Tensile test and the static strength of byssus network

In order to gain fundamental understanding of the global mechanical behaviour of the entire byssus network and its static strength, we assess the adhesion force between a clay substrate and the entire mussel (Mytilus edulis) using a tensile machine, for two loading modes, tension and shearing, in order to represent two extreme types of forces a mussel encounters (Fig. 1b). A series of snapshots of the deformation process of the byssus network under external loading is shown in Fig. 2a,c as well as corresponding force-extension curves depicted in Fig. 2b,d. Both force-extension curves feature two regimes before rupture, an initial linear regime and a subsequent stiffening regime. During the first regime, all byssus threads rotate towards the loading direction and more threads are subjected to tension. During the second regime, the elongation of aligned byssus threads contributes to the increasing strain, and the deformation of the soft proximal part is more significant than that of the stiff distal part. We repeat the measurements on different mussels similar in size and find strengths as 5.4±1.8 N (tension) and 5.1±2.8 N (shearing), where the measured force level agrees well with earlier test results1. The force is one order of magnitude larger than the mussel’s self-weight of 0.09±0.02 N without sea water between valves, or ~0.3 N with sea water between valves (Supplementary Table S1), ensuring its attachment in air at low tide. Yet, the strength is much smaller than the dynamic impact force, which is defined as the mean value of instant dynamic loading generated from waves on the fixed mussel at shores (measured to be ~17.8 N2). The impact force exceeds the strength of mussel threads and should make it impossible for the mussel to attach to bare rock, and make this an unlikely place to settle. However, this contradicts the observation that many isolated mussels are frequently found at exposed rocks and ships (Fig. 1a and Supplementary Fig. S1). Considering the fact that the dynamic loading is much larger than the static strength of byssus threads, the mechanical function of the byssus thread is more than simply affixing the mussel to the environment.

Figure 2: Experiments and force-extension curves for the byssus network. (a) Snapshots of the byssus network under the tension mode loading. Snapshots I–IV are taken at applied strain of 0, 0.66, 1.39 and 1.64, respectively. Scale bar, 10 mm. (b) Force-extension (f–ε) curve of the network corresponds to the snapshots in a, with each corresponding strain noted in the plot. (c) Snapshots of the byssus network under the shearing mode loading. Snapshots I–IV are taken under the strain of 0.74, 1.09, 1.40 and 1.76, respectively. Scale bar, 10 mm. (d) Force-extension curve of the network corresponds to the snapshots in c, with each corresponding strain noted in the plot. Full size image

Experiment-based computational modelling

The mussel with the network of byssus threads forms a complex structure, in which the link between architecture of the structure and the material properties of the threads are coupled. To gain insights into the mechanical behaviour of the byssus network under loading, we model the byssus network based on our own experimental measurements and literature data, and use the model to carry out a series of computational experiments to understand the mechanisms by which it can successfully deal with severe mechanical conditions. We model each byssus thread as three parts with distinct mechanical properties, that is, the proximal part, the distal part and the adhesion plaque as illustrated in Figs 1b and 3a. The tensile properties of the first two parts are fitted to experimental measurements1 as shown in Fig. 3b (details in the Methods section). Their differences are likely caused by the different ratio of secondary structures14,15. The umbrella-shaped byssus network model is built based on a single thread model with its shape described by the height–radius ratio (H/R). We model ten threads in the network as it is the maximum number of threads that deform together, which is interpreted from our own experiments, and as has been shown in literature1. For each thread of the length L, CL denotes the length of the distal part and (1−C)L the length of the proximal part, where C=80% corresponds to the natural composition of the byssus thread1,15,16. Owing to the lack of knowledge about the interacting strength between the mussel foot protein and clay, we perform tensile tests for single threads as illustrated in Fig. 3c and Supplementary Fig. S2, and obtain the detachment force as 0.4±0.1 N as shown in Fig. 3d. We also performed mechanical tests on several surfaces including natural rock and wood that are relevant to the substrates inhabited by mussels. We find that rocks yield a similar adhesion strength as clay. For wood, the strength fluctuates significantly as the wood substrate tends to rot in sea water. We combine these results to model each byssus thread (details in the Methods section and Supplementary Table S2) and apply tensile force to a single thread at the distal part in simulation to validate our model against experimental observations. Our model reproduces the experimental data well, and rupture occurs consistently at the plaque–substrate interface (Fig. 3e). The force-extension curve, as shown in Fig. 3f, is composed of a linear regime followed by a softening regime before failure at 0.4 N, in good agreement with experiments. These established agreements between simulation and experiment suggest that our model reflects the mechanical behaviour of byssus threads under loading. In addition to measuring the nonlinear elastic properties of the threads, we perform measurements to obtain the viscosity within a single byssus thread as shown in Fig. 4a–c (additional details provided in the Supplementary Method 1). This viscosity added to the particle interactions in the thread model yields a viscous force along the byssus thread. We test the mechanical responses of the thread under very different pulling rates and validate these results against experimental tensile tests under corresponding pulling rates15 (as shown in Supplementary Fig. S3). We use this model to assess how viscosity affects the mechanical response of the byssus network. To do this, we deform the entire byssus network by using pulling rates from 1 m to 10 m s−1 and find that the viscosity term has a negligible effect on the force-extension curves, as shown in Fig. 4d.

