Journal Preproof Uniaxial stretching mechanics of cellular flexible metamaterials Xudong Liang, Alfred J. Crosby
PII: DOI: Reference:
S23524316(20)300122 https://doi.org/10.1016/j.eml.2020.100637 EML 100637
To appear in:
Extreme Mechanics Letters
Received date : 2 October 2019 Revised date : 18 November 2019 Accepted date : 22 January 2020 Please cite this article as: X. Liang and A.J. Crosby, Uniaxial stretching mechanics of cellular flexible metamaterials, Extreme Mechanics Letters (2020), doi: https://doi.org/10.1016/j.eml.2020.100637. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. 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.
© 2020 Elsevier Ltd. All rights reserved.
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Uniaxial Stretching Mechanics of Cellular Flexible Metamaterials
Xudong Liang, Alfred J. Crosby*
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Polymer Science and Engineering Department, University of Massachusetts Amherst, Amherst, MA 01003, USA *
Correspondent author. Email:
[email protected]
Abstract
Cellular flexible metamaterials (CFMs) with alternating vertical and horizontal pores can exhibit nonlinear
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and tunable mechanical properties. While most of CFMs exhibit novel mechanical properties with elastic instabilities under compression, much less is known about their nonlinear responses under stretching. In this paper, we investigate the mechanics of CFMs under uniaxial stretching with combined experiments,
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numerical simulations, and analytical models. The internal structure design incorporates a socalled “soft mechanism”, where slender internal structures permit internal rotation of more rigid structures. The stressstrain responses and Poisson’s ratio are programmed by controlling the pore pattern (i.e. pore shape and
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ligament thickness). A 2D platechain model, which reduces the CFM into chains of rigid square plates connected by the flexible beams, is developed to study the relation between the elastic modulus of CFMs and pore patterns. An analytical model without fitting parameters is constructed to predict stressstrain responses at small and large stretching deformations, showing good agreement between experiment and
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theory.
Keywords: elastomer, flexible metamaterial, large deformation, mechanical properties, rotation, uniaxial stretching
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1. Introduction Cellular flexible metamaterials (CFMs) constitute a branch of mechanical metamaterials characterized by a welldefined pore pattern, which can exhibit mechanical properties and functionalities
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that differ from, and even surpass, those of the constituent materials [1,2]. Materials with cellular structures are ubiquitous in biological tissues and have inspired the design of synthetic materials. Cellular structures in bones [3,4] and shells [5,6] can provide flexibility to accommodate growth and offer protection from the surrounding environment. Furthermore, CFMs are found in engineering applications, including shapemorphing [7,8], energy conversion [911], damping and dissipation [1214] and wave propagation tuning [1518]. CFMs also find use in recent developments of mechanical devices for strain sensors [19,20],
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actuators and robots [2127]. Moreover, with recent advancements in additive manufacturing techniques, additional nonlinear features have been introduced in CFMs with constituent materials that permit large and nonlinear elastic deformation.[28,29].
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Elastic instabilities within CFMs have been widely exploited to create nonlinear and tunable mechanical properties [1,2]. When excessive compression is applied, the slender internal structures (e.g. ligament) become unstable and undergo buckling with dramatic deformation [30]. For an elastic material, the buckling instability can be both reversible and repeatable. Various CFMs have gained access to strongly
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nonlinear and large deformations due to the instabilitytriggered geometric reorganization [31,32]. Depending upon porosity [33], pore shape [34] and matrix geometry (e.g. 2D plate, sphere or cylinder) [3537], different types of pattern transformation can be triggered by the elastic instability [16,34]. In addition, both static [38] and dynamic localized deformation [17,18] have been reported in CFMs due to compressioninduced instabilities. Novel mechanical properties that are unprecedented in constituent
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materials, such as negative Poisson’s ratio [36,39] and negative incremental stiffness [40,41], have been realized in CFMs via elastic instabilities. While significant control and new responses have been demonstrated in compressing metamaterials, considerably fewer advances have been realized under tensile loading. One of the few examples of metamaterial designs for novel tensile responses incorporates a socalled “soft mechanism” [1], where
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slender internal structures permit internal rotation of more rigid structures. Rotational deformation of the nearly freelyhinged structures in CFMs requires relatively little energy compared to stretching [7,4248]. The rotationbased deformation and the elasticity of constituent materials collectively tune the stiffness of
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CFMs. Recent studies have shown that the geometry of the beamlike ligaments (e.g. thickness, curvature and crosssection shape) can control the macroscopic stressstrain responses in CFMs [31,49,50]. While considerable progress has been made in relating the CFM mechanical properties to internal structures based on full size or unit cell simulations [37,51], the nonlinear mechanical properties in soft mechanismbased CFMs, especially under large deformation, remain difficult to predict analytically.
To overcome previous challenges in designing and modeling soft mechanism CFMs, we discuss
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here a comprehensive experimental investigation coupled with numerical and analytical models designed to provide new physical insight into CFM behaviors under tension. Here, we investigate the mechanics of softmechanism based CFMs at small and large uniaxially stretching deformations and develop an
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analytical model that can accurately describe the complete uniaxial response without any fitting parameters. The CFMs investigated here consist of alternating vertical and horizontal pores in an elastomer sheet and are found to display tunable Poisson’s ratio and programmable stressstrain responses. We examine the deformation kinematics and the nonlinear mechanical responses with numerical simulations, and to
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rationalize the dependence of the mechanical properties upon the pore pattern, we formulate an analytical model that describes their uniaxial stretching response without requiring any adjustable parameters. This paper is organized as follows. In Section 2, the fabrication of CFMs is described and their mechanical response under uniaxial loading is characterized. A finite element (FE) model is constructed and used to provide insight into the strainsoftening and strainstiffening behaviors that are observed for
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different pore patterns. In Section 3, the kinematics at small and large deformations is discussed. Finally, the stressstrain responses of CFMs under uniaxial stretching are predicted at small and large deformations in Section 4. We conclude with a few remarks on the advantages and opportunities afforded by the tunable and programmable mechanical properties in the soft mechanism CFM.
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2. Tunable mechanical properties of CFMs 2.1 Experiments of uniaxially stretched CFMs CFMs with two different periodically distributed and orthogonally aligned pore patterns are shown
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in Fig. 1. The pores are characterized by a semiminor and semimajor axis a0 and b0, respectively, and the thickness of the ligaments between two adjacent pores, h. Circular pores with a0=b0, and elliptical pores with a0=0.5b0 are shown in Fig. 1a and Fig. 1b, respectively. CFMs are composed of repeating unit cells in dimensions of L0×L0, where L0=a0+b0+h. The length of unit cell is fixed at L0=5 mm, and the specimen is comprised of Nx×Ny=6×27 unit cells, where Nx and Ny are number of cells in the x and y directions, respectively. The tested CFMs have an initial length Ly=NyL0=135 mm, a width Lx=NxL0=30 mm, and a
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thickness w=1.6 mm. The configuration of the pore patterns is controlled by the parameters (a0, b0 and h), provided in Table 1. The pore pattern was lasercut (Universal Laser System, VSL 3.5) in a polyurethanecomposite sheet (PU 40A; McMasterCarr) at a 55 % power and an 8 % speed. The minimum scale of the
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pore pattern is limited by the resolution of the laser cutter (±0.25 mm), and the errors of the pore pattern parameters in the Table 1 are due to the variations of the pore shapes and ligament thicknesses. Table 1
h (mm)
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Design parameters for CFMs
a0, b0 (mm, mm)
(0.60±0.03, 3.91±0.06); (1.02±0.02, 3.48±0.06); (1.47±0.03, 2.91±0.03); (2.28±0.03, 2.19±0.04)
1.11±0.06
(0.50±0.03, 3.44±0.07); (0.98±0.02, 2.84±0.05); (1.34±0.05, 2.60±0.03); (1.97±0.02, 1.92±0.05)
1.60±0.05
(0.48±0.01, 2.78±0.06); (1.13±0.06, 2.32±0.07); (1.59±0.05, 1.91±0.06); (1.76±0.02, 1.64±0.03)
2.00±0.05
(0.61±0.01, 2.41±0.07); (1.12±0.05, 1.91±0.05); (1.57±0.03, 1.45±0.04)
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0.53±0.04
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Fig. 1 Cellular flexible metamaterials (CFMs) with periodic pore patterns. (a) Circular and (b) elliptical pores are fabricated in a polyurethanecomposite sheet. CFMs are stretched in the y direction, with the pin
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boundary conditions at the ends. The blue and red dashed box show the local configuration changes of CFMs under a global strain, increasing from 0 to 0.5. Part of the unit cells are marked with silver ink to show the local stretching and rotational deformation. The circular and elliptical pores are characterized with
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the semiminor and semimajor axis of a0 and b0, respectively, and the thickness of the ligaments between two adjacent pores is h. The scale bar is 5 mm.
