Second-Order Kinematic Invariants for the Design of Compliant Auxetic Symmetrical Structures

Abstract

Auxetic structures have great potential in modern engineering, and their design represents an emerging field in industrial applications. The accurate synthesis of the elements of such structures requires multidisciplinary approaches that combine kinematics and structural mechanics. Design methodologies are often based on complex time-consuming numerical methods and with considerable computational burden for exploring a large set of alternatives. The aim of the present work is to propose a novel method for designing symmetrical auxetic structures based on the use of pseudo-rigid mechanisms that can reproduce their nonlinear elasto-kinematic behavior with a limited set of parameters. For the definition of these pseudo-rigid mechanisms, the theory of kinematic invariants is proposed. It allows for the deduction of surrogate rigid-link mechanisms with a simpler structure but remarkable accuracy. This approach is an emerging method employed in the generic synthesis and analysis of compliant mechanisms, and, in this study, it is extended for the first time to support the design of auxetic structures. This paper describes the analytical process to deduce the design equations and discusses the example of an application to a symmetrical re-entrant structure, comparing the results with those of a numerical flexible multibody model, a finite element model, and with experimental tests. All the comparisons demonstrate the considerable potential of the proposed methodology, which can also be adapted to other types of auxetic structures.

1. Introduction

Auxetic structures are interesting components that exhibit unconventional deformation behavior when subjected to mechanical stresses and strains [1,2,3]. Unlike typical materials, when stretched longitudinally, auxetic structures thicken in one or more perpendicular, width-wise directions. Conversely, under uniaxial compression, they exhibit thinning in transverse directions. This counterintuitive behavior is a characteristic of auxetic embodiments. The concept of auxetic behavior was recognized at the beginning of the 20th century. However, intentional design and manufacturing of auxetic foams began in the early 1980s, marking a milestone in material science. Typically, auxetic structures mimic materials with a negative Poisson’s ratio. The Poisson’s ratio, defined as the negative ratio of transverse to longitudinal strain, quantifies how a material deforms laterally when stressed axially.

Auxetic structures are commonly referred to as auxetic materials; however, strictly speaking, it is not the material that has peculiar elastic characteristics but rather the macroscopic behavior of the resulting structure. Auxetic effects can be engineered into materials by modifying their internal microstructure. This involves accurate geometric adjustments to ensure that the structure deforms synergistically under the mechanical load or imposed displacement. The result is a range of counterintuitive deformation behaviors, such as thickening upon stretching, thinning upon compression, or even volume expansion under tensile stress [4,5]. These effects are particularly evident in macroscopic structures and can be replicated in microstructures, such as foams or bulk polymers. Thanks to material synthesis and scalability advances, auxetic structures have been applied across several industrial fields, including military, biomedical, aerospace, and textiles. Researchers have successfully adapted a wide variety of material types—polymers, metals, ceramics, and composites—to exhibit auxetic properties. Over the last 30 years, methods developed in academic and technical research centers have been effectively scaled up for industrial applications, making these materials widely accessible.

Research in auxetic structures is mainly focused on experimental studies [6,7,8,9,10] or, in general, on investigations by analytical or numerical methods of the elastic behavior of their implementations. The analytical approaches for studying these structures are mainly focused on the use of the beam theory [11,12,13,14], decomposing the auxetic structure into a set of beams connected to each other. Most numerical studies, however, make use of the well-known finite element theory [14,15,16,17]. In this perspective, auxetic structures have often been the subject of topological optimizations using numerical methods combined with finite element models [18,19,20,21,22]. In the scientific literature, there are also studies of more original approaches, such as that of elastic networks [23] or localized flexible joints [24,25,26]. Other studies, on the other hand, aim to treat auxetic embodiments as cellular or lattice structures [27,28] and, therefore, employ homogenization methods aimed at determining the anisotropic elastic characteristics of a single cell to then be able to calculate the elastic behavior of a structure composed of an ordered and conformal repetition of these units [29,30].

It is evident from an analysis of the scientific literature that the research is mainly focused on problems of the analysis of auxetic structures, while the contributions on the geometrical synthesis, or even on the full design starting from functional requirements, are limited. One of the main difficulties is the need to apply multidisciplinary approaches to the simultaneous study of kinematic and structural behavior, due to the elasto-kinematic behavior of these structures. To further complicate the investigations, there are aspects of geometric nonlinearity of the deformations that often occur in auxetic embodiments and cannot be neglected [31,32]. From this point of view, methodologies of the investigation of the compliant mechanisms seem to be very suitable for supporting the synthesis and design since the behavior of a single cell is very similar to that of a compliant mechanism [33]. These mechanisms are well known in the literature for obtaining their mobility from the localized or distributed deformation of their components rather than from the relative motion concentrated at kinematic constraints [34].

