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Mechanics of Materials

journal homepage: www.elsevier.com/locate/mechmat

Research paper

Mechanics of mutable hierarchical composite cellular materials

F. Ongaroa, E. Barbieri⁎,b, N.M. Pugno⁎,a,c,d

a School of Engineering and Materials Science, Queen Mary University of London, London, UKb Japan Agency for Marine-Earth Science and Technology (JAMSTEC), Department of Mathematical Science and Advanced Technology (MAT), Yokohama Institute forEarth Sciences (YES), 3173-25, Showa-machi, Kanazawa-ku Yokohama-city, Kanagawa, 236-0001, Japanc Laboratory of Bio-Inspired and Graphene Nanomechanics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento, Italyd Ket-Lab, Edoardo Amaldi Foundation, Italian Space Agency, Via del Politecnico snc, Rome, Italy

A R T I C L E I N F O

Keywords:CompositeCellular materialOrthotropyWinkler modelLinear elasticityDelosperma nakurenseKeel tissueHierarchy

A B S T R A C T

Cellular structures having the internal volumes of the cells filled with fluids, fibres or other bulk materials arevery common in nature. A remarkable example of composite solution is the hygroscopic keel tissue of the iceplant Delosperma nakurense. This tissue, specialized in promoting the mechanism for seed dispersal, reveals acellular structure composed by elongated cells filled with a cellulosic swelling material. Upon hydrating, thefiller adsorbs large amounts of water leading to a change in the cells’ shape and effective stiffness.

This paper, inspired by the configuration of the aforementioned hygroscopic keel tissue, deals with theanalysis of a two-dimensional honeycomb made of elongated hexagonal cells filled with an elastic material. Thesystem is treated as a sequence of Euler–Bernoulli beams on Winkler foundation, whose displacements are de-rived by introducing the classical shape functions of the Finite Element Method. The assumption of the Born rule,in conjunction with an energy-based approach, provide the constitutive model in the continuum form. It emergesa strong influence of the infill’s stiffness and cell walls’ inclination on the macroscopic elastic constants. Inparticular, parametric analysis reveals the system isotropy only in the particular case of regular hexagonalmicrostructure.

Even though a rigorous analysis of the keel tissue is well beyond our aim, the application of the theoreticalmodel to estimate the effective stiffness of such biological system leads to results that are in good agreement withthe published data, where the keel tissue is represented as an internally pressurized honeycomb. Specifically, anenergetic equivalence gives an explicit relation between the inner pressure and the filler’s stiffness. Optimalvalues of pressure and cell walls’ inclination also emerge.

Finally, the theory is extended to the hierarchical configuration and a closed form expression for the mac-roscopic elastic moduli is provided. It emerges a synergy of hierarchy and material heterogeneity in obtaining astiffer material, in addition to an optimal number of hierarchical levels.

1. Introduction

Cellular materials are commonly observed in nature (Gibson et al.,2010; Meyers et al., 2008; Altenbach and Oechsner, 2010; Gibson andAshby, 2001; Gibson, 2012). Due to their specific structural properties,they are very promising for engineering applications in a variety ofindustries including aerospace, automotive, marine and constructions(Wilson, 1990; Thompson and Matthews, 1995; Bitzer, 1994). As anexample, honeycombs are widely used in lightweight structures andsandwich panels because of their high bending stiffness and strength atlow weight.

Many authors extensively studied cellular materials and it would be

difficult to quote without omissions the vast literature flourished on themechanical modelling of such cellular structures in the last years.Noteworthy contributions such as Gibson and Ashby (2001),Gibson (1989) and Gibson et al. (1982) present a detailed discussion ofthe characteristics of many periodic cellular materials and providesimple relations between their density and equivalent mechanicalproperties through the application of beam theory. Other authors, likeOngaro et al. (2016b), Davini and Ongaro (2011), Kumar andMcDowell (2004), Burgardt and Cartraud (1999) and Chen et al. (1998)suggest alternative approaches to solve the crucial passage from dis-crete to continuum and to derive the constitutive model for the in-planedeformation of various two-dimensional microstructures by applying

https://doi.org/10.1016/j.mechmat.2018.05.006Received 22 September 2016; Received in revised form 21 May 2018; Accepted 23 May 2018

⁎ Corresponding authors at: Laboratory of Bio-Inspired and Graphene Nanomechanics, Department of Civil, Environmental and Mechanical Engineering, University of Trento, Trento,Italy; (N.M. Pugno) Department of Mathematical Science and Advanced Technology (MAT), Japan Agency for Marine-Earth Science and Technology (JAMSTEC); (E. Barbieri)

E-mail addresses: [email protected] (E. Barbieri), [email protected], [email protected] (N.M. Pugno).

Mechanics of Materials 124 (2018) 80–99

Available online 29 May 20180167-6636/ © 2018 Published by Elsevier Ltd.

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the energy equivalence. In addition, Warren and Byskov (2002),Wang and Stronge (1999), Dos Reis and Ganghoffer (2012) andDos Reis and Ganghoffer (2010) exploit a method based on the prin-ciples of structural analysis to obtain the homogenized continuummodel of the discrete lattice.

Although many efforts have been devoted to the prediction of theeffective properties of regular cellular materials with empty cells, in theliterature few investigations concern the characterisation of cellularstructures having the internal volumes filled with fluids, fibres or otherbulk materials as commonly happens in nature (Georget et al., 2003;Niklas, 1992; 1989; Van Liedekerke et al., 2010; Warner et al., 2000;Wu and Pitts, 1999; Zhu and Melrose, 2003; Mihai et al., 2015). Forexample, in the context of sandwich panels, D’Mello and Waas (2013)and Burlayenko and Sadowski (2010) present a finite element-basedtechnique to evaluate the structural performance of foam filled hon-eycombs. It emerges an increase in the load-bearing capacity of thematerial and an improvement in both the effective elastic propertiesand energy absorption due to the presence of the filler. More recently,Ongaro et al. (2016b) analyses the mechanics of a two-dimensionalfilled honeycomb by representing the microstructure as a sequence ofbeams on Winkler elastic foundation. The homogenized elastic moduliderived confirm, from a mechanical behaviour point of view, the ben-eficial effect due to the filling material. Other works,Harrington et al. (2011) and Guiducci et al. (2014), concerning thenature’s wonders of design, study the mechanics of the hygroscopic keeltissue of the ice plant Delosperma nakurense (Fig. 1) by representing it asa network of elongated cells internally pressurized. The ice plant, thatgrows in the arid regions of Africa, is a source of inspiration because ofits sophisticated origami-like movement mechanism for seed dispersal.The plant, in particular, has adopted its anatomy and material archi-tecture to the unfavourable environmental conditions by producing aspecial seed capsule to prevent the premature dispersion of the seeds. Inthe dry state, Fig. 1, five petal-like sections, the protective valves, coverthe seed compartment as a box-like lid. When it rains, the valves unfoldbackwards revealing a seed compartment composed by five seedchambers partitioned by five septa, Fig. 1. Within few minutes, most ofthe seeds are splashed out by the falling water (Lockyer, 1932). Whenthe capsule dries up, the valves return to the original position. Thespecialized organ promoting this movement is the hygroscopic keel,Fig. 1. In the dry state, this tissue consists of a network of elongatedcells filled with a ‘swelling cellulosic inner’ (CIL). If hydrated, the CILabsorbs large amounts of water giving rise to a change of the keel initialgeometry and stiffness, Fig. 1. In addition, experimental observations(Harrington et al., 2011) reveal that the filler contains a soft inclusionthat behaves like an elongated, thin septum partitioning the internalvolume of the cell. Consequently, the cell walls’ coupling effect due tothe presence of the filling material is compromised. Though manystudies experimentally investigated the morphology and composition ofthe keel, little is known about the relation between microstructure’sparameters and macroscopic mechanical behaviour (Guiducci et al.,2014).

Hierarchy is another way to enhance the mechanical properties oflightweight materials and structures.

Various authors studied structural hierarchy in biological systems(Gibson, 2012; Fratzl and Weinkamer, 2007; Mattheck and Kubler,1995; Pan, 2014; Gao, 2010; Chen and Pugno, 2013) and man-madematerials (Barthelat and Mirkhalaf, 2013; Sanchez et al., 2005; Fratzl,2007). Among others, Pugno and Chen (2011),Haghpanah et al. (2014), Ajdari et al. (2012), Fan et al. (2008) andTaylor et al. (2011) provide numerical and theoretical models, force orenergy based, to understand the role of hierarchy on the mechanicalbehaviour of cellular solids. All of them conclude that many desirableproperties, like stress attenuation, superplasticity, and increasedtoughness, are due to hierarchy. Conversely, for classical cellular ma-terials, the introduction of some levels of hierarchy is detrimental forthe specific stiffness. In spite of this, in the case of hierarchical archi-tectures of different types of fibre bundles, increasing the number ofhierarchical levels leads to an improvement in the material strength(Bosia et al., 2012).

Inspired by the previously introduced hygroscopic keel tissue, thispaper deals with a two-dimensional composite cellular material made ofelongated hexagonal cells, filled with an elastic medium. The studyprovides a theoretical model, based on the Born rule, that is able tounderstand the mechanics of the examined orthotropic configuration andis general enough to investigate the effects of adding some levels ofhierarchy. This work is organised in 6 sections, including this introduc-tion. Section 2 initially illustrates the mathematical formulation andmodelling technique while, in Section 3, the effective elastic constantsand constitutive equations are derived. Some considerations about theinfluence of the microstructure parameters, such as the stiffness of thefiller and the cell walls’ inclination, are presented in Section 4, as well asthe results of the application of the theoretical model to the biologicalkeel tissue. Despite a detailed investigation of the biological problem isbeyond our scope, it emerges that the elastic moduli obtained in thispaper agree with those proposed in the literature. Finally, in Section 5,the theory is extended to the hierarchical configuration and explicit ex-pressions for the macroscopic elastic moduli are derived. As a conclusion,Section 6 summarizes the main findings.

