Effective modulus of biological staggered nanocomposites with interface stress effect

Composites Part B 166 (2019) 701–709

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Composites Part B journal homepage: www.elsevier.com/locate/compositesb

Effective modulus of biological staggered nanocomposites with interface stress effect

T

Leiting Dong, Cezhou Chao, Peng Yan∗ School of Aeronautic Science and Engineering, Beihang University, Beijing, 100191, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Biological nanocomposites Staggered nanostructure Tension-shear chain model Interface stress effect Effective modulus

Biological composites with staggered matrix–platelet nanostructures, such as bone, teeth and nacre, possess superior mechanical properties. The interface property between mineral platelets and protein matrix at nanoscale is one of the key factors influencing effective properties of such nanocomposites. In this study, based on the Gurtin-Murdoch model, two tension-shear chain models (TSCMs) considering the interface stress effect are established to analyse the influence of interface stress effect on the effective Young's modulus of the staggered nanocomposites. One is the TSCM without tension region between two adjacent hard platelets, while another is with tension region. Two explicit formulae of effective Young's modulus of the nanocomposites are derived based on the two TSCMs. Meanwhile, an interface factor reflecting the combined influence of the platelet thickness and interface-platelet modulus ratio is also abstracted. The validity and accuracy of the two TSCMs are verified by comparing with the finite element simulations. Both the explicit formulae and the finite element simulations show that the interface stress effect has significant influence on the effective modulus, when the absolute value of the interface factor is large. For a given interface modulus, the interface effect on the effective modulus becomes more significant with decreasing the internal structure size of the nanocomposites to nanoscale.

1. Introduction Biological materials, such as bone, teeth and nacre, are nanocomposites of protein and mineral. Such biological composites possess superior mechanical properties, considering the weak constituents (protein and mineral) from which they are assembled [1,2]. For instance, the fracture strength and toughness of the biological composites are significantly higher than those of the mineral. Staggered proteinmineral nanostructures are observed in these biological composites [3]. It is revealed that this kind of special nanostructures make the composites insensitive to flaws at nanoscale [4], and the soft protein in the nanostructures contributes significantly to the toughness of the composites [5]. Inspired by the natural biological composites, biomimetic staggered composites are designed for higher load-bearing abilities [6,7] and better thermal shock resistance [8]. Stiffness or elastic moduli are important parameters of the biomimetic staggered composites, especially for the supporting and protecting functions. Many experiments showed that biological staggered nanocomposites with high stiffness can be designed in spite of weak constituents [9–11]. In fact, the Young's modulus of bio-composites can approach the upper limit defined by the Voigt model [9]. Therefore, it is necessary to reveal how the staggered ∗

nanostructures influence the effective moduli. For investigating the influence mechanism on effective moduli of the staggered composites, a range of models and analysis methods have been proposed. Gao and his co-workers [4,9] established a tension–shear chain model to predict the effective moduli of the staggered nanocomposites. Large aspect ratios and a staggered alignment of mineral platelets were thought to play important roles in the stiffness of biomaterials. Based on such a model, Liu et al. [12] put forward an analytical solution of the effective moduli of the biological nanocomposite structure with a perturbation method. The results show that the assumption of uniform shear stress distribution in the tension–shear chain model is valid with a smaller aspect ratio and a larger elastic modulus ratio of the nanostructure. Following the principle of minimum complementary energy, Zhang et al. [13] and Lei et al. [14] proposed an analytical solution of the effective moduli for the case of unidirectional platelets with arbitrary distribution. The results of this study confirmed that the distribution of platelets played a significant role in the effective moduli of a unidirectional nanocomposite. Lei et al. [15] and Qwamizadeh et al. [16] derived upper and lower bounds for the overall stiffness based on the principle of minimum potential/complementary energy, respectively. They compared effective moduli of different biocomposites with various physical and geometrical parameters including

Corresponding author. E-mail address: [email protected] (P. Yan).

https://doi.org/10.1016/j.compositesb.2019.03.001 Received 22 August 2018; Received in revised form 10 December 2018; Accepted 1 March 2019 Available online 06 March 2019 1359-8368/ © 2019 Elsevier Ltd. All rights reserved.

