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Research Papers

# Qualitative Analysis and Conceptual Design of Planar Metamaterials With Negative Poisson's RatioOPEN ACCESS

[+] Author and Article Information
Sree Kalyan Patiballa

Mem. ASME
Department of Industrial and
Enterprise Systems Engineering,
University of Illinois Urbana-Champaign,
Urbana, IL 61801
e-mail: patibal2@illinois.edu

Girish Krishnan

Mem. ASME
Department of Industrial and
Enterprise Systems Engineering,
University of Illinois Urbana-Champaign,
Urbana, IL 61801
e-mail: gkrishna@illinois.edu

1Corresponding author.

Manuscript received September 23, 2017; final manuscript received December 11, 2017; published online February 5, 2018. Assoc. Editor: Andreas Mueller.

J. Mechanisms Robotics 10(2), 021006 (Feb 05, 2018) (10 pages) Paper No: JMR-17-1321; doi: 10.1115/1.4038977 History: Received September 23, 2017; Revised December 11, 2017

## Abstract

This paper presents a new mechanics-based framework for the qualitative analysis and conceptual design of mechanical metamaterials, and specifically materials exhibiting auxetic behavior. The methodology is inspired by recent advances in the insightful synthesis of compliant mechanisms by visualizing a kinetostatic field of forces that flow through the mechanism geometry. The framework relates load flow in the members of the microstructure to the global material properties, thereby enabling a novel synthesis technique for auxetic microstructures. This understanding is used to qualitatively classify auxetic materials into two classes, namely, high-shear and low-shear microstructures. The ability to achieve additional attributes such as isotropy is shown to be related to the qualitative class that the microstructure belongs.

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## Introduction

Mechanical metamaterials are structures with extremal material properties like negative Poisson's ratio, negative thermal expansion coefficient, high bulk modulus, high shear modulus, negative compressibility, etc. These metamaterials have wide range of potential applications in structural [1], electronic [2], acoustic [3], bio-medical [4,5], and sporting [6] fields due to their unique properties. The most popular class of these metamaterials is auxetic (negative Poisson's ratio) materials. The wide interest in auxetic materials is due to their mechanical properties such as high indentation resistance, high shear resistance, and high fracture toughness. There are a wide variety of auxetics like molecular auxetics, cellular auxetics, auxetics polymers and others, each vary in their design, properties, and applications [7].

Design of auxetic metamaterials has been an active field of research in material design. There are several techniques proposed for the design of auxetic materials in the literature. The first attempt to develop and demonstrate the negative Poisson's ratio behavior was by Lakes [8] in foam structures. Almgren [9] proposed a methodology to obtain isotropic negative Poisson's ratio materials using a structure of rods, hinges, and springs. Cherkaev [10] obtained the negative Poisson's ratio materials as a two-phase composite with varying volume fractions of isotropic rigid and compliant phases. These designs were one-off solutions that relied on innovative sparks and deep insight of the designer to achieve realizable structures. Design of auxetic elastic microstructures gained pace with the advent of structural and topology optimization as proposed by Sigmund [11]. In this seminal paper, Sigmund poses the design problem as an inverse homogenization process. Since its publication, many researchers have used the structural optimization approach to attain microstructural geometries with prescribed effective elastic properties [1216]. A detailed review of microstructural design using topology optimization can be found in Ref. [17]. Even though numerical optimization methodology has its merits, it is computationally expensive and design solutions are not always practical. Recently, Andreassen et al. [18] demonstrated a way to obtain manufacturable three-dimensional elastic microstructures using topology optimization with the inclusion of manufacturing constraints. Though effective, the numerical intensiveness of the method renders it difficult to implement.

In this paper, we present an insightful technique to analyze and synthesize planar mechanical metamaterials for their linear homogenized elastic properties. The insight is obtained by visualizing a kinetostatic vector field of forces that flow through the geometry [19]. The nature and orientation of these force fields enable identification of certain members in the microstructure as transmitters (T) and certain others as constraints (C). Transmitters transmit energy from input to output, while constraints deform along a specified direction thereby storing energy. The theory of load flow has been formalized by the authors for qualitative analysis and design of compliant mechanisms in prior work [1921]. Here, we apply this technique to mechanical metamaterials and aim to correlate load flow orientations with stiffness and compliance under a particular loading condition. These will be then mapped to the homogenized material properties such as Poisson's ratio, bulk modulus and shear modulus. The homogenized material properties are computed using strain energy-based technique applying four different loading cases to obtain the components of elastic matrix. The background of strain energy-based homogenization (SEBH) and load flow-based design framework will be reviewed in Secs. 2 and 3.

In this paper, we further narrow our focus to understand the behavior of auxetic (negative Poisson's ratio) metamaterials in terms of load flow orientations. The load flow analysis of various microstructures to understand their behavior is presented in Sec. 4. Section 5 inverts this understanding to conceptualize feasible load flow paths to synthesize new auxetic microstructural topologies. We showcase challenges in the process, especially those that require compliance with predetermined boundary conditions, which limit the possible feasible load paths. Section 6 extends this understanding to solve two qualitative problems (1) auxetics with low shear modulus and (2) auxetics with high shear modulus. Furthermore, we show a relation between isotropy of the auxetic designs and the above two qualitative classes that the microstructure belongs. Section 7 presents conclusion and future work.

