PHYSICAL REVIEW E 85, 031901 (2012)

Bioinspired, mechanical, deterministic fractal model for hierarchical suture joints

Yaning Li,1,2 Christine Ortiz,1 and Mary C. Boyce2,*

1Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA2Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

(Received 3 September 2011; revised manuscript received 7 December 2011; published 1 March 2012)

Many biological systems possess hierarchical and fractal-like interfaces and joint structures that bear andtransmit loads, absorb energy, and accommodate growth, respiration, and/or locomotion. In this paper, anelastic deterministic fractal composite mechanical model was formulated to quantitatively investigate the roleof structural hierarchy on the stiffness, strength, and failure of suture joints. From this model, it was revealedthat the number of hierarchies (N ) can be used to tailor and to amplify mechanical properties nonlinearlyand with high sensitivity over a wide range of values (orders of magnitude) for a given volume and weight.Additionally, increasing hierarchy was found to result in mechanical interlocking of higher-order teeth, whichcreates additional load resistance capability, thereby preventing catastrophic failure in major teeth and providingflaw tolerance. Hence, this paper shows that the diversity of hierarchical and fractal-like interfaces and jointsfound in nature have definitive functional consequences and is an effective geometric-structural strategy to achievedifferent properties with limited material options in nature when other structural geometries and parameters arebiologically challenging or inaccessible. This paper also indicates the use of hierarchy as a design strategy toincrease design space and provides predictive capabilities to guide the mechanical design of synthetic flaw-tolerantbioinspired interfaces and joints.

DOI: 10.1103/PhysRevE.85.031901 PACS number(s): 87.10.Pq, 87.10.Kn, 68.35.Ct, 62.20.mm

I. INTRODUCTION

Hierarchical systems, which possess structure at multiplelength scales as well as fractal-like patterns where thehierarchy consists of an identical repeating geometry, are foundthroughout biology [1–4]. Nature employs fractal-like designsin many cases to serve a mechanical function. For example,gecko setae provide direction-dependent adhesion [5,6], spidercapture silk shows exceptional strength and elasticity [7,8],the fractal dimension of trabecular bone microarchitecture isfound to govern stiffness and strength [9,10], hard biologicaltissues provide flaw tolerance [4,11], and the cranial sutureof vertebrates with fractal-like morphology helps to distributeloads and to absorb energy to resist impact loading [12,13].Recently, fractals have been used to optimize mechanicaldesigns and to improve the mechanical efficiency of platesand shells [14,15].

In this paper, we focus on geometrically structured inter-faces called suture joints, which are composite mechanicalstructures that typically possess two interdigitating compo-nents (the teeth) joined by a thin more compliant interfaciallayer (the seam). They are found throughout nature and allowsegmentation of monolithic shells to bear and to transmit loads,to absorb energy, and to provide flexibility to accommodategrowth, respiration, locomotion, and/or predatory protection[12,13,16–19]. Two types of suture seam morphologies in-clude: (1) a single (first-order) repeating wave found, forexample, in the pelvic assembly of Gasterosteus aculeatus(the three-spined stickleback) [20] and in the cranial suturesof some vertebrates [12]; and (2) a hierarchical (higher-order)multiple wavelength pattern found, for example, in the craniumof the white tailed deer [21] and in the shells of ammonites[22–25]. Fractal analysis of the geometry of cranial and

*mcboyce@mit.edu

ammonite sutures is reported in the literature [26–31] toquantify their structural complexity. Additionally, we recentlydeveloped analytical and numerical models of first-ordertriangular and rectangular suture joints to assess the influenceof geometry on stiffness, strength, and failure mechanisms[32]. Building on this paper, here, we explore the role ofhierarchical design on the underlying fundamental mechanicsof suture joints using analytical and numerical approaches,which is a fascinating topic that is largely unexplored.

A two-dimensional composite deterministic self-similarfractal suture joint model is formulated to investigate therole of hierarchical geometry in tailoring stiffness, strength,and failure. This mechanical model considers the skeletalteeth material to exhibit rigid or, alternatively, to exhibitisotropic elastic mechanical behavior. The more compliantinterface material is taken as isotropic and elastic and isassumed to be bonded perfectly to the teeth. The elasticregime studied here is expected to be relevant biologically fortypical physiological loading conditions. Strength, toughness,and failure of the composite suture structures are assessedby assuming critical stress failure criteria for both the toothand the interface materials. The model captures the criticalmaterial and geometric features of many suture joints foundin nature as well as provides predictive capabilities to assistwith the tailored mechanical design of synthetic flaw-tolerantbioinspired suture joints.

This paper is organized as follows: in Sec. II, a hierarchicalsuture seam model is defined in a deterministic manner fora triangular-wave form; in Sec. III, under the assumptionof rigid teeth, the influence of the number of hierarchiesN and geometry on load transmission and suture stiffnessis explored quantitatively, which provides insights into theload transmission mechanisms and an upper bound for thestiffness; in Secs. IV and V, the assumption of tooth rigidityis relaxed, and the influence of the ratio of the elastic moduliof the teeth and the seam on the effective stiffness of the

031901-11539-3755/2012/85(3)/031901(14) ©2012 American Physical Society

YANING LI, CHRISTINE ORTIZ, AND MARY C. BOYCE PHYSICAL REVIEW E 85, 031901 (2012)

suture joints is evaluated; in Sec. VI, the failure mechanisms,strength, and fracture toughness of hierarchical suture jointsare studied. Finally, the benefits and potential applications ofthe hierarchical design are discussed, providing guidelines forthe development of potential bioinspired synthetic systems.

II. GEOMETRY OF A BIOINSPIRED DETERMINISTICFRACTAL MODEL FOR HIERARCHICAL

SUTURE JOINTS

While the fractal model developed here is generic and canbe applied broadly to hierarchical suture joints, it is particularlyrelevant to the ammonite shell septal suture, which forms thejunction between the septa of buoyancy chambers and theexternal wall of the phragmocone [22–25]. Ammonites, whichare extinct marine invertebrates in the subclass Ammonoideaof the class Cephalopoda, have hierarchical sutures, whichhave long been recognized as one of the most complexand rapidly evolving structures in the fossil record, showingconsistently increased suture complexity during evolution[25]. Representative hierarchical suture seams, best observedby polishing the outer layers of the shell, are shown inFig. 1(a).

In this paper, the hierarchical suture seam is modeled ina deterministic manner by iteratively superimposing a self-similar wave on each wave form of the former profile as shownin Fig. 1(b). By defining the center line of a straight interfacelayer as the zeroth-order suture line (L0), an N th-order sutureseam can be generated via successive superposition of N waveforms. Thus, N is also the total number of hierarchies ororder of the suture joint. Here, we focus on suture seamscharacterized by triangular-wave forms, given their presencein many natural systems [32]. Schematics of typical RVEs ofhierarchical suture joints of different order N are shown inFig. 1(c).

