A novel biomimetic material duplicating the structure and mechanics of natural nacre

Abstract

Nacre from mollusk shell is a high-performance natural composite composed of microscopic mineral tablets bonded by a tough biopolymer. Under tensile stress, the tablets slide on one another in a highly controlled fashion, which makes nacre 3000 times tougher than the mineral it is made of. Significant efforts have led to nacre-like materials, but none can yet match this amount of toughness amplification. This article presents the first synthetic material that successfully duplicates the mechanism of tablet sliding observed in nacre. Made of millimeter-size wavy poly-methyl-methacrylate tablets held by fasteners, this “model material” undergoes massive tablet sliding under tensile loading, accompanied by strain hardening. Analytical and finite element models successfully captured the salient deformation mechanisms in this material, enabling further design refinements and optimization. In addition, two new mechanisms were identified: the effect of free surfaces and “unzipping.” Both mechanisms may be relevant to natural materials such as nacre or bone.

Appendices

Appendix: RVE Calculations

A simplified analytical model was derived to capture the progressive locking of the RVE depicted in Fig. A1.

FIG. A1.

(a) Periodic RVE of the composite; (b) reduced RVE using symmetry; and (c) RVE deformation under uniaxial stress along the x axis.

A1. Boundary conditions

The RVE of the composite is periodic in the two in-plane directions. Representative volume elements were, therefore, used to capture its mechanics. Figure A1(a) shows an RVE of the material with dimensions. The model will capture uniaxial behavior of the composite, and for this reason the average shear strain was set to zero ( $\bar \gamma _{xy} > 0$ ). Periodic boundary conditions in displacements and tractions are enforced on all sides of the RVE:

$$\left\{ {\matrix{ {{u_x}\left( {{L_{\rm{o}}} + {L_{\rm{c}}},y} \right) = {u_x}\left( { - \left( {{L_{\rm{o}}} + {L_{\rm{c}}}} \right),y} \right) + 2{{{{\bar \varepsilon }}}_{xx}}\left( {{L_{\rm{o}}} + {L_{\rm{c}}}} \right)} \hfill \cr {{u_y}\left( {{L_{\rm{o}}} + {L_{\rm{c}}},y} \right) = {u_y}\left( { - \left( {{L_{\rm{o}}} + {L_{\rm{c}}}} \right),y} \right)} \hfill \cr {\vec t\left( {{L_{\rm{o}}} + {L_{\rm{c}}},y} \right) = \,\,\, - \vec t\left( { - \left( {{L_{\rm{o}}} + {L_{\rm{c}}}} \right),y} \right)} \hfill \cr } } \right.$$

((A1))

$$\left\{ {\matrix{ {{u_x}\left( {x,t} \right) = {u_x}\left( {x, - t} \right)} \hfill \cr {{u_y}\left( {x,t} \right) = {u_y}\left( {x, - t} \right) + 2{{{{\bar \varepsilon }}}_{yy}}t} \hfill \cr {\vec t\left( {x,t} \right) = \,\, - \vec t\left( {x, - t} \right)} \hfill \cr } } \right.$$

((A2))

The RVE was further reduced using symmetries about the x and y axes, as shown in Fig. A1(b). The associated boundary conditions are

$$\left\{ {\matrix{ {{u_x}\left( { - x,y} \right) = - {u_x}\left( {x,y} \right)} \hfill \cr {{u_y}\left( { - x,y} \right) = {u_y}\left( {x,y} \right)} \hfill \cr } } \right.\quad,$$

((A3))

$$\left\{ {\matrix{ {{u_x}\left( {x, - y} \right) = {u_x}\left( {x,y} \right)} \hfill \cr {{u_y}\left( {x, - y} \right) = - {u_y}\left( {x,y} \right)} \hfill \cr } } \right.\quad,$$

((A4))

Combining Eqs. (A1), (A2), (A3), and (A4) yields the periodic-symmetric boundary conditions:

$$\left\{ {\matrix{ {{u_x}\left( {0,y} \right) = 0} \hfill \cr {{u_x}\left( {{L_{\rm{o}}} + {L_{\rm{c}}},y} \right) = \left( {{L_{\rm{o}}} + {L_{\rm{c}}}} \right){{{{\bar \varepsilon }}}_x}} \hfill \cr {\vec t\left( {{L_{\rm{o}}} + {L_{\rm{c}}},y} \right) = - \vec t\left( { - \left( {{L_{\rm{o}}} + {L_{\rm{c}}}} \right),y} \right)} \hfill \cr } } \right.\quad,$$

