Universal composition–structure–property maps for natural ... · generality, we consider mno1, and thus 1ok

ARTICLE

Received 20 Sep 2014 | Accepted 3 Feb 2015 | Published 16 Mar 2015

Universal composition–structure–property maps fornatural and biomimetic platelet–matrix compositesand stacked heterostructuresNavid Sakhavand1 & Rouzbeh Shahsavari1,2,3

Many natural and biomimetic platelet–matrix composites—such as nacre, silk, and

clay-polymer—exhibit a remarkable balance of strength, toughness and/or stiffness, which

call for a universal measure to quantify this outstanding feature given the structure

and material characteristics of the constituents. Analogously, there is an urgent need to

quantify the mechanics of emerging electronic and photonic systems such as stacked

heterostructures. Here we report the development of a unified framework to construct

universal composition–structure–property diagrams that decode the interplay between

various geometries and inherent material features in both platelet–matrix composites and

stacked heterostructures. We study the effects of elastic and elastic-perfectly plastic

matrices, overlap offset ratio and the competing mechanisms of platelet versus matrix

failures. Validated by several 3D-printed specimens and a wide range of natural and synthetic

materials across scales, the proposed universally valid diagrams have important implications

for science-based engineering of numerous platelet–matrix composites and stacked

heterostructures.

DOI: 10.1038/ncomms7523

1 Department of Civil and Environmental Engineering, Rice University, Houston, Texas 77005, USA. 2 Department of Material Science and NanoEngineering,Rice University, Houston, Texas 77005, USA. 3 Smalley Institute for Nanoscale Science and Technology, Rice University, Houston, Texas 77005, USA.Correspondence and requests for materials should be addressed to R.S. (email: [email protected]).

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Natural materials such as nacre, spider silk, tendon and soon are primarily biocomposites that exhibit an excellentbalance of mechanical properties and multi-functionality

far beyond their constituents1,2. For instance, nacre is made of95% aragonite, a brittle mineral, but demonstrates a toughnessthat is B3,000 times larger than aragonite with the expense ofminor reduction in stiffness3. The key to success of thesebiocomposites is their nano/micrometre-size building blocks,which consist of hard inorganic platelets connected by a softorganic matrix in a brick-and-mortar pattern, known as platelet–matrix composites4,5. Numerous efforts are reported to decodethe complexities and synergies behind the interacting constituentsof natural and biological platelet–matrix composites with the goalof fabricating similar synthetic composites6–12. In this context,application of the well-known ‘shear-lag’ model13,14 has provento be essential in theoretically advancing the knowledge of loadtransfer in collagen fibrils15, nacre16 and nanotube-reinforcedpolymer composites17 to name a few. For instance, Zhang et al.18

studied mechanical properties of elastic platelet–matrixcomposites with different platelet distributions. With theassumption of uniform shear stress distribution, a criticalplatelet aspect ratio was reported that distinguishes the plateletversus matrix failures. Begley et al.19 lifted the assumption ofuniform shear stress and introduced different failure mechanismsfor composites with a plastic matrix and uniform distribution ofidentical platelets. More recent studies proposed characteristicoverlap lengths for elastic and plastic composites with uniformdistribution of identical platelets20.

Overall, given the material heterogeneity and structural varietyof numerous natural and biomimetic platelet–matrix composites,there is currently no unified understanding of the interplaybetween various geometers and inherent material characteristicsin controlling the balance of mechanical properties. The latter isessentially unexplored for composites made of dissimilar platelets.A prime example of such composites are stacked heterostructuresin emerging electronic and photonic devices that are made ofalternating two-dimensional (2D) sheets of nanometre sizeconnected by a network of weak van der Waals21–24 orcovalent bonds25 representing the soft matrix22 (Fig. 1b). Forexample, field-effect tunnelling transistors with high On/Offratios were recently fabricated by graphene (G) heterostructuressandwiching atomically thin hexagonal boron nitride (h-BN) ormolybdenum disulfide (MoS2) sheets acting as a vertical transportbarrier21. Similarly, h-BN/G/h-BN layers were fabricated aselectronic devices with high cutoff frequency24 and potentiallytunable and sizable bandgap via aligning G and h-BN in variousways26. Controlling the nanoscale mechanical properties ofstacked heterostructures is a fundamentally significant andchallenging topic, which remains largely unexplored27.

Here we facilitate a comprehensive understanding of themechanics of platelet–matrix composites and stacked hetero-structures in the shear mode via universal composition–structure–property maps. By considering both elastic andelastic-perfectly plastic matrices as well as platelets with offsetoverlap ratios, we identify a universal dimensionless characteristicquadruple, (lL, k, C, x), which governs the mechanical properties(stiffness, strength and toughness) of the platelet–matrixcomposites and stacked heterostructures consisting of bothsimilar and dissimilar platelets. Next, via mapping an extensivespectra of (lL, k, C, x) parameters into a set of 2D diagrams,we unify and infer important composition–structure–propertyrelationships in platelet–matrix composites. The universality ofthe these diagrams will be compared with a series of three-dimensional (3D)-printed specimens, and a wide range of existingnatural and biomimetic platelet–matrix composites as well asstacked heterostructures across several length scales and material

classes. Finally, we will study and compare the competingmechanisms between platelet and matrix failures as well as thefailure of platelet–matrix composites with stairwise and randomstaggering architectures.

ResultsElastic matrix with no overlap offset. The dominant deforma-tion mechanism of both platelet–matrix composite such as nacre(Fig. 1a) or a stacked heterostructure such as h-BN-graphene(Fig. 1b)23 can be represented by a 2D unit cell under shearloading as shown in Fig. 1c. In view of this common buildingblock, hereto we refer to ‘generalized platelet–matrix composite’as an archetype composite consisting of both conventionalplatelet–matrix composites and stacked heterostructures. First,we focus on elastic composites (C¼ 0) with varying plateletdissimilarity (k) and no overlap offset (x¼ 0.5). Then, the resultswill be extended to composites with elastic-perfectly plasticmatrices and platelets with varying overlap offset ratios (varyingC and x).

The 2D model of the generalized platelet–matrix composite ismade of two platelets with Young’s moduli, E1 and E2 and widthsof b1 and b2, and is subjected to stress s0 on the right end of thetop platelet. In addition, the system is clamped at the left end ofthe bottom platelet and traction free on all other dimensionsexcept the overlap length, L/2. The two platelets are connected bya thin film (matrix) with shear modulus G and thickness h. For aunit lateral dimension, the elastic stress distributions to thissystem are28

s1 xð Þ ¼ s0k2

sinh lkxð Þ� k2 � 1ð Þsinh lk L=2� xð Þð Þsinh lkL=2ð Þ þ k2� 1

h is2 xð Þ ¼ �ms0

k2sinh lkxð Þ� k2 � 1ð Þsinh lk L=2� xð Þð Þ

sinh lkL=2ð Þ � 1h i

t xð Þ ¼ lb1s0k

cosh lkxð Þþ k2 � 1ð Þ cosh lk L=2� xð Þð Þsinh lkL=2ð Þ

h i8>>><>>>:

ð1Þ

In equation (1), s1 and s2 are the axial stresses on the top andbottom platelets, respectively, t is the shear stress between the twoplatelets, m¼ b1/b2 and n¼E1/E2 are the ratios of the widths andYoung’s moduli of the two platelets, k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffimnþ 1p

is a unifiedparameter representing both the geometrical and elastic contrast ofthe two platelets and l ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiG= E1hb1ð Þ

pis a normalized parameter

denoting the interface properties (shear modulus and thickness).The above elastic solutions generalizes the common shear-lagproblem14,20,29 to non-identical platelets and dimensions.

