# Bulky auxeticity, tensile buckling and *deck-of-cards* kinematics emerging from structured continua

## Abstract

Complex mechanical behaviours are generally met in macroscopically homogeneous media as effects of inelastic responses or as results of unconventional material properties, which are postulated or due to structural systems at the meso/micro-scale. Examples are strain localization due to plasticity or damage and metamaterials exhibiting negative Poisson’s ratios resulting from special porous, eventually buckling, sub-structures. In this work, through *ad hoc* conceived mechanical paradigms, we show that several non-standard behaviours can be obtained simultaneously by accounting for kinematical discontinuities, without invoking inelastic laws or initial voids. By allowing mutual sliding among rigid tesserae connected by pre-stressed hyperelastic links, we find several unusual kinematics such as localized shear modes and tensile buckling-induced instabilities, leading to *deck-of-cards* deformations—uncapturable with classical continuum models—and unprecedented ‘bulky’ auxeticity emerging from a densely packed, geometrically symmetrical ensemble of discrete units that deform in a chiral way. Finally, after providing some analytical solutions and inequalities of mechanical interest, we pass to the limit of an infinite number of tesserae of infinitesimal size, thus transiting from discrete to continuum, without the need to introduce characteristic lengths. In the light of the theory of structured deformations, this result demonstrates that the proposed architectured material is nothing else than the first biaxial paradigm of *structured continuum*—a body that projects, at the macroscopic scale, geometrical changes and disarrangements occurring at the level of its sub-macroscopic elements.

### 1. Introduction

In the last years, for the growing cross-sector fertilization in the research fields of mechanics of materials, chemical engineering and biomechanics and as a consequence of emerging technologies and advances in additive manufacturing, a number of unprecedented scenarios have been glimpsed for designing new high-performance structures, revitalizing the interest of a wider scientific community in complex mechanical behaviours and, in turn, in non-standard modelling of continua and structures. In this framework, a large number of so-called *metamaterials* has been for instance proposed for applications ranging from the electromagnetic and optical fields to the mechanical and acoustic ones, these taking advantage of the possibility to rationally design composites exhibiting, among other, negative material parameters, multi-stable and programmable behaviours as well as optimized static and dynamic responses, by starting from standard constituents [1–4].

With particular reference to mechanics of metamaterials, it has been shown that ultra-high strength, toughness and stiffness can be, for example, obtained by combining geometrical and material size effects in extremely lightweight micro- and nano-lattices, provided that hierarchical or self-similar architectures across various length scales can be recognized [5,6]. Furthermore, it has been seen that proper configurations of the material microstructure allow to achieve macroscopic properties and deformation regimes characterized by reciprocity breaking (e.g. asymmetrical outputs under pushing and pulling [7]), twisting under uniaxial forces [8] as well as auxetic (i.e. negative Poisson’s ratio) responses [9–14], which are all features not ascertained in nature and also inaccessible to man-made materials until not so long ago.

Motivated by the vast variety of inputs and challenges posed by the current research in materials and by the opportunity to unveil some possible logics underlying the response of metamaterials, in this work, we exploit a simple—*ad hoc* designed—mechanism of mutual sliding between rigid tesserae, interconnected via pre-tensed hyperelastic cables, to obtain uniaxial and biaxial structures capable of offering simultaneously various uncommon mechanical responses. Multi-scale kinematics with local disarrangements [15], tensile buckling [16], instability-induced auxetic properties [17,18] related to chiral deformation modes arising from initially achiral (bulky and geometrically symmetrical) configurations, non-symmetrical responses to compressive and tensile loads as well as *deck-of-cards* deformation modes producing macroscopic shear [19] are in fact all replicated in the paradigmatic system studied here. In particular, we start from the case of a uniaxial multi-element macro-system comprising an arbitrary number of sub-units, so obtaining tunable critical buckling loads and stiffness if macroscopically stretched under tensile forces, reproducing instead overall shear as a sort of complementary version of the kinematics exhibited by Timoshenko beams if forces are applied orthogonally to the axis of the uniaxial system. Then, we generalize the results to biaxial ensembles thought as a mosaic of rigid tesserae elastically interacting with each other through nonlinear springs and derive analytical solutions for such architectured materials undergoing several spatially inhomogeneous and chiral deformation regimes under different selected boundary conditions. In this regard, figure 1 depicts an engineering solution for actually constructing the rigid skeleton of the system comprising mutually sliding tesserae and for connecting them by means of elastic links. Therein, it is shown a seven-by-seven prototype, built up via additive manufacturing [three-dimensional (3D) printing] technique, exhibiting both periodic kinematics—the simplest modes on which the theoretical modelling is focused in the present work—and further possible deformation modes in which inhomogeneities and localization mechanisms, such as shear bands, occur. The theory of structured deformations (SD) is then recalled to trace the kinematical response of the systems under analysis in the limit case in which they comprise an infinite number of infinitesimal constituents. Following this way we *de facto* propose a first paradigm of biaxially structured medium, demonstrating that a SD-based homogenization strategy—which at the first order does not require the introduction of any characteristic length—is the most effective way for transiting from discrete to continuum models by taking into account sub-macroscopic disarrangements that lead to non-standard macroscopic behaviours [15]. Finally, to validate the theoretical outcomes, numerical finite-element simulations and some qualitative experimental tests have been performed and shown synoptically.

