Nacre: an Orthotropic and Bimodular Elastic Material

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Available online at www.sciencedirect.com COMPOSITES SCIENCE AND TECHNOLOGY Composites Science and Technology 68 (2008) 1363–1375 www.elsevier.com/locate/compscitech

Nacre: An orthotropic and bimodular elastic material

K. Bertoldi a, D. Bigoni b,*, W.J. Drugan b,c

a Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA b Dipartimento di Ingegneria Meccanica e Strutturale, Universita` di Trento, Via Mesiano 77, I-38050 Trento, Italy c Department of Engineering Physics, University of Wisconsin-Madison, WI, 53706, USA

Received 30 August 2007; received in revised form 14 November 2007; accepted 29 November 2007 Available online 14 December 2007

Abstract

A new micromechanical model for nacre (mother-of-pearl) is proposed, based on an accurate description of the actual microstructure of the material. In the small-strain regime (where damage and fracture are excluded), it is shown via homogenization techniques that the mechanical behaviour of nacre is: (i) orthotropic, with a strong difference in directional stiffnesses; (ii) bimodular (different Young’s moduli in tension and compression). A simple closed-form analytical solution, and a highly-accurate boundary-elements-based numerical analysis, are developed to evaluate the macroscopic behaviour of nacre from the mechanical properties of its constituents. It is shown that the predictions of our simple analytical model are in excellent agreement with the highly-accurate numerical analysis and with exist- ing experimental data, and it is explained how our model can be further verified through new experiments. Importantly, we show that it is essential to account for nacre’s bimodularity and anisotropy for the correct interpretation of published (and future) experimental data. Ó 2007 Elsevier Ltd. All rights reserved.

Keywords: Mother-of-pearl; Biological composites; Masonry; Homogenization theory; Elasticity

1. Introduction While this microstructure may at least partially explain nacre’s high fracture toughness, it does not seem compati- The surprisingly excellent stiffness and toughness prop- ble with its high stiffness. The high fracture toughness erties of nacre [mother-of-pearl, the internal layer of many could be related to crack blunting/branching, as suggested mollusc shells (see for example Fig. 1) comprised of 95% by Almqvist et al. [1], where it is, however, concluded that aragonite, a mineral form of CaCO3, with only a few per- ceramics with a microstructure mimicking mother-of-pearl, cent of biological macromolecules] have been known since though superior to other ceramics, do not attain the frac- the experimental work of Currey [7,8]. The fact that these ture toughness of nacre. Damage and fracture mechanisms mechanical characteristics remain unchallenged by the cur- in nacre are, however, not completely understood. For rent ceramic materials has further stimulated research. instance, inelastic deformations have been considered by Nacre has a peculiar ‘brick-mortar’ microstructure (the Wang et al. [26], whereas Sarikaya et al. [22] advocate sev- so-called ‘stretcher bond’ in masonry nomenclature, see eral micromechanisms to explain the outstanding strength Figs. 2 and 3 and compare to Fig. 4), where stiff and flat of nacre: (a) crack blunting/branching, (b) micro-crack for- aragonite crystals, the ‘bricks’ (0.2–0.9 lm thick and with mation, (c) plate pullout, (d) crack bridging (ligament for- a mean transversal dimension ranging between 5 and mation) and (e) sliding of layers. On the other hand, from 8 lm), are connected and separated by nanoscale organic available experimental data (reported in Table 1), it is dif- interlayers (20–30 nm, see Table 1), the ‘mortar’ [23,24]. ficult to understand how a composite with a highly compli- ant organic matrix may exhibit such a high stiffness. * Corresponding author. Tel.: +39 0461 882507; fax: +39 0461 882599. Therefore, Schaffer et al. [23] and Song et al. [24] advocate E-mail addresses: [email protected] (K. Bertoldi), [email protected] the presence of mineral bridges, joining the aragonite plate- (D. Bigoni), [email protected] (W.J. Drugan). lets through the organic matrix. However, the existence of

Fig. 1. South African Abalone shell (Haliotis midae, purchased in dry condition at ‘Orientimport’, Torrepedrera, Italy; photos taken with a Nikon D200 digital camera at the University of Trento). Left: inner part; right: outer part.

Fig. 2. A rough fracture surface of the South African Abalone shell shown in Fig. 1 [micrographs taken with a Jeol 5500 scanning electron microscope (JEOL Inc., Peabody, Mass) at the University of Trento].

Fig. 3. A rough fracture surface of the South African Abalone shell shown in Fig. 1, etched by immersion in a 60% solution of HNO3 for 1 h. Center and right: two details of the figure on the left, showing the ‘brick-mortar’ structure of nacre. Note that separations where bricks abut are evident [micrographs taken with a Jeol 5500 scanning electron microscope (JEOL Inc., Peabody, Mass) at the University of Trento]. such mineral bridges has been disproved or at least not chain model’, originally proposed by Ja¨ger and Fratzl confirmed by Bruet et al. [3] and Lin and Meyers [16]. [11]. This is a simple model in which the organic matrix Prior to the present work, mechanical models were inca- is assumed on the one hand not to resist tension, but to pable of explaining the high stiffness of nacre and, addi- resist shear on the other. We will derive an improved ver- tionally, the anisotropy of the material (which appears sion of this model and employ it for tensile response paral- evident from consideration of the microstructure) has lel to the aragonite lamellae, for which we will show it to apparently not been modeled. Nukala and Simunovic be quite accurate. Ji and Gao [12] also note that the micro- [18] employ a discrete lattice where the aragonite platelets structure of nacre should imply anisotropic overall behav- are rigid, and use the shear-transmission hypothesis for the iour, but they do not model it. Katti et al. [14,15] measure organic matrix. The latter hypothesis has been advocated and predict (with a finite element method) a stiffness of by Ji and Gao [12], through the so-called ‘tension–shear 20 GPa for the organic matrix and conclude that the min- K. Bertoldi et al. / Composites Science and Technology 68 (2008) 1363–1375 1365

Fig. 4. An example of Roman masonry (Anfiteatro Flavio, built in the second half of the first century A.D. under Emperor Vespasiano at Pozzuoli, Naples. Photo by D. Bigoni). A detail is shown at right of the masonry in the upper left part of the figure at left.