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and compared in section 5.

volume fraction and moduli of constituents, and aspect ratio and alignment pattern of stiff reinforcements. Zhang and To Ref. [17] introduced the effect of tension region between two adjacent ends of the hard platelets into the tension-shear chain model. The results show that when the aspect ratio of platelets is small, the accuracy of tension-shear chain model for overall stiffness can be improved. Although above works have revealed the influence mechanism of the internal structures on the effective moduli of the biological staggered composites to a large extent, an important influence factor, that is the interface stress effect, has not been considered [4–6,8,9]. Studies have shown that with the increase of surface-to-volume ratio, the surface stress effect becomes significant [18]. Streitz et al. [19] pointed out that at the nanoscale, the surface stress can make atoms depart from the equilibrium positions in bulk macroscopic materials, and the change of interatomic distance has a great influence on the mechanical properties. Moreover, Gurtin and Murdoch [20,21] established a linearized surface stress–strain constitutive relation. The surface domain is assumed a mathematical layer of zero-thickness and has different elastic moduli from the bulk. Based on the Gurtin-Murdoch surface stress model, there are many studies showing that the surface or interface stress effect plays important roles in the effective moduli of nanocomposites [22,23]. Sun et al. [24] pointed out that the effective bulk modulus of a particlereinforced composite can be affected by the interface stress, with the composite spheres assemblage model [25]. Considering the surface/ interface stress effect, Duan et al. [26] derived the fundamental framework to predict the effective elastic moduli of heterogeneous solids including nanoinhomogeneities, based on the interface stress model and the generalized Young–Laplace equations. For fibrous nanocomposites with interface stress, Chen and Dvorak [27] investigated the macroscopic behaviour of solids containing circular cylindrical nanoinclusions. They found that Hill's and Levin's connections for effective moduli was dependent on the absolute size of the nanoinclusions. Moreover, Chen et al. [28] examined the overall thermoelastic properties of solids containing spherical inclusions with interface effects. Wang et al. [29] reviewed the advances in the surface stress effect in mechanics of nanostructured elements, including nanoparticles, nanowires, nanobeams, and nanofilms, and heterogeneous materials containing nanoscale inhomogeneities. Based on the Gurtin–Murdoch model and the Steigmann–Ogden model, Eremeyev [30] investigated the behaviour of nanostructured materials considering the inner regular and irregular surface thin coatings. Koutsawa et al. [24] generalized the mechanics of structure genome to model nanocomposites taking into account the surfaces/interfaces stress effect at nano-scale. Their numerical results show that the predicted effective elastic properties depend not only on the material properties and volume fractions, but also on the surface-to-volume ratio. From the above review, it can be seen that though the interface stress effect in some classes of nanocomposites has been studied, a study on the interface stress effect in an important class of nanocomposites, that is the staggered nanocomposites, has not been reported. Therefore, in order to reveal its influence mechanisms, an analytical solution of effective moduli of staggered nanocomposites considering the interface stress effect is desirable. In this paper, a tension-shear chain model considering the interface stress effect is developed to predict the effective modulus of biological staggered nanocomposites. The influence of the interface effect at the matrix–platelet interface is analysed with the model with/without considering the tension region between two adjacent ends of the hard platelets. The paper is organized as follows. In section 2, basic equations about the interface stress effect at nanoscale are outlined. A tension-shear chain model considering the interface stress effect is developed, and the explicit formula of the effective modulus is derived in section 3. A numerical model based on finite element method is also developed in section 4 for verification of the proposed tension-shear chain model. Finally, numerical results of effective modulus are given

2. Basic equations for the interface stress effect Atoms at a surface/interface experience a different local environment than those in the bulk of a material. As a result, the equilibrium position and energy of these atoms will, in general, be different from atoms positions and energies in the bulk. There are two types of interface stress in solids [26]. One is related to the deformation with equal tangential strains at the interface between two abutting solids. Another is related to the deformation with different tangential strains in the two solids, at the interface where slipping occurs [31]. As for the interface within materials discussed here, the first type of interface stress will be introduced into interface constitutive equations, displacement continuity and stress discontinuity conditions at the interface. Based on the Gurtin-Murdoch surface stress model [20,21], the relationship between surface/interface stress tensor σ s and strain tensor ε s [32] can be written as

σ s = Cs: ε s

(1)

where Cs is the surface/interface stiffness tensor. The super- and subscript “s” represents the surface/interface between materials, the same below. For an elastic and isotropic interface, the constitutive equation (1) can be written as

σ s = 2μs ε s + λ s (trε s) I

(2)

where λ s and μs are the interface moduli, and I is the second-order unit tensor in two-dimension. The effect of residual fields [26] is ignored in this study. The stress is discontinuous across the interface considered here. It is characterized by the generalized Young–Laplace equations [33] from the analysis of the mechanical equilibrium of the interface, which can be written as [26].

