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On optimal hierarchy of load-bearing biological materials, Zhang, Zhang, Gao, Proceedings of The Royal Society B, 2011


A novel quasi-self-similar hierarchical model is developed and addresses an important question concerning the hierarchical design principle of strong and tough biological materials, which provides insights in developing high-performance bioinspired structures. It is uncovered that, depending on the mineral content, there exists an optimal number of hierarchy level for maximized toughness, e.g., 6 levels for bone and 1-4 for nacre, agreeing well with experimental observations.

Scientific question:

What determines the number of structural hierarchy levels or is it the more levels the better in strong and tough biological materials?

Key of how:

By developing a quasi-self-similar mechanical model considering the flaw tolerance, strength criteria, and limited constituents, a bottom-up route determining the expressions of strength, modulus, and fracture energy as functions of number of structure hierarchy number is obtained. The present model is superior to the previous one in terms of considering the criteria of equal strength and efficient stress transfer and the same type matrix for all levels. The plots show clear dependences of fracture energy and flaw-tolerant size on the structure hierarchy number, with optimized values found.

Major points:

1. Multi-level structural hierarchy is one universal strategy for biological materials to fulfill functions and mechanical properties with high efficiency. Despite extensive characterization on the structure-property relationships, the underlying mechanisms explaining the size scale are recently illustrated, while what determine the number of the hierarchical levels remains unanswered.

2. Depending on different mineral content, shell nacre (95%), bone (45%), and mineralized tendon (15%) show hierarchical structures with 2-3, 7, and 4 levels in hierarchy, respectively.

3. There have been limited studies on biological structures from a global hierarchical point of view (e.g., size limit for the hierarchy for Gecko’s feet); for load-bearing biological materials, only one previous self-similar hierarchical model exists, which fail to explain the different numbers in hierarchy in different materials nor the experimentally measured strains at different hierarchical levels of bone. The model developed in this work bridges the gap.

4. The quasi-self-similar hierarchical model is designed and derived as the following:

At each level, rectangular hard inclusions are staggered in a soft matrix; the structure at the n-th level serves as the reinforcing inclusions at the (n+1)-th level. According to the accepted idea that the structural hierarchy is optimized for stiffness and toughness, and the limited selections of raw materials, a set of principles are used:

(1) principle of flaw-tolerance (to be insensitive to crack-like flaws requires the inclusion’s characteristic width to satisfy certain quantitative relationship with respect to (w.r.t.) the modulus, strength, and fracture energy, and multi-level properties can be calculated by previously derived recursive formulae), (2) criteria of equal strength and efficient stress transfer (the inclusion’s aspect ratio/length  needs to allow simultaneous failure of the matrix and the inclusion or efficient stress transfer (no larger than required length for inclusion’s linear increase of axial stress from constant matrix shear)), (3) limited selection of constituent materials (the matrix is the same type of protein (same shear modulus, shear stress, etc.) at all hierarchical levels, this is more realistic).

The volume fraction of inclusions is assumed fixed at each level. Inserting specific material properties of elastic modulus, shear modulus, strength, shear strength, and failure strains of matrix with two values (representing two types of materials) leads to the calculated Young’s modulus, strength, fracture energy/toughness, and flaw-tolerant size as functions of structural hierarchy number. The plots show the trends clear.

5. The modulus and strength of hierarchical composites decrease significantly with increasing number of hierarchy, while the toughness and flaw-tolerant size increase and then decrease with increasing number of hierarchy; thus an optimal number of hierarchy exists for each condition (failure strain or mineral content) in terms of maximum toughness. This is reasoned that the structural hierarchy allows the stress and deformation well partitioned between the inclusion and the matrix.

The predicted size scales and the optimal level of hierarchy are consistent with the experimental observations.

6. The present model predicts strains for the first three hierarchical levels (levels 0, 1, and 2); the ratio of the three strains is in good agreement with experimentally measured one of the mineral, fibril, and tissue.

This is a significant success.

7. For matrix failure strain of 0.35-1.00 representing mineralized tendon or bone, the optimized hierarchy number if 4-6, which agrees with the observation that tendon and bone show 4 and 7; for shell with failure strain of 0.35-1.00, structure hierarchy is 1-4, which is in agreement with the observed number of 2-3.

This is a clear superiority to the previous hierarchical model, which predicts monotonously increasing stiffness and toughness due to the use of ever-softer matrix with increasing hierarchy number, in that the present model considers more realistic situations and therefore higher agreement with practical measurement.

The present work is a classical one in understanding the biological design and mechanics for the development of next generation of high-performance composites.

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