To illustrate the geometric features of the RVEs of thehierarchical suture joints, the hierarchy map of the RVE of athird-order suture joint is shown in Fig. 2(a) as an example andthen is generalized to that of an N th-order joint in Fig. 2(b). Thenomenclature for a general N th-order suture takes hierarchylevel m = 1 to correspond to the longest wavelength λ1 andlargest amplitude A1 of the hierarchy [corresponding to L1 ofFig. 1(b)] with each successive level possessing a wavelengthand an amplitude that decreases in a self-similar manner (asmathematically specified later). Figure 2(a) shows that theeffective interface layer of the third-order RVE at hierarchicallevel [m = 1] is a second-order suture joint at a smaller scale,and the effective interface layer of this second suture joint is afirst-order suture joint. Generalizing, the hierarchy map of anN th-order suture joint is abstracted in Fig. 2(b). It can be seenthat the geometry of an N th-order suture joint is determinedby the total number of hierarchies N , the wave amplitudes Am

(m = 1 . . . N ), wavelengths λm, and the width of the effectiveinterface layer1 gm at each level of hierarchy m.

1Note gm is taken along the propagation direction of the wave lineat the mth hierarchy. gm equals the thickness of the effective interfacelayer at the mth hierarchy divided by cos θ .

(a)

2cm 1cm 1cm

(b)

(c)

N=3 …N=1 N=2

L1L2L3

L0

222

N=3N=1 N=2

FIG. 1. (Color online) A bioinspired model of a self-similarhierarchical suture seam: (a) suture lines of ammonite fossils showingdifferent levels of complexity, from left to right: Goniatitida [44],Craspedites nodiger [45], and Ammonoidea and (b) schematic ofthe hierarchical suture seam model; Lm represents the suture seam(interface) contour length at each level of hierarchy m = 1 . . . N ,where N represents the total number of hierarchies or order ofthe suture joint. The dotted black square represents a representativevolume element (RVE), and (c) RVEs of first-, second-, and third-order self-similar triangular suture joints (lighter gray and darker graycolors are used to distinguish the top and bottom rows of teeth; theinterfacial lines, green dash-dot-dot line, blue dashed line, and redsolid line, represent the first-, second-, and third-order suture lines,respectively).

To describe the relationships of these geometric parametersat different hierarchical levels, two nondimensional parame-ters, the suture complexity index [25] IN and the amplitudescaling factor αm (m = 1 . . . N − 1), are defined as

IN = LN

L0= LN

LN−1· · · Lm

Lm−1· · · L1

L0= iN ···im · · · i1 =

N∏1

im,

(1)

αm = Am+1

Am

(m = 1 · · · N − 1), (2)

where Lm is the contour length of an mth-order hierarchicalseam and im (m = 1 . . . N) is the incremental interdigitationindex for the mth superposition, defined by im = Lm

Lm−1. Hence,

L0 is the length of the zeroth-order suture line [see Fig. 1(b)],and LN is the total contour length of the suture seam [forexample, the L2 (when N = 2) and L3 (when N = 3) shownin Fig. 1(b)].

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BIOINSPIRED, MECHANICAL, DETERMINISTIC . . . PHYSICAL REVIEW E 85, 031901 (2012)

[m] [m+1]

(b)

…

The N th order suture joint

[1]

(a)

[2] [3]

The 3rd order suture joint: N=3

1A

2A

22g1g

1

3A

3g 3

[N]

…

[m] [m+1]

(b)

…

The N tNN h order suture joint

[1]

(a)

[2][2] [3]

The 3rd order suture joint: N=3

1A

2AA

22g1g

1

3A

3g 3

[N]NN

… mg

m

1+m

1+mA

1+mg

mA

[1] …

1g

1A NA

Ng N…

1

FIG. 2. (Color online) Hierarchy map of the RVEs of (a) a third-order suture joint and (b) an N th-order suture joint (the dotted interfacialregions represent the effective interfacial layer at different levels of hierarchy).

Due to our imposition of self-similarity, im and αm areconstants. By taking im = i and αm = α, three independentnondimensional parameters i, α, and N determine the ge-ometry of the suture seam. The suture complexity index isderived from Eq. (1) as IN = iN . For a triangular-wave sutureseam, i is related to the tip angle of the triangular tooth θ

via i = 1/ sin θ. From Eqs. (1) and (2), using the fractalbox-counting method [28] and taking the wavelength of an mthlevel suture line as a unit measure em, em = λm = λ1α

m−1, thefractal dimension of a suture seam is derived as

D = limm→∞ − log(Jm)

log(em)= 1 − log(i)

log(α)= 1 + log(sin θ )

log(α), (3)

where Jm is the number of em units over the entire length ofthe suture line Jm = Lm−1

em, where Lm = L0i

m.The log-log plot of em vs Jm, Fig. 3(a) (left), shows that the

self-similar triangular hierarchical suture seam has a constantfractal dimension D independent of N and, therefore, is adeterministic fractal. D is a function of θ and α with a valuebetween 1 and 2. D is larger when θ is smaller or α is larger asshown in Eq. (3) and its corresponding plot, Fig. 3(a) (right).A schematic of the box-counting method is shown in Fig. 3(b).The values of D of a large variety of ammonite suture seamshave been shown to lie in the range of ∼1.2–1.8 [33]. For some

physical systems, a deterministic fractal model has been shownto be a good approximation for the random fractal system withthe same fractal dimension [34].

Due to similarity, the wave number nw for each straightsegment is a constant at each hierarchy, related to θ and α

by nw = (2α sin θ )−1. By using Eq. (3), we obtain nw =0.5α−D = 0.5(sin θ )−D/(D−1). Examples of hierarchical sutureseams with the same nw = 2.5 and different θ and, therefore,different D are shown in Fig. 3(c).

In order to quantify the effective mechanical propertiesof hierarchical suture joints, the volume fraction of thecomponents in the composite interface region must be takeninto account. When the total volume fraction of the interfaceis the same for suture joints with different orders N , the lengthof the interface increases, hence, the width of the interfacelayer, at each level of the hierarchy gm, decreases and followsa power law relationship:gm = g1i

m−1 [Fig. 2(b)].As shown in Fig. 4(a), fundamental building blocks (i.e.,

the repeating geometry unit) of a self-similar hierarchicalsuture joint are RVEs of a first-order suture joint. Thesebuilding blocks are self-similar and are assembled through N

hierarchies to construct an N th-order hierarchical suture joint.By defining the tooth volume fraction of a single building blockas f (N = 1), the total tooth volume fraction of an N th-orderhierarchical suture joint fv (including minor teeth at all levels

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YANING LI, CHRISTINE ORTIZ, AND MARY C. BOYCE PHYSICAL REVIEW E 85, 031901 (2012)

(a)

(b)

(c)

FIG. 3. (Color online) (a) Fractal dimension relationships of ahierarchical suture seam; (b) schematics of the box-counting method;(c) examples of self-similar suture seams with the same wave numberat each level of hierarchy nw = 2.5 but different tooth angle θ andfractal dimension D.

of hierarchy) is related to f as

fv = 1 − (1 − f )N . (4)

Equation (4) is plotted in Fig. 4(b) where fv asymptoticallyapproaches 1 when N tends to infinity.

Thus, by taking the volume fraction into consideration, thescaling factor α and the suture complexity index i are related tof and θ via i = f

sin θand α = (f −1 − 1) sin θ , and the suture

width gm and wavelength λm at each level m are related tof by f = 1 − 2gm

λm. Hence, the mechanical model of the N th-

order hierarchical suture has two independent nondimensionalgeometric parameters: f and θ (or D).

