Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
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Bulky auxeticity, tensile buckling and deck-of-cards kinematics emerging from structured continua

S. Palumbo

S. Palumbo

Department of Structures for Engineering and Architecture, University of Napoli ‘Federico II’, Napoli, Italy

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A. R. Carotenuto

A. R. Carotenuto

Department of Structures for Engineering and Architecture, University of Napoli ‘Federico II’, Napoli, Italy

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A. Cutolo

A. Cutolo

Department of Structures for Engineering and Architecture, University of Napoli ‘Federico II’, Napoli, Italy

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D. R. Owen

D. R. Owen

Department of Mathematical Sciences and Center for Nonlinear Analysis, Carnegie Mellon University, Pittsburgh, PA, USA

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L. Deseri

L. Deseri

Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento, Italy

Department of Mechanical Engineering and Materials Science, University of Pittsburgh, Pittsburgh, PA, USA

Department of Mechanical Engineering and Department of Civil and Environmental Engineering, Carnegie Mellon University, Pittsburgh, PA, USA

Department of Nanomedicine, Houston Methodist Hospital, Houston, TX, USA

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and
M. Fraldi

M. Fraldi

Department of Structures for Engineering and Architecture, University of Napoli ‘Federico II’, Napoli, Italy

[email protected]

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Published:https://doi.org/10.1098/rspa.2020.0729

    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 [14].

    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 [914], 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.

    Figure 1.

    Figure 1. (a) Monge projections of the designed 3D structural unit and axonometric view of the sliding mechanism occurring among adjacent tesserae obtained through a ‘tongue and groove’ joint. (b) 3D printed seven-by-seven assembled structure in which adjacent rigid tesserae are doubly connected via sliders and pre-tensed hyperelastic rubber bands anchored to ad hoc designed mini-pillars. (c) Regular kinematics of the uniaxial structure resulting from tensile buckling (top) and shear (bottom) loading conditions. (d) Periodic kinematics exhibited by the biaxial structure when uniaxial and biaxial loads (left) and shear (right) forces are applied. Auxetic behaviour and deck-of-cards deformation mode can be observed in the two cases, respectively. (e) Examples of aperiodic kinematics, characterized by inhomogeneous distribution and localization of the deformation field that the biaxial system could generally undergo. (Online version in colour.)

    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 2a 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 2αL, α ∈ (0, 1). Under the action of the force F, the initially straight system buckles [16] by exhibiting the deformation mode shown in figure 2a, 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:

    Δ=2Ltanϕ,u=2L(secϕ1). 2.1
    On the other hand, the axial link is modelled as a pre-tensed hyperelastic band with length
    lp=2αL,0<α1, 2.2
    obtained by pre-stretching an undeformed element of (shorter) length l0 by a certain longitudinal pre-stretch λp in a way that
    l0=lpλp,λp1. 2.3
    When the system buckles, compatibility imposes the actual length l of the cable to be
    l=lp2+Δ2=2Lα2+tan2ϕ; 2.4
    therefore, the total elastic stretch that it sustains during the slider opening reads as
    λ=λϕλp,λϕ=llp=1+(tanϕα)2, 2.5
    with λϕ coherently going to 1 for vanishing ϕ.
    Figure 2.

    Figure 2. Sketch of the uniaxial (A) single-element and (B) multi-element systems under axial load, at their undeformed (A1,B1) and (structurally) deformed configurations (A2,B2,B3). The bars (in blue) are supposed to be rigid and to slide at their mutual interfaces, while the hyperelastic, eventually pre-stretched, links (in yellow) connect them longitudinally. In the case of the multi-element system, the periodic deformation (B2) and a generic bifurcation mode with possibly different interfaces’ slidings (B3) are illustrated: note that, independently from the way in which each module moves in the two cases, both the deformed structures exhibit the same axial displacement magnitude at the point of application of the external load. (Online version in colour.)