2. Problem statement: geometrical description and theoreticalmodelling of the discrete system

2.1. Geometrical description

In terms of crystallography, the configuration of the compositematerial considered here can be described as the union of two simplyshifted lattices (Fig. 2(a))

= ∈ = + ∈

= +

L n n n n L

L

x x l l

s

(ℓ) { : , with ( , ) }, (ℓ)

(ℓ),1

2 11

22

1 2 22

1

� �

(1)

with

= = +θ θ θl l(2 ℓ cos , 0), (ℓ cos , ℓ(1 sin ))1 2 (2)

the lattice vectors,

= θ θs (ℓ cos , ℓ sin ) (3)

Fig. 1. A schematic representation of the hygroscopic keel tissue of the ice plant Delosperma nakurense.

F. Ongaro et al. Mechanics of Materials 124 (2018) 80–99

81

the shift vector, ℓ and θ, respectively, the length (the lattice size) andangle of inclination of the cell walls.

2.2. Theoretical modelling

2.2.1. The discrete system continuum-springsThe discrete system is treated as a sequence of Euler–Bernoulli

beams on Winkler foundation, where a series of independent, linearelastic springs, the Winkler foundation, represent the material withinthe cells. In particular, each beam is supported by two sets of springs:the springs a, in the − η e

2 direction, and the springs b, in the η e2 (Fig. 3).

Being a rigorous analysis of the biological keel tissue a complexundertaking that does not coincide with the scope of our investigation,in the present paper the missing cell walls’ coupling effect caused by theseptum is modelled by anchoring the springs at the nodes of the latticeL3, defined by

= +L Ls(ℓ) 2 (ℓ).3 1 (4)

As illustrated in Fig. 2(a), the nodes of L3 are connected to the latticeL2 by means of line elements that, from a mechanical point of view, arerepresented as Euler–Bernoulli beams having stiffness much smallerthan the stiffness of the cell walls. Consequently, the energetic con-tribution of the beams composing the lattice L3 can be neglected withrespect to those composing the skeleton of the cells (i.e., the principallattices L1 and L2), introduced in the following section.

With reference to the equilibrium conditions of the springs’ an-chorage points it should be noted that the forces brought by the springsto such nodes balance with one another because of the symmetry of thehexagonal cells.

In particular, let us focus on the hexagonal cell illustrated in Fig. 4,where each beam is connected to the central point of the cell by closely-spaced elastic springs (i.e., the Winkler foundation). Note that, for easeof reading, in Fig. 4 the series of closely spaced springs are schemati-cally represented by a single spring connecting the beams to the centralpoint of the cell. As it can be seen, the symmetry of the hexagonal cellleads to a symmetric configuration of the springs. To make it moreclear, in Fig. 5 the two sets of symmetric springs (i.e., the springs arepresented in blue and the springs b represented in red) are enhanced.Let us now imagine to apply external forces to the cell, leading to ageneric deformation of the cell. Again, because of the symmetry of boththe hexagonal cell and the configuration of the springs, it emerges that,in terms of anchorage points (i.e., the central point of the cell), theforces brought by the springs balance with one another. Further detailsare provided in Section 2.4.

2.2.2. The Euler–Bernoulli beam on Winkler foundation elementIn the two-dimensional Euler–Bernoulli beam, each node has three

degrees of freedom, two translations and one rotation. Thus, the vectorof nodal displacements can be expressed as

= = u v φ u v φu u u[ ] [ ] .ei j

Ti i i j j j

T (5)

According to the Finite Element Method (FEM), the axial andtransverse displacements at every point within the beam are approxi-mated by

⎡⎣⎢

⎤⎦⎥

=u xv x

xΨ u( )( )

( ) ,e

(6)

with (0≤ x≤ ℓ) and

Fig. 2. Theoretical modelling of the composite hexagonal microstructure: (a) geometrical modelling, (b) the unit cell.

Fig. 3. The discrete system continuum-springs: (a) springs a, (b) springs b.

Fig. 4. The anisotropic hexagonal cell in the Winkler model with focus on thesprings’ anchorage point.


82

= ⎡⎣⎢

⎤⎦⎥

xx x

x x x xΨ( )

Ψ ( ) 0 0 Ψ ( ) 0 00 Ψ ( ) Ψ ( ) 0 Ψ ( ) Ψ ( )

1 4

2 3 5 6 (7)

is the shape functions matrix, whose components are

The elastic strain energy of the Euler–Bernoulli beam on Winklerfoundation element can be evaluated as the sum of three terms (Ongaroet al., 2016b; Dinev, 2012):

= + +w u k u Δu k Δu Δu k Δu12

( ) · 12

( ) · 12

( ) · .e e Tbe e e a T

wfe e a e b T

wfe e b, , , ,

(9)

The first is the elastic energy due to the axial and bending de-formations of the beam, the second and third are related to the elon-gation of the springs,

=Δu Δu Δu[ ] ,e aia

ja T, (10)

=Δu Δu Δu[ ] .e bib

jb T, (11)

In particular, for the beams 0–1, 0–2 and 0–3, the quantities in (10)and (11) are, respectively,

= ⎡⎣⎢

−−

⎤⎦⎥

= ⎡⎣⎢

−−

⎤⎦⎥

Δuu uu u Δu

u uu u, ,a b

10 61 6 1

0 41 4 (12)

= ⎡⎣⎢

−−

⎤⎦⎥

= ⎡⎣⎢

−−

⎤⎦⎥

Δuu uu u Δu

u uu u, ,a b

20 42 4 2

0 52 5 (13)

= ⎡⎣⎢

−−

⎤⎦⎥

= ⎡⎣⎢

−−

⎤⎦⎥

Δuu uu u Δu

u uu u, .a b

30 53 5 3

0 63 6 (14)

Finally, with obvious notation, kbe and k ,wf

e in turn, stand for thestiffness matrix of the classical Euler–Bernoulli beam and of the Winklerfoundation. In the FEM framework, their components are obtained byapplying the strain energy principle (Tsiatas, 2014). In particular,

∫∫=

⎧

⎨

⎪

⎩⎪

=

=

′ ′

″ ″

C x x dx i j

D x x dx i jk[ ]

Ψ ( )Ψ ( ) , , 1, 4,

Ψ ( )Ψ ( ) , , 2, 3, 5, 6,

0, otherwise

be

ij

i j

i j

0

ℓℓ

0

ℓℓ

(15)

and

∫=⎧⎨⎩

=K x x dx i jk[ ] Ψ ( )Ψ ( ) , , 2, 3, 5, 6,

0, otherwise,wfe

ijw i j0

ℓ

(16)

with Kw the Winkler foundation constant, =−

C E hνℓ 1

s

s2 and =

−D ,E h

νℓ 12(1 )s

s

32

respectively, the tensile and bending stiffness (per unit width) of thebeams, Es, νs, h, ℓ, in turn, Young’s modulus, Poisson’s ratio, thickness,

and length of the beams, while = ∂∂

′

x(·) (·)

and = ∂∂

″

x(·) (·)2

2 . Substituting

(8) into (15) and (16) leads to

=

⎡

⎣

⎢⎢⎢⎢⎢⎢⎢

−−−

−− − −

−

⎤

⎦

⎥⎥⎥⎥⎥⎥⎥

C CD D D D

D D D DC C

D D D DD D D D

k

/ℓ 0 0 /ℓ 0 00 12 /ℓ 6 /ℓ 0 12 /ℓ 6 /ℓ0 6 /ℓ 4 /ℓ 0 6 /ℓ 2 /ℓ

/ℓ 0 0 /ℓ 0 00 12 /ℓ 6 /ℓ 0 12 /ℓ 6 /ℓ0 6 /ℓ 2 /ℓ 0 6 /ℓ 4 /ℓ

be

ℓ ℓ

ℓ3

ℓ2

ℓ3

ℓ2

ℓ2

ℓ ℓ2

ℓ

ℓ ℓ

ℓ3

ℓ2

ℓ3

ℓ2

ℓ2

ℓ ℓ2

ℓ

(17)

and

=

⎡

⎣

⎢⎢⎢⎢⎢⎢

−−

−− − −

⎤

⎦

⎥⎥⎥⎥⎥⎥

K K K KK K K K

K K K KK K K K

k

0 0 0 0 0 00 13 /35 11 ℓ/210 0 9 /70 13 ℓ/4200 11 ℓ/210 ℓ /105 0 13 ℓ/420 ℓ /1400 0 0 0 0 00 9 /70 13 ℓ/420 0 13 /35 11 ℓ/2100 13 ℓ/420 ℓ /140 0 11 ℓ/210 ℓ /105

.

wfe

w w w w

w w w w

w w w w

w w w w

2 2

2 2

(18)

It should be noted that there are different approaches in evaluatingthe stiffness matrix of beam elements on elastic foundations(Tsiatas, 2014). The two main techniques are based on either the use ofapproximated shape functions (Janco, 2010; Kuo and Lee, 1994; Hosurand Bhavikatti, 1996; Chen, 1998) or the development of exact ones(Eisenberger and Yankelevsky, 1985; Sen et al., 1990; Razaqpur andShah, 1991). In the first case, both kb

e and kwfe are evaluated by

adopting the cubic polynomial shape functions typical of the Eu-ler–Bernoulli beam, listed in (8). In the second, the shape functions arederived by solving the governing differential equation of the Eu-ler–Bernoulli beam resting on Winkler foundation (Eisenberger andYankelevsky, 1985). However, despite the simplifications introduced,several existing studies dealing with a broad range of engineeringproblems (Janco, 2010; Tsiatas, 2014; Karkon and Karkon, 2016)conclude that the results of the numerical implementations based onthe approximated solution compare favourably to those obtained by theexact ones. Considering this and aiming to obtain a more mathemati-cally tractable problem, in this work the approximated approach is

Fig. 5. Equilibrium of the forces at the springs’ anchorage point: the two sets of symmetric springs (springs a represented in red, springs b represented in blue). (Forinterpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

⎜ ⎟

⎜ ⎟

= − = − ⎛⎝

⎞⎠

+ ⎛⎝

⎞⎠

= ⎛⎝

− ⎛⎝

⎞⎠

+ ⎛⎝

⎞⎠

⎞⎠

= = ⎛⎝

⎞⎠

− ⎛⎝

⎞⎠

= ⎛⎝

⎛⎝

⎞⎠

+ ⎛⎝

⎞⎠

⎞⎠

x x x x x x x x x

x x x x x x x x

Ψ ( ) 1ℓ

, Ψ ( ) 1 3ℓ

2ℓ

, Ψ ( )ℓ

2ℓ ℓ

ℓ,

Ψ ( )ℓ

, Ψ ( ) 3ℓ

2ℓ

, Ψ ( )ℓ ℓ

ℓ.

