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New insights into the Young's modulus of staggered biological composites
Materials Science and Engineering C 33 (2013) 603–607
Contents lists available at SciVerse ScienceDirect
Materials Science and Engineering C journal homepage: www.elsevier.com/locate/msec
New insights into the Young's modulus of staggered biological composites Benny Bar-On, H. Daniel Wagner ⁎ Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel
a r t i c l e
i n f o
Article history: Received 15 August 2012 Received in revised form 4 October 2012 Accepted 12 October 2012 Available online 23 October 2012 Keywords: Staggered structures Bio-mechanics Bio-composites Nano-composites
a b s t r a c t This communication presents a simplified “mechanics-of-materials” approach for describing the mechanics of staggered composite architectures, such as those arising in a variety of biological tissues. This analysis calculates the effective modulus of the bio-composite and provides physical insights into its elastic behavior. Simplified expressions for high- and low-mineralized tissues are then proposed and the effects of the mineral thickness ratio and aspect ratio on the modulus are demonstrated. © 2012 Elsevier B.V. All rights reserved.
1. Introduction
2. The mechanics of staggered biological composites: analysis
Staggered architectures are found at the micro- and nano-scales of a variety of biological tissues, and their stiffness significantly affects the macroscopic mechanical properties of tissues as a whole [1,2]. The mineralized collagen fibril, which is the building block of bones and teeth [3–5], can be viewed as a staggered array of stiff hydroxyapatite platelets embedded inside a compliant collagenous matrix. Similarly, the nacreous layer of a shell is made up of aragonite tiles joined together by a thin layer of protein — resulting in a highly stiff brick-and-mortar structure [6,7]. Jager and Fratzl [3] and Kotha et al. [8] were the first to propose approximated micro-mechanical models for the elastic deformations of such staggered architectures. Gao et al. [9] introduced a simplified model for describing the fundamental mechanics of such composites and proposed a compact formula for the effective modulus. Other works included also shear-lag effects in the analysis and showed improved expressions [10,11], which coincide with Gao's formalism for the case of a very soft matrix. These models, however, are all designed for the case of small axial spacing between the platelets and neglect the axial force carried by the matrix. Recently, Bar-On and Wagner [5,12,13] proposed an accurate compact formula for the effective modulus of staggered composite structures with a generic platelet architecture (the BW formula). In this communication we propose a simplified approach for the BW formula, provide additional physical insight and clarify the mechanical principles of staggering in biological composites.
A staggered biological composite is considered here as a twodimensional array of stiff platelet reinforcement with length Lp and height hp, forming a staggered structure inside a soft matrix (Fig. 1). Both the platelets and the matrix are considered isotropic with Young and shear moduli Ep and Gp, and Em and Gm, respectively. Each periodic unit of the array has length L, height h, and contains effectively two platelets, as shown in Fig. 1. For later convenience, we consider a ‘unit cell’ which is a quarter periodic unit, as also shown in Fig. 1. The platelet volume fraction within the staggered array, ϕp, is related to Lp, hp, L and h via:
⁎ Corresponding author. Tel.: +972 8 934 2594. E-mail address: [email protected] (H.D. Wagner). 0928-4931/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.msec.2012.10.003
ϕp ¼
2hp Lp : hL
ð1Þ
The staggered structure is conveniently characterized by three independent dimensionless geometric parameters: the platelet aspect ratio (ρp), and the thickness and length ratios (Δh and ΔL), defined as follows: ρp ¼
Lp ; hp
Δh ¼
2hp ; h
ΔL ¼
Lp : L
ð2Þ
Δh and ΔL represent the platelet volume fractions along and perpendicular to the staggering direction, noting that the overall platelet volume fraction is ϕp = ΔhΔL. For evaluating the stiffness of the bio-composite along the staggered direction, we focus on a unit cell, the lateral faces of which are subjected to an external force F per unit thickness (the out-of-plane direction), as shown in Fig. 2. This unit cell is divided into two types of sub-regions: sub-region 1 (SR1) which is asymmetric and sub-region 2 (SR2)
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B. Bar-On, H.D. Wagner / Materials Science and Engineering C 33 (2013) 603–607
Fig. 1. A schematic description of a staggered bio-composite. The periodic unit and unit cell are indicated by the blue and black dashed lines.
which is symmetric. Due to the external loading, the unit cell is effectively experiencing a longitudinal deformation and a transverse contraction, whereas other types of deformations (e.g. bending) are considered to be small in view of the overall symmetry of the periodic unit. The external force (F) applied to SR1 is split into platelet and matrix forces, fp and fm, such that (Fig. 3): F ¼ f p þ f m:
ð3Þ
SR2 is subjected to the fp and fm forces, applied on both sides by the corresponding elements in SR1 (Fig. 3). The edges of those platelets that end at the border between SR1 and SR2 are assumed not to adhere to the matrix at SR1 – similarly to the Cox model [14] – and are therefore free of forces. In the following, the longitudinal moduli of SR1 and SR2 (E1 and E2, respectively) are derived analytically, and the effective modulus of the unit cell is then composed via the inverse rule of mixtures, namely the classical Reuss model [15]. Starting with SR1 in which the shear deformations are assumed to be negligible, the axial deformations of the platelet and the matrix are considered to be uniform (δp = δm ≡ δ1), as shown in Fig. 4, and the overall strain of SR1 is εp = εm = ε1 = δ1/(Lp/2). By applying Hooke's law and using Eq. (3), fp and fm are easily calculated: fp ¼ F
Ep Δh ; 2E
ð4aÞ
fm ¼ F
Em ð2−Δh Þ 2E
ð4bÞ
Fig. 2. The unit cell of a staggered bio-composite divided into two types of sub-regions, SR1 and SR2, subjected to an external force F (per unit thickness).
