1 Introduction

Current research in metamaterials design is pushing to fill the gap between mathematical modeling and technological applications. Although both evolutionary selection in living organism and the past engineering scientifically based research have already promoted the conception of exotic metamaterials (the bone tissue is one example while woven fabrics gives another one) it is only a recent issue the systematic research of tailored materials having fixed well-determined a priori uses and applications. To meet all the requirements imposed by determined and well-specified applications it is needed to establish a designing procedure which involve the important step concerning the development of some predictive and computationally effective numerical tools. These tools will be then used to verify experimental measurements output and subsequently to design specifically adapted materials.

In this paper we focus on a specific, but in our opinion relevant, task: to compare the performances of a discrete model for pantographic lattices, sometimes also called pantographic sheets, with some interesting experiments. The interest in these structures resides in the exotic behavior that they have shown [3, 4] and their study seems promising. In particular pantographic structures:

  • are the actual realization of a (often disputed) continuum model: i.e. second gradient materials; indeed pantographic sheets are one of the first mechanical structures which have been proven, see [5], to need a second gradient models at a given macroscopic length-scale;

  • have been proven to have very promising properties in wave propagation, representing an example of effective wave-guides, see [6];

  • have shown promising toughness properties, which suggest that they could be fruitfully embedded in novel composite materials.

The comparison between an effective discrete model and some experiments shows, on one side, its robustness and, on the other side, the focus points which require strong attention when accurate results are desired.

In the following we briefly present the discrete model, Sect. 2, also mentioning the algorithm used to reconstruct the complete equilibrium path of the mechanical problem. Successively, in Sect. 3, we thoroughly discuss some experiments and the comparison with the corresponding numerical simulations. Finally, in Sect. 4, there are some concluding remarks and future perspectives.

2 Discrete Model for Pantographic Sheets

In this section we shortly describe the discrete Lagrangian model which we consider here to be possible model for planar pantographic structures. Their predictive performances will be analysed in the following Sect. 3. We limit ourselves to remark here that:

  • pantographic structures can be nowadays simply built using 3D printing technologies (see, e.g., Fig. 1) based on the concept first proposed in [4] where elastic pivots were realized with small size elastic cylinders interconnecting the two arrays of beams;

  • discrete models, see Fig. 2, are conceived by modelling interconnecting pivots as nodes linked each other by means of extensional (pairwise interaction, see Fig. 3a) and rotational springs, i.e. bending springs (triple interaction on a fiber, see Fig. 3b) and shear springs (triple interaction on a fiber of the same array and also on the nearest pivots on the other direction, see Fig. 3c).

Fig. 1
figure 1

Pantographic structure fabric built by using 3D printing technology

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Fig. 2
figure 2

Hencky-type mechanical model of a pantographic structure

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The kinematical description of the pantographic lattice model involves a finite configuration space. To be precise, the discrete model involves the introduction of a set of Lagrangian parameters specifying the position of all the material particles modelling the pivots. They are initially located in the nodes of the reference configuration and then they displace in the actual configuration. If we limit ourself to consider planar motions, only a set of 2N coordinates is sufficient (if N is the number of considered nodes, the generic of which has referential position given by \(P_{i,j}\), such a set of Lagrangian coordinates could be given by the corresponding actual position \(p_{i,j}\)).

The strain energies of the discrete model are the only kind of energy to be specified in hard devices deformations (in absence of relevant volume forces). The postulated expression for the Lagrangian discrete deformation energy \(W_{\mathrm {int}}\) (in terms of the Lagrangian coordinates \(p_{i,j})\) is completely defined specifying the contribution of each one kind of spring (see Figs. 2 and 3):

$$\begin{aligned} w_a&= \frac{1}{2}a (\ell -\ell _0)^2 \, , \end{aligned}$$
(1)
$$\begin{aligned} w_b&= b (\cos \beta + 1) \, , \end{aligned}$$
(2)
$$\begin{aligned} w_s&= \frac{1}{2}s (\sigma - \sigma _0)^2 \, , \end{aligned}$$
(3)