1 2

2 3

3

2 3

4 5

2 3

6

2 3

(8)


83

adopted (cf. Eq. (16)).

2.3. Elastic energy of the discrete system

For any given deformation, the elastic energy representative of thewhole discrete structure, W, can be evaluated from the analysis of theunit cell of the periodic array.

As illustrated in Fig. 2(b), the unit cell is composed by the centralnode 0 and the external nodes 1, 2, 3, 4, 5, 6, linked by the line ele-ments 0–1, 0–2, 0–3, treated as Euler–Bernoulli beams on Winklerfoundation, and 0–4, 0–5, 0–6, modelled as Euler–Bernoulli beams. Inthe global reference system (e1, e2), the beams are represented, re-spectively, by the vectors

= − = − = − = = − −

= −

b l s b l s b s b s b l s b

s l

, , , , ,

( )/2.1 1 2 2 3 4 5 1 6

2 (19)

The elastic energy W, in particular, is obtained by summing theelastic energies of the beams. However, as stated in Section 2.2.1, thecontribution of the beams composing the lattice L3, 0–4, 0–5, 0–6, isassumed to be negligible with respect to those composing the principallattices L1 and L2, 0–1, 0–2, 0–3. Consequently, in evaluating W, onlythe beams 0–1, 0–2, 0–3 will be considered.

Furthermore, as it can be seen in Fig. 2(b), the first node of eachbeam coincides with the central node 0, where it is imposed the balanceof forces and moments. This condition guarantees the equilibrium of theexamined cell and allows us to condense the degrees of freedom of 0,leading to

= =W W ju Δu Δu( , , ), 1, 2, 3,j ja

jb

(20)

with

= − = −Δu u u Δu u u[ ], [ ],a b1 1 6 1 1 4 (21)

= − = −Δu u u Δu u u[ ], [ ],a b2 2 4 2 2 5 (22)

= − = −Δu u u Δu u u[ ], [ ].a b3 3 5 3 3 6 (23)

2.4. Discussion

According to our method, it emerges that the elastic energy of thediscrete system is given by (cf. Eq. (20))

= =W W ju Δu Δu( , , ), 1, 2, 3.j ja

jb

(24)

In particular, the energetic contribution due to the Winkler foun-dation,

= =W W jΔu Δu( , ), 1, 2, 3,Winkler Winkler ja

jb

(25)

is a quadratic function of the elongation of the springs involving, asstated, the difference between the displacements of the end points ofthe beams and of the anchorage points. As it can be seen, the dis-placements of the nodes 4, 5 and 6 does not “directly” take part in thedescription of the system; they only contribute via the terms Δu j

a andΔu j

b.This can be verified by imagining to represent the composite ar-

chitecture in Fig. 2(a) as an hybrid system composed by one-dimen-sional (1D) beams and two-dimensional (2D) filler. With reference tothe unit cell in Fig. 2(b), the elastic energy of the hybrid 1D-2D con-figuration is the sum of the elastic energies of the beams, Wbeams, and ofthe filler, Wfiller:

= +−W W W .D D beams filler1 2 (26)

Wfiller, in particular, is given by

∫= ɛ σW dV12

,filler V fT

f (27)

with

=⎡

⎣⎢

⎤

⎦⎥ ← ⎡

⎣⎢⎤⎦⎥

=ɛ ϵ:ɛɛ2ɛ

ɛ ɛɛ ɛ :f f

1122

12

11 1212 22

(28)

and

=⎡

⎣⎢

⎤

⎦⎥ ← ⎡

⎣⎢⎤⎦⎥

=σσσσ

σ σσ σ T: : ,f f

112212

11 1212 22 (29)

respectively, the infinitesimal strain tensor, ϵf, and stress tensor, Tf,expressed in Voigt notation, Cf the stiffness tensor of the material withinthe cell, satisfying the generalised Hooke’s law

=σ ɛC .f f f (30)

For two-dimensional isotropic materials in plane-stress tensionalstate, Cf is defined by

=−

⎡

⎣

⎢⎢ −

⎤

⎦

⎥⎥

Eν

νν

νC :

1

1 01 0

0 0 (1 )/2,f

f

f

f

f

f

2

(31)

with Ef and νf, in turn, Young’s modulus and Poisson’s ratio of the filler.Accordingly, by substituting (30) into (27) and considering a uni-

tary width, =b 1, it emerges

∫= ɛ C ɛW dA12

,filler A fT

f f0 (32)

being =V bA0 and A0 the area of the examined cell (Fig. 2(b)).By discretizing the area A0 into a set of two-dimensional triangular

elements having nodes

− −− −− −− −− −− −

0 1 4,0 4 2,0 2 5,0 5 3,0 3 6,0 6 1 (33)

and by considering the so-called constant-strain triangular element(CST) frequently used in the Finite Element Method, (32) takes the form

∑==

W d k d12

,fillere

eT

e e1

6

(34)

with ke and =d u u u[ ] ,e i j mT respectively, the local stiffness matrix and

displacements vector of each triangular element of nodes i, j and m(Fenner, 1996). Specifically,

======

− −

− −

− −

− −

− −

− −

d u u ud u u ud u u ud u u ud u u ud u u u

[ ] ,[ ] ,[ ] ,[ ] ,[ ] ,[ ]

T

T

T

T

T

T

0 1 4 0 1 4

0 4 2 0 4 2

0 2 5 0 2 5

0 5 3 0 5 3

0 3 6 0 3 6

0 6 1 0 6 1 (35)

provide the displacements vector of the examined triangles.In terms of the global displacements vector, D, and stiffness matrix,

K, obtained by “summing” their local counterparts, (34) becomes

= =

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢⎢

⎤

⎦

⎥⎥⎥⎥⎥⎥

W D KD

uuuuuuu

K

uuuuuuu

12

12filler

T

T0123456

0123456 (36)

or, by splitting the vector D into the vectors =D u u u u[ ]T1 0 1 2 3 and

=D u u u[ ]T2 4 5 6 that, with reference to our model, represent the dis-

placements of the principal lattices, L1 and L2, and of the central points


84

of the cells (i.e., the springs’ anchorage points),

= ⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

⎡⎣⎢

⎤⎦⎥

WDD

K KK K

DD

12

,filler

T1

2

11 12

21 22

1

2 (37)

leading to

= + + +W D K D D K D D K D D K D12

( ),fillerT T T T1 11 1 1 12 2 2 21 1 2 22 2 (38)

with Kij obtained by partitioning K.When =ν 1/3,f value that coincides with Poisson’s ratio of the hy-

groscopic keel tissue considered in the present paper, it emerges thatthe elastic energy in (38) can be expressed as a quadratic function of thequantities −u u ,i j with =i 0, 1, 2, 3 and =j 4, 5, 6. Specifically,

= − − − −W W u u u u u u u u( , , , ),filler filler k l m n0 1 2 3 (39)

where =k 4, 5, 6, =l 6, 4, =m 4, 5 and =n 5, 6.As it can be seen, similarly to (25), in Eq. (39) the displacements of

the nodes 4, 5, 6 are not “directly” involved in the description of thesystem; their contribution is only related to the terms −u ui j that, inthe Winkler model, represent the elongation of the springs.

Also, by assuming that the elastic energy of the beams is the same inthe two considered models (i.e., hybrid system 1D–2D and Winklermodel), it can be concluded that, in the case of =ν 1/3,f

= + + +W D K D D K D D K D D K D12

( )fillerT T T T1 11 1 1 12 2 2 21 1 2 22 2

∑∼ = +W Δu k Δu Δu k Δu12

(( ) · ( ) · ).Winklere

e a Twfe e a e b T

wfe e b, , , ,

(40)

A final observation concerns the equilibrium conditions of thesprings’ anchorage points, i.e., the nodes 4, 5 and 6 (Fig. 2(b)).

As mentioned in Section 2.2.1, the symmetry of both the hexagonalcell and the configuration of the springs provide the equilibrium of theforces at the springs’ anchorages. This geometrically-based considera-tion can be verified by considering the equivalence between the hybridsystem 1D beams-2D filler and the system Euler–Bernoulli beams onWinkler foundation described above. In particular, when the hexagonalcell in Fig. 4 is subjected to a set of external forces leading to a genericdeformation of the cell, the reaction forces of the springs along thedirection ni take the form

= uf n K n n( )Δ ,i iT

w i i i (41)

where Kw is the stiffness matrix of the elastic foundation and Δui theelongation of the springs in the ni direction.

At the anchorage points (i.e., the central point of the cell), the sumof the springs’ reaction forces is expressed by

∑ ∑= = uf f n K n n( )Δanc i iT

w i i i (42)

or, by splitting the contribution of the two sets of springs,

∑ ∑= +u uf n K n n n K n n(( ) )Δ (( ) )Δ ,anc ia T

w ia

ia

ia

ib T

w ib

ib

ib

(43)

with uΔ ia and uΔ ,i

b respectively, the elongation of the springs a and ofthe springs b in the directions ni

a and nib (Fig. 4).

By taking into account the equivalence between the system 1Dbeams-2D filler and the Winkler model, it can be assumed

=u dΔ Δ ,i i (44)

being Δui and Δdi, respectively, the elongation in the ni direction in theWinkler foundation model and in the system 1D beams-2D filler. Fromclassical continuum mechanics,

= ϵd dn nΔ ( )i iT

f i i (45)

with di the original length of the cell in the ni direction and ϵf the in-finitesimal strain tensor (cf. Eq. (28)).