In equation (1), lkL is a key dimensionless parameterencompassing all the structural and elastic material informationof the generalized platelet–matrix composites with similar ordissimilar platelets. This parameter can be deconvoluted into twodimensionless parameters, lL and k, which control, respectively,the (non)uniformity and (a)symmetry of the elastic stressdistributions (Fig. 1d). A small (large) lL corresponds to moreuniform (non-uniform) shear stress distribution in the matrixand linear (non-linear) axial stresses in the platelets (Fig. 2).For identical platelets (that is, mn¼ 1 and thus k ¼

ffiffiffi2p

),equation (1) represent symmetric stress distributions with respectto L/4. For dissimilar platelets, that is, k 6¼

ffiffiffi2p

, equation (1)demonstrate asymmetrical stress distributions (Fig. 1d and Fig. 2).However, for varying m and n ratios, as long as the product of mn(or k) is constant, all the shear and axial stress distributions(equation (1)) remain unchanged. This concept is similar to thenotion of ‘transformed area’ in homogenizing the compositebehaviour of materials where both width and Young’s moduluscan be used interchangeably (with proper factors)30. Together theextent of the aforementioned asymmetry and non-uniformity instress distributions govern key mechanical properties, such as

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strength, toughness and stiffness, of the platelet–matrixcomposite. This will be discussed in the next section.

Strength. To understand the strength behaviour of the unit cell inFig. 1c, we note that by increasing the applied stress, s0, the

average shear stress, tave ¼R L

20 t xð Þdx= 0:5 Lð Þ, increases to satisfy

the force equilibrium, s0b1¼ 0.5taveL, until the shear stressreaches the maximum shear strength, tf, assuming the plateletsare strong enough such that the composite fails in the shear mode(the case of platelet failures will be discussed later). When mno1

(for example, top platelet is softer than the bottom one), themaximum shear stress occurs at x¼ L/2 (Fig. 1d), and when mn41, the maximum shear stress occurs at x¼ 0. Without loss ofgenerality, we consider mno1, and thus 1ok �

ffiffiffi2p

throughoutthis work. In this case, tf reads

tf ¼ t x ¼ L=2ð Þ

¼ sult0 b1l

kcoth lkL=2ð Þþ k2� 1

� �csch lkL=2ð Þ

� �1ok �

ffiffiffi2p

;

ð2Þ

90%Tcmax

90% �cmax

�fShear stress

Axial stress

b1 E1,�1

b2 E2,�2

�

�1

�2

�

0

0

= 2

GraphenehBN

k

3.2 3.6 ∞

1.4

1.2

1.0Strong

1.4

1.2

1.00 3 6 9

Tough

k

k

Stiff

1.2

201.0

1.4

30 50

90% Tcmax

2

2

k

k = 1.01

k =

k = 1.22

k = 1.01

k = 1.01

k = 1.22

k =

�Lcr

90% Ecmax

90% Ec max

�Lcr

�L

�Lcr

Tcmax

90%Tcmax

�0

�0

m�0

x

∞

1.5

1

0.5

0

0.8

0.6

0.4

0.2

0

10

8

6

4

2

010–1 100 101 102

Electrode

Graphene

Drain

Source

Top electrode

Nor

mal

ized

stif

fnes

s (E

c)N

orm

aliz

ed to

ughn

ess

(Tc)

Nor

mal

ized

str

engt

h (�

c)h

L / 2

f

Figure 1 | Mechanical properties in elastic platelet–matrix composites. (a) Structure of nacre along with a schematic model (adapted with permission

from20. Copyright 2012 American Chemical Society). (b) A cartoon of a 2D-stacked heterostructures along (adapted with permission from ref. 22) with a

transmission electron microscope image of 2D alternating h-BN and graphene sheets (adapted with permission from ref. 23 Copyright 2012 American

Chemical Society). (c) A 2D unit cell of the composites in a,b made of two platelets connected by a matrix (blue). (d) typical asymmetrical and non-

uniform shear and axial stress distributions along the overlap length of the unit cell in (c). The top platelet is assumed to be softer than the bottom one.

(e–g), Normalized strength, �sc ¼ sc

�tflB

� �, toughness, �Tc ¼ Tc

��t2f

GhB

�, and stiffness, �Ec ¼ Ec

�E1

b1B

� �, of the unit cell in c as a function of lL and k.

To avoid singularity, k starts from 1.01 (e) the solid black circles denotes the points where �sc becomes 90% �smaxc for each k. The inset represents the

‘strong’ region in the space of (lL,k). Owing to strength normalization to l, this parameter (l) is constant in the horizontal axis and only the overlap length,

L, is the varying parameter. (f) The dashed and solid lines show the points of �Tmaxc and 90% �Tmax

c , respectively. The inset represents the ‘tough’ region in the

space of (lL,k). (g) The black line denotes the points where �Ec becomes 90% �Emaxc . The inset represents ‘stiff’ region based on �Ec � 90 % �Emax

c .

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms7523 ARTICLE




where sult0 is the ultimate axial stress that can be applied to the top

platelet preceding interface failure at tf. By rearrangingequation (2) and use of the relation, sc ¼ sult

0 b1�ðb1þ b2þ hÞ,

the effective strength of the unit cell on interface failure becomes

sc ¼tf

lBk

A Lð Þ ; ð3Þ

where A(L)¼ coth (lkL/2)þ (k2� 1)csch(lkL/2) and B¼ b1þb2þ h. Figure 1e shows the normalized dimensionless strength,�sc ¼ sc � lB

tf, of the unit cell for varying characteristic overlap

lengths, lL, and a range of platelet contrasts between twoextremes, that is, k¼ 1 (dissimilar platelets) and k ¼

ffiffiffi2p

(identical platelets).By increasing k from 1 to

ffiffiffi2p

(reducing platelet dissimilarity),normalized strength increases for large lL. This patternindicates that the use of identical platelets maximizes the strengthcapacity. Furthermore, for all values of k, the normalizedstrength asymptotically saturates with lL to the upper limit ofstrength for elastic composites under matrix failure, that is,�smax

Y ¼ limlL!1 �sc ¼ k.

Toughness. Toughness is a crucial property for functionality ofseveral natural and synthetic platelet–matrix composites7,11. It isdefined as the amount of energy per volume a material absorbsbefore it fails2. Although related, this definition is different than

the classical definition of ‘fracture toughness’ with the unit ofPa

ffiffiffiffimp

. While many toughening mechanisms in natural materialstake place during crack propagations and/or interlockingmechanisms (so-called extrinsic mechanisms31), here we focuson toughening mechanisms associated with the strain energydensity of the system, that is, intrinsic mechanisms. For a lineargeneralized platelet–matrix composite in Fig. 1c (linear elasticplatelets and elastic matrix) toughness reads

Tc ¼1

0:5BLhZ L=2

0

t xð Þ2

2Gdxþ b1

Z L=2

0

s1 xð Þ2

2E1dx

þ b2

Z L=2

0

s2 xð Þ2

2E2dx

!:

ð4Þ

By substituting equation (1) into equation (4), taking the integralsand using of equation (2) for sult