where \(\ell \) and \(\ell _0\) are the actual and the reference length, respectively, of pantographic bar (i.e. the distance between two consecutive pivots along fibers’ direction), \(\beta \) is the angle between two consecutive pantographic bars (in the reference configuration this angle is \(\pi \)) and \(\sigma \) is the angle between two fibers starting from the same pivot (in the reference configuration this angle is related to \(\alpha _1\) or \(\alpha _2\) and \(\sigma _0\)). Furthermore, a, b and s are the axial, bending and shear stiffnesses of each one type of spring, respectively.

Fig. 3
figure 3

Kinematics of axial (a), bending (b) and shear (c) springs: reference (dashed line) and actual (continuous line) configurations

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Some remarks:

  1. i.

    The shear springs used for the discrete model, and depicted in Figs. 2 and 3, are actually four, having the same stiffness, for each node or pivot, one for each quadrant and having origin, in the reference configuration, at \(P_{i,j}\).

  2. ii.

    The bending deformation energy is expressed by means of the \(\cos \beta \) instead of the corresponding angle \(\beta \), these two possibilities are equivalent, at least in principle, but the first, avoiding the use of \(\arccos (\cdot )\) function, results more convenient from the computational point of view since it produces a more compact and effective code.

In order to have a complete solution of the considered equilibrium problem, i.e. the displacements (from which can be easily evaluated the structural reaction and the forces or couples exerted by each spring) a step-by-step procedure was implemented to reconstruct the complete equilibrium path of the pantographic sheet.

We briefly sketch here the basic ingredients of the procedure. The total energy of the pantographic structure can be computed in a straightforward manner simply adding the strain contribution of each spring. Formally we can write:

$$\begin{aligned} W(\mathbf {d}) = W_{\mathrm {int}} - W_{\mathrm {ext}} = \sum _{e} ( w_{a}+w_{b}+w_{s} ) - W_{\mathrm {ext}}\, , \end{aligned}$$
(4)

where e ranges on all the springs, extensional, bending and shear, and \(W_{\mathrm {ext}}\) is the work of the external loads and all quantities on the RHS depend on the vector \(\mathbf {d}\) which collects the nodal displacements of the pantographic lattice.

The equilibrium problem which we want to consider is a mixed one: we assume that the displacements of some particles are imposed and that some externally conservative forces are applied to the remaining particles. Let us therefore decompose \(\mathbf {d}\) into the pair composed by two vectors: the assigned displacements \(\mathbf {u}_{a}\) and the free displacements \(\mathbf {u}\). For notational aims, we will reorder \(\mathbf {d}\) to get the decomposition

$$ \mathbf {d=}\left( \mathbf {u},\mathbf {u}_{a}\right) \, . $$

Because of our assumption we have that \(W_{\mathrm {ext}}\) depends only on \(\mathbf {u}\).

The nonlinear system of equilibrium equations is obtained by imposing that the first variation of W vanish:

$$\begin{aligned} \mathbf {s}(\mathbf {u})-\mathbf {p(u})=\mathbf {0}\, , \end{aligned}$$
(5)

where \(\mathbf {p(u})\) is the vector which collects the Lagrangian components of external forces (which may be assumed to be dead loads, for instance, so that \(\mathbf {p}\) becomes independent of \(\mathbf {u}\)) and \(\mathbf {s} (\mathbf {u})\) is the vector of the internal forces (called also, in the context of structural mechanics, structural reaction), as defined by :

$$\begin{aligned} \mathbf {s}(\mathbf {u})=\frac{\mathrm {d} W_{\mathrm {int}}}{\mathrm {d}\mathbf {u}}\,,\qquad \mathbf {p(u})=\frac{\mathrm {d}W_{\mathrm {ext}}}{\mathrm {d}\mathbf {u}}. \end{aligned}$$
(6)

