Size effects in layered composites – Defect tolerance and strength optimization

Accepted Manuscript Size effects in layered composites – Defect tolerance and strength optimization Junjie Liu, Wenqing Zhu, Zhongliang Yu, Xiaoding Wei PII:

S0266-3538(18)30675-4

DOI:

10.1016/j.compscitech.2018.06.026

Reference:

CSTE 7282

To appear in:

Composites Science and Technology

Received Date: 22 March 2018 Revised Date:

22 June 2018

Accepted Date: 25 June 2018

Please cite this article as: Liu J, Zhu W, Yu Z, Wei X, Size effects in layered composites – Defect tolerance and strength optimization, Composites Science and Technology (2018), doi: 10.1016/ j.compscitech.2018.06.026. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Size Effects in Layered Composites – Defect Tolerance and Strength Optimization Junjie Liua, Wenqing Zhua, Zhongliang Yua, Xiaoding Weia, b,*

a

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Address:

State Key Laboratory for Turbulence and Complex System, Department of Mechanics

SC

and Engineering Science, College of Engineering, Peking University, Beijing 100871,

b

Beijing Innovation Center for Engineering Science and Advanced Technology, Peking

University, Beijing 100871, China *

M AN U

China

Corresponding Author:

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E-mail: [email protected]

1

ACCEPTED MANUSCRIPT Abstract: Mixture of hard and soft phases in a smart way makes strong and tough materials – this approach, inspired by natural composites, has been widely adopted by scientists and

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engineers. Behind it exist many interesting fundamental mechanics. In this study, we solve analytically the stress intensity factor for a crack propagating from the hard to the soft phase in a layered composite starting with the postulation on the crack profile

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inspired by the shear-lag model. Our analysis shows that when a crack extends from the

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hard phase into the soft one, the stress intensity factor amplifies first at the hard-soft interface and then declines quickly to zero as it progresses toward the next hard phase. This crack arresting mechanism works until a secondary crack initiates in the next hard layer and then merges with the main crack. The efficiency of the defect tolerance,

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measured by the effective strength of the layered composite, is found to exhibit strong size effects. Overall, the smaller the dimensions of two phases are, the more efficiently of the layered composites tolerate defects. Furthermore, if the dimension of one phase is

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given, there always exists a critical dimension of the other phase that optimizes the

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efficiency (or the composite strength). A relationship between the defect tolerance efficiency with the nanostructures which is analogous to the famous “Hall-Petch” and “inverse Hall-Petch” relationships for polycrystalline metals is found. The analysis in this work can be used to guide the micro- to nano-structure design for synthesizing innovative defect-tolerant composites. Keywords: Fracture; Crack arrest; Defect tolerance; Microstructures; Optimization 2

ACCEPTED MANUSCRIPT 1.

Introduction A promising solution to the trade-off between strength and toughness, well-known

in most bulk synthetic materials [1], is to assemble two phases (in brief, “hard” and

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“soft”) into a sophisticated architecture at micro- or nano-scales. Nature has given excellent lessons. For instance, abalone shell, spider silk, tooth and bone, are found to achieve the extraordinary balance of strength and toughness through ingenious

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assembly of hard and soft ingredients [2-9]. The soft phase is believed to play an

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important role in arresting the crack extending from the hard phase. The underline mechanism for the crack blockage in hybrid materials has been attracting much attention of researchers in solid mechanics. The first problem that had been investigated extensively is a crack normal to the interface in a bi-material. Researchers analyzed the

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stress intensity factor and the stress distribution around the crack tip [10-18]. Their analyses suggest that the stress singularity remains inversely proportional to the square root of the distance from the crack tip except for the crack terminating at the interface. It

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is also found that the stress intensity factor decreases as the crack grows in the media