0 , we obtain

Tc ¼t2

f

GhB

D Lð ÞA Lð Þ2

; ð5Þ

where D(L)¼ [k4 coth (lkL/2)þ (k2� 1)(lkL/2� 2 tan h (lkL/4))]/(lkL). For any given 1ok �

ffiffiffi2p

, Fig. 1f demonstrates the variationof normalized toughness, �Tc ¼ Tc � GB

t2f h, as a function of lL. As

illustrated in this figure, for a limited range of k, 1:2247ok �ffiffiffi2p

,�Tc has optimum values at certain lL, with k ¼

ffiffiffi2p

(identicalplatelets) having the maximum �Tc (see dashed lines). For other

h

h h

L/2

Shear stress

Axial stress 0

0

L/2

Shear stress

Axial stress

�2�2

�1�1

�2

�2�2�2

�1

�1�1�1

� �

��

� � �

�

0

0

L /2

2k =√

�L =1 �L =10 �L =100

�L =1 �L =10 �L =100

Shear stress

Axial stress

0

0

h

L/2

Shear stress

Axial stress

0

0

h

L/2

Shear stress

Axial stress

0

0

b1

b2

h

E2′�2

E1′�1b1

b2

b1

b2 E2′�2

E1′�1

E2′�2

E1′�1

b1 E1′�1 �0 �0 �0

�0

�0�0

�0�0

�0

�0

�0 �0�0�0�0

m�0m�0m�0

b1 E1′�1 b1E1′�1

b1 E1′�2 b1 b1E1′�2 E1′�2

L/2

Shear stress

Axial stress

0

0

k =1.15

�f�f

�f

�f�f �f� � �

Figure 2 | Influence of kL and k on uniformity and symmetry of stress distributions. (a) lL increases from left to right in platelet–matrix composites

made of similar and (b) dissimilar platelets. From left to right, uniform shear stress changes to non-uniform stresses along the overlap length, and the linear

axial stresses to non-linear stresses. This observation is valid for both (a,b) cases, corresponding to composites of k ¼ffiffiffi2p� �

and dissimilar platelets

(k¼ 1.15). Analogously, by changing the dimensionless parameter k, fromffiffiffi2p

to 1.15, we notice how stresses change from symmetric to asymmetric

distributions along the overlap length. This observation is valid for all ranges of lL.





values (1oko1.2247), the maximum �Tc occurs at the limit oflL-0, which is not practical; hence for any finite lL the actualtoughness is smaller than the maximum capacity. Interestingly,k¼ 1.2247 exactly corresponds to mn¼ 0.5. Therefore, thegeneralized platelet–matrix composites whose platelets have astiffness ratio or a width ratio of more than 2, or the combinationthereof such that mno0.5, do not have an optimum toughness.Thus, such composites cannot exhibit a high toughness whencompared with those with 1:2247ok �

ffiffiffi2p

. Furthermore, whenlL-N, all �Tc values reduce to an asymptotic level of (k2� 1)/2,indicating that toughness becomes vanishingly small forcomposites made of completely dissimilar platelets, that is, k-1.

Stiffness. An effective Young’s modulus, Ec, representing thestiffness of the generalized platelet–matrix composite, can becalculated from Tc ¼ s2

c

�2Ecð Þ. Thus, in view of equations (3)

and (5), Ec becomes

Ec ¼ E1b1

Bk2

2Dð6Þ

Figure 1g shows the normalized stiffness, �Ec ¼ Ec � BE1b1

, versus lL,

which eventually saturates to �Emaxc ¼ limlL!1 �Ec ¼ k2

k2 � 1ð Þ. For

composites with continuous platelets, �Emaxc is equal to the average

stiffness of the platelets. Note that this figure may imply thatidentical platelets result in lower stiffness. However, this is due tonormalization to E1, which is considered to be the softer plateletin this study. Indeed, for a unit cell to offer its maximum stiffnesscapacity, both platelets must be as stiff as possible.

Experimental validations. To validate the above theoreticalelastic derivations, we performed tensile tests on a series of 3D-printed though-model specimens made of distinct platelets andmatrices. We fabricated twelve prototypes with a wide range ofoverlap lengths, width ratios and three distinct materials, that is,A, B and C. (see Fig. 3, Supplementary Figs 1–3, SupplementaryTable 1, and Supplementary Methods). To have a comparativeadvantage, all the specimens are grouped as (a,a0), (b,b0), (c,c0),(d,d0), (e,e0), (f,f0) where within each group lL is identical, but ktakes the value of either

ffiffiffi2p

for specimens with identical platelets,or 1.155 for specimens with dissimilar platelets symbolized by theprime notations. Figure 3b shows the measured stress–strain plotsof all specimens. Details of pre-failure and post failure of tworepresentative specimens made of similar and dissimilar plateletsare depicted in Fig. 3c,d. On straining, the as-fabricated unit cells(snapshot I), the platelets undergo elastic deformations, while theinterface undergoes large shear deformation (snapshot II), owingto the soft material phase of the matrix (Supplementary Movie 1).Snapshot III demonstrates the onset of failure, which starts formthe two extreme ends due to the larger shear stresses at thesepoints (see, for example, Fig. 1d). Finally, the large shear stressesrapidly cover the entire overlap length and the unit cell fails(snapshot IV).

Figure 3e–g shows the measured strength, toughness andstiffness, which are extracted from Fig. 3b, and are normalizedusing factors identical to those detailed above (SupplementaryTables 2 and 3). The general trends and magnitude of thesenormalized experimental values match very well with those oftheoretical predictions (dashed lines).

Elastic-perfectly plastic matrix with no overlap offset. In manynatural and synthetic platelet–matrix composites, the matrix ishighly ductile and can undergo large plastic deformation. Hereplasticity is introduced by considering an elastic-perfectly plasticmatrix with a dimensionless parameter C ¼ gf � gy

gy, representing

the degree of matrix plasticity where gy denotes the yielding strainand gf represents the final strain of the matrix precedingthe failure. While the detailed calculations for the modifiedstrength and toughness are provided in the Supplementary Note 4(stiffness is identical to that of elastic solution), in essence,the matrix plasticity and dissimilarity of the platelets results inthree distinct failure mechanisms depending on geometry andmaterial properties: case 1) failure occurs at the right side, that is,g(x¼ L/2)¼ gf, before the left side reaches the yielding shearstrain, g(x¼ 0)ogy; therefore, there exists one plastic zone at theright side and one elastic zone (the rest) at the left side, case 2)failure occurs at g(x¼ L/2)¼ gf and the left side enters the plasticregime but never fails before the right side, that is, gyog(x¼ 0)ogf, while an elastic zone is present at the middle. Therefore,there are two plastic zones at the two ends and one elastic zone inthe middle and case 3) failure occurs at g(x¼ L/2)¼ gf whenthe whole overlap length enters the plastic regime, that is,gyog(xoL/2)ogf. Note that in the latter case, although the shearstress in the full overlap length is constant and equal to tf, theshear strain g has a non-linear distribution across the overlaplength (Fig. 4).

Each case requires a distinct set of equations to calculate thelengths of the elastic and plastic regions to obtain the values ofstrength and toughness. However, regardless of the failuremechanism, it turns out that the behaviours of toughness andstrength versus lL are similar to the fully elastic case, except thatthe ultimate values are increased by introduction of plasticity(Supplementary Fig 4 and Fig. 5c–f).

