Nanomechanics of silk: the fundamentals of a strong, tough and versatile material

Silk is a protein material and has a hierarchical organization that reaches from the atomic to the macroscale. Figure 1 describes the hierarchical organization of spider silk. At the molecular scale, silk is a sequence of amino acids which dictates the folding mechanism and the final molecular structure. Polyalalnine-rich regions fold into β-strands bonded by hydrogen bonds which form β-sheets. They are stacked into stiff β-sheet nanocrystals which are embedded in a disordered semi-amorphous phase derived from the folding of glycine-rich protein [6, 24–26]. This semi-amorphous phase constitutes the majority of silk and consists of disordered secondary structures. Along the fiber axis, the distribution of crystalline regions is heterogeneous [37]. Up a level of scale, the silk nanocomposites form a network that is nanoconfined into fibrils. The fibrils are bundled together into a fiber of a few micrometers which are bundled into fibers to finally form webs [6, 24–26]. For silkworm silk, the protein chain is organized into two fibroins which are bonded to each other. The pair of fibroins is coated in sericin glue to form the fiber [38, 39].

Spider silk's mechanical properties originate from the complex molecular structure of the amino acid sequence. The different types of silk have different sequences which result in different molecular structures and different mechanical properties to fulfill specific functions [40, 41]. However, the pattern between the different silks are similar. The proteins are composed mainly of repetitive amino acid motifs and bounded by non-repetitive N-termini and C-termini [42, 43]. The sequences of the termini are highly conserved, they remain the same for different silk types and spider species [8, 44]. They play a fundamental role in the storage of the spider protein in the glands and the self-assembly process for the formation of fibers [8, 43]. The common repeats are: (i) GPGGX/GPGQQ (where X is Q, L, Y or R [45]), (ii) GGX, (iii) poly-Ala/poly-Gly-Ala, and (iv) spacers [44, 46]. In particular, dragline silk from the spider N. Clavipes is known to be one of the strongest biomaterials, and is produced to build structural radial threads of webs. Dragline silk is produced by the major ampullate gland of the spider and is composed of two types of protein: major ampullate dragline silk protein 1 (MaSp1) and major ampullate dragline silk protein 2 (MaSp2) [6, 47, 48]. They represent 81% and 19% of dragline silk respectively [49]. The majority of the fiber is composed of MaSp1, while MaSp2 is found in clusters within the fiber [50, 51]. The major difference between MaSp1 and MaSp2 is the quantity of proline (P) residues which is 15% of MaSp2. There is no proline in MaSp1. The three principal repeats of MaSp1 are poly-Ala and poly-Gly-Ala and GGX. MaSp2 has the same repeats except that it has GPGGX domains instead of glycine-rich domains (GGX)[44]. The polyalanine-rich domains form stiff β-sheet nanocrystals responsible for the silk's strength. GGX motifs fold into 310-helix and GPGXX into β-spirals forming the semi-amorphous and disordered domain responsible for silk's elasticity [6, 50, 52]. The primary sequence controls the folding of the protein into its secondary structure which influences the mechanical and biological properties of silk. The behavior of silk can be predicted by understanding the role of each group of amino acids in the formation of the secondary structure (β-sheet, 310-helix, β-spirals) and their structural role (strength and elasticity). Thus, it is possible to control and improve silk's properties by modifying the protein sequence. For example, an increase of polyalanine domains would result in the formation of more β-sheet crystals, and consequently a stronger fiber. With only a few different amino acids, spiders can produce complex and versatile silks [16].