In order to construct a hierarchical suture joint, thegeometry of the building block needs to satisfy a condition thatthe interface of a building block must be able to accommodateat least one building unit in the successive hierarchy level.Thus, the wavelength of the (m + 1)th level interface mustbe less than the length of the slant edge of the mth levelteeth, i.e., λm+1 � λmf

2 sin θ(i.e. , nw � 1). Therefore, the two

independent geometric parameters f and θ need to satisfy

θ � θ∗ = sin−1

(f

2√

1 − f

). (5)

Equation (5) is plotted in Fig. 4(c) in which it showsthat, due to this geometric constraint, self-similar hierarchicalsuture joints with tip angle θ > θ∗ = sin−1( f

2√

1−f) do not

exist [Fig. 4(c), area 1]. For relatively smaller θ or larger f ,the constraint is relaxed. Furthermore, when f > f ∗ = 0.732(i.e., f ∗

2√

1−f ∗ = 1), the tip angle has no geometric constraint

(a)

(b) (c)

Region

Region 2

Reg

ion

3

FIG. 4. (Color online) Geometry of the mechanical self-similarhierarchical suture joint model, (a) schematics of the configuration ofa self-similar N th-order hierarchical triangular suture joint model,showing the relation between a building block (right hand side)and an overall structure [blue (upper) represents the top row ofteeth; green (lower) represents the bottom row of teeth; and pinkrepresents the interface layer], (b) the total volume fraction fv ofself-similar hierarchical suture joints as a function of the volumefraction of a building unit f and the number of hierarchies N , and(c) parametric space of the geometric constraint (the maximum tipangle 2θ∗ changing with f ) of this model (region 1: no hierarchicalsuture joint exists, region 2: geometric constraint, and region 3: noconstraint).

[Fig. 4(c), area 3] and can be any arbitrary value between 0◦and 180◦.

III. THE RIGID TOOTH COMPOSITE MODEL

Tensile loading within the suture plane and normal tothe suture axis is a prevalent physiologic loading condition[13,18,35] [Fig. 1(b)]. In this section, we determine the tensilestiffness of the self-similar hierarchical suture joint model forthe case of hierarchical sutures where the teeth are assumedto be rigid and only the interfacial material is deformable,which provides an upper bound for the suture stiffness. Underthis tensile load, tensile stress is transmitted from the majorteeth (i.e., the largest teeth or the first-order teeth) to minorteeth (i.e., smaller teeth in all other levels of hierarchy or thehigher-order teeth) and interface through tension and shear.Therefore, in this section, first, the tensile and shear stiffnessof the first-order suture joint is determined; and then, theload transmission mechanism is analyzed; finally, the tensilestiffness of an N th-order hierarchical rigid tooth model (RTM)is determined.

A. First-order suture joint model under tension and shear

Tension. For a first-order suture joint under tension perpen-dicular to the suture axis [Fig. 1(b)], the tension σ is balancedby interfacial shear stress τT and normal stress σT as shown inthe free body diagram of Fig. 5(a).

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BIOINSPIRED, MECHANICAL, DETERMINISTIC . . . PHYSICAL REVIEW E 85, 031901 (2012)

τSτ

SσSτ

Sσ

τ

2θ g

τ

Rigid tooth model (RTM): Shear

(b)

δSγ

gθ

(a)

2θ g

Rigid tooth model (RTM): Tensionσ

TτTσ

TτTσ

x

y

FIG. 5. (Color online) Kinematics and free body diagrams of the first-order RTM under (a) tension and (b) shear [lighter gray and darkergray colors are used to distinguish the top and bottom rows of teeth, pink (right interfacial layer), and light green (left interfacial layer) representthe interface layers experience tension and compression, respectively].

Force equilibrium, in the x and y directions, on the freebody diagrams of a tooth [Fig. 5(a)] yields [32]

σT = τT tan θ = σ sin2 θ, τT = σ sin θ cos θ. (6)

The strain energy URTMT of a RVE of the system due to the

applied tension can be expressed as

URTMT = 1

2

(f σ )2

ERTM

1

V = 1

2

(σ 2

T

EPS0

+ τ 2T

G0

)V0, (7)

whereERTM

1 is the effective tensile modulus of the first-ordersuture joint; EPS

0 and G0 are the plane strain modulus2 andshear modulus of the interface material, respectively; V is thetotal volume of the RVE; V0 is the volume of the interface mate-rial in the RVE, and V0

V= 1 − f (for first-order suture joints,

fv = f ). By substituting Eq. (6) into Eq. (7), the effectivetensile stiffness of the first-order suture joint is obtained

ERTM

1 = f 2

1 − f

(1

G0sin2 θ cos2 θ + 1

EPS0

sin4 θ

)−1

. (8)

From Eq. (8), we can see that, when 2θ < 90◦, G0

dominates, whereas, for 2θ > 90◦, EPS0 dominates.

Shear. When a first-order RTM suture joint is subjected tosimple shear, the applied shear stress at the tooth edges τ isbalanced by a uniform interfacial shear stress τS and tensileand compressive normal stress σS as shown in Fig. 5(b). Theteeth on the top and bottom rows experience a relatively rigidbody displacement δ as shown in Fig. 5(b). For small elasticdeformation, equilibrium together with the kinematic analysisof the interfacial segments yield,3

σS = ±EPS0 εS = ±EPS

0δ

g, τS = G0γS = G0

δ

gtan θ, (9)

where εS is the normal strain of the interface, γS is the shearstrain of the interface, and g is the width of the interface as

2EPS0 = E0

1−v20, E0 and v0 are the Young modulus and the Poisson

ratio of the interface material.3Note the deformation of the interface is zoomed in and is shown

in Fig. 5(b). It can be seen thatεS = δ cos θ

g cos θ= δ

gand γS = δ sin θ

g cos θ=

δ

gtan θ.

shown in Fig. 5(b). The total strain energy URTMS of an RVE

due to the applied shear can be expressed as

URTMS = 1

2G

RTM1

(δ

A

)2

V = 1

2

(σ 2

S

EPS0

+ τ 2S

G0

)V0, (10)

where GRTM

1 is the effective shear modulus of the first-ordersuture joint and A is the amplitude of the interface. Bysubstituting Eq. (9) into Eq. (10), the effective shear modulusis derived as

GRTM

1 = f 2

1 − f

(EPS

0

tan2 θ+ G0

). (11)

Equation (11) shows that the effective shear modulusGRTM

1of the first-order suture joint increases when the tip angle θ de-creases. Also, as expected, larger tooth volume fraction f and

larger stiffness of interface material correspond to largerGRTM

1 .

B. Tensile stiffness of the hierarchical RTM

The load transmission mechanism between two successivehierarchies is shown in Fig. 6. In a top-down view of the hierar-chy, the tensile load applied at the top hierarchy level (m = 1) istransmitted to the interface layer down through all hierarchies.The load transmitted to the teeth of all other hierarchies (m >

1) is the superposition of tension and shear. Following theconstruction process of a hierarchical suture joint, the effectiveinterface of the N th-order hierarchical suture joint is an(N − 1)th-order hierarchical suture joint as shown in Fig. 2(b).

