Emilio Barchiesi

Simon R Eugster

Francesco Dell'isola

François Hild

Large in-plane elastic deformations of bi-pantographic fabrics: asymptotic homogenisation and experimental validation

Keywords: variational asymptotic homogenisation, bi-pantographic fabrics, second gradient continua, additive manufacturing, local digital image correlation, Piola's ansatz, experimental mechanics

Bi-pantographic fabrics are composed of two families of pantographic beams and correspond to a class of architectured materials that are described in plane as second-gradient 2D-continua. On a discrete level, a pantographic beam is a periodic arrangement of cells and looks like an expanding barrier. The materialisation of a bi-pantographic fabrics made by Polyamide was achieved by additive manufacturing techniques. Starting from a discrete spring system, the deformation energy of the corresponding continuum is derived for large strains by asymptotic homogenisation. The obtained energy depends on the second gradient of the deformation through the rate of change in orientation and stretch of material lines directed along the pantographic beams. Displacementcontrolled bias extension tests were performed on rectangular prototypes for total elastic extension up to 25%.

Force-displacement measurements complemented by local digital image correlation analyses were used to t the continuum model achieving excellent agreement.

Introduction

Continuum modelling, i.e. spatially continuous formulations [START_REF] Harrison | Modelling the forming mechanics of engineering fabrics using a mutually constrained pantographic beam and membrane mesh[END_REF][START_REF] Andreaus | Numerical simulations of classical problems in two-dimensional (non) linear second gradient elasticity[END_REF][START_REF] Auray | A complete description of bi-dimensional anisotropic strain-gradient elasticity[END_REF][START_REF] Battista | Numerical investigation of a particle system compared with rst and second gradient continua: Deformation and fracture phenomena[END_REF][START_REF] Steigmann | The variational structure of a nonlinear theory for spatial lattices[END_REF], is routinely exploited to describe at macro length scales the collective behaviour of mostly periodic discrete systems, whose element-by-element micro-scale description [START_REF] Turco | a Fiber rupture in sheared planar pantographic sheets: Numerical and experimental evidence[END_REF][START_REF] Turco | b Large deformations induced in planar pantographic sheets by loads applied on bers: experimental validation of a discrete Lagrangian model[END_REF][START_REF] Turco | c Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Part I: numerical simulations[END_REF][START_REF] Turco | King post truss as a motif for internal structure of (meta) material with controlled elastic properties[END_REF][START_REF] Turco | Hencky-type discrete model for pantographic structures: numerical comparison with second gradient continuum models[END_REF] can get computationally challenging. Homogenisation procedures [START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium[END_REF][START_REF] Babu²ka | Homogenization approach in engineering[END_REF][START_REF] Allaire | Homogenization and two-scale convergence[END_REF][START_REF] Tartar | The general theory of homogenization: a personalized introduction[END_REF][START_REF] Yu | Variational asymptotic method for unit cell homogenization[END_REF] can be used to pass from a discrete to a continuous description. These procedures involve the denition of specic micromacro correspondences [START_REF] Dell'isola | The complete works of Gabrio Piola: volume I[END_REF], which enable to give a precise meaning to many features of the macro-model in terms of those of the micro-model. The last few decades have witnessed a high acceleration in the development of additive and subtractive techniques such as 3D-printing [START_REF] Golaszewski | Metamaterials with relative displacements in their microstructure: technological challenges in 3D printing, experiments and numerical predictions[END_REF]. Such techniques allow for micro-structure control at very small scales, which motivate the renewed interest in homogenisation [START_REF] Milton | a On the possible eective elasticity tensors of 2-dimensional and 3-dimensional printed materials[END_REF][START_REF] Milton | Towards a complete characterization of the eective elasticity tensors of mixtures of an elastic phase and an almost rigid phase[END_REF][START_REF] Abdoul-Anziz | Strain gradient and generalized continua obtained by homogenizing frame lattices[END_REF][START_REF] Barchiesi | Mechanical metamaterials: a state of the art[END_REF][START_REF] Cosmo | Acoustic Metamaterials Based on Local Resonances: Homogenization, Optimization and Applications[END_REF].

Pantographic structures [START_REF] Dell'isola | Designing a light fabric metamaterial being highly macroscopically tough under directional extension: rst experimental evidence[END_REF][START_REF] Giorgio | Buckling modes in pantographic lattices[END_REF][START_REF] Giorgio | Dynamics of 1D nonlinear pantographic continua[END_REF] are among the most straightforward examples of micro-structures whose continuum modelling gives a wealth of non-standard problems in the theory of higher-gradient [START_REF] Placidi | a A review on 2D models for the description of pantographic fabrics[END_REF][START_REF] Placidi | b Identication of two-dimensional pantographic structure via a linear D4 orthotropic second gradient elastic model[END_REF][START_REF]Numerical identication procedure between a micro-Cauchy model and a macro-second gradient model for planar pantographic structures[END_REF] and micromorphic continua [START_REF] Cuomo | Simplied analysis of a generalized bias test for fabrics with two families of inextensible bres[END_REF][START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium[END_REF][START_REF] Turco | Enhanced PiolaHencky discrete models for pantographic sheets with pivots without deformation energy: Numerics and experiments[END_REF][START_REF] Misra | Pantographic metamaterials show atypical Poynting eect reversal[END_REF], also of mathematical interest [START_REF] Eremeyev | Linear pantographic sheets: existence and uniqueness of weak solutions[END_REF]. Convenient discrete descriptions of pantographic structures have been obtained in the literature by Hencky-type modelling [START_REF] Turco | Enhanced PiolaHencky discrete models for pantographic sheets with pivots without deformation energy: Numerics and experiments[END_REF][START_REF] Turco | b Large deformations induced in planar pantographic sheets by loads applied on bers: experimental validation of a discrete Lagrangian model[END_REF][START_REF] Turco | c Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Part I: numerical simulations[END_REF].

The derivation of a 1D-continuum model being capable of describing the nite planar deformation of a discrete slender pantographic structure, referred to as pantographic beam, is presented in [START_REF] Barchiesi | A 1D Continuum Model for Beams with Pantographic Microstructure: Asymptotic Micro-Macro Identication and Numerical Results[END_REF]. The continuum model is deduced from a discrete one by applying a variational asymptotic procedure [START_REF] Alibert | Truss modular beams with deformation energy depending on higher displacement gradients[END_REF][START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium[END_REF][START_REF] Abdoul-Anziz | Strain gradient and generalized continua obtained by homogenizing frame lattices[END_REF][START_REF] Seppecher | Linear elastic trusses leading to continua with exotic mechanical interactions[END_REF]. Within the homogenisation process, the overall dimension of the system is kept xed, while the number of the periodically appearing subsystems called cells is increased, and the stinesses are appropriately scaled.

In [START_REF] Barchiesi | A 1D Continuum Model for Beams with Pantographic Microstructure: Asymptotic Micro-Macro Identication and Numerical Results[END_REF], the model of [START_REF] Alibert | Truss modular beams with deformation energy depending on higher displacement gradients[END_REF][START_REF] Seppecher | Linear elastic trusses leading to continua with exotic mechanical interactions[END_REF] has been generalized to the nite strain regime. Remarkably, the deformation energy density of such a 1D-continuum, [START_REF] Barchiesi | A 1D Continuum Model for Beams with Pantographic Microstructure: Asymptotic Micro-Macro Identication and Numerical Results[END_REF], does not only depend on the material curvature but also on the stretch gradient. Besides a more pedagogical presentation of such a continuum model, Barchiesi et al. [START_REF] Barchiesi | Pantographic beam: a complete second gradient 1D-continuum in plane[END_REF] addressed numerically the evaluation of dierences between the micro-and the macro-model in order to elucidate to what extent the continuum retains the relevant phenomenology of the discrete system. Special attention has been given to the dierence between the deformation energy of the micro-and the macro-model when the micro length scale tends to zero, i.e. the discretecontinuum error. This deviation gives a quantitative value to assess the quality of the approximation of the discrete by its continuous counterpart.

Bi-pantographic fabrics have been rst introduced by Seppecher et al. [START_REF] Seppecher | Linear elastic trusses leading to continua with exotic mechanical interactions[END_REF] as assemblies of discrete pantographic beams leading at macroscopic scale to second gradient materials [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][START_REF] Dell'isola | Cauchy tetrahedron argument applied to higher contact interactions[END_REF][START_REF] Dell'isola | A two-dimensional gradient-elasticity theory for woven fabrics[END_REF]. The corresponding deformation energy depends upon the rate of change in orientation and stretch of material lines directed along the pantographic beams. The aims of this work are the following ones. First, we want to generalize the homogenisation carried out in [START_REF] Seppecher | Linear elastic trusses leading to continua with exotic mechanical interactions[END_REF] in two respects. In particular, extensible elements and arbitrarily large strains are considered. Secondly, a possible design of bi-pantographic prototypes is sought, which is obeying the discrete model. Lastly, the derived results will be validated.

Addressing the above objectives leads to the following organisation of the paper. In Section 2, the discrete bi-pantographic structure is introduced followed by a homogenisation that is carried out by exploiting the results obtained for pantographic beams. In Section 3, we establish relations between quantities for the microscopic and macroscopic models, which go beyond Piola's micro-macro identication used throughout the homogenisation. Based on these relations, a non-standard bias extension test is then introduced for both models. Lastly, the nite element method employed to solve the continuum model is introduced with a special emphasis on the challenges arising from a weak mixed formulation. In Section 4, the design and manufacturing of a bi-pantographic prototype is reported together with the description of the experimental setup. The Digital Image Correlation (DIC) technique used to retrieve discrete displacement measures is also briey recalled. In Section 5, the tting of parameters by means of acquired experimental measures is presented and continuum is compared with experiments.

Heuristic homogenisation

The continuum is deduced by applying Piola's micro-macro identication procedure [START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium[END_REF][START_REF] Eugster | On the notion of stress in classical continuum mechanics[END_REF], which can be considered as a heuristic variational asymptotic procedure. The steps describing such a procedure can be sketched as:

(i) A family of discrete spring systems embedded in the two-dimensional Euclidean vector space E 2 , i.e. the micromodel with micro length scale ε > 0, is introduced generalized coordinates and energy contributions E ε are dened (ii) The kinematic descriptors of the continuum, i.e. the macro-model, are introduced as continuous functions with a closed subset of E 2 as their common domain these functions must be chosen such that their evaluation at particular points can be related to the generalized coordinates of the micro-model (iii) Formulation of the deformation energy of the micro-model E ε using the evaluation of the continuum descriptors at particular points, followed by a Taylor expansion of the energy with respect to the micro length scale ε (iv) Specication of scaling laws for the constitutive parameters in the micro-model followed by a limit process in which the energy of the continuum E is related to the micro-model by E = lim ε→0 E ε

Preliminaries

To ease the presentation, before addressing bi-pantographic structures, some preliminary computations related to pantographic beams are revisited. Pantographic beam discrete model. The assembly and kinematics of a discrete pantographic beam slightly generalizing that presented in [START_REF] Barchiesi | A 1D Continuum Model for Beams with Pantographic Microstructure: Asymptotic Micro-Macro Identication and Numerical Results[END_REF][START_REF] Barchiesi | Pantographic beam: a complete second gradient 1D-continuum in plane[END_REF] are sketched in Fig. 1. In the undeformed conguration, see Fig. 1(a), N cells are arranged upon a straight line along the direction of the unit basis vector e x ∈ E 2 . The total length L ∈ R of the undeformed pantographic beam accounts for N -1 cells, as depicted in Fig. 1(a). The cells are centred at the positions P i = iεe x for i ∈ {0, 1, . . . , N -1} with ε = L/(N -1). The basic i-th unit cell is formed by four extensional springs hinge-joined together at P i having length ε /(2 cos γ). Rotational springs, which are coloured in blue, red, and green in Fig. 1(d), are placed between opposite collinear and adjacent springs belonging to the same cell and between adjacent springs belonging to dierent cells. Note that extensional springs are rigid with respect to bending such that they can transmit torques. White-lled circles in Fig. 1 depict hinge constraints, requiring the end points of the corresponding springs to have the same position in space. We note that the assembly considered herein is a generalization of that studied in [START_REF] Barchiesi | A 1D Continuum Model for Beams with Pantographic Microstructure: Asymptotic Micro-Macro Identication and Numerical Results[END_REF], as the angle γ ∈ (0, π) between springs concurring at point P i from the right in Fig. [START_REF] Barchiesi | A 1D Continuum Model for Beams with Pantographic Microstructure: Asymptotic Micro-Macro Identication and Numerical Results[END_REF] is generally dierent from π /4. Moreover, further rotational springs, which are coloured in green in Fig. 1(d), are considered. When not otherwise mentioned, the indices i, µ and ν henceforth belong respectively to the following sets: i ∈ {0, 1, . . . , N -1}, µ ∈ {1, 2} and ν ∈ {D, S} 1 . The kinematics of the spring system is locally described by nitely many generalized coordinates. The coordinates are the positions p i ∈ E 2 of the points at position P i in the reference conguration and the lengths of the oblique deformed springs l µν i ∈ R. Various other kinematical quantities are considered to formulate the total potential energy in a most compact form. Applying the law of cosines, the angles ϕ µν i depicted in Fig. 1(c) are determined by the following relationships