Although the primary structure is essential for the formation of strong, tough and versatile silk, it has to be associated with a particular spinning process during which certain structural features are formed. The silk produced from recombinant protein silk has lower mechanical properties than naturally spun spider silk: the spinning process is fundamental [8, 35, 36]. Dragline silk spinning process occurs in the major ampulate glands. First, in the tail, specialized cells secrete spidroins or spider protein which are then stored in the sac or ampulla. The stored spidroin is a highly concentrated aqueous crystalline solution [53–56], called spinning dope (up to 50% w/v) [6, 8, 57]. Next, the dope undergoes elongation and shear flow in the S-shape tapered spinning duct [6, 8]. The elongational flow forces the spidroin to align and engenders the formation of β-sheet nanocrystals highly oriented along the fiber axis, where β-sheets are tightly stacked and bonded by hydrogen bonds. Throughout the spinning duct, the dope is subjected to a decrease in pH, from moderately basic to moderately acidic pH. Simultaneously, phosphate and potassium ions are added to the dope, while water, sodium and chloride ions are removed from the dope. Phase separation occurs and water is extracted. After dehydration the fiber is no longer soluble, from a chemical point of view [8, 58–60]. Finally, the already aligned spidroin fiber is further extended and aligned after exiting from the spigot. The natural spinning process is illustrated in figure 2, during which the emerging silk fiber is pulled and stretched. A valve located before the spigot helps restart the spinning process when fibers internally break [8]. In the spinning duct, the transition from spidroin to silk fibers is explained by two theories. The first theory considers the spinning dope to possess a liquid-crystalline phase characteristics. The second theory considers the formation of micelles due to the hydrophobic and hydrophilic domains of the spidroin. Respectively, they are inside of the micelles and outside in direct contact of the water solvent [57]. In both cases, shear flow is essential for the formation of aligned fibers. The N-termini and C-termini are crucial to avoid premature folding of the stored spidroin. At a low pH, C-termini have a molten globule state which has no tertiary structure but conserves its secondary structure elements. This suggests that C-termini might be at the origin of the ordered aggregation into fibrils [61]. As for the N-termini, transition from monomer to homodimer are more likely to occur in a low salt concentration environment [62].

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Recently, Lin et al [63] have investigated the crucial spinning processing conditions and design parameters of spidroin self-assembly, through mesoscopic modeling and experiments. The synthetic silks used for the study are composed of three main motifs: polyalanine including hydrophobic domains (at the origin of stiff β-sheet crystals), GGX including hydrophilic domains (at the origin of the semi-amorphous phase) and hydrophilic hexastidine fusion tag to facilitate purification. These micelles merge together into larger micelles after shear flow. Similarly, the β-sheet crystals are larger. The protein is elongated and aligned. Furthermore, homogeneous and continuous network along the fiber axis (or shear flow direction) is only obtained for longer chains. The optimal ratio is obtained for an intermediate ratio of hydrophobic and hydrophilic domains. Shear flow triggers the transition from isotropic to anisotropic cylindrical structures and leads to stronger and more robust fibers [4, 63].

The folding self-assembly mechanism has not been fully understood up until now. Giesa et al [64] have investigated the crucial role of shear flow on the transition from the protein sequence to secondary structures. Molecular dynamics simulations were applied to model the transition of Nephila clavipes MaSp1 protein into secondary structures experimentally determined for silk fibers. The shear stress needed to transform unfolding protein into β-sheets is between 300 and 700 MPa, which also corresponds to transition shear stress for protein self-assembly. The shear stress was estimated at the same time period taken for the assessment of the maximum amount of β-sheets figure 3(A(i)). The transition shear stress is lower than the failure shear stress (indicated by dashed lines; note that the shear stress is derived from the pulling force divided by the effective area). The time period corresponds to the maximum content of β-sheets secondary structures (figure 3(A(ii))). The following drop in shear stress indicates that the molecule enters a low energy state after β-sheets crystals formation. It is observed that the transition into β-sheets occurs at around 20% lower stress than the failure stress. The transition shear stress is independent of the pulling speed and the number of chain repeats [64]. The different stages of protein assembly are illustrated in figure 3(B). Silk dope is first stored in an unfolded state (Stage 1), shear flow then aids to fold the protein and creates aligned β-sheet crystals (Stage 2). The aligned protein was subjected to the presence of a water/ion solvent and vacuum environment, to determine the stability of the post-shear flow protein. Post-spun fibers' natural environment is the air and is considered to be an intermediate between vacuum and water/ion solvent environment. Indeed, as discussed before, ion exchange occurs in the spinning duct and the exited fiber is water-insoluble and stable [8, 58]. In vacuum (Stage 3a), the protein conserves its β-sheets structures independently of chain repeats, and all the proteins are stable. In solvent and in the presence of ions (Stage 3b), a disordered and unfolded spidroin-like state is recovered for short chain protein. The protein is water soluble. The smallest stable molecule is composed of a minimum of six polyalanine repeats, the β-sheets structures remain stable and the β-sheets content is relatively constant (figure 3(B))[64]. Recently, Giesa et al [64] suggested that the stability of β-sheets structures originates from the helices close in space in the spidroin disordered state.