    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 ψ~ and the axial force N~ can be, respectively, written as functions of the sole generic longitudinal stretch λ~

    ψ~=μ2[λ~2+λ~2ν(1+ν)ν]andN~=Aψ~λ~=μA(λ~λ~12ν), 2.6
    μ and ν being the Lamé shear modulus and the Poisson’s ratio2 of the elastic band’s material, respectively, while A is its nominal (undeformed) cross-sectional area [2022]. Hence, with reference to the cable in the buckled structure, one has
    ψ=ψ~|λ~=λandN=N~|λ~=λ. 2.7
    On this ground, the total potential energy of the system can be written as
    Π=VψFu, 2.8
    where V = A l0 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:
    ϕ=0,FandF=F^(ϕ)=μAα(λpλp12νλϕ22ν)secϕ=Nsecϕαλϕ,ϕ0, 2.9
    the latter providing the critical buckling load
    F^cr=limϕ0F^(ϕ)=μAα(λpλp12ν)=Npα. 2.10
    Herein, Np is the cable’s pre-tension, namely the axial force that it bears due to the sole pre-stretch
    Np=limϕ0N=N~|λ~=λp, 2.11
    according to equations (2.5) and (2.7)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 k0 on the pre-stretch is found
    k0=limu0uF^(ϕ(u))=Aμ[α2(λp2+2ν1)+2(ν+1)]2Lα3λp1+2ν, 2.12
    where the expression of the opening angle 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
    λpmin=[(1+2ν)(2α2+2ν)α2]12+2νandk0min=Aμ(1+ν)λpminLα(1+2ν). 2.13
    Finally, the study of the second variation shows that the trivial equilibrium solution (2.9)1 is stable for values of the applied force that are lower than 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.
    Figure 3.

    Figure 3. Trends of the normalized tangent stiffness at incipient opening (black curve—ordinate values on the left) and of the normalized critical load (grey curve—ordinate values on the right), obtained—for the single-element uniaxial structure—for an increasing value of the pre-stretch stored within the hyperelastic band and by setting α = 0.5 and ν → 0.5. The monotonic growing trend of the critical load can be here observed, along with the non-monotonic behaviour of the initial post-buckling stiffness, whose point of minimum of coordinates (λpmin,Lk0min/Aμ) is highlighted.

    Figure 4.

    Figure 4. Equilibrium bifurcation diagram for the elemental uniaxial system buckling under tensile load, obtained by setting α = 0.5 and ν → 0.5. Herein, solid lines identify stable paths, while the dashed tract indicates the (sole) unstable branch.

    (b) Multi-element system

    Let us now consider a uniaxial structure made of two rigid end bars, each of length Ln = L/n, and n − 1 rigid internal bars, each of length 2Ln, constrained as in figure 2b 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 2L. By imposing that the whole length 2L of the system is independent of n, the two rigid bars in each module have length Ln = 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 2B3. The prescribed kinematical (internal and external) constraints then requires that the displacement un at the point of application of the external load and the n relative sliding Δn(i) between the two adjacent endpoints of the bars in each module satisfy the following compatibility equations:

    un=u,i=1nΔn(i)=Δ, 2.14
    where u and Δ are the corresponding quantities defined for the one-element system in equation (2.1) and the index i{1,,n} identifies the i-th module. As a consequence, the lengths of the hyperelastic cables in the closed and buckled configurations, respectively, result as
    lp,n=2αLn=lpn,ln(i)=lp,n2+(Δn(i))2, 2.15
    lp being defined in equation (2.2), so that the stretch governing the change of configuration for each cable turns out to be
    λϕ,n(i)=ln(i)lp,n 2.16
    and the related total stretch can be written as
    λn(i)=λϕ,n(i)λp,n(i),λp,n(i)1, 2.17
    where λp,n(i) represents the elastic pre-stretch stored by the cable associated with the generic 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 l0,n(i), has to satisfy the following geometrical relation:
    l0,n(i)=lp,nλp,n(i). 2.18
    However, by imposing the sum of the cables’ rest lengths to be independent of n, as if one would ideally cut the same elastic band of length l0 used for the one-element system into n different segments of lengths l0,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 λp,n(i) must equate the reciprocal of the pre-stretch λp used in the single-element, that is
    i=1nl0,n(i)=l01ni=1n1λp,n(i)=1λp. 2.19
    In this regard, it is worth highlighting that λp also represents the pre-stretch that all the cables would sustain if equally pre-stretched, this hypothesis implying that l0,n(i)=l0,n=l0/n and λp,n(i)=λp,n=λp for any i{1,,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=1nVn(i)=V, 2.20
    Vn(i)=Al0,n(i) being the volume of the i-th undeformed cable, with Vn(i)=Vn=V/n for a uniformly pre-stretched structure.