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with a lower elastic modulus. Later, some researchers investigated the same problem by proposing an effective J-integral, i.e. the crack driving force, of a crack in a gradient material or near a bi-material interface and reached similar conclusions [19-21]. Nevertheless, these studies were carried out on the problem of two semi-infinite dissimilar elastic media, which differs from the periodic microstructures in realistic composites. Furthermore, these analyses usually involve coefficients that have to be 3

ACCEPTED MANUSCRIPT determined numerically (e.g., through finite element simulations). Later on, various experimental and numerical modeling techniques have been applied to explore the crack shielding mechanisms found in many layered biological

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composites [22-27]. In comparison, theoretical studies on this topic are limited. Recently, Fratzl et al. [28] extended the analysis by Simha et al. into the problem containing a normal crack in a composite with periodically varying Young’s modulus

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[20, 21]. They reported that a sufficiently large variation in Young’s modulus is

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required to arrest the crack. Kolednik et al. [29] proposed a semi-analytical formula for the strength of a composite containing short cracks through a periodical variation of mechanical properties. They claimed that the composite strength increases monotonically with the wavelength of the material property variation decreasing, and

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reaches the theoretical value as the wavelength reduces to a critical value. Hossain et al. developed a semi-analytical approach to investigate the effective toughness of heterogeneous media and reached similar conclusions [30]. Wang and Xia [31]

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proposed a semi-analytical model for weakly heterogeneous solids with the aid of finite

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element simulations to investigate the crack pinning by the soft phase. Nevertheless, most of these analyses still rely on more or less numerical calculations to determine some key coefficients.

In this work, we carried out fracture analysis on a normal crack in a composite

containing layered hard and soft phases. Starting with the crack profile estimation inspired by the shear-lag analysis, we derive analytically the energy release rate and the 4

ACCEPTED MANUSCRIPT stress intensity factor of the crack. The analytical model, validated by finite element analysis (FEA), reveals the mechanism for the soft layer arresting cracks. Further, our analytical model allows us to quantify the defect tolerance efficiency of the layered

2.

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composites, which is a close function of the constituents’ properties and sizes. Description of the problem

The configuration of the problem is shown in Fig. 1. A composite of unit thickness

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consisting of stacked continuous hard and soft layers (also called “tablets” and “matrix”)

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whose elastic moduli are E1 and E2, and layer widths are b and h, respectively. Herein, it is assumed that E1/E2 >> 1 to represent a big mismatch in the elastic moduli between hard and soft phases. In the middle exists a through crack perpendicular to the axial direction. The crack length is denoted as 2a0. We assume that the pre-existing crack

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spans the first hard layer (that is, 2a0 = b at the beginning). A uniform far-field tensile strain ε0 is applied on the composite along the axial direction to drive the crack tip into the soft layer. The far-field stress applied on the hard and soft layers are σ1 and σ2,

EP

respectively (and apparently σ1/σ2 = E1/E2 >>1). In this study, we focus on the Mode-I

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crack propagation mechanism in this layered system. In other words, the crack deflection at the interface between hard and soft phases (i.e., Mode-II crack) is not considered herein. This requires that at the interface the relation GI GII < Γ I Γ II holds, in which GI and GII are the energy release rates for Mode-I and Mode-II, and ΓI and ΓII are the fracture resistances for Mode-I and Mode-II, respectively [32]. Due to the symmetry, we take a quarter of this configuration and simplify it into a 2D problem 5

ACCEPTED MANUSCRIPT shown on the right side of Fig. 1. The crack span in the soft layer is denoted as a (apparently, a = a0 -‐b/2). The analysis below is conducted based on the coordinate system Oxy with its origin commoves with the crack tip as shown in Fig. 1. Fracture mechanics analysis

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3.

First, we consider the extreme occasion when the crack passes through the whole soft layer and reaches the next hard layer (i.e., a = h). Since E1/E2 >> 1, the soft layer (or

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matrix) is assumed in a pure shear state. In other words, the crack surface in the soft

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layer should be close to a line (see the blue dashed line in Fig. 1). Thus, through the shear-lag model analysis [33], when a = h, the profile of the crack surface, denoted by u(x), in the soft layer takes the form

u ( x) = −

bσ 1λ x for − h < x < 0 , 2G2

(1)

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where λ = 3G2 E1bh , and G2 = E2 ( 2 + 2ν 2 ) is the shear modulus of the soft phase. Now, we need to find out how the form of u(x) evolves before the crack reaches the next a < h).