(σ 2 − σ 1)⋅n = −∇s ⋅σ s

(3)

where n is the unit normal vector of the interface with a positive direction from material 1 to material 2. ∇s denotes the interface divergence of a tensor field σ s . σ 2 − σ 1 represents the discontinuity of the stress across the interface. The displacement is continuous across the interface, that is (4)

u1 = u2 = us

For plane stress problems considered here, according to Eq. (2), the relationship between interface stress σ s and interface strain ε s along the interface can be reduced as

σ s = Es ε s

(5)

The interface modulus Es can be non-positive [34], which can be expressed as

Es = 2μs + λ s −

λ s2 2μs + λ s

(6)

For plane stress problems with a flat interface along the y-direction illustrated in Fig. 1, the stress discontinuity condition (3) can be

Fig. 1. Flat interface between two different materials in plane stress problems. 702

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are assumed to be transversely uniform. Here the interface stress effect is introduced in the model as shown in Fig. 3 (b) and (c), in order to discuss its influence on the effective modulus. Based on above assumptions, and from the equilibrium condition, p the longitudinal normal stresses in the platelet σ22 and the interface s stress σ22 satisfy

L p s h + 2σ22 = 2τm ⎛ − y ⎞ σ22 ⎝2 ⎠

(9)

where τm is the shear loading applied to the interface from the matrix, and therefore is equal to the shear stress in the matrix. From the conp tinuity condition (8), the longitudinal normal strains in the platelet ε22 s and strains in the interface ε22 satisfy p s ε22 = ε22

(10)

Additionally, from the constitutive equations of the platelet and of the interface (5), p p s s σ22 = Ep ε22 , σ22 = Es ε22

(11)

where Ep is the Young's modules of platelets and Es is defined in Eq. (6). Therefore, it can be obtained that Fig. 2. The loading transfer path in the matrix–platelet nanocomposite structure with the interface stress effect.

p p ⎧ ⎪ σ22 = Ep ε22 = ⎨σs = E ε s = s 22 ⎪ 22 ⎩

reduced as

2 σαβ

(7) s σαβ

σ¯ =

and (α, β = 2,3) denote the stress tensor for material where 1, material 2 and the interface between two materials, respectively. The displacement continuity condition (4) at the interface can be reduced as uα1 = uα2 = uαs (α = 2,3)

uα1,

uα2

·2τm

( (

L 2

Es L ·2τm 2 Ep h + 2Es

) − y) −y

(12)

From Eq. (9), the longitudinal average stress σ¯ over the model in Fig. 3 can be calculated from the stresses at y = 0 , that is

2 1 s ⎧ σ23 − σ23 = −σ22,2 2 1 ⎨ σ33 − σ33 = 0 ⎩

1 σαβ ,

Ep Ep h + 2Es

p s (σ22 h + 2σ22 ) y=0

2(h + h m )

=

ρφτm 2

(13)

The contribution of the platelet and the matrix to the overall elongation of the nanocomposite, Δp and Δm can be written as

(8)

p Δp = ε¯22 L=

uαs

and where are the displacement vectors for material 1, material 2 and the interface, respectively.

1 1 ⋅ L2τm Ep h + 2Es 2

(14)

2γm h (1 − φ) φ

(15)

Δm = 2γm h m = 3. Tension-shear chain model considering the interface stress effect

where γm = τm/ μm is the shear strain of the matrix, and μm is the shear modulus. By combining Eqs. (14) and (15), the overall tensile strain ε¯ can be calculated as

Effective moduli of staggered nanocomposites are usually derived from the tension-shear chain model [9]. When the interface stress effect is additionally considered in this work, a schematic illustration of matrix–platelet nanostructures is shown in Fig. 2. ρ = L/ h is the aspect ratio of platelets. L and h are the length and thickness of hard platelets, respectively. h m is the thickness of soft matrix, and l m represents the distance between two adjacent ends of platelets. Let φ = h/(h + h m ) denote the thickness ratio, which is approximately equal to the volume fraction of platelets for l m ≪ L . Unidirectional loadings along the hard platelet orientation (longitudinal direction) as shown in Fig. 2 are considered. Then the Young's modulus in the longitudinal direction will be calculated with two tension-shear chain models in the following. One is the tension-shear chain model without tension region, and another is with tension region.