The paradigmatic structures used to highlight the proposed view could help in the future the design of novel metasurfaces and three-dimensional metamaterials exhibiting advanced mechanical features, by additionally offering the possibility to tailor them by properly playing with competition among geometrical parameters, elasticity and internal pre-stress.

### 2. Tensile buckling and stretching of uniaxial structures

#### (a) Single-element system

The response of the simply supported uniaxial structure illustrated in figure 2*a* is studied under the action of a tensile dead load *F* at one extremity. Here, the structure consists of two rigid bars, each of length *L*, interconnected by both a slider, which prevents relative rotation and separation normal to their interface while allowing mutual slip in the orthogonal direction, and a pre-stretched hyperelastic link, initially co-axial with the two bars and symmetrically fixed such as to cover a length 2*αL*, *α* ∈ (0, 1). Under the action of the force *F*, the initially straight system buckles [16] by exhibiting the deformation mode shown in figure 2*a*, which is fully characterized only by one degree of freedom, namely the rotation *ϕ* of the bars, as a consequence of the prescribed geometrical constraints.^{1} Hence, the relative sliding Δ at the interface between the bars and the displacement *u* occurring at the point of application of the external force can be obtained by virtue of the following compatibility relations:

*l*

_{0}by a certain longitudinal pre-stretch

*λ*

_{p}in a way that

*l*of the cable to be

*λ*

_{ϕ}coherently going to 1 for vanishing

*ϕ*.

Without loss of generality, if the hyperelastic cable is assumed to obey a compressible neo-Hookean hyperelastic law and the element is subject to uniaxial longitudinal stress regime, the strain energy density $\stackrel{~}{\psi}$ and the axial force $\stackrel{~}{N}$ can be, respectively, written as functions of the sole generic longitudinal stretch $\stackrel{~}{\lambda}$

*μ*and

*ν*being the Lamé shear modulus and the Poisson’s ratio

^{2}of the elastic band’s material, respectively, while

*A*is its nominal (undeformed) cross-sectional area [20–22]. Hence, with reference to the cable in the buckled structure, one has

*V*=

*A l*

_{0}is the volume of the undeformed band. By making Π stationary with respect to the opening angle

*ϕ*, i.e. by imposing ∂Π/∂

*ϕ*= 0, one obtains two possible equilibrium configurations for the analysed system, respectively, a straight one and a buckled one:

*N*

_{p}is the cable’s pre-tension, namely the axial force that it bears due to the sole pre-stretch

_{2}. This implies that the magnitude of the critical load turns out to be governed by both the cable’s geometry (the parameter

*α*indeed amplifying it through a hyperbolic trend) and the pre-stretch. In particular, the threshold value vanishes when no pre-tension is applied to the elastic cable, in such a case the system resulting unable to oppose to its incipient opening. Actually, a similar behaviour, characterized by the impossibility to resist external loads in the absence of internal pre-tension, is typically exhibited by tensegrity architectures, to which the elemental module under exam can be

*de facto*assimilated since it comprises a pre-tensed cable connected to a pre-compressed rigid bar ensuring the self-equilibrium of the overall system in the externally unloaded configuration [21,23,24]. In this sense, the studied structure can be somehow regarded as the simplest paradigm of tensegrity (equipped with a nonlinear hyperelastic cable) undergoing tensile buckling.

^{3}However, as shown in figure 3, while the critical load vanishes for a not pre-tensed structure and monotonically grows with the internal pre-stretch by essentially following a neo-Hookean type force-stretch relation, a more complex dependence of the initial tangent stiffness

*k*

_{0}on the pre-stretch is found

*ϕ*as a function of the displacement

*u*can be obtained by inverting the relation (2.1)

_{2}. In fact, this stiffness assumes finite value also in case of unitary pre-stretch and has a stationary point (a minimum) at

_{1}is stable for values of the applied force that are lower than ${\hat{F}}_{cr}$ and instead unstable for higher ones, while the non-trivial bifurcation path, characterized by the relationship (2.9)

_{2}between the magnitude of the external force and the opening angle, remains stable over the whole range of admissible rotations—say

*ϕ*∈ [0,

*π*/2 [ by ideally neglecting the finiteness of the slider’s height, which would reduce its travel—for any combination of the geometrical and constitutive parameters on which it depends and within their physical limits. The whole system’s equilibrium bifurcation diagram is reported in figure 4.