The tangent stiffness matrix is defined as the derivative of the structural reaction \(\mathbf {s}(\mathbf {u})\) with respect to the displacement vector \(\mathbf {u}\), in formulas:

$$\begin{aligned} \mathbf {K}_{T}(\mathbf {u})=\frac{\mathrm {d}\mathbf {s}(\mathbf {u})}{\mathrm {d}\mathbf {u}}=\frac{\mathrm {d}^{2} W_{\mathrm {int}}}{\mathrm {d}\mathbf {u}^{2}}, \end{aligned}$$
(7)

The solution of the nonlinear equilibrium system of Eq. (5) can be found by means of an incremental-iterative procedure based on the Newton–Raphson scheme. We will limit ourselves to the case of equilibrium paths depending only by the single parameter \(\mathbf {\lambda .}\) Starting from an estimated point of the equilibrium path \((\lambda _{j},\mathbf {u}_{j})\) verifying that the residue \(\mathbf {r}\) of Eq. (5) is

$$\begin{aligned} \left\| \mathbf {r(u}_{j}\mathbf {,\lambda }_{j}\mathbf {)}\right\| \mathbf {\le \ }\eta \, , \end{aligned}$$
(8)

i.e.with a pair being an \(\eta \)-approximate solution of the equilibrium condition (5), the iterative scheme, once the step \(\varDelta \lambda \) is fixed, is obtained by constructing the \(\eta \)-approximate solution \(\mathbf {(u}_{j+1}=:\mathbf {u}_{j}+\varDelta \mathbf {u}_{j} \, \mathbf {,\lambda }_{j+1}:=\mathbf {\lambda }_{j}+\varDelta \lambda \mathbf {)}\) by using the iteration scheme reported in Table 1 to compute \(\varDelta \mathbf {u}_{j}\).

Table 1 Scheme of a basic algorithm to compute a new point of the equilibrium path \((\lambda _{j+1}, \mathbf {u}_{j+1})\) given the previous \((\lambda _{j}, \mathbf {u}_{j})\)
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Further details on the Hencky-type model and on the strategy used to compute the complete equilibrium path are contained in [1], in addition [7,8,9] report a comparison of numerical simulations, also with second gradient numerical model, with experimental results in the case of fiber push-out and extensional and bending cases.

Fig. 4
figure 4

Drawing of the specimen for the fiber push-out test

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3 Comparison with Experiments

The reader will remark that very few parameters are postulated to characterize the discrete model. On the contrary a wealth of experimental data are nearly perfectly fitted using these few parameters. In [3] an identification of the parameters of the continuum model in terms of the discrete model was proposed, see [5, 10, 11].

Below we consider three experiments performed on specimen built by using 3D printing technologies. Each one test is characterized by a different orientation of the fibers. By referring to Fig. 2, the first one considers the case \(\alpha _1=\pi /4\), the second one \(\alpha _1=\pi /6\) and the last one \(\alpha _1=\pi /3\). We remark that in the first case the fibers are orthogonal, contrarily to the second and third case. Another remarkable difference regards the boundary conditions. More precisely, in the first test the same extensional displacements have been assigned only on two fibers of the small side (using a small bridge, see subsection 3.1) whereas in the other cases, see subsections 3.2 and 3.3, the displacements are assigned on all the fibers which intersect the small side.

Fig. 5
figure 5

Push-out test: sequence of deformations for \(\lambda =0\), 0.5 and 1

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Fig. 6
figure 6

Push-out test: particular of the deformation near the push-out fibers

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3.1 Fiber Push-Out Test

On the basis of the technical drawing reported in Fig. 4, a specimen was built, by using the 3D printing technology, in polyamide (PA 2200) by a SLS Formiga P100. The Young’s modulus for this material was estimated between 1.5 and 1.7 GPa following the rules of EN ISO 527 and EN ISO 178.

