Daria Scerrato

Ivan Giorgio

Nicola Luigi Rizzi

Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations

Keywords: Non-linear elasticity, Second gradient models, Woven fabrics

, is applicable to fiber lattices in which three dimensional bending, twisting and stretching are significant as well as a resistance to shear distortion, i.e. to the angle change between the fibers. Some relevant numerical examples are exhibited in order to highlight the main features of the model adopted: in particular buckling and post-buckling behavior of pantographic parabolic lattices is investigated. The fabric of the metamaterial presented in this paper has been conceived to resist more e↵ectively in the extensional bias tests by storing more elastic bending energy and less energy in the deformation of elastic pivots: a comparison with a fabric constituted by beams which are straight in the reference configuration shows that the proposed concept is promising.

Introduction

Design and synthesis of new materials that satisfy some required specific characteristics is a very attractive challenge that researchers have tackled since many years in di↵erent branches of Physics as Electromagnetism, Optics or Mechanics. Those assumptions which usually are accepted to be valid while modeling 'natural' materials lead to useful simplifications on which many engineering applications have been based up to now. In particular two-dimensional and three-dimensional continuum models have been formulated based on so-called Cauchy assumptions, which lead to the classical definition of stress and strain states. However, and based on purely logical considerations, already Gabrio Piola (see [START_REF] Dell'isola | The complete works of Gabrio Piola: Volume I -Commented english translation[END_REF][START_REF] Dell'isola | At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola[END_REF]) clearly proved that not all conceivable materials can be modeled under the simplifying assumptions put forward by Cauchy, Poisson and Navier (for a discussion of this point see e.g. [START_REF] Dell'isola | At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: An underestimated and still topical contribution of Gabrio Piola[END_REF][START_REF] Dell'isola | The postulations á la D'Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results[END_REF] and references there quoted).

Higher gradient continuum models for metamaterials

Already Piola considered the possibility to include in the deformation energy of three-dimensional continua together with the first gradient of placement also its second and possibly higher gradients, and he bases his argument on the eventual need to include in these models the description of long range interaction between the material particles. Piola's point of view has been recovered, many years later, for instance by Mindlin and Toupin [START_REF] Mindlin | Second gradient of strain and surface-tension in linear elasticity[END_REF][START_REF] Toupin | Theories of elasticity with couple-stress[END_REF]. When dealing with two-dimensional continua the classical models due to Love and Kirchho↵ do include higher gradient of transverse displacements as independent variables in the constitutive equation for deformation energy but, when considering tangential displacements, they restrict the attention to the particular case of dependence on first gradient of aforementioned tangential displacements. In the more recent papers [START_REF] Altenbach | On the linear theory of micropolar plates[END_REF][START_REF] Altenbach | On generalized Cosserat-type theories of plates and shells: a short review and bibliography[END_REF][START_REF] Altenbach | On the shell theory on the nanoscale with surface stresses[END_REF][START_REF] Eremeyev | The influence of surface tension on the e↵ective sti↵ness of nanosize plates[END_REF] and [START_REF] Dell'isola | A two-dimensional gradient-elasticity theory for woven fabrics[END_REF][START_REF] Giorgio | Buckling modes in pantographic lattices[END_REF][START_REF] Steigmann | Equilibrium of elastic nets[END_REF][START_REF] Steigmann | Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching[END_REF][START_REF] Steigmann | Theory of elastic solids reinforced with fibers resistant to extension, flexure and twist[END_REF] this last restriction is removed and so-called geodesic bending is taken into account for the determination of deformation energy. The more general models thus formulated allow for the theoretical framing of more sophisticated models which are able to describe the behavior of a large class of metamaterials, as those considered in the present paper.

Indeed when fabrics are constituted at micro-level by highly inhomogeneous materials and are formed by microscopically complex geometric patterns then the modeling assumptions accepted after Cauchy must be generalized if macroscopic homogenized models need to be introduced (see e.g. [START_REF] Dell'isola | The postulations á la D'Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results[END_REF]). More and more often such fabrics are attracting the attention of the researchers in material sciences: indeed so-called tailored or architectured or optimized or smart materials are more and more often being conceived and studied because of their specific and unconventional behavior (for more details on this subject see e.g. [START_REF] Vescovo | Dynamic problems for metamaterials: review of existing models and ideas for further research[END_REF][START_REF] Carcaterra | Macroscopic description of microscopically strongly inhomogenous systems: A mathematical basis for the synthesis of higher gradients metamaterials[END_REF][START_REF] Dell'isola | Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence[END_REF] and the references there cited).

The word metamaterials is a neologism which was constructed from the Greek word meta-, (meaning to go beyond ) composed with the Latin root materia. Metamaterials are materials designed and engineered in order to have properties which have not yet been observed in nature, which go beyond those materials which are already known. One has to remark, however, that if a property was not observed yet in nature may simply mean that nobody looked for it, due to the lack of suitable theoretical tools of investigation and modeling and that with a careful search one can find even natural materials having such an exotic property.

It seems to us that, in the context of Mechanical Sciences, this new concept focused on the design and synthesis of new materials, rather than on the analysis of common materials already employed, is quite little exploited when compared with what is done in the other fields of Physics already mentioned.

