## Abstract

The use of cellular elastomer substrates not only reduces its restriction on natural diffusion or convection of biofluids in the realm of stretchable electronics but also enhances the stretchability of the electronic systems. An analytical model of “zigzag” cellular substrates under finite deformation is established and validated in this paper. The deformed shape, nonlinear stress–strain curve, and Poisson’s ratio–strain curve of the cellular elastomer substrate calculated using the reported analytical model agree well with those from finite element analysis (FEA). Results show that lower restriction on the natural motion of human skin could be achieved by the proposed zigzag cellular substrates compared with the previously reported hexagonal cellular substrates, manifesting another leap toward mechanically “invisible” wearable, stretchable electronic systems.

## 1 Introduction

Significant progress has been achieved in materials [1–3] and mechanics [4–16] in offering the capability for stretchable electronics to be deformed into complex shapes without failure in functionality or structure. A recent direction of devising mechanically “invisible” skin-mounted stretchable electronics [17–19], which are ultrasoft and hardly detectable by skin through tactile sensation, demands a new class of compliant elastomer substrates. Like many biological materials, e.g., skin [20,21] and viscid spider silk [22,23], which show “J-shaped” stress–strain behaviors as a result of collagen microstructures, cellular elastomer substrates offer similar behavior [24,25]. The integration of stretchable electronics onto cellular substrates could achieve a similar “J-shaped” stress–strain response as in the biological materials, with a compliant mechanical behavior at low stretching strains and an increasingly stiff response at larger stretching strains to balance wear comfort and structural integrity, shows promising application potential in bio-integrated electronic microsystems and tissue engineering [26–31].

Apart from “J-shaped” stress–strain behavior and high permeability of biofluids [25,32], cellular substrates could achieve much larger stretchability compared with counterpart uniform solid substrates [33]. Some analytical models have been established to investigate the equilibrium and deformation compatibility of hexagonal cellular substrates under finite stretching [34–37]. The stretching behaviors of substrates affect the interfacial shear stress between substrate and human skin, which is one of the key considerations when improving the wear comfort of stretchable, epidermal electronics. The shear stress between hexagonal cellular substrates and human skin is usually higher than the skin sensitivity threshold under finite deformation [37]. In this paper, “zigzag” cellular substrates are investigated and engineered to achieve higher compliance than the previously reported hexagonal cellular substrates, inducing lower interfacial shear stress under finite deformation.

An analytical model to investigate the zigzag cellular substrates with finite deformation is presented in this paper. Results show that the stress–strain relationship calculated by the analytical model matches that from finite element analysis (FEA). The interfacial shear stress between zigzag cellular substrates and human skin is well below the skin sensitivity threshold (20 kPa) [38] in any stretching direction and is noticeably lower than that for the hexagonal cellular substrates [34,35].

## 2 Analytical Model

*l*is much larger than its width

*δ*. All beam segments in the zigzag substrate are assumed to have the same length

*l*in this study. The thickness of zigzag cellular substrates is assumed to be larger than the width of walls. For uniaxial stretching along

*x*-/

*y*-axis, the basic unit is denoted in the dashed rectangle in Fig. 1(a). 1/8 of the basic unit is chosen for analysis on account of symmetry as presented in Fig. 1(b). A local coordinate system ${x\xafi,y\xafi}$ is constructed as shown in Fig. 1(b). The $x\xafi$-axis is parallel to the beams of zigzag cellular substrate, which is also illustrated in Fig. 2. The stretching force per unit thickness acting on the 1/8 basic unit can be denoted as

*P*. The equivalent nominal stress is normalized by the initial modulus of substrate material

*E*

_{0}as

*α*

_{i}is the rotation angle between the local coordinate system and the global coordinate system (

*i*= 1 or 2 denotes beam

*AB*and

*BC*, respectively). The incompressible Mooney–Rivlin constitutive relation [34,39] gives the uniaxial stress–strain relationship $\sigma =E0f(\epsilon )$, where

*f*(

*ɛ*) is

*a*)

*b*)

*Q*is the shear force,

_{i}*N*is the axial force, and

_{i}*φ*

_{i}is the angle between the beam axial direction and $x\xafi$-direction. Let

*S*and

*s*represent the axial length of the beam segment in zigzag cellular substrate before and after stretching, respectively. The relationship between d

*s*and d

_{i}*S*is

_{i}*M*is the moment. The axial force

_{i}*N*and shear force

_{i}*Q*are related to

_{i}*P*and

*φ*

_{i}as

*a*)

*b*)

*E*

_{0}

*I*is the in-plane bending stiffness.

*AB*, the moment at point

*B*is denoted by

*M*

_{0}, and the moment condition at point

*B*is

*S*

_{1}= 0 to

*S*

_{1}=

*l*

*BC*, the moment condition at point

*C*is

*S*

_{2}= 0 to

*S*

_{2}=

*l*/2

*B*, of

*α*

_{1}−

*β*

_{1}as shown in Fig. 2(b). The local coordinates of points

*B*and

*C*can be obtained from $dx\xaf/dsi=cos\phi i$ and $dy\xaf/dsi=sin\phi i$, as

*a*)

*b*)

*B*and

*C*after deformation are related to their local coordinates by

*a*)

*b*)

*B*can be obtained based on the global coordinates of points

*B*and

*C*as

*a*)

*b*)

