NEUROMORPHIC ACOUSTIC SOURCE LOCALIZER
3.2 DESIGN GUIDELINES FOR BIOMIMETIC HARD MATERIALS
The remarkable performance of materials like bone and nacre is due to their sophisticated microstructure, in which high-aspect-ratio min-eral tablets are bonded between layers of organic materials and arranged in a brick-wall structure FIGURE 3.2 Tensile stress-strain behavior of nacre.
Adapted from Ref. 14.
known as a staggered arrangement [14, 23]; see Figure 3.3. This staggered arrangement of min-eral tablets is also found in other biological materials such as teeth, collagen fiber, spider silk, and cellulose fiber and is now considered a universal pattern [20], providing attractive com-binations of stiffness, strength, and toughness [23, 24]. Several analytical and numerical mod-els have therefore been proposed to predict the behavior of staggered structures [25, 26]. These models for stiffness, strength, and toughness are briefly reviewed next.
3.2.1 Stiffness
The tensile behavior of staggered structures has been investigated by a simple shear-tension chain model. The model is based on a small representative volume element (RVE) of the structure shown in Figure 3.4 [25]. The tablets have length L, thickness t, and overlap L/2.
These tablets are bonded between organic lay-ers of thickness ti.
Assuming an elastic, perfectly plastic behav-ior for the interface and a constant shear field along the interface, Kotha et al. [27] derived the following expression for the tensile modulus E of staggered composites:
where
Em is the Young’s modulus of the mineral, and Gi is the shear modulus of the interface.
A reasonable assumption for biological hard materials is that the organic interfaces are much (3.1) 1
E =
1+ ti
t 1 Em+2t
L tiγ
Gi
1+cosh(γL) sinh(γL)
,
(3.2) γ =
Gi Em
1 tit,
FIGURE 3.3 Staggered arrangement of mineral tablets. (a) scanning electron micrograph of nacre, (b) schematic of min-eralized collagen fibrils in bone (adapted from Ref. 25), and (c) scanning electron micrograph of tooth dentin. Adapted from Ref. 26.
3.2 DESIGN GUIDELINES FOR BIOMIMETIC HARD MATERIALS 63
softer than the mineral tablets (Gi Em) [14].
Equation (3.2) can then be rewritten as [28, 29]
where φ is the volume fraction of mineral tablets and ρ=L/t is the tablet aspect ratio. This expres-sion reveals the effects of microstructural param-eters on the modulus. Thus, for constant Gi, the elastic modulus of the material converges to φEm
when the mineral volume fraction φ increases to values near 1. Therefore, for high mineral con-centrations, the modulus of staggered compos-ites reaches its theoretical limit associated with a Voigt (uniform strain) composite.
3.2.2 Strength
Under tensile loading, staggered structures fail either at the interfaces (tablet pull-out fracture mode) or through the tablets (brittle fracture).
The latter should be prevented so that the tablets slide on each other and energy is dissipated through inelastic deformation at the interfaces.
Assuming tablet pull-out fracture mode and using a simple shear-tension load-transfer chain, we see that the strength of the composite σs can be written as [30]
provided that ti t. This expression shows that the strength of the composite is controlled by the shear strength τs of the interfaces and the aspect ratio ρ of tablets. Although strength increases with the aspect ratio, very high aspect
(3.3) 1
E ≈ 1
ϕEm +41 ρ2
1−ϕ ϕ2
1 Gi,
(3.4) σs =
1 2ρτs,
ratios may result in brittle fracture of the tablets, which is a detrimental failure mode. This limit to the aspect ratio of tablets is discussed in Section 3.2.5.
3.2.3 Toughness
Toughness is the most remarkable property of hard biological materials [31, 32]. Nacre, as an example, shows a toughness that is 3,000 times higher than that of its main constituent (arago-nite). Several experimental studies have identi-fied the main toughening mechanisms of these materials [14, 33]. Recently, it has also been theoretically demonstrated that the toughness achieved through the staggered arrangement of nacre is far greater than the toughness of both the mineral and organic mortar [26, 29].
Several powerful mechanisms exist in nacre to resist crack propagation and increase tough-ness. Crack deflection, crack bridging, and viscoplastic-energy dissipation in volumes of material around cracks (process zone) are the dominant toughening mechanisms. Bridging develops as a crack advances and occurs when mineral tablets are not completely pulled out.
The shear stresses between tablets therefore apply closure forces to the crack faces. The pro-cess zone, where the tablets slide onto one another, consists of two parts: the frontal zone and the wake. The frontal zone is the area in front of the crack tip experiencing inelastic deformation at the interfaces. Once the crack is advanced through the frontal zone, the stresses are released and some of the deformation is recovered, leaving a wake behind the crack tip, as depicted schematically in Figure 3.5. The energy is therefore dissipated through loading and unloading of inelastic interfaces. The effect of moisture on this inelastic behavior is also crucial; increasing the moisture plasticizes the organic molecules and increases their deform-ability [9].
