Mechanics of the hysteretic large strain behavior of mussel byssus threads

Abstract

Natural fibers are particularly interesting from a materials point of view since their morphology has been tailored to enable a wide range of macroscopic level functions and mechanical properties. In this paper, we focus on mussel byssal threads which possess a morphology specifically designed to provide a hysteretic yet resilient large strain deformation behavior. X-ray diffraction studies have shown that numerous natural fibers have a multi-domain architecture composed of folded modules which are linked together in series along a macromolecular chain. This microstructure leads to a strong rate and temperature dependent mechanical behavior and one which exhibits a stretch-induced softening of the mechanical response as a result of the underlying morphology evolving with imposed stretched. This paper addresses the development of a constitutive model for the stress–strain behavior of the distal portion of mussel byssal threads based on the underlying protein network structure and its morphology evolving with imposed stretched. The model will be shown to capture the major features of the stress–strain behavior, including the highly nonlinear stress–strain behavior, and its dependence on strain rate and stretch-induced softening.

Appendices

Appendix A: Derivation of the strain energy contribution due to unbending

Approximating the bent fiber to consist of two straight segments of length L ₁ and L ₂ connected by a circular bend of length ρ α (with α = α₁ + α₂), as shown in Fig. 12, a form for the force-stretch behavior of the unbending of a single fiber is derived. Both segments are assumed to be rigid, so that all the deformation is accommodated by unbending of the kinked region. In addition, the contour length of the kink is assumed to be fixed (i.e. stretching of the kink region is neglected), so that

$$ \rho \alpha=\rho_0 \alpha_0. $$

(A.1)

Figure 12 clearly shows that the angles α₁ and α₂ satisfy the condition

$$ L_1 \sin\alpha_1=L_2 \sin\alpha_2, $$

(A.2)

and, taking L >  > ρ α, the fiber stretching λ = r/r ₀ may be approximated as

$$ \lambda=\frac{L_1\cos\alpha_{1}+L_2\cos\alpha_{2}}{r_0}, $$

(A.3)

so that from Eqs. A.2 and A.3 α₁ and α₂ may be expressed as a function of λ as

$$ \alpha_1=\hbox{arcsin}\left[\frac{\sqrt{4 L_2^2 \lambda^2 r_0^2-(\lambda^2 r_0^2+L_2^2-L_1^2)^2}} {2 \lambda r_0 L_1}\right],\quad\alpha_2=\hbox{arccos}\left[\frac{\lambda^2 r_0^2+L_2^2-L_1^2}{2 \lambda r_0 L_2}\right]. $$

(A.4)

The strain energy of the fiber due to the unbending of the bend is given by

$$ w^u=\frac{1}{2}EI(\Delta\kappa)^2 \rho_0 \alpha_0, $$

(A.5)

where EI is the bending stiffness and Δκ denotes the change in curvature of the bend

$$ \Delta\kappa=\left(\frac{1}{\rho}-\frac{1}{\rho_0}\right)=\frac{\alpha- \alpha_0}{\rho_0 \alpha_0}. $$

(A.6)

Substitution of Eq. A.6 into Eq. A.5 yields

$$ w^u=\frac{EI}{2 \rho_0 \alpha_0}(\alpha-\alpha_0)^2, $$

(A.7)

so that the fiber force-stretch behavior is given by

$$ f^u=\frac{\partial w^u}{\partial r}=\frac{2 EI}{\alpha_0 \rho_0}\frac{\lambda r_0 (\alpha-\alpha_0)} {\sqrt{4 L_2^2 \lambda^2 r_0^2-(\lambda^2 r_0^2+L_2^2-L_1^2)^2}}. $$

(A.8)

Note that in case of a symmetrically bent fiber (i.e. L ₁ = L ₂), Eq. A.8 reduces to

$$ f^u={\frac{2 EI}{\alpha_0 \rho_0 r_0}\frac{\alpha_0-2\hbox{ arccos}\left(\lambda\cos {\frac{\alpha_0}{2}}\right)} {\sqrt{1-\left(\lambda\cos{\frac{\alpha_0}{2}}\right)^2}}}\cos\frac{\alpha_0}{2}. $$

(A.9)

Appendix B: Parameter identification for the constitutive model

The mussel byssus is a biological material and thus its mechanical properties are observed to vary, mostly influenced by the season, the temperature, the water salinity [41, 42]. Thus the identification of material parameters entering in the constitutive model is aleatoric and related to all these effects. However, Figs. 1 and 4 clearly show that the stress–strain behavior of the distal segment consists of three different regions (Fig. 20):

• an initial linear response: it is used to identify the value of the fiber Young’s modulus E;

• ‘yielding’ followed by stretch-induced softening: the ‘yielding’ points at different strain rates are used to identify the unfolding parameters x _u and α _u ;

• strain hardening: it permits us to identity the length of the fully unfolded fiber L _max.

The geometrical parameters θ₀, L ₁₀, L ₂₀, H ₀ and d have all been be measured with transmission electron microscopy and the values reported in Hassenkam et al. [8] are used.

Fig. 20

Nominal stress-engineering strain behavior in uniaxial tension of the distal region at extension rates of 10 mm/min and 1,000/mm. It consists of three different regions which used to identify the material parameters

To better facilitate the identification of the material parameters entering in the constitutive model additional tests are needed:

• additional tensile tests at multiple strain rates to better understand the rate-dependence of ‘yield’;

• Relaxation tests to better understand the sources of the rate-dependence;

• Unloading–reloading tests at higher strain rates to better understand the unloading/reloading process;

• Force test of a single macromolecule to validate the choice of material parameters.