ϕ 1D i = cos -1 p i+1 -p i 2 + l 1D i 2 -l 2S i+1 2 2l 1D i p i+1 -p i , ϕ 1S i = cos -1 p i -p i-1 2 + l 1S i 2 -l 2D i-1 2 2l 1S i p i -p i-1 , ϕ 2D i = cos -1 p i+1 -p i 2 + l 2D i 2 -l 1S i+1 2 2l 2D i p i+1 -p i , ϕ 2S i = cos -1 p i -p i-1 2 + l 2S i 2 -l 1D i-1 2 2l 2S i p i -p i-1 , (1)
while the angles ξ µ i depicted in Fig. 1(c) are determined by

ξ 1(2) i = cos -1 l 1(2)D i 2 + l 2(1)S i+1 2 -p i+1 -p i 2 2l 1(2)D i l 2(1)S i+1 . ( 2
)
For a ∈ E 2 , a = √ a • a corresponds to the norm induced by the inner product denoted by the dot. Note that ϕ µS 0 and ϕ µD N -1 cannot be determined by equations [START_REF] Harrison | Modelling the forming mechanics of engineering fabrics using a mutually constrained pantographic beam and membrane mesh[END_REF] and belong also to the set of generalized coordinates. Another restriction is that the choice of generalized coordinates holds only locally, as long as the angles ϕ 1D i and ϕ 2D i do not change sign. Throughout the derivation of the macro-model, it is assumed that the angles ϕ 1D i and ϕ 2D i remain in the range (0, π). This entails that ξ µ i ∈ (0, π). For the reduced index set i = {1, 2, . . . , N -2}, the angle between the two vectors p i -p i-1 and e x is denoted by ϑ i . Then the angle θ i between the vectors p i -p i-1 and p i+1 -p i reads

θ i = ϑ i+1 -ϑ i = tan -1 (p i+1 -p i ) • e y (p i+1 -p i ) • e x -tan -1 (p i -p i-1 ) • e y (p i -p i-1 ) • e x . (3)
Let us set θ 0 = θ 1 and θ N -1 = θ N -2 such that the deviation angles of two adjacent oblique springs from being collinear are given for the entire index set of i by

β 1 i = θ i + ϕ 1D i -ϕ 1S i , β 2 i = θ i + ϕ 2S i -ϕ 2D i . (4)
For the undeformed conguration, see Fig. 1(a), the following equalities are satised

l µν i = 1 2 cos γ ε , β 1 i = β 2 i = 0 , p i -p i-1 = ε . (5)
Letting the summations for i, µ and ν range over the above introduced sets {0, . . . , N -1}, {1, 2} and {D, S}, respectively, the micro-model deformation energy is dened as

E ε = k E 2 i µ,ν l µν i - 1 2 cos γ ε 2 + k F 2 i µ (β µ i ) 2 + k S 2 i µ (ξ µ i -π + 2γ) 2 (4) = k E 2 i µ,ν l µν i - 1 2 cos γ ε 2 + k F 2 i µ θ i + (-1) µ ϕ µS i -ϕ µD i 2 + k S 2 N -2 i=0 µ (ξ µ i -π + 2γ) 2 , (6)
where k E > 0 and k F , k S > 0 are the stinesses of the extensional and rotational springs, respectively. Boundedness of the deformation energy, both for the micro-model and for the macro-model is considered throughout this paper.

It is worth noting that, besides the rigid body modes, the set of admissible congurations dened by

l µν i = 1 2 cos γ ε , p i = p i-1 + Ke x , p 0 = P 0 , for K ∈ 0, 1 cos γ ε , (7)
also entails null deformation energy when k S = 0, i.e. when removing green springs in Fig. 1(d), and is referred to as extensional oppy mode [START_REF] Alibert | Truss modular beams with deformation energy depending on higher displacement gradients[END_REF]. Looking at the points p i , one observes uniform extension or compression. For the lengths l µν i of the oblique springs, the following asymptotic expansion is assumed

l µν i = 1 2 cos γ ε + ε 2 lµν i + o(ε 2 ) , (8)
where lµν i ∈ R. Inserting assumption (8) into the energy (6) leads to

E ε = k E 2 i µ,ν ε 2 lµν i + o(ε 2 ) 2 + k F 2 i µ θ i + (-1) µ ϕ µS i -ϕ µD i 2 + k S 2 i µ (ξ µ i -π + 2γ) 2 . (9)
Pantographic beam micro-macro identication. The slenderness of the discrete system makes it reasonable to aim for a one-dimensional continuum [START_REF] Steigmann | Variational theory for spatial rods[END_REF] in the limit of vanishing ε. The continuum is then parametrised by the arclength s ∈ [0, L] of the straight segment of length L connecting all points P i . The independent kinematic Lagrangian descriptors of the macro-model are assumed to be the functions χ : [0, L] → E 2 and lµν : [0, L] → R. The placement function χ places the 1D-continuum into E 2 and is best suited to describe the points p i ∈ E 2 of the discrete system on a macro-level. To take into account also the eect of changing spring lengths lµν i introduced in equation ( 8), the placement function is augmented by the four micro-strain functions lµν . The identication of the discrete system is possible with a one-dimensional continuum that is classied as a micromorphic continuum [START_REF] Germain | The Method of Virtual Power in Continuum Mechanics. Part 2: Microstructure[END_REF][START_REF] Forest | Nonlinear microstrain theories[END_REF][START_REF] Eremeyev | Foundations of micropolar mechanics[END_REF][START_REF] Altenbach | Cosserat-type rods[END_REF]. It is also convenient to introduce the functions ρ : [0, L] → R + and ϑ : [0, L] → [0, 2π) in order to rewrite the tangent vector eld χ to the deformed 1D-continuum as

χ (s) = ρ(s) [cos ϑ(s)e x + sin ϑ(s)e y ] , (10)
where prime denotes dierentiation with respect to the reference arc length s. Thus ρ corresponds to the norm of the tangent vector χ and is referred to as stretch. The current curve χ([0, L]) can, in general, have a length L 0 ρ ds dierent from L, as s is not an arc-length parametrisation for χ but for the reference placement χ 0 (s) = se x . Introducing the normal vector eld χ ⊥ (s) = ρ(s) [-sin ϑ(s)e x + cos ϑ(s)e y ], being rotated against χ (s) about 90 • in the anti-clockwise direction, the following results are found

ρ (s) = χ (s) • χ (s) χ (s) , ϑ (s) = χ (s) • χ ⊥ (s) χ (s) 2 . ( 11
)
In the sequel ρ and ϑ are called stretch gradient and material curvature, respectively. For Piola's micro-macro identication the generalized coordinates of the discrete system are related to the functions χ and lµν evaluated at

s i = iε as χ(s i ) = p i , lµν (s i ) = lµν i . (12)
For the asymptotic identication, the energy ( 9) is expanded in ε. The expansion of χ is given by

χ(s i±1 ) = χ(s i ) ± εχ (s i ) + ε 2 2 χ (s i ) + o(ε 2 ) . ( 13
)
Combining the asymptotic expansion [START_REF] Turco | c Non-standard coupled extensional and bending bias tests for planar pantographic lattices. Part I: numerical simulations[END_REF] with [START_REF] Babu²ka | Homogenization approach in engineering[END_REF] 2 and the expansion lµν (s i±1 ) = lµν (s i ) + o(ε 0 ), leads to

l µν i±1 = 1 2 cos γ ε + lµν (s i )ε 2 + o(ε 2 ) . (14)
In order to further expand [START_REF] Turco | King post truss as a motif for internal structure of (meta) material with controlled elastic properties[END_REF], the terms θ i , ϕ µS i -ϕ µD i and ξ µ i are expanded up to rst order (see App. A). For θ i according to equation ( 71)

θ i = ϑ (s i )ε + o(ε) . (15)
The dierences ϕ

1(2)S i -ϕ 1(2)D i
are given by equation (78) as

ϕ 1(2)S i -ϕ 1(2)D i = 4[ρ 2 -( 1 /2 cos 2 γ)]( l1(2)S -l1(2)D ) + ( 1 /cos γ)(ρ 2 ) + ( 2 /cos 2 γ)( l2(1)D -l2(1)S ) 4ρ( 1 /2 cos γ) ( 1 /cos 2 γ) -ρ 2 s=si ε + o(ε) . (16)
The angles ξ µ i are given by (80) as

ξ µ i = cos -1 1 - ρ 2 1 /2 cos 2 γ s=si + o(ε 0 ) . (17)
Substituting ( 15), ( 16) and ( 17) into (9) together with ρ(s i ) = χ (s i ) , the sought expansion of the micro-model energy E ε as a function of the kinematic descriptors χ and lµν reads

E ε = i k E ε 4 2 µ,ν lµν 2 + o(ε 0 ) + k S cos -1 1 - ρ 2 1 /2 cos 2 γ -π + 2γ + o(ε 0 ) 2 s=si + i k F ε 2 2 ϑ + 4[ρ 2 -( 1 /2 cos 2 γ)]( l1S -l1D ) + ( 1 /cos γ)(ρ 2 ) + ( 2 /cos 2 γ)( l2D -l2S ) 4ρ( 1 /2 cos γ) ( 1 /cos 2 γ) -ρ 2 + o(ε 0 ) 2 s=si + i k F ε 2 2 ϑ + 4[ρ 2 -( 1 /2 cos 2 γ)]( l2S -l2D ) + ( 1 /cos γ)(ρ 2 ) + ( 2 /cos 2 γ)( l1D -l1S ) 4ρ( 1 /2 cos γ) ( 1 /cos 2 γ) -ρ 2 + o(ε 0 ) 2 s=si . (18)
Let the parameters K E , K F , K S > 0 be constants, which do not depend on ε. Then they are related to the stinesses of each discrete system with micro length scale ε by a scaling law

k E = K E ε -3 , k F = K F ε -1 , k S = K S ε . (19)
Pantographic beam macro-model. The continuum limit is now obtained by letting ε → 0. The deformation energy for the homogenised macro-model becomes

E = L 0 K S cos -1 1 - ρ 2 1 /2 cos 2 γ -π + 2γ 2 + K E 2 µν lµν 2 ds + L 0 K F 2 ϑ + 4[ρ 2 -( 1 /2 cos 2 γ)]( l1S -l1D ) + ( 1 /cos γ)(ρ 2 ) + ( 2 /cos 2 γ)( l2D -l2S ) 4ρ( 1 /2 cos γ) ( 1 /cos 2 γ) -ρ 2 2 ds + L 0 K F 2 ϑ + 4[ρ 2 -( 1 /2 cos 2 γ)]( l2S -l2D ) + ( 1 /cos γ)(ρ 2 ) + ( 2 /cos 2 γ)( l1D -l1S ) 4ρ( 1 /2 cos γ) ( 1 /cos 2 γ) -ρ 2 2 ds . (20)
The basic properties of the energy are preserved during the asymptotic process. Both the energy of the micro-and the macro-model ( 6) and [START_REF] Abdoul-Anziz | Strain gradient and generalized continua obtained by homogenizing frame lattices[END_REF], respectively, are invariant under superimposed rigid body motions. Also the extensional oppy mode of the discrete model, see [START_REF] Turco | b Large deformations induced in planar pantographic sheets by loads applied on bers: experimental validation of a discrete Lagrangian model[END_REF], transfers to the continuum. Namely, if ρ = ϑ = lµν = 0 and ρ(s) = 1, then the deformation energy vanishes. When

K S = 0, if ρ = ϑ = lµν = 0, a constant stretch ρ(s) = K ∈ (0, 1 /cos γ)
can still be present without causing the deformation energy being dierent from zero.