Among the different approaches for the study of the behavior of compliant mechanisms, one of the most widely used is the definition of pseudo-rigid mechanisms. These surrogate mechanisms replicate the kinematic characteristics of displacement and forces using canonical rigid-body mechanisms that are properly synthesized [35]. In this way, it is possible to use the classical methodologies of rigid-body mechanism synthesis and design, extending them to flexible structures. The main problem is that the use of pseudo-rigid mechanisms often offers excellent approximations of the elasto-kinematic behavior in the range of small displacements, which may be fine for microstructures but is too approximate for macroscopic structures that are subjected to larger displacements [36]. The approximation of the behavior of a compliant mechanism subjected to large displacements would require a pseudo-rigid mechanism with many degrees of freedom, complicating the investigations [37]. To solve this problem, it has been proposed to adopt synthesized pseudo-rigid mechanisms starting from the study of higher-order kinematic invariants of the relative motion of bodies connected by flexure hinges [38,39,40,41]. These studies have brought the following to the definition of pseudo-rigid mechanisms: a limited number of degrees of freedom, able to approximate with sufficient accuracy, and the elasto-kinematic behavior of different types of embodiments [42,43,44,45].

Starting from this background, the aim of this work is to extend and apply the methodologies for the study of compliant mechanisms, adopting pseudo-rigid models deduced from the equivalence of the second-order kinematic invariants to the auxetic structures. The idea is to explore and propose a novel methodology for the analysis and, in particular, the synthesis of symmetrical auxetic structures composed of beam cells. This study focuses specifically on the re-entrant auxetic structures, which represent one of the simplest but most widely used layouts [5]. The objective is to deduce the design equations for supporting the selection of the dimensions of the compliant cells to be repeated in the global structure. The design starts from the choice of functional requirements, such as the ratio between longitudinal and transverse displacements, which is the main characteristic of auxetic structures. In this context, applying compliant mechanism methodologies can be of great help in defining a procedure suitable for the synthesis and design of auxetic structures undergoing large deflections.

This paper is organized as follows: In the first part, a brief recall of the pseudo-rigid modeling of compliant flexure hinge is presented to have a self-contained paper. In the second part, the second-order modeling of a generic re-entrant auxetic cell is described, the kinematic closed-loop equations are deduced, and the design procedure is presented. An example of application is discussed in the third part, and the proposed methodology is verified against a flexible multibody numerical approach, a finite element model, and an experimental campaign. The last part is dedicated to the discussion and conclusions.

2. Recall the Second-Order Pseudo-Rigid Model for a Nonlinear Straight Beam

In 2017, Valentini and Pennestrì [42] deduced the implementation of an advanced pseudo-rigid model capable of replicating the elasto-kinematic behavior of a flexible hinge composed of a thin beam connecting two massive bodies subjected to a pure bending moment. The model was deduced by observing that the diameter of the inflection circle [46,47] of the relative motion between the connected bodies can be considered constant for a wide range of deflection. Thanks to this finding, it is possible to consider a single-degree-of-freedom pseudo-rigid model that is composed of two bodies with two equal circles, each one fixed to one of the connected bodies, constrained to roll without slipping one on the other (Figure 1).

This means that one can use a pseudo-rigid model with a single degree of freedom for describing the nonlinear relative displacement of the connected bodies. This is a very relevant simplification for two main reasons. The first reason is that a single degree of freedom mechanism means a less complex structure and fewer equations to be solved. The second reason is that a single degree of freedom can lead to a completely determinate system for which the deformation following an imposed displacement can be determined without solving the elastic equilibrium equations but only by imposing the closure-loop equations of a generic rigid-body mechanism. The advanced pseudo-rigid mechanism can be implemented as an epicyclic arrangement as if the circles were gears meshing with each other. In the initial configuration of the undeformed beam, the circles touch each other at the midpoint of the beam itself, respecting the symmetry of the structure. The radius $r_c$ of these circles is proportional to the length $l$ of the flexible segment and can be approximated by the formula [42]:

$$r_c={\displaystyle \frac{\pi }{10}}l$$

(1)

In order to preserve the elastic compliance of the flexure hinge, the two bodies need to be connected by a torsion spring of stiffness $k$ [42]:

$$k={\displaystyle \frac{EI}{l}}$$

(2)

where $E$ is the Young’s modulus of the material, and $I$ the moment of inertia of the cross-section with respect to the deflection axis.