#### (b) Multi-element system

Let us now consider a uniaxial structure made of two rigid end bars, each of length *L*_{n} = *L*/*n*, and *n* − 1 rigid internal bars, each of length 2*L*_{n}, constrained as in figure 2*b* to allow relative sliding without separation along the *n* interfaces. Each of the *n* adjacent pairs is connected by a prestretched cable as in §2a and is referred to as an element (or module) of the uniaxial structure of length 2*L*. By imposing that the whole length 2*L* of the system is independent of *n*, the two rigid bars in each module have length *L*_{n} = *L*/*n*. Also, let us assume that an external tensile force *F* is applied along the axis of the system with values growing from zero up to a possible tensile buckling load, resulting in a macroscopic stretch of the structure due to the displacement jumps as shown in figure 2*B*3. The prescribed kinematical (internal and external) constraints then requires that the displacement *u*_{n} at the point of application of the external load and the *n* relative sliding ${\mathrm{\Delta}}_{n}^{(i)}$ between the two adjacent endpoints of the bars in each module satisfy the following compatibility equations:

*u*and Δ are the corresponding quantities defined for the one-element system in equation (2.1) and the index $i\in \{1,\dots ,n\}$ identifies the

*i*-th module. As a consequence, the lengths of the hyperelastic cables in the closed and buckled configurations, respectively, result as

*l*

_{p}being defined in equation (2.2), so that the stretch governing the change of configuration for each cable turns out to be

*i*-th module by assuming that each one can be differently pre-tensed. In this way, one also finds that the corresponding rest length, denoted as ${l}_{0,n}^{(i)}$, has to satisfy the following geometrical relation:

*n*, as if one would ideally cut the same elastic band of length

*l*

_{0}used for the one-element system into

*n*different segments of lengths ${l}_{0,n}^{(i)}$ in the respect of relation (2.18), one derives a condition according to which the arithmetic average of the reciprocal of the pre-stretches ${\lambda}_{p,n}^{(i)}$ must equate the reciprocal of the pre-stretch

*λ*

_{p}used in the single-element, that is

*λ*

_{p}also represents the pre-stretch that all the cables would sustain if equally pre-stretched, this hypothesis implying that ${l}_{0,n}^{(i)}={l}_{0,n}={l}_{0}/n$ and ${\lambda}_{p,n}^{(i)}={\lambda}_{p,n}={\lambda}_{p}$ for any $i\in \{1,\dots ,n\}$. Actually, the condition (2.19) guarantees that the total volume

*V*of deformable material is fixed, independent of

*n*, since, for a certain cross-sectional area

*A*of the cables, one has

*i*-th undeformed cable, with ${V}_{n}^{(i)}={V}_{n}=V/n$ for a uniformly pre-stretched structure.

On these bases, the total potential energy associated with the buckled multi-element structure, say Π_{n}, can be written as

*i*-th cable during the deformation, i.e. ${\psi}_{n}^{(i)}=\stackrel{~}{\psi}{|}_{\stackrel{~}{\lambda}={\lambda}_{n}^{(i)}}$.

#### (c) Energy, critical load and useful inequalities

By starting to consider the special case of a uniformly pre-stretched structure (i.e. ${\lambda}_{p,n}^{(i)}={\lambda}_{p}$), the above-mentioned energy results to be minimum when all the moduli undergo equal relative sliding (figure 2*B*2), that is, when the following equalities hold true:

*l*,

*λ*

_{ϕ}and

*λ*are given in equations (2.1)

_{1}, (2.4) and (2.5). Under these conditions, the function Π

_{n}reduces to the one associated with the deformation of the elementary system, i.e. Π, given in equation (2.8), and the rotation

*ϕ*is again the sole Lagrangian variable of the problem, which hence admits the same equilibrium states given by equations (2.9) and (2.10). This implies that, in this case, the number of structural constituents affects neither the occurrence of the buckling event nor the opening degree for a selected level of applied force (above the critical threshold). In order to show that, among all the possible bifurcation modes (exemplified in figure 2

*B*3), the most favourable one for a uniformly pre-tensed structure is that illustrated in figure 2

*B*2, characterized by uniform relative slidings, it is possible to appeal to the Jensen inequality

*x*

_{1}, …,

*x*

_{n}being elements of such domain. In particular, by setting

*ν*< 1/2 and the term related to the work done by the external load being the same in Π and Π

_{n}by virtue of the invariance of the displacement

*u*with respect to

*n*, highlighted in equation (2.14)

_{1}and illustrated in figure 2

*b*.

Equilibrium considerations finally allow to determine the reactions arising during buckling within the uniformly pre-tensed system: specifically, the internal sliders experience internal reactions that are vanishing momenta and normal tractions ${R}_{n}=(1-\alpha )\hat{F}(\varphi )\mathrm{cos}\varphi $, acting at the middle points of the interacting portions of their interfaces.

