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# Multi-layer auxetic surfaces with compliant mechanisms

Kazuki Hayashi, Romain Mesnil

► To cite this version:

Kazuki Hayashi, Romain Mesnil. Multi-layer auxetic surfaces with compliant mechanisms. Engineering Structures, 2025, 341, pp.120752. 10.1016/j.engstruct.2025.120752. hal-05148033

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Submitted on 7 Jul 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# Multi-layer auxetic surfaces with compliant mechanisms

Kazuki Hayashi<sup>a,b ID*</sup>, Romain Mesnil<sup>b ID</sup>

## ARTICLE INFO

**Keywords:**

Auxetic material

Computational geometry

Discrete differential geometry

Conformal mapping

Multi-layer structure

Kerf bending

## ABSTRACT

Despite its unique deployment mechanisms known as a negative Poisson's ratio, most existing studies on deployable auxetic structures have overlooked how to solve the dilemma between deployability and stability. In this study, we propose a method to improve both properties by introducing surface overlaying and joints with compliant mechanisms. We first derive open angles to achieve the target shape using conformal mapping. This derivation considers the combined effect of thickness and curvature, which contributes to a better surface approximation of multi-layer auxetic surfaces. Through the unique kerf joint system designed for the auxetic surfaces, we explain the importance of adjusting joint flexibility and stiffness in achieving the target shape. The simulation and physical experiment results of a unit cell show that double-layer auxetic surfaces effectively constrain in-plane torsional motion and efficiently transmit loads through in-plane forces like continuum shell structures. We further create a shell structure made of medium-density fiberboard with the proposed method and demonstrate its deployability and stability.

## 1. Introduction

Auxetic materials are a fascinating class of materials that exhibit negative Poisson's ratio, a property rarely found in natural materials. Unlike traditional materials such as metals, polymers, or woods, the term material in auxetic materials encompasses any substance that exhibits this unique deformation feature.

The geometric movement of auxetic materials has three major categories: elastic deformation, densification of unit cells, and rotational mechanism [1]. Among various geometric arrangements, rotating squares and rotating triangles, made of regular-shaped panels whose corners are connected like linkages, have rotation-dominant motion around the joints. Since they can be fabricated by assigning periodic cuts to a sheet material, they can cover a volume with a unique shade pattern [2]. To highlight their relative large cover area, We specifically call auxetic materials with such geometric pattern auxetic surfaces [3].

The application range of auxetic materials is versatile. Particularly, due to their negative Poisson's ratio combined with strain-hardening, auxetic materials are known to be advantageous as energy-absorbing or damping devices [4,5]. Furthermore, auxetic materials can be used as deployable structures that can achieve a target 3D shape from a 2D state. Konaković et al. [2] proposed a surface approximation method using auxetic surfaces with equilateral triangular linkages.

These linkages yield openings when stretched, which allows for achieving non-developable surfaces.

Despite active involvements of researchers in investigating the geometric and mechanical properties of auxetic materials, most existing studies consider a single stable configuration. Fewer studies have focused on their deployability between multiple configurations and how to stabilize the operational state, compared to other structural types (e.g., origami: [6]; kirigami: [7,8]). Note that some studies on auxetic surfaces have addressed this problem; for example, Konaković et al. [9] proposed auxetic surfaces that are deployed and stabilized via inflation or gravitational loading; some studies introduced a cut pattern that induces bi-stability due to strain energy barrier between initial and deployed configurations [3,10].

Nevertheless, there has not been a critical solution to balance two seemingly conflicting requirements of auxetic surface structures: flexibility for deployability and stiffness for serviceability. Deployable auxetic surfaces have traditionally relied on material hyperelasticity [3] or plastic deformability [11] to achieve joint flexibility, which limits their realization with traditional materials like wood.

In contrast, the auxetic surface must have sufficient stiffness to withstand operational loads once it is deployed and put into use. Although previous design methods for auxetic surfaces can precisely approximate target shapes, they are too flexible to withstand external forces. This poses a challenge: the “flexibility-stiffness dilemma”.