The fabricated CFMs are tested with a tensile tester (Instron 5564) under uniaxial stretching deformation. The specimen is clamped across a ligamentrich domain on both ends by pressurecontrolled
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grippers (Fig.1), allowing for an approximate pin boundary condition under uniaxial displacement. This boundary condition permits the unit cells on both ends to rotate uniformly under uniaxial stretching. To access the importance of this boundary condition on the measured parameters, we compared the stressstrain responses of the metamaterials gripped at the ligament and the plate regions (Appendix I). The fluctuations due to the gripping conditions decay quickly over a region much smaller than the length of the
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sample. Accordingly, the stressstrain responses are similar for the different boundary conditions (Fig. A1). The displacement and force are measured with a 50 N load cell with a 1 mN resolution. All tests are conducted under at a speed of 1.35 mm s1, which is equivalent to strain rate of 0.01 s1.
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The polyurethanecomposite elastomer is characterized through a uniaxial tensile test and described with an incompressible MooneyRivlin hyperelastic model,
W = C1 (l 12 + l 22 + l 32  3)+ C2 (l 12l 22 + l 22l 32 + l 32l 12  3),
(1)
where W is the strain energy density function, λi (i=1,2,3) is the principle stretch, and C1 and C2 are the material constants. The nominal stress under uniaxial stretching (λ1= λ, λ2= λ3=λ1/2) can be expressed as, (2)
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æ öæ 2C 2 ÷ 1 ö÷ ççl S = çç2 C1 + ÷ ÷. ÷ èç l øèç l 2 ø÷
We fit the experimental stressstrain curve with C1=0.15 MPa and C2=0.13 MPa, which agrees well with the measurements (Appendix II). The polyurethanecomposite elastomer is characterized with the small
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strain shear modulus μ=2(C1+C2)=0.6 MPa, and Poisson’s ratio ν0=0.49. The local deformation in the unit cell is distinct between the circular and elliptical patterns with global strain increasing from 0 to 0.5, as shown in Fig. 1. Part of the unit cells are marked with silver ink
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to show the stretching and rotational deformation. Stretching deformation is observed in the ligaments between the pores in CFMs with a circular pattern with a0=b0 (Fig. 1a), while CFMs with an elliptical pore with a0=0.5b0 show a rotation of the structures between the ligaments first, followed by the stretching of the ligaments with an increase of the global strain (Fig. 1b). The differences in the local deformation are clearly related to the geometry of the pore pattern and affect the mechanical properties of the CFM. The local deformation of the internal structures is delineated in Fig. 2a and will be discussed in details in the
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following section.
We track the deformation ux of CFMs in experiments when the uniaxial displacement uy is applied. We define the global strain as εy=uy/Ly, and the resulting horizontal strain εx=ux/Lx. In Fig. 2b and 2c, we plot the effective Poisson’s ratio, which is measured as νeff=εx/εy, as a function of the applied global strain
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εy. The effective Poisson’s ratios measured from experiments are plotted with solid squares and compared to predicted values from the developed analytical models, as discussed in the following section. For a fixed ligament thickness (h~0.5 mm), the effective Poisson’s ratio varies from 1 to 0.2 as the pore changes from
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an elliptical (a0
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thickness. In addition, the effective Poisson’s ratio changes with the global strain as the pores are deformed
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under the uniaxial stretching.
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Fig. 2 Tunable effective Poisson’s ratio in CFM under uniaxial stretching. (a) Schematics of the kinematics under uniaxial stretching. With increasing global strain, the pattern changes from an array of vertically and horizontally aligned elliptical pores (left) to circular ones (middle), which finally develops to vertically
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aligned elliptical pores (right). lp and lb are the diagonal length of the square rigid plate and the length of the flexible beam defined in the reduced platechain model, respectively. The effective Poisson’s ratio changes with the global strain with different pore shapes (b) and ligament thickness (c). The solid symbols and the lines represent the experimental and analytical results (Eq. (6)), respectively. The open circles in (b) and (c) represent the maximum strain for rotation that changes the pore from an elliptical to a circular
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shape predicted by the analytical models.
In Fig. 3a, we plot the nominal stressstrain curves for CFMs with different pore patterns (Table 1). The scaled nominal stress (defined as F/μwLx) increases with the ligament thickness and shows a
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dependence upon the shape of the pores. For a given ligament thickness, CFMs with elliptical pore patterns are initially compliant compared to the constituent material up to a critical strain, followed by an increase in stress with the global strain. The CFM with a circular pore shape deforms with a larger stiffness than the elliptical one, without a stiffening regime during stretching. The CFM shows a highly tunable stressstrain
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response under uniaxial stretching by varying the ligament thickness and pore shape.
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Fig. 3 Tunable stressstrain responses in CFM under uniaxial stretching. (a) The experimental stressstrain responses of CFMs with different pore shapes and ligament thicknesses. The pore shapes and ligament thicknesses are referred to the values in Table 1. (b) The stressstrain responses predicted by FEM with the
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same pore shape and ligament thickness as experiment. The model adopted in FEM is shown in the inset.
2.2 Finite element simulations of uniaxially stretched CFMs
To fully explore the relation between pore pattern and mechanical response, including the relationship between global force and displacement and local responses, we construct an FE model and validate it with the experimental measurements. Quasistatic nonlinear analysis of CFMs under uniaxial
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stretching is performed using the FE software ABAQUS/STANDARD (version 6.12) with the same configurations as in experiments (Table 1). In the FE simulations, a single layer of three unit cells is selected to mirror the mechanical response of the overall structure (inset, Fig. 3b). 2D planestrain simulations
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(ABAQUS element type CPE6H) are carried out using the incompressible MooneyRivlin hyperelastic model with material parameters based on the experimental measurements for the polyurethanecomposite elastomer. A symmetric boundary condition about the y axis is applied on the left boundary, while the right boundary is stressfree. To mimic the uniaxial loading in experiments, the bottom boundary is prescribed
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with a displacement, and the top boundary is constrained to be horizontal. As shown in Fig. 3b, the FE simulations capture the characteristic stressstrain responses in the experiments, particularly the dependence of the stressstrain curves upon the ligament thicknesses and pore shapes. To further explore the mechanics of the CFM, the validated FE models are adopted to systematically study the relation between pore pattern and stressstrain responses in the CFM. We
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differentiate the configuration of the pore pattern of the CFM with two dimensionless geometric parameters: e and ℎ̅, where e=1a0/b0 is the flattening number of the pore, and ℎ̅=1(a0+b0)/L0 is the dimensionless ligament thickness. By surveying the geometric parameters, we demonstrate the high tunability of the stressstrain responses in CFM. Fig. 4 presents the simulation results of the relation between the effective modulus (defined as F/μwLxεy) and the global strain for different pore patterns in CFM, where the ligament 1/27/2020
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thickness is fixed as ℎ̅ =0.1 (h=0.5 mm) and the pore shape changes gradually. The effective modulus decreases monotonically with the strain in the CFM with circular pores (e=0.0), similar to the strainsoftening behavior commonly observed in synthetic polymers at intermediate strains [52]. As e increases,
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the effective modulus changes nonmonotonically with the global strain with the appearance of the local rotation. The CFM is initially compliant with a low effective modulus compared to the constituent material, and then stiffens with an increase in the effective modulus. The growth of the modulus with strain represents a strainstiffening behavior, which is mostly found in biological materials and fiberreinforced polymers [53]. The correspondent stressstrain curves of the CFM are plotted in the inset of Fig. 4a for ℎ̅=0.1. The stressstrain curves transfer from concave to ‘J’ shape as e increases from 0 to 1, representing the transition
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from a strainsoftening to a strainstiffening behavior.