It is interesting to note that the structure of the pseudo-rigid mechanism does not depend on the structural or stiffness characteristics but only on the geometric features. The structural characteristics only influence the stiffness of the torsional elastic element. The model proposed by Valentini and Pennestrì has been used for the solution of several mechanical problems, demonstrating its validity even in the case of a non-pure, but still predominant, applied moment [48,49].

3. Second-Order Pseudo-Rigid Model of a Compliant Cell

The objective of the pseudo-rigid model of a compliant structure is the ability to replicate its deformation and force reaction also when subjected to large displacements using a reduced set of parameters and more straightforward equations.

Observing the structure of the cell of a re-entrant auxetic structure (Figure 2), it is well known from both experimental observations and numerical models available in the literature [5] that the deformation is localized in the four diagonal beams. The horizontal beams are considered the input and output ports where force and displacement are imposed or retrieved and are connected to the other cells of the structure. In the case of symmetrical loading (the most common case of application), the four beams deform symmetrically; therefore, the pseudo-rigid model’s definition may benefit from this simplification. As discussed in the previous section, the pseudo-rigid model of leaf-type slender structures can be built with an epicyclic embodiment as a very good tradeoff between simplicity (a single degree of freedom mechanism) and accuracy (the ability to reproduce very large deflections).

The pseudo-rigid epicyclic arrangement has the great advantage of reproducing the relative motion between the two ends considering a single degree of freedom (the rotation between the two circles) but has the limitation to be used only when the curvature of the beam has the same concavity (i.e., the midline of the beam does not have inflection points). Experimental observations and numerical models show that the diagonal beams of the re-entrant structure experience an inflection when the input ports are loaded (Figure 2, on the right). This means that we must complicate the arrangement by including two epicyclic arrangements in the series for each beam, thus considering each beam as split in two halves. With reference to Figure 3, a single beam is therefore split into three rigid parts connected by two pairs of rolling-without-slipping circles. Two torsion springs act between each couple of adjacent bodies. Due to the symmetric arrangement of the structure, endpoint A is constrained to translate along the y axis, and point C is constrained to translate along the x axis. For the same reason of symmetry, the rotations of the first and last segments are constrained as well. With this assumption, we can proceed to the computation of the degrees of freedom $f$ of the entire pseudo-rigid mechanism according to the Grübler’s count:

$$f=3n_b-2j_l-2j_g=3\cdot 3-2\cdot 2-2\cdot 2=1$$

(3)

where

$n_b$ is the number of moving bodies;

$j_l$ is the number of lower kinematic pairs;

$j_g$ is the number of rotational couplings (the rolling-without-slipping constraints).

Figure 3. Modeling a diagonal beam with a 3-link epicyclic arrangement with two pairs of rolling without slipping circles.

Figure 3. Modeling a diagonal beam with a 3-link epicyclic arrangement with two pairs of rolling without slipping circles.

Having a single degree of freedom means that the configuration of the pseudo-rigid mechanism is uniquely defined once the displacement of one of the input ports is assigned. This property is of extreme importance as it allows the displacements of the flexible structure to be determined using kinematic information only without having to build up and solve the elastica equations involving structural compliance. In other words, given an imposed displacement at the input port, the deflection of the beams and the displacement at the output port do not depend on the structural stiffness (cross-section geometry and material properties) and, therefore, do not depend on the presence of the torsion springs. This observation simplifies the procedure of synthesis and design of the auxetic structure, splitting the kinematics from the statics. On the other hand, the structural stiffness influences only the force to be applied to ensure the prescribed displacement but not the displacement itself.