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

    Πn=i=1nVn(i)ψn(i)Fu, 2.21
    where ψn(i) is the strain energy density stored by the i-th cable during the deformation, i.e. ψn(i)=ψ~|λ~=λn(i).

    (c) Energy, critical load and useful inequalities

    By starting to consider the special case of a uniformly pre-stretched structure (i.e. λp,n(i)=λp), the above-mentioned energy results to be minimum when all the moduli undergo equal relative sliding (figure 2B2), that is, when the following equalities hold true:

    Δn(i)=Δn=Δn,ln(i)=ln=ln,λϕ,n(i)=λϕ,n=λϕ,i{1,,n}, 2.22
    together with
    λn(i)=λn=λ,i{1,,n}, 2.23
    where Δ, l, λϕ and λ are given in equations (2.1)1, (2.4) and (2.5). Under these conditions, the function Πn 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 2B3), the most favourable one for a uniformly pre-tensed structure is that illustrated in figure 2B2, characterized by uniform relative slidings, it is possible to appeal to the Jensen inequality
    φ(i=1nxin)i=1nφ(xi)n, 2.24
    in general holding true for a function φ that is convex on its domain, the variables x1, …, xn being elements of such domain. In particular, by setting
    xi=Δn(i),φ()=(λplp,n2+()2ln,p)2+((λplp,n2+()2)/ln,p)2ν(1+ν)ν, 2.25
    the inequality (2.24) actually provides that
    ΠΠn|λp,n(i)=λp{Δn(1),,Δn(n)}, 2.26
    the selected function φ resulting convex for any Δn(i)>0 and −1/2 < ν < 1/2 and the term related to the work done by the external load being the same in Π and Πn by virtue of the invariance of the displacement u with respect to n, highlighted in equation (2.14)1 and illustrated in figure 2b.

    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 Rn=(1α)F^(ϕ)cosϕ, 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 Πn|Δn(i)=Δn, which results a function of the sole rotation ϕ. Then, solution of the equilibrium problem (Πn|Δn(i)=Δn)/ϕ=0 provides both the trivial configuration (i.e. ϕ=0,F) and the following non-trivial path:

    F=F^n(ϕ)=μAαni=1n[λp,n(i)(λp,n(i))12νλϕ22ν]secϕ, 2.27
    equilibrium bifurcation occurring at the critical threshold
    F^cr,n=limϕ0F^n(ϕ)=μAαni=1n[λp,n(i)(λp,n(i))12ν]=1αni=1nNp,n(i), 2.28
    and Np,n(i)=N~|λ~=λn(i) being the pre-tension within the 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 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, F^n(ϕ) and F^cr,n, respectively, reduce to the corresponding quantities F^(ϕ) and 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.:

    F^crF^cr,n{λp,n(1),,λp,n(n)}. 2.29
    In fact, by taking into account the previous assumptions and results, this inequality explicitly gives
    λpλp12ν1ni=1n[λp,n(i)(λp,n(i))12ν], 2.30
    which, by virtue of equation (2.19)2, also reads as
    [i=1n(λp,n(i))1n]1[i=1n(λp,n(i))1n]1+2ν1ni=1n[λp,n(i)(λp,n(i))12ν]. 2.31
    Such a relation is immediately demonstrated by invoking the Jensen’s inequality (2.24) and by setting xi=(λp,n(i))1 and φ=()1()1+2ν, this function being convex for any λp,n(i)>1 and −1/2 < 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 sn(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