Based on this load transmission mechanism and Eqs. (8)and (11), the effective tensile modulus and shear modulus ofthe second-order hierarchical suture joint can be determinedby the corresponding properties of the first-order hierarchicalsuture joint,

ERTM

2 = f 2

1 − f

(1

GRTM

1

sin2 θ cos2 θ + 1

ERTM

1

sin4 θ

)−1

,

GRTM

2 = f 2

1 − f

(E

RTM1

tan2 θ+G

RTM1

), (12a)

where ERTM

1 and GRTM

1 can be calculated from Eqs. (8) and(11). The effective tensile modulus and shear modulus of anN th-order hierarchical suture joint can be determined by the

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YANING LI, CHRISTINE ORTIZ, AND MARY C. BOYCE PHYSICAL REVIEW E 85, 031901 (2012)

LHS

LHS

LHS

LHS

RHS

RHS

RHS

RHS

th th th th-- - -th -

FIG. 6. (Color online) Load transmission mechanisms of the RTM between two successive hierarchies.

corresponding properties of the (N−1)th-order hierarchicalsuture joint following explicit iterative process whereby prop-erties of the N th-order joint are determined using propertiesof the (N−1)th-order joint as shown in Eqs. (12b),

ERTM

N = f 2

1 − f

(1

GRTM

N−1

sin2 θ cos2 θ + 1

ERTM

N−1

sin4 θ

)−1

,

GRTM

N = f 2

1 − f

(E

RTMN−1

tan2 θ+G

RTMN−1

). (12b)

Using the iterative Eqs. (12), the tensile stiffness and shearmodulus of the RTM hierarchical suture joints with an arbitraryorder N (>1) is obtained. Interestingly, when f > 0.618,

which corresponds to the golden ratio f 2

1−f> 1, thenG

RTMN >

GRTM

N−1 is guaranteed. The tensile behavior of hierarchical hardbiological tissues with arrays of bonded rigid rectangularcomponents were modeled with a similar iterative relation [4]where load transfer and deformation were more simply due tointerface shear [36].

To verify this load transmission mechanism and Eqs. (12a),finite element (FE) models of a first-order and a second-orderself-similar suture joint model are constructed with f = 0.667,and 2θ = 60◦ as shown in Fig. 7. It can be seen that Eqs. (12a)are very accurate, and remarkably, the addition of onehierarchy in the structure increases both the tensile andthe shear modulus by approximately 1 order of magnitude.Also, the piecewise spatially nonuniform strain (includingregions of both compression and tension) in the hierarchicalinterfaces significantly indicates a potential benefit of localnoncatastrophic failure (an energy dissipation mechanism) forthe hierarchical suture geometry where higher-order teeth canact sacrificially protecting lower-order teeth. The stress in themajor teeth and interfacial material will be uniform due to thetriangular geometry [32], indicating the maximum strength.

The iteration system of Eqs. (12a) and (12b) is plotted inFig. 8, showing how the hierarchy order influences the stiffnessof RTMs constructed by building blocks with different geome-tries. Figures 8(a) and 8(b) show that the nondimensional stiff-nesses for all combinations of geometric parameters are expo-nentially dependent on the hierarchy number N . The exponent

k(f,θ ) is plotted in Fig. 8(c) and is a function of the two inde-pendent geometric parameters of the building block f and θ ,

k(f,θ ) = d(

log ERTM

N

/E0

)dN

. (13)

Therefore, the effective stiffness of hierarchical RTMs canbe expressed as

ERTM

N = E0 × 10Nk(f,θ ). (14)

Equation (14) shows that, for the same building block, whenthe number of hierarchy N increases, the effective stiffnessincreases exponentially. Because the influence of N is ampli-fied by k(f,θ ) and k only is determined by the geometry of thebuilding block, we refer to k as a geometry amplification factor.

(a)

0.08

0.03

-0.02

maxP

222222222222222222

N=2

N=1

2θ=60o, f=0.667

100

101

102

103

104

100

101

102

103

104

E0 (MPa)

FE: N=1FE: N=2

RTM: N=1RTM: N=2

G (

MP

a)

|

0.10

0.05

0.00

maxP

N=1

N=2

100

101

102

103

104

100

101

102

103

104

FE: N=1FE: N=2

RTM: N=1RTM: N=2

E0 (MPa)

2θ=60o, f=0.667

E(M

Pa)

|

(b)

FIG. 7. (Color online) FE verification of Eqs. (12a) for first- andsecond-order RTM suture joints, under the loading cases of (a) shearand (b) tension.E andG refer to FE results (symbols), and the effective

shear and ERTM

N and GRTM

N calculated from Eqs. (12a), respectively,refer to the analytical results (solid and dashed lines).

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BIOINSPIRED, MECHANICAL, DETERMINISTIC . . . PHYSICAL REVIEW E 85, 031901 (2012)

FIG. 8. (Color online) The effective tensile stiffness of hierarchical suture joints predicted by RTM (a) influence of N on the effectivetensile stiffness when f = 0.8 and 2θ = 60◦, 90◦, and 120◦, respectively, (b) influence of N on the effective tensile stiffness when 2θ = 60◦

and f = 0.7, 0.8, and 0.9, respectively, and (c) the influences of f and θ on the geometry amplification factor k.

k(f,θ ) is obtained numerically using Eqs. (12a) and (12b)and is plotted in Fig. 8(c). Figures 8(a) and 8(c) show that k

increases rapidly when θ decreases. For RTMs with relativelylarge tip angles (θ > 90◦), θ has much less influence on k thanthose with relatively small tip angles (θ < 90◦). Also, for alltip angles, k increases when f increases.

IV. THE DEFORMABLE TOOTH COMPOSITE MODEL

A. First-order suture joint model under tension and shear

The RTM captures the main effects of increasing levelsof hierarchy on enhancing the stiffness and strength of suturejoints. However, when the tooth tip angle is small, deformationof the tooth must be taken into account as shown in thissection. Similar to the RTM, the strategy to develop themechanical hierarchical deformable tooth model (DTM) isto derive relationships for the shear and tension of a first-orderDTM first and then to develop models for the higher-ordersutures.

For a first-order DTM under tension, the total strain energyof its RVE can be written as

UDTMT = 1

2

σ 2

EDTM

1

V = 1

2

(σ 2

T

E0+ τ 2

T

G0

)V0 + 1

2

σ 2

E1V1, (15)

where V1 is the volume of the skeleton teeth V1 = V −V0.

Compared to Eqs. (8) and (15) now includes tooth strainenergy.

Similarly, by substituting Eq. (6) into Eq. (15), the effectivetensile stiffness of a first-order DTM is derived as

EDTM

1 = f 2E1

(1 − f )(

E1G0

sin2 θ cos2 θ + E1E0

sin4 θ) + f

. (16)

This same relationship was derived in Ref. [32] using akinematic analysis.

The shear behavior of the first-order DTM is more compli-cated than that of the first-order RTM due to the influence oftooth deformation. FE models of first-order suture joints witha wide range of tip angles (θ = 2.9◦, 5.7◦,11.3◦, 21.8◦, 38.7◦,and 58.0◦) and stiffness ratios (RS = E1

E0= 10,100,1000) were

constructed and were subjected to simple shear loading to gain

insight into the deformation mechanisns and then to compareto analytical models. These results are then compared with theresults of the RTM in Fig. 9(a).