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Both Giesa et al [64] and Lin et al [63] showed the importance of shear flow and molecular makeup in the spinning process. The results of these studies can contribute to the improvement of microfluidic devices and the design and production of silk-inspired protein materials. Moreover, silk and spinning-inspired protein show attractive advantages: the processing occurs in a mild environment (room temperature and aqueous solvent) and using just a few building blocks. A better understanding could lead to new bio-inspired materials using limited resources and low energy processing.

As shown in the preceding section, the primary structure, associated with a specific spinning process, determines the folding mechanism and the characteristics of the secondary structures. The main secondary structures are: β-sheets nanocrystals controlling the stiffness of the fiber and embedded in a semi-amorphous phase composed of 310-helix, β-spirals and β-turns [65–68]. Specifically, β-sheet nanocrystals are composed of tightly stacked and hydrogen-bonded β-sheets, formed by anti-parallel β-strands (these β-sheet crystals represent about 10%–15% of silk volume [15, 69],). Their dimensions are typically on the order of a few nanometers [69–71]. Under tension, the nanocrystals are under lateral loading and they form interlocking regions to transfer the load between chains, and β-sheet nanocrystals reinforce the fiber akin to cross-links in a polymer network. The structures ultimately fail at large loads and deformation [69].

The high strength of the nanocrystals is due to the cooperation of the hydrogen bonds between the β-strands and β-sheets. In earlier work [72] we studied the deformation under shear of the hydrogen bond of the β-strands and β-sheets, by applying a structural elastic model. Hydrogen bonds are considerably deformed within a small region of a size of a few bonds, in which the deformation and rupture of hydrogen bonds are delocalized, acting in concert [73]. When the cluster of hydrogen bonds are confined in small amounts so that they correspond to the size of cooperative deformation, the hydrogen bonds are used fully, at maximum utilization of the associated material volume [72]. Henceforth, the force applied on β-strands or β-sheets, bonded by hydrogen bonds, of characteristic length ${L}_{0}$ will remain the same even for larger length. The characteristic length corresponds to the cooperative length, within which the hydrogen bonds deform cooperatively [71, 74]

$$\begin{eqnarray}&&{L}_{0}=b\sqrt{{K}_{1}/{K}_{2}},\end{eqnarray} \tag{ 1 }$$

where b (0.35 nm [71]) is the distance between two hydrogen bonds, ${K}_{1}$ and ${K}_{2}$ are the stiffness of the β-sheet nanocrystals (41.5 N m⁻¹ [72]) and hydrogen bonds (2.66–8.43 N m⁻¹ [72]) respectively [71, 74]. Although weak and non-covalent, hydrogen bonds contribute to the strength of the crystal, by resisting to shear failure through cooperation. In addition, they can reform after breaking.