This specimen was tested by clamping the entire left side and assigning an increasing displacement u (parallel to the larger sides) to two fibers of the right side until its maximum \(u_{\max }\) was reached by using the MTS Bionix system strength machine selecting a velocity of about 5 mm/min. From the drawing, it is clear that the assigned displacement engages only on two fibers thanks to the small bridge on the right side. We remark the there is a small gap, i.e. 2.2 mm, between the fibers and the small bridge on the right side. The reasons of this gap will be better clarified in the following.

Figure 5 reports three pictures taken during the test execution and distinct for the displacement parameter \(\lambda =u/u_{\max }\). It has to be highlighted the particular effect reported in Fig. 6a which makes clear the necessity of the gap on the specimen.

Numerical simulation of this test was performed by assuming for the spring stiffnesses the same values already used in [7] for similar specimens. These values are reported in the first row of Table 2 and correspond to the value of the Young’s modulus \(E=1.6\) GPa (an intermediate value of the declared range) estimated by using the suggestion reported in [12].

Table 2 Spring stiffnesses consequent to different values of the Young’s modulus
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In Fig. 7a is reported the comparison between the structural reaction on the left side of the specimen both for the experiment, in black, and for the numerical simulation, in red. The uncertainties on E and the awareness that this parameter could hardly affects all the stiffness parameters, see e.g. [8, 9, 13] for some insights, suggested us for trying to improve the curve fitting simply modifying the Young’s modulus, and consequently the stiffness parameters of the springs. A surely better fitting was obtained using \(E=1.2\) GPa, see Fig. 7b, and the consequent stiffness parameters of the springs, see the second row of Table 2. For this value of E, and for the consequent springs stiffnesses, we also report in Fig. 8 both the strain energy, distinct for axial, bending and shear, as \(\lambda \) increases and the density of the structural reaction on the left side. Particularly remarkable is the presence of negative (red) values which indicate compressions on the most of the center of the side.

Fig. 7
figure 7

Push-out test: structural reaction for different values of the Young’s modulus

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Fig. 8
figure 8

Push-out test: energy, structural reaction and its density computed by using the Hencky-type numerical model

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Figure 9 reports the deformations as \(\lambda \) increases, in particular for the cases \(\lambda =0.25\), 0.5, 0.75 and 1. In grey is reported the reference configuration whereas colors shows the density of the achieved energy level.

Fig. 9
figure 9

Push-out test: deformations by using the Hencky-type numerical model

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Finally, we observe that the numerical simulation reproduces, see Fig. 6b, the same remarkable effect reported in Fig. 6a.

It has to be remarked that the used springs stiffnesses give quite satisfactory results for what concerns the structural reaction fitting. Conversely, looking at the contraction in the central part of the specimen, the numerical simulation appears more thick than that detected during in the experiment. The reason of this difference is probably due to a non precise choice of the constitutive parameters used in the numerical simulations.

Fig. 10
figure 10

Drawing of a pantographic structure with non-orthogonal fibers: \(\alpha _1=\pi /6\)

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Fig. 11
figure 11

Pantograph with non-orthogonal fibers (\(\alpha _1=\pi /6\)): sequence of deformations for \(\lambda =0\), 0.5, 1

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3.2 Pantograph with Non-orthogonal Fibers: \(\alpha _1=\pi /6\)

This test concerns the specimen depicted in Fig. 10. In this case the most remarkable thing is the lack of orthogonality between intersecting fibers.

Also in this case the specimen was built using the polyamide in the 3D printing process. In this case both the left and right side are clamped and an assigned displacement u, parallel to the larger side of the specimen, was imposed on the right side until the value \(u_{\max }=23.7\) mm. Using the same strength machine and the same assigned velocity (5 mm/min), the three pictures, distinct by the displacement parameter \(\lambda =u/u_{\max }\), were taken, see Fig. 11.