Figure 9 demonstrates that the inclusion of toothdeformability leads to a fundamentally different mechanicalbehavior relative to the rigid tooth system with RS and 2θ .In contrast to the RTM in which the shear modulus goes toinfinity when the tip angle decreases, the shear modulus of theDTM achieves a maximum at a tip angle 2θ∗ that is dependenton RS and 2θ [Fig. 9(a)]. The tip angle 2θ∗ correspondsto competition of two dominant deformation mechanisms:a moderate tooth coupled bending and shear occuringsimultaneously with deformation of the compliant interfacematerial in shear and tension or compression [Fig. 5(b)].Generally, when RS and 2θ decrease, tooth deformationbecomes more dominant, whereas, when RS and 2θ increase,the deformation of interfacial layers becomes more dominant,and the teeth tend to behave more like rigid bodies.

This trend is shown clearly by the contours of the maximumprincipal strain of various deformed FE suture joint modelsunder the same overall shear strain as shown in Fig. 9(b).For example, when RS = 1000 and 2θ = 43.6◦, teeth behaverigidly, and interface layers experience a quasiuniform stressstate; when RS = 10 and 2θ = 5.8◦, shear of teeth is thedominant deformation, and the inverse rule of mixtures isan accurate evaluation for mechanical properties; when RS =1000 and 2θ = 5.8◦, bending is the dominant deformationof teeth; for all other intermediate cases, the deformation ofteeth is a combination of shear and bending. For all cases,the maximum principal strains are located in the middle of theinterface layers, which could be a precursor to interface failure.

Extending the derivation of the RTM, the effect of de-formable teeth shown above can be captured quantitatively byan analytical model developed as follows. Two variables due tocoupling between shear and bending are introduced in Eq. (9):the average deflection δb and the rotation γb of deformableteeth. Thus, comparison of Fig. 9(a) (top) and Fig. 5(b) (left)yields

τs = G0

[(δ − δb)

gtan θ − γb

], σs = EPS

0(δ − δb)

g. (17)

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YANING LI, CHRISTINE ORTIZ, AND MARY C. BOYCE PHYSICAL REVIEW E 85, 031901 (2012)

103

102

4.9

2.5

0.0

(%)εmaxP

9.9

5.0

0.0

9.7

4.9

0.0

2.0

1.0

0.0

2.4

1.2

0.0

2.2

1.1

0.0

3.5

1.8

0.0

4.9

2.5

0.0

5.2

2.6

0.0

5.8o 22.6o 43.6o

0

1

E

E

2θ

(b)

2θ

�bδ

bγ103

102

44.99

2.5

0.0

(%)εmaxP

9.9

5.0

0.0

99.77

4.9

0.0

22.00

1.0

0.0

22.44

1.2

0.0

22.22

1.1

0.0

33.55

1.8

0.0

4.9

2.5

0.0

55.22

2.6

0.0

5.8o 22.6o 43.6o

0

1

E0

E1

2θ

(b)

2θ

�bδ

bbγb τ

sGxO

Deformable tooth model: Shear

(a)

0.6

0 30 60 90 120 150 1800

0.2

0.4

2θ (degrees)

FE: RTMRTMFE: RS=1000

FE: RS=100

DTM: RS=100

FE: RS=10

DTM: RS=10

G (

GP

a)

|

DTM: RS=1000

10

FIG. 9. (Color online) Comparison of the mechanical responses to simple shear of the first-order RTMs and DTMs. (a) Shear moduli asa function of the stiffness ratios and the tooth angles (lower) and the free body diagram of DTM accounting for the coupling effects amongshear, bending of a tooth, and the rigid tooth movement (upper) and (b) FE results of the maximum principal strain distribution as a functionof the tip angle and the stiffness ratio (in order to show the deformation clearly, a magnification factor of 5 is used).

To evaluate δb and γb, the RVE of a first-order DTM ismodeled as a cantilevered beam. An effective shear load isapplied at the end of this cantilevered beam as shown inFig. 9(a) (top). The equations governing the deformation ofthe beam under the coupled shear and bending are

GsAsv′s = E1Iv′′′

b , E1Iv′′′b = P, (18)

where E1 is the Young’s modulus of the tooth material; I is themoment of inertia of the cantilever beam such that I ≈ (λ−2g)3

12 ;Gs is the local shear modulus, which is calculated by the Voigtlaw Gs = G0(1 − f ) + G1f ; As is the effective shear areaof the cross section where, for a rectangular cross section,As = 2

3 (λ − 2g); P is the effective concentrated force at theend of the beam due to a shear stress τ ; where P = τ (λ − 2g).The general solution of Eq. (18) is

v(x) = vs(x) + vb(x), (19)

where vs is the deflection due to shear and vb is the deflectiondue to bending. The boundary conditions imply that v(0) = 0,v′

b(0) = 0, v′′b (A) = 0, and v′′′

b (0) = − PE1I

. Therefore, thetotal deflection can be solved as

v(x) = Px3

6EI+ PAx2

2EI+ Px

GsAs

. (20)

The average influence of tooth deformation on the kine-matics of the interface is evaluated in the middle of the RVEas

δb = v

(A

2

)= 5

48

PA3

EI+ PA

2GsAs

, (21)

γb = v′(

A

2

)= 3

8

PA2

EI+ P

GsAs

. (22)

Thus, the effective shear modulus of a first-order DTM isderived as

GDTM

1 = fτA

δ= f 2CE0E1

f CCbE0 + (1 − f )(1 + Cr )E1, (23a)

where

C = G0

E0+ EPS

0

E0(tan θ )−2, (23b)

Cb = 3E1

4Gs

+ 5

16(tan θ )−2, (23c)

and

Cr = 3G0

2Gs

+ 9

8

G0

E1(tan θ )−2. (23d)

When E1E0

→ ∞, Eq. (23a) degenerates into Eq. (11) as aconnection between DTM and RTM solutions. The parametersC, Cb, and Cr in Eq. (23a) are nondimensional and dependon θ and the tension and/or shear stiffness ratios of the twophases, as shown in Eqs. (23b)–(23d).

B. Tensile stiffness of the hierarchical DTM

The tensile stiffness of a hierararchical suture joint withdeformable teeth can be calculated using a similar iterativeapproach as that presented earlier for the hierarchical DTM.Therefore, for a second-order DTM,

EDTM

2 = f 2E1

(1 − f )(

E1

GDTM

1

sin2 θ cos2 θ + E1

EDTM

1

sin4 θ) + f

,

GDTM2 = f 2C2E

DTM1 E1

f C2Cb2E

DTM1 + (1 − f )

(1 + Cr

2

)E1

, (24a)

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whereEDTM

1 andGDTM

1 are the effective interfacial propertiesof the second-order suture joint and are obtained fromEqs. (16) and (23a)–(23d), respectively. The expressions ofthe nondimensional parameters C2,C

b2 , and Cr

2 in Eqs. (24a)are as follows:

C2 = GDTM

1

EDTM

1

+ (tan θ )−2, (24b)

Cb2 = 3E1

4GS2

+ 5

16(tan θ )−2, (24c)

Cr2 = 3G

DTM1

2GS2

+ 9

8

GDTM

1

E1(tan θ )−2, (24d)

where the effective local shear modulus GS2 of the second-order

RVE follows the Voigt law,

GS2 =G

DTM1 (1 − f ) + G1f. (24e)