Fig. 3. Diagram of forces acting on SR1 and SR2. fp and fm are the platelet and matrix forces at the edges of the sub-regions. The black dashed square represents a segment of SR2 of length dX, located at a distance X from the left edge. fp(+)(X) and fp(−)(X) are the local forces acting on the upper and lower cross sections, and τ is the shear stress within the matrix between the platelets.
where Ē is the effective modulus of SR1, calculated by using the Voigt model [15]: i F=ðh=2Þ 1 h E ¼ E1 ¼ ¼ Ep Δh þ Em ð2−Δh Þ : ε1 2
ð5Þ
Note that the effect of shear deformations is indeed negligible for long and slender platelets, which is the case in many biological arrays [12]. The analysis of SR2 is a bit more complicated due to variations in the platelet axial force within SR2 (from fp to zero), which involve non-negligible shear deformations. The local axial forces within the upper and lower platelets are denoted as fp(+) and fp(−) respectively (Fig. 3), and their longitudinal gradients are related to a shear stress within the matrix (τ) through the equilibrium relation (Fig. 3): ð Þ
df p ¼ τ: dX
ð6Þ
The X coordinate varies in SR2 from 0 (left edge) to Lp − L/2. τ is proportional to the shearing angle, assumed to be uniform across the matrix, and is given by:
τ ¼ Gm
uðþÞ −uð−Þ h=2−hp
ð7Þ
where u (+) and u (−) are the local displacements of the upper and lower platelets. The governing equations for fp(±) are extracted by a
Fig. 4. Schematic deformation patterns of SR1 (a) and SR2 (b). The deformation δ1 of SR1, δp of the platelet and δm of the matrix, are all equal. The deformation δ2 of SR2 is composed of the axial stretching δp of the platelets and the platelets sliding due to shear deformations δm within the matrix.
B. Bar-On, H.D. Wagner / Materials Science and Engineering C 33 (2013) 603–607
set of mathematical manipulations on Eqs. (6)–(7), resulting in the following relation (Appendix A): d2 f ðpÞ dx2
2 ðÞ −γ BW f p
1 2 ¼ − γ BW f p ; 2
γBW
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Gm Δh ¼ 2ρp 1− ð8Þ 2ΔL Ep 1−Δh
where x = X/(Lp − L/2) is a non-dimensional coordinate (0 b x b 1), and γBW is a shear-lag parameter, representing the build-up range of fp(±) from the edges of SR2. By solving Eq. (8) with the corresponding boundary conditions (Fig. 3), fp(±) becomes: ðÞ
f p ðxÞ ¼
fp 1 sinhðγBW xÞb coshðγBW xÞ þ 1 : tanhðγBW =2Þ 2
ð9Þ
Once fp(±) is found, the axial deformation of SR2 (δ2) can be evaluated. δ2 is dominated by two deformation mechanisms, the platelet stretching (δp) and a relative sliding between the platelets due to the matrix shearing (δm), as shown in Fig. 4. δp is obtained by integrating Eq. (9) along x and using Hooke's law. δm is proportional to the shear stress (τ) at the edge of SR2 (via Hooke's law); τ is extracted by combining Eqs. (6) and (9). The overall strain of SR2 is then (Appendix B): ε2 ¼
δp þ δm fp 1 1 ¼ : 1þ2 γBW tanhðγ BW =2Þ Lp −L=2 Ep hp
ð10Þ
The modulus of SR2 (E2) is calculated by the following relation (using Eqs. (4a) and (10)): F=ðh=2Þ γBW E: E2 ¼ ¼ ε2 cothðγ BW =2Þ þ ðγ BW =2Þ
As a final step, the moduli of SR1 (Eq. (5)) and SR2 (Eq. (11)) are combined by using the Reuss formula, resulting in a closed-form expression for the longitudinal modulus of the staggered bio-composite, named the BW formula: EBW ¼
A critical value of γBW exists (γBW ≈2.4), at which the moduli of SR1 and SR2 are equal. There, we have EBW =E1 =E2 =Ē, and EBW is not affected by variations of the relative amount of SR2 (through 2ΔL −1). For γBW b 2.4, E1 >E2, indicating that SR2 functions as a compliant adhesive between the SR1 parts. For these γBW values, an increase of 2ΔL −1 further reduces the modulus of the composite. When γBW >2.4, E1 b E2 and the modulus of the staggered array exceeds Ē. Here, higher 2ΔL − 1 ratios increase the value of EBW. 3. Example taken from the biological world Staggered bio-composites are typically made of slender reinforcements, for which the aspect ratio (ρp) – by being proportional to the shear-lag parameter (γBW) – has a significant effect on the stiffness of the composite (see Fig. 5). To illustrate this we consider two here staggered configurations — approximating the structure of highmineralized and low-mineralized biological tissues. Mineralized tissues are typically made of mineral platelets embedded in a proteinous matrix, for which Ep/Em ≈100. High-mineralized tissues consist of tightly packed staggered platelets, separated by a thin layer of matrix material. Here we consider a very large mineral content (ϕp ≈95%) and a very large SR2 portion (ΔL →1) (i.e. Δh =ϕp/ΔL ≈0.95) — representing the architectures of the nacreous layer of mollusk shells or the tooth enamel. For these high-mineralized configurations Ē/Ep ≈0.5 and γBW ≈ρp/4 (via Eqs. (5) and (8), respectively), the hyperbolic-cotangent in Eq. (12) approaches unity (slender reinforcement), and the modulus of highmineralized tissues (EHM) can be approximated as: EHM ≈
ð11Þ
2ð1−ΔL Þ 2ΔL −1 −1 γBW E: þ ¼ ð2ΔL −1Þ cothðγBW =2Þ þ γBW ð3=2−ΔL Þ E1 E2
ð12Þ This coincides with the BW formula in [5] by substituting ΔL =ϕp/Δh. Fig. 5 plots EBW/Ē as a function of γBW for selected 2ΔL −1 values, representing the relative amount of SR2 within the bio-composite. As seen, EBW exhibits different characteristics in the different γBW intervals.