Let us now dene the deformation energy density Ψ as the integrand of [START_REF] Abdoul-Anziz | Strain gradient and generalized continua obtained by homogenizing frame lattices[END_REF]. For the energy to be stationary, the necessary conditions are obtained by equating to zero the variation of the deformation energy functional [START_REF] Abdoul-Anziz | Strain gradient and generalized continua obtained by homogenizing frame lattices[END_REF] with respect to admissible variations in the independent kinematic descriptors. At this stage, only the variation with respect to lµν is carried out. This results in a linear system of four algebraic equations given by ∂Ψ/∂ lµν = 0 in which lµν are the unknowns. Introducing the abbreviations

C 1 = K F 2K F ρ 2 -1 /4 cos 2 γ (K E ρ 2 + 8K F ) , C 2 = K F 1 /cos 2 γ -ρ 2 K E ( 1 /4 cos 2 γ)ρ 2 -2K F ρ 2 -4K E ( 1 /16 cos 4 γ) , (21)
necessary conditions for equilibrium are that

lµD = 1 2 cos γ ρ ρ C 1 + (-1) µ-1 ϑ C 2 , lµS = 1 2 cos γ ρ [-ρ C 1 + (-1) µ ϑ C 2 ] . (22)
By substituting the results ( 22) into (20), a kinematic reduction is performed resulting in the deformation energy functional of the pantographic beam

E = L 0 K E K F ρ 2 cos 2 γ -1 ρ 2 cos 2 γ (K E -8K F cos 2 γ) -K E ϑ 2 + ρ 2 cos 2 γ (1 -ρ 2 cos 2 γ) [8K F + ρ 2 (K E -8K F cos 2 γ)] ρ 2 +K S cos -1 1 - ρ 2 1 /2 cos 2 γ -π + 2γ 2 ds , (23)
which merely depends on the placement function χ. The energy ( 23) is positive denite for 0 < ρ < 1/ cos γ and the complete second gradient χ of χ contributes to the deformation energy. Besides the term χ ⊥ • χ being related to the material curvature ϑ by means of (11) 1 , also the term χ • χ appears, which in turn is related to the stretch gradient ρ given by ( 23) 2 . It is also worth noting that, if ρ(x) = 1/ cos γ, then the term multiplying ϑ in ( 23) vanishes. Consequently, at point s = s 0 the beam undergoes a beam-to-cable transition, being curvature no more energetically penalized. At the same time, if ρ(s 0 ) = 1/ cos γ then the term multiplying ρ in ( 23) diverges. Therefore, boundedness of energy requires ρ (s 0 ) = 0.

Bi-pantographic fabrics discrete model

The assembly of a discrete bi-pantographic fabric is sketched in Fig. 2(b). The kinematics (and employed notation thereof) of discrete bi-pantographic fabrics is given by generalizing that of pantographic beams once the bipantographic structure is regarded as an assembly of two identical orthogonal families of parallel equi-spaced pantographic beams hinge joined at their intersection points. Thus, aimed at avoiding unwieldy pictures, we omit to show it in Fig. 2.

In the undeformed conguration, see Fig. 2(a), cells are arranged within the reference domain Ω upon straight lines in direction of the unit basis vectors e x , e y ∈ E 2 . The set Ω ⊆ R 2 is in general a non-simple reference domain with boundary ∂Ω being the disjoint union of N Ω ∈ N smooth line sets ∂Ω k , k ∈ [1; N Ω ], pairwise intersecting in distinct vertices belonging to the set [∂∂Ω]. A discussion on smoothness requirements for Ω is beyond the scope of this paper. For such a discussion the reader is referred to [START_REF] Auray | Analytical continuum mechanics à la HamiltonPiola least action principle for second gradient continua and capillary uids[END_REF]. The cells are centred at the positions P i,j = iεe x + jεe y see Fig. 2(b). The basic (i, j)-th unit cell see Fig. 2(c) is formed by eight extensional springs hinge-joined together at P i,j having length ε /(2 cos γ). Rotational springs which are coloured in blue, red, and green in Fig. 2(c) are placed between opposite collinear adjacent springs belonging to the same cell and between adjacent springs belonging to dierent cells. The kinematics of the spring system is locally described by nitely many generalized coordinates. The coordinates are the positions p i,j ∈ E 2 of the points at position P i,j in the reference conguration (equivalently one can consider the nodal displacements u i,j ∈ R 2 such that u i,j = p i,j -P i,j ) and the lengths of the oblique deformed springs l µν (i,j),α ∈ R, α ∈ x, y. The index α will be henceforth employed to distinguish quantities related to pantographic beams directed along e x (α = x) and e y (α = y). Various other kinematical quantities are introduced to formulate the total potential energy in a most compact form. Applying the law of cosines, the angles ϕ µν (i,j),α are determined by and

ϕ 1D (i,j),x = cos -1   p i+1,j -p i,j 2 + l 1D (i,j),x 2 -l 2S (i+1,j),x 2 2l 1D (i,j),x p i+1,j -p i,j   , ϕ 1S (i,j),x = cos -1   p i,j -p i-1,j 2 + l 1S (i,j),x 2 -l 2D (i-1,j),x 2 2l 1S (i,j),x p i,j -p i-1,j   , ϕ 2D (i,j),x = cos -1   p i+1,j -p i,j 2 + l 2D (i,j),x 2 -l 1S (i+1,j),x 2 2l 2D (i,j),x p i+1,j -p i,j   , ϕ 2S (i,j),x = cos -1   p i,j -p i-1,j 2 + l 2S (i,j),x 2 -l 1D (i-1,j),x 2 2l 2S (i,j),x p i,j -p i-1,j   , (24)
ϕ 1D (i,j),y = cos -1   p i,j+1 -p i,j 2 + l 1D (i,j),y 2 -l 2S (i,j+1),y 2 2l 1D (i,j),y p i,j+1 -p i,j   , ϕ 1S (i,j),y = cos -1   p i,j -p i,j-1 2 + l 1S (i,j),y 2 -l 2D (i,j-1),y 2 2l 1S (i,j),y p i,j -p i,j-1   , ϕ 2D (i,j),y = cos -1   p i,j+1 -p i,j 2 + l 2D (i,j),y 2 -l 1S (i,j+1),y 2 2l 2D (i,j),y p i,j+1 -p i,j   , ϕ 2S (i,j),y = cos -1   p i,j -p i,j-1 2 + l 2S (i,j),y 2 -l 1D (i,j-1),x 2 2l 2S (i,j),y p i,j -p i,j-1   , (25)
while the angles ξ µ (i,j),α become

ξ 1(2) (i,j),x = cos -1   l 1(2)D (i,j),x 2 + l 2(1)S (i+1,j),x 2 -p i+1,j -p i,j 2 2l 1(2)D (i,j),x l 2(1)S (i+1,j),x   , ξ 1(2) (i,j),y = cos -1   l 1(2)D (i,j),y 2 + l 2(1)S (i,j+1),y 2 -p i,j+1 -p i,j 2 2l 1(2)D (i,j),y l 2(1)S (i,j+1),y   . ( 26
)
Having used the law of cosines to determine ϕ µν (i,j),α , the choice of generalized coordinates holds only locally as long as the angles ϕ 1D (i,j),α and ϕ 2D (i,j),α do not change sign. Throughout the derivation of the macro-model, it is assumed that the angles ϕ 1D (i,j),α and ϕ 2D (i,j),α remain in the range (0, π). This entails that ξ µ (i,j),α ∈ (0, π). The angle between the two vectors p i,j -p i-1,j and e x is denoted by ϑ (i,j),x , while the angle between the two vectors p i,j -p i,j-1 and e y is denoted by ϑ (i,j),y . Then the angle θ (i,j),x between the vectors p i,j -p i-1,j and p i+1,j -p i,j becomes

θ (i,j),x = ϑ (i+1,j),x -ϑ (i,j),x = tan -1 (p i+1,j -p i,j ) • e y (p i+1,j -p i,j ) • e x -tan -1 (p i,j -p i-1,j ) • e y (p i,j -p i-1,j ) • e x , (27)
while the angle θ (i,j),y between the vectors p i,j -p i,j-1 and p i,j+1 -p i,j reads

θ (i,j),y = ϑ (i,j+1),y -ϑ (i,j),y = tan -1 (p i,j+1 -p i,j ) • e y (p i,j+1 -p i,j ) • e x -tan -1 (p i,j -p i,j-1 ) • e y (p i,j -p i,j-1 ) • e x . ( 28
)
The following relations hold true

β 1 (i,j),α = θ (i,j),α + ϕ 1D (i,j),α -ϕ 1S (i,j),α , β 2 (i,j),α = θ (i,j),α + ϕ 2S (i,j),α -ϕ 2D (i,j),α . (29)
Letting the summations for µ, ν and α range over the sets {1, 2}, {D, S} and {x, y}, respectively, and those for (i, j) over a set such that all energy contributions due to elastic elements in Ω are included in the subsequent formula, the micro-model deformation energy is dened as

E ε = k E 2 i,j α µ,ν l µν (i,j),α - 1 2 cos γ ε 2 + k F 2 µ β µ (i,j),α 2 + k S 2 µ (ξ µ (i,j),α -π + 2γ) 2 (29) = k E 2 i,j α µ,ν l µν (i,j),α - 1 2 cos γ ε 2 + k F 2 µ θ (i,j),α + (-1) µ ϕ µS (i,j),α -ϕ µD (i,j),α 2 + k S 2 µ (ξ µ (i,j),α -π + 2γ) 2 , (30)
with k E > 0 and k F , k S > 0 being the stinesses of the extensional and rotational springs, respectively. The summand in (6) for the sum over (i, j) will be henceforth denoted by Ψ i,j .

It is worth noting that, when k S = 0, besides the rigid body modes also the set of admissible congurations obtained as all possible combinations of 1) uniform shear, i.e. the angle between the centrelines of the two families of pantographic beams is uniform and ranging from 0 • to 180 • (pantographic beams are transformed rigidly, and hence this gives an innite family of oppy modes parametrised on a single parameter that is the above mentioned angle; when a bias rectangular specimen is considered i.e. bers form ±45 • with the sides this deformation mode corresponds to uniform extension/compression of the rectangle) and 2) extensional oppy mode of constituting pantographic beams entails null deformation energy. For more details on oppy modes in bi-pantographic structures the reader is referred to [START_REF] Seppecher | Linear elastic trusses leading to continua with exotic mechanical interactions[END_REF]. While each pantographic beam as well as pantographic fabrics (whose only nonrigid zero energy deformation mode is given by uniform macroscopic shear; pantographic beams are replaced by (extensible) Elasticae that cannot extend with zero energy) admits an innite family of extensional oppy modes parametrised over a single parameter (see equation ( 7)), the bi-pantographic structure admits an innite family of oppy modes parametrised over four parameters, see Fig. 11 in [START_REF] Seppecher | Linear elastic trusses leading to continua with exotic mechanical interactions[END_REF].

For the lengths l µν (i,j),α of the oblique springs, the following asymptotic expansion is assumed

l µν (i,j),α = 1 2 cos γ ε + ε 2 lµν (i,j),α + o(ε 2 ) , (31)
where lµν (i,j),α ∈ R. Inserting assumption (31) into the energy [START_REF] Turco | Enhanced PiolaHencky discrete models for pantographic sheets with pivots without deformation energy: Numerics and experiments[END_REF] leads to

E ε = α i,j k E 2 µ,ν ε 2 lµν (i,j),α + o(ε 2 ) 2 + k F 2 µ θ (i,j),α + (-1) µ ϕ µS (i,j),α -ϕ µD (i,j),α 2 + k S 2 µ (ξ µ (i,j),α -π + 2γ) 2 .
(32)

Bi-pantographic fabrics micro-macro identication

The two-dimensional extension of the discrete system makes it reasonable to aim for a two-dimensional continuum in the limit of vanishing ε. The independent kinematic Lagrangian descriptors of the macro-model are assumed to be the functions χ : Ω → E 2 and lµν α : Ω → R. The placement function χ places the 2D-continuum into E 2 and is best suited to describe the points p i,j ∈ E 2 of the discrete system on the macro-level. To take into account the eect of changing spring lengths lµν (i,j),α introduced in (8), the placement function is augmented by the eight microstrain functions lµν α . The identication of the discrete system with a two-dimensional continuum is also classied as a micromorphic continuum [START_REF] Germain | The Method of Virtual Power in Continuum Mechanics. Part 2: Microstructure[END_REF][START_REF] Forest | Nonlinear microstrain theories[END_REF][START_REF] Eremeyev | Foundations of micropolar mechanics[END_REF][START_REF] Altenbach | Cosserat-type rods[END_REF].