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hard layer (that is, 0

As the crack growing in the soft layer, the crack profile should gradually evolve

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into the final straight line given by Eq. (1). In addition, when the crack tip develops in the soft phase, the crack opening profile should take u ( x ) ∝

x , asymptotic to the

crack tip according to linear elastic fracture mechanics (LEFM) [34]. Therefore, when a < h, we assume that the crack profile in the soft layer follows a hyperbolic function expressed as

6

ACCEPTED MANUSCRIPT a u ( x) = C   h

(−x + h − a) − (h − a) 2

2

for − a < x < 0 ,

(2)

where C(a/h) is a coefficient varying with a/h. In addition, the crack profile in the

complete profile of the whole crack surface is

(3)

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 a 2 2 C  h  ( − x + h − a ) − ( h − a ) for − a ≤ x ≤ 0    u ( x) =  . a 2   2 C h − (h − a) for − a0 < x < −a   h 

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middle hard layer is assumed to maintain a horizontal line since E1/E2 >> 1. Thus, the

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We assume a virtual operation that applying a stress on the crack surface to close the crack slowly [35]. During this virtual process, the applied stress varies linearly from zero to the far-field stress. Thus, the work done per unit thickness in this process is U = 4∫

0

− a0

∫

u( x)

0

σ ( x ) du ( x ) dx = 2 ∫ σ ( x ) u ( x )dx , 0

− a0

(4)

yields

(−x + h − a) − (h − a) 2

2

 2  a  −a 2 dx + σ 1C   ∫ h − ( h − a ) dx  .(5)  h  − a0 

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σ E  a  0 U = 2 1 2 C   ∫  h  −a  E1

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where σ(x) is the applied stress on the crack surface. Substituting Eq. (3) into Eq. (4)

As E2/E1<<1, neglecting the first term in the above equation and noting a0 = a + b/2

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give

2

a a a U = bσ 1C   h 2   −   . h h h

(6)

Therefore, according to LEFM [34], the energy release rate Gr per unit thickness takes the form

7

ACCEPTED MANUSCRIPT  1 a 1 dC    2 −2 2 2     1 dU bσ 1  a  a   a   a   h  2  a  −  a   . (7) C  1 −   2   −    + Gr = =       2 da 2   h  h    h   h    a    h   h    d    h

= 0. That is, for –a < x < 0, the crack profile is approximately

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On the other hand, we expand the crack profile given in Eq. (2) into a power series at x

2  1   x x 1  a  . u ( x ) = C   2 ( h − a ) x 1 + ⋅ − ⋅  + ...   2 2 ( h − a ) 8  2 ( h − a )   h  

(8)

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According to Bazant and Planas [36], before the crack tip meets the interface between soft and hard phases (i.e. a < h), its profile takes the form

u ( x) =

2π E2′

KI

n ∞   x  x 1 + ∑ γ n    for − a < x < 0 ,  n=1  D  

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4

(9)

where KI is the stress intensity factor, and E2′ = E2 (1 −ν 2 2 ) for the plane strain condition in this study. When the crack tip meets the interface, the crack profile reduces

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to the form given by Eq. (1). Comparing Eq. (8) with Eq. (9), we can determine the coefficients γn and D. More importantly, the comparison suggests

π E2′ 2

a C  h

(h − a) .