Fig. 5. EBW/Ē as a function of γBW, for selected 2ΔL − 1 values representing the relative amount of SR2 within the composite. γBW ≈ 2.4 is a critical point where the modulus of SR2 is equal to the modulus of SR1. For γBW b 2.4 the modulus of SR2 is lower than the modulus of SR1, and vice versa. γBW is proportional to the platelet aspect ratio; slender platelets therefore increase the modulus of the composite.
605
1 E ðhigh mineralized tissuesÞ: 1=γBW þ 1=2
ð13Þ
By considering ρp ≈ 15 − 20 and ρp ≈ 50 − 100 as representative aspect ratio ranges for nacre and enamel, respectively, Eq. (13) approximates the modulus of the tissues as ENacre/Ep ≈ 0.65 − 0.7 and EEnamel/Ep ≈ 0.85 − 0.93, in good agreement with experimental data from the literature (e.g. [16,17]). Low-mineralized tissues can be characterized by several biocomposite architectures. For the current demonstration we consider a staggered configuration having thin platelets and a very large SR2 portion, as recently proposed for describing the small-scale mechanics of low-mineralized tendon [18]. ΔL →1 and Δh ≈0.2 (i.e. ϕp ≈0.2) are selected for the analysis, for which Ē/Ep ≈0.1 and γBW ≈ρp/30 — indicating a significant softening of both SR1 and SR2 compared to the high-
Fig. 6. The approximated moduli of high-mineralized and low-mineralized tissues, normalized by the modulus of the reinforcing platelet (EHM/Ep and ELM/Ep), plotted as a function of the mineral platelet aspect ratio ρp.
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B. Bar-On, H.D. Wagner / Materials Science and Engineering C 33 (2013) 603–607
mineralized configuration (see Fig. 5). Using Taylor expansion for the hyperbolic cotangent in Eq. (12), the modulus of low-mineralized tissues (ELM) is as follows: ELM ¼
1 E ðlow mineralized tissuesÞ: 2=ðγ BW Þ2 þ 2=3
ð14Þ
Fig. 6 plots EHM and ELM as a function of the reinforcement aspect ratio, using Eq. (13), Eq. (14) and the corresponding Ē and γBW parameters. The figure indicates that the modulus of bio-composite made up of thin platelets (low-mineralized) is much lower than that made up of thick platelets (high-mineralized). It is also shown that both EHM and ELM monotonically increase with ρp but the effect is more significant for ELM. Note that EHM → Ep for ρp ≫ 1, demonstrating the efficiency of slender staggered bio-composites.
By considering the balance of forces on the platelet segments as shown in Fig. 3, the longitudinal gradients of fp(±) are related to the shear stress within the matrix (τ) via: df ðpÞ ¼ τ: dX
Following Hooke's law for the matrix, τ is proportional to the shearing angle of the matrix via the shear modulus: τ ¼ Gm φ:
In conclusion, the modulus of staggered bio-composite is dominated by the moduli of SR1 and SR2 (E1 and E2), which are combined by the inverse rule of mixtures (Reuss model). E1is governed by Δh (Eq. (5)), E2 is governed by Δh and γBW (Eq. (11)), and the parameter ΔL determines the mixing ratio between E1 and E2. Higher values of Δh indicate that the platelets support a greater portion of the external loading — increasing the moduli of SR1 and SR2. A greater γBW value indicates that load transfer between the staggered platelets takes place closer to their edges — resulting in smaller deformations in SR2 and enhancing the modulus of SR2. Since γBW ∝ ρp and Δh ∝ hp, it can be concluded that thicker or slender platelets result in stiffer staggered biocomposite, as indeed is observed in stiff natural configurations such as highly-mineralized nacre and enamel. Acknowledgments We acknowledge support from the Israel Science Foundation (grant no. 1509/10) and from the G. M. J. Schmidt Minerva Centre of Supramolecular Architectures. This research was made possible in part by the generosity of the Harold Perlman family. H.D. Wagner is the recipient of the Livio Norzi Professorial Chair. Appendix A
uðþÞ −uð−Þ h=2−hp
σ
ðÞ
¼ Ep ε
ð Þ
:
ðA:1Þ
The local axial forces within the upper and lower platelets, fp(±), are: ðÞ
fp
¼σ
ðÞ
hp : 2
ðA:2Þ
The strains ε (±) are defined as the gradients of the platelet displacement fields (u (±)): ε
ð Þ
ðÞ
du : ¼ dX
τ ¼ Gm
By combining Eqs. (A.1)–(A.3), fp(±) and u (±) are related via; ðÞ fp
hp duðÞ : ¼ Ep 2 dX
uðþÞ −uð−Þ : h=2−hp
d2 f ðpÞ dX
2
¼
Gm 1 1 ðþÞ ð−Þ f −f p : Ep h=2−hp hp =2 p
ðA:9Þ
Transforming into a non-dimensional coordinate x = X/(Lp − L/2) (0 b x b 1), Eq. (A.9) transforms into: d2 f ðpÞ 2
dx
¼
Gm 2Lp −L 2Lp −L ðþÞ ð−Þ f p −f p : Ep h−2hp hp
ðA:10Þ