It is also convenient to introduce the functions ρ α : Ω → R + and ϑ α : Ω → [0, 2π) in order to rewrite the tangent vector eld ∂χ /∂α to deformed material lines oriented along e α in the reference conguration as

Thus ρ α corresponds to the norm of the tangent vector ∂χ /∂α to the deformed material lines directed along e α in the reference conguration, and it is referred to as α-stretch. Introducing the normal vector elds to deformed material lines directed, respectively, along e x and e y in the reference conguration

∂χ ∂x ⊥ (x, y) = ρ x (x, y) {-[sin ϑ x (x, y)] e x + [cos ϑ x (x, y)] e y } , ∂χ ∂y ⊥ (x, y) = ρ y (x, y) {-[sin ϑ y (x, y)] e y + [cos ϑ y (x, y)] e x } , (34)
being respectively rotated against ∂χ /∂x and ∂χ /∂y about 90 • in the anti-clockwise direction, it is found that

∂ρ x ∂x (x, y) = ∂χ ∂x (x, y) • ∂ 2 χ ∂x 2 (x, y) ∂χ ∂x (x, y) , ∂ϑ x ∂x (x, y) = ∂ 2 χ ∂x 2 (x, y) • ∂χ ∂x ⊥ (x, y) ∂χ ∂x (x, y) 2 , ∂ρ y ∂y (x, y) = ∂χ ∂y (x, y) • ∂ 2 χ ∂y 2 (x, y) ∂χ ∂y (x, y) , ∂ϑ y ∂y (x, y) = ∂ 2 χ ∂y 2 (x, y) • ∂χ ∂y ⊥ (x, y) ∂χ ∂y (x, y) 2 . ( 35
)
In the following ∂ρα /∂α and ∂ϑα /∂α are called α-stretch α-derivative and material α-curvature, respectively. For Piola's micro-macro identication the generalized coordinates of the discrete system are related to the functions χ and lµν α evaluated at (x i , y j ) = (iε, jε) as

χ(x i , y j ) = p i,j , lµν α (x i , y j ) = lµν (i,j),α . (36)
For the asymptotic identication, the energy (32) needs to be expanded in ε. The expansion of χ is given by

χ(x i±1 , y j ) = χ(x i , y j ) ± ε ∂χ ∂x (x i , y j ) + ε 2 2 ∂ 2 χ ∂x 2 (x i , y j ) + o(ε 2 ) , χ(x i , y j±1 ) = χ(x i , y j ) ± ε ∂χ ∂y (x i , y j ) + ε 2 2 ∂ 2 χ ∂y 2 (x i , y j ) + o(ε 2 ) . ( 37
)
Combining the asymptotic expansion [START_REF] Misra | Pantographic metamaterials show atypical Poynting eect reversal[END_REF] with (36) 2 , lµν x (x i±1 , y j ) = lµν x (x i , y j )+o(ε 0 ) and lµν y (x i , y j±1 ) = lµν y (x i , y j )+ o(ε 0 ), yields

l µν (i±1,j),x = 1 2 cos γ ε + lµν x (x i , y j )ε 2 + o(ε 2 ) , l µν (i,j±1),y = 1 2 cos γ ε + lµν y (x i , y j )ε 2 + o(ε 2 ) . (38)
In order to further expand (32), the terms θ (i,j),α , ϕ µS (i,j),α -ϕ µD (i,j),α and ξ µ (i,j),α need to be expanded up to rst order (see App. 6). For θ (i,j),α according to ( 71)

θ (i,j),α = ε ∂ϑ (i,j),α ∂α (x i , y j ) + o(ε) . (39)
The dierences ϕ

1(2)S (i,j),α -ϕ 1(2)D
(i,j),α are given by (78) as

ϕ 1(2)S (i,j),α -ϕ 1(2)D (i,j),α = 4[ρ 2 α -( 1 /2 cos 2 γ)]( l1(2)S α - l1(2)D α ) + ( 1 /cos γ) ∂(ρ 2 α ) ∂α + ( 2 /cos 2 γ)( l2(1)D α - l2(1)S α ) 4ρ α ( 1 /2 cos γ) ( 1 /cos 2 γ) -ρ 2 α (x,y)=(xi,yj ) ε + o(ε) .
(40) The angles ξ µ (i,j),α are given by (80) as

ξ µ (i,j),α = cos -1 1 - ρ 2 α 1 /2 cos 2 γ (x,y)=(xi,yj ) + o(ε 0 ) . (41)
Substituting ( 39), ( 40) and ( 41) into [START_REF] Eremeyev | Linear pantographic sheets: existence and uniqueness of weak solutions[END_REF] together with ρ α (x i , y j ) = ∂χ ∂α , the desired expansion of the micro-model energy E ε is derived as a function of the kinematic descriptors χ and lµν α as

E ε = i,j α k E ε 4 2 µ,ν lµν α 2 + o(ε 0 ) + k S cos -1 1 - ρ 2 α 1 /2 cos 2 γ -π + 2γ + o(ε 0 ) 2 (x,y)=(xi,yj ) + i,j α k F ε 2 2 ∂ϑ ∂α + 4[ρ 2 α -( 1 /2 cos 2 γ)]( l1S α -l1D α ) + ( 1 /cos γ) ∂(ρ 2 α ) ∂α + ( 2 /cos 2 γ)( l2D α -l2S α ) 4ρ α ( 1 /2 cos γ) ( 1 /cos 2 γ) -ρ 2 α + o(ε 0 ) 2 (x,y)=(xi,yj ) + i,j α k F ε 2 2 ∂ϑ ∂α + 4[ρ 2 α -( 1 /2 cos 2 γ)]( l2S α -l2D α ) + ( 1 /cos γ) ∂(ρ 2 α ) ∂α + ( 2 /cos 2 γ)( l1D α -l1S α ) 4ρ α ( 1 /2 cos γ) ( 1 /cos 2 γ) -ρ 2 α + o(ε 0 ) 2 (x,y)=(xi,yj ) . ( 42
)
Let the parameters K E , K F , K S > 0 be constants, which do not depend on ε. Then these constants are related to the stinesses of each discrete system with micro length scale ε by a scaling law

k E = K E ε -2 , k F = K F , k S = K S ε 2 . ( 43
)
2.4. Bi-pantographic fabrics macro-model

The continuum limit is now obtained by letting ε → 0 and considering the sum to turn into an integral according to i,j f (x i , y j )ε 2 ε→0 -→ Ω f dA, where f is a real valued function dened on Ω. Using (42) together with the scaling law [START_REF] Forest | Nonlinear microstrain theories[END_REF], the deformation energy for the homogenised macro-model becomes

E = Ω α K E 2 µ,ν lµν α 2 + K S cos -1 1 - ρ 2 α 1 /2 cos 2 γ -π + 2γ 2 dA + Ω α K F 2 ∂ϑ α ∂α + 4[ρ 2 α -( 1 /2 cos 2 γ)]( l1S α -l1D α ) + ( 1 /cos γ) ∂(ρ 2 α ) ∂α + ( 2 /cos 2 γ)( l2D α -l2S α ) 4ρ α ( 1 /2 cos γ) ( 1 /cos 2 γ) -ρ 2 α 2 dA + Ω α K F 2 ∂ϑ α ∂α + 4[ρ 2 α -( 1 /2 cos 2 γ)]( l2S α -l2D α ) + ( 1 /cos γ) ∂(ρ 2 α ) ∂α + ( 2 /cos 2 γ)( l1D α -l1S α ) 4ρ α ( 1 /2 cos γ) ( 1 /cos 2 γ) -ρ 2 α 2 dA . ( 44
)
Considerations on the above derived continuum limit analogous to those made in the previous subsection dealing with preliminary computations can be invoked. The above deformation energy is objective and discrete oppy modes transfer to the continuum after homogenisation. The above deformation energy is vanishing for χ(x, y) = [x + (ay + b)x]e x + [y + (cy + d)x]e y [START_REF] Seppecher | Linear elastic trusses leading to continua with exotic mechanical interactions[END_REF] when K S = 0. When a = c = d = 0, then χ represents uniform extension, while when a = c = 0 it describes uniform shear deformation, which is the only non-rigid zero energy deformation mode for pantographic fabrics, [START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium[END_REF]. The derived continuum limit as for pantographic fabrics inherits its orthotropicity from its bered structure at the micro-scale, i.e. it can be regarded as made by assembling two identical orthogonal families of (equispaced) parallel discrete pantographic beams. Let us now dene the deformation energy density Ψ as the integrand of [START_REF] Eremeyev | Foundations of micropolar mechanics[END_REF]. For the energy to be stationary, the necessary conditions are obtained by equating to zero the variation of the deformation energy functional [START_REF] Eremeyev | Foundations of micropolar mechanics[END_REF] with respect to admissible variations in the independent kinematic descriptors. First, only the variation with respect to lµν α is studied, and results in a linear system of eight algebraic equations given by ∂Ψ/∂ lµν α = 0 in which lµν α are the unknowns. Introducing the notations

C α 1 = K F 2K F ρ 2 α -1 /4 cos 2 γ (K E ρ 2 α + 8K F ) , C α 2 = K F 1 /cos 2 γ -ρ 2 α K E ( 1 /4 cos 2 γ)ρ 2 α -2K F ρ 2 α -4K E ( 1 /16 cos 4 γ) , (45)
necessary conditions for equilibrium are that

lµD α = 1 2 cos γ ρ α ∂ρ α ∂α C α 1 + (-1) µ-1 ∂ϑ α ∂α C α 2 , lµS α = 1 2 cos γ ρ α - ∂ρ α ∂α C α 1 + (-1) µ ∂ϑ α ∂α C α 2 . ( 46
)
By substituting the results ( 46) into (44), a kinematic reduction is performed and results in the deformation energy functional of the bi-pantographic structure

E = Ω α K E K F ρ 2 α cos 2 γ -1 ρ 2 α cos 2 γ (K E -8K F cos 2 γ) -K E ∂ϑ α ∂α 2 + ρ 2 α cos 2 γ (1 -ρ 2 α cos 2 γ) [8K F + ρ 2 α (K E -8K F cos 2 γ)] ∂ρ α ∂α 2 +K S cos -1 1 - ρ 2 α 1 /2 cos 2 γ -π + 2γ 2 dA (47)
which depends on the placement function χ only. Notice that, besides the term ∂χ ∂α ⊥

• ∂ 2 χ ∂α 2 being related to the material α-curvature ∂ϑα ∂α by means of [START_REF] Seppecher | Linear elastic trusses leading to continua with exotic mechanical interactions[END_REF], also the term ∂χ ∂α • ∂ 2 χ ∂α 2 appears which in turn is related to the α-stretch α-derivative ∂ρα ∂α given by equation [START_REF] Seppecher | Linear elastic trusses leading to continua with exotic mechanical interactions[END_REF]. A detailed derivation of Euler-Lagrange equations, essential and natural boundary conditions (BC's) as deduced from stationarity condition for energy functionals of the form Ω W (∇χ, ∇∇χ)dA as that in ( 47) is beyond the scope of this paper, and the reader is referred to [START_REF] Auray | Analytical continuum mechanics à la HamiltonPiola least action principle for second gradient continua and capillary uids[END_REF]. However, it is worth recalling that in such a case non-classical essential normal placement gradient BC's i.e. prescribing ∇χ(x, y) • n(x, y) = f (x, y) can be given at boundaries ∂Ω k 's n being the outwards pointing unit normal and essential placement BC's i.e. prescribing χ(x, y) = g(x, y) can be given at vertices belonging to [∂∂Ω], in addition to classical essential placement BC's at boundaries ∂Ω k .

Bi-pantographic fabrics linearisation of deformation energy

Let the vector valued displacement eld u be dened by u(x, y) = χ(x, y)-xe x -ye y . Then by the Piola's identication [START_REF] Barchiesi | Pantographic beam: a complete second gradient 1D-continuum in plane[END_REF] and by the denition of nodal displacements u i,j we have u(x i , y j ) = u i,j . From Taylor expansions it follows that

ϑ x =
and thus

∂ρ α ∂α = ∂ 2 u ∂α 2 • e α + o ∂u ∂α 0 . (51)
Hence, the energy (47) rewrites as (see equation (85) in App. A)

E = Ω K E K F cos 2 γ (1 -cos 2 γ) [8K F + K E -8K F cos 2 γ] ∂ 2 u ∂x 2 • e x 2 + ∂ 2 u ∂y 2 • e y 2 dA + Ω K E K F (cos 2 γ -1) cos 2 γ(K E -8K F cos 2 γ) -K E ∂ 2 u ∂x 2 • e y 2 + ∂ 2 u ∂y 2 • e x 2 dA + Ω α [4K S cot γ] ∂u ∂α • e α 2 + o ∂u ∂α 2 dA . (52)
For small strain hypothesis the remainder o( ∂u/∂α 2 ) in equation ( 52) can be neglected.

Computational aspects

In this section, the problem to be solved is introduced and solution methodologies employed for the macro-and micro-model are briey recalled.