For an N th-order DTM, the iterative system of equations isas follows:

EDTMN = f 2E1

(1 − f )(

E1

GDTM

N−1

sin2 θ cos2 θ + E1

EDTM

N−1

sin4 θ)

+ f,

GDTMN = f 2CNE

DTMN−1 E1

f CNCbNE

DTMN−1 + (1 − f )

(1 + Cr

N

)E1

, (25a)

where

CN = GDTM

N−1

EDTM

N−1

+ (tan θ )−2, (25b)

CbN = 3E1

4GsN

+ 5

16(tan θ )−2, (25c)

CrN = 3G

DTMN−1

2GsN

+ 9

8

GDTM

N−1

E1(tan θ )−2, (25d)

and

GsN =G

DTMN−1 (1 − f ) + G1f. (25e)

E E

E E

FIG. 10. (Color online) Tailorable nondimensional stiffness-to-density ratio of self-similar hierarchical suture joints (E in the y axes

refers to eitherERTM

N [Eqs. (12a) and (12b)] orEDTM

N [Eqs. (24a)–(24l) and (25)]). (a) Schematics of self-similar hierarchical suture joints incomparison, constructed through the method shown in Fig. 4(a) and the tailorability of the nondimensional stiffness-to-density ratio predictedby RTM and DTM, depending on: (b) number of hierarchy N (by taking f = 0.8, 2θ = 60◦, and RS = 10, 100, 1000, and 10 000 as examples),(c) stiffness ratio (by taking f = 0.8, 2θ = 60◦, and N = 1–3 as examples, the shaded pink region indicates the region of stiffness ratios formany biological systems and nanocomposites [38,39]), (d) tip angle 2θ (by taking f = 0.8, RS = 1000, and N = 1–4 as examples), (e) thevolume fraction of a building block f (by taking 2θ = 60◦, RS = 1000, and N = 1–4 as examples), and (f) tailorability of tensile stiffnesswhen the total volume fraction of teeth is fixed at fv = 0.98.

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YANING LI, CHRISTINE ORTIZ, AND MARY C. BOYCE PHYSICAL REVIEW E 85, 031901 (2012)

V. RESULTS OF THE RTM AND DTM MODELS: TENSILESTIFFNESS OF HIERARCHICAL SUTURE JOINTS

The numerical results of Eqs. (24a)–(24l) and (25) areshown in Fig. 10. In order to evaluate the effect of N forthe same composite composition, we define and plot thestiffness-to-density ratio [37], i.e., the effective stiffness perunit volume skeletal tooth material4 E /fvE0 as a function ofN as shown in Fig. 10(b).

As shown in Fig. 10(b), the RTM provides an upper boundto the DTM. As predicted by the DTM, the tensile stiffness-to-density ratio of a hierarchical suture joint increases withthe total number of hierarchies N , but this trend is limitedby the stiffness ratio. When N increases, the nondimensionalstiffness-to-density ratioE/fvE0 has an asymptotic upper limitdetermined by the stiffness ratio, which is approximately 10RS .When N > Nc (RS , θ , f ), any further increase in N causeslittle increase in the effective stiffness-to-density ratio, and Nc

varies approximately with log(RS), depending on the values ofRS , θ , and f . Nc increases when RS or θ increases and whenf decreases.

More importantly, when the stiffness ratio of the two phasesis in the range of 10–104 as shown in Fig. 10(c) (shadedinterfacial region), it is very efficient to increase the stiffnessof a hierarchical suture joint by increasing the number ofhierarchies N . Many biological systems or nanocompositeslie in this stiffness ratio range [38,39]. For a very smallstiffness ratio, the stiffness-to-density ratio cannot be increasedvery much (and can even be reduced slightly if RS is small)by increasing N . Moreover, when N increases, the stiffnesssignificantly increases for all θ and f , but the stiffness becomesless sensitive to θ and f . When N > 1, the sensitivity of the ef-fective stiffness to the geometric parameters θ and f is reducedsignificantly at small θ and large f [Figs. 10(d) and 10(e)].

When the total volume fraction is fixed, the influenceof N on the effective tensile stiffness can be evaluated viaEqs. (25a)–(25e) and (4). As shown in Fig. 10(f), whichindicates that, when the same composite composition is usedto construct a hierarchical suture joint, the effective stiffnessof the RTM significantly depends on its geometry, describedby the number of levels of hierarchy N and the tip angle θ .For the same tip angle θ , when N increases while holding thetotal volume fraction fv constant, the volume fraction f ofthe building block needs to decrease as shown in Fig. 4(b).Thus, when N increases, two mechanisms compete with eachother: (1) N is increased, which is the factor that increasesthe stiffness, however, (2) f is decreased, to keep the samefv , which is the factor that decreases the stiffness. Therefore,a peak stiffness is achieved for a certain N = N◦: whenN < N◦, the stiffness increases with N ; after N exceeds N◦,instead, the stiffness decreases with N . Generally, N◦ increaseswhen RS or θ increases and when f decreases. As shown inFig. 10(f), for very small tip angles (such as 2◦), increasing

4Note that the stiffness-to-density ratio of the suture joints is E/ρ =E/[(1 − fv)ρ0 + fvρ1] ≈ ρ1E/fv , where ρ is the density of the suturejoints, ρ0 is the density of the interface material, and ρ1 is the densityof the tooth material. Usually, ρ1 ρ0, then E /ρ ∝E /fν . So, wecan useE/fν to evaluate the stiffness-to-density ratio.

N from 1 to 2 does not increase the stiffness; this is becausethe influence of the competing mechanism, Eq. (2), exceedsthat of Eq. (1), which is because the shear stiffness is verysmall when θ is small; whereas, for modest to relatively largetip angles (2θ > 20◦), the stiffness increases sharply whenN increases, indicating increasing N to 2 is a very efficientway to increase the stiffness of hierarchical suture joints withmodest to relatively large tip angles.

VI. FAILURE MECHANISMS, STRENGTH,AND TOUGHNESS OF THE SELF-SIMILARHIERARCHICAL SUTURE JOINT MODEL

Here, we formulate a theoretical foundation for the failureof first-order and hierarchical suture joints and explore therole of geometry in tailoring failure mechanisms, strength,and toughness. Each material component (tooth and compliantinterface) is assumed to behave elastically (as previously) andpossesses a critical failure stress above which catastrophicfailure takes place (this theory can be extended easily tononlinear elastic-plastic materials). The tooth material isassumed to be bonded perfectly to the compliant interfacematerial. For hierarchical suture joints (N > 1), three potentialfailure mechanisms are identified: (I) catastrophic failure ofthe major (the first-order) teeth, (II) catastrophic failure ofthe second-order teeth; and (III) relative sliding or shearbetween the first- and the second-order teeth [Fig. 11(a)].Interfacial failure is not considered for hierarchical suturejoints because mechanical interlocking between higher-orderteeth prevents the interfacial fracture and/or flaws that developfrom propagating further, thereby preventing catastrophicfailure of the joint and providing flaw tolerance. Mechanism IIIis more likely to happen when the tooth material enters into theplastic regime. Here, we focus on failure mechanisms I and II.