The fractions in Eq. (A.10) can be expressed by the nonL 2h dimensional geometric relations in Eq. (2), ρp ¼ hpp ; Δh ¼ h p and Lp ΔL ¼ L , resulting in the following equations: d fp
dx2
1 2 ðþÞ ð−Þ ¼ γ BW f p −f p ; 2
γBW
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Gm Δh ¼ 2ρp 1− : ðA:11Þ 2ΔL Ep 1−Δh
Eq. (A.11) represents two coupled equations, governing the evolution of fp(±) along SR2. An additional relation between fp(±) can be obtained by adding the two relations in Eq. (A.5): df ðpþÞ df ðp−Þ þ ¼ 0: dX dX
ðA:12Þ
By integrating Eq. (A.12) over X and substituting the boundary conditions at X =0 as shown in Fig. 3, fp(+)(X = 0)= 0 and fp(+)(X = 0)= fp, the following relation is obtained: ðþÞ
ð−Þ
¼ f p:
ðA:13Þ
By substituting Eq. (A.13) into Eq. (A.11), the coupling between the two governing equations is released, resulting in independent equations for the fp(±) forces, as given in Eq. (8): 2 ð Þ
d fp
ðA:4Þ
ðA:8Þ
By derivation of Eq. (A.5), substituting Eq. (A.8) and expressing the displacement gradients by Eq. (A.4) we obtain:
fp þ fp ðA:3Þ
ðA:7Þ
where the denominator is the height of the matrix in SR2 (Fig. 2). Combining Eqs. (A.6) and (A.7) yields:
2 ðÞ
This appendix includes the derivation of Eq. (12), which is the governing equation for fp(±). The longitudinal stresses and strains within the upper and lower platelets (σ (±) and ε (±)) are related via Hooke's law:
ðA:6Þ
Since τ is assumed to be uniform across the matrix, φ can be expressed by the displacements of the platelets as follows: φ¼
4. Conclusions
ðA:5Þ
dx2
2
ð Þ
−γBW f p
1 2 ¼ − γBW f p ; 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Gm Δh : ðA:14Þ γBW ¼ 2ρp 1− 2ΔL Ep 1−Δh
B. Bar-On, H.D. Wagner / Materials Science and Engineering C 33 (2013) 603–607
ance of force on a platelet segment (Eq. (A.5)) and substituting fp(±) from Eq. (9), δm is found to be:
Appendix B This appendix includes the derivation of Eq. (12), representing the overall strain of SR2. The total deformation of SR2 can be considered as a summation of two contributions, the platelet stretching (δp) and a relative sliding between the platelets due to the matrix shearing (δm), as shown in Fig. 4. δp is extracted by integrating local strain along one of the platelets: 1 ð Þ δp ¼ Lp −L=2 ∫0 ε dx
ðB:1Þ
where Lp − L/2 is the length of SR2. Following Eqs. (A.1) and (A.2), ε(±) is related to fp(±) via: ε
ð Þ
¼
f ðpÞ 1 : hp =2 Ep
607
ðB:2Þ
By substituting Eq. (B.2) into Eq. (B.1), using the expression for fp(±) from Eq. (9) and integrating over x, δp is found to be:
δm ¼
h=2−hp fp τðx ¼ 1Þ ¼ Gm 2
1 h−2hp Gm 2Lp −L
!
γ BW : tanhðγBW =2Þ
Eq. (B.5) can be further simplified using the definition of γBW in Eq. (8), which results in: δm ¼
f p 2Lp −L 1 : Ep hp tanhðγ BW =2Þ
ðB:3Þ
δm is related to the shearing angle at the edge of SR2 (either φ(x = 0) or φ(x = 1)), multiplied by the height of the matrix in SR2: δm ¼ h=2−hp φðx ¼ 1Þ:
ðB:4Þ
Following Hooke's law for the matrix in Eq. (A.6), φ is proportional to the shear stress within the matrix (τ). By further considering a bal-
ðB:6Þ
The total strain of SR2 is calculated by combining δp and δm from Eqs. (B.3) and (B.6) and dividing by the length of SR2 (Lp − L/2, resulting in the expression in Eq. (10): δp þ δm fp 1 1 ¼ : ðB:7Þ 1þ2 ε2 ¼ γBW tanhðγ BW =2Þ Lp −L=2 Ep hp References
Lp −L=2 1 1 ðÞ f p Lp −L=2 ∫ f dx ¼ : δp ¼ hp =2 Ep 0 p Ep hp
ðB:5Þ
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
B. Ji, H. Gao, J. Mech. Phys. Solids 52 (2004) 1963–1990. P. Fratzl, R. Weinkamer, Prog. Mater. Sci. 52 (2007) 1263–1334. I. Jager, P. Fratzl, Biophys. J. 79 (2002) 1737–1746. H.S. Gupta, et al., Proc. Natl. Acad. Sci. U. S. A. 103 (2006) 17741–17746. B. Bar-On, H.D. Wagner, J. Biomech. 45 (2012) 672–678. M.A. Meyers, et al., J. Mech. Behav. Biomed. Mater. I (2008) 76–85. F. Barthelat, et al., Appl. Scan. Probe Methods XIII (2009) 17–44. S.P. Kotha, et al., Compos. Sci. Technol. 60 (2000) 2147–2158. H. Gao, et al., Proc. Natl. Acad. Sci. U. S. A. 100 (2003) 5597–5600. S.P. Kotha, et al., J. Mater. Sci. 36 (2001) 2001–2007. S. Zuo, Y. Wei, Acta Mech. Solida Sin. 20 (2007) 198–205. B. Bar-On, H.D. Wagner, J. Mech. Phys. Solids 59 (2011) 1685–1701. B. Bar-On, H.D. Wagner, Compos. Sci. Technol. 72 (2012) 566–573. H.L. Cox, Br. J. Appl. Phys. 3 (1952) 72. D. Hull, An Introduction to Composite Materials, Cambridge Press, 1981. A.P. Jackson, et al., Proc. R. Soc. Lond. B 234 (1988) 415–440. F. Barthelat, et al., J. Mech. Phys. Solids 55 (2007) 306–337. Z. Zhang, et al., Proc. R. Soc. Lond. B 278 (2011) 519–525.