Boundary value problem (non-standard bias extension test)

A rectangular specimen, i.e. N Ω = 4, with sides L =187 mm × =119 mm and ε=12.02 mm is considered, see Fig. 3. The geometric parameter γ is assumed to be equal to π/6. The following essential boundary conditions are considered u(x, y) = 0 at (x, y)

∈ ∂Ω 1 , u(x, y) = ūe ζ at (x, y) ∈ ∂Ω 3 , ū ∈ R [∇u(x, y)]n(x, y) = 0 at (x, y) ∈ ∂Ω 1 , [∇u(x, y)]n(x, y) = 0 at (x, y) ∈ ∂Ω 3 ( 53
)
which do not entail a oppy deformation mode. As the displacement eld u(x, y) is enforced to be constant along the boundaries ∂Ω 1 and ∂Ω 3 , then [∇u(x, y)]n ⊥ (x, y) is also vanishing along those boundaries. This, together with (53) 2 , implies that ∇u(x, y) = 0 at (x, y)

∈ ∂Ω 1 ∪ ∂Ω 3 . (54)
Equation ( 54) is equivalent to

ρ α (x, y) = 1 at (x, y) ∈ ∂Ω 1 ∪ ∂Ω 3 , ϑ α (x, y) = 0 at (x, y) ∈ ∂Ω 1 ∪ ∂Ω 3 . ( 55
)
Figure 3. Schematic drawing of the reference domain Ω considered in the boundary value problem for the macro-model.

To compare the micro-and macro-model, beyond the micro-macro identication [START_REF] Babu²ka | Homogenization approach in engineering[END_REF], the following micro-macro correspondences based on neglecting non-leading ε-terms in Taylor expansions of continuum quantities evaluated at discrete points shall be taken into account. For stretches and orientations of pantographic beams

ρ x (x i , y i ) ↔ p i+1,j -p i,j ε , ϑ x (x i , y j ) ↔ ϑ i,j = tan -1 (p i,j -p i-1,j ) • e y (p i,j -p i-1,j ) • e x ρ y (x i , y j ) ↔ p i,j+1 -p i,j ε , ϑ y (x i , y j ) ↔ ϑ i,j = tan -1 (p i,j -p i,j-1 ) • e x (p i,j -p i,j-1 ) • e y . ( 56
)
In addition, the micro-strains lµν α are related by

lµν α (x i , y j ) ↔ l µν (i,j),α ε . ( 57
)
The deformation energy density Ψ(x, y), which is the integrand of ( 47), is compared by the following relation

Ψ(x i , y j ) ↔ Ψ i,j . (58)
The shear angle is compared by the following relation

↔ π/2 -arccos (p i+1,j -p i,j ) • (p i,j+1 -p i,j ) p i+1,j -p i,j p i,j+1 -p i,j . (59)
micro-model macro-model u i,j = 0 for all (i, j) s.t. (x i , y j ) ∈ ∂Ω 1 u(x, y) = 0 for all (x, y) ∈ ∂Ω 1 u i,j = ūe ζ for all (i, j) s.t. (x i , y j ) ∈ ∂Ω 3 u(x, y) = ūe ζ for all (x, y) ∈ ∂Ω 3 u i+1,j = u i,j for all (i, j) s.t. (x i , y j ) ∈ ∂Ω 1 [∇u(x, y)]n(x, y) = 0 for all (x, y) ∈ ∂Ω 1 ∪ Ω 3 u i,j+1 = u i,j for all (i, j) s.t. (x i , y j ) ∈ ∂Ω 1 u i-1,j = u i,j for all (i, j) s.t. (x i , y j ) ∈ ∂Ω 3 u i,j-1 = u i,j for all (i, j) s.t. (x i , y j ) ∈ ∂Ω 3 Table 1. Boundary conditions for micro-and macro-model.

Last, in an analogous fashion the following micro-macro correspondences are dened on boundaries

∂u ∂x (x i , y j ) ↔ u i+1,j -u i,j ε and ∂u ∂y (x i , y j ) ↔ u i,j+1 -u i,j ε for all (x i , y j ) ∈ ∂Ω 1 ∂u ∂x (x i , y j ) ↔ u i,j -u i-1,j ε and ∂u ∂y (x i , y j ) ↔ u i,j -u i,j-1 ε for all (x i , y j ) ∈ ∂Ω 3 (60)
which, together with Piola's micro-macro identication [START_REF] Barchiesi | Pantographic beam: a complete second gradient 1D-continuum in plane[END_REF], are used to establish a correspondence between boundary conditions ( 53) for the continuum model and those for the discrete one. Such a correspondence is reported in Tab. 1.

Macro-model Finite Element formulation

A mixed nite element formulation is adopted for the solution of the macro-model. Let us dene the following augmented energy functional

Ẽ = NΩ k=1 ∂Ω k α µ α • [(∇u -M ) n ⊥ ] dA + Ω α λ α • M e α - ∂u ∂α + K E K F (ρ 2 α cos 2 γ -1)[κ α (M )] 2 ρ 2 α cos 2 γ (K E -8K F cos 2 γ) -K E + (ρ 2 α cos 2 γ)[ι α (M )] (1 -ρ 2 α cos 2 γ) [8K F + ρ 2 α (K E -8K F cos 2 γ)] +K S cos -1 1 - ρ 2 α 1 /2 cos 2 γ -π + 2γ 2 dA + ∂Ω1∪∂Ω3 (η • [∇u]n)dl + ∂Ω1 (γ • u) dl + ∂Ω3 [υ • (u -ūe ζ )] dl . ( 61
)
where M is an independent auxiliary eld that is weakly enforced by Lagrange multipliers µ α and λ α to be equal to ∇u [START_REF] Bersani | Seppecher P Lagrange multipliers in innite dimensional spaces, examples of application[END_REF], and

κ x (M ) = ∂(M ex) ∂x • [(1 + ∂u ∂x • e x , ∂u ∂x • e y ) T ] ⊥ (1 + ∂u ∂x • e x , ∂u ∂x • e y ) T 2 , ι x (M ) = ∂(M ex) ∂x • (1 + ∂u ∂x • e x , ∂u ∂x • e y ) T (1 + ∂u ∂x • e x , ∂u ∂x • e y ) T κ y (M ) = ∂(M ey) ∂y • [(1 + ∂u ∂y • e y , ∂u ∂y • e x ) T ] ⊥ (1 + ∂u ∂y • e y , ∂u ∂y • e x ) T 2 , ι y (M ) = ∂(M ey) ∂y • (1 + ∂u ∂y • e y , ∂u ∂y • e x ) T (1 + ∂u ∂y • e y , ∂u ∂y • e x ) T (62)
are α-curvature (κ α ) and α-stretch α-derivative (ι α ) expressed in terms of only rst spatial derivatives of the independent elds u and M . In such a way, the deformation energy ( 47) can be transformed into an augmented energy functional written in terms of rst spatial derivatives of the independent kinematic quantities. The discretisation of these quantities by the nite element method to solve the stationarity condition of such augmented energy functional does not require C 1 -continuous shape functions like those needed to solve the stationarity condition for the energy [START_REF] Bersani | Seppecher P Lagrange multipliers in innite dimensional spaces, examples of application[END_REF] in terms of the only unknown eld u. Let Ψ be the argument of integration over Ω in [START_REF] Sutton | Determination of Displacements Using an Improved Digital Correlation Method[END_REF]. Let Ψk be the argument of integration over ∂Ω k in [START_REF] Sutton | Determination of Displacements Using an Improved Digital Correlation Method[END_REF]. From the stationarity condition for the energy ( 61) is determined the weak form

0 = α NΩ k=1 ∂Ω k ∂ Ψk ∂ (∂u/∂α) • δ ∂u ∂α + Ψk ∂(M e α ) • δ(M e α ) + ∂ Ψk ∂µ α • δµ α dl + α Ω ∂ Ψ ∂ (∂u/∂α) • δ ∂u ∂α + ∂ Ψ ∂(M e α ) • δ(M e α ) + ∂ Ψ ∂λ α • δλ α dA + NΩ k=1 ∂Ω k ∂ Ψk ∂η • δη + ∂ Ψk ∂γ • δγ + ∂ Ψk ∂υ • δυ dl (63)
where δ(•) denotes the kinematically admissible variation of (•), which can then be solved numerically by a nite element code. The weak form package of the software COMSOL Multiphysics, which implements standard nite element techniques [START_REF] Abali | Strain gradient elasticity with geometric nonlinearities and its computational evaluation[END_REF][START_REF] Abali | Theory and computation of higher gradient elasticity theories based on action principles[END_REF], was used for the discretisation and the subsequent solution procedure. Essential boundary conditions in equation [START_REF] Misra | Micromechanical model for viscoelastic materials undergoing damage[END_REF] were not encoded within the basis functions but enforced by additional Lagrange multipliers (i.e. η α , γ and υ in equation ( 61)). In such a mixed formulation, normal displacement gradient line-boundary conditions (53) 2 are enforced in terms of the auxiliary eld M , while displacement line-boundary conditions (53) 1 are enforced in terms of the eld u. Quadratic Lagrangian polynomials were used as basis functions for the elds χ and M . All Lagrange multiplier elds were discretised by linear Lagrange polynomials. The mesh was Delaunay tessellated with maximum diameter size equal to 8.45 mm (see Fig. 3). Energy convergence of the solutions was successfully checked for the mesh-size tending to 0. Solution of each step i.e. for each ū was initialized by the solution of the previous one, considering for ū a constant step-size ∆ū equal to 1 mm. All specimens were designed in SolidWorks Computer Aided Design (CAD) software by sketching 2D proles and then using methods like extruding and lofting in order to produce solid shapes, see the technical drawings in Figs. 5, 6 and 10 (right). A full top-view of the manufactured specimen is shown in Fig. 7. The blue/red rotational springs in Fig. 2(c) and the adjacent extensional ones are fabricated as a whole by means of monolithic slender elements that are meant to predominantly bend (rotational spring) and (to a lesser extent) extend (extensional springs) in plane. Such elements are combined at extreme points by cylinders, which are meant to reproduce the green rotational springs of Fig. 2(c) by mainly twisting, and at middle points by hinge connections. They are shown in Fig. 8 (actual manufacturing on the left A and CAD modelling on the right B). As assumed above, the angle γ is equal to π/6, see Fig. 10 (right).

Each pantographic beam is made of two families of monolithic slender elements forming an angle 2γ and lying onto two dierent parallel planes. The two families of pantographic beams (whose centrelines form an angle of 90 • ) lying onto two dierent planes are hinge connected at intersection points, which is at the mid-point of the monolithic slender elements. The structure is then doubled in the out-of-plane direction by reection to avoid noticeable outof-plane movements, making it symmetric with respect to its middle plane, see Fig. 5 (bottom). Hinge axes are monolithic elements running through the full out-of-plane length of the structure.

Hard-device conditions given in row three and six of Tab. 1 are obtained by connecting the adjacent hinge axes in proximity of the gripping areas, see Fig. 9, with stocky rhomboidal elements, meant to be rigid with respect to other elements of the specimen for the considered load range.

Testing and data acquisition

An MTS Tytron 250 testing-device was used for the experiments. The total reaction-force was measured by a deviceown load cell, which is able to record axial forces in a range of ±250 N with an accuracy of 0.2 percent. Increasing displacements were prescribed horizontally on the right side of the specimen with a loading rate of 15 mm/min. The cross-head displacement was measured and monitored by a device-own encoder unit. Almost frictionless movement of the machine shaft was achieved by using an air-lm-bearing. External vibrations were avoided by placing the system on a massive concrete-substructure. Pictures of the surface during deformations were taken (0.5 pictures/second) by means of a Canon EOS 600D camera with a denition of 4272×2848 pixels. Each picture was synchronized with the recorded force-displacement data in real time. Regarding frictional dissipation due to PA2200 powder stuck in hinges, four loading-unloading cycles were performed for maximum prescribed displacements equal to 10 mm, 20 mm, 30 mm, 40 mm, and 50 mm, respectively. No out-of-plane movements of the specimen was observed. For all cycles, residual (negative compression) total reaction force following unloading was less than two percent of the total reaction force peak. Fig. 10(A) shows a picture of the deformed specimen.