<footnote><sup>a</sup> Department of Architecture and Architectural Engineering, Kyoto University, Kyotoodaigaku-katsura, Nishikyo, 6158540, Kyoto, Japan</footnote>

<footnote><sup>b</sup> Laboratoire Navier, UMR 8205, École nationale des ponts et chaussées, 6 et 8 avenue Blaise-Pascal Cité Descartes, 77455, Champs-sur-Marne, France</footnote>

<footnote>* Corresponding author at: Department of Architecture and Architectural Engineering, Kyoto University, Kyotoodaigaku-katsura, Nishikyo, 6158540, Kyoto, Japan.</footnote>

<footnote>E-mail addresses: hayashi.kazuki@archi.kyoto-u.ac.jp (K. Hayashi), romain.mesnil@enpc.fr (R. Mesnil).</footnote>
<figure><img src="image_3.png" /><figcaption>Fig. 1. Examples of conformal mapping. The middle is exponential mapping, and the right is Möbius transformation. These projections illustrate the angle-preserving property of conformal mapping.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 2. Relationship between uniform shape scaling and log-conformal factor. By uniform scaling of a conformal map, we can achieve conformal factors that are either larger or smaller than a specific value.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 3. Rotational mechanism of auxetic surfaces.</figcaption></figure>

Assembling multiple structural configurations to interlock their movement is an effective approach to stiffening deployable structures. Filipov et al. [12] showed that assembling a couple of origami tubes increases the stiffness by two orders of magnitude. Watson [13] developed a structural system consisting of planar slices interlocked via pairs of complementary slots placed along each intersection.

Torres [14] showed a physical prototype of a double-layer structure in which upper and lower faces interlock their in-plane rotational motion. Karunanidhi et al. [15] proposed a similar system for auxetic surfaces in which multiple layers interlock their deployment motion. However, the mechanical properties of multi-layer auxetic surfaces have not been fully uncovered.

Many studies have attempted to introduce compliant mechanisms to ensure the elastic deformation of auxetic materials. Lima and Paulino

[16] optimized the unit topology of micro and macrostructures to achieve an auxetic behavior with compliant mechanisms. Incorporating compliant mechanisms into the joints of linkages can be an effective approach to facilitate an auxetic behavior [17].

Kerf bending is also a type of compliant mechanism as it elastically deforms through a series of kerfs (cuts) in a piece of material. With the spread of computer numerical control (CNC) milling and laser cutters, kerf bending has been attracting attention as a new method to materialize bending-active structures made of wood.

In capturing the mechanical behaviors of kerf-bending structures, structural analysis is often performed by placing beam elements between kerfs. Chen et al. [18] used a beam element model to simulate
<figure><img src="image_6.png" /><figcaption>Fig. 4. Required rotation angle of connection for attaching two layers with prescribed deployment angle $\theta$.</figcaption></figure>

linear elastic axial, bending, twisting, and transverse shear deformations. Shadid et al. [19] also used beam elements to capture deformations of kerf-bending structures, and demonstrated that the beam element model can capture the dynamic response of kerf-bending structures. Zarrinmehr et al. [20] proposed a method for creating irregular Meander patterns on any polygon. They also varied the kerf patterns to control stiffness, albeit through a qualitative study.

To address the flexibility-stiffness dilemma of auxetic surfaces, this study proposes a new structural system: multi-layer auxetic surfaces with compliant mechanisms. The joints are created using compliant mechanisms with kerfs. This new type of structure allows for using panels made of traditional materials like wood, making auxetic materials accessible to a wider range of communities, from product design (e.g., lamp shade and furniture) to architectural applications (e.g., roof, shelter, and pavilion).

Section 2 describes a method to determine the positions of connections for surface overlaying using conformal mapping. Section 3 describes a design method of compliant mechanisms at the joints using kerf bending. Section 4 compares the deformation behaviors between single-layer and double-layer structures using eigenmode analysis. Section 5 validates the deployability and stiffness of the proposed structural system through physical prototyping. Section 6 is the concluding remarks, including the discussion of the applicability and limitations of this study.

## 2. Conformal flattening to determine connection positions for surface overlaying

### 2.1. Conformal flattening

Conformal mapping is a convenient technique for flattening a surface. This mapping preserves angles while changing the size across the surface (Fig. 1).