Fig. 4 Tunable strainsoftening and strainstiffening behaviors in the CFM. (a) The effective moduli predicted by the FE simulations with a fixed ligament thickness (ℎ̅=0.1) and different pore shapes. (b) The effective moduli predicted by the FE simulations with a fixed pore shape (e=0.5) and different ligament
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thicknesses. The correspondent stressstrain curves are shown in the insets.
The simulation results for the effective modulus with a fixed flattening number of the pore e=0.5 are plotted in Fig. 4b with the ligament thickness ℎ̅ varying from 0.05 to 0.5. Similarly, the CFM shows a transition between the strainsoftening and the strainsoftening behavior as the ligament thickness changes. 1/27/2020
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When ℎ̅ is larger than 0.3, the effective modulus monotonically decreases with the increase of the global strain, representing a strainsoftening behavior. Otherwise, the CFM shows a strainstiffening behavior with a small effective modulus at small strains, followed by an increase of the effective modulus at large strains.
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The correspondent stressstrain curves are plotted in the inset of Fig. 4b for e=0.5. The emergence of the tunable and nonlinear mechanical properties in experiments and numerical simulations is readily related to the pore pattern and the applied global strain in the uniaxially stretched CFMs. In the following, we rationalize such a dependence by constructing a 2D model to investigate the deformation kinematics and mechanical equilibrium under uniaxial stretching at small and large
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deformations.
3. Analytical model of CFMs under uniaxial stretching 3.1 Deformation kinematics of CFMs
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We propose a simplified kinematics analysis to describe the deformation and the changes of the pore pattern under uniaxial stretching. As shown in Fig. 2a, the pores are deformed by the coordinated rotation first, followed by elongation in the stretching direction when the maximum rotational angle θc is reached. The global strain related to the rotation is derived by considering the lines connecting the center
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of the ligaments along the x (or y) axis that are marked with red (or blue) dashed lines in Fig. 2a. The maximum strain for rotation is attained when the lines are fully stretched, which is related to the geometry c
of the pore pattern as εy=√𝐿20 + (𝑏0 − 𝑎0 )2 ⁄𝐿0 − 1. The rotation of unit cell under uniaxial stretching leads to an expansion of the CFM along the x c
axis when εy<εy. Due to the symmetry of the unit cell, the strains in both x and y axes satisfy εy=εy when c
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εy<εy (left, Fig. 2a). Such a mechanism gives rise to an auxetic behavior of the CFM under uniaxial stretching, c
with effective Poisson’s ratio ν=1. When the global strain εy=εy, the unit cell has rotated to the y axis, with the blue (red) dashed lines aligning to the y (x) axis. The pores deform to a circular shape, with a1=b1= (√𝐿20 + (𝑏0 − 𝑎0 )2 − ℎ)/ 2 (middle, Fig. 2a). Here, the deformation in the ligaments and the
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c
correspondent changes in the ligament thickness are neglected. When εy>εy, the aligned pore pattern prohibits rotation of the unit cell under stretching. Therefore, the pores deform to an elliptical shape aligning in the y axis, with a semiminor and major axis of a2 and b2 (right, Fig. 2a). The shape of the elliptical pores
( 2 1 b = ( 2 a2 =
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is related to the radius of the circular pores,
) L + (b  a )  h )(1 + (e  e )),
1
L20 + (b0  a 0 )  h (1  n (e y  e cy )), 2
2
2 0
2
0
0
(3)
c y
y
where ν is the Poisson’s ratio of the material with elliptical pores aligned in the y axis. ν can be obtained through a homogenization method of a 2D material with elliptical pores [54] and is related to the current 𝑎2 𝑏2
+
𝑏2
𝑎2
) 𝜈0 − 1) 𝛷 . Here the Poisson’s ratio of
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configuration of the pore pattern, 𝜈 = 𝜈0 − ((1 +
2
polyurethanecomposite is ν0=0.49, and Φ is the volume fraction of the pores and Φ=πa0b0/L0. By substituting a2 and b2 from Eq. (3) into the expression of ν, it is expressed as a function of the global strain
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εy,
n (e y ) =
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with
 g2 +
g 22  4 g1 g 3
2 g1
,
(4)
2
g1 = (n 0F  1)(e y  e cy )  (e y  e cy ), 2
g 2 = (n 0  n 0F + F )(e y  e cy ) + (n 0  3n 0F + F + 1)(e y  e cy ) + 1,
(5)
2
g 3 = n 0F (e y  e cy )  (n 0  3n 0F + F )(e y  e cy ) (n 0  3n 0F + F ).
as,
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Therefore, the effective Poisson’s ratio (νeff=εx/εy) of the CFM under uniaxial stretching can be expressed
n eff
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ìï  1 ïï ï = ïí e c ïï  y + n ïï e y ïî
， e y £ e cy 2 c ö . L20 + (b0  a 0 )  h æ çç e y ÷ c ÷ 1 ， e > e ÷ y y çç e ÷ ÷ L0 è y ø
12
(6)
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In Fig. 2b and 2c, the effective Poisson’s ratio from experiments with a controlled ligament thickness (h~0.53 mm) and a controlled semiminor axis of the pore (a0~0.55 mm) are compared with the predictions based on Eq. (6). As shown in Fig. 2b, the analytical predictions show a good agreement with
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experiments in metamaterial with circular pore patterns, as the ligament thickness is small (h/L0~0.1). In this small ligament limit, the trend of the development of Poisson’s ratio is captured by the model. The growing deviation in the elliptical pattern at large strain is due to finite deformation in the plate region, c
which is assumed to be rigid in our model. When εy<εy, the rotation of the unit cells gives rise to an auxetic behavior with νeff =1, which is independent of the geometric parameters of the pore pattern. The maximum c
strain for rotation εy is determined by the pore pattern and is plotted with open circles in Fig. 2b and 2c. c
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When εy>εy, the unit cell stops rotating and the ligaments connecting the neighboring unit cell are stretched by the increasing global strain. Since ν is positive for materials with aligned elliptical pores, the effective Poisson’s ratio starts to increase from 1. Similar transition of the effective Poisson’s ratio from 1 is
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observed with a fixed semiminor axis (a0~0.55 mm) of the pore (Fig. 2c). However, as the ligament thickness increases, the model predictions deviate more from the experiments. These deviations increase with the ligament thickness due to the increasing amount of deformation in the plate region, but the dependence of the Poisson’s ratio upon the geometry and uniaxial strain is still captured by our model. The
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proposed deformation kinematics reveals the dependence of the effective Poisson’s ratio upon the pore pattern and predicts the development of the pore pattern over the global strain.