Digital Image Correlation

The kinematic results described in the sequel were obtained via Digital Image Correlation (DIC). Digital Image Correlation consists in measuring displacement elds by registering pictures acquired during mechanical tests [START_REF] Sutton | Advances in Two-Dimensional and Three-Dimensional Computer Vision[END_REF][START_REF] Sutton | Image correlation for shape, motion and deformation measurements: Basic Concepts, Theory and Applications[END_REF][START_REF] Sutton | Computer Vision-Based, Noncontacting Deformation Measurements in Mechanics: A Generational Transformation[END_REF]]. Various approaches have been introduced, namely, local (i.e. subset-based) analyses [START_REF] Peters | Digital imaging techniques in experimental stress analysis[END_REF][START_REF] Sutton | Determination of Displacements Using an Improved Digital Correlation Method[END_REF][START_REF] Chu | Applications of Digital-Image-Correlation Techniques to Experimental Mechanics[END_REF], and global (e.g. nite element based) techniques [START_REF] Broggiato | Adaptive image correlation technique for full-eld strain measurement[END_REF][START_REF] Sun | Finite-element formulation for a digital image correlation method[END_REF][START_REF] Besnard | Finite-element displacement elds analysis from digital images: Application to Portevin-Le Chatelier bands[END_REF]. When dealing with pantographic structures, nite element based analyses have recently been performed at macroscopic [START_REF] Turco | Enhanced Piola-Hencky discrete models for pantographic sheets with pivots without deformation energy: Numerics and experiments[END_REF] and mesoscopic scales [START_REF] Dell'isola | Pantographic metamaterials: an example of mathematically-driven design and of its technological challenges[END_REF]. In the present case, the sought kinematics corresponds to the in-plane displacements of the hinges at positions p i,j of the bi-pantographic structure. The analysis of the displacement of these discrete points is performed via local DIC, i.e. using zones of interest or ZOIs [START_REF] Hild | Multi-Scale Displacement Field Measurements of Compressed Mineral Wool Samples by Digital Image Correlation[END_REF] centred on each hinge. The simplest approach seeks the rigid body translation of each considered ZOI, as originally performed in particle image velocimetry [START_REF] Barker | Measuring uid velocities with speckle patterns[END_REF][START_REF] Dudderar | Laser Speckle Photography in a Fluid Medium[END_REF][START_REF] Grousson | Study of ow pattern in a uid by scattered laser light[END_REF][START_REF]Scattering particle characteristics and their eect on pulsed laser measurements of uid ow: speckle velocimetry vs. particle image velocimetry[END_REF][START_REF]Twenty years of particle image velocimetry[END_REF]. Let f and g be the initial and current gray level images, respectively. For each ZOI, the correlation product

T (u) = Argmax υ ZOI f (x, y)g(x + υ • e x , y + υ • e y ) (64)
is maximized with respect to the rigid body translations υ ∈ R 2 . The computation of the correlation product can be performed in Fourier space (thanks to the shift/modulation property) via FFTs to speed up the calculations [START_REF] Hild | Digital Image Correlation[END_REF].

No subpixel resolution [START_REF] Hild | Multi-Scale Displacement Field Measurements of Compressed Mineral Wool Samples by Digital Image Correlation[END_REF] was sought in the present case since the expected displacements were very large when expressed in terms of pixels. Further, to account for the local angular variations between the beams connected by the hinges, the DIC calculations were performed incrementally, namely, for a series of pictures, the deformed picture of the n-th registration step becomes the reference picture of the n + 1-th step, and the corresponding displacement increment is cumulated with the previous ones to provide a Lagrangian estimation of the hinge displacements. Last, for each analysis, two passes were performed. The rst one used a rather large ZOI size (i.e. 100×100 pixels) to get a robust rst estimate. The second one utilised a smaller size (i.e. 50×50 pixels) to focus on the kinematic analysis about each hinge.

Results

The focus of this section is to present results obtained by the continuum model, and discuss how much they deviate from the experimental data. Owing to symmetry arguments (i.e. D4 [START_REF] Auray | A complete description of bi-dimensional anisotropic strain-gradient elasticity[END_REF] symmetry with respect to pantographic beam directions, symmetry of the specimen and boundary conditions with respect to specimen's axes) it is concluded that the following symmetries should be fullled by the continuum solution (analogous statements can be done for the discrete one) with the notation g(ζ, ς) standing for g[x(ζ, ς), y(ζ, ς)]

ϑ y (ζ, ς) = ϑ x (ζ, -ς), ρ y (ζ, ς) = ρ x (L -ζ, ς) ρ y (ζ, ς) = ρ x (ζ, -ς), ϑ y (ζ, ς) = ϑ x (L -ζ, ς) (65)
and

ϑ y (ζ, ς) = ϑ y (L -ζ, -ς), ρ y (ζ, ς) = ϑ y (L -ζ, -ς) . (66)
As in the considered problem either 0 < ϑ x (x, y) < π/2 and -π/2 < ϑ y (x, y) < 0 or 0 < ϑ y (x, y) < π/2 and -π/2 < ϑ x (x, y) < 0 this can be seen a posteriori by looking at Fig. 14) then the shear angle, which is null in the undeformed conguration and objective, is written in an easier way as π/2 -|ϑ x | -|ϑ y |. An analogous observation holds for the micro-model. The maximum prescribed displacement ū directed along ζ is equal to 50 mm. Parameters for the continuum (K F , K E and K S ) were found by tting three-curves (see Fig. 11 and cf. also [START_REF] Dell'isola | Large deformations of planar extensible beams and pantographic lattices: heuristic homogenization, experimental and numerical examples of equilibrium[END_REF] where the same quantities although dened for pantographic fabrics were used for tting). The rst one (Fig. 11 (left)) is the total reaction force along the direction ζ (determined by the load cell of the testing machine) vs ū (determined by the machine encoder unit). The second one (Fig. 11 (centre)) is the shear angle at pt. A (determined by DIC, see Fig. 3) vs ū. Finally, the third one (Fig. 11 (right)) is the shear angle at pt. B (determined by DIC, see Fig. 3) vs ū. The total reaction force acting on Ω 3 has been found for the continuum model by means of Castigliano's Theorem.

In order to check that computed Lagrange multipliers were consistent with the reaction force found by such a theorem, i.e. with energy conservation, one can compute the total reaction force acting on Ω 3 with the Lagrange multipliers as

- Ω3 υ • ζ dl. ( 67
)
This fact holds true for the numerical solution. Figure 12 compares the total reaction force along the direction ζ as computed by the continuum model using Lagrange multipliers and Castigliano's theorem vs ū. The forward nite dierence approximation of ∂E/∂ ū was computed with a step size for ū equal to ∆ū =1 mm. It is concluded that, up to nite dierence discretization errors, the results obtained with the two methodologies are consistent. Owing to symmetry arguments, the total reaction force along the direction ς as computed with Lagrange multipliers should be cancelling out. Also this fact holds true for the numerical solution. The tted values of the parameters for the continuum model are K F = 0.9 J, K E = 0.33 J, K S = 34 N•m -1 . Hence, the continuum model is capable to describe the considered experimental curves with only three constitutive parameters. The computed deformed conguration, i.e. χ(x i , y j ), is compared for dierent prescribed ū levels with experimentally measured data in Fig. 13. It is seen that experimental measurements by DIC and the continuum model agree very well. Experimental data, unlike the continuum model, exhibit a non-symmetry which is especially evident for ū = 40 mm and ū = 50 mm on the left. It is worth noting that only the use of homogenisation starting from a discrete model with a target model not chosen a priori allows such complex deformation energy to be recovered. The underlying family of discrete systems does not only lead to the deformation energy but also allows for a clear interpretation of non-standard boundary conditions that appear in this formulation. Contour plots of the y-stretch ρ y are shown in Fig. 14 for the continuum model. Figure 14 shows that the stretch is remarkably non-localized. This is due to pantographic beams being complete second gradient continua.

Let us quantify the sensitivity of the numerical simulation with respect to the application of non-standard zero normal displacement gradient boundary condition [∇u(x, y)]n(x, y) = 0 on Ω 1 ∪ Ω 3 . In Fig. 15, the quantities ρ y and ϑ y are plotted as functions of the local abscissa Φ of the boundary Ω 1 for the continuum model in both cases when zero normal displacement gradient boundary conditions are enforced and when they are not (ū = 50 mm). Especially

Conclusion and outlook

Bi-pantographic fabrics proved to have an extremely wide elastic range. This is possible because in such structures the total deformation is much greater than single-elastic-element deformations. Compatibly with boundary conditions and internal connection constraints, the elements arrange so as to minimize the total deformation energy by mimicking the wide variety of mechanisms corresponding to oppy modes. Some future outlooks of the present work are

• designing, experimenting and analysing a bi-pantographic system obeying the discrete model with k S = 0, which would mean that all cylinders connecting slender monolithic elements would be replaced by hinges, giving a purely second gradient material at macro-scale • studying the dynamics of bi-pantographic fabrics, which could be done by exploiting the results already obtained for pantographic beams [START_REF] Barchiesi | Wave dispersion in non-linear pantographic beams[END_REF] • studying out-of-plane deformations [START_REF] Giorgio | Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis[END_REF] of such metamaterials.

Acknowledgements. Authors thank P. Seppecher, G. Ganzosch and T. Lekszycki for stimulating discussions.

Appendix A

The terms θ i and ϕ µS i -ϕ µD i are expanded up to rst order by using the denitions (1) and (3) together with the expansions ( 13) and [START_REF] Tartar | The general theory of homogenization: a personalized introduction[END_REF]. According to [START_REF] Babu²ka | Homogenization approach in engineering[END_REF] and [START_REF] Allaire | Homogenization and two-scale convergence[END_REF] the vectors between two adjacent points p i are

p i+1 -p i = ε χ (s i ) + ε 2 χ (s i ) + o(ε) , p i -p i-1 = ε χ (s i ) - ε 2 χ (s i ) + o(ε) . (68)
The arguments of tan (

) 70
For a real valued function h(ε), tan -1 (h(ε)) = tan -1 (h(0)) + h (0) 1+h(0) 2 ε + o(ε). Since h i (0) = h i+1 (0), the rst terms in the Taylor series of both tan -1 expressions in (3) coincide

θ i = 1 1 + h i+1 (0) 2 h i+1 (0) - 1 1 + h i (0) 2 h i (0) ε + o(ε) (70) = 1 1 + χ (si)•ey χ (si)•ex 2 χ (s i ) • χ ⊥ (s i ) [χ (s i ) • e x ] 2 ε + o(ε) = χ (s i ) • χ ⊥ (s i ) χ (s i ) 2 ε + o(ε) (11)
= ϑ (s i )ε + o(ε) .

(

) 71
For the expansion (1), it is required to perform the expansion of the norm of a vector valued function a(ε), i.e. a(ε) = a(0) + a(0)•a (0) a(0)

ε + o(ε). Taking a(ε) to be the expansions appearing in the squared brackets of ( 68) and considering that ρ(s) = χ (s) ,

p i±1 -p i = ε χ (s i ) ± χ (s i ) • χ (s i ) χ (s i ) ε 2 + o(ε) = ε ρ(s i ) ± ρ (s i ) ε 2 + o(ε) . (72)
Consequently, the expansion of the squared expression of (72) reads

p i±1 -p i 2 = ε 2 χ 2 ± (χ • χ )ε + o(ε) s=si = ε 2 ρ 2 ± ρρ ε + o(ε) s=si . (73)
Using ( 14), ( 72) and [START_REF]Twenty years of particle image velocimetry[END_REF] in the argument of cos -1 of (1) 1,2 , h 1D(S) (ε) =

Expanding ϕ µS i -ϕ µD i with the help of ( 76), the rst term thereof cancels. Inserting the derivative with respect to ε evaluated at ε = 0 of ( 74) and ( 75

Using ( 14), ( 72) and ( 73) in the argument of cos -1 of (2), we can compute

h 1(2) (ε) = l 1(2)D i 2 + l
The angles ξ 1 and ξ 2 are thus expanded as

ξ µ i = cos -1 [h µ (0)] + o(ε 0 ) = cos -1 1 - ρ 2 1 /2 cos 2 γ s=si + o(ε 0 ) . (81)
Thus, for γ = π /6, ξ µ i = cos -1 1 -

3 2 ρ 2 s=si + o(ε 0 ) . ( 82
)
For the expansion of

K S cos -1 1 - ρ 2 α 1 /2 cos 2 γ -π + 2γ 2 (83)
in [START_REF] Bersani | Seppecher P Lagrange multipliers in innite dimensional spaces, examples of application[END_REF] with respect to ∇u required to get equation [START_REF] Angelo | The macroscopic behavior of pantographic sheets depends mainly on their microstructure: experimental evidence and qualitative analysis of damage in metallic specimens[END_REF], equation ( 50) is inserted into the following a cos -1 1 -(x + 1)

Figure 1 .

 1 Figure 1. Pantographic beam. (a) Undeformed conguration. (b) Generalized coordinates of i-th cell. (c) Deformed conguration with redundant kinematic quantities. (d) Force elements of a single cell.

Figure 2 .

 2 Figure 2. Bi-pantographic fabrics. (a) Domain Ω. (b) Undeformed conguration of (i, j)-th cell (including neighbouring elements). (c) Force elements of a single cell. (d) Deformed conguration of (i, j)-th cell (including neighbouring elements).

  y) = ρ x (x, y) {[cos ϑ x (x, y)] e x + [sin ϑ x (x, y)] e y } , ∂χ ∂y (x, y) = ρ y (x, y) {[cos ϑ y (x, y)] e y + [sin ϑ y (x, y)] e x } .

