. 2011;110:114303.
- Lin ST, Yuk H, Zhang T, Parada GA, Koo H, Yu CJ, Zhao XH. Stretchable hydrogel electronics and devices. Adv. Mater. In Press. 2015 DOI: 10.1002/adma.201504152.
- Lu NS, Yang SX. Mechanics for stretchable sensors. Curr. Opin. Solid State Mat. Sci. 2015;19:149–159.
- Mannsfeld SCB, Tee BCK, Stoltenberg RM, Chen CVHH, Barman S, Muir BVO, Sokolov AN, Reese C, Bao Z. Highly sensitive flexible pressure sensors with microstructured rubber dielectric layers. Nat. Mater. 2010;9:859–864. [PubMed]
- Sekitani T, Nakajima H, Maeda H, Fukushima T, Aida T, Hata K, Someya T. Stretchable active-matrix organic light-emitting diode display using printable elastic conductors. Nat. Mater. 2009;8:494–499. [PubMed]
- Sekitani T, Noguchi Y, Hata K, Fukushima T, Aida T, Someya T. A rubberlike stretchable active matrix using elastic conductors. Science. 2008;321:1468–1472. [PubMed]
- Sekitani T, Someya T. Stretchable, large-area organic electronics. Adv. Mater. 2010;22:2228–2246. [PubMed]
- Shyu TC, Damasceno PF, Dodd PM, Lamoureux A, Xu LZ, Shlian M, Shtein M, Glotzer SC, Kotov NA. A kirigami approach to engineering elasticity in nanocomposites through patterned defects. Nat. Mater. 2015;14:785–789. [PubMed]
- Someya T, Sekitani T, Iba S, Kato Y, Kawaguchi H, Sakurai T. A large-area, flexible pressure sensor matrix with organic field-effect transistors for artificial skin applications. Proc. Nat. Acad. Sci. USA. 2004;101:9966–9970. [PubMed]
- Song YM, Xie YZ, Malyarchuk V, Xiao JL, Jung I, Choi K-J, Liu ZJ, Park H, Lu CF, Kim R-H, Li R, Crozier KB, Huang YG, Rogers JA. Digital cameras with designs inspired by the arthropod eye. Nature. 2013;497:95–99. [PubMed]
- Song ZM, Wang X, Lv C, An YH, Liang MB, Ma T, He D, Zheng YJ, Huang SQ, Yu HY, Jiang HQ. Kirigami-based stretchable lithium-ion batteries. Sci. Rep. 2015;5:10988. [PMC free article] [PubMed]
- Su YW, Wu J, Fan ZC, Hwang K-C, Song JZ, Huang YG, Rogers JA. Postbuckling analysis and its application to stretchable electronics. J. Mech. Phys. Solids. 2012;60:487–508.
- Sun JY, Lu NS, Yoon J, Oh KH, Suo ZG, Vlassak JJ. Debonding and fracture of ceramic islands on polymer substrates. J. Appl. Phys. 2012;111:013517.
- Sun Y, Choi WM, Jiang HQ, Huang YG, Rogers JA. Controlled buckling of semiconductor nanoribbons for stretchable electronics. Nat. Nanotechnol. 2006;1:201–207. [PubMed]
- Wagner S, Lacour SP, Jones J, Hsu PHI, Sturm JC, Li T, Suo ZG. Electronic skin: architecture and components. Physica E. 2004;25:326–334.
- Widlund T, Yang SX, Hsu YY, Lu NS. Stretchability and compliance of freestanding serpentine-shaped ribbons. Int. J. Solids Struct. 2014;51:4026–4037.
- William FR, Leroy DS, Don HM. Mechanics of materials. John Wiley & Sons; New York: 1999.