Table 1

Available experimental data: Young’s moduli parallel and orthogonal to the lamellae, E1 and E2, of the aragonite platelets, Ep, and of the organic matrix, Eo; thickness of the organic layer, ho

Authors Material E1 ðGPaÞ E2 ðGPaÞ Ep ðGPaÞ Eo ðGPaÞ ho ðnmÞ Jackson et al. [10] Pinctada nacre 73 70 100 4 – Katti et al. [14] Red abalone – – 99.5 20 30 Wang et al. [26] Abalone 69 ± 7 66 ± 2 – – 20 Pearl oyster 77 ± 12 81 ± 4 – – 20 Bruet et al. [3] Trochus niloticus – – 92–79 – 30-300 Barthelat et al. [4] Red abalone – – 79 ± 15 2.84 ± 0.27 20-40 eral bridges have no effect on the nacre averaged stiffness. ple approach yields closed-form solutions to directly deter- Moreover, the fact that the material should be bimodular mine the four parameters defining the averaged properties (namely, to have a Young’s modulus lower in tension than of nacre from knowledge of the mechanical properties of in compression) has passed completely unnoticed the micro-constituents. The predictions of the simple (although this behaviour should be inferred from the Ja¨ger approach are found to be in excellent agreement with the and Fratzl [11] model; see, e.g., Fig. 2b of [12]) and is still numerical results obtained from a much more refined model. awaiting quantification. We note finally that although our model is based on hypoth- The goal of the present article is to provide a microme- eses obtained from the very recent paper by Lin and Meyers chanical model capable of explaining and quantifying all [16], our strategy of solution can be easily modified to the above-mentioned mechanical features. Employing the account for new features, should these be found in the future. detailed description of Lin and Meyers [16] allows us to Predictions of our model are shown to agree with exist- formulate a micromechanical model, which through stan- ing experimental data, and simple formulae are provided to dard homogenization techniques yields the macroscopic extract two of the four parameters defining our model from mechanical behaviour of nacre.1 This turns out to be: three (or four) point macroscopic bending experiments (without knowledge of the micromechanical properties). orthotropic, with a strong difference between normal Importantly, we show that it is essential to account for stiffnesses and shear stiffness; in particular, nacre is nacre’s bimodularity and anisotropy for correct interpreta- much less stiff under shear than under uniaxial tension tion of the experimental data. (in the two directions parallel and perpendicular to the aragonite platelets); 2. Mechanical model for nacre bimodular, i.e. having a Young’s modulus in tension smaller than in compression, in the direction parallel Our model is built on the very accurate and recent to the lamellae long boundaries. observations and data reported by Lin and Meyers [16] and is sketched in Fig. 5. In particular, the aragonite These results are obtained by developing a two-dimen- platelets are idealized as rectangular elastic isotropic sional, plane strain, micromechanical model of nacre blocks (with Lame´ moduli kp and lp) joined along the (Fig. 5), yielding averaged properties through a simple ana- lamellar ‘long’ boundaries by the organic matrix, also lytical approach and a standard numerical procedure (based taken to be linearly elastic and isotropic (with Lame´ mod- on a boundary element technique developed by us). The sim- uli ko and lo). This system is deformed under plane strain conditions. It is important to note that, along the short 1 We consider in this paper mechanical behaviour at small strains, so edges of the aragonite platelets (where these abut), Lin that damage leading to final failure will not be investigated here. and Meyers [16] find a simple contact without any organic 1366 K. Bertoldi et al. / Composites Science and Technology 68 (2008) 1363–1375

Fig. 5. Sketch of the model for the mechanical behaviour of nacre, where aragonite platelets are modeled as ‘bricks’ glued by a ‘mortar’ of organic matrix along the basal planes, but in unbonded unilateral contact along lateral surfaces. glue. We have idealized this as a unilateral contact (so With the two steps above, an average strain/average that, along the short edges, the platelets behave as per- stress relation is found, which, when compared to a fectly jointed in compression, whereas these are com- effective stress/strain relation of the type pletely disconnected under tension). This idealization is r ¼ E ; ¼ E1 r ; ð3Þ supported by: (i) direct observations by Lin and Meyers ij ijhk hk ij ijhk hk [16]; (ii) the fact that fracture surfaces do not cross arago- yields the determination of all independent material con- nite platelets, which are seen simply to slide one with stants defining the composite modulus tensor Eijhk. respect to the other [21,9]; (iii) the fact that the compressive strength was found to be 1.5 times the tensile strength Due to the peculiar microstructural geometry, it is well- by Menig et al. [17]; (iv) the fact that strains much larger known that masonry loaded in-plane follows an aniso- at the tensile than at the compressive surface have been tropic elastic description. Surprisingly, the fact that the found in three-point bending by Wang et al. [26]. Clearly, microstructure of nacre (which is almost identical to the unilaterality of contact introduces a nonlinearity in masonry) implies an anisotropic elastic macroscopic the model. We will see that this can be easily handled response appears never to have been modelled. For the at the cost of some approximations. microstructure of nacre, the effective elastic tensor Eijhk is orthotropic, which in a two-dimensional setting has only 2.1. Homogenization the following non-null components E E E m The above-introduced model of nacre is periodic and is E ¼ 1 ; E ¼ 2 ;E ¼ 1 21 ; 1111 1 m m 2222 1 m m 1122 1 m m very similar to typical models for masonry [19,2,5]. The goal 12 21 12 21 12 21 of homogenization theory (which has been thoroughly E2m12 E2211 ¼ ; E1212 ¼ E1221 ¼ E2121 ¼ E2112 ¼ G12: ð4Þ developed for periodic elastic media, see for instance San- 1 m12m21 chez-Palencia [20]) is to derive from a microstructure a mac- 1 The inverse tensor Eijhk has the following non-null roscopic response valid for an effective continuous components equivalent medium. To this purpose, a representative volume element is considered, which, due to the symmetry of 1 1 1 1 1 m21 1 m12 E1111 ¼ ; E2222 ¼ ; E1122 ¼ ; E2211 ¼ ; our structure, is selected as sketched in the detail of Fig. 5. E1 E2 E2 E1