The experience of the previous test suggested us to choose the springs stiffnesses using, as a fitting rule, the agreement between the deformations at the final stage both for the experiments and for the numerical simulations. This choice leads to the springs stiffnesses reported in Table 3.

Using these values for springs stiffnesses we obtained the results reported in Fig. 12 where we reported the energies and the structural reactions vs \(\lambda \) and the density of the structural reaction, for \(\lambda =1\), on the left side of the specimen. In Fig. 12b there is the comparison between the structural reaction evaluated during the experiment, in black, and that computed by the numerical simulation, in red. The closeness of the two curves it is remarkable.

Table 3 Stiffnesses of the springs
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Fig. 12
figure 12

Pantograph with non-orthogonal fibers (\(\alpha _1=\pi /6\)): energy, structural reaction and its density computed by using the Hencky-type numerical model

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Fig. 13
figure 13

Pantograph with non-orthogonal fibers (\(\alpha _1=\pi /6\)): deformations by using the Hencky-type numerical model

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Fig. 14
figure 14

Pantograph with non-orthogonal fibers (\(\alpha _1=\pi /6\)): deformations by using the Hencky-type numerical model

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Fig. 15
figure 15

Drawing of a pantographic structure with non-orthogonal fibers (\(\alpha _1=\pi /3\))

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Fig. 16
figure 16

Pantograph with non-orthogonal fibers (\(\alpha _1=\pi /3\)): sequence of deformations for \(\lambda =0\), 0.5 and 1

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Fig. 17
figure 17

Pantograph with non-orthogonal fibers (\(\alpha _1=\pi /3\)): energy, structural reaction and its density computed by using the Hencky-type numerical model

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Fig. 18
figure 18

Pantograph with non-orthogonal fibers (\(\alpha _1=\pi /3\)): deformations by using the Hencky-type numerical model

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Fig. 19
figure 19

Pantograph with non-orthogonal fibers (\(\alpha _1=\pi /3\)): deformations by using the Hencky-type numerical model

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The deformation history computed by using the numerical model is reported in Fig. 13 for \(\lambda =0.25\), 0.5, 0.75 and 1.

Finally we reported in Fig. 14 an overlapping of the picture taken at the final deformation of the experiment and that computed numerically which clearly shows the quality of the numerical model when an accurate fitting of the spring stiffnesses is preventively performed.

3.3 Pantograph with Non-orthogonal Fibers: \(\alpha _1=\pi /3\)

The last test concerns again a specimen made by the same polyamide of the previous tests and with non-orthogonal fibers but this time with a different orientation, see the drawing reported in Fig. 15.

Also in this case on the specimen was applied an assigned displacement on the right smaller side in the direction parallel to the larger sides until the value \(u_{\max }=74.7\) mm was reached. Three pictures were taken in the loading process (the strength machine and the velocity of the test are unchanged), see Fig. 16, corresponding to \(\lambda =0\), 0.5 and 1.

Numerical simulation of this experiment was performed using as springs stiffnesses those reported in Table 3. The results of the simulation are reported in Fig. 17, energies, structural reaction and density of structural reaction, and in Fig. 18, deformation history. We have to remark that in this case although there is a lack of closeness between the structural reaction given from the experiment and that computed numerically, mostly for values of \(\lambda >0.7\) (see Fig. 17b), the agreement on the whole set of displacement is again remarkable as can be observed by the overlapping of the two configurations at the final stage, see Fig. 19.

In our opinion this difference, unexpected if we consider the test with \(\alpha _1=\pi /6\), between the structural reaction evaluated in the experiment and computed numerically can be explained if we consider that in the \(\alpha _1=\pi /6\) test the ratio between the maximum assigned displacement and the long side, macro-strain, of the specimen is about 10% whereas the same quantity for the \(\alpha _1=\pi /3\) test is 53%. Looking again at Fig. 17b, if we consider only the first part of the curves (those corresponding approximatively to \(\lambda \le 0.4\)) then there is a remarkable agreement between them. We highlight that \(\lambda =0.4\) corresponds to a macro-strain of about 21% less than half of that imposed on the specimen (53%).