On the other hand, it is worth studying the response of the structure in the complementary case of differently pre-stretched elastic bands forced to undergo equal relative sliding during the buckling phase, in a way to preserve the previously considered periodic kinematics, for example by properly positioning external rollers at the end of each module. In such a case, the equalities (2.22) are still satisfied and the total potential energy is given by ${\mathit{\Pi}}_{n}{|}_{{\mathrm{\Delta}}_{n}^{(i)}={\mathrm{\Delta}}_{n}}$, which results a function of the sole rotation *ϕ*. Then, solution of the equilibrium problem $\mathrm{\partial}({\mathit{\Pi}}_{n}{|}_{{\mathrm{\Delta}}_{n}^{(i)}={\mathrm{\Delta}}_{n}})/\mathrm{\partial}\varphi =0$ provides both the trivial configuration (i.e. $\varphi =0,\hspace{0.17em}\mathrm{\forall}F$) and the following non-trivial path:

*i*-th cable. It must be highlighted that, in the limit case in which the system comprises an infinite number of units of infinitesimal length, the convergence of the critical load ${\hat{F}}_{cr,n}$ and its (eventual) finite limit value would generally depend on the value of the series on the right hand of equation (2.28) and therefore on the specific pre-stress distribution. As expected, under the hypothesis of uniformly pre-stressed structure, ${\hat{F}}_{n}(\varphi )$ and ${\hat{F}}_{cr,n}$, respectively, reduce to the corresponding quantities $\hat{F}(\varphi )$ and ${\hat{F}}_{cr}$ obtained for the single module and reported in equations (2.9)

_{2}and (2.10).

Additionally, it is possible to prove that, for a fixed number of moduli forming the structure, the uniformly pre-stretched system represents the one associated with the lowest bifurcation load as compared to those obtained by any different distribution of cables’ pre-stretches, i.e.:

_{2}, also reads as

*ν*< 1/2.

For sake of completeness, it should be said that, in general, the response of a uniaxial multi-element structure where the elastic links were non-uniformly pre-stretched and no external constraints were applied to force the system to open the sliders in a periodic way, energy minimization could lead to less elemental equilibrium bifurcation paths accompanied by localization of the deformation, i.e. different magnitudes of the slider opening along the structure axis. Although these more general kinematics are predicted by the proposed modelling approach and could be analysed numerically, a detailed discussion on these solutions is beyond the scope of the present work.

### 3. Complemental Timoshenko beam deformation of a uniaxial multi-element structure under normal force

Let us analyse the mechanical response of the cantilever multi-element system at the end of which a force orthogonal to its axis is applied, initial and deformed configurations being those illustrated in figure 5. Compatibility with internal and external constraints gives that the kinematics is governed by (generally different) vertical slips ${s}_{n}^{(i)}$ that occur at the interfaces between the rigid units. Such a motion induces an elongation of the elastic bands, which thereby reach stretch levels given by

*n*elastic links are equally pre-stretched by

*λ*

_{p}and ${l}_{n}^{(i)}$ is the

*i*-th cable’s final length. This implies that the total potential energy can be written as

*n*equations:

*s*is the maximum sliding of the system at the endpoint in which the force is applied. The equilibrium configuration hence provides equal relative sliding at any interface of two adjacent units, as shown in figure 5. The resulting solution is stable: this means that neither bifurcation of the equilibrium nor critical load occur in the present case and the structure deforms gradually as the force increases, as highlighted in figure 6 in terms of normalized force against sliding, the overall deformation recalling a mode complemental to the one of a Timoshenko beam undergoing pure shear. Finally, standard equilibrium equations written on the current configuration lead to find that, during the deformation, compressive reactions of constant magnitude ${R}_{n}=2\alpha L{\hat{F}}_{S}(s)/s$ arise at the centre of the overlying regions of the sliders’ interfaces, which are accompanied by axially decreasing bending moments ${M}_{n}^{(i)}=[2-(2i-1)/n]L{\hat{F}}_{S}(s)$.

### 4. Mechanics of biaxial multi-element structures

To investigate more general load cases and analyse the mechanical behaviour of enriched architectured materials, we start from the assembling of tesserae described above and generalize it in the simplest way, by assuming to periodically replicate in two mutually orthogonal directions the previously studied moduli to so have a sort of mosaic of rigid (square) pieces. By making reference to the figure 7, we in particular imagine that the whole structure is generated by the repetition—say along the horizontal and vertical directions—of an elemental unit comprising four square rigid tesserae, which can still mutually slide at the tessera-tessera interfaces, with adjacent elements interconnected each other by means of hyperelastic links. By indicating with *n* the number of such units along both the directions, the size of the tesserae is given by *L*_{n} = *L*/*n*, so that the overall system macroscopically appears as a square-shaped body with overall side of length 2*L*, regardless of how many sub-parts it is divided into. Then, similarly to the uniaxial case, in the reference configuration of the system the pre-tensed neo-Hookean elastic links connecting pairs of adjacent rigid elements have all a length *l*_{p,n}, according to equations (2.15)_{1} and (2.2), where *l*_{p} denoted the cables’ pre-stretched length in the simplest case of *n* = 1. Then, by imposing that all the tensed links are equally pre-stretched by *λ*_{p} ≥ 1, their rest lengths are provided by the following relation:

*l*

_{0}is defined in equation (2.3). As a consequence, the volume of each undeformed elastic band can be written as

*t*is the prescribed thickness of each elastic band while

*w*

_{n}represents its width, ideally obtained by dividing into

*n*equal parts the initial width

*w*of an originally unique elastic element. This assumption is necessary to guarantee that the total volume of deformable material employed to build up the whole structure, say $\overline{V}$, is kept constant independently from the number of units comprising it, namely $\overline{V}=4{n}^{2}{\overline{V}}_{n}=4A{l}_{0}$, with

*A*=

*tw*.