## Review: Homogenization Technique

Homogenization is an averaging process for computing the effective elastic properties of composites. In this section, we review the homogenization technique for planar linear-elastic microstructures. The most popular homogenization technique for the optimal design of material microstructures is asymptotic homogenization. This uses asymptotic double-scale expansion along with periodic boundary conditions to compute the effective elastic properties [22]. More details of asymptotic homogenization can be found in Refs. [12] and [23]. Recently, an alternate homogenization technique was proposed by Zhang et al. [24], known as SEBH. It evaluates the strain energy of the microstructure under four different loading conditions and equates this to the global strain energy, and thus the effective elastic properties of the homogenized material. In this paper, we will use the SEBH due to the simplicity and insightfulness associated with the process. A brief review of the Strain energy-based Homogenization technique [24,25] for two-dimensional periodic cellular structures is presented below.

###### Strain Energy-Based Homogenization.

Strain energy-based homogenization technique is based on the fact that the strain energy of the base cell and the strain energy of the homogenized medium are equivalent. Generalized Hooke's law for a two-dimensional orthotropic material relates stresses and strains using five elastic constants as shown in Eq. (1). We need to compute the effective elastic properties $C1111H, C1122H, C2211H, C2222H$, and $C1212H$. Homogenization is the process of finding these components of the elasticity matrix by analyzing local microstructure Display Formula

(1)${σ¯11σ¯22σ¯12}=[C1111HC1122H0C2211HC2222H000C1212H]{ε¯11ε¯222ε¯12}$

In SEBH, we apply four different loading cases for the calculation of the components of elasticity matrix. The load cases are shown in Fig. 1, and equations for the computation of elastic properties are given in Eqs. (2)(5). Equations (2) and (3) correspond to load cases of prescribed horizontal and vertical strains, respectively, as shown in Figs. 1(a) and 1(b). Equation (4) corresponds to load case of prescribed shear strain as shown in Fig. 1(c), and Eq. (5) corresponds to load case of prescribed biaxial strain as shown in Fig. 1(d)Display Formula

(2)$C1111H=2SEa$
Display Formula
(3)$C2222H=2SEb$
Display Formula
(4)$C1212H=2SEc$
Display Formula
(5)$C1122H=SEd−SEa−SEb$

The strain energy stored is indicative of the stiffness or flexibility under these boundary and loading conditions, which then correlate with high or low values of the homogenized elasticity constants. However, to qualitatively evaluate relative stiffness/flexibility in a microstructure, we will use the insights offered by load flow visualization.

## Review: Analysis of Compliant Mechanisms Through Load Flow Visualization

Most analysis methodologies seek to describe the behavior of compliant mechanisms by determining displacements at key locations. In Ref. [19], the authors proposed an alternate framework that analyzes a compliant mechanism by a field of forces that cause these displacements. This fictitious field of forces known as the transferred forces evaluated at any point in the mechanism in relation to the input applied forces. The nature and orientation of the transferred forces in a member determine its functionality. These are explained in further detail below.

###### Transferred Forces.

Consider a compliant displacement amplifying compliant inverter shown in Fig. 2(a). The input force $fi$ is applied at a point i and the inverted displacement output is obtained from point k. Let j be a point on the mechanism. From Ref. [19], the transferred force at point j denoted by $ftrj$ is an applied force that would yield the same displacement at j as would the input force at i. This transferred force can also be obtained by fixing point j and evaluating the reaction force due to the input force. The transferred force between two points in a continuum can be found from the terms of the compliance matrix. The compliance matrix relates the forces and displacement and is given by Display Formula

(6)$[uiuj]=[CiiCijCijTCjj][fifj]$

where $Cij$ are the components of the compliance matrix relating the input i and output ports j. The compliance matrix in general can be obtained by inverting the nonsingular global stiffness matrix. The relation between the transferred force and the force load is given in terms of the compliance matrix as Display Formula

(7)$ftrj=Cjj−1CijTfi TL=Cjj−1CijT$

where $TL$ is the load transfer matrix that maps the transferred force to the input force. For planar beam elements with three degrees-of-freedom per node, $TL$ is a 3 × 3 matrix. The reader is referred to Ref. [19] for a more detailed derivation of the transferred forces. The uniqueness of the compliance matrix terms under small deformation assumption renders a one–one mapping between the transferred force at a point and the applied input force. However, this formulation is valid for small deformation assumptions alone. Figure 2(a) plots the transferred force evaluated using Eq. (7) throughout the geometry.

###### Compliant Dyad: Fundamental Building Blocks for Compliant Mechanisms.

An advantage of the transferred force formulation for compliant mechanisms is the ability to decompose the topology into maximally independent building blocks. The transferred force between points i and j in Fig. 2(a) is dependent on members C1 and T1 alone and not on other members such as C2, T2, and C3. This is because the transferred force in j is evaluated by constraining all degrees-of-freedom at this point, which implies that the members below it do not exercise any influence on the transferred force at j. This has been more formally proved in Ref. [19]. Furthermore, the force transferred at point k is maximally dependent only on transmitter T2 and constraint C2. For T2 and C2 combination, the transferred force $ftrj$ is the input. Thus, the force transferred through a transmitter is dependent on itself and a corresponding constraint member, and is maximally decoupled from other members. This enables the compliant mechanism to be analyzed in a modular fashion in terms of building blocks composed of a transmitter and a constraint. We call this the transmitter-constraint set (TC), which in most cases is a compliant dyad.

