## 1. Introduction

Spider silks are biocompatible materials that present extraordinary mechanical properties in comparison to most of the natural and manmade fibers [

1,

2,

3,

4,

5,

6]. For this reason, they have been deeply investigated as potential materials in biomedical technologies, e.g., tissue engineering [

7,

8,

9,

10]. Spider silks are produced by different glands with different aims [

11,

12]. Because of this, they have evolved to have different mechanical properties, up to the species and individual gland that spins the silk [

13,

14,

15,

16]. Considering that more than 48,000 species of spiders exist, it is interesting to investigate, across different species and glands, the mechanical properties of the silks [

17,

18] (

Table 1). Usually, these are Young’s modulus, toughness modulus, ultimate strain and fracture strength, which are obtained by means of tensile tests, i.e., stress-strain curves [

19].

Exploring the mechanical properties of spider silks is difficult, because of their dependence on the testing parameters and their natural variability. Spinning forces [

20], intrinsic variability [

21], environmental humidity [

22], temperature [

23], possible contact with strong polar solvents (e.g., water) [

24] and strain rates [

19,

25] are examples of factors to take into consideration, and which have been investigated [

26].

In structural engineering, the role of defects has a huge importance in predicting the strength of materials [

27,

28]. Spider silks are self-assembly materials that, after their formation, could be affected by defects. Since the presence of these is strictly correlated with the size of the sample, thin fibers such as spider silk are expected to be intrinsically strong, but the length of the silk sample may affect its mechanical properties [

28,

29]. Thus, it is important to investigate the silk’s mechanical properties at different length scales. In this context, Weibull statistics have been widely used in past and recent years, to describe the statistical distribution of the ultimate strength of many natural or artificial materials (not necessarily brittles), including the silk of the spider

Argiope trifasciata [

21,

30]. The importance of this statistic is mainly due to its versatility in describing different phenomena, and the possibility of using it to obtain scaling laws for the analyzed material. The Weibull’s distribution [

31] is defined with its cumulative density function [thus

F(

x) represents the probability of the fracture strength being equal to or less than

x], as:

where, in our case,

x is the fracture strength of the material,

c is the shape parameter,

x_{0} is the scale parameter associated with the strength,

l is the length of the sample,

l_{0} is a characteristic length and

d is the topological dimension. The dimension

d expresses how the energy is dissipated during fracture, namely if it is dissipated in a volume (

d = 3), over an area (

d = 2) or along the length (

d = 1). Here, it can be considered as Weibull “fractal dimension”, 0 <

d < 3, which we expect to be close to 1 due to the mainly unidimensional nature of the fiber. Equally, it represents the dimension of the dispersion of defects in the material.

Fixing the argument of the exponential to a constant (

k) would result in the prediction of the scaling law, for the strength

x, as proportional (

$-{k}^{\frac{1}{c}}$) to

b, with

In this work, the mechanical properties and Weibull parameters of the silk of nine species, some of them still absent in the literature, were derived through tensile tests. Moreover, for the species Cupiennius salei, we studied scale and shape parameters at various lengths of the samples, and the effect of strain rates on the mechanical properties. We found that both length and strain rates affect the mechanical properties of spider silk, and Weibull’s statistics can help in their understanding.

## 2. Results

The mechanical properties and Weibull parameters of the silk of nine species of spiders were obtained by means of their stress-strain curves (

Figure S1,

Table 1), which showed the typical nonlinear constitutive law of the silk [

11]. Among them, the mechanical properties of the species

Ancylometes bogotensis,

Ceratogyrus marshalli,

Linothele fallax and

Phoneutria fera were absent in the literature. The other species analyzed in this work presented mechanical properties comparable with those in the literature [

32,

33,

34]. Furthermore, for the first time we report the Weibull parameters for these nine species. The scale parameter is related to the strength of the silk, and the ones obtained were comparable to the mean values of the analyzed silks’ strength. The shape parameters, on the other hand, can be separated into two groups. Shape parameters lower than 2 were found for the species

Ceratogyrus marshalli,

Grammostola rosea and

Linothele fallax. These belong to the infraorder of Mygalomorphae, which does not have major ampullate silk (the main component of the dragline). Shape parameters higher than 2 were found for all the other species, which belong to the infraorder of Araneomorphae and which have major ampullate silk.