*a*), for a specific location along beam

*AB*with angle

*β*

_{3}(the angle between

*x*-direction and the beam axial direction along beam

*AB*after deformation) ranging between

*β*

_{1}and

*α*

_{1}, the local coordinates can be obtained as

*b*)

*b*), for a specific location along beam

*BC*with angle

*β*

_{4}(the angle between

*x*-direction and the beam axial direction along beam

*BC*after deformation) ranging between

*β*

_{2}and $\alpha 2\u2212\alpha 1+\beta 1$, the local coordinates can be obtained as

*a*)

## 3 Results

Figure 3 compares the deformed shape predicted by the proposed analytical model and that by FEA, with *α*_{1} = 30 deg, *α*_{2} = 120 deg, *δ* = 0.1 *l*, and *l* = 0.3 mm at different stretching strains. Four-node shell elements are chosen for the cellular substrate in the commercial FEA software suite abaqus. The deformed shape calculated by the analytical model agrees well with that extracted directly from FEA.

The stress–strain curves predicted by FEA and the analytical model for the zigzag cellular substrates are demonstrated in Fig. 4. Figure 4(a) manifests the stress–strain curves with *α*_{1} = 30 deg, *α*_{2} = 120 deg for different widths. Figure 4(b) shows the stress–strain curves with *δ* = 0.1 *l*, *α*_{2}–*α*_{1} = 90 deg for different angles *α*_{1}. Figure 4(c) illustrates the stress–strain curves with *δ* = 0.1 *l*, *α*_{1} = 30 deg for different angle differences *α*_{2}–*α*_{1}. The results of parameter analysis show that a lower equivalent modulus can be obtained with a smaller width and a slenderer (larger lateral-parallel aspect ratio when uniaxially stretched; described by larger *α*_{1}) and wavier (longer curve length within the same span; described by larger (*α*_{2}–*α*_{1})) zigzag shape. The stress–strain responses calculated by the analytical model agree well with the FEA results.

Figure 5 presents the Poisson’s ratio–strain curves calculated by the analytical model and predicted by FEA for the zigzag cellular substrates. Poisson’s ratio versus stretching strain with *α*_{1} = 30 deg, *α*_{2} = 120 deg for different widths is shown in Fig. 5(a). The width has a noticeable effect on the Poisson’s ratio. Figure 5(b) illustrates Poisson’s ratio versus stretching strain with *δ* = 0.1 *l*, *α*_{2}–*α*_{1} = 90 deg for different angles *α*_{1}. The response of Poisson’s ratio versus stretching strain with *δ* = 0.1 *l*, *α*_{1} = 30 deg for different angle differences *α*_{2}–*α*_{1} is presented in Fig. 5(c). Results show that a slenderer (described by larger *α*_{1}) and wavier (described by larger (*α*_{2}–*α*_{1})) zigzag shape yields a lower Poisson’s ratio. The Poisson’s ratio–strain relations calculated by the analytical model perfectly match the FEA results.

*ψ*of the zigzag cellular substrate is calculated as [35,40]

The shear stress between the cellular substrates and the skin is computed using FEA where the cellular substrates are modeled with four-node shell elements and the skin with eight-node solid elements. Figures 6(c) and 6(d) gives the shear stress between the cellular substrate and the skin under 50% uniaxial stretching of the skin along *x*-direction. The maximum interfacial shear stress along the stretching direction between the skin and the zigzag cellular substrate (Fig. 6(d)) is 14.8 kPa, which is smaller than the human skin sensitivity of 20 kPa [38], suggesting smaller constraints on the skin than the hexagonal cellular substrate (Fig. 6(c)). The same phenomenon also holds when stretching along *y*-direction, as shown in Figs. 6(e) and 6(f). The interfacial shear stress distribution is insensitive to a moderate increase of cellular substrate thickness.

Figure 7 shows the dependence of the interfacial shear stress (on the skin) on the stretching direction (which is denoted by the angle between the stretching direction and the positive *x*-direction (in Fig. 6)), for hexagonal and zigzag cellular substrates. *S*_{13} and *S*_{23} are the interfacial shear stress components along and perpendicular to the stretching direction, respectively. Whatever the stretching direction, both *S*_{13} and *S*_{23} with the zigzag cellular substrate are lower than those with the hexagonal cellular substrate. The maximum shear stress between the zigzag cellular substrate and the skin is noticeably lower than the human skin sensitivity of 20 kPa for any stretching direction. Lower constraints on the natural movement of skin could be achieved using the zigzag cellular substrates.

## 4 Conclusions

An analytical model of zigzag cellular substrates under finite deformation is proposed. The deformed shape, nonlinear stress–strain curves and Poisson’s ratio–strain curves can be conveniently calculated by the reported analytical model and agree well with FEA predictions. The results of parameter analysis show that a smaller width and a slenderer (larger aspect ratio) and wavier (longer curve length within the same span) zigzag shape yield a lower equivalent modulus and a lower Poisson’s ratio for the zigzag cellular substrates. Whatever the stretching direction of skin, the skin-mounted zigzag cellular substrates could achieve lower constraints on the natural motion of human skin compared with those hexagonal cellular substrates, suggesting promising application opportunities toward mechanically “invisible” epidermal electronics.

## Acknowledgment

The authors gratefully acknowledge the support from the National Natural Science Foundation of China under Grant Nos. 11972059 and 11572023. Y. H. acknowledges the support from the National Science Foundation, USA (Grant No. CMMI1635443).