A compliant dyad building block as shown in Fig. 3 consists of two beams combined in series. The first beam (with length l1, second area moment I1, Young's modulus E) is the constraint, while the second beam (with length l2, second area moment I2, Young's modulus E) is the transmitter (T). The $TL$ matrix can be derived from Euler–Bernoulli beam mechanics as [19] Display Formula

(8)$TL=[0cotα−3(n2+n12 cos α)2l1n1(n1+n2)sin α01−3n12l1n1+2l1n200−n2(2(n1+n2))]$

where $n1=l2/l1$, and $n2=I2/I1$. The above matrix can be interpreted in the following:

1. (1)A horizontal input force $fix$ aligned along the axis of the constraint beam does not get transferred to the output. This is because the beam constraint resists any deformation due to an axial force, as indicated by the zeros in the first column.
2. (2)A vertical force $fiy$ gets transferred to the output as a force in a direction along the axis of the transmitter beam. The transferred force components can be expanded as Display Formula
(9)$fjtrx=cotα×fiyfjtry=fiymjtr=0tanβ=tanfiyfix=tanα$
There is no transferred moment for this force because $TL3,2=0$.
3. (3)An input moment applied manifests as transferred x and y direction forces at the output whose components are given by $TL1,3$ and $TL2,3$.
4. (4)An input moment is transferred as an output moment in the negative direction as indicated by the negative sign $TL3,3$.

Next, we relate the transferred force at the end of the dyad to its displacement. The displacement at any point can lie in a direction $±90 deg$ with respect to the direction of the transferred force. This is due to the positive definiteness of the stiffness matrix, implying that a force cannot perform negative work on the output. The semicircular band for a dyad at the input subjected to a Y-axis force is shown in Fig. 4(a). The actual displacement will lie in the intersection of the semicircular band and the freedom direction of the constraint at the output. From constraint-based design [2628], a beam is free to displace perpendicular to its axis. Thus a horizontal beam constraint shown in Fig. 4(b) yields a displacement perpendicular to its length. Now, to determine the output displacement we consider a simple, yet ubiquitous case where the transferred forces do not have any moment components. Equation (9) shows that the transferred force is oriented along the transmitter at an angle δ with respect to the horizontal. Thus, the direction of displacement of the output will lie in the intersection of the semicircular band and the freedom line of the constraint beam at the output.

## Load Flow Visualization of Metamaterial Microstructures

In this section, we will qualitatively understand the global material behavior of mechanical metamaterials by visualizing load flow in its microstructure. Strain energy-based homogenization relates the global material properties to four deformation modes for the microstructure as shown in Fig. 1. Load flow in the microstructure can then reveal whether the particular deformation mode is naturally permitted or is resisted by the microstructure geometry. This analysis can then provide qualitative assessment on three global quantities of interest: (a) Poisson's ratio, (b) bulk modulus, and (c) shear modulus for a planar material.

For the qualitative assessment, we follow two guidelines proposed in Ref. [29], which states that a particular deformation mode is restricted or constrained if (i) load flow in all the members (both transmitters and constraints) are predominantly axial and (ii) load flow in a transmitter conflicts due to the effect of two or more simultaneously acting loading conditions. These will be explained in detail with a few examples. We select two microstructures shown in Fig. 5, one having negative Poisson's ratio and the other having positive Poisson's ratio with a high bulk and shear modulus.

###### Analysis of Auxetic Microstructures.

A microstructure that exhibits negative Poisson's ratio is shown in Fig. 5(a) [25]. Since the material properties are orthotropic, we can consider a quarter structure as shown in Fig. 5(b). Furthermore, we consider four loading conditions to determine its material properties as shown in Fig. 1. The first boundary condition is used to evaluate stiffness along the X direction (C1111) as shown in Fig. 6(a), where a unit strain is applied along the X axis in the extreme right face of the microstructure. To better understand its deformation by drawing an analogy with compliant mechanisms, we replace the roller constraints with beam flexures corresponding to the boundary condition of the faces. For example, top left corner is constrained against both X and Y deflection, and is thus grounded. The top and the bottom faces are constrained against Y translation and thus constraint beams C3, C2, and C1, respectively, are added to denote this.

Qualitative analysis of elastic modulii C1111 and C2222: When a unit displacement is applied on the bottom right end of the geometry, we can visualize load flow in the mechanism by partitioning it into TC sets as seen from the example in Fig. 2. The first TC set consists of constraint C1 and transmitter T1. From Eq. (9), the load flow in the transmitter is axial. Load flow in constraint C1 is not depicted, but will be in predominantly transverse direction, permitting deformation. The load flow at the end of transmitter T1 is now the input for the second TC set comprised of C2 and T2 as shown in Fig. 6(c). Here, again from Eq. (9) load flow is an axial force along transmitter T2. This force is an input to TC sets T3 and C3. Since the ends of T3 and C3 are fixed, this point does not displace. Correspondingly, the load flow is axial in both members. Thus, the deformation in the microstructure under this loading condition is primarily due to the transmitters T1 and T2 and their corresponding constraints C1 and C2.