(10)

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KI =

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Combining Eqs. (7) and (10) and noticing Gr = K I2 E2′ , we have  a 1 −   h

a 1 dC    2 2 2  a a a π E2′ h   a    a  h         C  + C   1 −  . (11)  2  −    = 1 2bσ 1   h    h   a    h   h   h 2 2 d  a a  h  2   −    h h      

Noting that in the extreme occasion when a = h, we know that Eq. (2) should reduce to Eq. (1). Then, solving Eq. (11) yields 8

ACCEPTED MANUSCRIPT bλσ 1

a C  = h 2G2

a a 2  −   h h

2

2  E2′ hπλ   a   a    1 − ln 2 −        8G2   h   h    

,

(12)

(h − a) ,

(13)

π b 2σ 12 λ 2 E2′

Gr =

  a   a  2   E2′ hπλ   a   a  2   2 16G2  2   −    1 − ln  2   −     8G2   h   h      h   h   

and

2

2  E2′ hπλ   a   a    1 − ln 2 −        8G2   h   h    

(h − a) .

(14)

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4G2

a a 2  −   h h

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π bσ 1λ E2′

KI =

2

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Thus, Eqs. (7) and (10) can be rewritten as

To verify the analysis above, we conducted finite element analysis (FEA) using the commercial software ABAQUS/Standard 6.14. The FEA model (in plane strain condition) is shown in Fig. 2(a). The hard layer width is set b = 4 mm and the soft layer

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width is set h = 1 mm. The model length is 400 mm in order to apply a uniform far-field tensile strain of 1% along y-axis. The Poisson’s ratios for hard and soft phases are set 0.2 and 0.3, respectively. The elastic modulus of the hard phase E1 = 100 GPa, and the

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ratio E1/E2 varies from 50-100. A line crack is placed in the middle on the left, with the

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span in the soft layer, a, varying from 0.1 mm to 0.9 mm. CPE4 elements (four-node bilinear element for plane strain condition) were used for both hard and soft layers. Mesh sensitivity test suggests that the FEA result converges when the mesh size near the crack tip is less than 0.05 mm. Thus, we used an element size of 0.02 mm × 0.02 mm to mesh the regime around the crack tip. The stress intensity factor is obtained through a contour integral near the crack tip, and various integration contours with different

9

ACCEPTED MANUSCRIPT distances from the crack tip were used to ensure the integral is path independent. The results obtained from FEA are compared with the model prediction by Eq. (14). First, our analytical model agrees very well with FEA on the overall trend of the stress

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intensity factor with the crack length. Our analysis suggests that, at a/h =0, K I → ∞ . This implies that the crack grows easily when the crack just enters the soft layer. As the crack grows, the stress intensity factor drops quickly, showing evidently that the crack

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will arrest in the soft phase. Interestingly, the analysis suggests that K I → 0 when

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a h → 1 . This means that the main crack cannot keep growing continuously in the soft layer. Instead, it must stop at a distance very close to the interface, until the stress concentration in next hard layer causes a secondary crack. Then, this secondary crack will merge with the main crack so that the crack propagation continues. Interestingly,

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this crack propagation process revealed by our analytical model has also been reported by Wang and Xia through FEA simulations on the crack propagation in a two-phase laminated composite, and visualized in their tapered double-cantilever beam

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experiments on 3D printed heterogeneous specimens containing hard and soft phases

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[31].

To study the nucleation of the secondary crack, we need to investigate the

stress/strain concentration near the second soft-hard interface (i.e., at a =h). According to Bazant and Planas [36], the general form for the normal stress σyy along the crack line ahead of the crack tip (i.e., x > 0 and y = 0) reads

σ yy =

∞  n n 1 +  ∑ ( 2n + 1)( −1) γ n x  . 2π x  n =1 

KI

(15) 10

ACCEPTED MANUSCRIPT Substituting coefficients γn and K I obtained previously into the above equation and keeping the first three terms, we have the approximate expression for the normal stress near the crack tip as KI 2π x

 3  x  5  x 2  1 −  −    , for 0
(16)

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σ yy ≈

Apparently, the normal stress is discontinuous at the interface of the hard and soft

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phases (see Fig. 2(a)). In contrast, the normal strain εyy in two phases should be continuous at the interface. Thus, we choose εyy as the index for the secondary crack