The homogenization is performed in the following steps. 1 1 1 1 1 E1212 ¼ E1221 ¼ E2121 ¼ E2112 ¼ ; ð5Þ 4G12 The representative volume element (of volume V and external surface S having outward unit normal vector where m12, m21 are the two plane strain Poisson ratios, and E1, E2, G12 are the two plane strain elastic tensile moduli ni) is to be subjected to prescribed mean strain ij. To this purpose, the average strain theorem (e.g., [13]) and the plane strain shear modulus, subject to the symme- Z Z try condition 1 1 V u n u n S; ij ¼ ij d ¼ ð i j þ j iÞd ð1Þ m12 m21 V V 2V S ¼ ; ð6Þ E1 E2 (where ui is the displacement vector, ij the related strain) allows us to prescribe the mean strain in terms so that through the homogenization technique we must of an appropriate displacement on the boundary. identify four independent material constants. Note that in the particular case of isotropy, the four constants reduce The mean stress rij produced by the application of the mean strain must be (numerically or analytically) evalu- to the two plane strain constants ated on the representative volume element. The average E m stress theorem (e.g., [13]) E ¼ E ¼ ; m ¼ m ¼ ; ð7Þ Z Z 1 2 1 m2 21 12 1 m 1 1 rij ¼ rij dV ¼ riknkxj dS; ð2Þ in which E and m are the usual (three-dimensional) elastic V V V S constants (Young’s modulus and Poisson’s ratio). (where xj is the position vector and rij the stress tensor) The above four constants can be identiﬁed by subjecting allows us to work in terms of tractions on the boundary. the representative volume element to three independent K. Bertoldi et al. / Composites Science and Technology 68 (2008) 1363–1375 1367

Fig. 6. The three modes of deforming the representative volume element shown in Fig. 5 needed to determine the four constants comprising the eﬀective constitutive tensor, Eq. (4).

deformation components (which, employing the symme- culations, we derive it as follows. First, we consider the tries present, can be reduced to the three boundary dis- limiting case in which the composite material is consid- placements illustrated in Fig. 6), calculating the ered to be made up of inﬁnitely rigid (aragonite) platelets corresponding averaged stress components and comparing and soft (organic) layers. When pulled in tension, the to Eq. (3). material deforms as sketched in Fig. 7, left. Since the only deformation is the shearing of the soft layers, assuming uniform shearing of the soft layers2 yields the elastic mod- 2.2. A simple, closed-form homogenization model for nacre ulus as

To develop simple, closed-form formulae for the aniso- t ðLp bÞLpGo E1 ¼ : ð9Þ tropic and bimodular elastic response of nacre, we begin 4hoðho þ hpÞ by observing that simple mechanical considerations sug- where Lp P b P 0 is a length parameter, accounting for gest that the unilateral contact between aragonite platelets the fact that the uniform shearing solution violates mo- plays an important role only under tension aligned paral- ment equilibrium of the shearing stresses where the soft lel to the x1 axis. On the other hand, when subject to com- layers are separated. Therefore, parameter b allows us pression along the x1 axis, tension/compression parallel to to account for the fact that there is a small transition the x2 axis, and shear parallel to the axes, the unilaterality zone where r increases from zero to its full value, so of the contact plays a negligible role and the material 12 b Lp. behaves as a laminated medium composed of aragonite Second, we consider now the other limiting case, in and organic matrix layers. The accuracy of these assump- which the soft (organic) layers are absent (Fig. 7, right) tions was conﬁrmed by use of a more reﬁned numerical and the (aragonite) platelets are elastic and perfectly model. bonded to each other. Subjecting this structure to longitu-