4 Concluding Remarks

In [5, 10, 14] more or less rigorous homogenization results are presented, in the framework of linear elasticity: i.e. small deformations and quadratic deformation energies. The model presented here, instead, tries to model the behavior of real pantographic structures undergoing large displacements. In the proposed experiments, while the majority of the beams constituting the pantographic lattice are in the small deformation regime, we can however distinguish some boundary layers in which the involved beam elements undergo very large deformations (more that 5% of elongations, for instance, as remarked in [3]). These experimental evidence compelled us to introduce strongly nonlinear models in order to be able to design a priori pantographic sheet having tailored properties.

While the numerical simulations show a surprising agreement with experimental evidence, we feel that a rigorous basis on the homogenization results presented in [3] needs to be firmly established. We expect that \(\varGamma \)-convergence results can be now confidently formulated and conjectured, see [15]. Moreover we expect that the methods exploited in [16] could be adapted to get also a priori error estimates in the replacement process involved when passing from discrete to continuum models.

A final remark is needed: many cases of out of plane buckling of exotic pantographic sheets were observed. A phenomenological model proposed in [17, 18] has been successfully used to get qualitative predictions. However to get more general quantitative predictions an identification procedure involving discrete Lagrangian models with concentrated springs is needed, which applies to three-dimensional motion of two-dimensional pantographic sheets.

Future challenges concern:

  1. i.

    Although pantographic structures were conceived to give an example of second gradient metamaterial, see e.g. [19,20,21,22,23,24,25,26,27,28,29], the development of 3D printing technology allowed for the practical synthesis of such metamaterials. It deserves to be investigated how to improve the design of 3D printed fabrics in order to fully exploit the exotic behaviour of higher gradient metamaterials. We remark that, as seen in [3, 30], the behaviour of higher gradient continua shows many peculiarities which deserve a deeper experimental investigation.

  2. ii.

    The discrete nature of suitably designed beam lattices may be modelled also by means of more refined tools, see e.g. [31,32,33,34,35,36,37,38] for an in-depth description of NURBS interpolation or using the generalized beam theory, see [39, 40], this in order to design even more complex metamaterials also in the 3D case where could be efficiently used the Pipkin model described in [41] and in the review paper [42].

  3. iii.

    Another crucial point concerns the modelling of the breakdown evolution of pantographic sheets. Indeed some evidence has been already gathered about the onset and the evolution of failure. It is rather evident that have to be considered ruptures concerning both fibers and pivots. A first modelling effort to model such rupture phenomena was presented in [43] when the attention was limited to the rupture mechanism initiated by the rupture of a fiber, see also [13] for an insight on the modelling of fiber defects. In this context, surely deserve models able to consider the out-of-plane deformations and the related buckling phenomena, see [44,45,46,47,48] for a quick insight on this argument.

  4. iv.

    The experimental identification of the parameters of the discrete model, i.e. the stiffnesses of the springs, require a specific investigation (see [12]). In particular methods of best fitting must be coupled to those used in extended sensitivity analysis by adapting, for example, the tools described in [49] and exploited in [50,51,52,53,54,55,56], see also [57, 58] for a more specific application to the description of huge and innovative structures.

  5. v.

    Experimental evidence shows the onset of some vibration phenomena in some specific experimental conditions. Therefore, it is relevant the extension of modelling to dynamic regimes, which can be obtained following the methods presented in see [59, 60] and also in [61, 62].

  6. vi.

    The discrete Hencky-type model and the related numerical discretization technique could also be used to model granular media interactions, see [63], or generalized and micro-structured continua, see [19, 64,65,66,67] and, in particular, [68, 69] for applications in civil engineering and [70] in biomechanics.

  7. vii.

    In various experiments the contact between fibers was observed; if this kind of phenomenon has to be considered they could be interesting the guidelines reported in [71, 72].