To explore further mechanisms of deformation of β-sheets nanocrystals, large-scale molecular dynamic pull-out and bending simulations were reported that were aimed to determine the influence of crystal size under lateral load [69]. It has been established that for small β-sheets nanocrystals, deformations are controlled by shear and for large crystals, by bending. It is important to note that hydrogen bonds are in tension when the crystal is under bending (large size) which does not allow cooperation of hydrogen bonds. However for smaller crystals, hydrogen bonds work cooperatively to resist shear load. Bending to shear transition length is about 2.5 nm. In addition, because small crystals are stiff, under pull-out simulations crystals undergoes a stick-slip mechanism after the initial fracture. The stick-slip mechanism is as follows: the loaded β-sheet is pulled out of the crystal, breaking and reforming its hydrogen bonds (figure 4(B)). The peaks in the small crystal force-displacement plot show hydrogens bonds breaking and reforming and a substantial increase the total dissipated energy (figure 4(A(i))). The dependence of pull-out strength with crystal sizes has shown that small crystals show a strong and stiff response, which implies greater pull-out force and strength (figure 4(A(ii))). The variations of resilience and toughness with size show that smaller crystals dissipate more energy due to the stick-slip failure (figure 4(A(ii))). Critical nanoconfinement size is determined where strength (figure 4(A(ii))), resilience and toughness (figure 4(A(iii))) are at a maximum. The initial stiffness, breaking strength and toughness are maximized for a crystal of length of about 3 nm, in agreement with the previous result (figure 4(A)). The elastic energy storage before initial failure or resilience also increases with the decrease in crystal size. The estimated critical β-strand length (crystal width) is about 1–2 nm. Larger crystals are brittle and fail at a lower load resulting in a crack-like flaw, because of local failure of hydrogen bonds under tension (figure 4(C)). The study on larger crystals is relevant because under thermal excitation or other types of loading, the crystal may bend [24, 65, 69]. Nanoconfinement transforms weak hydrogen bonds into strong, tough and resilient β-sheets crystals.

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At the nanoscale, silk represents a two-phase composite organized in a semi-amorphous phase reinforced with stiff β-sheets nanocrystals. The phases provide extensibility and fracture strength respectively [68]. Stretching simulations on silk nanocomposites of MaSp1 and MaSp2 demonstrated the nonlinear behavior of spider silk. The characteristic force–displacement curve obtained from stretching of the nanocomposite follow four regimes (figure 5). First (i), the composite undergoes an initial stiff behavior [68]. The semi-amorphous phase stretch homogeneously until a yield point. At yield, the hydrogen bonds break resulting in the failure of the 310-helix, β-spirals and β-turns disordered secondary conformations [24, 65, 75]. (ii), The composite enters a softening regime [68]. The semi-amorphous phase unfolds along the pulling direction. The softening mechanism arises from the unraveling of the hidden length, unlocked from the rupture of the hydrogen bonds [24, 65, 75]. The decrease in β-turns content illustrates the unfolding of the secondary structures of the semi-amorphous phase. Simultaneously, the content in β-strand increases, which means that new hydrogen bonds are formed, leading to the creation of small β-sheet crystals in the semi-amorphous region (figure 6(B)) [24, 68]. In particular, the transitions from α-helices to β-sheets, have been observed at the chemical bond scale [16]. Colomban and Dinh [16] carried out a Raman analysis to determine the nanomechanics of silk fibers. As silk protein is a polyamide chain, they applied a controlled strain on the νN–H mode and measured the Raman shift generated. A perfect correlation was observed between the wavenumber shift-strain behavior of the hydrogen bonds at the nanoscale and stress–strain behavior of silk fibers at macroscale [16–18]. The softening regime (ii) is followed by (iii) a much stiffer regime [68]. The load is transferred from the fully extended semi-amorphous phase to the β-sheet nanocrystals [75]. Finally (iv), in the short softening regime preceding failure, the nanocomposite fails through a stick-slip mechanism: hydrogen bonds in the crystalline region break and lead to strands sliding, resulting in the separation of smaller intact nanocrystals [68, 75]. This specific mechanism is at the origin of silk's strength, extensibility and toughness.