t dinal loading and neglecting deformation of the layer near 2.2.1. Unilateral model: determination of E for tensile 3 1 the detachment zones we obtain loadings Ep The response to a tensile loading parallel to the x1 axis, Et ¼ ; ð10Þ 1 4 a þ 1 where the aragonite platelets are disconnected (where these Lp abut), can be obtained by borrowing and improving results where the length parameter a ðL =4 P a P 0Þ has been from Ja¨ger and Fratzl [11] (see [12], their Eq. (5)). This sim- p introduced. This is similar to the parameter b and permits ple model is based on the fact that under such an applied us to account for the fact that there is a small transition tensile stress, the mineral plates carry most of the tensile zone, at the unbonded platelet end, in which r increases load, while the organic matrix transfers the load between 11 from zero to its full value, so a L . When the transition the plates via shear. The eﬀective modulus of the composite p zone length is zero (i.e. a ¼ 0) the elastic modulus is that of for tensile loading along axis x , Et , can be expressed in our 1 1 the intact material. Clearly the behaviour of the actual notation (see Fig. 5)as[12] composite lies between these limiting cases. Combining, t Ep therefore, the two limiting cases as springs in series (i.e. E1 ¼ ; ð8Þ ho Ephpho summing the compliances) in proportion to the volume þ 1 4 2 þ 1 hp GoLp where Go denotes the shear modulus of the organic matrix, 2 Ep the Young’s modulus of the aragonite platelets, hp and It can be shown using the theorem of minimum potential energy that the assumption of uniform shearing yields an upper bound to Et , given by ho the thicknesses of the aragonite platelets and of the or- 1 Eq. (9) with b ¼ 0. ganic layer, and Lp the aragonite platelets’ length. 3 It can be shown using the theorem of minimum potential energy that With the goal of generalizing (8) so that its improve- the assumption of uniform longitudinal strain yields an upper bound to t ment agrees more closely with our accurate numerical cal- E1, given by Eq. (10) with a ¼ 0. 1368 K. Bertoldi et al. / Composites Science and Technology 68 (2008) 1363–1375

β/4

2α

Fig. 7. The Ja¨ger–Fratzl–Ji–Gao model in the limit cases of rigid platelets (left) and null thickness of the organic layers (right). Transition zones of lengths a and b have been introduced into the models (shown dashed).

c fraction of each material, where the platelet volume frac- 2.2.2. Layered model: determination of E1, E2, m12 and m21 for tion is compressive loadings When loaded in compression or under shear, nacre hp U ¼ ; ð11Þ behaves (following our simplifying assumptions) as a h þ h o p monolithic laminate material, composed of uniform iso- (U ¼ 1 corresponds to the limiting case shown at Fig. 7 tropic laminae of thicknesses hp and ho (see Fig. 5). right, while U ¼ 0 to the limiting case shown at Fig. 7 left) Homogenization of this type of material is standard and we obtain can be found for instance in Willis [27]. However, for 2 4 a þ 1 completeness, we provide a sketch of the approach here. 1 4hpð1 UÞ Lp : We assume 22 ¼ 12 ¼ 0and11 ¼ 11 ¼ constant in all t ¼ 2 þ ð12Þ E L ðL bÞG U UEp 1 p p o laminae (so that compatibility is satisfied). Uniform r11 Eq. (12) can be rewritten as and r22 are generated, the former different in each lamina (but trivially satisfying equilibrium) and the latter must be t Ep E1 ¼ ; ð13Þ required to be equal in each lamina (and equal to the ho þ 1 4 Ephpho þ 4a þ 1 mean value r ) by equilibrium considerations at the hp GoLpðLpbÞ Lp 22 interface. It follows that in a generic layer (having elastic which is a generalization of the Ja¨ger–Fratzl–Ji–Gao mod- constants E and m) el, in the sense that this model is retrieved when a ¼ b ¼ 0. Since parameter b=Lp is small, we can define a new E þ r mð1 þ mÞ r ¼ 11 22 : ð16Þ parameter c as 11 1 m2

c Ephpho 4b 4a ¼ þ ; ð14Þ Using Eq. (16) in the condition ¼ 0, valid for the unit L G L2 L L 22 p o p p p cell, we obtain so that a Taylor series expansion of Eq. (13) in b=L and p DE then use of Eq. (14) yields r m ð1 þ mÞð1 2mÞ 1 E ¼ 22 ¼ ; ð17Þ 2211 1 m E 1 m t Ep 11 ð Þ E1 ¼ ; ð15Þ ho Ephpho c þ 1 4 2 þ þ 1 where the bracket hi is used to denote the mean value of a hp GoL Lp p generic function w as follows again reducing to the Ja¨ger–Fratzl–Ji–Gao model, when h wðE ; m Þþh wðE ; m Þ c ¼ 0. Eq. (15) will be shown to yield a more accurate hwðE; mÞi ¼ o o o p p p : ð18Þ approximation than Eq. (8). ho þ hp K. Bertoldi et al. / Composites Science and Technology 68 (2008) 1363–1375 1369