Particularly when the conformal mapping maps the 2D manifold onto the 2D plane, this mapping is called conformal flattening. When a curved surface in the 3D space is mapped by conformal flattening, the

area metrics $g_{3D}$ and $\tilde{g}_{2D}$ at a certain point before and after flattening are generally related by

$$g_{3D} = \exp(2u) \cdot \tilde{g}_{2D} \quad (1)$$

where $u$ is called the log-conformal factor that defines the area distortion at an infinitesimal region; to reconstruct the original curved surface from the flattened state, a uniaxial strain of $\exp(u) - 1$ is necessary.

Fig. 2 illustrates the effect of uniform scaling of the flattened shape onto the log-conformal factors. If the flattened shape is uniformly scaled by $s$, the conformal factors are uniformly shifted by $-\log(s)$. This way, we can obtain a flat shape that requires only expansion or contraction to restore the original curved shape.

Among a variety of conformal flattening methods, the boundary-first flattening method is one of the most popular choices because of its computation speed and versatility (e.g., the ability to specify the boundary shape and cone singularities, etc.) [21].

### 2.2. Relation of log-conformal factors and deployment angle of auxetic materials

Fig. 3 shows a closed cell of an auxetic surface with an equilateral triangle pattern; two adjacent faces are connected at their corners with a joint. The deformability at the joint allows them to rotate around the corner during deployment. This rotation mechanism causes a negative Poisson's ratio, indicating that they expand or contract in the perpendicular in-plane direction in response to expansion or contraction in one direction.

If the panels are perfectly rigid, the area expansion is associated with the deployment angle $\theta \in [0, 2\pi/3]$: an open angle between two adjacent faces. Let the completely closed state, i.e., the left state of Fig. 3, be the initial configuration with zero strain; i.e., $u = 0$. Then, $\theta$ and $u$ are related by Eq. (2):

$$2 \cos(\pi/3 - \theta/2) = \exp(u) \quad (2)$$

### 2.3. Determination of connection positions

If $u$ at the centroid of each equilateral triangle is obtained by conformal mapping, the required open angle $\theta$ can be calculated using Eq. (2). The in-plane rotation of the two auxetic layers can be locked together by connecting them such that the upper and lower faces have opposite rotational directions (Fig. 4) [15]. The positions of the holes assigned to the equilateral triangular elements are determined by rotating them by $(\theta \pm \Delta\theta)/2$ around the center of gravity of the triangle, where $\Delta\theta$ is a modification term caused by material thickness and curvature. Bolts are joined so that they pass through each hole, thereby maintaining the opening angle corresponding to the target shape.

The combined effect of thickness and curvature is illustrated in Fig. 5. Suppose a plate material with a thickness of $t$. If a point on the target shape has a single curvature of $\kappa$ and the log-conformal factor of $u$, the modified log-conformal factors of the outer and inner layers, denoted as $u_{outer}$ and $u_{inner}$ respectively, hold the following relationship:

$$\exp(2u_{outer}) = \left(1 + \frac{t\kappa}{2}\right)^2 \exp(2u) \quad (3a)$$

$$\exp(2u_{inner}) = \left(1 - \frac{t\kappa}{2}\right)^2 \exp(2u) \quad (3b)$$

The required adjustment of the deployment angle $\Delta\theta$ for the case of single curvature is described in Fig. 6. This figure suggests that the combined effect of thickness and curvature cannot be overlooked in some practical cases. For example, to create a half-cylinder with a radius of 100 mm made of auxetic panels of $t = 1$ [mm], i.e., $t\kappa = 0.01$, and $\theta = \pi/3$, the value of $\Delta\theta$ is approximately one degree, which is not negligible for accurate shape approximation.
<figure><img src="image_7.png" /><figcaption>Fig. 5. Metric change due to the combined effect of thickness *t* and single curvature *κ*. The contact surface between the two layers is the middle plane, and the uniaxial metrics of mid-planes of inner and outer layers, i.e., inner and outer mid-planes, are scaled by 1 ± *t*/*κ*/2.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 6. Required adjustment of deployment angle $\Delta\theta$ due to the combined effect of thickness and curvature. The effect becomes larger as the deployment angle $\theta$, thickness $t$, and curvature $\kappa$ increase.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7. Simple beam representation of the joint.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Fig. 8. Notations for the kerfed joint design.</figcaption></figure>

The discussion above can be generalized to doubly curved surfaces. Application of the Steiner formula [22] allows us to determine how the area metric changes under an offset $\epsilon$ in the direction of the surface normal, as described below.

