A Nonlinear Mechanics Model of Zigzag Cellular Substrates for Stretchable Electronics

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].

Figure 3 compares the deformed shape predicted by the proposed analytical model and that by FEA, with α₁ = 30 deg, α₂ = 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 α₁ = 30 deg, α₂ = 120 deg for different widths. Figure 4(b) shows the stress–strain curves with δ = 0.1 l, α₂–α₁ = 90 deg for different angles α₁. Figure 4(c) illustrates the stress–strain curves with δ = 0.1 l, α₁ = 30 deg for different angle differences α₂–α₁. 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 α₁) and wavier (longer curve length within the same span; described by larger (α₂–α₁)) 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 α₁ = 30 deg, α₂ = 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, α₂–α₁ = 90 deg for different angles α₁. The response of Poisson’s ratio versus stretching strain with δ = 0.1 l, α₁ = 30 deg for different angle differences α₂–α₁ is presented in Fig. 5(c). Results show that a slenderer (described by larger α₁) and wavier (described by larger (α₂–α₁)) zigzag shape yields a lower Poisson’s ratio. The Poisson’s ratio–strain relations calculated by the analytical model perfectly match the FEA results.

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. S₁₃ and S₂₃ are the interfacial shear stress components along and perpendicular to the stretching direction, respectively. Whatever the stretching direction, both S₁₃ and S₂₃ 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.

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.

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).