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Research Papers

A Mechanics Model of Soft Network Materials With Periodic Lattices of Arbitrarily Shaped Filamentary Microstructures for Tunable Poisson's Ratios

[+] Author and Article Information
Jianxing Liu

AML,
Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China;
Center for Flexible Electronics Technology,
Tsinghua University,
Beijing 100084, China;
Center for Mechanics and Materials,
Tsinghua University,
Beijing 100084, China

Yihui Zhang

AML,
Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China;
Center for Flexible Electronics Technology,
Tsinghua University,
Beijing 100084, China;
Center for Mechanics and Materials,
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 February 11, 2018; final manuscript received February 13, 2018; published online March 2, 2018. Editor: Yonggang Huang.

J. Appl. Mech 85(5), 051003 (Mar 02, 2018) (17 pages) Paper No: JAM-18-1087; doi: 10.1115/1.4039374 History: Received February 11, 2018; Revised February 13, 2018

Soft network materials that incorporate wavy filamentary microstructures have appealing applications in bio-integrated devices and tissue engineering, in part due to their bio-mimetic mechanical properties, such as “J-shaped” stress–strain curves and negative Poisson's ratios. The diversity of the microstructure geometry as well as the network topology provides access to a broad range of tunable mechanical properties, suggesting a high degree of design flexibility. The understanding of the underlying microstructure-property relationship requires the development of a general mechanics theory. Here, we introduce a theoretical model of infinitesimal deformations for the soft network materials constructed with periodic lattices of arbitrarily shaped microstructures. Taking three representative lattice topologies (triangular, honeycomb, and square) as examples, we obtain analytic solutions of Poisson's ratio and elastic modulus based on the mechanics model. These analytic solutions, as validated by systematic finite element analyses (FEA), elucidated different roles of lattice topology and microstructure geometry on Poisson's ratio of network materials with engineered zigzag microstructures. With the aid of the theoretical model, a crescent-shaped microstructure was devised to expand the accessible strain range of network materials with relative constant Poisson's ratio under large levels of stretching. This study provides theoretical guidelines for the soft network material designs to achieve desired Poisson's ratio and elastic modulus.

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Figures

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Fig. 1

Geometric construction of the soft network materials with arbitrarily shaped curvy microstructures: (a) triangular network materials, (b) honeycomb network materials, and (c) square network materials. The images on the bottom show the configurations of the representative unit cells of different network materials.

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Fig. 2

Schematic illustration of the mechanics model for the arbitrarily shaped curvy microstructure and triangular network material: (a) internal forces and bending moment of a microstructure described by the curvilinear coordinate (S). (b) A simply supported beam microstructure subject to an axial force at the right end, and moments MA and MB at the two ends. (c) The free body diagram of the beam microstructure in (b). (d) A triangular soft network material subject to a uniform tensile stress along the horizontal stretching. (e) Free body diagrams of the three microstructures aligned in different directions in the triangular network. (f) and (g) Illustrations of the deformations in two representative unit cells of the triangular soft network material.

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Fig. 3

Schematic illustration of the mechanics model for the honeycomb and square network materials: (a) a honeycomb soft network material subject to a uniform tensile stress along the horizontal stretching, (b) free body diagrams of the three microstructures aligned in different directions in the honeycomb network, (c) a square soft network material subject to a uniform tensile stress along the horizontal stretching, and (d) free body diagrams of the four microstructures in the square network

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Fig. 4

Geometry of the zigzag microstructures: (a) schematic illustration and key geometric parameters of the zigzag microstructure, (b) geometric conditions of the normalized width (w¯=w/L0) and slopes (tanθ0) that should be satisfied to form the triangular, honeycomb and square network materials without any self-overlay, and (c) schematic illustration and key geometric parameters of the zigzag microstructure with a reduced width at the turning regions