In what follows, the mechanical response of the biaxial system is analytically investigated under uniaxial, biaxial and simple shear regimes, in the former cases additionally prescribing orthogonal rail-like constraints at the perimeter of the plane structure to reproduce the simplest (periodic) deformation modes due to uniform applied displacements (figure 7).

#### (a) Uniaxial loading regime

As illustrated in figure 7, with the aim to simulate uniaxial loading, the structure is assumed to be externally constrained by means of a series of rollers and a hinge at its left side and loaded by a normal tensile force *F* on the right one. Due to the nature of the internal constraints, this structure would behave as infinitely rigid under uniaxial compression, similarly to the uniaxial case, while it is able to buckle under uniaxial tension by exhibiting the regular deformation mode shown—for a more general biaxial regime—in figure 8. This is again governed by one Lagrangian variable, that is the rotation *ϕ* common to all the rigid tesserae, which allows one to adopt the same kinematical quantities Δ_{n}, *u*_{n} and *λ*_{ϕ,n} defined in equations (2.22) and (2.14). The whole stretch sustained by each elastic cable in the opened configuration thus results *λ*_{n} = *λ*_{ϕ}*λ*_{p}, which in turn induces a strain energy density ${\psi}_{n}=\stackrel{~}{\psi}{|}_{\stackrel{~}{\lambda}={\lambda}_{n}}$, so that the total potential energy of the system is

*auxetic*(with negative Poisson’s ratio) media, belonging to the wider class of the so-called

*metamaterials*. In fact, its elongation in the direction of the applied tensile load is accompanied by a transverse dilation (of the same magnitude as the longitudinal one), as clearly emerges from the chiral kinematics shown in figure 8. By calling $\overline{\nu}$ the overall Poisson’s ratio of the system, defined as the opposite in sign of the ratio between the relative length changes (macro-strains) of the generic module along the vertical (transverse) and horizontal (longitudinal) directions in case of uniaxial load, i.e.

#### (b) Biaxial loading regime

The mechanical response of the system under arbitrary biaxial loads is characterized by the same kinematics and constraints considered above for the uniaxial regime, as shown in figure 8. In present case, however, a horizontal force *F*_{h} = *F* is applied on the right side while tensile or compressive self-equilibrated vertical forces *F*_{v} = *ρF* act at the top and bottom sides, here being *F* > 0 and the dimensionless multiplier *ρ* > −1. Under these boundary conditions and with the applied constraints, the kinematics of the system is geometrically the same as that produced by the application of the sole tensile load (figure 8), while the total potential energy is obviously different and can be written as

*ϕ*is guaranteed by the undeformed state (i.e.

*ϕ*= 0) for any combination of applied loads as well as by a buckled configuration characterized by the force-rotation relation

_{2}significantly changes as a function of both magnitude and sign of the transverse forces. In particular, the condition

*ρ*> 0 identifies a situation in which the system buckles under biaxial tension, with the special case

*ρ*= 0 allowing to recover the uniaxial tensile buckling behaviour. On the other hand, the range −1 <

*ρ*< 0 identifies a transversal compression leading to an increase of the critical load up to a divergent value in correspondence of the inferior limit

*ρ*→ −1, when the system remains locked and then undeformed as effect of the competition between the two orthogonal loads. As a matter of fact, this might appear as a counterintuitive result, the tensile critical load being expected to decrease in presence of a transversely applied compressive force, as one would also have in the complementary case. This apparent paradox is however immediately explained if we recall that the system behaves as overall auxetic and thus a transverse tensile force that helps the longitudinal one in opening the sliders reduces the critical load, whereas a growing compressive force, which opposes the voids’ opening among the tesserae, increases the tensile buckling load up to infinity. Analogous behaviour can be expected in these unconventional (with negative Poisson’s ratio) materials, for instance in terms of stiffness, the ‘confining’ effect being inverted in the presence of biaxial load conditions.

#### (c) Shear loading regime and *deck-of-cards* deformation

A shear test can be simulated by constraining the bottom and top sides of the bi-dimensional structure by means of external hinges and rollers and by applying a resultant tangential force *F* on its top side, as sketched in figure 9. As an effect of these boundary conditions, the system deforms by following a *deck-of-cards* pattern where a relative sliding ${s}_{n}^{(j)}$ arises, in the horizontal direction, between tesserae placed at the top and bottom within the constituent moduli, *j* being hereafter used to refer to the units lying on the *j*-th row of the global grid (*n* × *n*) system. While the horizontally positioned elastic bands remain unperturbed by this kinematics, the vertical ones elongate, thus reaching an increased length ${l}_{n}^{(j)}$ so that their current stretches ${\lambda}_{n}^{(j)}$ read as

*n*equations:

*s*indicating the horizontal displacement on the top side of the whole structure and ${\hat{F}}_{S}(s)$ being given in equation (3.4). According to the uniaxial structure analysed in §3, no bifurcation of the equilibrium occurs and the overall force–displacement response of the system can be regarded as a nonlinear elastic one, accompanied by discrete forms of strain localization.