It also seems to us that the introduction of higher gradient models may be of help in the investigation and design of a large class of metamaterials, although we are aware of the fact that more generally microstructured continua [START_REF] Forest | Micromorphic approach for gradient elasticity, viscoplasticity, and damage[END_REF][START_REF] Forest | Nonlinear microstrain theories[END_REF] may be necessary: in this context, the results presented in [START_REF] Seppecher | Linear elastic trusses leading to continua with exotic mechanical interactions[END_REF][START_REF] Alibert | Truss modular beams with deformation energy depending on higher displacement gradients[END_REF][START_REF] Alibert | Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof[END_REF] prove rigorously that for a particular class of micro-fabrics the macro-models must be second gradient continua. The controversy about the relevance of higher gradient continua seems to have been solved by several results proving that many systems showing microscopic complexity can be modeled, at a suitably large scale, by higher gradient continua (see e.g. [START_REF] Alibert | Second-gradient continua as homogenized limit of pantographic microstructured plates: a rigorous proof[END_REF][START_REF] Camar-Eddine | Determination of the closure of the set of elasticity functionals[END_REF][START_REF] Carcaterra | Macroscopic description of microscopically strongly inhomogenous systems: A mathematical basis for the synthesis of higher gradients metamaterials[END_REF][START_REF] Cecchi | Heterogeneous elastic solids: A mixed homogenization-rigidification technique[END_REF][START_REF] Reis | Construction of micropolar continua from the asymptotic homogenization of beam lattices[END_REF][START_REF] Reis | Homogenized elastoplastic response of repetitive 2D lattice truss materials[END_REF][START_REF] Goda | Equivalent mechanical properties of textile monolayers from discrete asymptotic homogenization[END_REF][START_REF] Hans | Dynamics of discrete framed structures: a unified homogenized description[END_REF][START_REF] Pideri | A second gradient material resulting from the homogenization of an heterogeneous linear elastic medium[END_REF][START_REF] Rahali | Homogenization à la Piola produces second gradient continuum models for linear pantographic lattices[END_REF]). Actually an exhaustive review on the conceptual bases of higher gradient continuum theories may be found in [START_REF] Dell'isola | The postulations á la D'Alembert and á la Cauchy for higher gradient continuum theories are equivalent: a review of existing results[END_REF] while interesting applications are found in [START_REF] Placidi | A variational approach for a nonlinear one-dimensional damage-elasto-plastic second-gradient continuum model[END_REF][START_REF] Placidi | A variational approach for a nonlinear 1-dimensional second gradient continuum damage model[END_REF][START_REF] Rinaldi | A microscale second gradient approximation of the damage parameter of quasibrittle heterogeneous lattices[END_REF][START_REF] Yang | Higher-order stress-strain theory for damage modeling implemented in an elementfree Galerkin formulation[END_REF][START_REF] Yang | Higher-order continuum theory applied to fracture simulation of nanoscale intergranular glassy film[END_REF][START_REF] Seddik | A full Stokes ice flow model for the vicinity of Dome Fuji, Antarctica, with induced anisotropy and fabric evolution[END_REF][START_REF] Chiaia | Experimental analysis and modeling of two-way reinforced concrete slabs over di↵erent kinds of yielding supports under short-term dynamic loading[END_REF]. Remark also that nonlinear higher gradient elasticity is necessary also when the correct frame for continua having energetic boundaries is looked for (see e.g. [START_REF] Javili | Geometrically nonlinear higher-gradient elasticity with energetic boundaries[END_REF]).

Range of applicability for generalized continuum models

In technological applications many and di↵erent micro-fabrics are considered to form micro-architectured metamaterials. The di↵erent mechanical parts constituting these fabrics may be fibers, micro-beams, micro-plates or any other kinds of structural element. All considered structural elements may be constrained by suitable elastic or perfect constraints and their mechanical properties may be extremely di↵erent each other. A possible way for assembling fibers could be to weave them: in this case the constraint is obtained by means of friction forces whose e↵ectiveness may depend on the state of stress at the contact interface between di↵erent fibers. In this case a particular attention must be paid to frictional slip (as done for instance in [START_REF] Nadler | A model for frictional slip in woven fabrics[END_REF]) while other peculiar properties of several kind of composite materials [START_REF] Nikopour | Torsion of a layered composite strip[END_REF][START_REF] Nikopour | Concentrated loading of a fibre-reinforced composite plate: Experimental and computational modeling of boundary fixity[END_REF][START_REF] Selvadurai | Transverse elasticity of a unidirectionally reinforced composite with an irregular fibre arrangement: Experiments, theory and computations[END_REF][START_REF] Harrison | Modelling the forming mechanics of engineering fabrics using a mutually constrained pantographic beam and membrane mesh[END_REF] have also been taken into account.

