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Research Papers

Mechanics of Fractal-Inspired Horseshoe Microstructures for Applications in Stretchable Electronics

[+] Author and Article Information
Qiang Ma

Department of Engineering Mechanics,
Center for Mechanics and Materials,
AML,
Tsinghua University,
Beijing 100084, China

Yihui Zhang

Department of Engineering Mechanics,
Center for Mechanics and Materials,
AML,
Tsinghua University,
Beijing 100084, China
e-mail: yihuizhang@tsinghua.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received July 20, 2016; final manuscript received August 15, 2016; published online September 8, 2016. Editor: Yonggang Huang.

J. Appl. Mech 83(11), 111008 (Sep 08, 2016) (19 pages) Paper No: JAM-16-1364; doi: 10.1115/1.4034458 History: Received July 20, 2016; Revised August 15, 2016

Fractal-inspired designs represent an emerging class of strategy for stretchable electronics, which have been demonstrated to be particularly useful for various applications, such as stretchable batteries and biointegrated electrophysiological electrodes. The fractal-inspired constructs usually undergo complicated, nonlinear deformations under mechanical loading, because of the highly complex and diverse microstructures inherent in high-order fractal patterns. The underlying relations between the nonlinear mechanical responses and microstructure geometry are essential in practical applications, which require a relevant mechanics theory to serve as the basis of a design approach. Here, a theoretical model inspired by the mechanism of ordered unraveling is developed to study the nonlinear stress–strain curves and elastic stretchability for a class of fractal-inspired horseshoe microstructures. Analytic solutions were obtained for some key mechanical quantities, such as the elastic modulus and the tangent modulus at the beginning of each deformation stage. Both the finite-element analyses (FEA) and experiments were carried out to validate the model. Systematic analyses of the microstructure–property relationship dictate how to leverage the various geometric parameters to tune the multistage, J-shaped stress–strain curves. Moreover, a demonstrative example shows the utility of the theoretical model in design optimization of fractal-inspired microstructures used as electrophysiological electrodes, aiming to achieve maximum elastic stretchability for prescribed filling ratios. The results indicate a substantial enhancement (e.g., >4 times) of elastic stretchability by using fractal designs, as compared to traditional horseshoe designs. This study can serve as design guidelines of fractal-inspired microstructures in different stretchable electronic systems.

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Figures

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Fig. 2

Schematic illustration of the theoretical model and the results for equivalent flexibilities of the fractal horseshoe microstructures: (a) a freely suspended first-order horseshoe microstructure, clamped at the left end, and subject to an axial force N(1), a shear force Q(1), and a bending moment M(1), at the right end; (b) a freely suspended second-order horseshoe microstructure clamped at the left end, and subject to an axial force N(2), a shear force Q(2), and a bending moment M(2), at the right end, with a magnified view for its (k)th-unit cell; (c) dimensionless flexibility components versus the fractal order for the arc angle (θ = 180 deg) and number (m = 3) of unit cell; dimensionless flexibility components of the (d) second- and (e) fourth-order microstructures versus the arc angle for wide-ranging numbers (m) of unit cell; (f) the ratio of dimensionless flexibility components between neighboring orders T¯ij(n+1)/T¯ij(n) versus the arc angle for different fractal orders and fixed number (m = 8) of unit cell

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Fig. 1

Geometry of fractal horseshoe microstructures: (a) schematic illustration of the geometric construction of the fractal horseshoe microstructures, (b) schematic illustration of various geometric parameters for the third- and second-order microstructures, and (c) the critical arc angle (θ) to avoid self-overlap versus the number (m) of unit cell for fractal orders 2, 3, and infinity

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Fig. 3

Schematic illustration of the finite-deformation model and the results for the first-order horseshoe microstructures: (a) a simply supported horseshoe microstructure subject to a pair of axial forces at two ends; (b) schematic illustration on the deformation of a half of the horseshoe microstructure subject to a pair of axial forces at two ends; (c) normalized stress–strain curves for three different normalized widths (w¯=0.1, 0.3, and 0.5), and fixed arc angle (θ = 240 deg); (d) normalized stress–strain curves for three different arc angles (θ = 90 deg, 180 deg, and 240 deg), and fixed normalized width (w¯=0.2). The dashed-lines denote the critical strain.