Let $f$ be a smooth immersion with mean curvature $H$ and Gaussian curvature $K$. If $f^e$ is an offset surface of $f$, then the following area equation holds:

$$A(f^e) = A(f) + 2\epsilon H(f) + \epsilon^2 K(f) \quad (4)$$

Eq. (4) indicates that offsetting a smooth surface by $\epsilon$ results in the area change by a factor $1 + 2\epsilon H + \epsilon^2 K$.

Therefore, substituting $\epsilon = \pm t/2$, the log-conformal factors of two offset surfaces, denoted as $u_{\text{outer}}$ and $u_{\text{inner}}$ respectively, are expressed as

$$\exp(2u_{\text{outer}}) = \left(1 + tH + \frac{t^2}{4}K\right) \exp(2u) \quad (5a)$$

$$\exp(2u_{\text{inner}}) = \left(1 - tH + \frac{t^2}{4}K\right) \exp(2u) \quad (5b)$$

Note that the mean curvature and Gaussian curvature at a mesh vertex can be easily computed using a method such as Ref. [23].

In short, deployment angles obtained by Eq. (2) where $u$ is replaced by $u_{\text{outer}}$ or $u_{\text{inner}}$ are equivalent to $\theta \pm \Delta\theta$, which considers the combined effect of thickness and curvature and contributes to better surface approximation in actual cases.

# 3. Structural design of joints with compliant mechanism

## 3.1. Elastic flexibility constraint

The joints of auxetic surfaces must be flexible enough during the deployment process. Existing examples ensured joint flexibility relying on hyperelasticity of rubber-like materials [3], plastic deformability of materials such as metals [11] or fabrics [24], or special joint devices ensuring mechanism [25]. On the other hand, this study derives the necessary conditions for the joints to stay within their elastic limit.

Suppose a beam model, as shown in Fig. 7, with Young's modulus $E$ and the second moment of inertia $I_z = t d^3 / 12$ representing the joint. Although a geometrical relationship holds for $\theta, l$, and $d$, these values are treated independently in the following. In the classic beam theory, the curvature $\kappa$ and the bending moment $M$ in the beam hold the following relationship:

$$\kappa = \frac{M}{EI_z} \quad (6)$$

Since the deformation angle $\theta$ is the integral of the curvature over the beam length $l$, $\theta$ under the pure bending condition is expressed as

$$\theta_z = \int_0^l \frac{M}{EI_z} dx = \frac{Ml}{EI_z} \quad (7)$$

The required bending moment $M_t$ to cause the deformation angle $\theta$ is computed as

$$M_t = \frac{EI_z\theta_z}{l} \quad (8)$$

Considering that the bending stress is the bending moment over the section modulus $t d^2 / 6$, $\sigma_b$, the bending stress subject to the required bending moment $M_t$, is written as

$$\sigma_b = \frac{Ed\theta_z}{2l} \quad (9)$$

To ensure that the deployment of auxetic materials is within the elastic range, this bending moment should not exceed the material's allowable strength $\bar{\sigma}$. Therefore, the necessary condition for the non-dimensional material parameter $E/\bar{\sigma}$ and beam shape parameter $l/d$ is written as

$$\frac{l}{d} > \frac{\theta}{2} \cdot \frac{E}{\bar{\sigma}} \quad (10)$$

It is readily understood that Eq. (10) is a very strict condition. For example, even if the target deployment angle is $\pi/15$, 10% of the geometrical limit, $l/d$ must be more than 3.6 for general polymers, e.g., polyethylene, whose $E/\bar{\sigma}$ is approximately 35 [26]. The beam shape must be much more slender for more brittle materials like wood or medium-density fiberboard (MDF).

## 3.2. Flexible joints through kerfs

To augment their flexibility, this study develops “kerf joints”, a small part of the structure that connects two elements with a compliant mechanism produced by kerfing. Although the same system was already presented in our previous work [27], this study further investigates the mechanical properties of the kerf joints in more detail.