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The importance of nanoconfinement of β-sheet crystals is identified again when associated with a semi-amorphous phase. In the stress–strain plots of different crystal size nanocomposites, the composite with small crystals (3 nm) shows large strength, toughness and a stick-slip mechanism (figure 6(A)). Only for small β-sheet crystals can the composite fully take advantage of the extensibility of the semi-amorphous phase through unfolding. Thus, nanoconfinement contributes vastly to the extensibility and energy dissipation capacity of silk [75].

In addition to stick-slip (shear) failure and brittle (bending) failure [69], it has been hypothesized that the strain stiffening behavior (stiffer regime (iii) described above) may be caused by β-sheet crystals unfolding [76]. Using molecular dynamics calculations, a recent study [70] proposed a mechanism for silk's strain stiffening behavior following failure of β-sheet crystals. Residual strengths resulting from two types of crystal failure mechanisms were compared: lateral separation of two β-sheets (corresponding to a stick-slip mechanism) and unfolding of the β-sheet. The residual strength was calculated from electrostatic and van der Waals force. As the residual strength of the unfolding of the β-sheet is significantly higher for any deformation state, the study proposed the following deformation mechanism: first, the β-sheet crystals fail by stick-slip giving rise to smaller crystals or just the separation of one β-sheet. If the failure strength of the smaller crystals is greater than that of the original crystal, they participate in the stiffening mechanism. Second, the β-sheets unfold. When completely unfolded and stable, they participate in a secondary stiffening regime. The straightened β-sheets perform as a fiber reinforcing the nanocomposite (figure 7) [70].

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In conclusion, at the nanoscale, silk is a two-phase nanocomposite that consists of the nanoconfined β-sheet and semi-amorphous phase, which combined, provide strength, extensibility and toughness. Cooperative deformation of hydrogen bonds transform weakness into strength within the crystalline regions.

Silk's upper-scale hierarchical level is reflected in macroscopic silk architectures, such as spider webs (in 2D or 3D), or silkworm cocoons. Silk macroscale structures were likely adapted through evolution to ensure characteristic functions. For instance, viscid spider silk is used to trap prey whereas dragline silk is used for framework of the web. Out of the numerous types of spider webs (cob webs, brushed sheet webs...), orb web is the most familiar and studied [77, 78]. As web construction is energy costly, spiders minimize material use while exploiting fully the unique properties of the different silks and securing durability of the web [79, 80]. Orb web architectures are organized in radial dragline silk (which properties are discussed previously) and spiral viscid or sticky silk [10, 15].

Recently, Qin et al [79] have investigated the mechanical performance of a 3D-printed synthetic elastomeric spider web to determine the influence of structural design and material distribution in web mechanics and design. Spider web experiments are difficult to repeat and compare, because webs are unique and testing requires destruction of webs. Consequently, testing on a 3D-printed web-inspired structure and simulations on its model were carried out. Under point loading, e.g. prey impact or spider weight, it is preferable to distribute uniformly silk in all the fibers in order to optimize strength. Radial and spiral threads as well should have the same diameter/thickness. On the other hand, for a distributed loading, e.g. wind loading, the web is the strongest when radial threads are thicker permitting sacrificial failure of thinner spiral threads. Defects are easily repairable. Furthermore, these different geometries are observed in nature as spiders optimize their use of silk through web architecture. For example, centimeter-scale spider webs (garden spider webs) are optimized for local load: the spiral and radial threads have similar diameter. Meter-scale spider webs (Caerostris darwini webs) are optimized for wind load as they cover a larger surface: radial threads are thicker [4, 79].