In the literature many di↵erent generalized continuum models have been proposed for mechanical systems including inextensible and extensible fibers: see e.g. [START_REF] Steigmann | Equilibrium of elastic nets[END_REF][START_REF] Steigmann | Theory of elastic solids reinforced with fibers resistant to extension, flexure and twist[END_REF][START_REF] Giorgio | Buckling modes in pantographic lattices[END_REF][START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: Heuristic homogenisation, experimental and numerical examples of equilibrium[END_REF][START_REF] Dell'isola | Plane bias extension test for a continuum with two inextensible families of fibers: a variational treatment with Lagrange Multipliers and a perturbation solution[END_REF] and references there cited. To our knowledge, however, it has not been addressed yet the problem of studying the deformation of second gradient plates having two families of extensible material curves having nonvanishing referential curvature and being capable of storing deformation energy when their curvature is changing. Some relevant results in the formulation of the needed theories can be found in [START_REF] Eremeyev | Local symmetry group in the general theory of elastic shells[END_REF][START_REF] Eremeyev | Material symmetry group of the non-linear polar-elastic continuum[END_REF][START_REF] Eremeyev | Material symmetry group and constitutive equations of micropolar anisotropic elastic solids[END_REF] where local symmetry properties for elastic generalized shells are studied and in [START_REF] Altenbach | On the linear theory of micropolar plates[END_REF][START_REF] Altenbach | On generalized Cosserat-type theories of plates and shells: a short review and bibliography[END_REF][START_REF] Altenbach | On the shell theory on the nanoscale with surface stresses[END_REF][START_REF] Eremeyev | The influence of surface tension on the e↵ective sti↵ness of nanosize plates[END_REF] where some theories of plates and shells with microstructure are presented. Remark that the problems addressed in the present paper are static. A dynamical analysis of second gradient plates needs to be developed and for the first results in this context presented the reader is referred e.g. to [START_REF] Dell'isola | Elastic pantographic 2D lattices: a numerical analysis on static response and wave propagation[END_REF][START_REF] Dell'isola | Pantographic 2D sheets: Discussion of some numerical investigations and potential applications[END_REF][START_REF] Frischmuth | Free vibrations of finite-memory material beams[END_REF][START_REF] Soubestre | Non-local dynamic behavior of linear fiber reinforced materials[END_REF]. Besides, considering that the system under study is very light, applications in which there is a fluid-structure interaction could show an unexpected behavior which seems worthy of study (see, e.g., [START_REF] Pagnini | Reliability analysis of wind-excited structures[END_REF][START_REF] Nguyen | Aeroelastic instability and wind-excited response of complex lighting poles and antenna masts[END_REF][START_REF] Carassale | E↵ects of free-stream turbulence and corner shape on the galloping instability of square cylinders[END_REF][START_REF] Pagnini | Model reliability and propagation of frequency and damping uncertainties in the dynamic along-wind response of structures[END_REF][START_REF] Pagnini | Serviceability criteria for wind-induced acceleration and damping uncertainties[END_REF] for more details on this issue). It has also to be remarked that non-standard and exotic dynamical behavior can be described in some particular micromorphic continua [START_REF] Placidi | Euromech 563 Cisterna di Latina 17-21 March 2014 Generalized continua and their applications to the design of composites and metamaterials: A review of presentations and discussions[END_REF] and in multiphysics metamaterials, as those conceived to exploit piezoelectric transduction see e.g. [START_REF] Rosi | Control of sound radiation and transmission by a piezoelectric plate with an optimized resistive electrode[END_REF][START_REF] Porfiri | Circuit analog of a beam and its application to multimodal vibration damping, using piezoelectric transducers[END_REF][START_REF] D'annibale | Linear stability of piezoelectric-controlled discrete mechanical systems under nonconservative positional forces[END_REF][START_REF] Pagnini | The three-hinged arch as an example of piezomechanic passive controlled structure[END_REF] and the references there cited.

Finally it has to be remarked that microscopically complex systems are not designed by engineers only: indeed Nature, and in particular evolution, produced many tissues whose microscopic fabric is very complex: some e↵orts are being directed towards the formulation of generalized continuum models in this context: some relevant works are [START_REF] Andreaus | A 2-D continuum model of a mixture of bone tissue and bio-resorbable material for simulating mass density redistribution under load slowly variable in time[END_REF][START_REF] Federico | Elasticity and permeability of porous fibre-reinforced materials under large deformations[END_REF][START_REF] Tomic | Poroelastic materials reinforced by statistically oriented fibresnumerical implementation and application to articular cartilage[END_REF][START_REF] Giorgio | The influence of di↵erent geometries of matrix/sca↵old on the remodeling process of a bone and bioresorbable material mixture with voids[END_REF][START_REF] Goda | A 3D elastic micropolar model of vertebral trabecular bone from lattice homogenization of the bone microstructure[END_REF][START_REF] Grillo | Growth, mass transfer, and remodeling in fiber-reinforced, multiconstituent materials[END_REF][START_REF] Grillo | Remodelling in statistically oriented fibre-reinforced materials and biological tissues[END_REF][START_REF] Lekszycki | A mixture model with evolving mass densities for describing synthesis and resorption phenomena in bones reconstructed with bio-resorbable materials[END_REF][START_REF] Placidi | On the role of grain growth, recrystallization and polygonization in a continuum theory for anisotropic ice sheets[END_REF][START_REF] Seddik | A full Stokes ice flow model for the vicinity of Dome Fuji, Antarctica, with induced anisotropy and fabric evolution[END_REF].