The parameters for kerfing are described in Fig. 8. The kerf bending parameters consist of the depth of kerf bending part $h$, the kerf interval $b$, and the number of intervals $n$. The compressive zone of bending caused by the deployment contains wedge-shaped cuts, which reduces extra forces due to contact up to half of the limit deployment angle: $\theta_z = \pi/3$.

For simplicity, we set the following assumption:

* $h$ is larger enough than $b$ so that the kerfed joint is modeled as a set of linear beam elements along the load path.

* The kerf joint endpoints are subject to in-plane bending moment $\Gamma$ only, and no other external forces are considered.

* The total length of beam elements corresponding to the kerf joint is $2nh + (2n - 1)b$, which is slightly shorter than the actual line length of the center load path.

The in-plane deformation angle $\theta_z$ of a beam subject to in-plane bending moment is generally computed as

$$\theta_z = \oint \frac{M_z(s) \bar{M}_z(s)}{EI_z} ds \quad (11)$$

Substituting $M_z(s) = \Gamma$, $\bar{M}_z(s) = 1$, and $I_z = b^3 t / 12$ to Eq. (11) leads to

$$\theta_z = \frac{24nh + (24n - 12)b\Gamma}{Eb^3t} \quad (12)$$

Since $\theta_z \in [0, 2\pi/3]$ is a known value determined from a conformal map of the target surface, the maximum bending stress is computed as

$$\sigma_b = \frac{1}{4n \frac{h}{b} + 4n - 2} E \theta_z \quad (13)$$

Similar to Eq. (10), to stay the material within the elastic range, the shape parameters and the material properties can be related as follows [27]:

$$4n \frac{h}{b} + 4n - 2 > \theta \cdot \frac{E}{\bar{\sigma}} \quad (14)$$
<figure><img src="image_11.png" /><figcaption>Fig. 9. Auxetic material made of paper with slits. The deployment results in a rugged shape, which implies that too-thin auxetic materials cannot achieve and maintain an intended shape due to low torsional and out-of-plane bending stiffnesses at the joints.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. Notations for the stiffness constraint of box-shaped kerfed material.</figcaption></figure>

This inequality constraint can be satisfied in realistic scenarios. Polymer materials become feasible for realizing elastic auxetic materials owing to kerf joints without strict constraints on the kerf parameters *n* and *h/b*. Although using wood materials might be more challenging than using polymers, increasing the value of *n* makes wood materials feasible for the joint design. Similarly to the inequality condition (10), using a target surface with low curvatures, i.e., small *θ*, can relax the flexibility constraint (14).

Since both sides of Eq. (14) are dimensionless, the shape requirement for the kerf joints remains unchanged under uniform scaling. Thus, once the in-plane deployability of a panel with a designed cut pattern is verified using a small model, the cut pattern can be scaled up for larger structural applications.

### 3.3. Stiffness constraint

To approximate the target surface shape accurately, the torsional and out-of-plane bending stiffnesses at the joints should be high enough compared to the in-plane stiffness. Otherwise, the deployment results in a rugged shape as shown in Fig. 9 without any correspondence with the target surface, as reported in Ref. [28].

This subsection derives the stiffness constraint at the kerf joints, which ensures sufficient relative torsional and out-of-plane stiffnesses to the in-plane rotational stiffness. Fig. 10 illustrates the box-shaped kerfed material used in this discussion.

Based on the unit force theorem, the deformation angles, $\theta_x$, $\theta_y$, and $\theta_z$, in the global coordinate system are

$$ \theta_x = \oint \frac{M_y(s) \bar{M}_y(s)}{EI_y} ds + \oint \frac{M_x(s) \bar{M}_x(s)}{GJ} ds \quad (15) $$

$$ \theta_y = \oint \frac{M_y(s) \bar{M}_y(s)}{EI_y} ds + \oint \frac{M_x(s) \bar{M}_x(s)}{GJ} ds \quad (16) $$