Using Eq. (17) in Eq. (16) and averaging, we obtain axial tension or compression, a criterion for dividing strain DEspace into compression and tension subdomains is needed r E m 2 ð1 þ mÞð1 2mÞ 1 11 : (something analogous to a yield criterion in strain space E1111 ¼ ¼ 2 þ 11 1 m 1 m Eð1 mÞ plasticity), and continuity of the stress/strain law should ð19Þ be enforced (see [6] for details). We do not pursue this sub- Finally, we impose a mean strain whose only non-zero ject further here. component is 22. Analogously to the previous case, we obtain for a generic layer (defined by E and m) 2.3. Results from a boundary-elements-based numerical technique r22m r11 ¼ ; ð20Þ 1 m Since via the averaging theorems we can work with dis- so that placements and tractions on the surface of the representa- 2 1 tive volume element, a numerical boundary element r22 1 m 2m E2222 ¼ ¼ : ð21Þ technique seems to be particularly appropriate. We have 22 Eð1 mÞ used a general-purpose Fortran 90 code for two-dimen- 4 c sional analysis. In the code, linear (constant) shape func- Hence, the effective elastic constants E1, E2, m12 and m21 can be determined by employing Eq. (4) as tions for the displacements (tractions) at the boundary are assumed, and the material has been taken to be linear 2 2 Eohoð1 m ÞþEphpð1 m Þ Ec p o ; elastic, with Lame´ moduli k and l. The unit cell (see 1 ¼ 2 2 ðho þ hpÞð1 mpÞð1 moÞ Fig. 5) has been discretized along boundaries parallel to 2 2 EoEpðho þ hpÞ½Eohoð1 mpÞþEphpð1 moÞ the x1 axis employing 50 elements and along the boundaries E2 ¼ ; C parallel to the x2 axis using 20 elements for the aragonite E E ðh þ h Þð1 þ m Þð1 þ m Þ½m h ð1 m Þþm h ð1 m Þ platelets and 10 elements for the organic matrix layer. Fol- m ¼ o p o p p o o o p p p o ; 21 C lowing Sarikaya et al. [22], the thickness of the aragonite mohoð1 mpÞþmphpð1 moÞ plates and of the interlayers have been chosen respectively m12 ¼ ; ðho þ hpÞð1 mpÞð1 moÞ as hp ¼ 0:5 lm and ho ¼ 20 nm, while the length of the ð22Þ plates has been selected as Lp ¼ 5 lm. where In the numerical analyses, both the aragonite plates and the organic interphase have been modelled as linear elastic 2 2 2 2 C ¼ hohp½Epð1 þ moÞ ð1 2moÞþEoð1 þ mpÞ ð1 2mpÞ materials, having Young’s moduli Ep ¼ 100 GPa and Eo ranging between 0 and 10 GPa to cover the range of values þ 2EoEpmompð1 þ mpÞð1 þ moÞ 2 2 2 2 experimentally measured (see Table 1) (the Poisson ratio þ EoEpðho þ hpÞð1 mpÞð1 moÞ: ð23Þ has been taken equal to 0.33 for both materials). The unilaterality of the contact has been accounted for by using a small gap where platelets abut and confirming that this gap 2.2.3. Layered model: determination of G12 In order to determine the effective shear modulus of the opens under tensile applied loading, while no gap was used composite, we consider again the monolithic layered model under compressive loading. Results of the analysis are reported in Figs. 8 and 9, considered before and we prescribe a mean strain 12, together with the predictions of the simple analytical for- requiring r12 ¼ r12 in all layers. We obtain mulae, Eqs. (22) and (25) and the Ja¨ger–Fratzl–Ji–Gao 1 r12 1 1 þ m model, Eq. (8), and our improved version, Eq. (15), with G12 ¼ E1212 ¼ ¼ ; ð24Þ 12 2 E c ¼ 0:26 Lp and labelled ‘Proposed model’. In Fig. 8 the effective moduli (normalized by Ep) are reported versus which becomes the normalized Young’s modulus of the organic matrix, Eo=Ep, while the same effective moduli are reported versus EoEpðho þ hpÞ G12 ¼ : ð25Þ the thicknesses ratio h =h (see Fig. 5)inFig. 9. The numer- 2½E h ð1 þ m ÞþE h ð1 þ m Þ o p p o o o p p ical simulations reported in Figs. 8 and 9 have been per- Formulae (15), (22), and (25) represent the results of our formed under both the assumptions of disconnection simple homogenization approach; these allow a relation (tensile loading) and perfect contact (compressive loading) between the micromechanical geometry and properties and at the unilateral contact between aragonite platelets. The the macroscopic, effective behaviour. As a simple check, note results show clearly that the unilaterality of the contact that in the special case Eo ¼ Ep ¼ E and mo ¼ mp ¼ m, Eqs. only plays a significant role for tension parallel to x1 axis (22) and (25) yield the correct plane strain isotropic con- E E E= m2 m m m= m2 stants: 1 ¼ 2 ¼ ð1 Þ, 12 ¼ 21 ¼ ð1 Þ, and 4 G ¼ E=ð2 þ 2mÞ. This code has been developed at the Computational Solids & 12 Structural Mechanics Laboratory of the University of Trento, whose It should be noted that in order to use the bimodular executable is available at: http://www.ing.unitn.it/dims/laboratories/ model that we propose for stress states different from uni- comp_solids_structures.php. 1370 K. Bertoldi et al. / Composites Science and Technology 68 (2008) 1363–1375

E/Ec 1 1p 0.8 E/E2p t 0.8 E/E 1p 0.6

0.6 0.4 0.4 BEM continuous platelets BEM separated platelets Analytical compression loading 0.2 0.2 Proposed model Jager-Fratzl-Ji-Gao

0 0 E/Eo p E/Eo p 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1

0.8 0.5 G/G12 p 21

0.4 0.6

0.3 0.4 0.2

0.2 0.1

0 0 E/Eop E/Eop 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1

Fig. 8. Eﬀective elastic moduli calculated numerically and from the simple formulae (8), (15) (with c ¼ 0:26 Lp), (22) and (25).