Elastic-perfectly plastic matrix with overlap offset. So far, allour results were based on platelets of length. L with an equaloverlap length of L/2 on both sides, a so-called uniformlystaggered distribution. Other staggering patterns are alsoobserved in nature15. As a first step towards non-uniform plateletdistributions, here we introduce a dimensionless parameter0oxr0.5, denoting the overlap offset ratio between theplatelets. In this case, the platelet length of L is divided intotwo parts of length xL and (1–x)L as shown in Fig. 5a.Equation (7) shows the derived strength, toughness andstiffness for any given quadruple of lL,k,C and x (seeSupplementary Notes 5 and 6 for details):

sc ¼ tflB fs lL; k;C; xð Þ

Tc ¼ t2f

GhB fT lL; k;C; xð Þ

Ec ¼ E1b1B fE lL; k; xð Þ

8><>: ð7Þ

Here fs, fT and fE are the dimensionless functions representingnormalized strength, toughness and stiffness. Using x¼ 0.5,equation (7) degenerates to a system with uniformly staggereddistribution, and as x approaches zero the offset increases. Ouranalysis shows that for any triple of (lL, k, C), strength, toughnessand stiffness are higher or equal (depending on lL) in systemswith x¼ 0.5 than those with xo0.5, and they get lower as x-0(Fig. 5b–f). Our results corroborate on the findings of Zhanget al.18 (but further extend them to plasticity of the matrix,dissimilarity of the platelets and non-uniform shear distributionsin the overlap region (Supplementary Fig. 5). As shown in Fig. 5,stiffness changes minimally with varying x and is not a functionof C. Strength gets larger as the degree of plasticity increases andsaturates slower as x decreases. Figure 5e,f shows that toughnessis maximized for composites with uniformly staggered platelets(x¼ 0.5). However, as lL-N, stiffness, strength and toughnessof composites with varying x approach those of x¼ 0.5, in linewith common intuition that both xlL and (1–x)lL becomeinfinite as lL-N.





Universal composition–structure–property maps. Having fullycharacterized the elastic and elastic-perfectly plastic responses ofthe composite within a non-uniform staggering distribution ofplatelets, here we aim at developing a unified map decoding thenon-intuitive synergies between structures and materials thatconfer mutually inclusive or exclusive mechanical properties inthe generalized platelet–matrix composite. In view of Fig. 1e(C¼ 0 and x¼ 0.5), to effectively obtain the maximum strength

capacity of the unit cell for each specific k, we define the com-posite ‘strong’ if its characteristic overlap length, lL, is equal orlarger than a critical value, lLcr, at which the normalized strengthhas reached at least 90% of its saturation value (black dots inFig. 1e). This definition does not imply that composites withlLolLcr are not strong; instead, it means that they have not yetutilized their maximum theoretical capacities. The blue inset inFig. 1e clearly illustrates this definition in a 2D diagram for all

b

�

c

de

f

c ′d ′

e ′

f ′

b ′a ′ a

� (M

Pa)

III

III

IV

� (M

Pa)

3D-fabricated unit cells 60

50

40

30

20

10

00

8

00 0.05 0.1 0.15

0.01 0.02 0.03 0.15

Strength

Ductility

a

b

f

a ′

4.8

cm

30.48 cm

b ′

f ′

c ′

cde

d ′e ′

�

III 5 cm

IVIII

As-fabricated At elastic strain

Onset of failure After failure

As-fabricated

At elastic strain Onset of failure

After failure

I

II III

IV

k = √k = 1.15

2

2.0

1.5

1.0 b d e

a bc d

ef

b ′

a ′

a

c ′

d ′c e ′

a ′b ′

c ′d ′

e ′ f ′b ′

c ′

e ′d ′

f ′

f

c

bd

e

a ′a

f ′

f

0.5

0.0

0.8

0.6

0.4

0.2

0.0

5.0

3.8

2.5

1.3

0.00.1 1 10 0.1 1 100.1 1 10

λL λL λL

Nor

mal

ized

str

engt

h (�

c)

Nor

mal

ized

toug

hnes

s (T

c)

Nor

mal

ized

stif

fnes

s (E

c)

Figure 3 | Tensile experiments on 3D-fabricated specimens made of similar and dissimilar platelets. (a) Planar representation of 12 3D-printed

specimens. Specimens a–f are made of similar platelets k ¼ffiffiffi2p� �

while specimens a0–f0 are made of dissimilar platelets (k¼ 1.15). In each group, the

platelet–matrix materials are distinct. (b) The measured effective stress–strain plots of the specimens in a. The inset shows the zoom-in plots of the

specimens a and a0 , which exhibit significant ductility. (c,d) Typical snapshots of the pre-and post-failure during tensile experiments. The arrows indicate

the locations of maximum shear stresses. (c) Refers to the specimen b representing a unit cell with similar platelets and (d) refers to the specimen a0

representing a unit cell with dissimilar platelets. (e–g) normalized strength, toughness and stiffness of the specimens in a versus characteristic overlap

length lL. For comparison, theoretical predictions (dashed lines) are also given. The error bars are due to uncertainty in the Young modulus of the base

material (E1), which corresponding change l (horizontal error bars) and normalization factors (vertical error bars). The normalization factor of �Tc does not

depend on E1.





values of k and relevant lLcr. As k increases, the normalizedstrength requires a larger lLcr to reach its respective 90% of theupper limit. Hence, given the platelet elastic moduli, width ratios(k) and interface properties (l), this diagram identifies theminimum required platelet length to make a strong compositethat can exhibit at least 90% of its maximum theoretical strengthcapacity, �smax

c .Similarly, we consider the unit cell ‘tough’ for any given k if its

lL in Fig. 1f is within a critical range for which the normalizedtoughness is at least 90% of its maximum theoretical capacity,�Tmax

c . In Fig. 1f, the parameters �Tmaxc and 90% �Tmax

c arerepresented by dashed and solid lines, respectively, and the

yellow diagram in the inset demonstrates the ‘tough’ region for allvalues of k. Analogous to the definition of ‘strong’, we considerthe unit cell ‘stiff’ for any given k if its lL in Fig. 1g is equal orlarger than a critical value, lLcr, for which the normalizedstiffness is at least 90% of its maximum theoretical capacity, �Emax

c .Finally, by overlapping insets of Fig. 1e–g, and extending the

above analysis to various degrees of matrix plasticity (Ca0) andvarious platelet offset ratios (xa0.5), we construct a set ofuniversal diagrams representing normalized strength, toughnessand stiffness in the space of (lL, k) applicable to all natural andbiomimetic platelet–matrix composites and stacked heterostruc-tures (Fig. 6). The advantage of such diagrams is that one can

Elastic Plastic

�0�0 �0

�′′

Top Bottom

Fai

lure

Elastic Plastic

Top Bottom

Fai

lure

Pla

stic

Case 1 Case 2

�0b1 E1′σ1h

b2 E2′σ2

τ

c

a b=

−

=

τf

τfτf

τf

τf

τf�y

�y

�f �y�y

�f

1 C

C

G

Case 3

Plastic

Top Bottom

Fai

lure

L/2

m�0

m�0m�0m(�0–�′) m(�0–�′′) m(�0–�′) m(�0–�′)

L1 L1

L3L2

L3 L2L2

x x ′ x ′x

x ′xx ′′

�′�′

�′

L/2 L/2

d k = 1.4

k = 1.1

k = 1.2

Case 3

Case 2

Case 1

4C = 5

C = 5

C = 2

�L �L

�L �L

C = 2

3

2

1

010–1 100 101 102 10–1 100 100 102

10–1 100 101 102 10–1 100 101 102

8

10

6

4

2

0

3

4

2

1

0

4

5

3

2

1

0

e

k = 1.4

k = 1.1

k = 1.2

Case 3 Case 1

Case 2

f

Case 3

Case 2

Case 1

k = 1.4

k = 1.1

k = 1.2Case 1

Case 3

Case 2

g

k = 1.4

k = 1.1

k = 1.2

Nor

mal

ized

str

engt

h (�

c)