$$ \theta_z = \oint \frac{M_z(s) \bar{M}_z(s)}{EI_z} ds \quad (17) $$

Note that the moments $M_x$, $M_y$, $M_z$, $\bar{M}_x$, $\bar{M}_y$, $\bar{M}_z$ are computed in the local axes; bending behaviors around the local x, y, and z-axes correspond to torsion, out-of-plane bending, and in-plane bending, respectively. When applying a bending moment in the direction of deformation angle with the same magnitude $\Gamma$, Eqs. (15)–(17) are rewritten as

$$ \theta_x = \frac{nh\Gamma}{EI_y} + \frac{(n-1)b\Gamma}{GJ} \quad (18) $$

$$ \theta_y = \frac{(n-1)b\Gamma}{EI_y} + \frac{nh\Gamma}{GJ} \quad (19) $$

$$ \theta_z = \frac{nh\Gamma + (n-1)b\Gamma}{EI_z} \quad (20) $$

To ensure that the in-plane deformation is dominant during deployment, the flexibility ratios $\theta_z/\theta_x$ and $\theta_z/\theta_y$ must be much larger than 1. From thickness *t* and the interval between kerfs *b*, the cross-sectional properties are determined as

$$ I_y = bt^3/12 \quad (21) $$

$$ I_z = b^3 t/12 \quad (22) $$

$$ J = \alpha \max\{b, t\} (\min\{b, t\})^3 \quad (23) $$

$$ \alpha = \frac{1}{3} - 0.21 \frac{\min\{b, t\}}{\max\{b, t\}} \left( 1 - \frac{(\min\{b, t\})^4}{12(\max\{b, t\})^4} \right) \quad (24) $$

The shear modulus *G* and Young's modulus *E* can be related with Poisson's ratio *v*

$$ G = \frac{E}{2(1 + v)} \quad (25) $$

Substituting Eqs. (21)–(25), the flexibility ratios are expressed as

$$ \frac{\theta_z}{\theta_x} = \begin{cases} \frac{\frac{h}{b} + \frac{(n-1)}{n}}{\frac{bh}{t^2} + \frac{(n-1)(1+v)}{6na}} & \text{if } t > b \\ \frac{\frac{h}{b} + \frac{(n-1)}{n}}{\frac{bh}{t^2} + \frac{(n-1)(1+v)b^2}{6nat^2}} & \text{else} \end{cases} \quad (26) $$

$$ \frac{\theta_z}{\theta_y} = \begin{cases} \frac{\frac{b}{h} + \frac{n}{(n-1)}}{\frac{b^3}{ht^2} + \frac{n(1+v)}{6(n-1)a}} & \text{if } t > b \\ \frac{\frac{b}{h} + \frac{n}{(n-1)}}{\frac{b^3}{ht^2} + \frac{n(1+v)b^2}{6(n-1)at^2}} & \text{else} \end{cases} \quad (27) $$

Figs. 11 and 12 describe the values of $\theta_z/\theta_x$ and $\theta_z/\theta_y$ varying *t/b*, *h/b*, and *n*, respectively. From these figures, we can draw the following observations:

* The thickness *t* should be larger than the kerf interval *b* to make the in-plane deformation more dominant than torsional and out-of-plane bending deformations.
<figure><img src="image_13.png" /><figcaption>Fig. 11. Parametric study on the value of $\theta_x/\theta_s$. (a): $v = 0.0$, (b): $v = 0.25$, (c): $v = 0.5$.</figcaption></figure>

· Increasing the values of *h/b* and *n* reduces torsional deformation but increases out-of-plane bending deformation.

· A material with a low Poisson's ratio is preferable in order to constrain torsion and out-of-plane bending of kerf joints.

Since the discussion above is made based on dimensionless parameters, the relationship between stiffness and kerf shape is invariant under uniform scaling. However, as the size of the structure increases, the influence of self-weight cannot be ignored. For large-scale structures, it is necessary to separately evaluate how much the structure deflects due to its own weight and whether that deflection falls within acceptable limits.

## 4. Effect of surface overlaying

Fig. 13 illustrates the unit models for evaluation of the stiffening effect of overlaying. These models do not include kerf joints and hold

their open shape without stretching. In the double-layer model, fixing the overlayed shape is modeled by kinematic coupling of three nodes on each pair of faces.