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nucleation in the next hard layer. Yet, under the plane strain condition, it is difficult to solve for εyy in the soft phase with only σyy unless the Poisson’s ratio of soft phase is zero. Thus, we first look at the normal strain distribution in the specific case where ν2 = 0, which is expressed as:

E2

2 =0

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ε yy =

 3  x  5  x 2  1 −  −   . 2π x  4  h − a  32  h − a  

KI ν

(17)

Fig. 3(a) compares the normal strain distributions obtained by FEA with those predicted

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by Eq. (17) for a/h = 0.8. It is found that this approximate solution predicts very

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accurately the stress field close to the crack tip. However, this approximate solution underestimates εyy in the location very close to the next hard layer. This is because when the crack tip is very close to the soft-to-hard interface (i.e., a/h ~ 1), the first two terms in the right-hand-side of Eq. (17) are accurate enough to describe the asymptotic strain field near the crack tip. The underestimation shown in Fig. 3(a) mainly comes from the third term. Therefore, we propose to modify Eq. (17) by multiplying the third term by a 11

ACCEPTED MANUSCRIPT function δ ( a h ) which equals to 1 when a/h <<1 (i.e., the crack tip is away from the soft-to-hard interface) and approaches to 0 when a/h

1 (i.e., the crack tip is very 1

close to the soft-to-hard interface). For simplicity, we choose δ ( a h ) = (1 − a h ) 2 in

ε yy =

1 2   2 3 5 x x a       1 −  −   1 −   .  4 32 h − a h − a 2π x      h  

KI ν E2

2 =0

(18)

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becomes

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this study (note that the form of δ ( a h ) is not unique) so that the above equation

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As shown in Fig. 3(b), the distribution of εyy predicted by the above equation agrees very well with FEA results. What we are mostly interested in is actually the tensile strain at the next soft-hard interface, i.e. ε yy

x =( h − a )

, where the secondary crack will

nucleate. Thus, for ν2 = 0, the tensile strain εyy at the interface is

E2

2 =0

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ε yy |x =( h -a ) =

1   2 a 1 5    − 1 −   . 2π ( h − a )  4 32  h    

KI ν

(19)

Fig. 3(c) shows a good comparison between the predicted εyy and those obtained from

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FEA at the interface. For the cases when ν2 ≠ 0, ε yy = σ yy E2 does not hold. However,

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additional FEA calculations show that the tensile strain at the interface is not sensitive to the value of ν2, as shown in Fig. 3(d). Thus, in the following discussion, the effect of the Poisson’s ratio of the soft phase on the tensile strain along the soft-hard interface is neglected, and Eq. (19) is used to estimate the strain state at the interface for any value of ν2.

4.

Strength optimization and its size effects 12

ACCEPTED MANUSCRIPT The criterion for the nucleation of the secondary crack is set as: when

ε yy

x =( h − a )

= ε f , damage starts in the next hard layer close to the interface and grows into

a secondary crack (where εf is the critical tensile strain of the hard phase). On the other

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hand, at the moment when this criterion is met, we know that the main crack requires Gr = Γ2, where Γ2 is the fracture resistance of the soft phase. Thus, Eq. (19) can be

1   2 1 5 a    ε yy |x =( h -a ) = − 1 −   .  4 32  h   2π E2 ( h − a )  

Γ2

(20)

= ε f yield the maximum crack length

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The above equation together with ε yy

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rewritten as

x =( h − a )

within the soft phase before the secondary crack nucleates in the next hard layer. This critical crack length reads

(

)

  2, 

(21)

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 64 ac = h 1 −  5+32 α h 

in which α = 2π E2ε 2f Γ 2 .

x =( h − a )

=εf :

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ε yy

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On the other hand, setting Gr = Γ2 in Eq. (13) yields the far-field stress when

a  4G2 f  c  Γ h , σ 1,max = 2 E2′ bλ π ( h − ac )