### 5. Some *in silico* simulations and experimental tests

The analytical formulation presented above excludes the opportunity to observe non-symmetrical configurations that could occur, for example, in the presence of non-uniform pre-tension throughout the system. Therefore, with the aim to include the effect of inhomogeneities in the model, *in silico* experiments were further conducted. All the analyses were performed by reconstructing the seven-by-seven 3D printed structure with the aid of the finite-element (FE) commercial code ${\text{ANSYS}}^{\circledR}$ [25]. In particular, the tesserae were modelled by employing SHELL181 elements interconnected by sliders (MPC184 elements) at the interfaces, while LINK180 elements were used for meshing the elastic bands. Without loss of generality, in order to provide numerical simulations at finite strains and large displacements, the links were modelled according to the Hencky hyperelastic model by activating the nonlinear geometry option providing true strain. In particular, figure 10*a* reports the post-critical path followed by the system in the uni-axial regime, showing a good agreement between the numerical results and analytical predictions provided by the theoretical model when the Hencky Law [23] is employed for the bands. Furthermore, representative experiments on a three-by-three elementary prototype were conducted to observe characteristic opening modes under both shear and bi-axial conditions (figure 10*b*).

Finally, the FE model allowed us to impose more complex distributions of pre-stress, whose effects could not be traced via analytical solutions. Non-uniform pre-stretch was prescribed to the structure in different manners. By way of example, three representative cases were analysed, that is (i) a piece-wise constant pre-stretch in which only one-half of the system is pre-tensed; (ii) a striped alternation of elastically pre-stretched and unloaded cables and (iii) a random pre-stress distribution. In this way, aperiodic and localized deformations were observed, as illustrated in figure 10*c*.

### 6. From discrete to continuum via structured deformations: kinematics of *structured continua*

The above described uniaxial and biaxial structures, made of an arbitrary finite number of discrete elements, have shown how the competition of sliding mechanisms, pre-stress and elastic interaction among constituents (i.e. rigid tesserae) lead to, at the macroscopic scale, concurrent and somehow unprecedented responses such as strain localization and bulky auxeticity, without invoking inelastic assumptions, chiral or porous geometries, provided that displacement discontinuities are properly incorporated in the models. Now, it is natural to explore what happens if the number of discrete elements becomes infinite while their size tends to zero. This can be effectively achieved by adopting the so-called first-order SD theory, which gives a unique way to pass from discrete to continuum without loss of information about displacements and their spatial gradients, allowing one to interpret the classical (smooth) macroscopic motion of a body as the projection of non-classical (piecewise-smooth) deformations occurring at the sub-macroscopic scale [15,19,26–30]. We will call these media—resulting from an initially discrete structure through a limit procedure and preserving internal discontinuities (disarrangements)—*structured continua*.

#### (a) Uniaxially structured continua under tensile load

The kinematics of the uniaxial multi-element system under tensile load can be described in the light of the SDs by extending a procedure already recently proposed by some of the authors [15]. In particular, the straight and buckled configurations of the system, illustrated in figure 2*B*1 and 2*B*2, can be, respectively, recognized as the so-called virgin and deformed configurations introduced in the SD theory, the first identifying the co-existence of both macroscopically and sub-macroscopically undeformed states of the body and the second one its actual deformed state, in which both a global (macroscopic) deformation and a deformation at the level of the elementary constituents occur. Hence, each material point of the body in the virgin configuration can be identified by a position vector ${\mathbf{x}}_{0}={x}_{1}{\hat{\mathbf{e}}}_{1}+{x}_{2}{\hat{\mathbf{e}}}_{2}$, where ${\hat{\mathbf{e}}}_{1}$ and ${\hat{\mathbf{e}}}_{2}$ are the unit vectors of the selected biaxial Cartesian coordinate system having the *x*_{1}-axis coinciding with the body’s axis and **x _{0}** varies within the domain $\mathcal{D}=[0,2L]\times [-T/2,T/2]$,

*T*indicating the thickness of the rigid bars. On the basis of the SD theory, the motion undergone by the rigid constituents when the whole system moves from the virgin to the deformed configuration can be mapped, at the sub-macroscopic scale, by a sequence of approximating functions

**g**

_{n}that, by virtue of geometrical arguments, read as

*n*+ 1 rigid bars comprising the body, $H(\bullet )$ is the Heaviside function,

*δ*

_{pq}is the Kronecker symbol and

**Q**indicates the rotation tensor

*n*. Then, in the limit of continuum body, namely when the number

*n*of elements goes to infinity, the approximating functions and the related deformation gradients, respectively, provide the following classic macroscopic motion

**g**and the deformation without disarrangements

**G**, which together define the first-order SD (

**g**,

**G**):

**G**, coherently with the basic principles of the SD theory, need not represent the gradient of any macroscopic motion and coinciding with the gradient of

**g**only in absence of geometrical disarrangements (such as slips and/or formations of voids) at the discrete scale. In this regard, the classic deformation gradient associated with the macroscopic mapping