In Fig. 14, we can observe a clear difference in the in-plane eigenmodes between the single-layer and double-layer unit cells. The eigenmodes reveal that the single-layer structure deforms by changing its open angles, whereas the double-layer structure retains its open angles after deformation.

As a result, double-layer auxetic surfaces will transfer loads through membrane forces. Furthermore, considering that the shape of an auxetic surface is greatly influenced by open angles, double-layer auxetic surfaces are expected to possess improved stiffness for maintaining the deployed shape.

Fig. 15 illustrates our analysis of the out-of-plane eigenmodes. In the double-layer case, the desired dome-like and saddle-like eigenmodes are distinctly isolated. This suggests that the double-layer structure
<figure><img src="image_14.png" /><figcaption>Fig. 12. Parametric study on the value of $\theta_x/\theta_y$. (a): $v = 0.0$, (b): $v = 0.25$, (c): $v = 0.5$.</figcaption></figure>

facilitates smooth, doubly-curved deformations, mitigating ruggedness during deployment and service phases.

## 5. Physical model validation

Owing to the flexible compliant mechanism at the joints, we successfully deployed an auxetic surface structure made of an MDF panel ($t = 3$ [mm]) into a spherical shape, as shown in Figs. 16 and 17. In this example, the target shape is a spherical cap with a height of 56.2 mm and a boundary circle with a diameter of approximately 226 mm. The auxetic materials are composed of equilateral triangles of 45 mm in size. The circular boundary shape is enforced by fitting the model into a circular cut-out MDF panel, and no fixing devices are used on the boundary.

In the single-layer case, the surface only reaches a maximum height of 22 mm, which is quite far from the target height. Meanwhile, we achieve much better accuracy with the double-layer structure attached

with M1.2 bolts, as the middle plane reaches a maximum height of approximately 56 mm, very close to our target. This result shows that the double-layer structure provides a much more accurate approximation of the target shape.

The mechanical behavior of the auxetic surfaces is investigated using the system shown in Fig. 18. A center part of the auxetic surfaces is loaded by a robot arm equipped with a 3D-printed cone attachment made of acrylonitrile butadiene styrene (ABS). The deformed shape is observed by measuring the position of reflective markers applied across the auxetic surface. The force at the tip of the robot arm is measured by a force sensor. The robot arm enforces the downward displacement of 10 mm from the initial state, i.e., Fig. 16.

Fig. 19 shows the difference of the history of captured forces between single-layer and double-layer structures. Although the double-layer structure uses twice as much MDF material as the single-layer structure, significant improvement in stiffness is observed. Since the single-layer structure maintains its convex shape through contact with
<figure><img src="image_15.png" /><figcaption>Fig. 13. Unit models for comparing deformation behaviors. The material is assumed to be a polymer with Young's modulus $E = 2800$ [N/mm²] and Poisson's ratio $v = 0.35$.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 14. Low-lying in-plane eigenmodes of a unit cell, where those corresponding to rigid-body motion are disregarded. $\lambda_1, ..., \lambda_5$ are the eigenvalues of these modes, with $\lambda_1$ normalized. The open angle between adjacent faces changes significantly for the single-layer case, but is relatively maintained for the double-layer case.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 15. Low-lying out-of-plane eigenmodes of a unit cell, where those corresponding to rigid-body motion are disregarded. For the single-layer case, the dome-like shape (Mode 3) suffers from a lack of clear dominance due to contributions from undesired twisting motions at the joints (Modes 1, 2, 4 and 5). On the other hand, for the double-layer case, the dome-like deformation (Mode 1) and the saddle-like deformations (Modes 2 and 3) are dominant.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>**Fig. 16.** Difference of the maximum height between single layer (left) and double layer (right) cases. A 10 mm downward prescribed displacement was applied to each loaded point.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>**Fig. 17.** Deployment simulation using the method of Konaković et al. [2] (left) and physical model (right).</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Fig. 18.** Experiment setup. The auxetic surfaces are loaded by a robot arm equipped with a cone-shaped attachment at the tip. The motion tracking system captures the displacements. The force sensor measures the forces at the tip of the attachment.</figcaption></figure>

adjacent elements, friction is dominant in resisting external forces. A sudden decrease in the applied force is observed in the first half of the loading process, probably because some contact parts are subject to forces beyond the static friction limit.