By substituting (44) into (42) and splitting the contribution of the

springs a and of the springs b as in Eq. (43), it emerges

∑∑

=

+

ϵ

ϵ

d

d

f n K n n n n

n K n n n n

(( ) )(( ) )

(( ) )(( ) ) .

anc ia T

w ia

ia T

f ia

ia

ia

ib T

w ib

ib T

f ib

ib

ib

(46)

Finally, by observing that =d dia

ib and that = −n n ,i

aib Eq. (46)

provides

=f 0,anc (47)

relation that coincides with the equilibrium of the forces at the springs,anchorage points. It should be noted that the above equation is valid forany deformation of the cell, both symmetric and not-symmetric.

3. The homogenized model

3.1. Elastic energy

It is possible to express W in a continuum form by introducing theaffine interpolants of the nodal displacements and microrotations, u(·)and φ (·), and by assuming that in the limit ℓ→ 0 the discrete variables(uj, φj) previously introduced can be expressed by

= + ∇ = + ∇ =φ φ φ ju u u b b, , 1, 2, 3.j j j j0 0 (48)

The terms u0 and φ0 stand for the values of u(·) and φ (·) at the centralnode of the cell in the continuum description, while bj are the vectorsformerly defined. Substituting (48) into (20) and dividing the expres-sion that turns out from the calculation by the area of the unit cell,

= +A θ θ2ℓ cos (1 sin )02 (Fig. 2(b)), give the strain energy density in

the continuum approximation w. Similarly to Ongaro et al. (2016b) andDavini and Ongaro (2011), in the limit ℓ→ 0 it emerges the in-dependency of w by the microrotation gradients, φ ,α, that scale with firstorder in ℓ. Accordingly,

= −w w ω φ(ɛ , ɛ , ɛ , ( )),11 22 12 (49)

with = +u uɛ ( )αβ α β β α12 , , and = −ω u u( )1

2 1,2 2,1 the infinitesimal strainsand the infinitesimal rotation of classical continuum mechanics. Moredetails are given in Appendices A and B.

3.2. Constitutive equations

The stress–strain relations of the equivalent continuum take theform

=

+ + + −

+ ++ +

+

+−

=+ − +

+ +

+−

= =+ + + −+ +

++ + + +

+ +

+

= − =−

+

= + = +

σ

C c c D c C D s s C D f

D s sf c D C s

K c f ff f

σC C s f c C s D s D f

c c D C sK f c cf

f f

σ σD c C sf s s D sf

f c C c D sfD c C sf D C c D s f

f c C c D sfK f

cf f

σ σD ω φ

c sf

σ σ σ σ σ σ

(ɛ (24 ( ℓ 48 )) (ɛ ( ℓ 12 )

ɛ (12 (1 2 ))))ℓ (24 ℓ (1 2 ))

(ɛ ɛ )104

,

( ɛ ℓ (ɛ ( ℓ 12 ) 12 ɛ ))ℓ(24 ℓ (1 2 ))

(ɛ / ɛ )104

,

3 ɛ ( ( ℓ (4 ( 3) 3) 24 ))2ℓ (2 ℓ 3 (4 3))

3 ɛ (4 ( ℓ (2 1) 3 ) 4 ℓ 12 )2ℓ (2 ℓ 3 (4 3))

ɛ208

,

9 ( )ℓ (3 4 )

,

, ,

w

w

sym sym

w

skw skw

sym skw sym skw

11

ℓ 114

ℓ2

ℓ2

ℓ2

22 ℓ2

ℓ 0

11 ℓ2

02

ℓ ℓ2 2

11 1 22 2

0 3

22ℓ ℓ 22

2 20

211 ℓ

2ℓ ℓ 22 0

2ℓ ℓ

2 2

22 4 11 2

0 3

12 21ℓ 12

2ℓ

20

2ℓ 0

30 ℓ

2 2ℓ 0

ℓ 124

ℓ2

0 ℓ ℓ2 6

ℓ2

02

30 ℓ

2 2ℓ 0

12 5

0 3

12 21ℓ

30

12 12 12 21 21 21

(50)

with σγδsym and σ ,γδ

skw in turn, the symmetric and skew-symmetric part of


85

the Cauchy stress tensor defined by

= ∂∂∇

σA

Wu

1 .0 (51)

Note that, to simplify the notation, in (50) c and s stand, respec-tively, for cos θ and sin θ while =f f θ θ(cos , sin )i i are polynomial ex-pressions listed in Appendix B.

3.3. Elastic constants

Mathematical manipulations provide the elastic constants of thelimit problem, given by

⎜ ⎟

=+ + + +

+ + + +

++ + + +

= −+ + + −

+ + + +

=+ + + +

+ + + + + +

++ + + + + +

= −+ + + −

+ + + + + +

= ⎛⎝

+ − ++ +

+ ⎞⎠

Ec K vc λ f E v f K s λE λ E f vf f K

f K v λ c s λf E λ c sλ K f vc

f K v λ c s λf E λ c s

νc K vf λ c s λ λ E f

K vf λ c s λf E λ c s f s

EλE λ E f vf f K K c λ E f K vf sf c K vf λ c s λ f E s c c c

λ K vc ff K vf λ c s λ f E s c c c

νK vf λ c s λ λ f E

K vf λ c s λ E f s c c c

Gf c

λ E c λ f λ sf fv λ f s c

K ff

*( ((4 )/ (2 1)) 4 ((104 )/( ) ))

4( (2 2 1) 104 ( ))

2( (2 2 1) 104 ( )),

* ( (2 2 1) 104 ( 1) )(2 2 1) 104 ( ) /

,

*4 ((104 )/( ) ) (4 (2 1))4 ( (2 2 1) 104 ( (3 2 ) 2 ))

2 ( (2 2 1) 104 ( (3 2 ) 2 )),

* (2 2 1) 104 ( 1)(2 2 1) 104 ( (3 2 ) 2 )

,

* 1416

104 ( (2 ) 2 )( (4 3) 8 )

,

w s w s s w

w s

w

w s

w s

w s

s s w w s w

w s

w

w s

w s

w s

s w

1

2 38 6

2 310 0 7

92 2 2

102 2 2

2 26

5

92 2 2

102 2 2

12

22

2 2 2 211

42 2 2

02 2 2

11

2

310 0 7

2 38 6

2

0 12 2 2 3

32 2 4 2

2 2 36

0 12 2 2 3

32 2 4 2

212

2 2 2 211

12 2 2 3

32 2 4 2

0

3 2 213

20 12

20

211

10

(52)

with =λ h/ℓ, = −v ν(1 ),s2 = =c θ s θcos , sin and =f f θ θ(cos , sin )i i

the expressions in Appendix B. Also, with obvious notation, E ν*, *1 12 andE ν*, *2 21 denote, in turn, Young’s modulus and the corresponding Pois-son’s ratio in the e1 and e2 direction, G* the shear modulus. As ex-pected, the macroscopic elastic moduli derived satisfy the classical re-

lation revealing the system isotropy, =+

G Eν

* *2(1 *)

, with

= ≡E E E* * *1 2 and = ≡ν ν ν* * *,12 21 only in the particular case = ∘θ 30 .

4. Discussion

4.1. The hygroscopic keel tissue: Comparison with other authors

As stated, the present work is inspired by the hygroscopic keel tissueof the ice plant. This biological tissue reveals a cellular microstructurecomposed by elongated hexagons filled with the CIL. If hydrated, theCIL adsorbs large amount of water, leading to a change of the cells’shape and, consequently, to the macroscopic stiffness.

As a matter of fact, let us consider the compact expression of thestress–strain relations derived in Section 3.2

⎡

⎣

⎢⎢

⎤

⎦

⎥⎥

=⎡

⎣⎢⎢

⎤

⎦⎥⎥

⎡

⎣⎢

⎤

⎦⎥

σσσ

C C CC C CC C C

ɛɛ2ɛ

,

sym

sym

sym

11

22

12

11 12 13

21 22 23

31 32 33

1122

12 (53)

with Cij the components of the effective stiffness tensor previously ob-tained and reported here for ease of reading

= + + + ++ +

+

=+

+ ++

= = −+ +

−

=+ + + −

+ +

++ + + +

+ +

+

= = = =

C C c c D c C D s s D s sf c D C s

K cff f

CC C s D c f

c c D C sK f c

f f

C C C cs C Dc D C s

K cff f

CD c C sf s s D sf

f c C c D sfD c C sf D C c D s f

f c C c D sfK f

cf fC C C C

(24 ( ℓ 48 ) (12 (1 2 )))ℓ (24 ℓ (1 2 ))

104,

( ℓ 12 )ℓ(24 ℓ (1 2 ))

/104

,

( ℓ 12 )ℓ(24 ℓ (1 2 )) 104

,

3 ( ℓ (4 ( 3) 3) 24 )2ℓ (2 ℓ 3 (4 3))

3 (4 ( ℓ (2 1) 3 ) 4 ℓ 12 )2ℓ (2 ℓ 3 (4 3))

208,

0.

w

w

w

w

11ℓ

4ℓ

2ℓ

2ℓ

2ℓ

2

02

ℓ ℓ2 2

1

0 3

22ℓ ℓ

2 2ℓ

20

2ℓ ℓ

2 24

0 3

12 21ℓ ℓ

2ℓ

2ℓ ℓ

2 22

0 3

33ℓ ℓ

20

2ℓ 0

30 ℓ

2 2ℓ 0

ℓ4

ℓ2

0 ℓ ℓ2 6

ℓ2

02

30 ℓ

2 2ℓ 0

5

0 3

13 23 31 32

(54)

It emerges a strong influence of the inclination of the cell walls θ(Fig. 2), via the terms = =c θ s θcos , sin and the polynomials

=f f θ θ(cos , sin )i i .Before addressing a parametric analysis to investigate further this

influence, let us verify the adopted modelling technique by comparingthe proposed results with the available data in the literature.