The presented method features also the possibility of describing buckling and post-buckling phenomena, as in its deformation energy some non-quadratic terms depending on some deformation energies are introduced. The buckling and post-buckling analysis performed here is purely numerical: we are aware of the fact that only via suitable analytical or semi-analytical studies (those presented in [START_REF] Ruta | A beam model for the flexural-torsional buckling of thin-walled members with some applications[END_REF][START_REF] Pignataro | The e↵ects of warping constraints on the buckling of thin-walled structures[END_REF][START_REF] Pignataro | E↵ects of warping constraints and lateral restraint on the buckling of thin-walled frames[END_REF][START_REF] Rizzi | The e↵ects of warping on the postbuckling behaviour of thin-walled structures[END_REF][START_REF] Rizzi | On the postbuckling analysis of thin-walled frames[END_REF][START_REF] Gabriele | On the imperfection sensitivity of thin-walled frames[END_REF][START_REF] Rizzi | Initial postbuckling behavior of thin-walled frames under mode interaction[END_REF][START_REF] Aminpour | A one-dimensional continuum with microstructure for single-wall carbon nanotubes bifurcation analysis[END_REF], [START_REF] Luongo | Mode localization in dynamics and buckling of linear imperfect continuous structures[END_REF][START_REF] Luongo | On the e↵ect of twist angle on nonlinear galloping of suspended cables[END_REF][START_REF] Carassale | Non-linear discrete models for the stochastic analysis of cables in turbulent wind[END_REF] and [START_REF] Challamel | Nonlocal equivalent continua for buckling and vibration analyses of microstructured beams[END_REF][START_REF] Wang | Hencky bar-chain model for buckling and vibration of beams with elastic end restraints[END_REF][START_REF] Challamel | Analytical length scale calibration of nonlocal continuum from a microstructured buckling model[END_REF] seem to us relevant in our context) it will become possible a complete classification of such behaviors.

Experimental and numerical characterization of higher gradient constitutive parameters

In order to use the introduced second gradient model to get e↵ective predictions of considered metamaterials (as done in [START_REF] Dell'isola | Designing a light fabric metamaterial being highly macroscopically tough under directional extension: first experimental evidence[END_REF]) one has to identify macroscopic constitutive parameters in terms of the specific micro-structure under consideration. In this paper we have used for obtaining such identification the semi-analytical results presented in [START_REF] Placidi | Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coe cients[END_REF]. We are aware that this analysis needs to be improved and generalized. We intend, in future investigations, to use to this aim several numerical and experimental methodologies: the most relevant in this context seem to be e.g. those presented in [START_REF] Turco | Is the statistical approach suitable for identifying actions on structures?[END_REF][START_REF] Solari | A numerical algorithm for the aerodynamic identification of structures[END_REF] where the problem of the identification of macro-properties of structures is addressed or those in [START_REF] Dietrich | Problems of identification of mechanical characteristics of viscoelastic composites[END_REF][START_REF] Lekszycki | Modelling and identification of viscoelastic properties of vibrating sandwich beams[END_REF] where viscosity e↵ects are introduced in the picture. Remark that pantographic structures considered here include small elements in which a relatively larger amount of deformation energy may be stored: therefore, experimental non invasive detection of damage methods based on dynamic features as natural frequency, eigen-modes may be used (see e.g. [START_REF] Vescovo | Theoretical and experimental dynamic analysis aimed at the improvement of an acoustic method for fresco detachment diagnosis[END_REF]) together with dynamic characterization and vibration absorption methods (see e.g. [START_REF] Carcaterra | Trapping of vibration energy into a set of resonators: Theory and application to aerospace structures[END_REF][START_REF] Roveri | Vibration absorption using non-dissipative complex attachments with impacts and parametric sti↵ness[END_REF][START_REF] Roveri | Damage detection in structures under traveling loads by Hilbert-Huang transform[END_REF][START_REF] Carcaterra | An iterative rational fraction polynomial technique for modal identification[END_REF][START_REF] Carcaterra | Tire grip identification based on strain information: Theory and simulations[END_REF]) or even the more sophisticated impact analysis (see e.g. [START_REF] Andreaus | Friction oscillator excited by moving base and colliding with a rigid or deformable obstacle[END_REF][START_REF] Andreaus | Soft-impact dynamics of deformable bodies[END_REF][START_REF] Andreaus | Dynamics of SDOF oscillators with hysteretic motion-limiting stop[END_REF][START_REF] Andreaus | Experimental and numerical investigations of the responses of a cantilever beam possibly contacting a deformable and dissipative obstacle under harmonic excitation[END_REF]).