With this important premise, the relationship between the input and output ports of the compliant cell can be studied using the mechanism closure loop equations for canonical rigid-link mechanisms. With reference to Figure 4, let us consider a symmetrical re-entrant cell with height $h$ and diagonal beam inclination $\alpha$. Due to symmetry, as already discussed, we can study the behavior of the cell focusing only on a quarter and imposing boundary constraints. Let us also prescribe a vertical displacement $\delta$ at input point A that produces a horizontal resulting displacement $\gamma$ at output point C. The closure loop equations can be written as follows:

$$A{A}_0+A_{0}D+D{D}_0+D_0{C}_0+C_{0}C+CA=0$$

(4)

The moduli of the vectors can be computed from the undeformed geometry of the cell as follows:

$$\begin{array}{l}‖A{A}_0‖=‖C_{0}C‖={\displaystyle \frac{l}{2}}-r_c=\left({\displaystyle \frac{5-\pi }{10}}\right)l=al\\ ‖A_{0}D‖=‖D_0{C}_0‖=2r_c={\displaystyle \frac{\pi }{5}}l=bl\\ ‖D{D}_0‖=l-2r_c=\left({\displaystyle \frac{5-\pi }{5}}\right)l=2al\end{array}$$

(5)

where

$L={\displaystyle \frac{2h}{\sin \alpha }}$ is the entire length of the diagonal beam;

$l={\displaystyle \frac{L}{2}}$ is the half-length of the diagonal beam;

$r_c={\displaystyle \frac{\pi }{10}}l$ is the radius of each circle of the epicyclic arrangements that is related to the length of the corresponding link, according to the findings in [42].

The vector Equation (4) can be projected along x and y axes and, taking into account Equation (5), becomes two scalar relationships:

$$\begin{array}{l}al\cos {\vartheta }_1+bl\cos {\vartheta }_{c1}+2al\cos {\vartheta }_2+bl\cos {\vartheta }_{c1}+al\cos {\vartheta }_3-\left(L\cos \alpha +\gamma \right)=0\\ al\sin {\vartheta }_1+bl\sin {\vartheta }_{c1}+2al\sin {\vartheta }_2+bl\sin {\vartheta }_{c1}+al\sin {\vartheta }_3-\left(L\sin \alpha -\delta \right)=0\end{array}$$

(6)

Considering the geometrical constraints due to symmetry, we can deduce the following:

$$\cos {\vartheta }_1=\cos {\vartheta }_3=\cos \alpha$$

(7)

The rolling-without-slipping condition between each of the two pairs of circles in contact can be enforced using the well-known Willis’ equation for ordinary gearing with a gear ratio equal to −1 (external gears):

$$\begin{array}{l}\left({\vartheta }_1-{\vartheta }_{c1}\right)=-\left({\vartheta }_2-{\vartheta }_{c1}\right)\\ \left({\vartheta }_2-{\vartheta }_{c2}\right)=-\left({\vartheta }_3-{\vartheta }_{c2}\right)\end{array}$$

(8)

By taking into account Equation (7), Equation (8) can be simplified as follows:

$${\vartheta }_{c1}={\vartheta }_{c2}={\displaystyle \frac{\alpha +{\vartheta }_2}{2}}$$

Therefore, Equation (6) can be rewritten in a compact form as follows:

$$\begin{array}{l}aL\cos \alpha +bL\cos {\displaystyle \frac{\alpha +{\vartheta }_2}{2}}+aL\cos {\vartheta }_2-\left(L\cos \alpha +\gamma \right)=0\\ aL\sin \alpha +bL\sin {\displaystyle \frac{\alpha +{\vartheta }_2}{2}}+aL\sin {\vartheta }_2-\left(L\sin \alpha -\delta \right)=0\end{array}$$

(9)

In practical applications, since the functionality of the cell is the achievement of auxetic behavior, we can assume the auxetic behavior $aux$ as the ratio between the output and input displacements:

$$aux={\displaystyle \frac{\gamma }{\delta }}$$

(10)

By considering an input displacement as a fraction of the height of the cell (i.e., $\delta =\kappa h$), Equation (9) becomes the following:

$$\begin{array}{l}a\cos \alpha +b\cos {\displaystyle \frac{\alpha +{\vartheta }_2}{2}}+a\cos {\vartheta }_2-\left(\cos \alpha +{\displaystyle \frac{\kappa }{2}}\cdot aux\cdot \sin \alpha \right)=0\\ a\sin \alpha +b\sin {\displaystyle \frac{\alpha +{\vartheta }_2}{2}}+a\sin {\vartheta }_2-\left(1-{\displaystyle \frac{\kappa }{2}}\right)\sin \alpha =0\end{array}$$

(11)

which represents a system of two scalar equations with two scalar unknowns, ${\vartheta }_2$ and $\alpha$. From the kinematic design point of view, the variable of interest is the angle $\alpha$, which is a geometrical parameter of the re-entrant structure.