Contents lists available at SciVerse ScienceDirect
Materials Science and Engineering C journal homepage: www.elsevier.com/locate/msec
New insights into the Young's modulus of staggered biological composites Benny Bar-On, H. Daniel Wagner ⁎ Department of Materials and Interfaces, Weizmann Institute of Science, Rehovot 76100, Israel
a r t i c l e
i n f o
Article history: Received 15 August 2012 Received in revised form 4 October 2012 Accepted 12 October 2012 Available online 23 October 2012 Keywords: Staggered structures Bio-mechanics Bio-composites Nano-composites
a b s t r a c t This communication presents a simplified “mechanics-of-materials” approach for describing the mechanics of staggered composite architectures, such as those arising in a variety of biological tissues. This analysis calculates the effective modulus of the bio-composite and provides physical insights into its elastic behavior. Simplified expressions for high- and low-mineralized tissues are then proposed and the effects of the mineral thickness ratio and aspect ratio on the modulus are demonstrated. © 2012 Elsevier B.V. All rights reserved.
1. Introduction
2. The mechanics of staggered biological composites: analysis
Staggered architectures are found at the micro- and nano-scales of a variety of biological tissues, and their stiffness significantly affects the macroscopic mechanical properties of tissues as a whole [1,2]. The mineralized collagen fibril, which is the building block of bones and teeth [3–5], can be viewed as a staggered array of stiff hydroxyapatite platelets embedded inside a compliant collagenous matrix. Similarly, the nacreous layer of a shell is made up of aragonite tiles joined together by a thin layer of protein — resulting in a highly stiff brick-and-mortar structure [6,7]. Jager and Fratzl [3] and Kotha et al. [8] were the first to propose approximated micro-mechanical models for the elastic deformations of such staggered architectures. Gao et al. [9] introduced a simplified model for describing the fundamental mechanics of such composites and proposed a compact formula for the effective modulus. Other works included also shear-lag effects in the analysis and showed improved expressions [10,11], which coincide with Gao's formalism for the case of a very soft matrix. These models, however, are all designed for the case of small axial spacing between the platelets and neglect the axial force carried by the matrix. Recently, Bar-On and Wagner [5,12,13] proposed an accurate compact formula for the effective modulus of staggered composite structures with a generic platelet architecture (the BW formula). In this communication we propose a simplified approach for the BW formula, provide additional physical insight and clarify the mechanical principles of staggering in biological composites.
A staggered biological composite is considered here as a twodimensional array of stiff platelet reinforcement with length Lp and height hp, forming a staggered structure inside a soft matrix (Fig. 1). Both the platelets and the matrix are considered isotropic with Young and shear moduli Ep and Gp, and Em and Gm, respectively. Each periodic unit of the array has length L, height h, and contains effectively two platelets, as shown in Fig. 1. For later convenience, we consider a ‘unit cell’ which is a quarter periodic unit, as also shown in Fig. 1. The platelet volume fraction within the staggered array, ϕp, is related to Lp, hp, L and h via:
⁎ Corresponding author. Tel.: +972 8 934 2594. E-mail address: [email protected] (H.D. Wagner). 0928-4931/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.msec.2012.10.003
ϕp ¼
2hp Lp : hL
ð1Þ
The staggered structure is conveniently characterized by three independent dimensionless geometric parameters: the platelet aspect ratio (ρp), and the thickness and length ratios (Δh and ΔL), defined as follows: ρp ¼
Lp ; hp
Δh ¼
2hp ; h
ΔL ¼
Lp : L
ð2Þ
Δh and ΔL represent the platelet volume fractions along and perpendicular to the staggering direction, noting that the overall platelet volume fraction is ϕp = ΔhΔL. For evaluating the stiffness of the bio-composite along the staggered direction, we focus on a unit cell, the lateral faces of which are subjected to an external force F per unit thickness (the out-of-plane direction), as shown in Fig. 2. This unit cell is divided into two types of sub-regions: sub-region 1 (SR1) which is asymmetric and sub-region 2 (SR2)
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B. Bar-On, H.D. Wagner / Materials Science and Engineering C 33 (2013) 603–607
Fig. 1. A schematic description of a staggered bio-composite. The periodic unit and unit cell are indicated by the blue and black dashed lines.