As summarised in Table 1, the comparison is established by com-paring the Cij constants of the present paper with those suggested inGuiducci et al. (2014), where the keel tissue, represented as a pres-surized diamond-shaped honeycomb, is analyzed by Finite Elementhomogenization and theoretical modelling based on the Born rule.Specifically, four cell configurations are considered, characterised bydifferent values of θ and inner pressure, p. Notwithstanding the diversestrategies adopted, Table 1 reveals that the agreement is generallygood. The discrepancies that emerge in some cases are due to the dif-ferent cells’ shape considered: diamond-shaped cells inGuiducci et al. (2014), elongated hexagons in the present work. Also,neglecting the compromised cell walls’ coupling effect could be another

Table 1A practical application to the keel tissue of the ice plant. Comparison between the results of the present paper and those of Guiducci et al. (2014).

Guiducci et al. (2014)

Es=1 GPa, νs=0.3, h/ℓ=0.07

p (MPa) C22 (GPa) C11 (GPa) C33 (GPa) C12 (GPa) C21 (GPa)

0 0.1÷ 0.3 0.002 0.004÷ 0.012 0.028 0.0282.5 0.03 0.020÷ 0.027 0.03÷ 0.086 0.020÷ 0.026 0.023÷ 0.0265 0.025 0.03÷0.05 0.03÷ 0.086 0.015 0.0156 0.02 0.03÷ 0.04 0.02÷ 0.096 0.02 0.02

PresentEs=1 GPa, νs=0.3, h/ℓ=0.07

θ(°) Kw (MPa) C22 (GPa) C11 (GPa) C33 (GPa) C12 (GPa) C21 (GPa)

75 0 0.15 0.002 0.0035 0.025 0.02548 15.27 0.020 0.020 0.04 0.018 0.01847 33 0.019 0.046 0.057 0.018 0.01846 41.1 0.02 0.05 0.054 0.016 0.016


86

source of dissimilarities. As mentioned in Section 2.1, a rigorous ana-lysis of the biological keel tissue is beyond our aim. However, fromTable 1 it emerges that the proposed theory could be applied in biologyto study the mechanics of composite tissues having a not-regular hex-agonal microstructure.

In addition, in Table 1 the values of the Winkler foundation con-stant, Kw, are obtained by the energetic equivalence described inAppendix C. In particular, it emerges that Kw, expressed by

⎜ ⎟

=⎛⎝

− + ⎞⎠

+

⎛⎝ +

+−

− ⎞⎠

+ ⎛⎝ −

− ⎞⎠

K pp θ θ θ θ

θ θ θ θ( )

2 8 sin cos17

1 cos 1 sin

4 sin6 2

4 cos6 2

1 cos 4 cos6 2

1,w 3 3 2 2

(55)

is a function of the pressure, p, and cell walls’ inclination, =θ θ p( ).One question that arises is if there exist an optimal value of p, p ,

that maximises the area of the hexagonal cell, A0, given by

= +A p θ p θ p( ) 2ℓ cos ( )·(1 sin ( )),02 (56)

with ℓ and θ(p), in turn, the length and inclination of the cell walls.As illustrated in Fig. 6, A0 attains the maximum at ≡ = ∘θ θ 30 and,

according to the analysis of Guiducci et al. (2014), the correspondingvalue of p is given by ≈p 15 MPa. It should be noted that the outcomeof the analysis is not affected by the particular value of cell walls’ length

assumed in Fig. 6, =ℓ 1 mm.Finally, a schematic representation of this smart mechanism is

shown in Fig. 7. In the dry state, at zero pressure, the tissue is composedby elongated cells characterised by high values of θ and minimumabsorption (Fig. 7(a)). When it starts raining, the filler absorbs moreand more large amounts of water, leading to an increase in the innerpressure and, consequently, to a decrease in θ. In particular, decreasingθ provides an increase in A0 (cf. Fig. 6), as well as an increase in theabsorption (Fig. 7(b)). At = ∘θ 30 , the stationary condition of maximumabsorption is reached (Fig. 7(c)). Then, when the rain stops, the pres-sure inside the cells decreases, as the absorbed water starts to evaporate(Fig. 7(d)). It follows an increase in θ and a decrease in A0, until theoriginal configuration is restored (Fig. 7(e)).

4.2. Parametric analysis

From the expressions in (54) it is clear that the macroscopic me-chanical behaviour is strongly affected by the microstructure’s geome-trical and mechanical properties.

Assuming lignified cell walls as in the keel tissue, with =E 1s GPaand =ν 0.3s (Guiducci et al., 2014), this section investigates the influ-ence of the infill’s stiffness, Kw, and cell walls’ inclination, θ, in theeffective stiffness. In particular, two different cases are considered:slender beams, with =h/ℓ 0.01, Fig. 8, and thick beams, with =h/ℓ 0.1,Fig. 9. As it can be seen, Figs. 8(a) and 9(a) suggest that when Kw isfixed, an increase in θ leads to a decrease in the C11 constant, that ismore significant in the case of =h/ℓ 0.1 (Fig. 9(a)). Conversely, forfixed Kw, increasing the cell walls’ inclination provides an increase inC22 (Figs. 8(b) and 9(b)). This is not surprising since the smaller theangle θ, the more elongated in the e1 direction will be the resulting cell.Consequently, the smaller θ, the higher C11. Similarly, increasing θyields a more and more elongated cell in the e2 direction and a moreand more higher C22. In addition, Figs. 8(a), 9(a) and 8(b), 9(b) showthat, for fixed θ, to high values of − −K E E(10 , 10 )w s s

1 2 corresponds anhigher initial value of both C11 and C22.

Regarding the constant C33, from Figs. 8(e) and 9(e) it emerges thatwhen Kw is fixed, an increase in θ leads to an increase in C33, that ismore evident for high values of − −K E E(10 , 10 )w s s

1 2 .In terms of the cross stiffness components, C12 and C21, Figs. 8(c),

9(c) and 8(d), 9(d) reveal that increasing θ provides a fast initial in-crease followed by a gradual decrease. In contrast to what would be

Fig. 6. Optimal value of θ, with =ℓ 1 mm: area of the cell.

Fig. 7. The smart mechanism of the hygroscopic keel tissue: (a) dry state, (b) when it starts raining, the filler absorbs water leading to an increase in the absorptioncapability, (c) stationary condition, maximum absorption, (d) the rain stops and the water absorbed starts to evaporate, until (e) the original configuration is restored.


87

expected, for small values of θ the presence of the filling material doesnot stiffen the structure. Also, by comparing the curves correspondingto slender beams (Fig. 8(c) and (d)) and thick beams (Fig. 9(c) and (d)),it can be said that this peculiar behaviour is geometry-related. Thisresult could be of interest in practical applications as a strategy to de-sign a new more mechanically efficient material or to improve existingones.

As in classical orthotropic materials, it emerges C11≠ C22 and

=C C12 21. Regardless the values of h/ℓ, only in the particular case= ∘θ 30 the equivalence =C C11 22 holds true. This, as expected, reveals

the system isotropy.

5. Hierarchical extension

A hierarchical material can be defined as a material that containsstructural elements which themselves have structure (Lakes, 1993;

Fig. 8. The influence of Kw and θ in the effective stiffness constants in the case of =h/ℓ 0.01: (a) C11, (b) C22, (c) C12, (d) C21, (e) C33.


88

Ongaro et al., 2016a).This work, in particular, deals with a hierarchical composite cellular

material having n levels of hierarchy and an elongated hexagonal mi-crostructure with filled cells at all levels (Fig. 10). Similarly toSection 2, the Euler–Bernoulli beam on Winkler foundation elementrepresent the skeleton of the cells, the −n( 1)th level. Again, the elasticsprings are imagined to be anchored at the nodes of the lattice L3,modelled as a sequence of Euler–Bernoulli beams much less stiff than

the principal ones (cf. Section 2.1).

5.1. Effective elastic constants

Let us focus on the nth level structure of Fig. 10. By assuming thatthe size of the microstructure of each cell wall, the −n( 1)th level, isfine enough to be negligible with respect to the nth level, each cell armcan be treated as a continuum having the elastic moduli derived in

Fig. 9. The influence of Kw and θ in the effective stiffness constants in the case of =h/ℓ 0.1: (a) C11, (b) C22, (c) C12, (d) C21, (e) C33.


89

Section 3.3. Consequently, the effective elastic constants of the nth levelstructure in the continuum form are

=+ + + +

+ + + +

++ + + +

= −+ + + −

+ + + +

=+ + + +

+ + + + + +

++ + + + + +

= −+ + + −

+ + + + + +

=⎛

⎝⎜

+ − +

+ ++

⎞

⎠⎟

Ec K v c λ f E v f K s λ E λ E f vf f K

f K v λ c s λ f E λ c s

λ K f v c

f K v λ c s λ f E λ c s

νc K v f λ c s λ λ E f

K v f λ c s λf E λ c s f s

Eλ E λ E f vf f K K c λ E f K v f s

f c K v f λ c s λ f E s c c c

λ K v c f

f K v f λ c s λ f E s c c c

νK v f λ c s λ λ f E

K v f λ c s λ E f s c c c

Gf c

λ E c λ f λ s f f

v λ f s c

K ff

ˇ ˇ ( ˇ ˇ ˇ ((4 ˇ ˇ ˇ)/ ˇ ˇ ˇ (2ˇ 1)) 4 ˇ ˇ ((104 ˇ ˇ ˇ )/( ˇ ˇ ) ˇ ˇ ))

4(ˇ ˇ ˇ (2 ˇ ˇ 2ˇ 1) 104 ˇ ˇ ˇ ( ˇ ˇ ˇ ))ˇ ˇ ˇ ˇ ˇ

2(ˇ ˇ ˇ (2 ˇ ˇ 2ˇ 1) 104 ˇ ˇ ˇ ( ˇ ˇ ˇ )),

ˇˇ ( ˇ ˇ ˇ (2 ˇ ˇ 2ˇ 1) 104 ˇ ( ˇ 1) ˇ ˇ )