(there is a small effect in the shear case). Moreover, it is (taken positive for tension and negative for compression) clear from Figs. 8 and 9 that the agreement with the simple so that the normal stress and strain components in the theory, Eqs. (15), (22) and (25), is excellent. direction ni are Note also that formula (15), employed with c ¼ rnn ¼1;nn ¼ niijnj ¼ninjEijhknhnk: ð28Þ 0:26 Lp, clearly improves upon that of Ji and Gao [12], Eq. (8), based on the Ja¨ger and Fratzl [11] Since in plane strain ni becomes a function of the incli- model. nation angle h with respect to the x1 axis, we can define It is important at this point to quantify the degree of the elastic moduli in tension and compression, respectively, anisotropy and bimodularity of the material, since these EtðhÞ and EcðhÞ,as two properties have not previously been modelled for nacre. 1 1 Et h ; Ec h : To this purpose, we report in Fig. 10 the tension/compres- ð Þ¼ t ð Þ¼ c ð29Þ t c nnðhÞ nnðhÞ sion ratio of the effective Young’s moduli E1=E1 (equal to 1 when bimodularity is absent) and in Fig. 11 the ratios Therefore, a polar plot of Eqs. (29), obtained by using t c E1=EAv, E1=EAv, E2=EAv, G12=GAv, where Eqs. (15), (22) and (25) in Eq. (5), allows a visual quantification of the elastic anisotropy. This is reported in Fig. 12, Et þ Ec þ E mt þ mc þ m after normalization by Ec . Since the graphs would reduce E ¼ 1 1 2 ; m ¼ 12 12 21 ; 1 Av 3 Av 3 to unit circles in the case of isotropy and equal behaviour in tension and compression, the strong degrees of material EAv GAv ¼ ; ð26Þ anisotropy and bimodularity are evident. 2ð1 þ m Þ Av Although parametric investigations are reported in the so that the ratios are equal to 1 in the case of isotropy and graphs of Figs. 10 and 11, the ratio Eo=Ep usually (see Table equal behaviour in tension and compression. 1) takes values between the two cases considered in Fig. 12, To better quantify the anisotropy and bimodularity namely, 0.01 and 0.04. We can, therefore, conclude that: effects, we introduce a (unit norm) uniaxial stress in the the ratio Et =Ec ranges between 0.5 and 0.8, showing that direction of the unit vector n as 1 1 i bimodularity is an important effect (a fact previously not rij ¼ninj; ð27Þ understood, and, therefore, not tested); K. Bertoldi et al. / Composites Science and Technology 68 (2008) 1363–1375 1371

1 1 E/E2p c E/E1p 0.8 0.8

0.6 t E/E1p 0.6 0.4

0.4 0.2

h/oph h/oph 0 0.04 0.08 0.12 0.16 0.2 0 0.04 0.08 0.12 0.16 0.2

1 0.6 G/G12 p 21 BEM continuous platelets 0.8 0.5 BEM separated platelets Analytical compression loading 0.6 0.4 Proposed model Jager-Fratzl-Ji-Gao

0.4 0.3

0.2 0.2

0 0.1 h/oph h/oph 0 0.04 0.08 0.12 0.16 0.2 0 0.04 0.08 0.12 0.16 0.2

Fig. 9. Eﬀective elastic moduli calculated numerically and from the simple formulae (8), (15) (with c ¼ 0:26 Lp), (22) and (25). Modulus Eo has been taken equal to 4 GPa.

tc 0.8 E/E11

0.76 tc 0.6 E/E11 0.72

0.68 0.4 0.64 BEM 0.6 0.2 Analytical E/Eop 0.56 0.01 0.02 0.03 0.04

0 E/Eop 0 0.02 0.04 0.06 0.08 0.1

Fig. 10. Bimodularity of the material, quantiﬁed by the tension/compression ratio of the eﬀective elastic moduli. A detail in the relevant range of parameters is given at right.

c both the ratios E1=EAv and E2=EAv lie near 1, (a fact con- 3. Comparison with experimental data sistent with experimental results by Jackson et al. [10] and Wang et al. [26]); To compare the predictions of our model with available the ratio G12=GAv ranges between 0.3 and 0.6, showing a experimental data, we refer to Table 1, where data taken strong diﬀerence in degree of anisotropy between the from the literature are reported. shear and Young’s moduli (a fact noticed also by Jack- Before discussing the model’s predictions, we note that son et al. [10]). the experimental results reported in Table 1 have been 1372 K. Bertoldi et al. / Composites Science and Technology 68 (2008) 1363–1375

3 1

G/G12 Av

0.8

2 c 0.6 E/E1Av

0.4 1 BEM 0.2 Analytical E/E2Av t E/E1Av 0 0 E/Eop E/Eop 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1

Fig. 11. Anisotropy of the material, quantified by the effective elastic moduli normalized by the directionally-averaged moduli defined in Eq. (26).

1 1

0.8 0.8

Eop /E =0.01 Eop /E =0.04 0.6 0.6

c c 0.4 0.4 E( )/E1

c c 0.2 E( )/E1 0.2 t c E(t )/E c E( )/E1 1 0.2 0.4 0.6 0.8 1 0.2 0.4 0.6 0.8 1

c Fig. 12. Polar graphs of the effective elastic moduli in tension and compression, normalized by E1, revealing the strong anisotropy and bimodularity of nacre. (In the case of isotropy and in the absence of bimodularity, a single unit circle would be present in each case.) pffiffiffi mainly determined via three-point bending tests. These ð1 þ cÞ2Px u ðx Þ¼ 1 ð3ll 3l2 x2Þ; x 2½0; l tests are usually interpreted by employing the linear theory 2 1 t 3 0 0 1 1 0 2E1h of unimodular isotropic elasticity, but, in light of our find- pffiffiffi ð1 þ cÞ2Pl ings, these should instead be interpreted with formulae u x 0 3x l x l2 ; x l ; l=2 2ð 1Þ¼ t 3 ð 1ð 1Þ 0Þ 1 2½ 0 valid for bimodular, orthotropic elasticity. These are derived 2E1h in Appendix A, and summarized here: ð32Þ