Since failure mechanisms of first-order suture joints(N = 1) are different than those of hierarchical suture joints(N > 1), the failure analysis of each is derived separately asfollows.

A. Failure mechanisms and strength of first-ordersuture joints (N = 1)

For first-order suture joints, potential failure mechanismsthat are considered include: (I) failure of the major teethleading to catastrophic failure of the joint and (II) failureof the compliant interfacial material also leading to catas-trophic failure of the joint. The normal stress σT and shearstress τT in the interface layer are related to the tensilestress [30] in the teeth via Eqs. (6). Thus, the stress stateof the interfacial material is [σT ,v0σT ,τT ], and the stressstate of the tooth material is [σ,0,0]. A failure criterion ofmaximum principal stress is used for both materials. Wedefine the tensile strength of tooth material to be σ

f

1 andthe tensile strength of interfacial material to be σ

f

0 . Whenσ = σ

f

1 , tooth failure occurs; when the maximum principal

stress σP0 = (1+v0)σT

2 +√

[ (1−v0)σT

2 ]2 + (τT )2 of the interfacial

material satisfies σP0 = σ

f

0 , the first-order suture joints fail byinterface failure. Hence, the effective strength of the first-order

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BIOINSPIRED, MECHANICAL, DETERMINISTIC . . . PHYSICAL REVIEW E 85, 031901 (2012)

(a)

(c) (d)

(b)

FIG. 11. (Color online) Failure mechanisms, strength, and toughness of hierarchical suture joints. (a) Schematics of three failuremechanisms, (b) parametric spaces of failure mechanisms I and II for the first-order suture joints and hierarchical suture joints (the bluecurves are the failure mechanism indicator H of hierarchical suture joints; the black curve is the failure mechanism indicator H1), (c) tailorablestrength changing with θ , c, and N when fixing fv = 0.98, and (d) tailorable toughness changing with θ , c, and N when fixing fv = 0.98 andRS = 100.

suture joints (N = 1) according to each mechanism is asfollows:

SI[1] = f σ

f

1 (tooth failure),(26)

SII[1] = σ

f

1

cH1(θ )(interface failure),

where SI[1] is the effective strength of the first-order suture

joints when tooth failure occurs, SII[1]is the effective strength of

the first-order suture joints when interface failure occurs and

H1(θ ) = sin2 θ

[1 + v0

2+

√(1 − v0

2

)2

+ tan−2 θ

], (27)

and c is the strength ratio of the two materials where

c = σf

1

σf

0

. (28)

The function H1(θ ) provides the failure mechanism indica-tor of the first-order suture joints and is plotted in Fig. 11(b).If H1(θ ) < 1

c, tooth failure occurs, otherwise, interface failure

occurs. If H1(θ◦) = 1c, the critical tip angle θ◦ corresponds

to the transition between the two failure mechanisms. Whenθ < θ◦, tooth failure occurs, and when θ > θ◦, interface failureoccurs. The tensile strength S[1] of the first-order suture jointsis determined by the strength of the two materials σ

f

1 and σf

0and the volume fraction of teeth f and tip angle θ . Based onEq. (26), S[1] is the smaller value of SI

[1] and SII[1], expressed as

S[1](σ

f

1 ,σf

0 ,f,θ) = min

[SI

[1],SII[1]

]. (29)

B. Failure mechanisms and strength of the hierarchicalsuture joints (N > 1)

Failure in the higher-order sutures begins by assessing thestress state in the higher-order teeth. The relationships between

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YANING LI, CHRISTINE ORTIZ, AND MARY C. BOYCE PHYSICAL REVIEW E 85, 031901 (2012)

the tensile stress in the first-order teeth σ[1], the tensile stressof the second-order teeth σ[2], and the shear stress between thefirst-order and the second-order teeth τ[2] are as follows:

τ[2] = σ[1]sin 2θ

2f, σ[2] = σ[1]

sin2 θ

f+ σb, (30)

where σb is the local maximum normal stress in the second-order teeth, generated by tooth bending [similar to what isshown Fig. 8(a)] due to the shear stress τ[2]. σb is expressed as

σb = f τ[2]6A2

f λ2= f τ[2]

3

tan θ, (31)

where A2 and λ2 are the amplitude and wavelength of thesecond-order teeth. It is worth noting that σb only exists forfailure mechanisms I and II of hierarchical suture joints. Forthe first-order suture joints, the term σb does not exist. Also,when the tooth material fails by relative sliding in the plasticregion between major and minor teeth, the term σb disappearsin Eq. (30), and therefore, a higher strength is expected for thismechanism.

Thus, the critical stress state of the second-order teethis [σ[2],σ

tt[2],τ[2]] where the component is σ tt

[2] = σ[1]cos2 θ

f. If

the failure criterion of maximum principal stress is usedfor tooth material, when σ[1] = σ

f

1 , then failure mechanism

I occurs; if the maximum principal stress σP[2] = σ[2]+σ tt

[2]

2 +√(σ[2]−σ tt

[2]

2 )2 + (τ[2])2 of the second-order teeth satisfies σP[2] =

σf

1 , then failure mechanism II occurs.Therefore, the effective strength of hierarchical suture joints

(N >1) according to each mechanism is as follows:

SI[N] = f σ

f

1 (mechanism I),(32)

SII[N] = f σ

f

1

H (f,θ )(mechanism II),

where SI[N] is the effective strength of hierarchical suture joints

when mechanism I occurs, SII[N] is the effective strength of

hierarchical suture joints when mechanism II occurs, and

H (f,θ ) = 0.5

(1

f+ 3 cos2 θ

)+

√0.25

(sin2 θ − cos2 θ

f+ 3 cos2θ

)2

+(

sin θ cos θ

f

)2

. (33)

If we define that the strength of the hierarchical suture jointscorresponds to catastrophic failure, then the strength S[N] ofthe hierarchical suture joints (N > 1) is determined by thetooth material strength σ

f

1 , and the geometric parameters ofthe building block (i.e., f and θ ) and is expressed as

S[N](σ

f

1 ,f,θ) = min

[SI

[N],SII[N]

](N > 1). (34)

The function H (f,θ ) is the failure mechanism indicator ofhierarchical suture joints. If H (f,θ ) > 1, failure mechanismII occurs, otherwise, failure mechanism I occurs. However,H (f,θ ) is always larger than 1 for all values of f andθ as shown in Fig. 11(b), which means that, under theassumption of material elasticity for the hierarchical suturejoints, catastrophic failure will never occur in mechanism I.By fixing the total volume fraction fv of the tooth material, theeffective strength S[N] of hierarchical suture joints (N > 1) andthat of the first-order suture joints S[N] are nondimensionalizedby fvσ

f

1 and are plotted and are compared in Fig. 11(c).

C. Fracture toughness of the first-orderand hierarchical suture joints

We define the nondimensional tensile fracture toughness� as the effective tensile toughness of the suture jointsnormalized by the tensile toughness of tooth material, whichis expressed as

� = (S)2(σ

f

1

)2

E1

EDTM

N

, (35)

where S is the effective strength of the suture joint, referringto S[1] in Eq. (29) (when N = 1) and S[N] in Eq. (34) (when

N > 1). The influences of hierarchy N and geometry on tensilefracture toughness of the first-order and hierarchical suturejoint model are plotted and are compared in Fig. 11(d) whenthe total volume fraction fv is fixed at 0.98 and the stiffnessratio is fixed at 100.