Although silk's properties are usually obtained through tensile testing of the fibers, Koski et al [81] determined the stiffness tensors of the fibers of an orb web using non-invasive, non-destructive Brillouin light scattering method. The stiffness of the dragline radial, viscid spiral silk threads and the silk junctions were mapped on the orb web. When hydrated, spider silk is subjected to supercontraction; it shortens its length by 50% and increases by more than 40% the stiffness of the fibers. Even though water droplets apply a gravity load, they stiffen the web considerably and the deflection of the web can decrease by 17% [77, 81].

While a spider web's robustness rests in part on its discrete architecture, a silkworm cocoon's remarkable properties are hypothesized to rely on its multi-layered nonwoven structure. Cocoons are spun from a single and continuous fiber that can reach 1 km in length. Layers and intersecting intra-layer fibers are bonded by sericin glue, which can be considered as the matrix of the composite. A cocoon is a three-dimensional laminated composite composed of a porous matrix reinforced by randomly arranged and oriented fibers. Cocoon performance highly depends on its porosity and bonding connectivity [82–86]. Understanding cocoon geometry and mechanics can result in new bio-inspired designs for materials, for instance, in random fiber and particulate composites [82].

Further understanding of the interplay between silk properties and macroscale structural optimization of webs could provide structural solutions for engineering, especially for resilient and lightweight design. In particular, robustness of webs and their underlying mechanics will be discussed in the following section.

Silk fibers not only have excellent mechanical strength and toughness but also are extremely robust against defects. Indeed, they perform remarkably well even though it has been observed that silk fibers often have defects such as cavities cracks, surfaces, and tears [87–89]. Defects can be very large (several hundreds of nanometers) for a micrometer size fiber and they create stress concentrations which could trigger fiber failure [90].

Spider silk fibers are composed of a bundle of fibrils defined by a three-dimensional two-phase composite. To understand how interplay between β-sheet crystals and semi-amorphous phase at the nanoscale influences and helps to resist fracturing at the macroscale, Giesa et al [90] applied a mesoscale spring-model to connect length-scales (figure 8(C)). The springs follow the same four-regime nonlinear behavior of the two-phase composite derived from nanomechanics (figure 5(A)). Various loading cases were investigated to describe the mechanical response of fibrils under different loadings (tension and shear) and defects. The crack size is 50% of the length (L) or width (H) of the fibril sample figure 8(A)). Fracture strengths were measured for the different cases and different fibril widths H, and were compared to fracture strengths of defect-free silk fibers obtained experimentally (figure 8(D)). Fracture stresses and strains increase considerably as the width H decreases. When fibril size H converges toward the critical length-scale H* = 50 ± 30 nm, failure stresses and strains also converge toward the experimental data of defect-free fibers. The large crack does not impact the mechanical response of fibrils, if H = H*. When H reaches H* = 50 ± 30 nm, there is a drastic increase of material entering regime (iv) in which the β-sheet crystals fail through stick-slip. After this threshold, 100% of the material contribute to resisting failure through the stick-slip mechanism, as seen in figure 8(B)). As a result, when fibrils are nanoconfined at H = H*, they behave as defect-free fibers by distributing stresses homogeneously. The entire fibril works at resisting fracture. Through nanoconfinement of fibrils, macroscale properties can be traced back to molecular scale properties (stick-slip and unfolding mechanisms).

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In nature, spider webs are rarely intact as they are subjected to numerous threats, such as wind load, impact from debris and prey, resulting in failure of threads. Despite having defects, they remain functional for hosting their spiders and capturing prey [91].