While experimental evidence is the ultimate check for every modeling e↵ort, also in the context of microscopically complex fabrics, it can be useful to get quantitative and qualitative results about their behavior by resorting to micro-models and intensive numerical simulations, based on simpler mechanical models valid at lower length scales. In this context isogeometric numerical analysis (see e.g. [START_REF] Turco | A three-dimensional B-spline boundary element[END_REF][START_REF] Cazzani | Isogeometric analysis of plane-curved beams[END_REF][START_REF] Cazzani | Constitutive models for strongly curved beams in the frame of isogeometric analysis[END_REF][START_REF] Greco | An implicit G1 multi patch B-spline interpolation for Kirchho↵-Love space rod[END_REF][START_REF] Greco | An isogeometric implicit G1 mixed finite element for Kirchho↵ space rods[END_REF]) or other numerical methods (see e.g. [START_REF] Cazzani | Numerical aspects of coupling strongly frequency-dependent soil-foundation models with structural finite elements in the time-domain[END_REF][START_REF] Cazzani | On some mixed finite element methods for plane membrane problems[END_REF][START_REF] Cazzani | A four-node hybrid assumed-strain finite element for laminated composite plates[END_REF][START_REF] Turco | Elasto-plastic analysis of Kirchho↵ plates by high simplicity finite elements[END_REF][START_REF] Greco | On the force density method for slack cable nets[END_REF][START_REF] Greco | Consistent tangent operator for an exact Kirchho↵ rod model[END_REF][START_REF] Cesarano | Humbert polynomials and functions in terms of Hermite polynomials towards applications to wave propagation[END_REF]) have been successfully applied to very similar mechanical problems. Remark that recently some alternative methods ( see e.g. [START_REF] Della Corte | Referential description of the evolution of a 2D swarm of robots interacting with the closer neighbors: perspectives of continuum modeling via higher gradient continua[END_REF][START_REF] Koh | Molecular dynamics simulation of a solid platinum nanowire under uniaxial tensile strain: Temperature and strain-rate e↵ects[END_REF][START_REF] Misra | Reliability analysis of drilled shaft behavior using finite di↵erence method and Monte Carlo simulation[END_REF]) based on generalized cellular automata calculations have been proposed which seems suitable to describe numerically the time evolution of higher gradient continua.

Organization of the paper and the main result presented

In this paper, we want to explore the possibility of designing new fabric sheets with a particular arrangement of the fibers to obtain specific and uncommon mechanical features di↵erent from the usual woven fabrics in which the fibers are straight lines.

Herein, we focus on the following key idea: to use the fibers having a parabolic form in the reference configuration and resisting to variations of curvature. In this way we intend to exploit the benefit of the greater resistance given by curved beams to improve the extensional strength of the designed metamaterial. The model used here in order to describe this class of fabrics employs two-dimensional second gradient continuum theory of elastic surfaces to model three-dimensional placements and deformations of fibered pantographic sheets: this model has been recently developed by Steigmann and dell'Isola [START_REF] Steigmann | Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching[END_REF]. The results which we present indicate that the same amount of the same material can be re-organized at micro-level in order to form micro-structures whose extensional resistance is nearly one order of magnitude greater.

The paper is organized as follows: • Section 2 describes the features of the model employed in this paper; more specifically in subsect. 2.1 the kinematics of the considered 2D continuum is specified, together with its material symmetry class; in subsect. 2.2 the deformation measures are decomposed in order to take into account the material symmetry properties of a parabolic pantographic sheet; in subsect. 2.3 a suitable deformation energy is postulated. All equilibrium configurations will be determined by minimization methods. • Section 3 shows the most relevant numerical simulations which we have obtained. In particular, we show that: i) the force exerted in extensional bias test (in a suitably chosen optimal direction relative to fiber orientation) in parabolic pantographic sheet is larger than in the case of micro-structure formed by straight lines; ii) interesting out-of-plane buckling and post-buckling phenomena may occur beyond suitable thresholds in extension tests; iii) interesting wrinkling out-of-plane shapes are formed in the case of imposed planar shear and compression boundary displacements. The numerical integration scheme must be based on intensive application of Argyris planar finite element, as the space in which the minimization problem for second gradient energies is formed by the set of functions having integrable second order weak derivatives. This is the reason for which isogeometric methods (applied in very similar contexts in [START_REF] Greco | An implicit G1 multi patch B-spline interpolation for Kirchho↵-Love space rod[END_REF][START_REF] Greco | An isogeometric implicit G1 mixed finite element for Kirchho↵ space rods[END_REF][START_REF] Cazzani | Isogeometric analysis of plane-curved beams[END_REF]) seems suitable to supply a very e cient numerical tool. • Section 4 concludes the paper by indicating the novel properties of considered parabolic pantographic sheets. They include: i) higher extensional resistance in specific directions, ii) relatively low mass/resistance ratio, iii) localized patterns of deformation energy and iv) capability of producing specific wrinkling out-of-plane patterns.

2D Pantographic sheets with initially parabolic fibers 2.1. Kinematics

We consider a plane sheet formed from two families of fibers that initially are curved and lie parallel to the coordinate lines of a two-dimensional orthogonal coordinate system, i.e. confocal parabolas. We treat the sheet as a 2D continuum, so that introducing the parabolic coordinates {', } every line in which ' or are constant in the initial rectangular domain B is regarded as a fiber. Specifically, in a Cartesian coordinate system {X 1 , X 2 }, the fibers are defined by the curves of constant '