(22)

in which a f c  h

  ac  = 2   h

2   ac   E2′ hπλ   ac − ln  2     1 − 8G2   h     h

2   ac    −     .   h   

(23)

Eq. (22) represents the maximum far-field stress applied on the hard phase when the 13

ACCEPTED MANUSCRIPT main crack is still obstructed by the soft layer. Beyond this point, the second crack will nucleate in the next hard layer and merge with the main crack. Therefore, noting that E2/E1 << 1, the effective strength of the composite is approximately

b σ 1,max b+h

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σs

a  f c Γ 2 E1 h = Θ , b ac  h  1 − 1 +  h  b

(24)

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where Θ = 4 (1 −ν 2 ) 6π for the plane strain condition. With the critical crack length

we simplify the above equation as -1

a  h  Γ 2 E1  σs = Θ  1 − c 1 +  . b h  b  

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given by Eq. (21), numerical calculations suggest that f(ac/h) ~ 1 for nearly all h. Thus,

(25)

Substituting Eq. (21) into the above equation, we rewrite it in a dimensionless form −1

(26)

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5 + 32 h  h  σs = 1 +  , 8 b  b

where h = α h , b = αb are the dimensionless soft and hard layer widths, respectively;

(

2π E1E2 ε f Θ

)

is the dimensionless composite strength, which quantifies

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and σ s = σ s

the efficiency of the defect tolerance from the layered micro- and nano-structures.

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Eq.(26) clearly underscores the relationship between the layered composite strength with constituents’ properties (i.e. E1, E2, Γ2, εf and so on), and particularly their dimensions (i.e. h and b). To further explore how the sizes of two phases affect the effective strength, we inspect Eq. (26) by plotting the function in 3D as shown in Fig. 4(a). The 3D plot highlights evidently the size effects of the composite strength. Overall, the smaller the 14

ACCEPTED MANUSCRIPT constituents’ dimensions are, the higher strength the composite can achieve. In other words, well-designed microstructures at the nanoscale would make composite materials insensitive to cracks/defects. This damage tolerance of materials arousing at small

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scales is consistent with previous studies [28, 29, 37]. Second, at a given hard layer width b , the composite strength exhibit an optimization with respect to the soft layer width, as shown in Fig. 4(b). Solving ∂σ s ∂h = 0 , we get the critical soft layer width

5 512

(

)

25 + 1024b − 5 .

(27)

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hopt = b −

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that optimizes σ s

As shown in Fig. 4(c), for a large hard layer width, the optimal soft layer width approaches to the same value as the hard layer. Yet, as the size of the hard layer b reduces, the ratio hopt b drops accordingly. For example, at b = 0.1 , the composite

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strength optimizes when the soft layer width is approximately 39% of the hard layer. In addition, for a given soft layer width h , the material strength also shows an optimization with respect to the hard layer width b (see Fig. 4(d)). Solving

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∂σ s ∂b = 0 yields the optimal ratio ( b h ) = 1 . Fig. 4(d) shows that the composite h

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first strengthen when b decreases from a large value and then weakens when b is smaller than the given soft layer width h . Particularly, when b >> h , Eq. (26) suggests that the composite strength is inversely proportional to the square root of the hard layer width (i.e., σ s ∝ 1

b ), as highlighted by the slope of −1 2 in the log-log

plot in Fig. 4(d). Although based on different mechanisms, the interesting trend of the material strength vs. nanostructures found in this work is similar to the well-known

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ACCEPTED MANUSCRIPT Hall-Petch and inverse Hall-Petch relationships for metals (that is, the flow strength first shows an inversely proportional relationship with the square root of the grain size and then decreases when the grain size is smaller than a certain threshold) [38-40]. Although

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the mechanisms of the interactions between dislocations and grain boundaries (which can be treated as the interfaces between adjacent grains) are quite different than the mechanism found herein for the layered composites. The similar trend of material

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strength vs. microstructure in both systems suggests that they might share some features

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in strengthening approach.