    λn(i)=λs,n(i)λp,λs,n(i)=ln(i)lp,n,ln(i)=lp,n2+(sn(i))2, 3.1
    if one assumes that the n elastic links are equally pre-stretched by λp and ln(i) is the i-th cable’s final length. This implies that the total potential energy can be written as
    Πn=Vn(i=1nψn(i))F(i=1nsn(i)), 3.2
    so that equilibrium solution is the one satisfying the following set of n equations:
    Πnsn(m)=0,m{1,,n}Vnψn(m)sn(m)F=0,m{1,,n}, 3.3
    from which
    sn(i)=sn=sn,i{1,,n},F=F^S(s)=Aμλps2αL[1(λp4α2L2+s22αL)22ν], 3.4
    where 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 Rn=2αLF^S(s)/s arise at the centre of the overlying regions of the sliders’ interfaces, which are accompanied by axially decreasing bending moments Mn(i)=[2(2i1)/n]LF^S(s).
    Figure 5.

    Figure 5. Sketch of the uniaxial n-element system under vertical load: undeformed configuration (top) and (structurally) deformed configuration (bottom). (Online version in colour.)

    Figure 6.

    Figure 6. Plot of the solution, in terms of normalized force versus sliding, for a uniformly pre-stretched multi-element uniaxial structure under orthogonal load, obtained by setting α = 0.5 and ν → 0.5.

    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 Ln = L/n, so that the overall system macroscopically appears as a square-shaped body with overall side of length 2L, 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 lp,n, according to equations (2.15)1 and (2.2), where lp 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 λp ≥ 1, their rest lengths are provided by the following relation:

    l0,n=lp,nλp=l0n, 4.1
    where l0 is defined in equation (2.3). As a consequence, the volume of each undeformed elastic band can be written as
    V¯n=twnl0,n=twl0n2,wn=wn, 4.2
    where, under the hypothesis of rectangular cross section, t is the prescribed thickness of each elastic band while wn 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 V¯, is kept constant independently from the number of units comprising it, namely V¯=4n2V¯n=4Al0, with A = tw.
    Figure 7.

    Figure 7. (a) Generalization of the uniaxial multi-element system to a biaxial (n × n) ‘chocolate bar-like’ case under uniaxial load. Square tesserae (in blue) are assumed to be rigid with interfaces that allow sliding. Adjacent tesserae are all connected through hyperelastic, eventually pre-stretched, links (in yellow) both along the horizontal and the vertical directions. (b) To obtain regular (spatially homogeneous) kinematics in case of uniaxial and biaxial tests and properly transfer the loads to the structure, the midpoints of the sides of the tesserae at the boundary of a real system are thought to be constrained to slide along a rigid track at which the forces are applied, as actually ad hoc built up for the 3D printed (3 × 3) prototype shown in the picture. (Online version in colour.)

    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 Δn, un and λϕ,n defined in equations (2.22) and (2.14). The whole stretch sustained by each elastic cable in the opened configuration thus results λn = λϕλp, which in turn induces a strain energy density ψn=ψ~|λ~=λn, so that the total potential energy of the system is