- Xu F, Lu W, Zhu Y. Controlled 3D buckling of silicon nanowires for stretchable electronics. Acs Nano. 2011;5:672–678. [PubMed]
- Xu S, Yan Z, Jang K-I, Huang W, Fu HR, Kim J, Wei ZJ, Flavin M, McCracken J, Wang R, Badea A, Liu YH, Xiao DQ, Zhou GY, Lee J, Chung HU, Cheng H, Ren W, Banks A, Li XL, Paik U, Nuzzo RG, Huang YG, Zhang YH, Rogers JA. Assembly of micro/nanomaterials into complex, three-dimensional architectures by compressive buckling. Science. 2015;347:154–159. [PubMed]
- Xu S, Zhang YH, Cho J, Lee J, Huang X, Jia L, Fan JA, Su YW, Su J, Zhang HG, Cheng HY, Lu BW, Yu CJ, Chuang C, Kim T.-i., Song T, Shigeta K, Kang S, Dagdeviren C, Petrov I, Braun PV, Huang YG, Paik U, Rogers JA. Stretchable batteries with self-similar serpentine interconnects and integrated wireless recharging systems. Nat. Commun. 2013;4:1543. [PubMed]
- Xu S, Zhang YH, Jia L, Mathewson KE, Jang KI, Kim J, Fu HR, Huang X, Chava P, Wang RH, Bhole S, Wang LZ, Na YJ, Guan Y, Flavin M, Han ZS, Huang YG, Rogers JA. Soft microfluidic assemblies of sensors, circuits, and radios for the skin. Science. 2014;344:70–74. [PubMed]
- Yang SX, Chen YC, Nicolini L, Pasupathy P, Sacks J, Su B, Yang R, Sanchez D, Chang YF, Wang PL, Schnyer D, Neikirk D, Lu NS. “Cut-and-Paste” manufacture of multiparametric epidermal sensor systems. Adv. Mater. 2015;27:6423–6430. [PubMed]
- Yang SX, Su B, Bitar G, Lu NS. Stretchability of indium tin oxide (ITO) serpentine thin films supported by Kapton substrates. Int. J. Fracture. 2014;190:99–110.
- Yao SS, Zhu Y. Nanomaterial-enabled stretchable conductors: strategies, materials and devices. Adv. Mater. 2015;27:1480–1511. [PubMed]
- Yoon J, Baca AJ, Park S-I, Elvikis P, Geddes JB, III, Li L, Kim RH, Xiao J, Wang S, Kim T-H, Motala MJ, Ahn BY, Duoss EB, Lewis JA, Nuzzo RG, Ferreira PM, Huang Y, Rockett A, Rogers JA. Ultrathin silicon solar microcells for semitransparent, mechanically flexible and microconcentrator module designs. Nat. Mater. 2008;7:907–915. [PubMed]
- Zhang YH, Fu HR, Su YW, Xu S, Cheng HY, Fan JA, Hwang K-C, Rogers JA, Huang YG. Mechanics of ultra-stretchable self-similar serpentine interconnects. Acta Mater. 2013a;61:7816–7827.
- Zhang YH, Fu HR, Xu S, Fan JA, Hwang KC, Jiang JQ, Rogers JA, Huang YG. A hierarchical computational model for stretchable interconnects with fractal-inspired designs. J. Mech. Phys. Solids. 2014;72:115–130.
- Zhang YH, Huang YG, Rogers JA. Mechanics of stretchable batteries and supercapacitors. Curr. Opin. Solid State Mat. Sci. 2015a;19:190–199.
- Zhang YH, Xu S, Fu HR, Lee J, Su J, Hwang K-C, Rogers JA, Huang YG. Buckling in serpentine microstructures and applications in elastomer-supported ultra-stretchable electronics with high areal coverage. Soft Matter. 2013b;9:8062–8070. [PMC free article] [PubMed]
- Zhang YH, Yan Z, Nan KW, Xiao DQ, Liu YH, Luan HW, Fu HR, Wang XZ, Yang QL, Wang JC, Ren W, Si HZ, Liu F, Yang LH, Li HJ, Wang JT, Guo XL, Luo HY, Wang L, Huang YG, Rogers JA. A mechanically driven form of Kirigami as a route to 3D mesostructures in micro/nanomembranes. Proc. Nat. Acad. Sci. USA. 2015b;112:11757–11764. [PubMed]
- Zhu SZ, Huang YJ, Li T. Extremely compliant and highly stretchable patterned graphene. Appl. Phys. Lett. 2014;104:173103.

PubMed Central Canada is a service of the Canadian Institutes of Health Research (CIHR) working in partnership with the National Research Council's national science library in cooperation with the National Center for Biotechnology Information at the U.S. National Library of Medicine(NCBI/NLM). It includes content provided to the PubMed Central International archive by participating publishers. |