Last, it is worth to point out that, for a very large hard layer width, the average size of the pre-existing cracks/flaws in a real composite, in general, will saturate to a certain value rather than equals to b as assumed in this study. Thus, in this case, Eq. (26) is just

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the measurement of the efficiency of the soft layer blocking the crack. The material strength, instead, should be determined by the far-field stress that drives the pre-existing flaws.

Conclusions

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5.

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In this work, we carried out fracture analysis on a pre-existing crack in a composite containing stacked soft and hard layers. By approximating the crack profile using a hyperbolic function inspired by the shear-lag model solution, the energy release rate and the stress intensity factor for the crack developing into the soft layer are derived. Our analysis suggests that the stress intensity factor drops quickly from an extremely high value (at the moment when the crack first extends from the hard phase to the soft phase) 16

ACCEPTED MANUSCRIPT to zero during the crack development (when it approaches the next hard phase). This trend of the stress intensity factor, validated by FEA simulations, reveals an interesting fracture process in layered composites – the main crack cannot grow continuously in the

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soft layer. Rather, the soft phase arrests the main crack until the stress concentration in next hard layer initiates a secondary crack. Then two crack merges and the fracture proceeds. Obtaining the approximate stress and strain distributions in front of the main

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crack and proposing a strain criterion for the secondary crack initiation, we get the

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analytical formula for the efficiency of the defect tolerance. Quantified by the effective composite strength, the defect tolerance efficiency is found to be greatly affected by the properties and dimensions of the hard and soft phases. In summary, the composite strength (or the defect tolerance efficiency) shows prominent size effects with respect to

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the hard and soft layer widths that are analogous to the famous “Hall-Petch” and “inverse Hall-Petch” relationships for polycrystalline metals. The analysis in this work will provide important insights into the micro- to nano-structure design for synthesizing

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novel defect-tolerant composites.

Acknowledgements

The authors greatly appreciate the support by National Key Research and

Development Plan of China (Grant No. 2016YFC0303700), the National Natural Science Foundation of China (Grants No. 11772003 and No. 51701005), and the National “Young 1000 Talents” Program of China. 17

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ACCEPTED MANUSCRIPT Figure Captions Fig. 1. Schematic of the composite containing stacked hard and soft layers. A through crack is placed in the middle perpendicular to the axial direction along which a uniform

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far-field strain ε0 is applied. The blue dashed line on the right subfigure represents the crack profile in the soft layer when 2a0 = (b+2h) or a = h. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this

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article.)

Fig. 2. Validation of the fracture analysis using FEA. (a) Distribution of σyy obtained by FEA for the case where E1/E2 = 100, a = 0.5 mm. (b) Comparison of the stress intensity factor values given by FEA to the analytical predictions at varying crack lengths for

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different values of E1/E2.

Fig. 3. Comparison between analytical predictions with FEA results (E1/E2 = 100,

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ν1=0.2 and ν2=0 are employed in all FEA simulations shown in this figure unless stated

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otherwise). (a) Distribution of ε yy in front of the crack predicted by Eq. (17) is compared with the FEA result. (b) Distribution of ε yy in front of the crack predicted by Eq. (18) is compared with the FEA result. (c) Normal strain at the next interface, that is

ε yy

x =( h − a )

, predicted by Eq. (19) is compared with FEA results at various crack lengths.

(d) Series of FEA calculations show that ε yy ratio of the soft layer. 20

x =( h − a )

is not sensitive to the Poisson’s

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Fig. 4. Size effects of the composite strength. (a) 3D plot shows how the dimensionless strength σ s varies with the dimensionless widths ( h and b ) of soft and hard layers.

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(b) Plot shows how the composite strength σ s varies with the soft layer width h for

b = 1. (c) Plot of the optimal ratio hopt b vs. b . (d) Log-log plot shows how σ s

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varies with b for h = 1. When b >> 1, a slope of -1/2 in the curve is highlighted.

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