When a crack advances across the direction of the tablets, as shown in Figure 3.5, the effect FIGURE 3.4 Representative volume element of a
stag-gered composite when loaded in tension (Umax represents the maximum cohesive displacement).
of bridging and process zone can then be esti-mated for the steady-state case; thus [29]
where J is the mode-I fracture toughness of the composite, Umax is the maximum cohesive dis-placement as depicted in Figure 3.4, and Ji is the toughness of the interface. In this model, debonding is assumed to happen after the ulti-mate shear strain of the organic ulti-material at the interface is exceeded. The ultimate shear strain is the maximum shear strain that the organic mate-rial at interfaces can withstand.
The values for toughness predicted by this model were found to be in good agreement with experimental values for red-abalone nacre [29].
This shows that the toughness of nacre can be analytically explained by the effect of bridging (3.5) J=
ρ
2. 5−(Umax/ L) (1/ρ) (E/τs)Ji,
and process zone. Equation (3.5) also shows that: (i) staggered arrangement amplifies the toughness of interfaces, (ii) increasing the aspect ratio positively affects the toughness, and (iii) composites made of stiff inclusions and soft interfaces have enhanced toughness (term E/τs) [29]. These models for stiffness, strength, and toughness can greatly help the designers choose the best materials and microstructural parame-ters in order to tailor and optimize the perfor-mance of biomimetic staggered composites.
3.2.4 Strain Hardening at the Interfaces
The foregoing models show the effects of shear strength and shear modulus of interfaces on the behavior of staggered composites. However, in natural composites like nacre, the shear strength of the interfaces is not constant and increases with increases in shear strain. This strain hard-ening at the interfaces causes progressive tablet sliding (Figure 3.6a), which is one of the most important deformation mechanisms of biologi-cal hard materials like nacre and is the origin of toughening mechanisms such as viscoelastic energy dissipation at process zone.
Progressive sliding prevents strain localiza-tion and spreads the deformalocaliza-tion through large volumes of material, thereby providing high levels of strain and therefore improving the energy absorption properties of the material (because this energy is the area under the stress-strain curve). Figures 3.6b and c show how the incorporation of wavy tablets improves the load transfer [14]. In the case of flat tablets, the load is transferred between the tablets only by shear stresses. For wavy tablets, tablet sliding generates transverse tensile and compression stresses, which contribute to the load transfer, increase the resistance over sliding, and ate hardening. The organic material itself gener-ates hardening if the shear resistance of organic material increases with shear strain. The choice of organic material is therefore crucial for FIGURE 3.5 Schematic of a crack advancing in a
stag-gered composite, where λ represents the bridging length and a represents the crack advance. Adapted from Ref. 30.
3.2 DESIGN GUIDELINES FOR BIOMIMETIC HARD MATERIALS 65
improving the load transfer in biomimetic materials.
3.2.5 Size Effects
In staggered composites, the flow of stress is such that the interfaces are under shear while mineral inclusions are under tension (tension-shear chain). Therefore, the mineral inclusions should resist high levels of tensile stress in order to prevent brittle fracture. Brittle materials are sensitive to initial flaws, which, for example, include organic molecules embedded in the mineral crystals during the biomineralization process [34, 35]. These organic molecules are much softer than the mineral and act as cracks within the material.
For a cracked brittle inclusion, the condition for failure is governed by the Griffith criterion:
where σmf is the fracture strength of the mineral, γs is the surface energy, h is the thickness of the mineral tablet, and the parameter α depends on the geometry of the crack. Based on the Griffith criterion, Gao et al. [26] showed that the strength of the inclusions increases when they are made smaller, because they can only contain small defects. In theory, inclusions smaller than a critical size of 30 nm [26] have a strength (3.6) σmf =αEmφ, φ=
γs
Emh,
approaching the theoretical strength of the material. Interestingly, the size of mineral inclu-sions in hard biological materials like bone and tooth is on the same order [36, 37]. This suggests that nanometer inclusions in these materials maximize their fracture resistance.
Although in nacre the tablets are in the micrometer range, their small size still confers on them high strength. For example, Bekah et al.
[30] found that the aspect ratio must be small enough so that an assumed edge crack extend-ing halfway through the tablet is prevented from propagating further. This condition is given by:
where KIC is the mode-I fracture toughness of the tablets. This expression suggests that by decreasing the thickness of the tablets, the maxi-mum allowable aspect ratio in the structure increases. Increasing the aspect ratio is desirable because it improves the performance of materi-als with staggered structure, as indicated by Eqs. (3.1)–(3.3).
Bekah et al. [30] also argued that junctions in the staggered composites act as crack-like features when the material is loaded in tension.
Thus, decreasing the tablet thickness results in a decrease in the size of these crack-like fea-tures and therefore decreases the resulting stress-intensity factor KI. Computing this
(3.7) ρ <0. 56KIC
τs
√t,
FIGURE 3.6 (a) Schematic of progressive tablet sliding in staggered composites. Load-transfer mechanisms in (b) flat tablets and (c) wavy tablets (θ is the dovetail angle).
stress-intensity factor and setting the condition KI < KIC gives the soft-wrap condition for pre-venting the fracture of tablets:
The soft-wrap condition also shows that decreasing the thickness of the tablets improves the fracture resistance of the structure. Rather than improving the fracture resistance of the tablets, reducing the size of building blocks also increases the number of interfaces and therefore increases the energy dissipation during loading and unloading of organic material, improving fracture resistance. These recent developments in the understanding of design aspects of these materials can greatly assist the optimization of biomimetic hard and stiff materials.