which is symmetric. Due to the external loading, the unit cell is effectively experiencing a longitudinal deformation and a transverse contraction, whereas other types of deformations (e.g. bending) are considered to be small in view of the overall symmetry of the periodic unit. The external force (F) applied to SR1 is split into platelet and matrix forces, fp and fm, such that (Fig. 3): F ¼ f p þ f m:
ð3Þ
SR2 is subjected to the fp and fm forces, applied on both sides by the corresponding elements in SR1 (Fig. 3). The edges of those platelets that end at the border between SR1 and SR2 are assumed not to adhere to the matrix at SR1 – similarly to the Cox model [14] – and are therefore free of forces. In the following, the longitudinal moduli of SR1 and SR2 (E1 and E2, respectively) are derived analytically, and the effective modulus of the unit cell is then composed via the inverse rule of mixtures, namely the classical Reuss model [15]. Starting with SR1 in which the shear deformations are assumed to be negligible, the axial deformations of the platelet and the matrix are considered to be uniform (δp = δm ≡ δ1), as shown in Fig. 4, and the overall strain of SR1 is εp = εm = ε1 = δ1/(Lp/2). By applying Hooke's law and using Eq. (3), fp and fm are easily calculated: fp ¼ F
Ep Δh ; 2E
ð4aÞ
fm ¼ F
Em ð2−Δh Þ 2E
ð4bÞ
Fig. 2. The unit cell of a staggered bio-composite divided into two types of sub-regions, SR1 and SR2, subjected to an external force F (per unit thickness).
Fig. 3. Diagram of forces acting on SR1 and SR2. fp and fm are the platelet and matrix forces at the edges of the sub-regions. The black dashed square represents a segment of SR2 of length dX, located at a distance X from the left edge. fp(+)(X) and fp(−)(X) are the local forces acting on the upper and lower cross sections, and τ is the shear stress within the matrix between the platelets.
where Ē is the effective modulus of SR1, calculated by using the Voigt model [15]: i F=ðh=2Þ 1 h E ¼ E1 ¼ ¼ Ep Δh þ Em ð2−Δh Þ : ε1 2
ð5Þ
Note that the effect of shear deformations is indeed negligible for long and slender platelets, which is the case in many biological arrays [12]. The analysis of SR2 is a bit more complicated due to variations in the platelet axial force within SR2 (from fp to zero), which involve non-negligible shear deformations. The local axial forces within the upper and lower platelets are denoted as fp(+) and fp(−) respectively (Fig. 3), and their longitudinal gradients are related to a shear stress within the matrix (τ) through the equilibrium relation (Fig. 3): ð Þ
df p ¼ τ: dX
ð6Þ
The X coordinate varies in SR2 from 0 (left edge) to Lp − L/2. τ is proportional to the shearing angle, assumed to be uniform across the matrix, and is given by:
τ ¼ Gm
uðþÞ −uð−Þ h=2−hp
ð7Þ
where u (+) and u (−) are the local displacements of the upper and lower platelets. The governing equations for fp(±) are extracted by a
Fig. 4. Schematic deformation patterns of SR1 (a) and SR2 (b). The deformation δ1 of SR1, δp of the platelet and δm of the matrix, are all equal. The deformation δ2 of SR2 is composed of the axial stretching δp of the platelets and the platelets sliding due to shear deformations δm within the matrix.
B. Bar-On, H.D. Wagner / Materials Science and Engineering C 33 (2013) 603–607
set of mathematical manipulations on Eqs. (6)–(7), resulting in the following relation (Appendix A): d2 f ðpÞ dx2
2 ðÞ −γ BW f p
1 2 ¼ − γ BW f p ; 2
γBW
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Gm Δh ¼ 2ρp 1− ð8Þ 2ΔL Ep 1−Δh
where x = X/(Lp − L/2) is a non-dimensional coordinate (0 b x b 1), and γBW is a shear-lag parameter, representing the build-up range of fp(±) from the edges of SR2. By solving Eq. (8) with the corresponding boundary conditions (Fig. 3), fp(±) becomes: ðÞ
f p ðxÞ ¼
fp 1 sinhðγBW xÞb coshðγBW xÞ þ 1 : tanhðγBW =2Þ 2
ð9Þ
Once fp(±) is found, the axial deformation of SR2 (δ2) can be evaluated. δ2 is dominated by two deformation mechanisms, the platelet stretching (δp) and a relative sliding between the platelets due to the matrix shearing (δm), as shown in Fig. 4. δp is obtained by integrating Eq. (9) along x and using Hooke's law. δm is proportional to the shear stress (τ) at the edge of SR2 (via Hooke's law); τ is extracted by combining Eqs. (6) and (9). The overall strain of SR2 is then (Appendix B): ε2 ¼
δp þ δm fp 1 1 ¼ : 1þ2 γBW tanhðγ BW =2Þ Lp −L=2 Ep hp
ð10Þ
The modulus of SR2 (E2) is calculated by the following relation (using Eqs. (4a) and (10)): F=ðh=2Þ γBW E: E2 ¼ ¼ ε2 cothðγ BW =2Þ þ ðγ BW =2Þ
As a final step, the moduli of SR1 (Eq. (5)) and SR2 (Eq. (11)) are combined by using the Reuss formula, resulting in a closed-form expression for the longitudinal modulus of the staggered bio-composite, named the BW formula: EBW ¼
A critical value of γBW exists (γBW ≈2.4), at which the moduli of SR1 and SR2 are equal. There, we have EBW =E1 =E2 =Ē, and EBW is not affected by variations of the relative amount of SR2 (through 2ΔL −1). For γBW b 2.4, E1 >E2, indicating that SR2 functions as a compliant adhesive between the SR1 parts. For these γBW values, an increase of 2ΔL −1 further reduces the modulus of the composite. When γBW >2.4, E1 b E2 and the modulus of the staggered array exceeds Ē. Here, higher 2ΔL − 1 ratios increase the value of EBW. 3. Example taken from the biological world Staggered bio-composites are typically made of slender reinforcements, for which the aspect ratio (ρp) – by being proportional to the shear-lag parameter (γBW) – has a significant effect on the stiffness of the composite (see Fig. 5). To illustrate this we consider two here staggered configurations — approximating the structure of highmineralized and low-mineralized biological tissues. Mineralized tissues are typically made of mineral platelets embedded in a proteinous matrix, for which Ep/Em ≈100. High-mineralized tissues consist of tightly packed staggered platelets, separated by a thin layer of matrix material. Here we consider a very large mineral content (ϕp ≈95%) and a very large SR2 portion (ΔL →1) (i.e. Δh =ϕp/ΔL ≈0.95) — representing the architectures of the nacreous layer of mollusk shells or the tooth enamel. For these high-mineralized configurations Ē/Ep ≈0.5 and γBW ≈ρp/4 (via Eqs. (5) and (8), respectively), the hyperbolic-cotangent in Eq. (12) approaches unity (slender reinforcement), and the modulus of highmineralized tissues (EHM) can be approximated as: EHM ≈
ð11Þ
2ð1−ΔL Þ 2ΔL −1 −1 γBW E: þ ¼ ð2ΔL −1Þ cothðγBW =2Þ þ γBW ð3=2−ΔL Þ E1 E2
ð12Þ This coincides with the BW formula in [5] by substituting ΔL =ϕp/Δh. Fig. 5 plots EBW/Ē as a function of γBW for selected 2ΔL −1 values, representing the relative amount of SR2 within the bio-composite. As seen, EBW exhibits different characteristics in the different γBW intervals.