Figure 3: Experimental and computational tests of a single thread. (a) Configuration and geometric characteristics of a byssus thread network model. The grey bead represents the centre of mass of the mussel. (b) Tensile property of the proximal part and the distal part of a single thread. (c) Schematic figure to illustrate how we test the single fibre’s adhesion force in experiment. Scale bar, 4 mm. (d) Force-extension curve of a single thread under tension obtained in the experimental tensile test. (e) Computational snapshots of the single thread under tension, fracture at the adhesion plaque governs the failure of the thread. Scale bar, 2 mm. (f) Computational result of the force-extension of the single thread model corresponds to the snapshots in e, with each corresponding strain noted in the plot. Full size image

Figure 4: Viscosity and mechanical response. (a) A mussel thread is kept in a thin tube (inner diameter, 0.6 mm) filled with sea water, scale bar, 4 mm. (b) The thread and tube is mounted on a tensile machine while the thread is in tension while the upper clamp moves at a constant strain rate of 0.02 s−1. (c) We slightly deform the thread to reach the stress (σ) of 7.4 MPa, stop loading and record the stress relaxation as a function of time. The residue stress converges at a 7.1 MPa level. (d) Simulation results of the force-strain curves of the byssus network under different pulling rates and modes. We test the model with a geometry as shown in Fig. 3a with (w/) and without (w/o) considering the viscosity within a single thread. The model is stretched both in tension and shearing modes with 1 and 10 m s−1 loading speeds (strain rate of 33 and 333 s−1), which are very relevant to the impact speeds (up to 3.5 m s−1) on mussels. Overlapping of the curves (with and without the viscosity term) before the peak force suggests that the viscosity term has a negligible effect on the mechanical response of the byssus network under loading. Full size image

Dynamic response under impact loading

Impact loading applied to the mussel is modelled as an instantaneous initial velocity of the mussel’s body, reflecting that it accounts for 99.5% of the total mass of the system. According to the experiment, the measured impact force of 17.8 N yields an instantaneous velocity of 2.0 m s−1 at the center of mass of the mussel where the byssus threads attach to (see the Methods section for details). We test the instantaneous velocity both in the vertical direction for the tension mode and along the substrate for the shearing mode, as shown in Fig. 5a,c, respectively. We find that the proximal part undergoes significant deformation as the mussel moves apart while the distal part remains in a relaxed state with zero tensile force. The tensile force in the proximal part is rather sensitive to the movement of the mussel from its equilibrated position as it reaches a peak within 0.01 s. However, the tensile force in the distal part at that moment is only halfway to reaching the peak force, as shown in Fig. 5b,d. The peak force in the distal part is much smaller than that of the proximal part, and reaches 80% in the tension mode and only 53% in the shearing mode. The proximal part connected to the mussel effectively buffers the impact to the adhesion plaque. Therefore, we can understand the role of the proximal part as a safety belt of the byssus network, as it is critical to absorb the impact loading. We increase the initial speed stepwise in the simulations, and find that the maximum impact speed for failure is 3.3 m s−1 for tension and 3.5 m s−1 for shearing. Those initial velocities correspond to a maximum impact force of 48.5 N, which is over nine times that of the static strength. This value agrees well with the highest impact forces on mussels that have been recorded at shores and reported in the literature2.