Nor

mal

ized

str

engt

h (�

c)N

orm

aliz

ed to

ughn

ess

(Tc)

Nor

mal

ized

toug

hnes

s (T

c)

Figure 4 | Failure mechanisms and mechanical properties for composites with plastic matrix. (a) Schematic view of a unit cell exhibiting tensile stress

on the platelets and shear stress in the matrix. (b) Plasticity in the matrix is considered using an elastic-perfectly plastic shear deformation model. The

parameter C represents the degree of plasticity, which is the ratio of the extra allowable strain of the matrix to the yielding strain. (c) Three different shear

failure mechanisms are possible: case 1 with one elastic and one plastic region, case 2 with two plastic regions and one elastic region in the middle and

case 3 where the full overall overlap length is in the plastic region. In all three cases, failure starts from the right end (free end of the stiffer platelet).

(d,e) Normalized strength, �sc ¼ sc

�tflB

� �, and (f,g) normalized toughness, �Tc ¼ Tc=

�t2f

GhB

�, for two degrees of matrix plasticity, C¼ 2 and C¼ 5. Three values

of dissimilarity ratios (k) are shown for each C. The colour of the plots in denote different failure mechanisms occur at different overlap lengths, lL.





map all the important structural, material and mechanicalinformation (both in terms of overlapping optimum propertiesand their relative values) in simple 2D plots and infer thefollowing:

Given a (lL, k) pair, a platelet–matrix composite may beinclusively/exclusively strong, tough (in the context of intrinsictoughening mechanisms) and/or stiff depending on the degree ofmatrix plasticity (C) and overlap offset ratio (x).

Generally speaking, compared with the effect of offset ratio (x),the increase in matrix plasticity (C) mainly accounts for increasedabsolute values of the strength and toughness with relatively smallchanges in the boundaries of overlapping region. On the otherhand, the increase in offset ratio (x-0)—for a constant matrixplasticity (C)—results in small alterations in absolutes values(compared with the effect of plasticity) but significant changes inthe boundaries of overlapping regions, particularly enablingoverlapping optimum properties for composites with dissimilarplatelets ko

ffiffiffi2p� �

.While strength and stiffness exhibit similar trend versus lL

(both saturate with lL), stiffness saturation happens at a muchslower pace than strength, especially for platelets with largestiffness and/or thickness contrasts (k-1). However, as x-0and/or C increases, the critical characteristic length (lLcr) of

strength moves to the right, thus increasing the chance of overlapbetween the optimally strong and stiff regions, or even strong,tough and stiff regions.

Contrary to stiffness and strength, toughness maximizes at afinite lL. Increasing matrix plasticity and/or offset ratio increasesthe characteristic length (lLcr) of toughness and also widens therange of this lLcr, thereby increasing the chance of finding anoverlapping region where strength, toughness and stiffness arebalanced.

The white areas in Fig. 6 are physically admissible but indicatethat the composite has not yet exhibited its optimum toughnessand/or strength capacities.

The quadruple lL, k, C and x are dimensionless universalcharacteristic parameters, making their absolute values and theirinterplays (borderlines) in Fig. 6 applicable to all generalizedplatelet–matrix composites made of similar and/or dissimilarplatelets. This is a critical advantage of these diagrams and thesignificant result of this paper, which can be used as a guidingtoolbox for knowledge-based synthesis of numerous biomimeticcomposites and stacked heterostructures.

The intensity of the colours in Fig. 6 represents the relativevalues of the properties. For overlapping regions (for example,tough and strong), the colour intensity is the arithmetic average

b1′E1

b2′E2

�0�� 1–�

�0�1

1–�

�21–�

(1 – �)L

� = 0.5

� = 0.5

� = 0.5

� = 0.1

� = 0.1

� = 0.1

� = 5 � = 5

�L

�L �L

h

5 k = 1.01

k = 1.01

k = 1.22

k = √2

k = 1.01

k = 1.22

k = √2

k = 1.01

k = 1.22

k = √2

k = √2

k = 1.01

k = √2

k = 1.01

k = √2

k = 1.01

k = √2

C = 8

C = 8C = 8

C = 8

C = 2

C = 2 C = 2

C = 2

C = 0

C = 0C = 0

C = 0

4

6Platelet failure

Matrix failure

Platelet failure

Matrix failure4

2

0

15

10

5

0 0

2

4

6

8

10

6

4

2

0

3

2

1

010–1 100 101 102

10–1 100 101 102 10–1 100 101 102

10–1 100 101 102 10–1 100 101 102

Nor

mal

ized

stif

fnes

s (E

c)

Nor

mal

ized

str

engt

h (�

c)N

orm

aliz

ed to

ughn

ess

(Tc)

Nor

mal

ized

toug

hnes

s (T

c)N

orm

aliz

ed s

tren

gth

(�c)

x x

��1

��2

a b

c d

e f

Figure 5 | Mechanical properties for composites with plastic matrix and overlap offset. (a) Schematic view of representative unit cell with non-uniform

overlap (offset ratio x). (b) Normalized stiffness, �Ec ¼ Ec

�E1

b1B

� �, is not a function of C and is plotted for two values of x¼0.5 and x¼0.1. (c,d), Normalized

strength, �sc ¼ sc

�tflB

� �, The dash lines correspond to j¼ 5 as an example defining the boundary of platelet and matrix failures. (e,f), Normalized

toughness, �Tc ¼ Tc=t2

fG

hB

� , are shown for varying lL, k and C. Each colour represents a specific C and x but varying k from 1 to

ffiffiffi2p

.





of the relative overlapping properties. Although approximated(that is, distinct mechanical properties cannot be added together),Fig. 6 qualitatively predicts the relative values of the optimummechanical properties in the space of (lL, k, C, x). To predict theabsolute value of the properties, one can take the optimum (lL, k)in Fig. 6 and plug them into Fig. 5 to read (interpolate) thenormalized values of the properties for a given (C, x) and thensimply obtain the absolute values by multiplying them to thenormalization factors. Alternatively, one can directly plug theoptimum (lL, k) into Supplementary equations (73), (75) and(77) to obtain the exact absolute values of stiffness, strength andtoughness, respectively.

In essence, the borderlines in diagrams of Fig. 6 representwhether the generalized platelet–matrix composite is able toachieve the maximum capacity for a certain property (or a set ofproperties) given the space of (lL, k, C, x), and the intensity of thecolours represents the relative normalized values of the mechan-ical properties. To better understand these features and test our

results, Fig. 7 shows the diagram with C¼ 0 and x¼ 0.5 that issuperimposed with the 12 (lL, k) pairs of our 3D-printedspecimens, validating our predicted theoretical capacities vis-a-visexperiments (Supplementary Table 2). For instance, specimen clL ¼ 7:49; k ¼

ffiffiffi2p� �

lies in the green region of Fig. 7 indicatingoptimum strength and toughness; in accord with the highestmeasured sexp

c ¼ 50 MPa and Texpc ¼ 602 KPa.