    ΠU,n=4n2V¯nψnFu. 4.3
    By solving the equilibrium problem for this case, one finds the explicit solution in terms of force versus rotation as
    F=F¯U(ϕ)=4μAα(λpλp12νλϕ22ν)secϕ=4F^(ϕ),ϕ0, 4.4
    that is valid for loads overcoming the critical threshold
    F¯U,cr=limϕ0F¯U(ϕ)=4μAα(λpλp12ν)=4F^cr, 4.5
    F^(ϕ) and F^cr being the analogous quantities obtained for the uniaxial problems. This result implies that the tensile buckling load for the considered biaxial system is not affected by the number of the sub-units and that, regardless of this parameter, the discussion on the equilibrium bifurcation path and on the system stability is essentially the same as one would do for the case illustrated in figure 4. Independently from the already discussed details on stability and equilibrium characterizing the response of the system at hand, it is interesting to observe that the structure exhibits a mechanical behaviour that is typical of the 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 ν¯ 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.
    ν¯=[δ(2Ln)]v/(2Ln)[δ(2Ln)]h/(2Ln). 4.6
    and noting that in the case at hand the two length variations have the same magnitude in the two directions, that is
    [δ(2Ln)]v=[δ(2Ln)]h=2Ln(secϕ1), 4.7
    we find that the overall system shows the highest degree of auxeticity admissible for a homogeneous and isotropic elastic material (the lower bound of the Poisson’ratio), i.e. ν¯=1, which arises here, for the first time, from a bulky (non-porous) and symmetrical (non-chiral) body.
    Figure 8.

    Figure 8. On the top, a scheme of the biaxial (n × n) system under combined horizontal (tensile) and vertical (tensile, red-coloured, or compressive, green-coloured) loads, shown in both the undeformed (left) and deformed (right) configurations. Overall, the deformation kinematics is governed by the sole rotational degree of freedom, this being representative for both the cases of biaxial forces and the uniaxial tensile load (obtainable by setting ρ = 0). The latter regime is reproduced in the bottom part of the figure by pulling by hand the 3D printed prototype. Theoretical and real deformed configurations highlight the predicted auxeticity of the bulky (initially without voids) material. (Online version in colour.)

    (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 Fh = F is applied on the right side while tensile or compressive self-equilibrated vertical forces Fv = 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

    ΠB,n=4n2V¯nψn(Fh+Fv)u. 4.8
    Its stationarity with respect to the angle 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
    F=F¯B(ϕ)=F¯U(ϕ)1+ρ,ϕ0,withF¯B,cr=limϕ0F¯B(ϕ)=F¯U,cr1+ρ, 4.9
    The stability of the system equilibrium recalls the one illustrated in figure 4 for the tensile test. However, as expected, the (horizontal) critical load (4.9)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 sn(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 ln(j) so that their current stretches λn(j) read as

    λn(j)=λs,n(j)λp,λs,n(j)=ln(j)lp,n,ln(j)=lp,n2+(sn(j))2, 4.10
    the unknown sliding amounts sn(j) being in principle different along the vertical direction. Therefore, the total potential energy is
    ΠS,n=2nV¯n(j=1nψS,n(j))F(j=1nsn(j)), 4.11
    where ψS,n(j)=ψ~|λ~=λn(j), and the associated equilibrium problem consists in the following set of n equations:
    ΠS,nsn(m)=0,m{1,,n}2nV¯nψS,n(m)sn(m)F=0,m{1,,n}, 4.12
    which admits the unique (stable) solution
    sn(j)=sn=sn,j{1,,n},F=F¯S(s)=AμλpsαL[1(λp4α2L2+s22αL)22ν]=2F^S(s). 4.13
    This highlights a uniform relative sliding of the tesserae along the structure’s height obeying the force-displacement relation F¯S(s), s indicating the horizontal displacement on the top side of the whole structure and 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.
    Figure 9.

    Figure 9. Loads and kinematical response of the biaxial (n × n) system under simple shear (top). The deformation is here governed by mutual horizontal sliding—without rotation—of adjacent tesserae, finally producing the maximum (horizontal) displacement at the top side. At the bottom, a photo of the 3D-printed prototype undergoing the shear mechanism. (Online version in colour.)

    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 ANSYS® [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 10a 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 10b).

    Figure 10.

    Figure 10. (a) Response of the seven-by-seven structure equipped with Hencky-type elastic bands under uni-axial tension: comparison of FE and theoretical outcomes. (b) Kinematics of a three-by-three elementary prototype under bi-axial and shear conditions experimentally reproduced. (c) Aperiodic kinematics of the FE model under uni-axial regime for non-uniform pre-stretch distributions, i.e. pre-tension confined in a region of the system, anisotropic and randomly distributed pre-stress. (Online version in colour.)