ε¯ =

Δp + Δm L

(16)

Finally, according to Eqs. (13) and (16), the overall Young's modulus E¯ can be obtained as

4(1 − φ) 1 ε¯ 1 = = + E¯ σ¯ (1 + k ) φEp μ m φ2ρ 2

(17)

where

k=

2Es Ep h

(18)

The interface factor k is introduced to account for the interface effect at the interface between soft matrix and the sides of the platelets.

3.1. Tension-shear chain model without tension region 3.2. Tension-shear chain model with tension region The tension-shear chain model without considering the tension region [9] is illustrated in Fig. 3(a). In this model, it is assumed that the platelets are with large aspect ratio and are much harder than the matrix, and then the bearing capacity of the tension region between two adjacent hard platelets is neglected. As a result, the load in the longitudinal direction can only be transferred via shear in matrix. Moreover, the shear strain and stress in the matrix are assumed to be piecewise uniform, and the longitudinal normal strain and stress in the platelets

For more general cases, the tension region should be considered. The tension-shear chain model with tension region [17] is illustrated in Fig. 4 (a). The interface stress effect between the sides of hard platelets and soft matrix is illustrated in Fig. 4 (b) and (c). Similarly, the shear strain and stress in the matrix are assumed to be piecewise uniform, and the longitudinal normal strain and stress in the platelets are assumed to be transversely uniform. 703

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Fig. 3. Mechanical equilibrium in the tension-shear chain model without considering the tension region. (a) Staggered nanocomposites with interface stress. (b) The interface between the side of platelets and matrix. (c) The equilibrium condition of force in the tension-shear chain model without tension region.

Distinct from the TSCM without tension region, tensile stress σm should be applied to the end of the hard platelets [17]. Therefore, the tensile stress σm can be written as

2Em γm h m σm = φl m

σ¯ =

p s + hσm ) y = 0 (σ22 h + 2σ22

2(h + h m )

=

ρφτm + φσm 2

(22)

The contribution of the platelet to the overall elongation of the nanocomposite, Δp can be written as

(19) p Δp = ε¯22 L=

L2τm + 2Lhσm 2(Ep h + 2Es )

where Em is the Young's modules of soft matrix. Based on the equilibrium condition, the relationship between p s normal stress in the platelet σ22 and the interface stress σ22 can be expressed as

Therefore, combining Eqs. (22) and (23) and Eqs. (15) and (16), the overall Young's modulus E¯ can be obtained as

L p s σ22 h + 2σ22 = 2τm ⎛ − y ⎞ + hσm ⎝2 ⎠

4(1 − φ) 1 ε¯ 1 = = + E¯ σ¯ (1 + k ) φEp αμm φ2ρ2

(20)

Ep Ep h + 2Es

⋅⎡2τm ⎣

Es ⋅⎡2τ Ep h + 2Es ⎣ m

( (

L 2 L 2

) − y ) + hσ

− y + hσm ⎤ ⎦ m⎤

⎦

(24)

where α = 1 + 8(1 + νm ) h m /(ρφl m ) is the tension region factor. And νm is the Poisson's ratio of soft matrix. In this case, the lateral displacement in present model is assumed unconstrained, which is different from the situation in Ref. [17].

Moreover, combining Eqs. (10), (11) and (20), the normal stress of the platelet and the interface is able to be obtained as p p ⎧ σ22 = Ep ε22 = ⎪ ⎨σs = E ε s = s 22 ⎪ 22 ⎩

(23)

3.3. Discussion of the two formulae of effective modulus

(21)

The longitudinal average stress σ¯ over the model in Fig. 4 can be calculated from the stresses at y = 0 , that is

Now compare the two formulae of effective modulus (17, 24) obtained by the two tension-shear chain models, with and without 704

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Fig. 4. Mechanical equilibrium in the tension-shear chain model considering the tension region. (a) Staggered nanocomposites with interface stress. (b) The interface between the side of hard platelets and soft matrix. (c) The equilibrium condition of force in the tension-shear chain model with tension region.