3.2 Platechain model for CFMs
To model the development of force in the soft mechanismbased CFM under uniaxial stretching,
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we adopt a platechain model that reduces the CFM into a structure of rigid plates connected by the flexible beams. The flexible beams act as hinges connecting the rigid plates, undergoing stretching, bending and shearing deformations, while the rigid square plates only rotate under uniaxial stretching [55]. Therefore,
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the elastic energy of the CFM is concentrated in the flexible beams. The derivation of the energy density
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and governing equations for the mechanical equilibrium are outlined in the followings.
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Fig. 5 (a) Schematic of the simplified platechain model of the CFM under uniaxial stretching. CFMs are reduced as a structure of rigid plates connected by flexible beams and are subjected to an extension of δ with a corresponding external force F. lp and lb are the diagonal length of the square rigid plate and the c
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length of the flexible beam, respectively. When εy<εy, the CFM adopts the initial configuration with an c
alternating vertical and horizontal pores pattern. When εy>εy, the CFM adopts a different configuration with tilted rigid square plates representing the vertically aligned pore pattern. (b) The rigid plates undergo translational displacement un and rotational displacement θn. The stretching, bending and shearing deformations in the flexible beams are described by the longitudinal strain εn, the bending angle θnθn+1 and
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the tilt angle φn, respectively.
In Fig. 5a, we show the schematic of the platechain model for the CFM, where Ny rigid square plates are connected by flexible beams, deforming under the uniaxial displacement δ with the corresponding force F. Given that the CFM is stretched homogeneously along the horizontal direction, we neglect finite
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size effect in the x direction and consider a model of a chain of Ny rigid square plates. Under uniaxial stretching, the rigid plates align with the y axis, following a mirror symmetry in the y direction. Therefore, the displacements of the rigid plate can be represented by its center of mass translational displacement un
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and the rotational displacement with respect to the y axis θn (Fig. 5b). We define the counterclockwise direction as the positive direction for θn in our analysis. Therefore, the rotational displacement of neighboring rigid plates (θn and θn+1) are in opposite signs under uniaxial stretching. For the flexible beams, the deformation is represented by the longitudinal strain εn, the bending angle between the two ends θnθn+1 and the tilt angle of deformed beam with respect to y axis, φn (Fig. 5b). To obtain the energy density of the metamaterial, we express the deformation of the flexible beams in terms of displacements of the rigid plates,
j
n
u n+ 1  u n L0
,
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en =
= 
l p (qn + qn + 1 ) 2 lb
(7)
,
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where lb is the length of the flexible beam and lp is the diagonal length of the square rigid plate. In addition, the uniaxial displacement δ is expressed in terms of the rigid plate displacements, d = u1  u N y +
lp 2
(q 1
)
qN y .
(8)
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Therefore, the strain energy of the metamaterial with Ny rigid plates connected by the flexible beams (one chain in Fig. 5a) is,
2ö æk ö÷ qn + qn + 1 Cb Cs æ 2 2 çç l ÷ ç ÷ U s = å ç (L0 e n ) +  j n÷ + (qn  qn + 1 ) + çç ÷ ÷ ÷ è ø 2 2 2 2 ÷ ç n= 1 è ø Ny 1
Ny
å
n= 1
Cb 2
2
(2qn ) ,
(9)
where kj, Cb and Cs are the stretching, bending and shearing stiffness of the flexible beam, respectively. The
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first sum represents the strain energy of the vertically aligned beams, including the stretching, bending and shearing strain energy, and the second one is the strain energy of the horizontally aligned beams that only undergo bending deformation. The free energy of the metamaterial Π is obtained by considering the geometric constraint of the uniaxial stretching (Eq. (8)),
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2ö æk ö÷ qn + qn + 1 Cb Cs æ 2 2 çç l ÷ ç ÷+ P = å ç ( L0 e n ) +  j n÷ (qn  qn + 1 ) + çç ÷÷ ÷ è ø 2 2 2 2 ç n= 1 è ø÷ Ny 1
æ lp + F ççd  u1  u N y q1  qN y çè 2
(
)
(
Ny
å
Cb
n= 1
2
2
(2qn )
(10)
ö ÷ ÷ , ÷ ÷ ø
)
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where F is the Lagrange multiplier associated with the geometric constraint and corresponding to the external force due to the uniaxial stretching. Considering the geometric relations between the flexible beams and the rigid plates in Eq. (7), Π is expressed as a functional of un, θn and F,
2 æ ö ö Cb Cs æ 2 2 2÷ çç k l çç1 + l p ÷ ÷ ÷ u u + q q + q + q + ÷ ( ) ( ) ( ) å çç 2 n + 1 n n n+ 1 n+ 1 ÷ ÷ ÷ n ÷ 2 8 çè lb ø ç n= 1 è ø
Ny 1
æ + F ççd çè
(u
1
)
 uNy 
lp 2
(q 1
qN y
ö ÷. ÷ ÷ ÷ ø
)
Ny
å
n= 1
Cb 2
2
(2qn )
(11)
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P (u n , qn , F ) =
We applied EulerLagrange theorem on the free energy functional to obtain the mechanical equilibrium equations by prescribing ∂Π/∂un=0, ∂Π/∂θn=0 and ∂Π/∂F=0. The derivative over the
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translational displacement un results in,
u 2  u1 =  F k l ,
2 u n  u n  1  u n + 1 = 0, n Î [2, N y  1],
(12)
u N y  u N y  1 =  F kl .
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The derivative over the rotational displacement θn yields, 2
ö lp Cs æ çç1 + l p ÷ ÷ F, (q1 + q2 ) = (q2  5q1 )+ ÷ ç ÷ 4C b è lb ø 2C b 2
ö Cs æ çç1 + l p ÷ ÷ (q + 2qn + qn + 1 ) = (qn 1  6qn + qn + 1 ), n Î [2, N y  1], ÷ ÷ n 1 4 C b çè lb ø
(13)
2
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ö lp Cs æ çç1 + l p ÷ ÷ q N y  1 + q N y = q N y  1  5q N y F. ÷ ç ÷ 4C b è lb ø 2C b
(
) (
)
By combing Eq. (12), the derivative over the Lagrange multiplier F results in,
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F=
æ ö ççk d + k l l p q  q ÷ ÷ l Ny 1 ÷. ÷ 2 ø ( N y  1) èç 1
(
16
)
(14)
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Eqs. (12) ~ (14) are the governing equations for the mechanical equilibrium of the metamaterial under uniaxial stretching. By solving the equilibrium displacements for the rigid plates, the deformation of the flexible beams and the forcedisplacement relations for the metamaterial can be obtained. In the
pro of
following sections, we solve the governing equations at small and large deformations under uniaxial stretching. This approach provides a framework for discussing the dependence of the metamaterial moduli upon the geometric parameters of the pore pattern, as well as the competition between the rotationbased and the stretchingbased deformation in the metamaterial.