2X 2 = X 2 1 ' 2 ' 2 (1)
and the curves of constant

2X 2 = X 2 1 2 + 2 (2)
When a deformation occurs, the material particle that initially is at the point X = (X 1 , X 2 ) 2 B goes to the point in 3D space whose place is indicated by the map r(X 1 , X 2 ) : B ⇢ R 2 7 ! R 3 . By introducing the components of displacement u i along the three unit vectors of the Cartesian coordinate system {e i }, we can express the placement map as

r(X ↵ ) = X ↵ e ↵ + u i (X ↵ )e i (3)
with Latin indexes ranging from 1 to 3 and Greek indexes from 1 to 2. The derivatives of r are denoted by

a ↵ = r ,↵ (4)
The deformation gradient F = rr, thus can be write as

F = a ↵ ⌦ e ↵ (5)
Therefore, the Cauchy-Green deformation tensor is given by

C = F > F = C ↵ e ↵ ⌦ e (6)
As a result, the strain tensor E, in terms of its components, becomes

E ↵ = 1 2 (C ↵ ↵ ) with C ↵ = r i,↵ r i, (7)
with ↵ the Kronecker delta. Here we consider also the second gradient of the deformation, rF = rrr, i.e. the third-order tensor rF = F ,↵ ⌦ e ↵ , in order to describe the fiber curvatures and twist [START_REF] Dell'isola | A two-dimensional gradient-elasticity theory for woven fabrics[END_REF][START_REF] Steigmann | Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching[END_REF].

Fiber decompositions

Let {L(X), M (X)} be orthogonal families of unit vectors in the plane of B defining the fiber directions in the reference configuration. Assuming the fibers to be material curves with no relative slipping and tied together at their points of intersection, we can represent the fibers directions after the deformation with the families of unit vectors {l(X), m(X)} given by

l = F L, µm = F M (8)
where and µ are the fiber stretches. As a result, we may use {l, m} spanning the deformed tangent plane at the material point X to define the fiber shear angle by

sin = l • m (9)
For a parabolic net, from Eq. ( 1) we obtain

' 2 = 2 (kXk X 2 ) (10)
choosing the root of Eq. ( 1) that is always positive, and then we may evaluate its gradient

r' = 1 ' ✓ X kXk e 2 ◆ (11)
and, therefore, the vectors L(X) and M (X) as

L(X) = r' kr'k and M (X) = e 3 ⇥ L(X) (12)
Employing the fiber decomposition proposed in [START_REF] Steigmann | Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching[END_REF], the gradient of deformation may be represented as F = l ⌦ L + µm ⌦ M (13) and thus the Cauchy-Green deformation tensor is

C = 2 L ⌦ L + µ 2 M ⌦ M + µ sin (L ⌦ M + M ⌦ L) (14)
while the second gradient of the deformation can be written as [START_REF] Steigmann | Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching[END_REF] rrr

= (g l + K L n) ⌦ L ⌦ L + (g m + K M n) ⌦ M ⌦ M + ( + T n) ⌦ (L ⌦ M + M ⌦ L), (15)
with

g l = 2 ⌘ l p + (L • r ) l, g m = µ 2 ⌘ m q + (M • rµ) m (16) and = (L • rµ) m + µ m q = (M • r ) l + µ l p, (17)
in which ⌘ l and ⌘ m are the geodesic curvatures of the deformed fibers, l and m are the so-called Tchebychev curvatures, and

p = n ⇥ l, q = n ⇥ m and l ⇥ m = | cos |n (18)
define the orthogonal directions of the fibers on the deformed surface, while

K L = 2  l , K M = µ 2  m and T = µ⌧, (19)
where  l and  m are the normal curvatures of the deformed fibers and ⌧ measures the twist of the deformed surface. These are non-zero if the deformation is such as to generate a curvature of the surface in 3D space. Accordingly, they describe those parts of the fiber curvatures that can be attributed to surface flexure, whereas the geodesic curvatures represent the components of fiber curvatures in the tangent planes of the deformed surface.

Strain Energy

In this paper, as done in [START_REF] Giorgio | Pattern formation in the three-dimensional deformations of fibered sheets[END_REF], we postulate the first term of the elastic stored energy as follows

W I (" L , " M , J) = 1 2 Y L " 2 L + 1 2 Y M " 2 M G LM [ln(J) + 1 J] (20)
where Y L , Y M and G LM are positive material constants and suitable strain measures (see e.g. [START_REF] Steigmann | Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching[END_REF][START_REF] Giorgio | Pattern formation in the three-dimensional deformations of fibered sheets[END_REF]) are employed

" L = E ↵ L ↵ L = 1 2 2 1 " M = E ↵ M ↵ M = 1 2 µ 2 1 J = kL ↵ M r ,↵ ⇥ r , k = k l ⇥ µmk (21)
Indeed, " L and " M are measures of fiber extension along L and M directions, respectively and J is the area stretch. The second energy term may be assumed as follows

W II = 1 2 (A L |g l | 2 + A M |g m | 2 + A | | 2 + k L K 2 L + k M K 2 M + k T T 2 ), (22)
Therefore, a simple strain-energy function incorporating the curvilinear orthotropic symmetry associated with the initial fiber geometry is

W = W I (" L , " M , J) + W II (g l , g m , , K L , K M , T ) (23)

Numerical Examples

In this section, we show some numerical examples employing the model sketched above and proposed in [START_REF] Steigmann | Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching[END_REF], adopting a rectangular domain whose edges are in ratio 1:3 and consisting of a parabolic fiber net; in all the cases analyzed, it is assumed that the samples have the same arrangement of the fibers unless otherwise specified. The FE analysis is performed by using COMSOL Multiphysics, a software flexible enough to allow us to insert any kind of non-standard strain energies not necessarily included in its libraries. Specifically, we utilize Eq. ( 23) which is characterized by a term depending on the second gradient of displacement. For this reason, we adopt the Argyris element which is an element of class C 1 and thus, particularly suitable to approximate the solution of the problem under study.