Similarly, we can interpret the loading condition leading to evaluating stiffness in the Y direction (C2222) as shown in Figs. 6(d)6(f). The corresponding boundary condition introduces a new set of constraints C1 and C2, while rendering the bottom right corner fixed. Here, transmitter T1 and constraint C2 do not translate as their load flow behavior indicate pure axial forces. The deformation is primarily in T1 and C1. These two loading conditions can be related to Young's modulus of the material.

Bulk modulus: Bulk modulus is evaluated under the boundary condition shown in Fig. 7(a). Two faces (top and right) are subjected to unit strain. The bottom face is constrained against Y translation and hence is replaced by two parallel constraint beams C2 and C3 as shown in Fig. 7(b). Similarly, the left face is constrained against X deflection. For the sake of qualitative understanding of the load flow in this case, we will consider the two strain loading cases independently. Applying the X strain alone on the right face leads to the load flow as shown in Fig. 7(c). Applying Y strain alone on the top face leads to load flow in the same directions as the previous case. Thus, the two loading conditions complement each other, thus qualitatively lowering the bulk modulus.

Shear modulus: The boundary and loading conditions for evaluating the shear modulus are shown in Fig. 8(a). Just as the bulk modulus case, we have two strain loading conditions that will be considered separately. Once again, for understanding of the load flow behavior better, we replace the boundary conditions with constraint beam flexures. For both boundary conditions, it is seen that the load flow directions in the transmitters and constraints are nonconflicting leading to a qualitatively low shear modulus. More importantly, transmitter T2 in Fig. 8(c) and transmitter T1 in Fig. 8(d) do not have a constraint beam flexure and thus deform without resistance.

###### Analysis of Positive Poisson's Ratio Microstructure.

In summary, we make the following correlation between load flow in the microstructure and its global elastic properties:

1. (1)A material with high bulk or shear modulus demonstrates conflict of load flow orientations in their transmitters, when independent unit strains are applied according to Fig. 1(a).
2. (2)A material with negative Poisson's ratio demonstrates load flow behavior such that any applied strain in positive X (or Y) direction is transferred as a corresponding force in the positive Y (or X) direction.
3. (3)A negative Poisson's ratio always leads to a low bulk modulus.

###### Qualitative Design Problems for Planar Auxetic Microstructures.

Poisson's ratio of a microstructure can be varied by varying the relative values of shear and bulk modulus while keeping Young's modulus of the base material constant [30]. From our load-flow-based understanding, we observed the dependence of the bulk modulus with Poisson's ratio—a negative Poisson's ratio always yields a low Bulk modulus. Thus, planar auxetic microstructures can be qualitatively classified into

1. (1)Auxetics with low shear modulus.
2. (2)Auxetics with high shear modulus.

The response to shear loading will be qualitatively assessed based on whether the load flow contradicts in the transmitters of the structure. This will be seen in Secs. 5 and 6.

## Synthesis of Planar Auxetic Microstructures

In this section, we will present guidelines for the de novo synthesis of microstructures exhibiting auxetic or negative Poisson's ratio behavior. The emphasis is on the conceptual design, with details such as geometry and cross section dimensions left to a subsequent refinement process. Based on our understanding of the load flow characteristics in auxetic materials from Sec. 4, our problem statement will be presented as

Design a microstructure geometry such that any input strain in the axial, or positive X axis leads to a corresponding transferred force with a component in the transverse or positive Y axis, and vice versa

The design methodology will be presented using several steps similar to Ref. [20], which are elaborated with an example.

• Step I: Establish the design domain. Determine semicircular bands at the input and output corresponding to their required displacement directions.

• The design domain is shown in Fig. 10(a). We arbitrarily choose the input to be the bottom right end, while in general it could be at any point along the right face (R). Any displacement in the positive X direction here will lead to a load transfer at an angle $±90 deg$, which is indicated by the semicircular band SB1 in Fig. 10(b). This condition arises due to the positive definiteness of the stiffness matrix at this point. Due to the negative Poisson's ratio (or auxetic) requirement, the displacement at the output, which is any point on the top face (T) must be in the positive Y direction. We arbitrarily choose the top left end as the output, in which case the load transferred must be along a semicircular band SB2 about this direction.

• Step II: Determine feasible load paths between input and output.

• Load paths are a collection of transmitters with axial load flow in them. The first most obvious solution is to check whether a direct connection between the input and output is feasible. Such a direct connection would have its load flow oriented toward the input as shown in Fig. 10(b), which would not correspond to any direction in SB2. This is true even if the input and output were moved along their respective edges (R and T). Thus, the only possible solution is to bifurcate the load path and generate two transmitters T1 and T2 shown in Figs. 10(c) and 10(d). Here, T1 can have load flow oriented toward SB1 and T2 toward SB2. Next, we design constraints at the intersections of T1 and T2 that establish these load flow directions.

• Step III: Determine truncated bands at the intersection of the transmitters

• To enforce the desired load flow directions in the transmitters, we need to orient constraint beams at the intersection of T1 and T2. There are only a limited constraint orientations that are possible. The set of feasible freedom direction of the constraint is determined by the intersection of modified semicircular bands at the input and output ($SB1a$ and $SB2a$, respectively), which are now determined about the transmitter orientation [20]. For the load path shown in Fig. 10(c), the truncated band SB3 is very narrow. Furthermore, there is no possible constraint direction within this band that blends seamlessly with the roller constraints of edge B as shown in Fig. 10(c). Thus, we explore other ways to increase the span of the truncated band SB3. One possible solution is to change the output to a different location in the top face as shown in Fig. 10(d).