3.3 Platechain model at small strains
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At small strains, the metamaterial maintains the initial configuration of the pore pattern (Fig. 5a). The elasticity of the metamaterial derived from the mechanical equilibrium equations is governed by the stiffness of the flexible beams, including the stretching stiffness kl, the bending stiffness Cb and the shearing
is derived as [56], kl =
Ew
2h
p
L0  h
32 2 Ew
,
5
Cs =
9p e 2
32 2 Ew 3p
2
 1
æ ö æa 0 ö 2 çç ÷ h ( L0  h ) 2 çç L20 + h (L0  h )÷ , ÷ ÷ ÷ ÷ èç ø çè 2 ø 3 2

1
urn a
Cb =
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stiffness Cs. The stiffness of the flexible beams based on the geometry of the pore and the ligament thickness
3 1 æ 8 h 2 ( L0  h ) 2 çç L0 çè 3 2p
 1
ö h (L0  h )÷ ÷ ÷ ø
(15) 2
æa 0 ö ç ÷ , ÷ ÷ ççè 2 ø
where E is the Young’s modulus and E=2(1+ν0)μ. In addition, the length of the flexible beam at the initial state is lb=2a0, beyond which the crosssection area of the beam becomes comparable to the square plates
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(Fig. 2a). Therefore, the diagonal length of the square plates is lp=√2(L0lb) based on the geometrical parameters of the initial pore pattern. By substituting the beam stiffness (Eq. (15)) into the equations of the mechanical equilibrium (Eqs. (12) – (14)), we solve for the displacements of rigid plates (un and θn) and the correspondent reaction force F under the uniaxial displacement δ. To obtain the analytical solution of the metamaterial under small strains uniaxial stretching, we consider a continuum limit of the rotation angle 1/27/2020
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field 𝜃̃ (𝑦). Using a Taylor expansion of 𝜃̃(𝑦 ± 𝐿0 ) and follow the sign convention for rotation angle, we have 𝜃𝑛±1 = −𝜃̃ (𝑦) ∓ 𝐿0
̃ 𝜕𝜃 𝜕𝑦
−
2̃ 𝐿20 𝜕 𝜃
2 𝜕𝑦2
. Substituting into Eq. (13), the continuum limit of the equilibrium
equation for the rotation angle is,
8
¶ y2
2
𝑙
pro of
(a  1) L20 ¶ 2 q% % = q,
(16)
𝐶𝑠 𝑝 where 𝑎 = 4𝐶 (1+ ) . The dimensionless value α represents the competition between the shearing and the 𝑙 𝑏
𝑏
bending stiffness of the flexible beam. When α >1, the continuum limit of the rotation angle is solved by satisfying the boundary conditions at θ1 and θNy (see Appendix III for details),
(L 1)
qn =
(
 1
(a  1) 8

) 1)L (L
 L n 1
Fl p
Ny 3
(a 
)
 1 2C b
,
(17)
) . When α≤1, the solution of the rotation angle can be deduced from Eq. (13),
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where L = exp
(a + 5 )(L
Ny 1
Ny n
re
n 1
( 1)
ìï Fl 2 (a + 5 )C , n= 1 p b ïï ï qn = í 0, n Î [2, N y  1] . ïï ïïî  Fl p 2 (a + 5 )C b , n = N y
(18)
urn a
It is noted that the by taking the limit of α=1 in Eq. (17), we can recover the solution in Eq. (18) at α=1, indicating that the distribution of the rotation angle changes continuously with α. In addition, the reaction force in the metamaterial can be rewritten as a function of uniaxial displacement δ by substituting Eq. (17) and (18) into Eq. (14),
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 1 ìï é ù k l l p2 ïï ê ú ,a £ 1 ïï k l d ê( N y  1) + 2 (a + 5 )C b úúû êë ïï F= í ,  1 ïï 2 Ny 1 é ù k l L 1 ïï k d ê N  1 + l p ú ,a > 1 ) ïï l ê( y 2 C b g (L ) úúû êë ïî
(19)
where 𝑔(𝛬) = (𝛼 + 5)(𝛬𝑁𝑦 −1 − 1) − (𝛼 − 1)𝛬(𝛬𝑁𝑦 −3 − 1). The longitudinal strain εn in the flexible beams is derived by combining Eq. (7), (12) and (19),
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 1
ù k l l p2 d éê ú N 1 + (ê y ) L0 ëê 2 (a + 5 )C b úûú
,a £ 1  1
k l l p2 L N y  1  1 ù d éê ú ,a > 1 N 1 + ( y ) L0 êëê 2 C b g (L ) úûú
,
(20)
pro of
ìï ïï ïï ï e n = ïí ïï ïï ïï ïî
In Fig.6a, we plot the scaled maximum rotation angle (θmax/(δ/L0)) of the rigid plates for different pore patterns under small strains uniaxial stretching. The maximum rotation angle increases with the flattening number e, starting from 0 in the circular pore. The rigid plates in the metamaterial with a circular pore pattern will not rotate during uniaxial stretching. It also increases with decreasing ligament thickness, due to the rotational deformation incurred by the ligaments’ increased flexibility. In the inset of Fig. 6a, we
re
plot the distribution of the rotation angle in the metamaterial, which has a fixed ligament thickness ℎ̅=0.1 (h=0.5 mm) and pore shapes (in terms of e) based on experimental parameters. The rotational angles in the neighboring rigid plates are in opposite signs, as indicated by the platechain model. The rotation angle
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reaches its maximum (or minimum) value at the ends of the metamaterial and decreases from the edges to center during uniaxial stretching. The maximum rotation angle also decreases with the flattening number e, as the pore shape changes from elliptical to circular. For a pore shape with e=0.04, the metamaterial only undergoes stretching deformation as the rotation angle θn=0 with α<1. Otherwise, the dimensionless value
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α>1, and the rigid plates rotate under uniaxial stretching. In Fig. 6b, the scaled maximum longitudinal strain (εmax/(δ/(Ny1)L0)) for the flexible beams is plotted for different pore patterns under small strains uniaxial stretching. The scaled strain decreases with increasing flattening number e, starting from 1 for the circular pore pattern. This trend is in accord with the rotationfree rigid plates observed in the circular pore pattern with e=0 (Fig. 6a). Meanwhile, the maximum
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strain also increases with ligament thickness, as the ligament becomes more rigid to bending deformation. Based on the equilibrium equation of the plate displacement (Eq. (12)), the longitudinal strain of the flexible beams is homogeneously distributed in the metamaterial. As shown in the inset of Fig. 6b, the scaled longitudinal strain decreases from 1, and the flattening number of the pore is based on experimental
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parameters. In addition, there is a finite longitudinal strain even for e~1 in the metamaterial. This finding is due to the finite bending stiffness of the ligaments, causing both stretching and bending deformation in the
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pro of
flexible beams under the tension.
Fig. 6 Deformation of the metamaterial predicted by the platechain model under small strains uniaxial stretching. (a) The maximum rotation angle of the rigid plates increases with e for different ligament
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thicknesses. The distribution of the rotation angles for ℎ̅=0.1 is shown in the inset. (b) The maximum strain in the flexible beams decreases with the e for different ligament thicknesses. The distribution of the strain
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ℎ̅=0.1 is shown in the inset.
To derive the tangential modulus of the metamaterial at small strains uniaxial stretching, we return to the forcedisplacement relation (Eq. (19)) and define the tangential modulus as Ey=ΔFLy/(μwLxΔδ). Here, Δδ and ΔF are the incremental displacement and the corresponding force. For small strains, ΔF/Δδ=F/δ. Therefore, the tangential modulus at small strains (𝐸𝑦0 ) is,
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 1 ìï ù k l l p2 ïï N y k l éê ú ,a £ 1 ïï ê( N y  1) + 2 (a + 5 )C ú w b ê ú ï ë û E y0 = ïí ,  1 ïï 2 Ny 1 é ù N k k l  1ú ïï y l ê N  1 + l p L ,a > 1 ) ïï w ê( y 2 C b g (L ) úûú ê ë ïî
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(21)
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In Fig. 7a, we plot the tangential moduli of the metamaterial predicted by the platechain model and compare with the FE simulations and experiment results (at εy=0.05). The tangential moduli decrease as e increases from 0 to 1, which is related to the transition from the stretchingbased deformation to the
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rotationbased deformation in the metamaterial. When e=0, the metamaterial is deformed by the longitudinal stretching in the flexible beam with a negligible rotation, and the elasticity of the metamaterial is determined by the stretching stiffness of the flexible beams kl. As e increases, the longitudinal strain decreases while the rotation angle increases as shown in Fig. 6. The tangential modulus is governed by the bending stiffness of the beams Cb as e approaching 1. Moreover, the tangential moduli increase when ℎ̅ grows from 0.1 to 0.4, as the stiffness of the flexible beams are proportional to ℎ̅ (Eq. (15)). For the pore
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shape with an intermediate value of e, the tangential modulus is controlled by the stretching, bending and shearing stiffness in the flexible beams, giving rise to a tunable mechanical response. Compared with the FE simulations (open symbols) and experiment results (solid symbols), the model shows a good agreement
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in the circular pore pattern (e < 0.4) metamaterial with small ligament thickness. Due to the simplifications we have made in the geometry and the material properties, the absolute agreement is not achieved in the current model. The deviations at the intermediate value of e are related to elastic deformation in the plate structures in experiments, which are assumed to be rigid in the platechain model. In addition, the deviation
urn a
also increases with the ligament thickness as the stiffness of the flexible beam predicted by Eq. (15) is less
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accurate for thick beams [56].