Fig. 20 illustrates the interpolated displacements over the auxetic. Radial basis function (RBF) interpolation is used here, and we use a Gaussian function $\exp(-(\beta r)^2)$ with $\beta = 10^{-1.4}$ as the RBF. In response

to the external load, the center part of the single-layer auxetic surface horizontally pushes the adjacent elements. Consequently, a more spread displacement field is observed for the single-layer case, which might be another cause of poor surface approximation accuracy. On the other hand, the displacement field is more concentrated around the loaded point for the double-layer case, demonstrating its stable performance in maintaining the initial shape.

## 6. Conclusion

This study tackled the conflicting design requirement for auxetic surfaces intended for use as deployable structures, specifically the flexibility-stiffness dilemma. To augment deployment flexibility, we developed a special compliant mechanism at the joints. Simultaneously, in order to uphold and stiffen the deployed state, we overlayed two surfaces to offset their in-plane rotation.

In addition to elucidating the strict elastic flexibility constraint for the joints, we determined the stiffness of the compliant mechanism induced by kerf bending. Parametric studies on rotational flexibility in three directions revealed that the shape parameters for compliant mechanisms should be carefully chosen to ensure dominant in-plane rotation during deployment.

For the deployment of a flat auxetic surface to a target shape, we calculated the required rotation angles for each face using conformal geometry processing. In this calculation, we accounted for the combined effect of thickness and curvature and analytically demonstrated that this effect is not negligible in most practical design scenarios. The computed rotation angles were utilized to determine the connection positions for surface overlaying.

The case studies showed that combining the two strategies effectively resolves the dilemma. Kerf joints transform MDF from a
<figure><img src="image_21.png" /><figcaption>**Fig. 19.** History of captured forces at the loaded point. The z-direction is vertical, and the x and y-directions are horizontal. The double-layer structure required approximately three times more load to cause the same displacement, demonstrating its stiffness.</figcaption></figure>

challenging material for auxetic surfaces to a viable option by relaxing the required material constraints. With its capacity to transmit load without significant open angle changes, the double-layer auxetic surface demonstrates a strong stabilization effect to achieve and maintain the target shape.

While many investigations to uncover the mechanical property were conducted, some technical challenges remain for practical application. Firstly, the flexibility-stiffness dilemma becomes more challenging for large-scale structures, because the required deployment force and stiffness generally increase as the structural scale increases. Secondly, the joints are weak points in the structure and risk breaking due to stress concentration from self-weight or external loads. Particularly for the service phase of the structure, the kerf joints need to be designed to withstand various static and dynamic loads. Thirdly, although we analytically derived the stiffness of the kerf-bending joint, it remains challenging to balance its in-plane auxetic motion, out-of-plane bending motion, and twisting motion under practical loading scenarios.

To address these limitations, future studies will involve optimization of kerf shapes to better compromise between the flexibility-stiffness dilemma, evaluation of the fatigue property and strength, and detailed parametric study and physical experiment of kerf bending joints to understand their actual effect on the deformation behavior.

### CRediT authorship contribution statement

**Kazuki Hayashi:** Writing – original draft, Visualization, Validation, Methodology, Investigation, Conceptualization.
**Romain Mesnil:** Writing – review & editing, Methodology, Conceptualization.

### Declaration of competing interest

The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: Kazuki Hayashi reports financial support was provided by The Kajima Foundation. Kazuki Hayashi reports financial support was provided by Japan Society for the Promotion of Science. Kazuki Hayashi reports financial support was provided by Japan Science and Technology Agency. If there are other authors, they declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

This research is supported by The Kajima Foundation Long-term Research Program, Japan Society for the Promotion of Science (JSPS) Grant No. 24K17392 and The Japan Science and Technology Agency (JST), Core Research for Evolutionary Science and Technology (CREST) Grant No. JPMJCR1911. We would like to thank Krittika Walia for setting up the experiment environment.

### Data availability

Data will be made available on request.
<figure><img src="image_22.png" /><figcaption>Fig. 20. Interpolated displacement colormap around when the maximum downward enforced displacement (10 mm) is applied.</figcaption></figure>

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