**g**is

**M**as:

**g**

_{n}and by the associated deformation gradients $\mathrm{\nabla}{\mathbf{g}}_{n}$—projects, at the macroscopic level, the mapping

**g**and its gradient $\mathrm{\nabla}\mathbf{g}$, which essentially provide a shear-like deformation accompanied by axial elongation and transversal contraction of magnitude governed by the rotation

*ϕ*occurring at the sub-macroscopic scale (figure 11

*a*). On the other hand, the tensor fields

**G**and

**M**represent the continuum quantities revealing the non-standard nature of the deformation, where

**G**keeps track, at the continuum scale, of the local rotation of the bars, while

**M**captures the disarrangements occurring in the form of slips at the internal sliders.

Additionally, the kinematical tensor $\mathbf{K}={(\mathrm{\nabla}\mathbf{g})}^{-1}\mathbf{G}$ is generally introduced in the SD theory, since it allows for the multiplicative decomposition of a generic SD as sequence of two special SDs: in the order, (**i**, **K**)—with $\mathbf{i}(\bullet )$ indicating the identity mapping—that carries the body from the virgin to a reference configuration characterized by a purely sub-macroscopic deformation with no macroscopic effects, followed by the classical (purely macroscopic) deformation $(\mathbf{g},\mathrm{\nabla}\mathbf{g})$, mapping the body from the reference to the actual deformed configuration, i.e.

**K**reads as

**G**and $\mathrm{\nabla}\mathbf{g}$ obey, for the whole possible range of rotation of the bars, the so-called accommodation inequality prescribed by the SD theory. This requires that, over the entire domain where the SD is defined, the following order relation holds true:

*de facto*represents a compatibility condition avoiding matter interpenetration during the motion, despite, on the contrary, formation of local voids is instead admitted. For the case at hand, no voids arise during the non-classic deformation process, the emerging void fraction

*η*being indeed given by

#### (b) Uniaxially structured continua under shear

Let us now employ the SD theory to describe the deformation of the uniaxial structure in response to a point-force applied orthogonally to its axis, in the continuum limit. With reference to the figure 5 and to the solution (3.4)_{1}, providing the invariance of the relative sliding along the system axis at equilibrium, the sequence of approximating functions accounting for the kinematics of the problem at hand can be written as

**g**and

**G**:

**g**,

**G**) at the continuum level. Then, the classical deformation gradient turns out to be

**M**coincides with the classic gradient of the macroscopic displacement.

These outcomes highlight that the piece-wise (non-uniform) rigid translation experienced by each elementary constituent at the sub-macroscopic scale (tracked by **G** at the continuum level) and characterized by disarrangements induced by mutual sliding of the units (accounted for by **M**), results in an overall shear deformation at the macroscopic continuum level (figure 11*b*). Furthermore, both at the level of the infinitesimal units and at the whole-body scale, the volume is preserved during the motion:

*η*= 0) and the purely microscopic deformation being described by the tensor

*a*) whereas the orthogonal point-load produces a shear deformation to finally give a sort of complementary version of the Timoshenko-beam kinematics (figure 11

*b*), the uniaxially structured continuum giving naturally a generalized uniaxial model of a polar (Cosserat-like) medium.

#### (c) Biaxially structured continua under uniaxial and biaxial loads

In the light of SDs, the kinematics of the biaxial system shown in figure 8, which deforms under uniaxial or biaxial loads passing from an initially virgin (without voids) to a buckled (with voids) configuration, can be described as follows. Since in the virgin state each material point has position ${\mathbf{x}}_{0}={x}_{1}{\hat{\mathbf{e}}}_{1}+{x}_{2}{\hat{\mathbf{e}}}_{2}$, with $\{{x}_{1},{x}_{2}\}\in \overline{\mathcal{D}}=[0,2L]\times [0,2L]$, the approximating functions moving it to its actual position in the deformed configuration at the sub-macroscopic scale can be written as

**Q**is the rotation tensor given in equation (6.2). Then, the sequence of deformation gradients deriving from such approximating functions is

**g**,

**G**) of the biaxial body results

*c*). Note that the

*equibiaxial*deformation is the response of the structure independently from the presence of uniaxial or biaxial loads.

^{4}Then, the classical deformation gradient results given by

**M**and the kinematical tensor

**K**, respectively, take the forms

*c*), this implying—independently from the deformation level—a Poisson’s ratio equal to −1. This is in accordance with the value derived for the engineering parameter $\overline{\nu}$ introduced in equation (4.6), which, in the continuum limit, recovers the classical definition of Poisson’s ratio adopted for continua. Noteworthy, such a result reveals a possible way to build up newly conceived architectured media showing an extreme degree of auxeticity by starting from bulky (densely packed), stereologically homogeneous and isotropic initial configurations. This would

*de facto*allow one to overcome some conventional limiting features of auxetic metamaterials, which are instead generally forced to be characterized by anisotropy induced by chiral geometries of the microstructure or by porous architectures with high void fractions that penalize the overall stiffness.