Fig. 5. EBW/Ē as a function of γBW, for selected 2ΔL − 1 values representing the relative amount of SR2 within the composite. γBW ≈ 2.4 is a critical point where the modulus of SR2 is equal to the modulus of SR1. For γBW b 2.4 the modulus of SR2 is lower than the modulus of SR1, and vice versa. γBW is proportional to the platelet aspect ratio; slender platelets therefore increase the modulus of the composite.
605
1 E ðhigh mineralized tissuesÞ: 1=γBW þ 1=2
ð13Þ
By considering ρp ≈ 15 − 20 and ρp ≈ 50 − 100 as representative aspect ratio ranges for nacre and enamel, respectively, Eq. (13) approximates the modulus of the tissues as ENacre/Ep ≈ 0.65 − 0.7 and EEnamel/Ep ≈ 0.85 − 0.93, in good agreement with experimental data from the literature (e.g. [16,17]). Low-mineralized tissues can be characterized by several biocomposite architectures. For the current demonstration we consider a staggered configuration having thin platelets and a very large SR2 portion, as recently proposed for describing the small-scale mechanics of low-mineralized tendon [18]. ΔL →1 and Δh ≈0.2 (i.e. ϕp ≈0.2) are selected for the analysis, for which Ē/Ep ≈0.1 and γBW ≈ρp/30 — indicating a significant softening of both SR1 and SR2 compared to the high-
Fig. 6. The approximated moduli of high-mineralized and low-mineralized tissues, normalized by the modulus of the reinforcing platelet (EHM/Ep and ELM/Ep), plotted as a function of the mineral platelet aspect ratio ρp.
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B. Bar-On, H.D. Wagner / Materials Science and Engineering C 33 (2013) 603–607
mineralized configuration (see Fig. 5). Using Taylor expansion for the hyperbolic cotangent in Eq. (12), the modulus of low-mineralized tissues (ELM) is as follows: ELM ¼
1 E ðlow mineralized tissuesÞ: 2=ðγ BW Þ2 þ 2=3
ð14Þ
Fig. 6 plots EHM and ELM as a function of the reinforcement aspect ratio, using Eq. (13), Eq. (14) and the corresponding Ē and γBW parameters. The figure indicates that the modulus of bio-composite made up of thin platelets (low-mineralized) is much lower than that made up of thick platelets (high-mineralized). It is also shown that both EHM and ELM monotonically increase with ρp but the effect is more significant for ELM. Note that EHM → Ep for ρp ≫ 1, demonstrating the efficiency of slender staggered bio-composites.