((A5))

$$\left\{ {\matrix{ {{u_y}\left( {x,0} \right) = 0} \hfill \cr {{u_y}\left( {x,t} \right) = t{{{{\bar \varepsilon }}}_y}} \hfill \cr {\vec t\left( {x,t} \right) = - \vec t\left( {x, - t} \right)} \hfill \cr } } \right.\quad,$$

((A6))

These conditions imply that the edges of the RVE can deform, but in doing so they must remain straight and parallel to their initial location. To impose uniaxial tension, a uniform displacement is imposed along the right boundary. The sliding of the tablets is expected to provide the large extensions of the RVE, so that axial strains in the tablets are neglected:

$${{{\bar \varepsilon }}_x} \approx {{{u_{\rm{S}}}} \over {{L_{\rm{o}}} + {L_{\rm{c}}}}}\,.$$

((A7))

Note that owing to Poisson’s effect and other mechanisms described below, the RVE may expand or contract in the transverse direction.

In the following sections, the mechanics of the RVE is examined in more depth. Stresses and strains are assumed to be uniform in the overlap region of the tablets ( $\sigma _x^{{\rm{(O)}}},\sigma _y^{{\rm{(O)}}},\varepsilon _x^{{\rm{(O)}}},\varepsilon _y^{{\rm{(O)}}}$ ) and in the core regions ( $¡gma _x^{{⤪{(C)}}},¡gma _y^{{⤪{(C)}}},⋴repsilon _x^{{⤪{(C)}}},⋴repsilon _y^{{⤪{(C)}}} $ ). The balance between these strains and stresses results from interface kinematics, stress transfer at the interface, elasticity of the tablets, and boundary conditions.

A2. Kinematics at the interface

The geometry of the dovetail is such that sliding the tablets may generate expansion in the y direction. This expansion is combined with the strains in the tablets to generate the transverse displacement at the upper edge of the RVE:

$${u_y}\left( {x,t} \right) = t{{\varepsilon }}_y^{\left( {\rm{O}} \right)} + {u_{\rm{S}}}\tan {{\theta,}}$$

((A8))

and for small θ,

$${u_y}\left( {x,t} \right) = t{{\varepsilon }}_y^{\left( {\rm{O}} \right)} + {u_{\rm{S}}}{{\theta }}\,{\rm{, }}$$

((A9))

Because of the symmetric-periodic boundary conditions (A6), the expansion along y must be uniform on the upper boundary (y = t) of the RVE, so that

$${{{\bar \varepsilon }}_y} = {{\varepsilon }}_y^{\left( {\rm{C}} \right)} = {{\varepsilon }}_y^{\left( {\rm{O}} \right)}{{{u_{\rm{S}}}} \over t}{\rm{ + \theta }}{\rm{. }}$$

((A10))

A3. Stress transfer at the interface

The key mechanism for the composite is the load transfer at the interface in the overlap region. Let N and T be the forces normal and tangential to the interface in the overlap region, respectively. They are connected through Coulomb friction by

$$T = fN\,.$$

((A11))

The x and y components of the contact force can then be written with the small angle approximation as

$${R_x} \approx N{{\theta }} + T\,,$$

((A12))

$${R_y} \approx N + T{{\theta }}\,{\rm{.}}$$

((A13))

Combining Eqs. (A7), (A10), and (A11) gives the longitudinal force as function of the transverse force:

$${R_x} = {{{{\theta }}\,{\rm{ + }}\,f} \over {1 + f{{\theta }}}}{R_y}\,{\rm{.}}$$

((A14))

In turn, these forces generate stresses in the overlap region:

$${{\sigma }}_x^{\left( {\rm{O}} \right)} \approx 2{{{R_x}} \over t}\,\,,$$

((A15))

and

$${{\sigma }}_y^{\left( {\rm{O}} \right)} \approx - {{{R_y}} \over {{L_{\rm{o}}} - {u_{\rm{s}}}}}\,\,,$$

((A16))

Note that as the tablets slide on one another, the contact area decreases. Note also that R_x induces a tensile (positive) stress along the x direction, whereas R_y induces a compressive (negative) stress along the y direction. Combining the Eqs. (A14), (A15), and (A16) provides

$${{\sigma }}_x^{\left( {\rm{O}} \right)} = - {2 \over t}{{{{\theta + }}f} \over {1 - f{{\theta }}}}\left( {{L_o} - {u_{\rm{s}}}} \right){{\sigma }}_y^{\left( {\rm{O}} \right)}\,\,,$$

((A17))

Equation (A17) shows that the tensile stress in the material is provided by a friction term and augmented by a “locking term” proportional to the dovetail angle. The effect of friction is magnified by the dovetail angle through the term f θ at the denominator.