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Fig. 7 Tangential moduli of the metamaterial at small strains. (a) Effective tangential moduli predicted by the platechain model are compared with the FE simulations (open symbols) and experiment results (solid symbols) at εy=0.05. The dimensionless ligament thickness ranges from 0.1 to 0.4 (solid lines) (b) Contour
pro of
of tangential moduli of the metamaterial with different geometric parameters of the pore pattern (e and ℎ̅) under small strains. The color stands for the tangential moduli scaled by the shear modulus of the material. The blue dashed line represents the curves of α=1, which separates the configurations of the metamaterial generating stretching or stretchingrotational deformation under uniaxial stretching at small strains.
The contour of the tangential moduli at small strains with respect to the geometric parameters of
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the pore pattern (e and ℎ̅) is plotted in Fig. 7b. The color stands for the scaled tangential moduli (𝐸𝑦0 /μ), for different geometric parameters of the pore pattern. As shown by the contour lines of the scaled tangential moduli (black lines), a prescribed tangential modulus can be achieved through different pore shapes and
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ligament thicknesses. In addition, the flexible beams can switch from a stretching deformation without rotations (θn=0) to a combination of stretching, bending and shearing deformations (θn≠0), depending on 2
𝑙
𝐶𝑠 𝑝 the value of α. The dimensionless value 𝑎 = 4𝐶 (1+ ) represents competition between the shearing and the 𝑙 𝑏
𝑏
urn a
bending stiffness of the flexible beam. When α>1, the shearing stiffness is large enough to support shearing deformation generated by the rotation in the metamaterial that decreases from the edges to the center (Fig. 6a inset). However, the shearing stiffness is not able to support shear deformation caused by the same rotation in the metamaterial when α<1. Therefore, only stretching deformation in the flexible beams satisfies the mirror symmetry in the y direction and the boundary conditions of the metamaterial under
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uniaxial stretching. The geometric parameters of the pore pattern satisfying α=1 is plotted with a blue dashed line in Fig. 7b. It separates the metamaterial that undergoes stretching or stretchingrotational deformation under small strains uniaxial stretching. Based on the predictions from the platechain model, the tangential modulus of the metamaterial can be programmed through the geometric parameters of the pore pattern. Multiple pore patterns (the pore shape and ligament thickness) and different deformation
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modes (stretching or rotationstretching deformation) in the metamaterial can be adopted to program the tangential modulus under uniaxial stretching, making the mechanical properties in the metamaterial highly
pro of
tunable.
3.4 Platechain model at large strains
c
When the uniaxial stretching is larger than the maximum strain for rotation (εy>εy), the metamaterial adopts a new configuration with pores aligning with the y axis (Fig. 2a). To obtain the analytical expression of the tangential modulus of the metamaterial at large strains, we adopt the platechain model with rigid vertically aligned square plates (Fig. 5a). Based on the deformation kinematics proposed in Section 3.1, the
re
deformed configurations can be predicted based on the global strain εy and the geometric parameters of the pore pattern (e and ℎ̅) of the metamaterial. The stiffness of the flexible beams in the deformed metamaterial with a pore pattern aligned with the y axis is derived as (see Appendix IV for derivations), a2 h
Ew
b22
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k lc (e y ) =
C
c s
(e y ) =
2 Ew
(e y ) =
2 Ew
urn a
C
c b
p
9p
3p
,
5 2
12 3 2 2 2
æh ö çç ÷ ÷ , çè b ÷ ÷ 2ø
3 2 12 2 2
æh ö çç ÷ ÷ . çè b ÷ ÷ 2ø
a b
(22)
3 2
a b
c
In the deformed metamaterial at large strains (εy>εy), the pores align with the y axis and prohibit the rigid plates rotating with the uniaxial stretching. Therefore, the flexible beams in the metamaterial only can undergo stretching deformation, with θn=0 and φn=0, n Î [1, Ny]. The force increment ΔF in a
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metamaterial is related to the incremental displacement Δδ as,
D F = k lc (e y )
Dd Ny  1
.
(23)
The length of the flexible beam is lb=2b2 as the ligaments are greatly stretched and elongated under large strains (Fig. 2a), and the diagonal length of the rigid plates is lp=√2h based on the geometrical
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parameters of the deformed pore pattern at large strains. Substituting stretching stiffness of the flexible beam in Eq. (22) into Eq. (23), the tangential modulus of the deformed metamaterial at large global strains c
(εy>εy) can be expressed as,
NyE
a2 h
(N y  1)p
b22
.
pro of
E y (e y ) =
(24)
The tangential modulus of the deformed metamaterial can be obtained by the deformed pore shape (a2 and b2) predicted by Eq. (3) – (5) and the global strain εy. By solving Eq. (24) numerically, the tangential moduli of the deformed metamaterial are plotted with respect to e with ℎ̅ in Fig. 8a. The scaled moduli increase with the ligament thickness as the stiffness of the flexible beam at large strains is proportional to ℎ̅.
re
Moreover, the tangential moduli only slightly increase as the e changes from 0 to 1 for all the ligament thicknesses, due to the similar aligned pore patterns of the metamaterial under a large global strain (Fig. 1), regardless of the initial pore shape. Compared to the FE simulations (open symbols) and the experiment
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results (solid symbols), the model agrees well for the metamaterial with the circular pore pattern (e~0) and small ligament thickness (ℎ̅<0.4). The stiffening of the tangential modulus as e approaches unity in the experiments is not well described in the model, partially due to the straindependent modulus of the material
urn a
and the finite deformation in the square plate at large strains that are not considered in the analytical model. In Fig. 8b, we also plot the contour of the tangential moduli with respect to the geometric parameters of the pore pattern (e and ℎ̅) in the metamaterial. The tangential moduli of the deformed metamaterial show a strong dependence of the ligament thickness, regardless of the pore pattern before loading. As the ligament thickness varies from 0 to 0.5, the tangential moduli are still highly tunable, ranging from 0 to 0.8μ. The metamaterial under large strains uniaxial stretching can be programmed via the design of the ligament
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thickness to control its mechanical properties.
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Fig. 8 Tangential moduli of the metamaterial at large strains (εy=0.5). (a) The predicted tangential moduli
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for the dimensionless ligament thickness ranging from 0.1 to 0.4 (solid lines) are compared with the FE simulations (open symbols) and experiment results (solid symbols). (b) Contour of tangential moduli of the metamaterial with different pore shapes and ligament thicknesses at large strains (εy=0.5). The color stands
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for the tangential moduli scaled by the shear modulus of the material.