ˇ ˇ ˇ (2 ˇ ˇ 2ˇ 1) 104 ˇ ˇ ˇ ( ˇ ˇ ˇ ) ˇ / ˇ,

ˇ 4 ˇ ˇ ((104 ˇ ˇ ˇ )/( ˇ ˇ ) ˇ ˇ ) ˇ ˇ (4 ˇ ˇ ˇ ˇ ˇ ˇ (2ˇ 1))

4ˇ ˇ ( ˇ ˇ ˇ (2 ˇ ˇ 2ˇ 1) 104 ˇ ˇ ˇ (ˇ (3 2ˇ ) 2ˇ ˇ ))ˇ ˇ ˇ ˇ ˇ

2ˇ ( ˇ ˇ ˇ (2 ˇ ˇ 2ˇ 1) 104 ˇ ˇ ˇ (ˇ (3 2ˇ ) 2ˇ ˇ )),

ˇˇ ˇ ˇ (2 ˇ ˇ 2ˇ 1) 104 ˇ ( ˇ 1) ˇ ˇ

ˇ ˇ ˇ (2 ˇ ˇ 2ˇ 1) 104 ˇ ˇ ˇ (ˇ (3 2ˇ ) 2ˇ ˇ ),

ˇ 1416ˇ ˇ

104 ˇ ˇ ( ˇ (2 ˇ ˇ ) 2 ˇ ˇ ˇ ˇ )

ˇ ( ˇ (4ˇ ˇ 3) 8 ˇ )

ˇ ˇˇ ,

w w w

w

w

w

w

w

w w w

w

w

w

w

w

w

12 3

8 62 3

10 0 7

92 2 2

102 2 2

2 25

5

92 2 2

102 2 2

122

22 2 2 2

11

42 2 2

02 2 2

11

2

310 0 7

2 38 6

2

0 12 2 2 3

32 2 4 2

2 2 36

0 12 2 2 3

32 2 4 2

212

2 2 2 211

12 2 2 3

32 2 4 2

0

3 2 213

20 12

28

211

10

(57)

with E νˇ , ˇ1 12 and E νˇ , ˇ2 21 Young’s modulus and corresponding Poisson’sratio in the e1 and e2 direction, respectively, G the shear modulus. Inaddition, =λ hˇ ˇ/ℓ, = =c θ s θˇ: cos ˇ, ˇ: sin ˇ with h θˇ, ℓ, ˇ, in turn, the thick-ness, length and inclination of the cell walls (Fig. 10), Kw the Winklerconstant, = −v νˇ (1 ˇ), E and ν Young’s modulus and Poisson’s ratio ofthe beams in the longitudinal direction (Chen and Pugno, 2013; Pugnoand Chen, 2011) obtained in Section 3.3. The polynomials

=f f θ θˇ ˇ (cos ˇ, sin ˇ)i i are derived by substituting θ for θ into the expres-sions listed in Appendix B. It should be noted that the previous notation,

(·)ˇ for (·)(n) and (·)ˇ for −(·) ,n( 1) is introduced to simplify the relations andfacilitate reading.

5.2. The stiffness-to-density ratio

The stiffness-to-density ratio takes the form

=+

=+

=+

Eρ

Ea ρ b ρ

Eρ

Ea ρ b ρ

Gρ

Ga ρ b ρ

ˇˇ

ˇ

ˇ ˇ ˇ ˇ,

ˇˇ

ˇ

ˇ ˇ ˇ ˇ,

ˇˇ

ˇ

ˇ ˇ ˇ ˇ,

f f f

1 1 2 2

(58)

with ρ and E ,1 E ,2 G, in turn, the density and the effective elastic con-stants of the nth level structure previously defined. In particular, asexplained in Appendix D, ρ is given by

⎜ ⎟ ⎜ ⎟= ⎛⎝

+ −+

⎞⎠

+ ⎛⎝ +

⎞⎠

= +ρ c s λc s

ρ λc s

ρ a ρ b ρˇ 2 ˇ (1 ˇ) 3 ˇ

2 ˇ (1 ˇ)ˇ 3 ˇ

2 ˇ (1 ˇ)ˇ ˇ ˇ ˇ ˇ ,f f

(59)

where ρf and ρ are the density of the filling material, the first, and ofthe cell walls, the second, at level n.

5.3. Parametric analysis and optimal values

Based on the above formulation, this section aims at understandinghow the microstructure’s parameters affect the macroscopic elasticmoduli in the case of structural hierarchy. The analysis involves a three-level hierarchical honeycomb having a elongated hexagonal micro-structure with filled cells at all levels and such that the self-similarcondition (Pugno and Chen, 2011)

= = =λ λ θ θ i, , 1, 2, 3,i i( ) ( ) (60)

holds true. The hypothesis that the density of the filling material, ρ ,fi( ) is

the same at all levels leads to

= = =ρ ρ α ρ i, 1, 2, 3,fi

f s( )

(61)

with =α 0.4, 0.2, 0.1, 0 for assumption. In addition, the lignified cellwalls of the starting element, the level 0 in Fig. 10, have Young’smodulus =E 1s GPa, Poisson’s ratio =ν 0.3,s density =ρ 1400s kg/m3

(Gibson and Ashby, 2001). The Winkler constant, derived inAppendix D, is expressed by

= = =K K α E i4 35

, 1, 2, 3.wi

w s( ) 3

(62)

As Fig. 11 shows, for fixed Kw the stiffness-to-density ratio, E ρ/ ,1(3) (3)

E ρ/ ,2(3) (3) G(3)/ρ(3), is strongly affected by the inclination of the cell walls

θ, as in the not-hierarchical case. In particular, increasing the values ofθ leads to an increase in E ρ/2

(3) (3) and to a decrease in E ρ/1(3) (3). This is

explained by the fact that the higher θ, the more elongated in the e2direction will be the cell. Also, it emerges that the cell-filled config-uration is generally stiffer than the hollow one ( =K 0w ), especially inthe case of high values of Kw ( − E10 ,s

1 − E10 s2 ). However, for high values

of θ, Fig. 11(a) illustrates that the composite configuration with highvalues of Kw is not the best solution in terms of E ρ/1

(3) (3). The reason isthat, firstly, the macroscopic stiffness in the e1 direction is more andmore smaller by increasing the cell walls’ inclination. Secondly, fillingthe cells provides not only a stiffer material but also a higher value ofthe density. Regarding E ρ/ ,2

(3) (3) analogous considerations apply

Fig. 10. The hierarchical composite cellular material.


90

(Fig. 11(b)). That is to say, high values of Kw leads to a stiffer materialwhen θ>24°, since small values of θ result in a hierarchical config-uration characterised by cells strongly elongated in the e1 directionand, consequently, by smaller values of E2

(3).As expected, =E ρ E ρ/ /1

(3) (3)2(3) (3) only in the case = ∘θ 30 .

Finally, in the cell-filled configuration, in contrast to the standardhierarchical material (Pugno and Chen, 2011; Bosia et al., 2012), in-creasing the number of hierarchical levels leads to an increase in thespecific stiffness (Fig. 12) and an optimal number of levels alsoemerges.

In the practical context, these findings could suggest a method toobtain a stiffer composite material via structural hierarchy and couldassist the designer in the selection of the geometric and mechanicalcharacteristics of the microstructure.

6. Conclusions

Composite cellular materials have been credited with significantlyimproving the mechanical behaviour of hollow structures. However, inthe literature a small number of analytical techniques has been pro-posed to predict the effective properties of filled cellular materials,especially in the case of not-regular microstructures.

This paper, inspired by the keel tissue of the ice plant Delospermanakurense, deals with the analysis of a composite honeycomb composedby elongated cells filled with an elastic material. By modelling thecomposite hexagonal microstructure as a sequence of Euler–Bernoullibeams on Winkler foundation and by applying an energy-based tech-nique, the constitutive equations and elastic moduli in the continuumapproximation are derived. It emerges a strong influence of the cellwalls’ inclination and of the filler’s stiffness on the effective elasticconstants.

Fig. 11. The influence of Kw and θ in the stiffness-to-density ratios of a three-level hierarchical composite in the case of =h/ℓ 0.01: (a) Young’s modulus in the e1direction, (b) Young’s modulus in the e2 direction, (c) shear modulus.


91

The application of the theoretical model to the keel tissue of the iceplant, in conjunction with a comparison with the available data in theliterature, reveals the validity of the proposed modelling approach.Despite the simplifications introduced to obtain a mathematicallytractable problem, the present work could be useful to gain some in-sights into the mechanics of biological and bio-inspired structures.

The theory is also extended to the hierarchical configuration and aclosed-form expression for the effective elastic moduli and specificstiffness is provided. From the parametric analysis developed, itemerges that increasing the hierarchical levels leads to an increase inthe specific stiffness and an optimal number of levels also exists.

Acknowledgement

NMP is supported by the European Commission under the GrapheneFlagship Core 2 grant No. 785219 (WP14 “Polymer Composites”) andFET Proactive “Neurofibres” grant No. 732344 as well as by the ItalianMinistry of Education, University and Research (MIUR) under the“Departments of Excellence” grant L.232/2016.

EB’s work was supported by JSPS KAKENHI Grant NumberJP18K18065.

Appendix A

The elastic energy of the Euler–Bernoulli beam is the sum of three terms:

= + +w u k u Δu k Δu Δu k Δu12

( ) · 12

( ) · 12

( ) · .e e Tbe e e a T

wfe e a e b T

wfe e b, , , ,

(A.1)

The first,

u k u12

( ) · ,e Tbe e

(A.2)

is related to the axial and bending deformations of the classical elastic beam, while the second and the third,

Δu k Δu Δu k Δu12

( ) · , 12

( ) · ,e a Twfe e a e b T

wfe e b, , , ,

(A.3)

are related to the Winkler foundation and, in particular, to the elongation of the springs a, the first, and of the springs b, the second (Fig. 14).The elastic energy of the unit cell, W, derives from that of the beams composing the skeleton of the cells: 0–1, 0–2, 0–3. Also, imposing the

balance of forces and moments in 0 and condensing the corresponding degrees of freedom, provides

=W W u u u Δu Δu Δu Δu Δu Δu( , , , , , , , , ).a a a b b b1 2 3 1 2 3 1 2 3 (A.4)

Then, the assumption that in the limit ℓ→ 0 the discrete variables (uj, φj) can be written as

= + ∇= + ∇ =φ φ φ j

u u u bb , 1, 2, 3,

j j

j j

0

0

(A.5)

provides the continuum description of the discrete structure. The terms u0 and φ0 are the values of u(·) and φ (·) at the central point of the cell in thecontinuum description and in what follows, to simplify the notation, they will be denoted with u and φ . Finally, substituting (A.5) into (A.4) givesthe strain energy of the unit cell as a function of the fields u and φ .