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Fig. 4

Schematic illustration of the finite-deformation model for the second-order fractal horseshoe microstructures: (a) stage I—unraveling the second-order microstructure, in which the entire microstructure is modeled by an equivalent first-order horseshoe structure; and (b) stage II—unraveling each first-order microstructure that is extracted from the fully unraveled second-order one at the end of stage I

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Fig. 6

Theoretical and FEA results of normalized stress–strain curves and tangent modulus–strain curves for fractal horseshoe microstructures: (a) normalized stress–strain curves (with the stress in logarithmic scale) for fractal order from 1 to 3; for different strain ranges, those stress–strain curves appear in linear scale for strains between (b, c, e) 0–170%, (d, f) 150–550%, and (g) 500–1400%; (h) corresponding tangent modulus–strain curves (with the tangent modulus in logarithmic scale). The arc angle, normalized width, and number of unit cell are fixed as θ = 240 deg, w¯=0.2, and m = 8, respectively. The dashed-lines denote the critical strain.

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Fig. 7

Key mechanical properties versus the geometric parameters: (a) theoretical results of critical strain versus arc angle for various fractal orders and numbers of unit cell; (b) theoretical and FEA results of the normalized elastic modulus Et−initial(n,n)/[Es(w/R(n))2] versus arc length for fractal orders from 1 to 3; (c, d) theoretical results of the modulus ratio Et−initial(n,i−1)/Et−initial(n,i) versus geometric parameters (arc angle and number of unit cell); (e, f) theoretical results of the normalized tangent modulus Et−cr(n)/Es at the largest critical strain versus geometric parameters (arc angle and equivalent width) for first- and second-order microstructures

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Fig. 5

Theoretical predictions and optical images of deformed configurations for two second-order fractal horseshoe microstructures, with arc angles of (a) θ = 180 deg and (b) θ = 240 deg, under at different levels of applied strain. The normalized width and unit cell number are fixed as w¯=0.15 and m = 5. The scale bars in (a) and (b) are 5 mm.

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Fig. 9

Theoretical and FEA results of normalized stress–strain curves for the generalized second-order fractal horseshoe microstructures: normalized stress–strain curves for (a) a widerange of θ(1), with m(2) = 8 and θ(2)=180 deg; (b) a wide range of angle combinations, (θ(1),θ(2))=(219 deg,120 deg), (204 deg,150deg), (180deg,180deg), and (141deg,210deg), with m(2) = 8; and (c) a wide range of m(2), with θ(1)=θ(2)=180deg. The normalized width is fixed as w¯=0.2. The dashed-lines denote the critical strain.

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Fig. 10

(a) Schematic illustration of a representative EP electrode constructed with the generalized second-order fractal horseshoe microstructures. Theoretical and FEA results on the ratio of maximum strain to normalized width versus applied strain for (b) a wide range of w¯, with m(2) = 8, and θ(1)=θ(2)=180 deg; (c) a wide range of θ(1), with w¯=0.2, m(2) = 8, and θ(2)=180 deg; (d) a wide range of θ(2), with w¯=0.2, m(2) = 8, and θ(1)=180 deg; and (e) a wide range of m(2), with w¯=0.2, and θ(1)=θ(2)=180 deg. The dashed-lines denote the critical strain.

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Fig. 11

Elastic stretchability and filling ratio for the generalized second-order fractal horseshoe microstructures: (a) elastic stretchability and (b) filling ratio versus m(2), with θ(1)=θ(2)=210 deg; (c) elastic stretchability and (d) filling ratio versus θ(1), with m(2) = 6 and θ(2)=210 deg; (e) elastic stretchability and (f) filling ratio versus θ(2), with m(2) = 6 and θ(1)=210deg. A wide range of normalized width from 0.1 to 0.6 is considered.

Grahic Jump Location
Fig. 12

Design optimization of the generalized second-order fractal horseshoe microstructures and traditional first-order microstructures for uses as EP electrodes: (a) the maximum elastic stretchability versus filling ratio of EP electrodes with the generalized second-order fractal designs and traditional first-order designs; (b) optimal geometries of EP electrodes with the generalized second- and first-order microstructures for three representative filling ratios; (c) distribution of the maximum principal strain in the deformed configurations, for the optimal designs of EP electrodes (15% filling ratio) with the generalized second- and first-order microstructures

Grahic Jump Location
Fig. 8

Theoretical and FEA results of normalized stress–strain curves for the second-order fractal horseshoe microstructures: (a) normalized stress–strain curves for a wide range of arc angle, and fixed normalized width of w¯=0.2 and (b) normalized stress–strain curves for a wide range of normalized width, and fixed arc angle of θ0 = 240 deg. The number of unit cell is fixed as m = 8. The dashed-lines denote the critical strain.

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