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, 278 (1705), 519-25

On Optimal Hierarchy of Load-Bearing Biological Materials

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On Optimal Hierarchy of Load-Bearing Biological Materials

Zuoqi Zhang et al. Proc Biol Sci.

Abstract

Load-bearing biological materials such as shell, mineralized tendon and bone exhibit two to seven levels of structural hierarchy based on constituent materials (biominerals and proteins) of relatively poor mechanical properties. A key question that remains unanswered is what determines the number of hierarchical levels in these materials. Here we develop a quasi-self-similar hierarchical model to show that, depending on the mineral content, there exists an optimal level of structural hierarchy for maximal toughness of biocomposites. The predicted optimal levels of hierarchy and cooperative deformation across multiple structural levels are in excellent agreement with experimental observations.

Figures

Figure 1.
Figure 1.
Typical hierarchical structures of load-bearing biological materials: (a) bone [7,16], (b) mineralized tendon fibre [18], and (c) shell [21]. (Figures have been modified based on those from the corresponding papers cited here.)
Figure 2.
Figure 2.
A quasi-self-similar hierarchical material. Every level structure consists of slender hard inclusions (blue) aligned in a parallel staggered pattern in the soft matrix (yellow). The aspect ratio of the inclusions varies from level to level. The inclusions at the (n + 1)-th level are made of the staggered microstructure at the n-th level.
Figure 3.
Figure 3.
Bottom-up design route for quasi-self-similar hierarchical materials.
Figure 4.
Figure 4.
Variations of (a) Young's modulus, (b) strength, (c) toughness, and (d) size of an N-level quasi-self-similar hierarchical material. (a,b) Turquoise circles, equal-strength criterion; blue triangles, combined criterion, Θp = 35%; red squares, combined criterion, Θp = 100%. (c,d) Purple pentagons, equal strength criterion, Θp = 35%; turquoise circles, equal strength criterion, Θp = 100%; blue triangles, combined criterion, Θp = 35%; red squares, combined criterion, Θp = 100%.
Figure 5.
Figure 5.
Plots of strain ratios formula image (red circles) and formula image (green squares) for the first three levels of hierarchy in mineralized tendon fibre against experimental data [11]. Error bars are standard errors of the mean. Solid lines, theoretical, one soft matrix; dotted lines, theoretical, two soft matrices.
Figure 6.
Figure 6.
Stress transfer in the tension-chain model of biocomposites under uniaxial tensile load. The soft matrix (yellow) is modelled as an idealized plastic material. The dotted line depicts the general constitutive behaviour of protein.

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