In particular, the aforementioned quantities are (Figs. 13–15):– Beam 0–1Discrete system

= ⎡⎣⎢

−−

⎤⎦⎥

= ⎡⎣⎢

−−

⎤⎦⎥

φ φ φ φΔuu u

Δuu u

, .a b1

1 6

1 6 11 4

1 4 (A.6)

In the continuum description,

= + ∇ = + ∇ =φ φ φ iu u u b b, , 1, 6, 4,i i i i (A.7)

that, substituted in (A.6), lead to

Fig. 12. Stiffness-to-density ratio vs levels of hierarchy, optimal value in the case of =h/ℓ 0.01 and = ∘θ 30 .


92

Fig. 13. The triplet of elastic beams with focus on springs: (a) beam 0–1, (b) beam 0–2, (c) beam 0–3.

Fig. 14. The two sets of springs connecting the triplet of elastic beams: (a) springs a, (b) springs b.

Fig. 15. The bfi vectors.


93

= ⎡⎣⎢

∇ − ∇∇ − ∇

⎤⎦⎥

= ⎡⎣⎢

∇ − ∇∇ − ∇

⎤⎦⎥φ φ φ φ

Δuu b u b

b bΔu

u b u bb b

, .a b1

1 6

1 61

1 4

1 4

(A.8)

– Beam 0–2Discrete system

= ⎡⎣⎢

−−

⎤⎦⎥

= ⎡⎣⎢

−−

⎤⎦⎥

φ φ φ φΔuu u

Δuu u

, .a b2

2 4

2 4 22 5

2 5 (A.9)

Continuum description

= + ∇ = + ∇ =φ φ φ iu u u b b, , 2, 4, 5,i i i i (A.10)

and

= ⎡⎣⎢

∇ − ∇∇ − ∇

⎤⎦⎥

= ⎡⎣⎢

∇ − ∇∇ − ∇


Δuu b u b

b bΔu

u b u bb b

, .a b2

2 4

2 42

2 5

2 5

(A.11)

– Beam 0–3Discrete system

= ⎡⎣⎢

−−

⎤⎦⎥

= ⎡⎣⎢

−−

⎤⎦⎥

φ φ φ φΔuu u

Δuu u

, .a b3

3 5

3 5 33 6

3 6 (A.12)

Continuum description

= + ∇ = + ∇ =φ φ φ iu u u b b, , 3, 5, 6,i i i i (A.13)

and

= ⎡⎣⎢

∇ − ∇∇ − ∇

⎤⎦⎥

= ⎡⎣⎢

∇ − ∇∇ − ∇


Δ uu b u b

b bΔ u

u b u bb b

, .a b3

3 5

3 53

3 6

3 6

(A.14)

Finally, the vectors bi are (Fig. 15):

= − == − = − −= − = −

b l s b sb l s b s lb s b s l

, ,, ,

, ( )/2.

1 1 4

2 2 5 1

3 6 2 (A.15)

Appendix B

B.1. Energy

The strain energy density in the continuum form defined in Section 3.1 is expressed by

=+ + + ++ + +

++ +

+ ++ − +

+ +

++ + + + + +

+ + + +

+− + + + + + +

+ + + +

+−

+ ++ − + + − +

+ + +

++ + + +

+ + +

++ + + − +

+ + +

++ + + + + −

+ + +

++ + + − +

+ + +

wC c c D D s s c C D s

s c D C sC s c D C s

c c D C sC c D C s

c D C sD C c D s s c D C s s

c s C c D s sD c D s s C s s s s

c s C c D s sD ω φ

c s sK c s s s s

s s sK c s s s s

s s sK c s s s s

s s sK c s s c s s s s

c s s sK s s s s s

c s s s

ɛ (24 12 (1 2 ) ( ℓ 48 ))2ℓ(1 )(24 ℓ (1 2 ))

ɛ (1 )(12 ℓ )2 ℓ(24 ℓ (1 2 ))

ɛ ɛ ( 12 ℓ )24 ℓ (1 2 )

ɛ 3 (4 ℓ 12 (1 ) 4 (3 ℓ (1 2 (1 ))))2 ℓ (1 )(2 ℓ 3 (3 4 (1 )))

ɛ 3 ( ( 24 (1 ) ℓ (3 4 (1 )(3 ))))2 ℓ (1 )(2 ℓ 3 (3 4 (1 )))

9 ( )ℓ (3 4 (1 ))

ɛ ɛ ( 1352 (9412 (1901 8 (8 1851 ))))104(1 )(347 484 452 )

ɛ (11518 (13520 (23761 24 (540 617 ))))208(1 )(347 484 452 )

ɛ (20280 (9464 (9721 8 ( 1604 1851 ))))208(1 )(347 484 452 )

ɛ ( (10969 8 (1958 1851 )) 6 (4158 (2063 8(44 617 ) )))208 (1 )(347 484 452 )

ɛ (35114 (22836 (21585 8 ( 2222 1851 ))))208 (1 )(347 484 452 )

,

w

w

w

w

w

112

ℓ4

ℓ ℓ2 2 2

ℓ2

ℓ2

2ℓ ℓ

2 2

222

ℓ2

ℓ ℓ2 2

2ℓ ℓ

2 211 22 ℓ ℓ ℓ

2

2ℓ ℓ

2 2

122

ℓ ℓ6 2

ℓ2 2 4

ℓ ℓ2

3ℓ

2 2ℓ

122

ℓ2

ℓ ℓ2 2

3ℓ

2 2ℓ

ℓ2

311 22

2

112

2

222

2

122 4 2

2

122 2

2 (B.1)

with Kw the Winkler foundation constant, =−

C E hνℓ 1

s

s2 and =

−D ,E h

νℓ 12(1 )s

s

32 respectively, the tensile and bending stiffness (per unit width) of the beams,

Es, νs, h, ℓ and θ, in turn, Young’s modulus, Poisson’s ratio, thickness, length and inclination of the cell walls. Also, to simplify the notation, =c θcosand =s θsin .

In the case of regular hexagonal microstructure, = ∘θ 30 , (B.1) takes the form


94

=

+ + + − ++

+−

++ + +

wC D C C D C D C

D CD ω φ K

(ɛ ɛ )( ℓ 36 ℓ ) 2ɛ ɛ ( ℓ 12 ℓ ) 96 ℓ ɛ4 3 ℓ (12 ℓ )

3 ( )3 ℓ

(305(ɛ ɛ ) 544 ɛ 66 ɛ ɛ )1664 3

.w

112

222

ℓ2 4

ℓ ℓ2

11 22 ℓ2 4

ℓ ℓ2

ℓ ℓ2

122

3ℓ ℓ

2

ℓ2

3112

222

122

11 22

(B.2)

B.2. Constitutive equations and elastic constants

The polynomial expressions =f f θ θ(cos , sin )i i introduced in Section 3.2 are:

= += + + + += + − − += + += − + + += − + + + + + +

+ − + += + + + += + + + += + + + +

+ + + + + − += + = + = += + + + = + + + +

f sf s s s sf s s s sf s sf s s s cf s s s s s s s c

s s s scf s s s sf s s f c f c s s ff f s s s

s s s s s sf s f f s f f s s f

f s s c c f s s s s

1 ,11518 (13520 (23761 24 (540 617 ))),

( (8 (1851 8) 1901) 9412) 1352,347 484 452 ,( ( (8 (1851 1604) 9721) 9464) 20280) ,(((8(1851 2222) 21585) 22836) 35114) (8(1851 1958) 10969)

6((8 (44 617 ) 2063) 4158) ,25, 688 (9464 (24, 093 4 (581 8207 ))),

(1 ) 2 (1 ) ,11, 518 2 (1 ) 33, 852 98, 719

2 (53, 046 (59, 222 (9464 (9721 8 ( 1604 1851 ))))),(1 ) , (1 ) , ( 1) ,(8( 1) 4) 4 , 4( 1) ( 3) 3,

0

1

2

32

42

52 4

2

6

72 2

12

42

2

8 42 2

3

9 4 103

4 11 3

124 6

132 (B.3)

with =c θcos and =s θsin .In particular, for regular hexagonal microstructure, = ∘θ 30 ,

= = = − = == = = = == = = =

f f f f ff f f f ff f f f

3/2, 107, 055/4, 11, 583/4, 702, 321, 165/16,35, 802, 77, 571/2, 53, 703/2, 249, 561/2, 963, 495/32,9477/4, 1053/2, 117/16, 57/4.