A4. Tablet elasticity

The RVE model is composed of two elastic blocks (modulus E and Poisson’s ratio ν). In this simplified analytical model, the stresses and strains are assumed to be uniform in the core area and in the overlap area (although they might take different values in these two areas). The elasticity in the tablets is governed by Hooke’s law, in-plane stress conditions

$${{\varepsilon }}_y^{\left( {\rm{C}} \right)} = {1 \over E}\left[ {{{\sigma }}_y^{\left( {\rm{C}} \right)}\, - {{\nu \sigma }}_y^{\left( {\rm{O}} \right)}} \right]\,\,,$$

((A18))

and for the overlap area,

$${{\varepsilon }}_y^{\left( {\rm{O}} \right)} = {1 \over E}\left[ {{{\sigma }}_y^{\left( {\rm{O}} \right)}\, - {{\nu \sigma }}_x^{\left( {\rm{O}} \right)}} \right]\,\,,$$

((A19))

A5. Fastener

The tablets are held in place by transverse fasteners (bolts) running through the core regions. Initially the bolt may be tightened by a distance Np, where N is the fraction of turn given to the nut and p is the screw pitch (defined as the axial distance between two consecutive threads). In addition, the bolt will expand or contract according to the transverse deformations of the RVE. The total strain in the bolt is, therefore, ${{Np} \over t} + \varepsilon _y^{{\rm{(C)}}}$, and the force carried by each bolt is

$${F_{\rm{b}}} = {{{A_{\rm{t}}}{E_{\rm{b}}}} \over t}\left( {{N_p} + t{{\varepsilon }}_y^{\left( {\rm{C}} \right)}} \right)\,\,,$$

((A20))

where A_t is the tensile area of the bolt, E_b is the elastic modulus of the bolt, and ε_b is the strain in the bolt.

A6. RVE force balance

Along the x axis, the applied axial stress is simply given by

$${{{\bar \sigma }}_x} = {{\sigma }}_x^{\left( {\rm{C}} \right)}\,.$$

((A21))

The axial stress is transmitted in the core region through a thickness t, whereas it is transmitted through the overlap region through a thickness t/2. Therefore, one can write

$${{\sigma }}_x^{\left( {\rm{O}} \right)} \approx 2{{\sigma }}_x^{\left( {\rm{C}} \right)}\,.$$

((A22))

Along the y axis balancing the forces gives

$$w\left( {{L_{\rm{O}}} - {u_{\rm{S}}}} \right){{\sigma }}_y^{\left( {\rm{O}} \right)} + {L_{\rm{C}}}w{{\sigma }}_y^{\left( {\rm{C}} \right)} = - {{{A_{\rm{t}}}{E_{\rm{b}}}} \over t}\left( {Np + t{{\varepsilon }}_y^{\left( {\rm{C}} \right)}} \right)\,\,.$$

((A23))

A7. Result

Equations (A10), (A17), (A18), (A19), (A22), and (A23) form a system of six equations that can be solved for the six unknowns ( $\sigma _x^{{\rm{(O)}}},\sigma _y^{{\rm{(O)}}},\varepsilon _y^{{\rm{(O)}}},\sigma _x^{{\rm{(C)}}},\sigma _y^{{\rm{(C)}}},\varepsilon _y^{{\rm{(C)}}}$ ). Of particular interest is the axial stress as a function of the axial strain:

$${{{{{{\bar \sigma }}}_x}} \over E} = {{{{\kappa }}{{\phi } {\rm{ + }}\left( {1 + {{\kappa }}} \right){{\alpha }}{{{{\bar \varepsilon }}}_x}{{\theta }}} \over {{1 \over {{{\alpha }}\left( {{{\beta }} - {{{{\bar \varepsilon }}}_x}} \right)}}\left[ {{{1 - {{{{\bar \varepsilon }}}_x}} \over {1 - {{\beta }}}} + {{\kappa }}} \right]{{1 - {{\theta }}} \over {{{\theta }} + f}} + {{\nu }}\left[ {1 + 2{{\kappa }}} \right]}}}\,\,\,,$$

((A24))

with the nondimensional tablet aspect ratio $\alpha > {L \over t},$ overlap ratio $⌆ta = {{L_{⤪{0}} } ⩈er L},$ the nondimensional bolt stiffness $⦎ppa = {{A_{⤪{t}} } ⩈er {wL}}{{E_{⤪{b}} } ⩈er E}{1 ⩈er {1 - ⌆ta }},$ and the nondimensional bolt tightening $ϕ = {{Np} ⩈er t}.$ This result highlights the effect of the structure material parameters on the tensile response of the material.