In what follows, the above formulation is recast in a non-dimensional form by normalizing the elastic energy [START_REF] Cazzani | Numerical aspects of coupling strongly frequency-dependent soil-foundation models with structural finite elements in the time-domain[END_REF] with respect to a reference sti↵ness while the lengths are normalized with respect to the shorter edge. Non-dimensional quantities are denoted by a superimposed tilde.

The constitutive parameters assumed in the current analysis are listed below: ỸL = 100, ỸM = 100, GLM = 0.2, ÃL = kL = 0.01, ÃM = kM = 0.01, Ã = kT = 0.1 [START_REF] Cecchi | Heterogeneous elastic solids: A mixed homogenization-rigidification technique[END_REF] In the first case, we examine the standard bias extension test in which one of the shorter side is fixed and on the other a uniform displacement is imposed which is equal to 0.8 and orthogonal to the same side. In particular, Fig. 1 displays the arrangement of the fibers in the reference configuration (Fig. 1a), the equilibrium shape of the sample after the deformation and the new disposition of the net (Fig. 1b), the measure of the shear strain (Fig. 1c), and the second gradient energy (Fig. 1d). The plot of in Fig. 1c shows the presence of two distinct areas separated by a transition zone because of the presence of a second gradient energy term. This behavior is quite standard for a bias extension test of fabric sheets (see e.g. [START_REF] Giorgio | Pattern formation in the three-dimensional deformations of fibered sheets[END_REF]). However, in the common bias test performed on sheets with straight In the bias test considered, the displacement imposed on the sample under test along the direction of X 1 -axis, due to the particular arrangement of the parabolic fibers, induces a compression of the straight fibers parallel to the vector e 2 and therefore a buckling phenomenon occurs in correspondence of a critical displacement. This last-mentioned has been evaluated as ũ1 = 0.8984. Fig. 2 displays the buckling mode related to this critical displacement; the colors indicate the out-of-plane component of displacement, ũ3 . To determine an equilibrium shape related to the buckling mode, we take geometrical and mechanical imperfections into account by imposing on short sides the additional boundary condition on the derivatives of the displacement out-of-plane of the pantographic sheet and, in particular we set ũ3,↵ ⌫ ↵ = 2 ⇥ 10 4 , where ⌫ is the unit vector normal to the edge and on the plane determined by the vectors e 1 and e 2 .

In order to explore the features of the fiber arrangement considered, we compare two kind of samples: one constituted by a straight and orthogonal lattice of fibers and the other characterized by a parabolic net as it has been already analyzed. Specifically, we investigate the behavior of these two arrangements in the cases of a bias extensional test and a shear displacement imposed.

In the former case, we plot the equilibrium shapes for three imposed displacements along the direction of the X 1 -axis, ũ1 = {0.31, 0.62, 0.85}; in Fig. 3, the colors indicate the distribution of the shear angle while in Fig. 4 they are related to the total strain energy density. We can observe that the maximum value of the shear angle is almost the same for the two fiber dispositions, but the stored strain energy is much greater in the case of parabolic fibers. Similar considerations apply in the latter case when one short edge is fixed and a displacement is imposed in the direction of the X 2 -axis on the opposite side (see Fig. 5). Finally, for a quantitative comparison we show the overall constraint reactions by varying the imposed displacement in the two tests under examination (see Fig. 9) and once again, it is confirmed that the arrangement of the parabolic net is much sti↵er than the one with straight fibers.

Afterwards, the standard bias extension test is applied to a specimen with a di↵erent initial arrangement of the fibers (see Fig. 6). The considered sheet is deformed by fixing it at one shorter edge and assigning a uniform displacement of amplitude 1 at the opposite boundary so as to move away these two sides. Fig. 6 exhibits from left to right the fiber pattern prior to deformation, the equilibrium shape of the sample after the deformation and the new disposition of the net, the measure of the shear strain , and the second gradient energy. Similarly to the previous case, two main zones kept separate from transition regions can be noticed, i.e. one with a fiber shear angle close to zero and the other with an angle increase of about 75 degrees (see Fig. 6). This time, it is much more evident that the localization of the second gradient energy occurs along the transition regions.