Figures 11(c) and 11(d) show that the first-order suturejoints may fail for either of the two failure mechanisms (I forsmall θ and II for large θ ), whereas, hierarchical suture jointsonly fail for mechanism II. Also, the tensile strength andfracture toughness of the first-order suture joints depend on thestrength ratio c, decreasing with increasing values of c, whilethe tensile strength and fracture toughness of hierarchicalsuture joints (N > 1) do not depend on c. If the same com-posite compositon (the same fv) is used to construct either afirst-order suture joint or a self-similar hierarchical suture joint,when the tip angle θ is small, the tensile strength and fracturetoughness of the first-order suture joints are larger than that ofhierarchical suture joints, and the strength and tensile fracturetoughness of different order hierarchical suture joints are quiteclose to each other. However, for relatively large tip angles,the fracture strength and fracture toughness are increasedsignificantly by increasing N from 1 to a value larger than 1.For hierarchical suture joints (N > 1), when N increases, thefracture toughness increases, but the strength decreases.

VII. CONCLUSIONS

In this paper, we construct an elastic deterministic fractalcomposite mechanical model for interdigitating hierarchicallystructured interfaces and suture joints to quantitatively investi-gate the role of structural hierarchy on the mechanical stiffness,strength, and failure of suture joints. Suture joints in nature

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BIOINSPIRED, MECHANICAL, DETERMINISTIC . . . PHYSICAL REVIEW E 85, 031901 (2012)

allow for segmenting monolithic shells into more compliantstructures to accommodate biological functions, such asgrowth, respiration, locomotion, and/or predatory protection[32,40]. The elastic regime studied here is expected to be bio-logically relevant for typical physiological loading conditions.

The use of hierarchy in natural suture joints is an effectivegeometric-structural strategy to achieve different propertieswith limited material options in nature. The hierarchical struc-ture is employed to increase the design space when other struc-tural geometries and parameters (such as the tooth tip angle)are biologically challenging or inaccessible. This paper alsoprovides predictive capabilities to guide the mechanical designof synthetic flaw-tolerant bioinspired interfaces and joints.

A. Mechanical tailorability and amplification

The key finding of this paper is the capability of the hierar-chy in a linear elastic system to tailor and to amplify mechan-ical behavior (e.g., stiffness, strength, and fracture toughness)nonlinearly and with high sensitivity over a wide range ofvalues (orders of magnitude) for a given volume and weight.

Stiffness. It is found that, if the teeth of a hierarchical suturejoint are rigid sufficiently relative to the more compliant in-terlayer, the effective tensile stiffness increases exponentiallywith N and is enhanced further by a geometry amplificationfactor k(f,θ ). This is because the mechanical properties of theeffective interface are amplified via the hierarchical geometry.However, if tooth deformation is considered, the increase instiffness with the increase in hierarchy is limited by the stiff-ness ratio between the skeletal teeth and the interfacial materialRS . When N > log RS, no significant improvement can beachieved by further increasing N [Fig. 10(a)]. Remarkably,as shown in Figs. 10(c)–10(e), when the stiffness ratio of thetwo phases is in the range of 10–104 (many biological systemslie in this stiffness ratio range [38,39]), the stiffness of thehierarchical suture joint is very sensitive to the number ofhierarchies. Therefore, for the biological systems (or syntheticsystems with stiffness ratios in the same range), it is veryefficient to increase the stiffness of the suture joint by increas-ing the number of hierarchies N . However, when the totalvolume fraction of skeletal teeth is fixed, there is an optimal N◦(depending on RS and θ ), beyond which, no further stiffeningcan be achieved by increasing N as shown in Fig. 10(f).

Strength and fracture toughness. Generally, for the samecomposite compositon, there are two efficient ways to increasethe tensile strength and fracture toughness: using the first-ordersuture joints (N = 1) and decreasing θ , promoting tooth fail-ure; for relatively large θ , using the higher-order hierarchicalsuture joints (N > 1) to increase the effective strength ofthe effective interface, promoting failure in the minor teeth[Fig. 2(b)]. For relatively large θ , when N increases, thefracture toughness increases, but the strength-to-density ratiodecreases because of the local normal stress, arisen by thecoupled bending effect of minor teeth under shear. In biologicalsystems, plasticity will also play a role in toughness, andincluding postyield deformation and dissipation would furtherenhance the effects on strength and on toughness presentedhere. Hence, for hierarchical suture joints, the postyieldingstrength and fracture toughnesses are expected to be improvedsignificantly when N > 1.

B. Interlocking of higher-order teeth and flaw tolerance

The continuity of the teeth at different hierarchies along theeffective interface provides a substantial advantage in reducinginterfacial failure. Increasing hierarchy was found to result inmechanical interlocking of higher-order teeth, which createsadditional load resistance capability and local nonuniforminterfacial stress distributions with regions of compressionor tension (Fig. 7), thereby preventing catastrophic failure inlower-order teeth (a sacrificial mechanism) and providing flawtolerance, providing more recovery time for the self-healingprocess of either biological or engineering systems [41,42].

C. Relevance to natural systems

Hence, the results shown in this paper demonstrate the greatsensitivity of mechanical behavior to hierarchy indicating thatthe diversity of hierarchical and fractal-like interfaces andjoints found in nature have definitive functional consequences.Hierarchy is one geometric strategy that nature used to addressthe limited material option. The use of hierarchy as a designparameter in biological systems likely is employed when otherstructural geometries and parameters (for example, θ ) arebiologically challenging or inaccessible. This model may alsofacilitate a better understanding of the evolution of biologicalsutures.

(a) The pelvic suture of Gasterosteus aculeatus (the three-spined stickleback [17,20]) exhibits a first-order suture N = 1with a spatially graded range of θ (∼2◦–60◦). As previouslyshown [32], G. aculeatus utilizes small θ to provide increasedstiffening near the trochear joint.

(b) The turtle carapace suture [40] exhibits a complexthree-dimensional suture joint with two morphologies in twocross sections [40]: a first-order suture in the sagittal plane anda fractal one in the transversal plane within the suture region.

(c) Vertebrate cranial sutures [12,13,18,19] possess mor-phological evolution in different stages of growth. For ex-ample, the cranial suture of humans develops from a simplerelatively straight line to very complex first-order and evenfractal-like patterns as one matures [43]; the cranial sutureof mammals with horns, such as the white tailed deer [21]exhibit more complex patterns than that of humans to improvethe mechanical properties to resist occasional impact loadingtransmitted via the horns.

(d) Ammonite septal sutures [22–25], which are hierarchi-cal (N = 1–3) or fractal, with low θ [compared with (a)]are observed to increase complexity continuously throughseveral mass extinctions [25], implying that the ammoniteshave adapted and have evolved toward a functional design.

ACKNOWLEDGMENTS

We gratefully acknowledge support of the US Armythrough the MIT Institute for Soldier Nanotechnologies (Con-tract No. DAAD-19-02-D0002), the Institute for CollaborativeBiotechnologies through Grant No. W911NF-09-0001 fromthe US Army Research Office, and the National SecurityScience and Engineering Faculty Fellowship Program (GrantNo. N00244-09-1-0064).

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