• Step IV: Determine constraint directions that correspond to the microstructure boundary conditions

• It is clear from the geometry that the only constraint possible is the restriction against Y motion, which is shown in Fig. 10(e) as constraint C1. Its direction of freedom coincides with a feasible direction in the truncated band SB3. Furthermore, we add a constraint C2 at the output to ensure pure Y translation.

• Step V: Assemble the symmetric quarter sections of the microstructure.

• The assembled microstructure is shown in Fig. 10(f). A prototype of this microstructure was fabricated as a 2 × 2 array using additive manufacturing as shown in Fig. 11. A positive X deflection of $32 mm$ yielded a positive Y deflection of around $15 mm$, thus demonstrating negative Poisson's ratio. Also, numerical simulations for the variation of Poisson's ratio with strain are conducted in abaqus with plain stress assumption. The results depicted in Fig. 12 show a smaller increase of νxy and a rapid decrease of νyx with positive strain. The trends match with deformation of auxetic honeycombs [31].

• In the process illustrated so far, there are a number of decisions that the designer had freedom to choose. A different, yet feasible choice of (a) number of load paths, (b) input and output locations, and (c) constraint locations will yield different microstructure designs. In Figs. 13(a)13(h), we demonstrate various microstructure geometries obtained using the load flow-based design framework that potentially exhibit auxetic behavior. The first two microstructure topologies consist of two load paths similar to Fig. 10, but with different inputs, while the last two topologies consist of three load paths.

###### Qualitative Classification of the Solutions.

For the five auxetic microstructure designs presented (Figs. 10 and 13), we will attempt to classify them into two qualitative classes introduced in Sec. 4.3. Since all the designs exhibit negative Poisson's ratio, the only distinguishing feature is the shear modulus.

The analysis for qualitative shear modulus for two contrasting designs (Figs. 13(c) and 13(g)) is shown in Fig. 14. The analysis of load flow for shear is similar to the examples in Fig. 9. Figure 13(c) mechanism exhibits complementing load flow in its transmitter T2 (Figs. 14(c) and 14(d)). Furthermore, the nature of load flow is not completely axial because the transmitter is not properly constrained. This yields flexibility and consequently a qualitatively low shear modulus. On the other hand, a similar analysis of Fig. 13(g) mechanism shows clear axial and contradicting load flow in its transmitters T2 and T3. We thus expect qualitatively high shear modulus for this case. Using similar load flow-based inferences, it was found that all the microstructures with the exception of Fig. 13(g) could be classified as auxetic low shear microstructures. In Sec. 6, we will relate these qualitative classes to secondary attributes such as isotropy.

## Obtaining Isotropy

Most of the auxetic microstructural designs obtained in the literature are anisotropic in nature [32]. Even though anisotropic designs can achieve higher negative Poisson's ratios, their directional dependence is an impediment when being deployed in practical applications. Researchers have previously alluded to the difficulty involved in obtaining the isotropic designs [15,32]. Even with computational tools such as topology optimization, it is a challenging task to impose isotropic constraints and obtain manufacturable designs [15]. In this section, we formulate a shape optimization framework to render the conceptual microstructures isotropic. The problem statement with the inclusion of isotropic constraints [18] can be shown as in Eqs. (10)(13)Display Formula

(10)$minimizeXi:(ν−ν*)2$
Display Formula
(11)$s.t:(C1111−C2222)2≤ε$
Display Formula
(12)$[(C1111−C11222)−C1212]2≤ε$
Display Formula
(13)$Lbi≤Xi≤Ubi$

Here, ν is Poisson's ratio of the microstructure obtained from numerical homogenization, $ν*$ is the desired Poisson's ratio, and ε is a small numerically insignificant quantity. The isotropic constraints require that the C1111 and C2222 be close to each other. Furthermore, the shear term C1212 is required to be very close to half the difference between C1111 and C1122 terms. This implies that the isotropy constraints reduce the number of elastic constants to two, with shear modulus being related to Poisson's ratio and Bulk modulus. Here, the design variables Xi's are coordinates of the points and also the width of each beam as shown in Fig. 15(a).

We ran this optimization formulation with the objective of the desired Poisson's ratio $ν*=−1$ on the conceptual designs obtained in Figs. 13(a), 13(c), 13(e), and 13(g) and observed that microstructural designs in Figs. 13(a), 13(c), and 13(e) could not satisfy the isotropic constraints mentioned in Eqs. (11) and (12), even after an extensive design space search. Only the microstructure Fig. 13(g) could satisfy both the posed constraints to give an isotropic auxetic design as shown in Fig. 15(b). A possible explanation here is because of the high qualitative shear modulus of Fig. 13(g) mechanism. Equation (12) requires the shear modulus C1212 to have the same order of magnitude as Young's modulus C1111. For microstructures composed of thin slender beams, this implies that the shear modulus must be qualitatively high, a condition matched by Fig. 13(g)Display Formula

(14)$CunoptH=[1.0000−0.73490−0.73490.57860000.0341]$
Display Formula
(15)$CoptH=[1.0000−0.93920−0.93921.00000000.9696]$

The homogenized elasticity matrices of the unoptimized and optimized designs of Figs. 15(a) and 15(c) are shown in Eqs. (14) and (15), respectively. We can clearly see that the optimized design satisfies the isotropic constraints [33].