    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 10c.

    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 [15,19,2630]. 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 2B1 and 2B2, 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 x0=x1e^1+x2e^2, where e^1 and e^2 are the unit vectors of the selected biaxial Cartesian coordinate system having the x1-axis coinciding with the body’s axis and x0 varies within the domain D=[0,2L]×[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 gn that, by virtue of geometrical arguments, read as

    gn(x0)=h=0n{Qx0+2Lnh[(secϕcosϕ)e^1sinϕe^2]}[H(x1Ln(2h1+δh0))H(x1Ln(2h+1δhn))], 6.1
    where h{0,,n} is an index that identifies one of the n + 1 rigid bars comprising the body, H() is the Heaviside function, δpq is the Kronecker symbol and Q indicates the rotation tensor
    Q=cosϕe^1e^1sinϕe^1e^2+sinϕe^2e^1+cosϕe^2e^2. 6.2
    Accordingly, the sequence of deformation gradients describing the deformation of the discrete units can be written as
    gn=Q, 6.3
    this being uniform along the entire body and independent of 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(x0)=limngn(x0)=Γx0,Γ=secϕe^1e^1sinϕe^1e^2+cosϕe^2e^2,G=limngn=Q. 6.4
    Herein, the limits are given in the sense of uniform convergence and the tensor 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
    g=Γ, 6.5
    so that, in general gG and their difference providing the so-called disarrangement tensor M as:
    M=gG=ΓQ=sinϕ(tanϕe^1e^1e^2e^1), 6.6
    which accounts for the deformation amount due to sub-macroscopic disarrangements. In more detail, the obtained results show that, in the limit in which the multi-element structure represents a continuous body made of an infinite number of rigid elements of infinitesimal length, the kinematics of its discrete units under tensile load—described by the approximating functions gn and by the associated deformation gradients gn—projects, at the macroscopic level, the mapping g and its gradient 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 11a). 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.
    Figure 11.

    Figure 11. Macroscopic deformations of uniaxially and biaxially structured continua in the studied cases. Note that, for instance in the uniaxial case, the combination of the deformation modes due to stretching (tensile) and shear (orthogonal to the beam axis) loads recover naturally a polar (Cosserat-like) continuum that generalizes the Timoshenko model. (Online version in colour.)

    Additionally, the kinematical tensor K=(g)1G 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 i() 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 (g,g), mapping the body from the reference to the actual deformed configuration, i.e.

    (g,G)=(g,g)(i,K). 6.7
    In the present case, the explicit form of the tensor K reads as
    K=I+tanϕe^2e^1, 6.8
    where I=e^1e^1+e^2e^2 is the identity tensor. Finally, it is worth noting that G and 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:
    0<detGdetg, 6.9
    which 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
    η=1detK=1detGdetg=0, 6.10
    the overall process resulting hence isochoric (detg=detG=detK=1).

    (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

    gn(x0)=h=0n(x0+hsne^2)[H(x1Ln(2h1+δh0))H(x1Ln(2h+1δhn))]. 6.11
    Therefore, the sequence of related deformation gradients results
    gn=I, 6.12
    each bar undergoing a rigid vertical (i.e. orthogonal to the axis of the structure) translation whose magnitude depends on its position along the system. In the limit of an infinite number of elements, such sequences uniformly converge to the following vector and tensor fields g and G:
    g(x0)=limngn(x0)=x0+s2L(x0e^1)e^2,G=limngn=I, 6.13
    in this way providing the SD (g, G) at the continuum level. Then, the classical deformation gradient turns out to be
    g=I+M,M=s2Le^2e^1, 6.14
    where the disarrangement tensor 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 11b). Furthermore, both at the level of the infinitesimal units and at the whole-body scale, the volume is preserved during the motion:

    detG=detg=detK=1, 6.15
    the rate of void here coherently vanishing (η = 0) and the purely microscopic deformation being described by the tensor
    K=IM. 6.16
    Interestingly, if we look synoptically at both the results obtained in stretching and shear for the uniaxial structure at its structured continuum limit, we can note that the tensile buckling load produces a kinematics that combines beam elongation with Timoshenko-like shear (the cross sections being no longer orthogonal to the beam axis after the deformation) (figure 11a) whereas the orthogonal point-load produces a shear deformation to finally give a sort of complementary version of the Timoshenko-beam kinematics (figure 11b), 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 x0=x1e^1+x2e^2, with {x1,x2}D¯=[0,2L]×[0,2L], the approximating functions moving it to its actual position in the deformed configuration at the sub-macroscopic scale can be written as

    gn(x0)=h,k=0n{Qx0+2Ln[(secϕcosϕ)(he^1+ke^2)+sinϕ(ke^1he^2)]}[H(x1Ln(2h1+δh0))H(x1Ln(2h+1δhn))][H(x2Ln(2k1+δk0))H(x2Ln(2k+1δkn))], 6.17
    where h{0,1,,n} and k{0,1,,n} are two translation indexes and Q is the rotation tensor given in equation (6.2). Then, the sequence of deformation gradients deriving from such approximating functions is
    gn=Q, 6.18
    which recalls the uniaxial tensile buckling case. In the continuum limit, the SD (g, G) of the biaxial body results
    g(x0)=limngn(x0)=secϕx0,G=limngn=Q, 6.19
    this indicating that the local rotation of the infinitesimal rigid tesserae produces, at the macroscopic scale, an uniform equi-biaxial elongation of the whole body (figure 11c). 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
    g=secϕI, 6.20
    while the disarrangement tensor M and the kinematical tensor K, respectively, take the forms
    M=sinϕ[tanϕI+e^1e^2e^2e^1],K=cosϕ[cosϕI+sinϕ(e^1e^2+e^2e^1)]. 6.21
    As a consequence, it is found that the SD in (6.19) allows for the respect of the accommodation inequality (6.9) over the full range of geometrically admissible rotations. Also, a void fraction arises during the motion, which increases with the opening degree
    η=sin2ϕ. 6.22
    In addition, with reference to the special case in which the described kinematics is related to an uniaxial tensile regime, one can observe that the auxetic behaviour previously highlighted for the discrete structure produces, in the limit of infinite units of infinitesimal size, a bulky structured continuum body macroscopically exhibiting a negative Poisson’s ratio. The gradient g in (6.20) indeed shows that equal and uniform principal stretches occur in both the loading and transverse directions (figure 11c), 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 ν¯ 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

    gn(x0)=k=0n(x0+ksne^1)[H(x2Ln(2k1+δk0))H(x2Ln(2k+1δkn))],gn=I. 6.23
    The deformation at the continuum level is thus provided by the pair of fields g and G such that
    g(x0)=limngn(x0)=x0+s2L(x0e^2)e^1,G=limngn=I. 6.24
    Consequently, the classical deformation gradient g, the disarrangements tensor M and the microscopic deformation tensor K are, respectively, given by
    g=I+M,M=s2Le^1e^2,K=IM, 6.25
    which show that the local rigid translation of the elementary tesserae projects a simple shear as macroscopic deformation, characterized by no bifurcation of the equilibrium (i.e. no critical load occurs) and an overall force–displacement relation in a form of a nonlinear elastic law whose initial stiffness modulus depends on the distribution of the initial pre-stress into the elastic links (figure 11d) [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.

    Figure 12.

    Figure 12. (a) Example of granular medium made of an assembly of packed particles that produces dilatancy associated with shear (re-edited from [32]). (b) Macro-view (top) and micro-view (bottom) of an example of translation-induced shear/dilatation coupling described via SDs theory (re-edited from [19]). (c) Macro-view (top) and micro-view (bottom) of the kinematical coupling between shear and contraction exhibited by the proposed uniaxial paradigm when undergoing tensile buckling. (d) Example of a combined shear/dilatation deformation mode obtained through the proposed 3D printed biaxial model. (Online version in colour.)

    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.

    Published by the Royal Society. All rights reserved.

    References