By the observation of Equation (11), it is important to underline that the solution of the system (and, therefore, the configuration of the pseudo-rigid assembly) depends on the value of the chosen parameter $\kappa$, and the auxetic characteristic actually depends on the amplitude of the imposed displacement. It is due to the intrinsic geometrical nonlinearity of the assembly, which undergoes large displacement. Equation (11) is provided in the actual form without simplification or linearization in order to benefit from the description of the elasto-kinematic behavior using second-order pseudo-rigid embodiments.

After the computation of the elasto-kinematic congruent solution, it is possible to move to the structural design, estimating force $F_{\delta }$, which is needed to produce the prescribed displacement $\delta$. We can impose that the virtual work of this force for the prescribed displacement is equal to the elastic energy stored by the two torsion springs acting between links $A{A}_0$ and $D{D}_0$ and between links $D{D}_0$ and $C_{0}C$, respectively:

$$F_{\delta }\cdot \delta =2k{\left({\vartheta }_2-\alpha \right)}^2$$

(12)

where, according to Equation (2), the stiffness coefficient of the springs is the same and equal to the following:

$$k={\displaystyle \frac{EI}{l}}$$

(13)

Solving Equation (12) for $F_{\delta }$ and taking into account Equation (13), we obtain the following:

$$F_{\delta }={\displaystyle \frac{2EI\sin \alpha {\left({\vartheta }_2-\alpha \right)}^2}{h\delta }}$$

(14)

4. Design Procedure and an Example of Application

With reference to Figure 5, which shows a flow chart of the proposed design of the auxetic cell, the design procedure of the re-entrant cell can be split between the elasto-kinematic part and the structural part. The kinematic design starts with the knowledge of the design constraints (i.e., the height of the cell $h$) and the design requirements (the auxetic behavior $aux$ and the nominal input displacement $\delta$ at which the desired auxetic behavior is expected). Then, by solving Equation (11), it is possible to find the value of $\alpha$, the length $l$ of the beams, and the angle ${\vartheta }_2$ of the pseudo-rigid intermediate links, completing the synthesis. After the elasto-kinematic synthesis, the design moves to the structural part in which the cross-section geometries and material properties can be chosen in order to fulfill the desired stiffness and resistance requirements. They may be chosen as design assumptions. The amplitude of the required force for producing the prescribed vertical displacement can be assessed using Equation (14), where the values $\alpha$ and ${\vartheta }_2$ are the output variables of the elasto-kinematic design. Also, the structural design may be accomplished by prescribing the required force and finding the cross-section dimensions according to Equation (14). Having decoupled the kinematic design from the structural one allows this double independent approach, which can be chosen according to the cases of interest.

The last step of the design is the structural verification of the integrity of the structure that can be achieved by computing the mechanical stress at the maximum displacement and comparing to the fatigue limit of the material.

It is important to underline that the design of auxetic structures can include one or more optimization phases, with a single- or multi-objective function. However, the aim of the paper is to show how it is possible to use the advanced pseudo-rigid mechanisms that, in recent years, have been used in the synthesis and analysis of compliant mechanisms, even for auxetic structures. Therefore, optimization phases are not included in the design procedure that is deliberately simplified, but the proposed methodology can also be successfully used in such approaches.

In order to show the potential and the straightforwardness of the proposed simplified design methodology, the following section presents an example of the synthesis of a compliant re-entrant symmetrical auxetic cell given the ratio between the input and output displacements as follows:

$$aux={\displaystyle \frac{\gamma }{\delta }}=1.06$$

(15)

It means that, given an imposed overall vertical negative displacement of $\delta$ (a compression), the output port should have a negative displacement of $\gamma =1.06\cdot \delta$ (a contraction). The auxetic behavior should be ensured when applying a displacement of $\delta =0.05h$. The example has been chosen for its significant auxetic behavior and provides the opportunity to check the accuracy in a large nonlinear case. In particular, with reference to

Table 1. Model parameters of the designed auxetic re-entrant cell.