To demonstrate the universality of the composition–structure–property maps in Fig. 6, we overlaid the diagram with x¼ 0.5with several existing natural and biomimetic platelet–matrixcomposites as well as stacked heterostructures and let C go from 0to 2, representing fully elastic and 200% plastic matrix (Fig. 7).We used the uniform (x¼ 0.5), versus non-uniform, staggeringsince xa0.5 leads to lower mechanical properties (see Fig. 5),thus likely less common staggering pattern for a generaldescription of natural and biomimetic composites and stackedheterostructures. Then, by calculating the (lL, k) pairs for a widerange of materials with geometrical and material characteristics

0.5C

, Pla

stic

ity d

egre

e0.3 0.2 0.1

T

0 k

2

5

8

S

S

S

S

S

S

S

S

S

S

S

S

SF

SF

SF

SF

SF

SF

SF

SF

S

S

ST

ST

T

T

S

S

ST

ST

T

S

S

S

STF

F

F

STF

STF

STF

STF

STF

Tough (T)0 0.5 1

Strong, tough (ST)Strong, tough, stiff (STF)Strong (S)

Strong, stiff (SF)Stiff (F)

0 0.5 1 0 0.5 1

1.4

1.3

1.2

1.1

10 10 20 30 40 ∞

1.4

1.3

1.2

1.1

10 10 20 30 40 ∞

1.4

1.3

1.2

1.1

10 10 20 30 40 ∞

1.4

1.3

1.2

1.1

10 10 20 30 40 ∞

1.4

1.3

1.2

1.1

10 10 20 30 40 ∞

1.4

1.3

1.2

1.1

10 10 20 30 40 ∞

1.4

1.3

1.2

1.1

10 10 20 30 40 ∞

1.4

1.3

1.2

1.1

10 10 20 30 40 ∞

1.4

1.3

1.2

1.1

10 10 20 30 40 ∞

1.4

1.3

1.2

1.1

10 10 20 30 40 ∞

1.4

1.3

1.2

1.1

10 10 20 30 40 ∞

1.4

1.3

1.2

1.1

10 10 20 30 40 ∞

1.4

1.3

1.2

1.1

10 10 20 30 40 ∞

1.4

1.3

1.2

1.1

10 10 20 30 40 ∞

1.4

1.3

1.2

1.1

10 10 20 30 40 ∞

1.4

1.3

1.2

1.1

1 0 10 20 30 40 ∞

�, Overlap ratio

k k k

kk

�L �L �L �L

k k

kk k k

kk k k

Figure 6 | Universal composition–structure–property maps. The coloured regions correspond to optimum mechanical properties, which are

inclusively/exclusively strong (S), tough (T) and/or stiff (F) depending on the value of characteristic length (lL), platelet dissimilarity (k), matrix plasticity

(C) and platelet overlap offset ratio (x). For a given (k,C,x), the borderlines correspond to a critical lL, for which a property (for example, T) or a set of

properties (e.g., ST) is 90% of its maximum theoretical capacity. Given the values of the universal dimensionless quadruple (lL, k, C, x), one can readily

determine to what extent the material has attained or sanctified its mutually exclusive (inclusive) mechanical properties. The intensity of the colours

represents the relative value of the properties. To predict the absolute values of the properties, see text. Compared with x, the increase in C mainly accounts

for increased absolute values of the properties with relatively small changes in the boundaries. On the other hand, the increase in offset ratio (x-0) results

in small alterations in absolutes values (compared with the effect of C) but significant changes in the boundaries of overlapping regions, particularly

enabling overlapping optimum properties for composites with dissimilar platelets koffiffiffi2p� �

.





from atomic scale to microscale (Supplementary Table 1), weidentified their precise locations in Fig. 7. For example, nacre,consisting of platelets of a few hundred nanometres tomicrometre dimensions lL ¼ 4:6; k ¼

ffiffiffi2p� �

, and silk, consistingof platelets of nanometre range lL ¼ 9:3; k ¼

ffiffiffi2p� �

, lie in the‘strong, tough’ regions, consistent with the previous studies32,33.By mimicking nacre, nanoscale clay–PVA (polyvinyl alcohol)composites are reported to be ultrastrong and stiff9, consistentwith the ‘stiff and strong’ region of the diagram of Fig. 7lL ¼ 12:7; k ¼

ffiffiffi2p� �

.The universality of the map of Fig. 7 can be also verified with

composites made of dissimilar platelets. For instance, a bilayercement-PVA nanocomposite connected by a network of hydro-gen bonds has been shown to exhibit high toughness but at theexpense of lower strength than the maximum capacity28,consistent with the prediction of Fig. 7 (lL¼ 1.7, k¼ 1.251).Stacked heterostructures such as h-BN-graphene or graphene-MoS2 are another important class of composites, which arecomposed of dissimilar platelets with strong in-plane bondingnetworks but weak interplanar van der Waals bonding matrices.Although there are no experimentally measured values (to ourknowledge) on the composite strength, stiffness and toughnessvalues of such heterostructures, Fig. 7 predicts these compositestructures to be on the right extreme, that is, ‘stiff and strong’,

owing mainly to the high stiffness of the platelets and largeoverlap domain (lLc26). However, by reducing lL, for example,via lowering the overlap length, these multifunctional stackedheterostructures can even exhibit toughness, thereby preventingor hampering their typical brittle fracture.

Note that the solid borderlines of various optimum propertiesin Fig. 7 corresponds to elastic response (C¼ 0). By consideringthe plasticity of the matrix, these borderlines typically move to theright of the diagram. As an example, the dashed lines in Fig. 7corresponds to C¼ 2 (200% ratio of the plastic to elastic strain),which are easily distinguishable compared with C¼ 0. One canalso use this strategy to further include extrinsic and hierarchicaltoughening mechanisms, and interlocking phenomena in Fig. 7and quantify the relative values and locate the positions of themodified borderlines with respect to the intrinsic tougheningmechanisms.

Platelet and/or matrix failure. So far our analyses were based onmatrix failure. If the platelets are not strong enough, the com-posite might fail from the platelets before the matrix reaches themaximum shear stress. Let j1

lb1and j2

lb2represent the ratio of

the strength of the top and bottom platelets, respectively, to the

yielding strength of the matrix, tf, that is, s1ftf¼ j1

lb1and s2

ftf¼ j2

lb2.

Strong

Strong

&stiffTou

ghStro

ng& to

ugh

c ′ d ′ e ′ f ′b ′a ′

fedc

ab

Clay–PVA

Alumina–PMMA

1.4

1.3

1.2

k

1.1

10 10 20

�L

30 ∞

Spider silk

GraphiteNacreGraphene paper

Collagen

Cement–PVA

Alumina–chitosan

hBN–graphene

Graphene–MoS2

Silumin–alumina

MoS2-Ti

Graphene–3hBN

Graphene–MoS2

GMoS2

MoS2

MoS2

Graphene

Ti

Tough (T)Strong, tough (ST)

Strong (S)Strong, stiff (SF)

0 0.5 10 0.5 1

CB

Figure 7 | Overlay of composition–structure–property diagram with natural and biomimetic composites. The lines (for elastic composites with C¼0)

and dash lines (for plastic composites with C¼ 2) correspond to the critical lL, defining the boundaries of achieving 90% of strength (blue line),

toughness (orange and green lines) and stiffness (magenta line). These lines move to the right as plasticity increases. Intensity of the colours shows the

relative values of the properties for C¼0. To predict the absolute values of the properties, see text. The (k, lL) parameters are universal dimensionless

parameters, thus their absolute values are applicable to many real platelet–matrix composites and stacked heterostructures. This important feature is

evident from overlaying several natural and biomimetic platelet–matrix composites as well as stacked heterostructures onto the above diagram. This is a

significant result of this paper and can be used as a guiding toolbox for knowledge-based synthesis of numerous biomimetic composites as well as

emerging heterostructures. The letters corresponds to the 3D-printed specimens. The predicted locations of these specimens match with their

experimentally measured properties. The (lL, k) data of natural and synthetic platelet–matrix composites and stacked heterostructures are calculated from

the literature7,9,10,21–23,25,27,28,33,43–56. Snapshots of the natural and biomimetic composites are adapted with permission from ref. 20 Copyright 2012

American Chemical Society, adapted with permission from ref. 23 Copyright 2012 American Chemical Society and adapted with permission from refs

9,10,21,22,25,33,57,58. PMMA, poly(methyl methacrylate).