By considering the balance of forces on the platelet segments as shown in Fig. 3, the longitudinal gradients of fp(±) are related to the shear stress within the matrix (τ) via: df ðpÞ ¼ τ: dX
Following Hooke's law for the matrix, τ is proportional to the shearing angle of the matrix via the shear modulus: τ ¼ Gm φ:
In conclusion, the modulus of staggered bio-composite is dominated by the moduli of SR1 and SR2 (E1 and E2), which are combined by the inverse rule of mixtures (Reuss model). E1is governed by Δh (Eq. (5)), E2 is governed by Δh and γBW (Eq. (11)), and the parameter ΔL determines the mixing ratio between E1 and E2. Higher values of Δh indicate that the platelets support a greater portion of the external loading — increasing the moduli of SR1 and SR2. A greater γBW value indicates that load transfer between the staggered platelets takes place closer to their edges — resulting in smaller deformations in SR2 and enhancing the modulus of SR2. Since γBW ∝ ρp and Δh ∝ hp, it can be concluded that thicker or slender platelets result in stiffer staggered biocomposite, as indeed is observed in stiff natural configurations such as highly-mineralized nacre and enamel. Acknowledgments We acknowledge support from the Israel Science Foundation (grant no. 1509/10) and from the G. M. J. Schmidt Minerva Centre of Supramolecular Architectures. This research was made possible in part by the generosity of the Harold Perlman family. H.D. Wagner is the recipient of the Livio Norzi Professorial Chair. Appendix A
uðþÞ −uð−Þ h=2−hp
σ
ðÞ
¼ Ep ε
ð Þ
:
ðA:1Þ
The local axial forces within the upper and lower platelets, fp(±), are: ðÞ
fp
¼σ
ðÞ
hp : 2
ðA:2Þ
The strains ε (±) are defined as the gradients of the platelet displacement fields (u (±)): ε
ð Þ
ðÞ
du : ¼ dX
τ ¼ Gm
By combining Eqs. (A.1)–(A.3), fp(±) and u (±) are related via; ðÞ fp
hp duðÞ : ¼ Ep 2 dX
uðþÞ −uð−Þ : h=2−hp
d2 f ðpÞ dX
2
¼
Gm 1 1 ðþÞ ð−Þ f −f p : Ep h=2−hp hp =2 p
ðA:9Þ
Transforming into a non-dimensional coordinate x = X/(Lp − L/2) (0 b x b 1), Eq. (A.9) transforms into: d2 f ðpÞ 2
dx
¼
Gm 2Lp −L 2Lp −L ðþÞ ð−Þ f p −f p : Ep h−2hp hp
ðA:10Þ
The fractions in Eq. (A.10) can be expressed by the nonL 2h dimensional geometric relations in Eq. (2), ρp ¼ hpp ; Δh ¼ h p and Lp ΔL ¼ L , resulting in the following equations: d fp
dx2
1 2 ðþÞ ð−Þ ¼ γ BW f p −f p ; 2
γBW
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Gm Δh ¼ 2ρp 1− : ðA:11Þ 2ΔL Ep 1−Δh
Eq. (A.11) represents two coupled equations, governing the evolution of fp(±) along SR2. An additional relation between fp(±) can be obtained by adding the two relations in Eq. (A.5): df ðpþÞ df ðp−Þ þ ¼ 0: dX dX
ðA:12Þ
By integrating Eq. (A.12) over X and substituting the boundary conditions at X =0 as shown in Fig. 3, fp(+)(X = 0)= 0 and fp(+)(X = 0)= fp, the following relation is obtained: ðþÞ
ð−Þ
¼ f p:
ðA:13Þ
By substituting Eq. (A.13) into Eq. (A.11), the coupling between the two governing equations is released, resulting in independent equations for the fp(±) forces, as given in Eq. (8): 2 ð Þ
d fp
ðA:4Þ
ðA:8Þ
By derivation of Eq. (A.5), substituting Eq. (A.8) and expressing the displacement gradients by Eq. (A.4) we obtain:
fp þ fp ðA:3Þ
ðA:7Þ
where the denominator is the height of the matrix in SR2 (Fig. 2). Combining Eqs. (A.6) and (A.7) yields:
2 ðÞ
This appendix includes the derivation of Eq. (12), which is the governing equation for fp(±). The longitudinal stresses and strains within the upper and lower platelets (σ (±) and ε (±)) are related via Hooke's law:
ðA:6Þ
Since τ is assumed to be uniform across the matrix, φ can be expressed by the displacements of the platelets as follows: φ¼
4. Conclusions
ðA:5Þ
dx2
2
ð Þ
−γBW f p
1 2 ¼ − γBW f p ; 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 Gm Δh : ðA:14Þ γBW ¼ 2ρp 1− 2ΔL Ep 1−Δh
B. Bar-On, H.D. Wagner / Materials Science and Engineering C 33 (2013) 603–607
ance of force on a platelet segment (Eq. (A.5)) and substituting fp(±) from Eq. (9), δm is found to be:
Appendix B This appendix includes the derivation of Eq. (12), representing the overall strain of SR2. The total deformation of SR2 can be considered as a summation of two contributions, the platelet stretching (δp) and a relative sliding between the platelets due to the matrix shearing (δm), as shown in Fig. 4. δp is extracted by integrating local strain along one of the platelets: 1 ð Þ δp ¼ Lp −L=2 ∫0 ε dx
ðB:1Þ
where Lp − L/2 is the length of SR2. Following Eqs. (A.1) and (A.2), ε(±) is related to fp(±) via: ε
ð Þ
¼
f ðpÞ 1 : hp =2 Ep
607
ðB:2Þ
By substituting Eq. (B.2) into Eq. (B.1), using the expression for fp(±) from Eq. (9) and integrating over x, δp is found to be:
δm ¼
h=2−hp fp τðx ¼ 1Þ ¼ Gm 2
1 h−2hp Gm 2Lp −L
!
γ BW : tanhðγBW =2Þ
Eq. (B.5) can be further simplified using the definition of γBW in Eq. (8), which results in: δm ¼
f p 2Lp −L 1 : Ep hp tanhðγ BW =2Þ
ðB:3Þ
δm is related to the shearing angle at the edge of SR2 (either φ(x = 0) or φ(x = 1)), multiplied by the height of the matrix in SR2: δm ¼ h=2−hp φðx ¼ 1Þ:
ðB:4Þ
Following Hooke's law for the matrix in Eq. (A.6), φ is proportional to the shear stress within the matrix (τ). By further considering a bal-
ðB:6Þ
The total strain of SR2 is calculated by combining δp and δm from Eqs. (B.3) and (B.6) and dividing by the length of SR2 (Lp − L/2, resulting in the expression in Eq. (10): δp þ δm fp 1 1 ¼ : ðB:7Þ 1þ2 ε2 ¼ γBW tanhðγ BW =2Þ Lp −L=2 Ep hp References
Lp −L=2 1 1 ðÞ f p Lp −L=2 ∫ f dx ¼ : δp ¼ hp =2 Ep 0 p Ep hp
ðB:5Þ
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
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