Indeed, simulations on orb webs with different defects (removal of threads) not adjacent to the loaded thread have shown that defects do not impact web behavior and failure mechanisms [10]. For all webs, failure is only localized near the loaded thread. When loading is applied on radial threads, web deflection and damage are larger than in the spiral thread loading case. Moreover, webs fail at a considerably larger load when load is applied to radial threads. This suggests that spiral threads are non-structural; made out of viscid/sticky silk, their role is to catch prey. The localized damage of webs under local loading can be traced back to their molecular structure and nanomechanics in particular, the underlying nonlinear behavior. When a local load is applied to a radial thread (and prior to failure), all the radial threads reach the onset of yield stress. At this point, only the loaded thread enters the unfolding of the semi-amorphous phase regime (regime (ii)) at a low stress. As shows the bar plot figure 9(A(i)) under local loading on a radial thread, nonlinear behavior leads to stress concentration in the loaded thread only, and the other threads stay at the yield stress. Then the silk enters the third regime: load is transferred to β-sheet crystals and the silk stiffens. As soon as the failure stress is reached, β-sheet crystals, located in the loaded region of the fiber, fail. Only the loaded thread fails (figure 9(A(i))) [10, 92]. To show that a spider web's robustness results from its nonlinear stiffening behavior, Cranford et al [10] also compared the previous case to the response of webs with the same geometry but composed of silks with different stress–strain properties: linear elastic and elastic-perfectly plastic (softening behavior). For both new cases, the damage area increased compared with the nonlinear behavior case, especially for the web made of softening behavior silk. This is due to the transfer of load to the adjacent radial fibers (linear elastic case) (figure 9(A(ii))) or even to the entire web (softening case) (figure 9(A(iii))). Although they fail at a higher loading force thanks to the cooperation of all the threads in resisting load and deflect less, they lose in robustness when compared to a web made of nonlinear stiffening silk (figure 9(A(ii)(iii))) [10, 92].

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Under wind loading the three types of webs fail at 60 m s⁻¹ wind speed. For the natural silk web, the deflection is controlled by dragline radial silk threads, because they are stiffer than viscid spiral threads. Webs deflect similarly for wind speed lower than 10 m s⁻¹. However, for higher wind speed, the nonlinear stiffening behavior is a disadvantage for web performance. Their webs have a considerably larger deflection because of the softening regime (unfolding) of dragline silk, making the radial threads deflect more (figure 9(D)) [10, 92]. Larger deflection means more risk for the web to be caught in vegetation or walls to which they are fixed.

In the same way that nonlinear stiffening increases the flaw-tolerance of fibers [93], nonlinear stiffening of silk also increases the robustness of spider webs. Similarly to the previous case study, a power-law with a controlling parameter N (N > 1 for nonlinear stiffening) is used, so that $\sigma \propto {\varepsilon }^{N}$ ($\sigma$ and $\varepsilon$ are the stress and strain, respectively). As α increases, there is a transition from softening to stiffening material (figure 9(B)). For spider webs, quantized fracture mechanics can be applied instead to describe their failure behavior. The scaling law is derived [10, 92], as

$$\begin{eqnarray}&&{\rm{\Omega }}(\alpha )=1-{S}^{2\alpha }\end{eqnarray} \tag{ 2 }$$

with ${\rm{\Omega }}$ the structural robustness defined by fraction of intact web after failure, S is a system-dependent constant and is the fraction of damaged web associated with linear elastic behavior. Here, $\alpha =\tfrac{1}{(\mathrm{1+1}/N)}$ and $\alpha$ is a parameter describing the behavior of silk: linear elastic ($\alpha =\mathrm{1/2}),$ nonlinear stiffening ($\alpha \gt \mathrm{1/2}),$ or nonlinear softening ($\alpha \lt \mathrm{1/2}).$ According to the scaling law, structural robustness increases with $\alpha ,$ which means that a nonlinear stiffening behavior increases spider web robustness (figure 9(C)) [10, 92].

The macroscale spider web robustness originates from failure of sacrificial elements (threads) due to highly localized failure, which is possible because of nonlinear stiffening. Similarly, silk threads are also extremely resilient because of this particular behavior. As for nonlinear stiffening, this characteristic is due to the interplay between stiff nanocrystals and semi-amorphous phase, which comes from the very protein sequence. Silk's resilience originates from its hierarchical organization.