interface effect can be significant, which will be shown in the numerical examples in Section 5. From the two formulae, it can also be seen that the influences of both the interface modulus and the platelet modulus are only reflected by (1 + k ) Ep in Eqs. (17) and (24). Therefore, if we let Ep' = (1 + k ) Ep , the composite parameter Ep′ is an equivalent modulus of both the interface and the platelet. That is, when Ep′ is the same, the effective modulus E¯ will be same for any combination of the interface modulus Es and the platelet modulus Ep . As a special case, a composite with the interface modulus Es and the platelet modulus Ep can be equivalent to a composite with the platelet modulus Ep′ and without interface effects. Considering that this equivalent modulus is abstracted by an approximate model (the tension-shear chain model), the equivalency will be verified in the numerical examples in Section 5.

considering the tension region. It can be seen that when the tension region factor α → 1, formula (24) is reduced to formula (17). That is, for given φ (or h/ h m ) and large ρ (=L/ h) , when l m / h m is meanwhile large enough, the influence of the tension region can be neglected. Furthermore, when the aspect ratio ρ (=L/ h) of platelets is large enough, the effective modulus calculated by both Eq. (17) and Eq. (24) approaches a limit value (1 + k ) φEp , which is only contributed by the interface effect and elastic modulus of platelets. This conclusion will be verified in the numerical examples in Section 5. From Eqs. (17) and (24), it can be seen that the influence of the interface effect on effective modulus is only represented by the interface factor k . Note that the interface modulus Es can be negative, thus k can be negative. The effective modulus increases with the absolute value of a positive k , while decreases with that of a negative k . In other words, the interface effect can either increase or decrease the overall modulus of the nanocomposites. When k → 0 , the formulae of effective modulus (17) and (24) are reduced to those obtained by the tension-shear chain models without considering the interface effect [9,17]. In this case, Es/ Ep → 0 or h → ∞, that is the interface modulus is negligible comparing with the platelet modulus, or the internal structure size of the composites is large enough. When the nanocomposites internal size (represented by h for given φ ) is small enough, the influence of the

4. Finite element method based on unit cell model In order to verify the validity and accuracy of the two tension-shear chain models considering the interface effect, a finite element method (FEM) based on unit cell model is presented for comparison in the following. The key techniques are modelling the interface effect and prescribing the periodic boundary conditions. 705

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4.1. Fictitious truss element for interface

Y Y ⎧ U y 2 − U y 1 = ε¯y Lz ⎪ Y2 Y1 ⎪ Uz − Uz = γ¯yz Lz ⎨ U yZ2 − U yZ1 = γ¯ L y yz ⎪ ⎪ UzZ2 − UzZ1 = ε¯z L y ⎩

Consider the matrix–platelet nanocomposite structure with interface effects illustrated in Fig. 1. For the plane stress problem considered here, the matrix and platelet domains are discretized into 4-node quadrilateral elements. The interface considered here can be seen as a bar, thus can be discretized into two-node segments. The stiffness matrix of the two-node segment [35] can be written as

[K ] =

Es ⎡ 1 − 1⎤ −1 1 ⎥ Le ⎢ ⎦ ⎣

(26)

− where in the first equation, is the displacement difference in y-direction between the two opposite edges Y2 and Y1, and the notations in the rest three equations in Eq. (26) are similar to those in the first equation. ε¯y, ε¯z , and γ¯yz represent the average normal strain along ydirection, the average normal strain along z-direction and the average shear strain in the yz plane, respectively. Traditionally, in order to obtain the effective modulus, the average strain is applied to the unit cell, and then the average stress is calculated. Thereby the effective stiffness matrix of the unit cell is solved. However, the “dummy” node method [39] can be used to calculate the effective modulus directly by applying a unidirectional force to the “dummy” node and then calculating its displacement. The “dummy” nodes A1 and A2 are introduced to make their displacements denote the deformation of the unit cell, which can be written as ε¯y Lz = U yA1, γ¯yz Lz = UzA1, γ¯yz L y = U yA2 , ε¯z L y = UzA2 , where U yAα and UzAα represent the displacements of Aα (α = 1, 2) along y-direction and z-direction, respectively. These constraints are implemented using “Equation” type of constraints and “dummy” nodes in ABAQUS. Additional constraints, U yA0 = UZA0 = 0 and U yA2 = 0 , are applied to the corner node A0 and the “dummy” node A2 , to constrain rigid-body translations and rotations, where U yA0 and UZA0 are the displacements of A0 along y-direction and zdirection, respectively. After prescribing above periodic boundary conditions, the unidirectional tensile load along y-direction is applied to the “dummy” node A1. The average normal stress σ¯y over the unit cell along the loading direction can be written as U yY2