Parameter	Symbol	Value	Design/Assumption
height of the cell	$h$	20.0 mm	assumption
auxetic ratio	$aux$	1.06	assumption
amplitude motion ratio	$\kappa$	0.2	assumption
angle of the beams	$\alpha$	1.03 rad	design result
length of the beams	$l$	11.7 mm	design result
section shape of the beams	-	rectangular	assumption
thickness of the beams	$t$	0.8 mm	assumption
transverse length of the beams	$q$	10 mm	assumption
Young’s modulus	$E$	1.8 GPa	assumption
Poisson’s ratio	$\nu$	0.37	assumption
force to be applied	$F_{\delta }$	14.57 N	design result

, which lists all the assumed and computed parameters, we assume the height $h$ of the entire cell as 20 mm. The solution of Equation (11) gives $\alpha =1.03$ rad and a ${\vartheta }_2=0.52$ rad. The evaluation of Equation (14) gives $F_{\delta }=14.57$ N.

5. Verification

The accuracy of the proposed model has been verified using three different methods. The first one is an alternative numerical model based on the use of the discrete flexible modeling approach to take into account the nonlinear elasto-kinematic behavior of the flexible beams of the cell. This approach is often used for simulating the behavior of slender structures due to an appropriate trade-off between simplicity and accuracy [50]. The second one is a nonlinear finite element model developed using Timoshenko beams, and the third method is an experimental verification using a 3D-printed structure. In all cases, the verification is made by imposing a vertical displacement of the structure and measuring the corresponding transverse displacement. The verifications using the flexible multibody model and the finite element model are focused on half of a single auxetic cell, benefiting from the symmetry of loads and displacements. In contrast, the experimental verification involves an entire model with 3 × 3 repetitions of cells to compare the full behavior. The dimensions and properties of the cell are those presented in Table 1 and discussed in Section 4.

5.1. Verification Through a Flexible Multibody Model and a Finite Element Model

Half a cell is composed of two inclined flexible beams. In the discrete flexible multibody model, each beam is modeled using a decomposition with 10 beam elements. A planar model is preferred due to the expected overall motion of the structure. According to this strategy, each beam is decomposed into smaller rigid segments whose position and attitude (pose) are described by three-degree-of-freedom nodes. Adjacent nodes are connected by three-degree-of-freedom linear springs simulating the multi-dimensional elastic compliance between adjacent parts. A stiffness matrix mathematically expresses the three-degree-of-freedom linear springs:

$$\left[K_{ij}\right]=\left[\begin{array}{ccc}{\displaystyle \frac{EA}{d}}& 0& 0\\ 0& {\displaystyle \frac{12EI}{d^3\left(1+\varphi \right)}}& -{\displaystyle \frac{6EI}{d^2\left(1+\varphi \right)}}\\ 0& -{\displaystyle \frac{6EI}{d^2\left(1+\varphi \right)}}& -{\displaystyle \frac{\left(4+\varphi \right)EI}{d\left(1+\varphi \right)}}\end{array}\right]$$

(16)

$$\varphi ={\displaystyle \frac{12EI\chi }{GAd}}$$

(17)

where $E$ is the Young’s modulus of the material, $A$ is the cross-section area, $d$ is the distance between adjacent nodes, $I$ is the cross-section’s area moment of inertia, $\chi$ is the shear correction factor, and $G$ is the shear modulus of the material.

The multi-dimensional force field between adjacent nodes can be computed as follows:

$$F_{ij}=\left[K_{ij}\right]{\left\{\begin{array}{ccc}\mathrm{\Delta }x& \mathrm{\Delta }y& \mathrm{\Delta }\vartheta \end{array}\right\}}^T$$

(18)

where ${\left\{\begin{array}{ccc}\mathrm{\Delta }x& \mathrm{\Delta }y& \mathrm{\Delta }\vartheta \end{array}\right\}}^T$ is the vector of the relative displacement and rotation between adjacent connected nodes. Therefore, the nonlinear behavior is then considered as the summation of smaller linear contributions.

Figure 6a shows the second-order epicyclic pseudo-rigid model of the half cell in the undeformed condition. The two central links are connected at the common end (point C) with a fixed joint, which constrains all the three degrees of freedom of the coincident points. The ends of the first and last links (A and B) are then constrained to move along the y axis and are subjected to an imposed symmetrical displacement of amplitude $\delta$ (the upper point is moved downward, and the lower point is moved upward). Due to the symmetry of the model, the displacement of point C is constrained along the x axis (no parasitic motion).

Figure 6b shows the discrete flexible multibody model of the half-cell developed using 10 elements for each flexible beam. Also, in this case, the two beams are connected at the common end (point C) with a fixed joint, and the two other ends (A and B) are constrained to move along the y axis and are subjected to an imposed symmetrical amplitude displacement.