#### (d) Biaxially structured continua under shear load

Let us finally consider the *deck-of-cards* motion exhibited by the biaxial system that distorts in a shear mode in response to prescribed forces applied as illustrated in figure 9. In such a case, the sequences of approximating functions and the corresponding gradients describing the horizontal rigid sliding of the tesserae are

**g**and

**G**such that

**M**and the microscopic deformation tensor

**K**are, respectively, given by

*d*) [19,31].

### 7. Conclusion

In this work, novel paradigms of uniaxially and biaxially structured bodies have been proposed, by theoretically studying and prototyping multi-element systems made of rigid tesserae interconnected via internal sliders and pre-tensed hyperelastic links. Thanks to the proper design of the internal constraints and to the possibility of tailoring geometrical and elastic parameters, the conceived systems are able to offer programmable and multiple unconventional mechanical behaviours, naturally gathering different properties met or postulated in metamaterials. In particular, the presented structures can exhibit asymmetrical behaviours under compressive and tensile loads and can provide a tensile buckling response characterized by critical threshold and post-buckling stiffness tunable by simply modulating the level and the distribution of the internal cables’ pre-stress. Importantly, in the biaxial case, the tension-induced instability phenomenon is additionally related to a mechanism of chiralization of the internal structure accompanied by local disarrangements during the (stable) bifurcation mode, which lead to an extreme auxetic response (i.e. Poisson’s ratio tending to −1) of the initially bulky and geometrically symmetrical architectured material. Furthermore, macroscopic shear kinematics are obtained as a result of piecewise-smooth *deck-of-cards* deformation modes, which—under non-periodical distribution of internal pre-tension or non-symmetrical boundary conditions—provide more complex mechanisms like shear bands and strain localization. Numerical finite-element simulations and some qualitative experimental tests have been also performed to validate the theoretical outcomes. Finally, starting from the description of these phenomena in discrete structures and by means of limit procedures, first examples of uniaxially and biaxially *structured continua* have been proposed in the light of the first-order SD theory. It is so demonstrated that this powerful framework allows one to properly manage the transition from discrete to continuum without invoking characteristic lengths, to recover in a natural way polar models, as well as to reproduce a number of concurrent non-standard mechanical behaviours, from strain localization—without introducing dissipation such as damage or plasticity—to auxeticity that does not require porous matrices. Figure 12 additionally shows that kinematics involving shear/dilatation coupling as typically exhibited by granular media, can be also traced by means of the proposed paradigms. For all these reasons, it is felt that SDs could be the best candidate to conceive and design new classes of materials with enriched hierarchical and multi-scale discontinuous kinematics, which classical continuum theories and related standard homogenization processes would not admit.

### Ethics

The authors declare that all the ethical issues were respected.

### Data accessibility

This article has no additional data. However, further data can be attained by replacing other parameters in the proposed model.

### Authors' contributions

All the authors contributed equally to the paper. In particular, S.P. developed the model and wrote the paper; S.P. and A.R.C. performed the analyses; A.C. verified the mechanical behaviour through numerical models and conceived with M.F., S.P. and A.R.C. the three-dimensional structures to be constructed by 3D printing; M.F. conceived the idea, advised the model development, wrote and reviewed the paper; D.R.O. and L.D. provided the theory of structured deformations, reviewing and editing the final draft.

### Competing interests

We declare we have no competing interests.

### Funding

The work was supported by the grant nos PRIN-20177TTP3S, PON-ARS01_01384 and the ‘Departments of Excellence’ grant no. L. 232/2016 from the Italian Ministry of Education, University and Research (MIUR), by the grant no. PON-AIM1849854-1 and by the grant FET Proactive (Neurofibres)grant no. 732344 from the European Commission (EC).

## Acknowledgements

S.P., L.D. and M.F. gratefully acknowledge the support of the Italian MIUR through the grant nos PRIN-20177TTP3S and PON-ARS01_01384. A.R.C. acknowledges the support of the grant no. PON-AIM1849854-1. L.D. also acknowledges the support of the Italian MIUR through the ‘Departments of Excellence’ grant L. 232/2016 and the support of the EC through the FET Proactive (Neurofibres) grant no. 732344.

## Footnotes

1 It is worth noting that the structure would not deform under axial compression, it hence behaves as an infinitely rigid bar in such a case. For this reason, although deforming tesserae might be conceived in a more general condition, the sole response under tensile load is analysed in the present work.

2 This stress-stretch relation is actually physically consistent only for Poisson’s ratios *ν* such that −1/2 < *ν* < 1/2.

3 Actually, we deal with a system that obeys a sort of inverted tensegrity principle: the structure is in fact in a state of self-equilibrium with touching compressed elements (the rigid tesserae) and non-convergent tensed cables (the nonlinear elastic links), whereas in classical tensegrity configurations the rigid struts are floating and the pre-tensed cables touch and converge at the struts’extremities.

4 It is worth to recall that, if the forces were applied in a non-uniform way at the object boundary or equivalently no constraints were imposed to transfer uniformly the loads to the tesserae, spatially inhomogeneous deformation modes, both at the discrete and the continuum level, could arise and therefore more complex non-equibiaxial and eventually macroscopically non-symmetrical (e.g. distorted) configurations could be expected.

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