For three-point bending of a beam of span l, thickness h so that the vertical displacement d at x1 ¼ l0, i.e. where and loaded by a force per unit depth P, the deflected the vertical force P is applied, is shape of the neutral axis is ffiffiffi p 2 2 2 pffiffiffi d ð1 þ cÞ l0ð3l 4l0Þ ð1 þ cÞ2Px ð3l2 4x2Þ 3Px ¼ : ð33Þ u x 1 1 1 ; x 0; l=2 ; P 8Et h3 2ð 1Þ¼ t 3 þ 1 2½ 1 16E1h 5G12h ð30Þ It is clear from Eqs. (31) and (33) that using three or t c (where c ¼ E1=E1, and c ¼ 1 corresponds to equal behav- four-point bending tests we can only identify two iour in tension and compression) so that the mid-span ver- parameters tical displacement d divided by the applied load P is pffiffiffi ffiffiffi 2 p 2 3 ð1 þ cÞ d ð1 þ cÞ l 3 l ; and G12; Et ¼ t þ : ð31Þ 1 P 16E1 h 10G12 h For four-point bending of a beam of span l, thickness h so that the full exploration of bimodularity and orthotropy requires use of at least one additional test (such as for in- and loaded at x1 ¼ l0 by a force per unit depth P (neglecting shear deformation), the deflected shape of stance a direct tension or compression test or, perhaps, the neutral axis is nanoindentation). K. Bertoldi et al. / Composites Science and Technology 68 (2008) 1363–1375 1373

both based on a sound micromechanical approach. We 1.2 E/E c = 0.6 1 p have also provided simple, closed-form formulae to deter- 1.0 0.8 1.0 mine two of our four macroscopic moduli directly from 0.8 macroscopic three and four-point bending tests on the 0.6 composite material. This provides a direct check of our 0.4 constitutive model, since its macroscopic moduli are com-

0.2 pletely determined from the constituents material moduli and the composite microstructure. The comparison with E/Et 0.2 0.4 0.6 0.8 1 1p available experimental data is excellent, and we expect that future experiments will further validate our model. Fig. 13. Relation between the Young’s moduli in the direction of the x1 axis obtained by incorporating, and without incorporating, bimodularity, t respectively E1 and E1, Eq. (34). Results are given for different values of Acknowledgements t c bimodularity c ¼ E1=E1, and the gray region indicates the parameter range of interest. Financial support of MURST-Cofin 2003 (prot. 2003082105_002) (KB) and MURST-Cofin 2005 (prot. Moreover, the effect of low shear modulus G12 only 2005085973_002) (DB) is gratefully acknowledged. WJD plays a role for sufficiently high values of h=l. The experi- acknowledges support from Italian Ministry Research ments performed by Jackson et al. [10] refer to very low Grant ‘‘Rientro dei cervelli MIUR 26/1/2001”. values of h=l, so that shear deformation should be negligible. However, bimodularity should play an important role Appendix A. Bending of an orthotropic, bimodular beam in those experiments, and, therefore, the Young’s modulus deformed in plane strain obtained by Jackson et al. must be corrected before comparing it to the prediction of our model. This correction Let us consider a bimodular beam in plane strain sub- can be obtained by considering Eq. (31) without shear jected to bending, for example, three-point bending as illus- deformation, with c ¼ 1, as used by Jackson et al., and trated in Fig. 14. Since only normal strains are involved for with c < 1, as found by us. The result is the relationship bending, bimodularity reduces to switching between two 4Et possible behaviours as a function of the sign of the strain E 1 ffiffiffi ; 1 ¼ p 2 ð34Þ and stress. It is, therefore, expedient to introduce a piece- ð1 þ cÞ wise nonlinearity for certain quantities. We denote, for where E1 is the Young’s modulus in the direction of the x1 instance, the elastic modulus in the direction parallel to e axis, assuming equal behaviour in tension and compres- the x1 axis by E1, meaning sion. It should be noted that there is no difference between e t c E1 ¼ E Hð11ÞþE Hð11Þ; ðA:1Þ the Young’s modulus in the direction of the x2 axis found 1 1 by us and by Jackson et al., since bimodularity does not where for conciseness bars over the symbols have been play a role in that case. t c dropped, so that E1 and E1 are the two composite plane Eq. (34) is plotted in Fig. 13, for the three values strain Young’s moduli in tension and compression, respec- c ¼f1; 0:8; 0:6g. The grey zone denotes the parameter tively, and H is the Heaviside step function. An analogous range of interest. We observe that the correction is use of variables with a superposed tilda will be made in the significant and that we find values of E1=Ep in the range following. 0.7–0.8, in agreement with the experimental data reported For plane strain bending, it is assumed that the resultant in Table 1. force in the direction of the x1 axis is null Z h=2 4. Conclusions e 11ðx2ÞE1 dx2 ¼ 0; ðA:2Þ h=2 Within the framework of the small-strain behaviour of nacre (mother-of-pearl), we have accounted for the first time for the facts that the material response is P orthotropic, with low shear stiffness; bimodular (lower Young’s modulus in tension than in compression). x 1 h