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Fig. 5

Theoretical and FEA results of the linear Poisson's ratio and elastic modulus for triangular network materials: (a) linear Poisson's ratio (ν) of the triangular network material versus the width ratio (w2/w1) for a wide range of microstructure slopes (tanθ0) and a fixed length ratio (L2/(L1+L2) = 0.1), (b) linear Poisson's ratio (ν) versus the length ratio (L2/(L1+L2)) for a wide range of width ratio (w2/w1) and a fixed microstructure slope (tanθ0 = 1.0), and (c) normalized linear elastic modulus (E/Es) versus the microstructure slopes (tanθ0) for a wide range of width ratio (w2/w1) and a fixed length ratio (L2/(L1+L2) = 0.1)

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Fig. 6

Theoretical and FEA results of the linear Poisson's ratio and elastic modulus for honeycomb network materials: (a) linear Poisson's ratio (ν) of the honeycomb network material versus the width ratio (w2/w1) for a wide range of microstructure slopes (tanθ0) and a fixed length ratio (L2/(L1+L2) = 0.1), (b) linear Poisson's ratio (ν) versus the length ratio (L2/(L1+L2)) for a wide range of width ratio (w2/w1) and a fixed microstructure slope (tanθ0 = 1.0), and (c) normalized linear elastic modulus (E/Es) versus the microstructure slopes (tanθ0) for a wide range of width ratio (w2/w1) and a fixed length ratio (L2/(L1+L2) = 0.1)

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Fig. 7

Theoretical and FEA studies on the square network materials under uniaxial stretching in two different boundary conditions: (a) schematic illustration of the two different boundary conditions, (b) linear Poisson's ratio (ν) versus the width ratio (w2/w1) in the two different boundary conditions, for the square network material with fixed microstructure slope (tanθ0 = 1.0) and length ratio (L2/(L1+L2) = 0.1), (c) linear FEA result on the undeformed and deformed configurations of a square network material under 10% uniaxial stretching, (d) schematic illustration of a variant design with a symmetric geometry, and (e) linear Poisson's ratio (ν) of the square network material with symmetric microstructures versus the width ratio (w2/w1) for a wide range of microstructure slopes (tanθ0) and a fixed length ratio (L2/(L1+L2) = 0.1)

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Fig. 8

Theoretical and FEA results of the linear Poisson's ratio for square network materials under uniaxial stretching along the diagonal (45 deg) direction and the elastic modulus along two directions: (a) Linear Poisson's ratio (ν) versus the width ratio (w2/w1) for a wide range of microstructure slopes (tanθ0) and a fixed length ratio (L2/(L1+L2) = 0.1). (b) Linear Poisson's ratio (ν) versus the length ratio (L2/(L1+L2)) for a wide range of width ratio (w2/w1) and a fixed microstructure slope (tanθ0 = 1.0). (c) Normalized linear elastic modulus (E/Es) of versus the microstructure slopes (tanθ0) for a wide range of width ratio (w2/w1) and a fixed length ratio (L2/(L1+L2) = 0.1). (d) Similar results of the normalized linear elastic modulus (E/Es) for the square network material along the 0 deg directions.

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Fig. 9

Design of a crescent-shaped microstructure for the network material: (a) schematic illustration and key geometric parameters of the crescent-shaped microstructure, (b) geometric construction of the triangular network material with crescent-shaped microstructures connecting the nodes, (c) linear Poisson's ratio (ν) versus the microstructure slopes (tanθ0) for a wide range of normalized curve radius (R¯), and (d) critical strain (εcr) of the crescent-shaped microstructure as a function of the microstructure slope (tanθ0), for a wide range of normalized curve radius (R¯)

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Fig. 10

Tunability of the linear Poisson's ratio of triangular network material with zigzag and crescent-shaped microstructures: (a) and (b) Geometries of the representative unit cell in the network materials with zigzag and crescent-shaped microstructures that offer deterministic Poisson's ratio ν = 0 and ν = −0.1, respectively. (c) Critical strain (εcr) of the six representative microstructures in (a). (d) Poisson's ratio (ν) versus the applied tensile strain (εx) for four of the representative triangular network materials in (a). (e) and (f) Similar results for the representative structures in (b).

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