0 1 2 3 4

5 6 7 8 9

10 11 12 13 (B.4)

Accordingly, the constitutive equations and elastic moduli of Sections 3.2 and 3.3 take the form:

= =+ + −

++ +

= =+ + −

++ +

= =+

+

= − =−

= + = +

σ σC D C C D C

D CK

σ σC D C C D C

D CK

σ σ D CD C

K

σ σD ω φ

σ σ σ σ σ σ

( ℓ 36 )ɛ ( ℓ 12 )ɛ2 3 ℓ(12 ℓ )

(305ɛ 33ɛ )832 3

,

( ℓ 36 )ɛ ( ℓ 12 )ɛ2 3 ℓ(12 ℓ )

(305ɛ 33ɛ )832 3

,

48 ɛ2 3 ℓ(12 ℓ )

17 ɛ52 3

,

3 ( )ℓ

,

, ,

sym w

sym w

sym sym w

skw skw

sym skw sym skw

11 11ℓ2 2

ℓ ℓ 11 ℓ2 2

ℓ ℓ 22

ℓ ℓ2

11 22

22 22ℓ2 2

ℓ ℓ 22 ℓ2 2

ℓ ℓ 11

ℓ ℓ2

22 11

12 21ℓ ℓ 12

ℓ ℓ2

12

12 21ℓ

3

12 12 12 21 21 21 (B.5)

and

= ≡ =− + + − +− + − + +

= ≡ =+ − − −+ − + +

=+ − +

+ −

E E EK ν λE λ K ν λ E

ν λ K ν λ λ E

ν ν νλ K ν λ λ Eλ K ν λ λ E

Gλ K ν λ E

λ ν

* * * (13 (1 ) 32 )(17(1 ) (1 ) 104 )2 3 (1 )(305(1 ) (1 ) 416( 3 ) )

,

* * * 33(1 ) (1 ) 416 ( 1)305(1 ) (1 ) 416 (1 3 )

,

* 17(1 ) (1 ) 104104 3 (1 )(1 )

.

w s s w s s

s w s s

w s s

w s s

w s s

s

1 2

2 2 2 3

2 2 2 3

12 21

2 2 2

2 2 2

2 2 3

2 2 (B.6)

Appendix C

As stated, an energetic equivalence provides a suitable relation between the Winkler foundation constant of the present work, Kw, and thehydrostatic pressure p of Guiducci et al. (2014).

First of all, let us focus on a single cell and let us consider its elastic energy,Wc, obtained by summing the contribution of the walls,Ww, and of thefilling material, Wf:

= +W W W .c w f (C.1)

In particular,

= ⎧⎨⎩

= += +

WW W WW W W

Winkler modelpressurized cell,c

c Winkler w beams f Winkler

c pressurized cell w walls pressure

, , ,

, , (C.2)


95

with Ww, beams, Wf, Winkler and Ww, walls, Wpressure, in turn, the elastic energies of the cell walls and of the filling material in the case of Winklerfoundation model, Fig. 16(a), and pressurized cell (Guiducci et al., 2014), Fig. 16(b). By assuming

≡W W ,w beams w walls, , (C.3)

the energetic equivalence

≡W Wc Winkler c pressurized cell, , (C.4)

takes the form

≡W W .f Winkler pressure, (C.5)

The first term, Wf, Winkler is the sum of the elastic energies of the three series of springs in the directions n1, n2, n3:

∑= ⎛

⎝⎜

⎞

⎠⎟

=

W bΔU K ΔU12

· ,f Winkleri

iT

w i,1

3

(C.6)

where ΔUi is the elongation of the springs in the ni direction, b the width,

= ⎡⎣⎢

⎤⎦⎥

KKK0

0ww

w (C.7)

the stiffness matrix of the elastic foundation, Kw the Winkler constant. Also, Wpressure is related to the change in the volume of the cell and itsexpression, given in Guiducci et al. (2014), is

= − − = − + + −W p V VV

p b((1 ɛ )(1 ɛ ) 1) ,pressure0

011 22 (C.8)

with p the inner pressure, =V V p( ) and = =V V p( 0),0 respectively, the volume of the cell in the deformed and undeformed configuration, b thewidth. From classical continuum mechanics, the strains = pɛ ɛ ( )ij ij take the form

= = =ϵp p dd

in nɛ ( ) ( ) Δ , 1, 2, 3,ij iT

f ii

i (C.9)

with Δdi the elongation in the ni direction, ϵf(p) the infinitesimal strain tensor,

= = + =d d θ p d θ pℓ 2 2 sin ( ) , 2ℓ cos ( ),1 3 2 (C.10)

and θ(p) the inclination of the cell walls in the deformed configuration (Fig. 16). In addition, the assumption

= =U d iΔ Δ , 1, 2, 3i i (C.11)

provides, in view of (C.9),

= =ϵ p Ud

in n( ) Δ , 1, 2, 3,iT

f ii

i (C.12)

leading to

= =ϵU p d in nΔ ( ( ) ) , 1, 2, 3.i iT

f i i (C.13)

Substituting (C.13) into (C.6) and taking into account (C.5), gives

∑ = + + −=

ϵ ϵd p p d p p pn n K n n n12

( ( ) ) ( ( ) ) ((1 ɛ ( ))(1 ɛ ( )) 1).i

i iT

f iT

w i iT

f iT

i1

3

11 22(C.14)

From standard mathematical manipulations, it follows

Fig. 16. Practical application to the keel tissue of the ice plant. Equivalence between (a) the Winkler foundation model and (b) the pressurized cell (Guiducci et al.,2014).


96

=− + + + +

+ + +K

p p p θ p θ pθ p θ p p θ p p θ p p

( (1 ɛ ( ))(1 ɛ ( )) 1)2 cos ( )(1 sin ( ))2 2 sin ( ) (sin ( ) ɛ ( ) cos ( ) ɛ ( )) cos ( )ɛ ( )

,w11 22

211

222

222

2 (C.15)

being

= − = −pθ p

θp

θ pθ

ɛ ( )sin ( )

sin1, ɛ ( )

cos ( )cos

1110

220 (C.16)

obtained from classical continuum mechanics and simple geometrical considerations. In the above relations, = =θ θ p( 0)0 stands for the inclinationof the cell walls in the undeformed configuration and its approximated value, 75°, is given in Guiducci et al. (2014). By considering this and inserting(C.16) into (C.15), it emerges

⎜ ⎟

=⎛⎝

− + ⎞⎠

+

⎛⎝ +

+−

− ⎞⎠

+ ⎛⎝ −

− ⎞⎠

K pp

θ p θ pθ p θ p

θ p θ pθ p

θ p( )

28 sin ( )cos ( )

171 cos ( ) 1 sin ( )

4 sin ( )6 2

4 cos ( )6 2

1 cos ( )4 cos ( )6 2

1,w 3 3 2 2

(C.17)

where the values of θ(p) are derived from Guiducci et al. (2014).

Appendix D

D.1. Density

Let us focus on the 0th order level structure in Fig. 11. From the rule of mixtures, the density of this composite configuration, ρ(0), is given by

= + −ρ f ρ f ρ(1 ) ,f s(0) (0) (0) (0)

(D.1)

with =f V V/f tot(0) (0) (0) the porosity,V f

(0) andV ,tot(0) in turn, the volume of the filling material and of the entire cell, ρs and ρf

(0) the density of the cell walls,the first, and of the filler, the second. In particular,

= = + −+

fA b

A bθ θ λ

θ θ2 cos (1 sin ) 3

2 cos (1 sin ),f

tot

(0)(0)

(0)

(0) (0) (0)

(0) (0) (D.2)

where Atot(0) and Af

(0) are, on order, the total area of the cell and of the filling material, b the width, =λ h /ℓ(0) (0) (0) the ratio between the thickness andlength of the walls. Accordingly,

⎜ ⎟ ⎜ ⎟= ⎛⎝

+ −+

⎞⎠

+ ⎛⎝ +

⎞⎠

ρ θ θ λθ θ

ρ λθ θ

ρ2 cos (1 sin ) 32 cos (1 sin )

32 cos (1 sin )f s

(0)(0) (0) (0)

(0) (0)(0)

(0)

(0) (0) (D.3)

or, to simplify the notation,

= +ρ a ρ b ρ ,f s(0) (0) (0) (0)

(D.4)

with

= + −+

=+

a θ θ λθ θ

b λθ θ

2 cos (1 sin ) 32 cos (1 sin )

, 32 cos (1 sin )

.(0)(0) (0) (0)

(0) (0)(0)

(0)

(0) (0) (D.5)

Regarding the density of the first level structure ( =n 1), ρ(1), let us assume that the length of scale of the cell walls’ microstructure is muchsmaller than the cell wall itself. So, as done in Section 5.1, a continuum having density ρ(0) approximates each cell arm. As a consequence,

= +ρ a ρ b ρf(1) (1) (1) (1) (0)

(D.6)

where ρf(1) is the density of the filling material, a(1), b(1) are derived by substituting θ(1) and =λ h /ℓ(1) (1) (1) for θ(0) and λ(0).

Finally, analogous calculations provide the density in the case on n levels of hierarchy:

= + −ρ a ρ b ρ ,n nfn n n( ) ( ) ( ) ( ) ( 1)

(D.7)

with ρfn( ) and −ρ ,n( 1) in turn, the density of the filler and of the cell walls, a(n) and b(n) obtained as before.

D.2. Winkler foundation constant as a function of the filler’s Young’s modulus

In Section 5.3, the hypothesis that the density of the filling material, ρ ,fi( ) is the same at all levels provides

= = =ρ ρ α ρ i, 1, 2, 3,fi

f s( )

(D.8)

with α a positive constant depending on the material inside the cells. For simplicity, let us assume that the filler is a standard cellular material withhexagonal microstructure, as commonly happens in nature (Gibson and Ashby, 2001). Thus, the classical relations (Gibson and Ashby, 2001)

= = =ρ

ρλ

EE

λ i23

, 43

( ) , 1, 2, 3fi

s ffi f

i

s ffi

( )

,

( )( )

,

( ) 3

(D.9)

provide its (effective) Young’s modulus, E ,fi( ) and density, ρ ,f

i( ) as a function of the cell walls’ properties, i.e., the thinness ratio, λ ,fi( ) the density, ρs, f,

and Young’s modulus, Es, f.


97

By taking into account the energetic equivalence in Ongaro et al. (2016b),

= =K E i85 3

, 1, 2, 3,wi

fi( ) ( )

(D.10)

together with the assumption

= =ρ ρ E E, ,s f s s f s, , (D.11)

simple mathematical manipulations give

=⎛

⎝⎜

⎞

⎠⎟ =K E

ρ

ρi4 3

5, 1, 2, 3,w

is

fi

s

( )( ) 3

(D.12)

a suitable relation between the Winkler constant, K ,wi( ) and the filler’s density. Finally, in view of (D.8),

= = =K K α E i4 35

, 1, 2, 3.wi

w s( ) 3

(D.13)

In particular, four values of α are considered: 0.4, 0.2, 0.1, 0, leading to = −K E10 ,w s1 − E10 ,s

2 − E10 ,s3 0, respectively.

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