Figure 6c shows the finite element model of the half-cell developed using 100 Timoshenko beam elements for each flexible beam. The finite element model was built with a structured mesh, which makes it easier to verify the congruence. Since we are more interested in displacements rather than stresses, we chose a convergence criteria based on nodal displacement rather than on stresses. With this discretization, the error on nodal displacement is lower than 10⁻⁴ mm, which is more than acceptable for the required accuracy. This level of accuracy has also been used in other comparisons between flexible multibody models and finite element models [51].

Also, in this case, the two beams are connected at the common end (point C) with a fixed joint, and the two other ends (A and B) are constrained to move along the y axis and are subjected to an imposed symmetrical amplitude displacement.

Figure 6d presents the superimposition of the three models when a large opposite vertical displacement is applied to the A and B points.

The output of all the three models is the relationship between the applied displacement $\delta$ and the computed displacement $\gamma$ of point C along the x axis, as a direct measurement of the transverse contraction and, therefore, of the auxetic behavior.

5.2. Verification Through Experimental Tests

The third verification was conducted with an experimental test bench. An auxetic structure of 3 × 3 cells, whose dimensions are reported in Table 1, was printed with a Zortrax M300 plus printer (by Zortrax S.A., Olsztyn, Poland) using Z-ABS material and maximum infill. Values of material properties in Table 1 refer to the datasheet provided by the manufacturer. The printing direction was aligned to the transverse axis (z axis of Figure 3). The structure was placed on a rigid plane in contact with the bottom surface, and a controlled displacement was applied to the upper surface by means of the end-effector of a UR10e cobot (Figure 7). To reduce friction effects, small rollers are placed at the interface between the structure and the bottom plane and between the structure and the cobot end-effector. The lateral displacement was measured using an analogic centesimal gauge with a resolution of 0.01 mm in contact with the central lateral cell in order to reduce the contribution of boundary constraint effects. The displacement was applied with six equally spaced steps from 0 to $2{\delta }_{\max }$, recording the corresponding contraction $\gamma$. Measurements were repeated five times to increase statistical significance. Real-time communication with the cobot was implemented using a Python 3.6 script that includes the UR_RTDE library (the library and its documentation, including examples, can be downloaded at https://sdurobotics.gitlab.io/ur_rtde/ (accessed on 22 November 2024)) for imposing controlled movement using the Real Time Data Exchange protocol. The displacement was applied using the MOVEL command, which prescribes an interpolation in the Cartesian space. Although the cobot has a load cell placed at its end-effector, evaluating the actuation forces was not considered significant because of the effects of boundary deformations and contact friction, which would have made the results between the experiments and the three simulative models not directly comparable.

5.3. Comparison and Discussion

Figure 8 shows the graphical comparison among the relationships ${\displaystyle \raisebox{1ex}{$\gamma $}\!\left/ \!\raisebox{-1ex}{$h$}\right.}\left({\displaystyle \raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$h$}\right.}\right)$ computed with the second-order epicyclic pseudo-rigid model, the discrete flexible multibody model, the finite element model and measured with the experimental setup. The results of the experimental tests are the mean values among the five repetitions, and error bars are also included. The first observation concerns the nonlinear behavior between vertical displacement ${\displaystyle \raisebox{1ex}{$\delta $}\!\left/ \!\raisebox{-1ex}{$h$}\right.}$ and transverse displacement ${\displaystyle \raisebox{1ex}{$\gamma $}\!\left/ \!\raisebox{-1ex}{$h$}\right.}$, thus confirming the necessity of employing models that can take into account the geometric nonlinearities of the beams. It can also be observed that the three numerical models show completely comparable results. The pseudo-rigid model can approximate the displacement of the discrete flexible multibody model in the entire range of motion with a maximum difference value of 0.002 (less than 4%) at maximum imposed displacement. The discrete flexible model and the finite element model show differences lower than 0.5%, confirming that, for this type of simulation, discrete flexible models can represent a gold standard. It is also interesting to note that the difference between the curves does not increase as the imposed displacement increases but shows alternate behavior. In the first range of motion, the second-order pseudo-rigid model tends to overestimate the transverse displacement, while, in the last phase, it tends to underestimate it. This means that the movement can be approximated with better accuracy because the plots are closer.