In the next example, we impose a relative rotation and translation to the opposite shorter boundaries in order to cause bending, stretching and twisting in three dimensions. In more detail, we fix one edge and assign the following displacement field on the other edge 8 > < > :

ũ1 = 0.3 ũ2 = s 1 2 (cos # 1) ũ3 = s 1 2 sin # ( 25
)
where s is a parameter which varies from 0 to 1 and # is a rotation angle with respect to the longitudinal axis of rectangle, here assumed to be equal to ⇡/3. Fig. 7 shows: the equilibrium shape (Fig. 7a), where colors indicate the out-of-plane component of displacement, ũ3 and the fiber pattern is highlighted; the fiber shear angle (Fig. 7b); the first gradient energy (Fig. 7c); and the second gradient energy (Fig. 7d). Also in this case, we can observe a localization of both energies of first and second gradient near the largest parabolic fibers. We complete this gallery of examples with a case of buckling in which compressive and shear displacements, respectively ũ2 = 0.2 and ũ1 = 0.5, are imposed on one of the long sides; the opposite edge is fixed, and the short sides are left free. In addition, on the moving long side we assign the extraconstraint: ũ3, ⌫ = 1 ⇥ 10 4 s, where = {1, 2}, s is a parameter which varies from 0 to 1 and ⌫ is the unit vector normal to the edge and on the plane determined by the vectors e 1 and e 2 . Fig. 8 exhibits the fiber pattern and the equilibrium shape (Fig. 8a), where colors indicate the out-of-plane component of displacement, ũ3 ; the fiber shear angle (Fig. 8b); the first gradient energy (Fig. 8c); and the second gradient energy (Fig. 8d). It should be noted that at the central area of the sample a geodesic buckling appears in the plane of the fabric sheet.

Conclusions

In modern engineering there are three features which are more and more frequently demanded to novel materials: i) the capability to resist in an elastic way in large deformation regimes, ii) the capability to resist to applied load also when damage phenomena start to occur, iii) the capability of localizing deformation energy so that the parts of the system to be checked in order to assess its integrity are determined a priori.

In this paper, we try to prove the applicability and feasibility of the following concept: given a mass of an elastic material it is possible to arrange it in order to form a network of beams connected by cylinders (playing the role of elastic pivots) to get a fabric which is able to undergo large deformations remaining in elastic regimes and is capable to sustain externally applied loads even when some damage phenomenon occurs.

The presented analysis did not consider any model for damage onset and evolution: however we could verify that in the most relevant deformation patterns the conceived fabric actually shows high concentration of deformation energy: it is therefore likely that damage onset will be localized in these regions. Future investigations will address the relevant related modeling issues. An aspect of the micro-structure introduced here concerns the shape of the involved beams in the stress-free reference configuration: we have assumed it is parabolic. Indeed, we assumed that a parabolic system of coordinated in the reference configuration characterizes the material symmetries of the considered fabric. The enhanced bending deformation of such fibers (when comparing the performances of the present fabric with that constituted by straight lines) produces a greater sti↵ness in extensional bias test without changing the capability of undergoing large deformations in elastic regimes (see Fig. 9). Indeed to use interconnected fibers having a parabolic form in the reference configuration allows us to exploit the benefit of the greater resistance to deformation given by pre-curved beams. The designed metamaterial results to have an improved extensional strength. The model used here in order to describe parabolic pantographic sheets is based on the second gradient continuum theory of elastic surfaces which can undergo three-dimensional large placements and deformations recently developed by Steigmann and dell'Isola [START_REF] Steigmann | Mechanical response of fabric sheets to three-dimensional bending, twisting, and stretching[END_REF].

As a byproduct of performed numerical simulations, we prove that many wrinkling shapes of considered pantographic sheets are assumed in equilibrium conditions even when purely plane boundary displacements are imposed. These buckling phenomena are expected but not yet described in the literature together with their post-buckling evolution. We intend to systematically investigate these phenomena by means of the perturbative methods described in [START_REF] Piccardo | On the contribution of Angelo Luongo to Mechanics: in honor of his 60th birthday[END_REF].

A more di cult problem consists in looking for optimized micro-structures which are able to perform some assigned tasks: we claim that it can be of use the introduction of generalized continuum models in this kind of investigations.

( a )Figure 1 .

 a1 Figure 1. Bias extension test -case I.

Figure 2 .

 2 Figure 2. Bias extension test -case I: buckling shape related to the critical displacement.

Figure 3 .

 3 Figure 3. Comparison between straight and parabolic fibers: bias extensional test with imposed displacement {0.31, 0.62, 0.85}. The colors indicate the shear angle .

Figure 4 .

 4 Figure 4. Comparison between straight and parabolic fibers: bias extensional test with imposed displacement {0.31, 0.62, 0.85}. The colors indicate the total strainenergy density.

Figure 5 .

 5 Figure 5. Test with shear displacement imposed: Equilibrium shapes related to straight fibers (left), parabolic fibers (right). The colors indicate the total strainenergy density.

Figure 6 .

 6 Figure 6. Bias extension test -case II, in the given order: reference and actual fiber patterns; the fiber shear angle ; second gradient energy.

( a )

 a Equilibrium shape and fiber pattern (b) Fiber shear angle (c) First gradient energy (d) Second gradient energy

Figure 7 .

 7 Figure 7. Test with stretching (ũ 1 = 0.3) and twist (# = 60 degrees)

( a )

 a Equilibrium shape and fiber pattern (b) Fiber shear angle (c) First gradient energy (d) Second gradient energy

Figure 8 .Figure 9 .

 89 Figure 8. Example of buckling with shear (ũ 1 = 0.5) and compressive (ũ 2 = -0.2) displacement imposed.

Acknowledgments. The authors thank Professors David Steigmann and Francesco dell'Isola for helpful comments and advice through the study.