4 .

 4 Materials and methods 4.1. Manufacturing Specimens were 3D-printed using a Selective Laser Sintering (SLS) procedure. Polyamide powder was used as raw material. Possible use of metallic powders is to be investigated[START_REF] Spagnuolo | Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments[END_REF][START_REF] Spagnuolo | Phenomenological aspects of quasi-perfect pivots in metallic pantographic structures[END_REF][START_REF] Angelo | The macroscopic behavior of pantographic sheets depends mainly on their microstructure: experimental evidence and qualitative analysis of damage in metallic specimens[END_REF]. A picture obtained by optical microscopy showing the granularity of the printed Polyamide is presented in Fig.4. Modelling at lower scales taking into account such a granular structure[START_REF] Misra | Micromechanical model for viscoelastic materials undergoing damage[END_REF][START_REF] Misra | Granular micromechanics model for damage and plasticity of cementitious materials based upon thermomechanics[END_REF][START_REF] Misra | Thermomechanics-based nonlinear rate-dependent coupled damage-plasticity granular micromechanics model[END_REF][START_REF] Misra | Theoretical nonlinear response of complex single crystal under multi-axial tensile loading[END_REF]] might be considered in future investigations.

Figure 4 .

 4 Figure 4. Micrograph of a monolithic slender element of the prototype showing granularity of printed Polyamide.

Figure 5 .

 5 Figure 5. Technical drawing of the designed bi-pantographic prototypes. Top-view (top) and prole view (bottom). All lengths are expressed in mm.

Figure 6 .

 6 Figure 6. Section A-A indicated in Fig. 5 (bottom). All lengths are expresses in mm.

Figure 7 .

 7 Figure 7. Full top-view of bi-pantographic prototype manufactured according to the technical drawings in Figs. 5, 6 and 10 (right).

Figure 8 .

 8 Figure 8. Manufactured specimen (A) and CAD design (B). Zoomed view of 1. monolithic slender elements corresponding by means of bending and extension to blue/red rotational springs and to the adjacent extensional springs of Fig. 2(c) (black arrows), 2. cylinders materialising by means of torsion the green rotational springs of Fig. 2(c) (green arrow), and 3. hinges connecting monolithic slender elements at middle points (red arrow).

Figure 9 .

 9 Figure 9. Realization of boundary conditions listed in Tab. 1. Manufactured specimen (A) and zoomed view of its CAD design (B).

Figure 10 .

 10 Figure 10. Deformed specimen (A) and zoomed view (B) of the top-left corner of Fig. 5 (top).

Figure 11 .

 11 Figure 11. Total reaction force along the direction ζ with changed sign (N) vs prescribed displacement ū (mm) along the direction e ζ (left), shear angle at pt. A ( • ) vs prescribed displacement ū (mm) along the direction e ζ (centre), and shear angle ( • ) at pt. B vs prescribed displacement ū (mm) along the direction e ζ (right).

Figure 12 .Figure 13 .

 1213 Figure 12. Total reaction force along the direction ζ with changed sign (N) vs ū (mm) as computed by the continuum model using Lagrange multipliers and Castigliano's theorem. The forward nite dierence approximation of ∂E/∂ ū has been computed with a step size for ū equal to ∆ū =1 mm.

Figure 14 . 1 1 Figure 15 .

 141115 Figure 14. y-stretch ρ y as computed by the continuum model. Abscissas and ordinates are expressed in mm.

( 69 )

 69 It is noted that hi (0) = h i+1 (0) = [χ (s i ) • e y ] / [χ (s i ) • e x ]. Moreover h i+1 (0) = -h i (0) = 1 2[χ • e x ] 2 [(χ • e y )(χ • e x ) -(χ • e x )(χ • e y )] • e x ] 2 χ • (e y ⊗ e x -e x ⊗ e y ) • χ s=si = χ (s i ) • χ ⊥ (s i ) 2 [χ (s i ) • e x ] 2 .

ε 2 ρ 2 -. ( 75 ) 1 -

 2751 ρρ ε + ε( 1 /cos γ)( l1D(S) -l2S(D) ) + o(ε)2ε 2 1 /2 cos γ + l1D(S) ε + o(ε) ρ -ρ ε 2 + o(ε) s=si = ρ 2 + ε ( 1 /cos γ)( l1D(S) -l2S(D) ) -ρρ + o(ε) ( 1 /cos γ)ρ + ε 2 l1D(S) ρ -( 1 /2 cos γ)ρ + o(ε) expansions of the arguments of cos -1 appearing in (1) 3,4 read h 2S(D) (ε) = ρ 2 + ε ( 1 /cos γ)( l2S(D) -l1D(S) ) -ρρ + o(ε) ( 1 /cos γ)ρ + ε 2 l2S(D) ρ -( 1 /2 cos γ)ρ + o(ε)s=si All functions are of the form h µν (ε) = [a + εb µν + o(ε)] / [c + εd µν + o(ε)] with h µν (0) = a/c and (h µν ) (0) = (b µν c -d µν a)/c 2 . The angles ϕ µν i are thus expanded as ϕ µν i = cos -1 [h µν (0)] -ε h µν (0) 2 (h µν ) (0) + o(ε) .

ρ 2 2 - 2 s=si 3 (

 2223 ( l1S -l1D ) + ( 1 /2 cos γ)ρρ -( 1 /2 cos 2 γ)( l1S -l1D + l2S -l2D ) ρ( 1 /2 cos γ) √ 2 ( 1 /cos 2 γ)-ρ 2 2 s=si ε + o(ε) = ρ[ρ 2 -( 1 /2 cos 2 γ)]( l1S -l1D ) + ( 1 /2 cos γ)ρ 2 ρ + ρ( 1 /2 cos 2 γ)( l2D -l2S ) ρ 2 ( 1 /2 cos γ) √ 2 ( 1 /cos 2 γ)-ρ 2 2 s=si ε + o(ε) .(77)In the same manner the expansion for the dierence in angles of the oblique springs indexed by µ = 2 is obtained. Moreover, the previous expressions are simplied to giveϕ ( 1 /2 cos 2 γ)]( l1(2)S -l1(2)D ) + ( 1 /cos γ)(ρ 2 ) + ( 2 /cos 2 γ)( l2(1)D -l2(1)S ) 4ρ( 1 /2 cos γ) ( 1 /cos 2 γ) -ρ 2 -2 /3)( l1(2)S -l1(2)D ) + ρρ + 2 / √ l2(1)D -l2(1)S ) ρ 4 /3 -ρ 2 s=si ε + o(ε) .

2( 1 )S i+1 2 -p i+1 -p i 2 2l 1 = ε 2 1 / 2 1 / 2

 12211212 cos 2 γ -ρ 2 + ρρ ε + ε( 1 /cos γ)( l1(2)D + l2(1)S ) + o(ε) 2ε 2 1 /2 cos γ + l1(2)D ε + o(ε) 1 /2 cos γ + l2(1)S ε + o(ε) s=si = cos 2 γ -ρ 2 + ε ρρ + ( 1 /cos γ)( l1(2)D + l2(1)S ) + o(ε) 1 /2 cos 2 γ + ε( 1 /cos γ)( l1(2)D + l2(1)S ) + o(ε) s=si .

  -1 in (3) are written as functions of εh i+1 (ε) = (p i+1 -p i ) • e y (p i+1 -p i ) • e x (68) 1 = χ (s i ) • e y + ε 2 χ (s i ) • e y + o(ε) χ (s i ) • e x + ε 2 χ (s i ) • e x + o(ε) , h i (ε) = (p i -p i-1 ) • e y (p i -p i-1 ) • e x (68) 2 = χ (s i ) • e y -ε 2 χ (s i ) • e y + o(ε) χ (s i ) • e x -ε 2 χ (s i ) • e x + o(ε).


Nicolas Passat

Benoît Naegel

AN EXTENSION OF COMPONENT-TREES TO PARTIAL ORDERS

Keywords: Mathematical morphology, componenttrees, filtering, partial orders, colour imaging

An extension of component-trees to partial orders

Nicolas Passat, Benoît

INTRODUCTION

The component-tree models some structural characteristics of an image by considering its "binary slices" obtained by threshold operations at successive levels. It has been defined in the theoretical framework of mathematical morphology and involved, in particular, in the development of morphological operators [START_REF] Breen | Attribute openings, thinnings, and granulometries[END_REF][START_REF] Salembier | Anti-extensive connected operators for image and sequence processing[END_REF].

Thanks to efforts devoted to the efficient computation of component-trees [START_REF] Salembier | Anti-extensive connected operators for image and sequence processing[END_REF][START_REF] Najman | Building the component tree in quasi-linear time[END_REF], or their use in complex knowledge handling procedures, they have been considered for the design of various kinds of grey-level image processing methods, including image and video segmentation [START_REF] Salembier | Anti-extensive connected operators for image and sequence processing[END_REF][START_REF] Jones | Connected filtering and segmentation using component trees[END_REF], image registration [START_REF] Mattes | Tree representation for image matching and object recognition[END_REF] or image compression [START_REF] Salembier | Anti-extensive connected operators for image and sequence processing[END_REF].

We propose to explore new ways for the use of componenttrees by not only considering images taking their values in completely-ordered sets (i.e. "grey-level" images), but more generally in partially-ordered ones, such as colour images, for instance. A data structure and some strategies for filtering such images are described. They are, in particular, compliant with the classical approaches proposed in the grey-level case. Experimental results enable to illustrate the soundness and usefulness of the approach.

The article is organised as follows. In Sec. 2, theoretical background notions are provided. In Sec. 3, the definition of the component-tree, and classical ways for using it in filtering procedures are recalled. In Sec. 4, the proposed extension of component-trees to images taking their values in partiallyordered sets is described. Experimental results and perspectives will be found in Sec. 5.

THEORETICAL BACKGROUND

Let n ∈ N * . Let E ⊂ Z n (generally, E = n i=1 [0, d i -1] with (d i ) n i=1 ∈ (N * ) n ).
Let V be a finite set. Let ≤ be an order relation on V . We suppose that there exists an infimum, noted ⊥, for (V, ≤). Let I : E → V (we also note I ∈ V E ), I is a (discrete) image, taking its values in V . If (V, ≤) is a completely-ordered (resp. partially-ordered) set, then we that I is a grey-level image (resp. a label image).

Let x, y ∈ E. We say that x and y are adjacent if d(x, y) ≤ ǫ for a well-chosen distance d(., .) on E and a nonnegative value ǫ (with ǫ = 1 and d(., .) = . -. 1 (resp. d(., .) = . -. ∞ ), we retrieve the classical (2n)-adjacency

(resp. (3 n -1)-adjacency) associated to Z n ).
Let X ⊆ E. Let x, y ∈ X. We say that x and y are connected (in X), and we note x ∼ y, if there exists a sequence (x k ) l k=1 (l ≥ 1) of elements of X such that x 1 = x, x l = y and x k , x k+1 are adjacent for all k ∈ [1, l -1]. Note that ∼ is an equivalence relation on X. The connected components of X are the equivalence classes of X w.r.t. ∼, i.e. the elements of the quotient set X/∼ (noted C[X] in the sequel).

Let v ∈ V . Let X v : V E → P(E) be the threshold operator defined by

X v (I) = {x ∈ E | v ≤ I(x)} for all I ∈ V E . Let v ∈ V and X ⊆ E. We define the cylinder func- tion C X,v : E → V by C X,v (x) = v if x ∈ X and ⊥ otherwise. Based on this definition, a discrete image I ∈ V E can be expressed as I = v∈V C Xv(I),v = v∈V X∈C[Xv(I)] C X,v
, where is the pointwise supremum of a set of functions, i.e.

I(x) = sup v∈V {C Xv(I),v (x)} (= max v∈V {C Xv(I),v (x)} as V finite). Let K = v∈V (C[X v (I)] × {v}). Let ⊆ K be the relation defined by (X, v X ) ⊆ K (Y, v Y ) if X ⊂ Y or (X = Y and v Y ≤ v X ). The inclusion relation ⊆ K is a partial order on K.
By abuse of notation, in the sequel, we will sometimes note X ∈ K for (X, v X ) ∈ K, and ⊆ for ⊆ K .

COMPONENT-TREES

In this section, we assume that (V, ≤) is a completely-ordered set. In particular, we identify (V, ≤) and Let 1 (b-f) for an illustration of this assertion). Note that, in particular, we necessarily have

([0, |V | -1], ≤). (a) (b) (c) (d) (e) (f) v = v = v = v =
v 1 ≤ v 2 ∈ V . Let B 1 , B 2 ⊆ E be the binary images defined by B k = X v k (I) for k ∈ {1, 2}. Let C 2 ∈ C[B 2 ] be a connected component of B 2 . Then, there exists a (unique) connected component C 1 ∈ C[B 1 ] of B 1 such that C 2 ⊆ C 1 (see Fig.
B 2 ⊆ B 1 .
Based on these considerations, it easily derives that the Hasse diagram of the partially-ordered set (K, ⊆) is a tree (i.e. a connected acyclic graph), the "root" of which is the supremum sup(K, ⊆) = X ⊥ (I) = E. This tree is called the component-tree of I (see Fig. 1(g)).