The displacements measured by the experimental test rig are always slightly lower than those estimated by the three numerical models. This is reasonable, considering that this difference may be due to the boundary-constraining effects that the upper and lower rows of cells apply to the middle cells where the displacement is measured. However, the differences that can be evaluated from the plots are 0.018 with respect to the discrete flexible multibody model/finite element model (about 3%) and 0.035 with respect to the second-order pseudo-rigid model (less than 7%). They both reach the maximum in correspondence to the maximum imposed vertical displacement.

Figure 9 presents the graphical comparison between the forces computed to impose the prescribed vertical displacement between the second-order pseudo-rigid model, the discrete flexible multibody model, and the finite element model. The forces refer to the contribution of only half the cell, as all the models include only half the cell for symmetry.

Also, in this case, the nonlinear behavior is clearly observed, which provides a softening of the stiffness of the cell as the vertical displacement progresses. In a similar way to what is observed for displacements, the discrete flexible multibody model and the finite element model show almost identical behavior (differences are below 0.5%). The second-order pseudo-rigid model tends to overestimate the force necessary to impose the displacement in the first phase of motion and then tends to underestimate it in the last phase. The maximum differences between the curves are always observed in correspondence with the greatest imposed displacement, reaching a difference of 0.6 N (about 4% with respect to the maximum value of the force of the discrete flexible multibody model/finite element model). However, for the forces, it is also possible to conclude that the approximation obtained with the second-order pseudo-rigid model is fully comparable with the results of both the discrete flexible model and the finite element model.

6. Conclusions

This work presents a methodology for designing auxetic structures based on the use of pseudo-rigid mechanisms deduced from the theory of kinematic invariants applied to the compliant mechanisms. The auxetic structure under investigation is a repetition of symmetrical re-entrant cells composed of inclined straight beams. The results of the investigation have shown that it is possible to replicate the elasto-kinematic behavior through a surrogate rigid-body mechanism composed of epicyclic arrangements with an overall single degree of freedom and with kinematic characteristics independent of the structural ones. This represents an extension of the modeling of flexible hinges using the equivalence of kinematic invariants to beam-like structures that also exhibit concavity changes, effectively breaking them down into several equivalent sub-mechanisms.

This approach has allowed a substantial simplification in the synthesis of structures for which the relationship between longitudinal displacement and transverse displacement is prescribed, allowing us to deduce design formulas that can easily account for the effects of geometric nonlinearity when large displacements occur. The proposed approach was verified for a case study by comparing the results with those of two multi-degree-of-freedom models developed using the discrete flexible multibody approach and the finite element approach, respectively. The proposed methodology was also verified against the measurement with an experimental campaign. All the comparisons have shown that the model and the proposed equations are able to describe in an accurate way both the displacements of the structure and its stiffness. The differences between the numerical models and the experimental results are always below 4%, a value considered more than acceptable in the design of flexible cellular structures.

There are two main limitations of the proposed methodology. The first concerns, like all design approaches based on pseudo-rigid mechanisms, the adequacy of the model limited to situations in which the combination of applied loads is the same as that considered in the deduction of the equations. This can be justified considering that the elasto-kinematic structures have a kinematic behavior also dependent on the applied loads, opposite to mechanisms with rigid links. The second limitation concerns the need for a preliminary study aimed at understanding the actual shapes of beam deformation necessary to choose the most suitable epicyclic embodiment. However, we are convinced that these limitations do not affect the effectiveness of the proposed model.

The proposed approach, although developed in detail for a specific auxetic symmetrical re-entrant structure, is believed to be extended to other cellular structures composed of beam elements since the proposed pseudo-rigid subsystem allows for the description of the deformation of a straight beam under large displacements in a general way. This emphasizes the possibility of using the methodologies to analyze and synthesize compliant mechanisms to design cellular and lattice structures, even with auxetic behaviors.

One of the most promising future directions for the use of the proposed methodology is the optimization of the geometric and elastic parameters of the auxetic cell. From this point of view, having simplified but accurate equations that describe the elasto-kinematic behavior of the structure will allow the use of computationally efficient models that can adapt both to the design of experiment studies and to optimization procedures based on more advanced techniques. Another future direction of considerable interest is that of the extension of the methodology for cells with more complex topological structures for which advanced elasto-kinematic models will have to be revised and adapted in an appropriate manner, maintaining a sufficient degree of accuracy and simplicity in deduction. A third direction of future research activities is the possibility of including the proposed model within interactive virtual simulations or entire digital twins of auxetic structures that require models with high computational efficiency to guarantee solutions to structural problems in real time.