## Conclusion and Future Work

This paper presents a first-of-its-kind treatment of the analysis and design of planar auxetic microstructures using load flow visualization. Load flow visualization has been previously employed to obtain insight into the working of compliant mechanisms by identifying maximally decoupled building blocks known as TC sets that constitute its topology. In this paper, we analyze microstructure behavior using load flow in the TC sets and relate their qualitative material properties such as Poisson's ratio, bulk modulus, and shear modulus. This insight has been used to synthesize novel topologies for negative Poisson's ratio microstructures.

An important contribution of this paper was to identify two fundamental classes of problems that auxetic microstructures can be classified into based on their qualitative shear response, which was informed by the load flow behavior. A link was shown to relate practical attributes such as material isotropy to the shear modulii, a larger value demonstrating a greater inclination to achieve isotropy. Such an insight has never been presented earlier, and its implications will be studied in detail in future work.

In general, design of metamaterials using structural optimization is an intensive and computationally expensive process. The framework presented in this paper divides the design process into two phases: (a) conceptual design phase, where we obtain the conceptual geometry using the load flow behavior and (b) shape optimization of the obtained geometry to meet the required elastic properties. The latter phase was demonstrated to obtain isotropic elastic properties, and will be studied in detail in future work.

Apart from these, we plan to extend the analysis and design method to three-dimensional microstructures and aperiodic metamaterials.

## Acknowledgements

The authors thank Professor Kai James from the Aerospace department at University of Illinois, Urbana-Champaign for providing helpful suggestions.

## Funding Data

• National Science Foundation (Grant No. CMMI-1454276).

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Cadman, J. E. , Zhou, S. , Chen, Y. , and Li, Q. , 2013, “ On Design of Multi-Functional Microstructural Materials,” J. Mater. Sci., 48(1), pp. 51–66.
Andreassen, E. , Lazarov, B. S. , and Sigmund, O. , 2014, “ Design of Manufacturable 3D Extremal Elastic Microstructure,” Mech. Mater., 69(1), pp. 1–10.
Krishnan, G. , Kim, C. , and Kota, S. , 2013, “ A Kinetostatic Formulation for Load-Flow Visualization in Compliant Mechanisms,” ASME J. Mech. Rob., 5(2), p. 021007.
Krishnan, G. , Kim, C. , and Kota, S. , 2010, “ Load-Transmitter Constraint Sets—Part II: A Building Block Method for the Synthesis of Compliant Mechanisms,” ASME Paper No. DETC2010-28819.
Krishnan, G. , Kim, C. , and Kota, S. , 2010, “ Load-Transmitter Constraint Sets—Part I: An Effective Tool for Visualizing Load Flow in Compliant Mechanisms and Structures,” ASME Paper No. DETC2010-28810.
Hassani, B. , and Hinton, E. , 1998, “ A Review of Homogenization and Topology Optimization—I: Homogenization Theory for Media With Periodic Structure,” Comput. Struct., 69(6), pp. 707–717.
Guedes, J. , and Kikuchi, N. , 1990, “ Preprocessing and Postprocessing for Materials Based on the Homogenization Method With Adaptive Finite Element Methods,” Comput. Methods Appl. Mech. Eng., 83(2), pp. 143–198.
Zhang, W. , Dai, G. , Wang, F. , Sun, S. , and Bassir, H. , 2007, “ Using Strain Energy-Based Prediction of Effective Elastic Properties in Topology Optimization of Material Microstructures,” Acta Mech. Sin., 23(1), pp. 77–89.
Mehta, V. , Frecker, M. , and Lesieutre, G. A. , 2012, “ Two-Step Design of Multicontact-Aided Cellular Compliant Mechanisms for Stress Relief,” ASME J. Mech. Des., 134(12), p. 121001.
Blanding, D. K. , 1999, Exact Constraint: Machine Design Using Kinematic Principles, ASME Press, New York.
Awtar, S. , and Slocum, A. H. , 2007, “ Constraint-Based Design of Parallel Kinematic XY Flexure Mechanisms,” ASME J. Mech. Des., 129(8), pp. 816–830.
Hopkins, J. B. , and Culpepper, M. L. , 2010, “ Synthesis of Multi-Degree of Freedom, Parallel Flexure System Concepts Via Freedom and Constraint Topology (FACT)—Part I: Principles,” Precis. Eng., 34(2), pp. 271–278.
Satheeshbabu, S. , and Krishnan, G. , 2016, “ Qualitative Mobility Analysis of Wire Flexure Systems Using Load Flow Visualization,” ASME J. Mech. Rob., 8(6), p. 061012.
Prawoto, Y. , 2012, “ Seeing Auxetic Materials From the Mechanics Point of View: A Structural Review on the Negative Poisson's Ratio,” Comput. Mater. Sci., 58, pp. 140–153.
Wan, H. , Ohtaki, H. , Kotosaka, S. , and Hu, G. , 2004, “ A Study of Negative Poisson's Ratios in Auxetic Honeycombs Based on a Large Deflection Model,” Eur. J. Mech. A, 23(1), pp. 95–106.
Shan, S. , Kang, S. H. , Zhao, Z. , Fang, L. , and Bertoldi, K. , 2015, “ Design of Planar Isotropic Negative Poissons Ratio Structures,” Extreme Mech. Lett., 4, pp. 96–102.
Ogden, R. W. , 1997, Non-Linear Elastic Deformations, Dover Publication, Mineola, NY.
View article in PDF format.