Then, under platelets and/or matrix failures, the effective strengthand toughness of the generalized platelet–matrix composite withan elastic-perfectly plastic matrix and offset overlap ratio can bewritten as

sc ¼tf

lBmin �S0;jf g

Tc ¼t2

f

GhB

xD xLð Þþ 1� xð ÞD 1� xð ÞLð Þð Þmin1

AðxLÞ2;j2

k2

�ð8Þ

In equation (8), j¼min{j1, j2} and �S0 is the normalizedcomposite strength under matrix failure including the effect ofmatrix plasticity. Figure 8 illustrates the behaviour of equation (8)versus lL for various values of j, and also the conditions at whichthe platelet and the matrix fail simultaneously. For example,considering strength, the following criteria clearly describes thematrix and/or platelets failure mechanisms (see also Fig. 5c,d)

�S0oj! matrix shear failure�S0 ¼ j! Simultaneous failure of the matrix and platelet�S04j! platelet failure

8<:

ð9ÞFor toughness, the situation is more complex depending on(lL, k, C, x). In essence, the ultimate value of strength (toughness)for varying lL is the minimum of strength (toughness) under thematrix and/or platelet failures. This is represented by red arrows(for the example of j¼ 1) in Fig. 8, which show the path oftoughness and strength versus lL considering platelet and/ormatrix failures. These analyses are further extended to include

platelet failures when considering stairwise and random stag-gering distribution of the platelets and our results indicate thatwhile the former may improve the composite properties, the lattertypically lead to lower mechanical properties (SupplementaryFigs 6–9). Nevertheless, a rough estimation based on sf¼ 0.1Eand tf¼ 0.1G and assuming a large plasticity of the matrix (thatis, C¼ 8) shows that in majority of the natural and syntheticplatelet–matrix composites and stacked heterostructures �S0oj,indicating that the matrix failure is dominant (SupplementaryTable 4). We also obtained the detailed stress–strain trajectories,demonstrating more insights into the transitions from linearelasticity to one- and/or two-sided plasticity, to full plasticity andeventually rupture (Fig. 9, Supplementary Figs 10 and 11).

DiscussionThe key novelty of this work lies in reducing the complexity andnon-intuitive synergies of the platelet–matrix systems to a few(four) universal dimensionless parameters and in providingguidelines as to how these universal characteristic parameterscan be modulated to optimize conflicting mechanical propertiesof such composites. From this perspective, the condensation ofour results in the form of ‘Universal Composition–Structure–Property Design maps’, containing only four characteristicparameters, is a significant first step in the field, which placesthe platelet–matrix composites in equal footing with nanocrystal-line metallic systems and alloys in the current applications ofstructure–property relationships to significantly alter the materialdesign landscape. Broadly, the innovative concepts, strategies andtechniques of this work (specially the notion of universal maps),although casted in the context of platelet–matrix architectures,have important implications for understanding and mimickingseveral other fascinating damage-tolerant natural composites,which are similarly made of hard and soft building blocksbut arranged in different architectures (for example, criss-crossdecussation in tooth enamel34, Bouligand arrangement in

� = 1.5

2a b

c d

1.5

1

0.5

0.3

0.2

0.1

0

0.8

0.6

0.4

0.2

0

0

2

1.5

1

0.5

010–1 100 101 10210–1 100 101 102

10–1 100 101 10210–1 100 101 102

� = 1.5

� = 1.22� = √

� = 1.5

� = 1

� =

� = 1� = 0.5

�Lcr

�Lcr

C = 0√2

√2

k =

√

� = 0.5C = 0

2k =

� = 0.1C = 0k = 1.1

Nor

mal

ized

str

engt

h (�

c)N

orm

aliz

ed to

ughn

ess

(Tc)

Nor

mal

ized

toug

hnes

s (T

c)N

orm

aliz

ed s

tren

gth

(�c)

� = 1

� = 0.1C = 0k = 1.1

� = 1

� = 1.1

� = 1.5

�L�L

�Lcr�=

1

�Lcr�

=1

�Lcr�

=1

�=

1.1

�Lcr

�Lcr�

=1

�=

1.1

�=

√2�L

cr�=

√2

Figure 8 | Mechanical properties considering both matrix and platelet

failures. (a,b) Normalized strength, sc

� tflB

� �, (c,d) Normalized toughness,

Tc

.t2

fG

hB

� for an elastic platelet–matrix composite. (a,c) correspond to

composites with identical platelets and no overlap offset. (b,d) represent an

example of a composite with dissimilar platelets and overlap offset ratio.

Toughness under platelet failure varies with lL,k,C and x. The coloured solid

lines (green for x¼0.5 and magenta for x¼0.1) show toughness and

strength based on matrix failure. The black dashed lines denote toughness

and strength under platelet failure scenarios for different ratios of platelet to

matrix strengths, represented by j. The blue dashed lines show lL at which

the platelet and the matrix fail simultaneously. The red arrows show an

example (for j¼ 1) path of ultimate toughness and strength versus lL,

which is equal to the minimum of toughness and strength values based on

both the matrix and platelet failures.

Nor

mal

ized

str

engt

h (�

c)

Initialyielding

Nor

mal

ized

str

engt

h (�

c)N

orm

aliz

ed s

tren

gth

(�c)

Nor

mal

ized

str

engt

h (�

c)

Normalized strain (�c) Normalized strain (�c)

Normalized strain (�c)

Initialyielding

Full plasticity(case 3)

Fullplasticity(case 3)

Partialtwo-sidedplasticity(case 2)Partial

two-sidedplasticity(case 2)

Partialone-sidedplasticity(case 1)

ElasticElastic

5k = 1.1C = 1

k = √2C = 1

k = √2C = 8

k = 1.1C = 8

�L =

10

λL =

10

�L =

10

λL =

5

�L =

10

�L =

5 �L = 5

�L = 5

4

3

2

1

00 1 2 3 4 5

5

4

3

2

1

00 1 2 3 4 5

5

4

3

2

1

00 1 2 3 4

Normalized strain (�c)

5

5

4

3

2

1

00 1 2 3 4 5

Figure 9 | Trajectories of stress–strains. (a–d) Stress �sc ¼ sc

�tflB

� �� –

strain �ec ¼ ec

�tfG lh� ��

plots of composites for different k, C and lL (in all

cases, x¼0.5). Each figure includes two plots for lL¼ 10 (top) and lL¼ 5

(bottom). Different colours show different stages of matrix deformation in

the composite: black (elastic), gold (partial one-sided plasticity, case 1),

blue (partial two-sided plasticity, case 2) and magenta (full plasticity,

case 3).





exoskeleton of anthropod2, helicoidal matrix of hammer-like dactylclubs of the stomatopods35 and so on). Furthermore, one canemploy the novel concept and strategies of this work to createuniversal design maps for other synthetic material classes such ascermets, metallic glasses, combinatorial and particulate oxides, boronnitride layered composites and emerging high entropy alloys36–42.