(25)

where Le is the length of segments. Comparing Eq. (25) with the stiffness matrix of the truss element in conventional finite element method, it can be seen that Es is equal to the product of the Young's modulus and cross-sectional area of a truss. Note that the interface stiffness can be negative [36], thus the stiffness matrix in Eq. (25) can be negative definite. As a result, the interface effect cannot be simulated directly with truss elements in conventional finite element codes [35]. Therefore, we introduce a user-defined material into the FEM simulation, which is implemented by using a user subroutine UMAT in the commercial FEM software ABAQUS [37].

4.2. Unit cell model and periodic boundary conditions The matrix-platelet nanocomposite illustrated in Fig. 2 can be seen as a doubly-periodic structure, thus a unit cell shown in Fig. 5 can be taken as a representative volume element. Four edges of the unit cell are denoted by Y1, Y2 , Z1 and Z2 . The lengths of edges Y1 and Z1 in the rectangular unit cell are L y and Lz , respectively. After partitions of the unit cell into finite elements, periodic boundary conditions are further imposed [38–40]:

σ¯y =

U yY1

F yA1 Ly

(27)

is the force carried on the “dummy” node A1 along y-direcwhere tion. Note that σ¯z = 0 . As a result, the average normal strain ε¯y along the loading direction can be calculated as

F yA1

ε¯y =

U yA1 Lz

(28)

According to the average field theory, the effective Young's modulus E¯ of the composite can be obtained as

E¯ =

σ¯y ε¯y

(29)

The effective modulus obtained from Eq. (29) by FEM will be compared with the prediction from the two approximate formulae (17) and (24) by tension-shear chain models. 5. Numerical results and discussion Some numerical examples are given in the following for verifying the validity and accuracy of the two presented tension-shear chain models. Influences of the interface effect and relevant internal structural parameters on the effective modulus of the nanocomposites are also shown and discussed. The FEM simulations provide reference results for comparison and verification. As shown in Fig. 5(b), the twodimensional plane stress element CPS4 is used for the meshes of both the hard and soft regions. The aspect ratio of elements is 1. The twodimensional truss element T2D2 is used for the interface. Moreover, the mesh sensitivity has been checked in the FEM model, and the mesh has been proved fine enough to get convergent results. In the following numerical examples, typical material properties are

Fig. 5. FEM model for the nanocomposite. (a) A unit cell of the matrix–platelet structure. (b) Finite element discretization of the unit cell. 706

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Fig. 7. Variation of normalized effective modulus E¯ / Ep with the nanocomposites internal size (represented by the platelet thickness h ) for different interface modulus Es : a comparison of the predictions from the two tension-shear chain models (TSCMs) with and without tension region (TR) with the FEM simulation results.

Fig. 8. Effective modulus E¯ of nanocomposites predicted by FEM versus the interface modulus Es at three different given equivalent moduli of the platelet and interface, Ep' = 50, 100, 200 GPa .

chosen from reference [41]. The Young's moduli of hard platelets and soft matrix are Ep = 100 GPa and Em = 280 MPa , respectively. The Poisson's ratios of platelets and matrix are taken as νp = 0.27 and νm = 0.4 , respectively. Thus, the shear modules of matrix is μm = 100 MPa and Ep = 1000μm . Additionally, h m = l m is assumed in the following so that the tension region factor can be simplified as α = 1 + 8(1 + νm )/(ρφ) . The interface properties are usually obtained by atomistic calculations [34,42]. However, due to lack of such work on biological materials discussed here, we assume the elastic modulus of the interface to be within a range, i.e., Es = −100~100 N/m as an approximation [42]. Following the work [43], these values are reasonable estimates for some metal and ceramic surfaces.

Fig. 6. Normalized effective modulus E¯ / Ep of the nanocomposites versus platelet aspect ratio ρ for different interface factors k : a comparison of the predictions from the two tension-shear chain models (TSCMs) with and without tension region (TR) with the FEM simulation results for three different thickness ratios: (a) φ = 50% ; (b) φ = 80% ; (c) φ = 95% .