4. Analytical model for stressstrain responses
The stressstrain relation in the metamaterial under uniaxial stretching is obtained by integrating
urn a
tangential modulus Ey with respect to the global strain εy. Based on the platechain model, the tangential c
moduli for both small strains (εy~0) and large strains (εy>εy) deformations are predicted based on the geometric parameters of the pore pattern (e and ℎ̅). For a first order approximation, the tangential moduli c
between the small and large strains limit (0< εy< εy) are assumed to change linearly,
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E y (e y ) = (1  e y ) E y0 + e y E yc ,
(25) c
where 𝐸𝑦0 and 𝐸𝑦𝑐 are the tangential moduli at small deformation (Eq. (21)) and large deformation with εy=εy (Eq. (24)), respectively. A nonlinear analysis of the tangential moduli at the intermediate strain (0<εy<εcy) could improve the accuracy of the predicted stressstrain responses, but this extension is beyond the scope
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of the current paper. By integrating the tangential modulus Ey with respect to the global strain εy, the stress of the metamaterial during uniaxial stretching is, , e y £ e cy ey
ò (N e cy
NyE
a2 h
.
(26)
pro of
2 ìï ïï e E 0 + e y E c  E 0 ( y y) ïï y y 2 s y = ïí 2 ïï c 0 (e cy ) ïï e y E y + (E yc  E y0 ) + 2 ïï î
y
 1) p
b22
d e y ,e y > e
c y
The stressstrain responses of the metamaterial at small and large deformations can be predicted based on Eq. (26). In Fig. 9, we plot the stressstrain curves predicted by Eq. (26), and corresponding effective stiffness for ℎ̅=0.1 in the inset. Similar to experiments, the model predicts the initial compliant (compared to the constituent material) responses of the metamaterial under uniaxial stretching, followed by c
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an increase in the stiffness after the maximum strain for rotation (εy) is reached when the pores are in an elliptical shape. For the circular pore pattern, the flexible beams in the metamaterial undergo the stretching deformation from the beginning, leading to a negligible compliant regime. The model also captures the
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transition of the stressstrain responses in experiments as the pore shape changes from a circular to elliptical shape. Moreover, the model predicts a transition from strainsoftening to strainstiffening behaviors as e increases from 0.0 to 0.9, showing a similar response predicted by the FE simulations in Fig. 4a. Although
urn a
absolute agreement with the experiments is not achieved, the model does successfully the deformation mechanism and the corresponding stressstrain responses in the metamaterial are indeed well captured by our model. In addition, the analytical model predications are freefrom any fitting parameters, as the predictions are scaled with the smallstrain shear modulus (μ) measured from the experiments. Therefore, the developed theoretical model captures the relation between the geometric parameters of the pore pattern
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and the stressstrain responses in the metamaterial under uniaxial stretching, providing insight into how the pore patterns program the mechanical properties of synthetic materials.
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Fig. 9 Stressstrain responses of the metamaterial predicted by the analytical model for ℎ̅=0.1. The inset
depending on the pore shapes.
7. Conclusions
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shows the effective moduli predicted by the model, with strainsoftening and strain stiffening behaviors
urn a
Tunability in Poisson’s ratio, stressstrain responses, strainsoftening and strainstiffening behaviors in synthetic material is highly desirable in engineering applications, driving the development of metamaterials that can exhibit properties and functionalities differing from, and even surpassing, the constituent materials. CFMs with alternating vertical and horizontal pores display nonlinear and tunable mechanical properties under uniaxial stretching. By combining experiments, numerical simulations, and
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analytical models, we investigated the mechanics of the metamaterial at small and large deformations and developed an analytical model to predict the mechanical responses under uniaxial stretching based on the pore geometry and the material constituent properties. The tunable effective Poisson’s ratio, from a negative auxetic response to a positive value, is related to the rotation of the internal structures in the metamaterial.
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The programmable stressstrain responses, including strainsoftening and strainstiffening behaviors, are governed by the competition between rotationbased deformations and the stretchingbased deformations. Without adjustable parameters, the proposed analytical model predicts the characteristic stressstrain
pro of
responses at small and large deformations in the metamaterial, showing good agreements with experiments and FE simulations. This study provides a design guideline for tailoring mechanical properties and programming performance of CFMs in applications, ranging from new actuators [20, 57, 58], stretchable sensors [19] and phononic band gap materials [16].
Acknowledgement
contract/grant number W911NF1510358.
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Appendix I. Effect of boundary conditions
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This work is supported by the U. S. Army Research Laboratory and the U. S. Army Research Office under
To explore the effect of boundary conditions in the deformation behavior, we performed experiments with the metamaterials gripped at both the ligament and the plate regions and compared the stressstrain responses. As shown in Fig. A1a, the stressstrain responses are similar in both boundary
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conditions. The fluctuations due to the boundary conditions decay within several plates, and the
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deformation at the central region is similar for the different boundary conditions (Fig. A1b).
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Fig. A1 (a) Stressstrain responses in metamaterials with different boundary conditions. (b) Control of boundary conditions via the gripping region. BC1: the plate region. BC2: the ligament region.
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Appendix II. MooneyRivlin model for polyurethane composite The constitutive response of the polyurethane composite is assumed to follow the MooneyRivlin hyperelastic model of Eq. (1) in the main text. To obtain the material constants C1 and C2, we plotted the measurements of S/(λλ2) over 1/λ as shown in Fig. A2a. By fitting the experiment data with a linear function, the slope and intercept are 2C2 and 2C1, respectively. In Fig. A2b, the stressstrain curve predicted by the MooneyRivilin model with the fitting parameters C1 and C2 shows a good agreement with the
urn a
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experiment.
Fig. A2 Fitting of the stressstrain response of polyurethanecomposite with the MooneyRivlin model. (a) Determination of the material constants C1 and C2 with leastsquare method. (b) Comparison of the stress
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strain responses between the experiment and MooneyRivlin model prediction.
Appendix III. Continuum solutions for the platechain model As the uniaxial stretching is applied in the metamaterial through a long chain of rigid plates (Ny=27 in the experiment) and the gradient of the rotation angle in the neighboring plates is relatively small, we adopt a continuum field to describe the distribution of the rotation angle,
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qn = q%( y ),
(A1)
2 ¶ q% L0 ¶ 2 q% qn ± 1 = q%( y ± L0 ) =  qn m L0 . ¶y 2 ¶ y2
the rotation angle is,
pro of
By substituting Eq. (A1) into Eq. (13) in the main text, the continuum limit of the equilibrium equation for
(a  1) L20 ¶ 2 q% % = q, ¶ y2
8
where a =
Cs 4 Cb
2
(1 + ) L1 l1
(A2)
. As a result, when α>1, the continuum solution of Eq. (A2) is,
(A3)
re
æ ö æ ö ÷ ÷ çç çç y  y ÷ ÷ % q ( y ) = A1 exp ç + A2 exp ç , ÷ ÷ ÷ ÷ çç L (a  1) 8 ÷ ç ÷ L a 1 8 ÷ ÷ ç ( ) è 0 ø è 0 ø
where A1 and A2 are the constants to be determined by the boundary conditions of the two ends (at θ1 and θNy),
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(a + 5 )q1 + (a  1)q2 = FL1 (2C b ), (a  1)qN
y 1
(A4)
+ (a + 5 )qN y =  FL1 (2 C b ),
Substituting Eq. (A3) into the boundary conditions Eq. (A4) and adopt the condition of odd plate number
urn a
in experiments, Ny=27, the rotational angle is solved as,
(L 1)
n 1
qn =
where L = exp
(
 1
(a  1) 8
( 1)
(a + 5 )(L
Ny 1

Ny n
) 1)L (L
 L n 1
(a 
FL1 Ny 3
)
 1 2C b
,
(A5)
) . Substituting Eq. (A5) into Eq. (19), we can express the rotational angle of rigid
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plates as a function of the uniaxial displacement δ, n 1
qn =
( 1)
(
k l L1 L
Ny n
 L n 1
(
2 ( N y  1)C b g (L ) + k l L12 L
where 𝑔(𝛬) = (𝛼 + 5)(𝛬𝑁𝑦 −1 − 1) − (𝛼 − 1)𝛬(𝛬𝑁𝑦 −3 − 1).