Using the above definitions and notations, the componenttree -as a data structure -can be described as follows. Definition Let I ∈ V E be a grey-level image. The componenttree of I is the rooted tree (K, L, R) defined by

R = E = sup(K, ⊆), (1)
L = {(X, Y ) ∈ C[X v (I)] × C[X v+1 (I)] | Y ⊆ X}. (2)
The elements K, R and L are the set of the nodes, the root and the set of the edges of the tree, respectively (in the sequel, we will use the usual terminology associated to trees). In Fig. 1(g), K is the set of all white rectangles (R is the one located at the highest level) and L is visualised by the set of all black edges (linking the elements of each pair). Component-trees enable the storage -at each node -of elements of information, also called attributes, related to the binary connected component associated to the node. For instance, in Fig. 1(h), the size (i.e. the cardinal) of the connected components has been added at their corresponding nodes 1 .

Component-trees, equipped with attributes related to their image, are generally considered for filtering operations con- 1 Here, the attribute is a single numerical value. It is however possible to consider any kind of (quantitative/qualitative and scalar/vectorial) attributes, provided they can be conveniently formalised. sisting in "pruning" the branches of the tree, based on a chosen predicate ρ on K, indicating if the attribute of a node satisfies a given criterion. In case of non-increasing criteria, various pruning policies can be considered (see [START_REF] Urbach | Connected shape-size pattern spectra for rotation and scale-invariant classification of gray-scale images[END_REF]):

(i) Min: A node N ∈ K is removed if ¬ρ(N ) or the parent node P ∈ K of N has been removed; (ii) Max: A node N ∈ K is removed if ¬ρ(N ) and all its children C ∈ K have been removed; (iii) Direct: A node N ∈ K is removed if ¬ρ(N ).
Once the set K has been reduced to a set K ρ ⊆ K according to one of these three policies, a reconstructed image I ρ of I can be defined as I ρ = (X,vX )∈Kρ C X,vX . Filtering and reconstruction results of these strategies are illustrated on a 1-D example in Fig. 2.

EXTENSION TO PARTIAL ORDERS

In this section, we now assume that (V, ≤) is a partiallyordered set and that (V, ≤) has an infimum, noted ⊥ (otherwise we enrich (V, ≤) with such an infimum).

Let

I ∈ V E be a label image. Let v ∈ V . Let B ⊆ E be the binary image defined by B = X v (I). Let C ∈ C[B]
be a connected component of B. By opposition to the case of completely-ordered sets (Sec. 3), there may exist several andv i is maximal w.r.t. ≤ for these properties.

values v i ≤ v (i ∈ N), such that C ⊆ C i ∈ C[X vi (I)],
As in Sec. 3, the relation ⊆ is still a partial order on K, but the Hasse diagram of the partially-ordered set (K, ⊆) is not necessarily a tree. This derives from the fact that -using tree terminology -a node of the diagram can be the child of several parent nodes (and not only one). Note that such a diagram (called component-graph in the sequel) has however a supremal element (which can be assimilated to its "root"), namely X ⊥ (I) = E, while it can possibly have several minimal elements.

(c) (d) (e) (f) (g) (h) (2,2,2) (0,0,0) (0,0,1) (0,0,2) (0,1,0) (1,0,0) (2,0,2) (1,0,2) (0,1,2) (1,1,0) (2,1,2) (i)

Component-graphs

Let I ∈ V E be a label image. The component-graph of I is the "rooted" oriented graph (K, L, R) defined such that (K, L) is the Hasse diagram of the partially-ordered set (K, ⊆ ), and R = sup(K, ⊆) = X ⊥ (I) = E. Keeping the same terminology as previously, K, R and L are the set of the nodes, the root and the set of the edges of the graph, respectively. Fig. 3 illustrates a label image I (a), some of its threshold images (b-h), and its component graph (i): K is the set of all rectangles (R is the black one) and L is visualised by the set of all black edges (linking the elements of each pair).

Similarly to component-trees, attributes can be stored at each node of a component-graph. The preservation/removal of nodes of the graph w.r.t. such attributes can then also lead to filtering strategies. However, the less restrictive properties of component-graphs (by opposition to those of componenttrees) do not enable to straightforwardly apply the filtering strategies designed for component-trees, described in Sec. 3. To tackle this problem, we propose hereafter a two step filtering procedure.

Filtering strategy

Let I ∈ V E be a label image and (K, L, R) be its componentgraph (the image I of Fig. 3(a) will be considered for illustration). Let ρ be a predicate on K. The following strategy is devoted to the filtering of I by "pruning" its component-graph w.r.t. ρ.

Step 1: Node selection

Let K ′ = K \ {R}. In case of non-increasing criteria (notice that with increasing ones, all policies are equivalent), the following policies can be considered to select a subset K ρ ⊆ K of nodes to be preserved in the component-graph: of the parent node P ∈ K of N has been removed;

(i) Min 1 : A node N ∈ K ′ is removed if ¬ρ(N ) or at least one (2,2,2) (0,0,0) (0,0,1) (0,0,2) (0,1,0) (1,0,0) (2,0,2) (1,0,2) (0,1,2) (1,1,0) (2,1,2) (0,0,0) (0,1,0) (1,0,0) (1,1,0) (0,0,0) (0,1,0) (1,0,0) (0,1,2) (1,1,0) (2,1,2) (0,0,0) (0,0,1) (0,0,2) (0,1,0) (1,0,0) (2,0,2) (1,0,2) (0,1,2) (1,1,0) (2,1,2) (0,0,0) (0,1,0) (1,0,0) (2,0,2) (0,1,2) (1,1,0) (2,1,2) (a) (b) (c) (d) (e)
(ii) Min 2 : A node N ∈ K ′ is removed if ¬ρ(N ) or all the parent nodes P ∈ K of N have been removed; (iii) Max: A node N ∈ K is removed if ¬ρ(N ) and all its children C ∈ K ′ have been removed; (iv) Direct: A node N ∈ K is removed if ¬ρ(N ).
Note that with the Min 1 and Min 2 (resp. Max) policy(ies), K ρ has to be computed in a top-down (resp. bottom-up) fashion. Moreover, with the first three policies, we should assume that R ∈ K ρ (otherwise, we would obtain a -useless -trivial set K ρ = ∅). More formally, with the Min 1 (3), Min 2 (4), Max (5), and Direct (6) policies, we have (recursively)

K ρ = {R} ∪ {N ∈ K ′ | ρ(N ) ∧ ∀(P, N ) ∈ L, P ∈ K ρ }, (3)
K ρ = {R} ∪ {N ∈ K ′ | ρ(N ) ∧ ∃(P, N ) ∈ L, P ∈ K ρ }, (4)
K ρ = {R} ∪ {N ∈ K ′ | ρ(N ) ∨ ∃(N, C) ∈ L, C ∈ K ρ },(5) K ρ = {N ∈ K | ρ(N )}. (6)
Once K ρ has been computed, a new set of edges L ρ can straightforwardly be defined by building the Hasse diagram of (K ρ , ⊆). Examples of graphs (K ρ , L ρ ) obtained with these different policies are illustrated in Fig. 4.

Step 2: Coherence recovery

By considering the Min 1 and Min 2 (resp. Max, Direct) policy(ies) proposed above, in the case of a completely-ordered set, we obviously retrieve the "classical" Min (resp. Max, Direct) policy defined in Sec. 3. The proposed approach is then compliant with the case of component-trees. However, in the considered case of a partially-ordered set (V, ≤), it may sometimes be impossible to reconstruct an image I ρ = (X,vX )∈Kρ C X,vX defined by I ρ (x) = max (X,vX )∈Kρ {C X,vX (x)} for all x ∈ E, since such a maximum may be undefined, due to the partiality of (V, ≤). As an example, in Fig. 4, the graph of (b) directly leads to a "correct" image I ρ , while it is not the case for those of (c-e) since the lowest nodes of the graph correspond to intersecting connected components associated to non-comparable values.

In order to deal with this issue, a coherence recovery procedure has to be proposed to define unambiguously max (X,vX )∈Kρ {C X,vX } for all x ∈ E. This procedure consists in adding to the current graph (K ρ , L ρ ) a set of nodes (actually removed during the previous step) and their associated edges, such that the resulting corrected graph (K c ρ , L c ρ ) enables the generation of a well-defined image. Practically, this can be done by computing iteratively -and until stability -K c ρ (initialised to K ρ ) as follows. Choose a point x ∈ E and two nodes

N i = (X i , v Xi ) ∈ K c ρ (i ∈ {1, 2}) (with N 1 ⊆ N 2 , N 2 ⊆ N 1 ) such that x ∈ X 1 ∩ X 2 and ∀N = (X, v X ) ∈ K c ρ \ {N i }, x ∈ X ⇒ (N ⊆ N i ). Then, there exists N = (X, v X ) ∈ K \ K c ρ such that x ∈ X ⊆ X 1 ∩ X 2 and v X1 , v X2 ≤ v X . Choose such a node N and set K c ρ = K c ρ ∪ {N }. Once K c ρ has been com- puted, L c
ρ is obtained by computing the Hasse diagram of (K c ρ , ⊆).

Note that the element N may be non-unique at a given iteration, leading to a possibly non-deterministic process, and thus various results (in the case where (V, ≤) is a lattice, some deterministic strategies can however be developed ; this will be considered in further works). Examples of corrected graphs and their associated image are available in Fig. 4(c-e).

EXPERIMENTAL RESULTS AND PERSPECTIVES

We have introduced an extension of the notion of componenttrees (namely the component-graphs) to the case of partiallyordered sets, enabling in particular to deal with colour images.

To illustrate the soundness of this new concept, a denoising example is proposed on a colour image (see Fig. 5). Considering a predicate ρ based on the size of the connected components of the nodes, a two-step component-graph filtering procedure (with the ≤ and ≥ partial orders on the RGB space, successively) is applied with a Direct policy on an image (a) corrupted with 10% (b) and 15% (c) random noise. It can be observed that the result images (e,f) present visual qualities quite similar to the original one (a,d).

This simple example (presented here due to space limitations) is -of course -not the only nor the main application field of component-graph approaches: more sophisticated ones (applied on both real colour images and "synthetic" ones [START_REF] Ronse | Morphology on label images: flat-type operators and connections[END_REF] obtained from hierarchical classification processes, for instance) will be presented in further works. From a methodological point of view, we will also develop this preliminary study by extending classical concepts of the component-trees such that the Subtractive or Viterbi policies, and by developing alternative coherence recovery strategies.

Fig. 1 .

 1 Fig. 1. (a) A grey-level image I : [0, 9] 2 → [0, 4]. (bf) Threshold sets X v (I) for v from 0 (b) to 4 (f). (g) The component-tree of I; its successive levels correspond to increasing threshold values v. (h) The same tree, enriched by an attribute (the size of the connected component of each node).

Fig. 2 .

 2 Fig. 2. (a) A 1-D grey-level image I and (f) its componenttree. (b) (Fictive) non-increasing attributes associated to the nodes of I. (c-e) Filtering results I ρ obtained by processing the component-tree w.r.t. a predicate ρ true if and only if the attribute of the node is higher than 15: (c) Min, (d) Max and (e) Direct policies.

Fig. 3 .

 3 Fig. 3. (a) A label image I : [0, 10] 2 → V (where V ⊂ [0, 2] 3 is composed of 11 distinct values depicted in (i)) visualised as an RGB image. (b-h) Some threshold images X v (I) (from (b) to (h), v = (0, 0, 2), (0, 1, 0), (1, 0, 0), (0, 1, 2), (1, 1, 0), (2, 0, 2), (2, 2, 2). (i) The component graph of I.

Fig. 4 .

 4 Fig. 4. (a) I and its component-graph, with nodes verifying a given predicate ρ (grey-background). (b-e) Light-grey background: graph (K ρ , L ρ ) obtained with the Min 1 (b), Min 2 (c), Max (d), and Direct (e) policies; light/dark-grey background: corrected graph (K c ρ , L c ρ ).

Fig. 5 .

 5 Fig. 5. (a,d) Original image (Lenna, c Playboy). (b,c) Noisy image (10% and 15% random noise corruption, respectively). (e-f) Filtered images obtained from (b,c) by removing the connected component of size smaller than 10 pixels.