## References

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Ziolkowski, R. W. , Jin, P. , and Lin, C.-C. , 2011, “ Metamaterial-Inspired Engineering of Antennas,” Proc. IEEE, 99(10), pp. 1720–1731.
Lee, S. H. , Park, C. M. , Seo, Y. M. , Wang, Z. G. , and Kim, C. K. , 2009, “ Acoustic Metamaterial With Negative Modulus,” J. Phys.: Condens. Matter, 21(17), p. 175704. [PubMed]
Evans, K. E. , and Alderson, A. , 2000, “ Auxetic Materials: Functional Materials and Structures From Lateral Thinking!,” Adv. Mater., 12(9), pp. 617–628.
White, L. , 2009, “ Auxetic Foam Set for Use in Smart Filters and Wound Dressings,” Urethanes Technol. Int., 26(4), pp. 34–36.
Sanami, M. , Ravirala, N. , Alderson, K. , and Alderson, A. , 2014, “ Auxetic Materials for Sports Applications,” Procedia Eng., 72, pp. 453–458.
Mir, M. , Ali, M. N. , Sami, J. , and Ansari, U. , 2014, “ Review of Mechanics and Applications of Auxetic Structures,” Adv. Mater. Sci. Eng., 2014 p. 753496.
Lakes, R. , 1987, “ Foam Structures With a Negative Poisson's Ratio,” Science, 235(4792), pp. 1038–1041. [PubMed]
Almgren, R. F. , 1985, “ An Isotropic Three-Dimensional Structure With Poisson's Ratio–1,” J. Elasticity, 15(4), pp. 427–430.
Cherkaev, A. V. , 1995, “ Which Elasticity Tensors are Realizable?,” ASME J. Eng. Mater. Technol., 117(4), pp. 483–493.
Sigmund, O. , 1994, “ Materials With Prescribed Constitutive Parameters: An Inverse Homogenization Problem,” Int. J. Solids Struct., 31(17), pp. 2313–2329.
Bendsøe, M. P. , and Kikuchi, N. , 1988, “ Generating Optimal Topologies in Structural Design Using a Homogenization Method,” Comput. Methods Appl. Mech. Eng., 71(2), pp. 197–224.
Sigmund, O. , 2000, “ A New Class of Extremal Composites,” J. Mech. Phys. Solids, 48(2), pp. 397–428.
Xia, L. , and Breitkopf, P. , 2015, “ Design of Materials Using Topology Optimization and Energy-Based Homogenization Approach in Matlab,” Struct. Multidiscip. Optim., 52(6), pp. 1229–1241.
Neves, M. M. , Rodrigues, H. , and Guedes, J. M. , 2000, “ Optimal Design of Periodic Linear Elastic Microstructures,” Comput. Struct., 76(1), pp. 421–429.
Vogiatzis, P. , Chen, S. , Wang, X. , Li, T. , and Wang, L. , 2017, “ Topology Optimization of Multi-Material Negative Poisson's Ratio Metamaterials Using a Reconciled Level Set Method,” Comput.-Aided Des., 83, pp. 15–32.
Cadman, J. E. , Zhou, S. , Chen, Y. , and Li, Q. , 2013, “ On Design of Multi-Functional Microstructural Materials,” J. Mater. Sci., 48(1), pp. 51–66.
Andreassen, E. , Lazarov, B. S. , and Sigmund, O. , 2014, “ Design of Manufacturable 3D Extremal Elastic Microstructure,” Mech. Mater., 69(1), pp. 1–10.
Krishnan, G. , Kim, C. , and Kota, S. , 2013, “ A Kinetostatic Formulation for Load-Flow Visualization in Compliant Mechanisms,” ASME J. Mech. Rob., 5(2), p. 021007.
Krishnan, G. , Kim, C. , and Kota, S. , 2010, “ Load-Transmitter Constraint Sets—Part II: A Building Block Method for the Synthesis of Compliant Mechanisms,” ASME Paper No. DETC2010-28819.
Krishnan, G. , Kim, C. , and Kota, S. , 2010, “ Load-Transmitter Constraint Sets—Part I: An Effective Tool for Visualizing Load Flow in Compliant Mechanisms and Structures,” ASME Paper No. DETC2010-28810.
Hassani, B. , and Hinton, E. , 1998, “ A Review of Homogenization and Topology Optimization—I: Homogenization Theory for Media With Periodic Structure,” Comput. Struct., 69(6), pp. 707–717.
Guedes, J. , and Kikuchi, N. , 1990, “ Preprocessing and Postprocessing for Materials Based on the Homogenization Method With Adaptive Finite Element Methods,” Comput. Methods Appl. Mech. Eng., 83(2), pp. 143–198.
Zhang, W. , Dai, G. , Wang, F. , Sun, S. , and Bassir, H. , 2007, “ Using Strain Energy-Based Prediction of Effective Elastic Properties in Topology Optimization of Material Microstructures,” Acta Mech. Sin., 23(1), pp. 77–89.
Mehta, V. , Frecker, M. , and Lesieutre, G. A. , 2012, “ Two-Step Design of Multicontact-Aided Cellular Compliant Mechanisms for Stress Relief,” ASME J. Mech. Des., 134(12), p. 121001.
Blanding, D. K. , 1999, Exact Constraint: Machine Design Using Kinematic Principles, ASME Press, New York.
Awtar, S. , and Slocum, A. H. , 2007, “ Constraint-Based Design of Parallel Kinematic XY Flexure Mechanisms,” ASME J. Mech. Des., 129(8), pp. 816–830.
Hopkins, J. B. , and Culpepper, M. L. , 2010, “ Synthesis of Multi-Degree of Freedom, Parallel Flexure System Concepts Via Freedom and Constraint Topology (FACT)—Part I: Principles,” Precis. Eng., 34(2), pp. 271–278.
Satheeshbabu, S. , and Krishnan, G. , 2016, “ Qualitative Mobility Analysis of Wire Flexure Systems Using Load Flow Visualization,” ASME J. Mech. Rob., 8(6), p. 061012.
Prawoto, Y. , 2012, “ Seeing Auxetic Materials From the Mechanics Point of View: A Structural Review on the Negative Poisson's Ratio,” Comput. Mater. Sci., 58, pp. 140–153.
Wan, H. , Ohtaki, H. , Kotosaka, S. , and Hu, G. , 2004, “ A Study of Negative Poisson's Ratios in Auxetic Honeycombs Based on a Large Deflection Model,” Eur. J. Mech. A, 23(1), pp. 95–106.
Shan, S. , Kang, S. H. , Zhao, Z. , Fang, L. , and Bertoldi, K. , 2015, “ Design of Planar Isotropic Negative Poissons Ratio Structures,” Extreme Mech. Lett., 4, pp. 96–102.
Ogden, R. W. , 1997, Non-Linear Elastic Deformations, Dover Publication, Mineola, NY.