5.1. Validity and accuracy of the two models Variations of the normalized effective modulus E¯ / Ep with the platelet aspect ratio ρ are depicted in Fig. 6 for three different thickness ratios (φ = 50%, 80%, 95%). The interface factor is in a range 707

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k = −0.5 ∼ 0.5. The predictions from the tension-shear chain model without tension region (TSCM without TR) and with tension region (TSCM with TR) are compared with the FEM simulation results. It can be seen from Fig. 6 that the predictions from the two tensionshear chain models are consistent with the FEM simulation results. On the whole, the prediction from the TSCM with TR is more close to the FEM simulation results when the aspect ratio ρ is small, while the prediction from the TSCM without TR is more close when ρ is large. The effective modulus increases with the interface factor k increasing from −0.5 to 0.5. The effective moduli predicted by both TSCMs and FEM approach limit values when the platelet aspect ratio ρ is large enough, and the trend is more obvious when thickness ratios φ is higher, which verifies the conclusion drawn from the explicit formulae in Section 3.

with the finite element simulations. It is also shown that the equivalent modulus from the TSCM with/without tension region is accurate. By using such an equivalent modulus, a nanocomposite with the interface stress effect can be equivalent to another nanocomposite without the interface effect. Both the explicit formulae and the finite element simulations show that the interface effect has significant influence on the effective moduli, when the absolute value of the interface factor is large. Considering that the interface modulus can be positive or negative, the interface effect can either increase or decrease the effective modulus of the nanocomposites. For a given interface modulus, the interface effect on the effective modulus becomes more significant with decreasing the internal structure size of nanocomposites to nanoscale.

5.2. Size effect on effective modulus

Acknowledgement

The effective moduli of staggered nanocomposites without the interface effect are size-independent [4,9,17,44]. However, by observing the explicit formulae derived from the TSCMs for the nanocomposites with the interface effect, it can be seen that the effective modulus is dependent on the platelet thickness h when the interface modulus Es ≠ 0 . In order to display such a size effect, the variation of effective modulus E¯ with the nanocomposites internal size (represented by the platelet thickness h for given φ = 95% and ρ = 100 ) is depicted in Fig. 7, for three different interface moduli, Es = −50, 0, 50 N/m . The predictions from the two TSCMs are also compared with the FEM simulation results. It can be seen from Fig. 7 that the size effect on effective modulus is considerable for Es = ± 50 N/m when the platelet thickness h is less than 20 nm, and is more significant with decreasing the internal size. On the other hand, when the internal size is large enough, the effective modulus approaches the value without considering the interface effect (Es = 0 ). In this case, the size effect can be neglected, as we do for conventional composites.