Figure 5: Dynamic response of the byssus thread network after impact. (a) Simulation snapshots of the deformation of the byssus network under the impact loading by giving the mussel an initial velocity of 2 m s−1 in the tension direction. Scale bar, 20 mm. (b) Sum of the force within the byssus threads at the proximal part and that at the distal part as functions of time for the tension mode. In this plot, we denote the time when each subfigures I to III in a is taken in simulation. It is noted that the mussel stretched by the byssus network decelerates, and then moves in the opposite direction as shown in III. (c) Simulation snapshots of the byssus network under the impact loading by giving the mussel an initial velocity of 2 m s−1 in the shearing direction. Scale bar, 20 mm. (d) Sum of the force within the byssus threads at the proximal part and that at the distal part as functions of time for the shearing mode. In this plot, we denote the time when each subfigure I to III in c is taken in simulation. Full size image

Energy absorption and dynamical strength by structure

In order to understand the underlying mechanism, and specifically how the material distribution in the byssus thread (that is, the ratio of stiff material versus the entire thread, here denoted by C) is adapted to dynamic loading, we systematically alter the ratio between the distal and proximal parts and the geometry of the byssus network, and investigate how those characteristics affect the dynamic response of the byssus network under impact loading. The total energy of the system (the sum of kinetic energy and potential energy) as a function of time before the mussel reaches maximum deformation is shown in Fig. 6a,b for the tension and shearing modes, respectively. The energy dissipation for byssus threads with a proximal part is significantly higher than the byssus thread composed of a pure distal part (C=1). We measure the time when the total energy decreases to half of its initial value as shown in Fig. 6c,d, where C=1 corresponds to the slowest total energy dissipation while decreasing C corresponds to faster energy dissipation. The transition between C=0.5 and 1 contributes mostly to the increment of the rate of energy dissipation. We repeat the measurement of the decay time of the kinetic energy to its half value (Fig. 6c,d and Supplementary Fig. S4) and find that a larger C leads to a greater advantage in terms of its ability to decrease the kinetic energy. This shows that the combination of those two materials benefits the energy absorption of the byssus network as the distal part quickly transits the instant kinetic energy into potential energy, while the proximal part dissipates the kinetic energy by viscosity. For pure elasticity, the peak force is proportional to where K eff is the effective stiffness of the network and m is the mass of the mussel while for pure damping the peak force is proportional to vη eff where η eff is the effective coefficient of viscosity, which depends on the thread diameter and the value of water viscosity. The reaction force experienced by the mussel is proportional to (detailed derivation in the Methods section)

Figure 6: Composition and structure affect the energy and force. (a,b) History of the total energy (E) of the byssus network under impact loading before the maximum displacement of the mussel. (c,d) Time constants ( ) when the kinetic energy (E ke ) and the total energy (as the sum of E ke and potential energy (E pe ) as E=E ke +E pe )decrease to half of their initial values. Continue curves are obtained by exponential fit. The results summarized in (a–d) are for the byssus network with H/R=0.6. (e,f) Normalized reaction force, given by the ratio of the peak force exert to the mussel (f mussel ) and the maximum peak force on the adhesion plaque (f π ), is shown for different C and H/R of the network. The continue curves are fitted results as given by equation (8) (with parameters summarized in Supplementary Table S3). For each H/R, the C value that gives the minimum force is coloured in grey in the graph. a, c and e are results of the tension mode, and b, d and f are results of the shearing mode. Full size image