This work proposes a new dimensionless characteristicquadruple, (lL, k, C, x), to quantify the stiffness, strength andtoughness of platelets-matrix composites and stacked hetero-structures with an elastic and elastic-perfectly plastic matrices aswell as platelets with non-uniform staggering distributions. Giventhe value of (lL, k, C, x), one can readily determine to what extentthe composite has attained or sacrificed its mutually exclusive(inclusive) mechanical properties and also readily predict theabsolute values of the mechanical properties. By mapping anextensive range of lL, k, C, x values into a set of 2D diagrams, weconstructed a universal framework that offers a simple, yetpowerful tool to decode the non-intuitive synergies betweenstructures, materials and properties of generalized platelet–matrixcomposites. Validated by several 3D-printed specimens, and awide spectrum of existing natural and biomimetic composites aswell as stacked heterostructures across different scales andmaterial classes, the proposed diagrams provide a deep under-standing on how to intelligently utilize the maximum theoreticalcapacities for simultaneously improving strength, toughnessand/or stiffness, and to design new heterostructures andplatelet–matrix composites with contrasting platelet character-istics. This work opens up several new opportunities to furtherextend the proposed diagrams to include and quantify the effectof locking mechanisms43, stacked multi-heterostructures22,extrinsic and hierarchical toughening processes31 and so on andidentify new boundaries and overlaps in multi-phase multi-functional materials.

MethodsSynthesis. All specimens used in the study were printed at Stratasys, in Minne-sota, USA using an Objet Connex500 multi-material 3D printer. In a single print,we fabricated composites made of two base materials with strongly contrastingmaterial properties (Fig. 3a). All platelets were made of VeroWhitePlus, while thematrices were made of either TangoBlackPlus 27ShoreA (for specimens a,a0 ,b,b0) orTangoBlack 95ShoreA (for the rest). These materials are all proprietary acrylic-based photopolymer, and we will henceforth refer to them as Material A, MaterialB and Material C, respectively. In all specimens, the lighter phase represents thestiffer constituent and was printed with Material A, and the darker phase, repre-senting the compliant constituent, was printed with Material B or C. Figure 3aindicates the relative planar dimensions of the test specimens and SupplementaryTable 1 shows all the detailed structural dimensions. To print the composites, weused a dual material jetting technology allowing two distinct materials to be printedsimultaneously. Each material resides in cartridges and is funneled through a liquidsystem that is connected to the printing block consisting of eight printing heads.Two printing heads, each containing 96 nozzles with 50 mm diameters, are used forprinting each of the two base materials, while the remaining printing heads arereserved for printing a support material. An ultraviolet light follows the printingheads to immediately cure the printed materials, thus letting new layers to beprinted instantaneously. We printed 12 specimens covering a wide range of overlaplengths and width ratios as detailed in Supplementary Table 1. Owing to an in situcuring of the printed materials, the Materials A and B (and A and C) adhere toeach other perfectly, that is, no interfacial debonding or sliding on loading, con-sistent with our theory and confirmed by the experiments (Supplementary Movie1). To avoid undesirable bending, and ensure pure shear deformations in thematrix and axial deformations in the platelets, we printed symmetric specimens bymirroring the unit cells as shown in Supplementary Fig. 3. In view of the definitionof k ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffimnþ 1p

, the platelet contrast can be introduced via m or n or both. Owingto the current limitation of the dual material jetting technology, which can printonly two materials at a time, we introduced platelet contrast via varying the widthratios of the platelets (m) while using identical material (Material A) for allplatelets, that is, constant n (Supplementary Table 1).

Tensile experiments. For optimal griping of the specimens in the tensile testingapparatus, we attached (printed) two small uniform strips, made of Material A, tothe two ends of the specimens (Supplementary Fig. 3). We tested the 3D-printedspecimens in an Instron 4500 Universal Testing Machine with a 100-kN static load

cell and applied displacement boundary conditions until failure. The load capacityof the grips is 100 kN and the spring stiffness of the entire testing device far exceedsthe stiffness of our specimens. We attached our specimens firmly in the grips andcentre the middle platelet of the geometries with the force applied through the viceaction grips to ensure pure shear in the matrix and axial deformations in theplatelets. We did this to emulate the boundary conditions applied in our theoreticalderivations. In the tensile test, we employed a displacement rate of 2.5 mm min� 1.

Base materials. Selections of various Objet printable materials with different bulkmechanical properties are already available. The referenced data of Stratasysindicates that of the materials available for printing, the three employed in thisstudy represent a strong contrasting mechanical behaviour: Materials A being thehardest, Material B being the softest and Material C being slightly stiffer thatMaterial B. However, there is no previous data on the shear modulus, G, of thesematerials or the interfacial shear strength, tf, between them. To obtain G and tf forthe platelet–matrix system of A–B, we fabricated three identical test specimenssimilar to the specimen ‘a’ but with a short overlap length (3 cm) to have (almost)uniform shear along the overlap length (Fig. 2). Next via straining the specimenswith the same procedure as detailed above, we obtained the force–displacement(F� d) plots. Then, the shear modulus of Material B and interfacial shear strengthof the A–B interface were approximated by G ¼ t=g ¼ F

0:5 Lt

� ��ðd=hÞ ¼ 1:32�

0:05 MPa and tf ¼ Fmax0:5Lt

� �¼ 2:51 � 0:07 MPa, respectively. Using a similar pro-

cedure, we found G¼ 6.73±0.28 MPa and tf¼ 6.93±0.3 MPa for the Material Cand the A–C interface, respectively. The Young modulus of Material A (E1) isreported to be 2.50±0.5 GPa from the manufacturer and was used in this study.The modulus of the most compliant material, B, is three orders of magnitude lowerthan the stiffer material, A, by a factor of E 1900. In view of this large factor,we only considered the uncertainty in the E1 as the source of error bars.

See Supplementary Note 1 for analysis of elastic deformations,Supplementary Note 2 for details on composite stiffness, SupplementaryTable 1, Supplementary Note 3 and Supplementary Figs 1–3 for experimentaldetails, Supplementary Note 4 for analysis of plastic deformations, SupplementaryNote 5 and 6 for studying the effects of overlap offset, Supplementary Note 7 forstudying platelet versus matrix failures, Supplementary Note 8 and 9 for analysis ofstairwise and random staggering, Supplementary Note 10 for derivation of stress–strain plots, Supplementary Note 11 and Supplementary Fig. 12 for analysis ofvolume fraction.

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AcknowledgementsThis work was supported in part by the Department of Civil and EnvironmentalEngineering at Rice University and in part by the National Science Foundation grantnumber 1235522 and 1346506.

Author contributionsR.S. designed the research; N.S and R.S performed the theoretical and experimentalresearch; N.S. analysed the data; and N.S. and R.S. wrote the paper.

Additional informationSupplementary Information accompanies this paper at http://www.nature.com/naturecommunications

Competing financial interests: The authors declare no competing financial interests.

Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/

How to cite this article: Sakhavand, N. and Shahsavari, R. Universal composition–structure–property maps for natural and biomimetic platelet–matrix composites andstacked heterostructures. Nat. Commun. 6:6523 doi: 10.1038/ncomms7523 (2015).






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Universal composition–structure–property maps for natural ... · generality, we consider mno1, and thus 1ok