The authors thankfully acknowledge the support from the National Natural Science Foundation of China (grant No. 11502069) and the National Key Research and Development Program of China (No. 2017YFA0207800). References [1] Meyers MA, Chen PY, Lin AYM, Seki Y. Biological materials: structure and mechanical properties. Prog Mater Sci 2008;53(1):1–206. [2] Chen Q, Pugno NM. Modeling the elastic anisotropy of woven hierarchical tissues. Compos B Eng 2011;42(7):2030–7. [3] Gao H. Application of fracture mechanics concepts to hierarchical biomechanics of bone and bone-like materials. Int J Fract 2006;138(1–4):101–37. [4] Gao H, Ji B, Jager IL, Arzt E, Fratzl P. Materials become insensitive to flaws at nanoscale: lessons from nature. Proc Natl Acad Sci USA 2003;100(10):5597–600. [5] Anup S, Sivakumar SM, Suraishkumar GK. Influence of viscoelasticity of protein on the toughness of bone. J Mech Behav Biomed Mater 2010;3(3):260–7. [6] Zhang Z, Zhang YW, Gao H. On optimal hierarchy of load-bearing biological materials. Proc Biol Sci 2011;278(1705):519–25. [7] Ghazlan A, Ngo TD, Tran P. Influence of interfacial geometry on the energy absorption capacity and load sharing mechanisms of nacreous composite shells. Compos Struct 2015;132:299–309. [8] Lei HJ, Liu B, Wang CA, Fang DN. Study on biomimetic staggered composite for better thermal shock resistance. Mech Mater 2012;49:30–41. [9] Ji B, Gao H. Mechanical properties of nanostructure of biological materials. J Mech Phys Solids 2004;52(9):1963–90. [10] Barthelat F, Espinosa HD. An experimental investigation of deformation and fracture of nacre–mother of pearl. Exp Mech 2007;47(3):311–24. [11] Kim Y, Kim Y, Lee T-I, Kim T-S, Ryu S. An extended analytic model for the elastic properties of platelet-staggered composites and its application to 3D printed structures. Compos Struct 2018;189:27–36. [12] Liu G, Ji B, Hwang K-C, Khoo BC. Analytical solutions of the displacement and stress fields of the nanocomposite structure of biological materials. Compos Sci Technol 2011;71(9):1190–5. [13] Zhang ZQ, Liu B, Huang Y, Hwang KC, Gao H. Mechanical properties of unidirectional nanocomposites with non-uniformly or randomly staggered platelet distribution. J Mech Phys Solids 2010;58(10):1646–60. [14] Lei HF, Zhang ZQ, Liu B. Effect of fiber arrangement on mechanical properties of short fiber reinforced composites. Compos Sci Technol 2012;72(4):506–14. [15] Lei HJ, Zhang ZQ, Han F, Liu B, Zhang YW, Gao HJ. Elastic bounds of bioinspired nanocomposites. J Appl Mech 2013;80(6):061017. [16] Qwamizadeh M, Lin M, Zhang Z, Zhou K, Zhang YW. Bounds for the dynamic modulus of unidirectional composites with bioinspired staggered distributions of platelets. Compos Struct 2017;167:152–65. [17] Zhang P, To AC. Highly enhanced damping figure of merit in biomimetic hierarchical staggered composites. J Appl Mech 2014;81(5):051015. [18] Sharma P, Ganti S, Bhate N. Effect of surfaces on the size-dependent elastic state of nano-inhomogeneities. Appl Phys Lett 2003;82(4):535–7. [19] Streitz FH, Cammarata RC, Sieradzki K. Surface-stress effects on elastic properties. I. Thin metal films. Phys Rev B 1994;49(15):10699–706. [20] Morton E, Gurtin AIM. A continuum theory of elastic material surfaces. Arch Ration Mech Anal 1975;57(21):291–323. [21] Morton E, Gurtin AIM. Surface stress in solids. Int J Solids Struct 1978;14(6):431–40. [22] Chen Q, Wang G, Pindera M-J. Homogenization and localization of nanoporous composites - a critical review and new developments. Compos B Eng 2018;155:329–68. [23] Wang G, Chen Q, He Z, Pindera M-J. Homogenized moduli and local stress fields of unidirectional nano-composites. Compos B Eng 2018;138:265–77. [24] Sun L, Wu Y, Huang Z, Wang J. Interface effect on the effective bulk modulus of a particle-reinforced composite. Acta Mech Sin 2004;20(6):676–9. [25] Hashin Z. The elastic moduli of heterogeneous materials. J Appl Mech 1962;29(1):143–50.

5.3. Equivalency between the influences of hard platelet modulus and of interface modulus From the two explicit formulae (Eq. (17) from the TSCM without TR and Eq. (24) from the TSCM with TR) of effective modulus, the influences of platelet modulus and of interface modulus can be treated as an equivalent, by using the equivalent modulus defined by Ep′ = (1 + k ) Ep . Such an equivalency is shown in Fig. 8 at three different equivalent moduli, for h = 4 nm , φ = 50% and ρ = 50 . The accuracy of the two equivalent modulus is also compared. It can be seen that if the equivalent modulus (Ep′ = (1 + k ) Ep ) is at a fixed value, the effective modulus keeps almost constant, though the interface modulus Es varies. 6. Conclusions Two tension-shear chain models (TSCMs) with considering the interface stress effect are established to predict the effective elastic modulus of biological staggered nanocomposites containing hard platelets in soft matrix. One is the tension-shear chain model without tension region between two adjacent hard platelets, while another is with tension region. Two explicit formulae of effective Young's modulus of nanocomposites are derived based on the two TSCMs. Meanwhile, an interface factor reflecting the combined influence of the platelet thickness and interface-platelet modulus ratio is also derived. From the explicit formulae, an equivalent modulus reflecting the combined influence of platelet and interface moduli is abstracted. By introducing these parameters, the relations between the effective modulus and the internal structure\material properties are greatly simplified. A finite element model considering the interface stress effect is also presented, based on the unit cell model. Numerical examples show that the predictions of effective moduli by the two TSCMs are consistent 708

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