To characterize this material from the knowledge of the x2 microstructure, we have provided: l a simple analytical model, and a highly-accurate numerical analysis, Fig. 14. Bending of a bimodular beam. 1374 K. Bertoldi et al. / Composites Science and Technology 68 (2008) 1363–1375 pffiffiffi and that the applied bending moment per unit thickness M ð1 þ cÞ2Px ð3l2 4x2Þ u x 1 1 : A:12 is given by 2ð 1Þ¼ t 3 ð Þ Z 16E1h h=2 e M ¼ 11ðx2ÞE1x2 dx2; ðA:3Þ Since our material has a low shear modulus G12, it may h=2 be important to evaluate the contribution to the elastica of where h is the height of the beam. The strain component 11 shear deformability. To this purpose, we follow Jouraw- is assumed to vary linearly along the direction of the x2 ski’s analysis (which by the way originated to take into axis, so that from Eqs. (A.2) and (A.3) we obtain account the high compliance of wood under shearing, ffiffiffi p 2 [25]), thus finding for the mean shear stresses in a bimodu- 3M½hð1 cÞþ2x2ð1 þ cÞ 11ðx2Þ¼ t 3 ; ðA:4Þ lar (plane strain) beam 2E1h pffiffiffi pffiffiffi c Et Ec 3T 2 2 where ¼ 1= 1. Therefore, the neutral axis is defined by r12ðx2Þ¼ f4h c ½ð c 1Þh 2ð c þ 1Þx2 Hð11ðx2ÞÞ pffiffiffi 8h3c h c 1 pffiffiffi pffiffiffi x ¼ pffiffiffi ; ðA:5Þ cð c þ 1Þðh 2x2Þ½ð c 3Þh 2 2 c þ 1 pffiffiffi 2ð c þ 1Þx2Hð11ðx2ÞÞg; ðA:13Þ so that its position is independent of the applied moment. From the bimodular, two-dimensional stress–strain rela- where T is the shear force per unit thickness. tionship for an orthotropic material, we find Equating the strain energies in the beam calculated from external load and from internal stresses (or, in other words, r11 r11~m12 using Clapeyron’s theorem) we obtain for the shear 11 ¼ e ; and 22 ¼ e : ðA:6Þ E1 E1 deformability Integration provides the displacement field in the form @u 6T 2 ¼ ; ðA:14Þ @x 5hG u1ðx1; x2Þ¼11ðx2Þx1; 1 12 ffiffiffi p 2 so that (within the usual approximations) shear deforma- 3M zx2½hð1 cÞþx2ð1 þ cÞ e u2ðx1; x2Þ¼ m12 þ v0ðx1Þ; tion is unaffected by bimodularity. 2Et h3 1 Thus, accounting for shear deformation, integration of ðA:7Þ the elastica gives Eqs. (30) and (31). where 11ðx2Þ is given by Eq. (A.4). Since the shear strains must be zero, we find that References ffiffiffi p 2 2 3ð1 þ cÞ M z x1 [1] Almqvist N, Thomson NH, Smith BL, Stucky GD, Morse DE, v0ðx1Þ¼ þ v1ðx1Þ; ðA:8Þ Et h3 2 Hansma PK. Methods for fabricating and characterizing a new 1 generation of biomimetic materials. Mater Sci Eng C 1999;1:37–43. and, in order to have continuity of the displacement at the [2] Anthoine A. Derivation of the in-plane elastic characteristics of neutral axis, we get masonry through homogenization theory. Int J Solids Struct pffiffiffi 1995;32:137–63. 2 t c [3] Bruet BJF, Qi HJ, Boyce MC, Panas R, Tai K, Frick L, et al. 3M zð1 þ cÞ ðm12 m12Þ v1ðx1Þ¼ c Hð11ðx2ÞÞ: ðA:9Þ Nanoscale morphology and indentation of individual nacre tablets 8cE1h from the gastropod mollusc Trochus niloticus. J Mater Res The displacement components of the neutral axis are 2005;20:2400–19. [4] Barthelat F, Li CM, Comi C, Espinosa HD. Mechanical properties of given by nacre constituents and their impact on mechanical performance. J Mater Res 2006;21:1977–86. u1 ¼ 0 ffiffiffi ffiffiffi [5] Cecchi A, Sab K. A multi-parameter homogenization study for p 2 2 p 2 2 t 3M½4ð1 þ cÞ x1 þð1 þ cÞ h m12 modeling elastic masonry. Eur J Mech-A/Solids 2002;21:249–68. u2 ¼ ; ðA:10Þ 8Et h3 [6] Curnier A, He Q-C, Zysset P. Conewise linear elastic materials. J 1 Elasticity 1995;37:1–38. so that the elastica is given by [7] Currey JD. Mechanical properties of mother of pearls in tension. Proc pffiffiffi Royal Soc Lond 1977;196:443–63. @2u 3Mð1 þ cÞ2 [8] Currey JD. The mechanical properties of biological materials. In: 2 ¼ : ðA:11Þ @x2 t 3 34th Symposium of the society for experimental biology. Cambridge 1 E1h University Press; 1980. For a simply-supported beam of span l loaded by two [9] Evans AG, Suo Z, Wang RZ, Aksay IA, He MY, Hutchinson JW. P Model for the robust mechanical behavior of nacre. J Mater Res concentrated loads per unit thickness (plane strain 2001;16:2475–84. four-point bending), integration of the elastica yields [10] Jackson AP, Vincent JFV, Turner RM. The mechanical design of (neglecting shear deformation) Eq. (32) and thus Eq. (33). nacre. Proc Royal Soc Lond 1988;234:415–40. For a simply-supported beam of span l loaded by a cen- [11] Ja¨ger I, Fratzl P. Mineralized collagen fibrils: a mechanical model tral concentrated load P per unit thickness (plane strain with staggered arrangement of mineral particles. Biophys J 2000;79:1737–46. three-point bending), integration of the elastica yields [12] Ji B, Gao HJ. Mechanical properties of nanostructures of biological (neglecting shear deformation) material. J Mech Phys Solids 2004;52:1963–90. K. Bertoldi et al. / Composites Science and Technology 68 (2008) 1363–1375 1375

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