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)
Ny 1
)
 1
d,
(A6)
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In the following, we discuss effect of odd and even plate number Ny in the distribution of the rotation angle. For an even plate number Ny, the rotational angle θn satisfying the boundary conditions (A4) is,
qn =
(
(
2C b (a + 5 ) L
( + 1)
FL1 L
Ny 1
Ny n
) 1)L (L
+ L n 1
))
.
pro of
n 1
( 1)
(a 
Ny 3
(A7)
+1
The correspondent tangential modulus of the metamaterial for an even plate number is,
(A8)
re
 1 ìï ù k l L12 ïï N y k l éê ú N  1) + ,a £ 1 ïï ê( y ú w 2 a + 5 C ( ) ê ú b ï ë û E y0 = ïí ,  1 ïï Ny 1 2 é ù N k k L L + 1 ïï y l ê N  1 + l 1 ú ,a > 1 ) ïï w ê( y 2 C f L ( ) úúû êë b ïî
where 𝑓(𝛬) = (𝛼 + 5)(𝛬𝑁𝑦 −1 + 1) − (𝛼 − 1)𝛬(𝛬𝑁𝑦 −3 + 1) . Fig. A3 plots the normalized tangential moduli of the metamaterial as a function of the pore flattening number e for odd and even plate numbers,
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when the ligament thickness ranging from 0.1 to 0.4. The even plate number are 26 and 28, which are similar to the one in experiment Ny=27. As shown in Fig. A3, the tangential moduli of the metamaterial are not affected by the odd and even plate number when e<0.9. As e approaches 1, the metamaterial with even
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plate number shows a much softer response than the odd one. For even values of Ny, the local rotation of each plate θn and the globally applied deformation δ/(NyL0) are of the same order, and the tangential modulus of the metamaterial is completely determined by the rotational stiffness of the flexible beams. For odd values of Ny, the counterrotation for each plate cancels in the leading order, resulting in a finite stretch deformation in the metamaterial (Fig. 6). The difference between the deformations with odd and even values
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of Ny, leads to the changes in the modulus for metamaterials with e~1. However, the effect of the plate number in the tangential modulus only becomes obvious when e>0.9, which is beyond the current experiments limit (e≤0.88). Therefore, the effect of the odd and even plate number is not substantial in the metamaterial studied here.
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Fig. A3 Effect of the odd and even rigid plate number in tangential moduli at small strains. The moduli of metamaterials with odd (Ny=27) and even (Ny=26 and 28) square plate number show a negligible difference when e < 0.9.
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Appendix IV. Stiffness of flexible beams at large strains
When εy>εy, the pores in the metamaterial align and elongate along the y axis and the ligament structure between the pores are deformed by the forces (F and V) and moments (M), as shown in Fig. A4.
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Here we model the ligament as a flexible beam with varying crosssection area, and derive the analytical expressions the stretching, bending and shearing stiffness of the ligament at large deformation. We focus on the small ligament thickness limits (ℎ̅<<1) and assume the ligament thickness remains unchanged under uniaxial stretching.
As shown in Fig. A4, the yaxis is placed at the center of the flexible beam, with its origin located
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at the narrowest part. The width of the flexible beam is written as,
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H ( y) = h +
32
a2 y b22
2
,
(A9)
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where a2 and b2 are the semiminor and semimajor axis of the pores. In addition, we neglect the higher order term of O((y/b2)2) in Eq. (A9). The stretching stiffness kl can be obtained by applying the uniaxial forces F at the ends of the flexible beam (inset, Fig. A4). The longitudinal displacement Δu is derived as,
ò
b2
 b2
Fdy
. Ew (h + a 2 b2 2 y 2 )
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Du =
(A10)
where E is the elastic modulus of the elastomer. The integral on Eq. (A10) can be obtained exactly,
u =
b22 a2 h
F Ew
arctan
( ) a2 y 2 b22 h
b2
~ − b2
F Ew
( ) , as h<
1 2
2
is defined as kl=F/Δu, is written as,
1 2
re
ö Ew æ çç a 2 h ÷ ÷. kl = 2 ÷ p çè b ÷ ø
(A11)
2
The bending deformation in the flexible beam is derived by applying a pair of moments M at its
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ends as indicated in Fig. A4. The bending deformation of ligament is governed by, d 2v dy
2
=
12 M 2
Ew (h + a 2 b2 2 y 2 )
,
(A12)
where v is the deflection of the center line under bending. Integrating Eq. (A12) with the boundary
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conditions v(0)=0 and dv(0)/dy=0, the deflection of the flexible beam is,
v( y) =
æ ö æ a y ö 3b ÷ M ç 9 b2 y y2 ÷ çç 2 ÷ 2 ÷. çç arctan + ÷ 5 2 1 2 ç ÷ ÷ 2h a b y + b h ÷ çè b2 h ø Ew çèç 2 a 2 h 2 ÷ 2 2 2 ÷ ø
(A13)
The bending stiffness of the flexible beam is defined as Cb=M/Δγb, where Δγb is the angle between the ends
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under bending deformation, and it is related to the deflection, Δγb=(v(b2)v(b2))/2b2. Based on Eq. (A13) and assuming the small ligament thickness, h<
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5
æ h ö2 ÷ Cb = a 2 b2 çç ÷ . ÷ çè b ø ÷ 9p 2 2 Ew
33
1 2
3 2
(A14)
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The shearing deformation is derived by applying a pair of forces V at the ends of the flexible beam as indicated in Fig. A4. The deformation of ligament is governed by, d 2v
=
12V ( b2  y ) 2
Ew (h + a 2 b2 2 y 2 )
.
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dy
2
(A15)
Integrating Eq. (A15) with the boundary conditions that v(0)=0 and dv(b2)/dy=0, the deflection of the flexible beam is,
v( y) =
æ ö 3 æ a yö 3b24 a 2 y 2 + b2 hy ÷ V ç 3b2 ç 2 ÷ ÷. ÷ çç arctan çç + ÷ 3 2 2 3 ÷ ÷ çè b2 h ÷ Ew ççè 2 a 2 h 2 2 a h a b y + b h ÷ ÷ ø 2 2 2 2 ø
(A16)
The shearing stiffness of the ligament is defined as Cs=Vb2/Δγs, where Δγs is the angle between the two ends
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under shearing deformation, and it is related to the deflection, Δγs=(v(b2)v(b2))/2b2. Based on Eq. (A16) and assuming the small ligament thickness, h<
æ h ö2 ÷ Cs = a 2 b2 çç ÷ . ÷ çè b ø ÷ 3p 2 3 2
1 2
(A17)
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2 Ew
Fig. A4 Schematic of the metamaterial at large deformation. Inset: ligament structure between the vertically aligned elliptical pores. The forces F, V and moment M induce the stretching, shearing and bending deformation in the ligament, respectively.
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10/2/2019 Dear Editors:
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The authors declare no conflict of interests. Sincerely,
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Alfred J. Crosby Professor, Polymer Science & Engineering Department CoDirector, Center for Evolutionary Materials University of Massachusetts Amherst
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Xudong Liang, Ph. D. Polymer Science & Engineering Department University of Massachusetts Amherst