## Figures

Fig. 1

Different load cases for SEBH: (a) prescribed horizontal strain, (b) prescribed vertical strain, (c) prescribed shear strain, and (d) prescribed bi-axial strain

Fig. 2

Definition of load flow using the concept of transferred force in a displacement amplifying compliant inverter. (a) Force applied at i produces the same deformation at point j as a transferred force ftrj acting at j. (b) The deformed profile of the mechanism. (c) The transferred force in TC sets is maximally decoupled from other members.

Fig. 3

A compliant dyad as a TC set

Fig. 4

Output displacement of a dyad building block: (a) semicircular band capturing the possible directions of displacement and (b) actual displacement determined by the intersection of the semicircular band and the freedom direction of the constraint beam

Fig. 5

(a) Microstructure geometry for auxetic material [25], (b) corresponding quarter cell geometry used for analysis, and (c) corresponding deformation of the microstructure geometry, (d) microstructure geometry exhibiting positive Poisson's ratio [13,15], (e) corresponding quarter cell geometry for analysis, and (f) corresponding deformation of the microstructure geometry

Fig. 6

Load flow analysis in evaluating X direction stiffness: (a) boundary condition and applied strain for X axis, (b) substituting boundary conditions with beam constraints, and (c) load flow analysis on individual TC sets. Similarly, (d)–(f) analyzes load flow for evaluating stiffness in the Y direction.

Fig. 7

Load flow analysis in evaluating bulk modulus and Poisson's ratio: (a) boundary condition and applied strain, (b) substituting boundary conditions with beam constraints, (c) load flow analysis on individual TC sets for applying X strain alone, and (d) load flow analysis on individual TC sets for applying Y strain alone

Fig. 8

Load flow analysis in evaluating shear modulus: (a) boundary condition and applied strain, (b) substituting boundary conditions with beam flexures, (c) load flow analysis on individual TC sets for applying X strain alone, and (d) load flow analysis on individual TC sets for applying Y strain alone

Fig. 9

Load flow analysis in evaluating shear modulus of a microstructure in Figs. 5(c) and 5(d): (a) boundary condition and applied strain for shear modulus analysis, (b) substituting boundary conditions with beam constraints, (c) load flow analysis on individual TC sets for applying X strain alone, (d) load flow analysis on individual TC sets for applying Y strain alone, and (e)–(h) load flow analysis to evaluate bulk modulus

Fig. 10

Steps involved in the design of auxetic microstructure using the load flow framework

Fig. 11

Fabricated prototype of the auxetic microstructure from Fig. 10(f): (a) undeformed microstructure and (b) a positive X direction displacement at the right face shows a clear positive Y direction deformation in the top face

Fig. 12

Variation of Poisson's ratio with strain (a) shows for νxy and (b) shows for νyx

Fig. 13

Design of alternative microstructure topologies from load flow-based framework and their deformations demonstrating auxetic behavior

Fig. 14

Qualitative analysis of shear modulus using load flow visualization: (a)–(d) auxetic design with low shear modulus and (e)–(h) auxetic design with high shear modulus

Fig. 15

(a) is the initial geometry, (b) is the optimized quarter cell geometry with isotropy, and (c) is the full microstructure geometry with isotropy

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