Figure 1 and

Table 2 show the results related to the dragline silk of

Cupiennius salei, tested at different strain rates (0.08 mm/s, 0.10 mm/s, 0.11 mm/s, 0.15 mm/s and 0.17 mm/s). The strain at break, displayed in

Figure 1a (

p-values and

d_{c} coefficient in

Table S1), was significantly higher at the strain rate 0.11 mm/s than that at 0.08 mm/s, 0.10 mm/s and 0.17 mm/s (with a medium effect size, ES). Respectively, the obtained values were 0.21 ± 0.15, 0.21 ± 0.12, 0.29 ± 0.15, 0.23 ± 0.17 and 0.20 ± 0.11.

For the strength, displayed in

Figure 1b (

p-values and

d_{c} coefficient in

Table S2), we observed significantly higher values at the strain rate 0.15 mm/s with respect to all the others (with a large ES). Respectively, the obtained values were 288 ± 241 MPa, 289 ± 218 MPa, 253 ± 217 MPa, 510 ± 311 MPa and 259 ± 168 MPa.

Similarly, for the Young’s modulus (

Figure 1c,

p-values and

d_{c} coefficient in

Table S3), we observed a significantly higher value for the strain rate of 0.10 mm/s, with respect to the 0.08 mm/s strain rate (with a small ES). The strain rate of 0.15 mm/s presented the highest Young’s modulus by a significant degree, compared to 0.08 mm/s (very large ES), 0.10 mm/s (medium ES), 0.11 mm/s (very large ES) and 0.17 mm/s (medium ES). The 0.11 mm/s strain rate gave lower values, compared to 0.08 mm/s (with a large ES), 0.10 mm/s (with a large ES) and 0.17 mm/s (with a large ES). Finally, the strain rate of 0.17 mm/s gave lower Young’s modulus values with respect to the strain rate of 0.10 mm/s (with a very small ES). Respectively, the obtained values were 6.5 ± 3.9 GPa, 8.8 ± 6.5 GPa, 3.5 ± 2.8 GPa, 13.5 ± 7.0 GPa and 8.6 ± 5.4 GPa.

The toughness modulus (

Figure 1d,

p-values and

d_{c} coefficient in

Table S4) measured at 0.15 mm/s was significantly higher compared to all the others (with a medium ES). Respectively, the obtained values were 36 ± 41 MJ/m

^{3}, 45 ± 46 MJ/m

^{3}, 37 ± 35 MJ/m

^{3}, 76 ± 63 MJ/m

^{3} and 37 ± 36 MJ/m

^{3}.

Figure 2 and

Table 3 show the results relative to the analyses of the mechanical properties and Weibull parameters at different lengths (0.55 cm, 0.75 cm, 1.0 cm, 1.25 cm and 1.5 cm). We did not observe any statistically significant differences in the strain at break, Young’s modulus or toughness modulus for the fibers tested at different lengths. For the strain at break, we measured respectively 0.23 ± 0.11, 0.20 ± 0.15, 0.25 ± 0.14, 0.22 ± 0.10 and 0.21 ± 0.17. For the Young’s modulus, we obtained respectively 8.0 ± 4.1 GPa, 7.6, ± 5.3 GPa, 6.1 ± 3.8 GPa, 7.1 ± 4.2 GPa and 7.5 ± 2.5 GPa. For the toughness modulus, we measured respectively 71 ± 40, 65 ± 41, 60 ± 35, 70 ± 45 and 51 ± 39.

On the other hand, in

Figure 2a and

Table S5, it is possible to notice a slight decrease of the diameter for longer samples. In particular, the diameter value for the 0.55 cm length was greater than the one for 0.75 cm (with a medium ES), and greater than the ones at 1.0 cm, 1.25 cm and 1.5 cm (with a large ES). Moreover, the diameter values for the samples 0.75 cm long were higher than those of the 1.0 cm long samples (with a medium ES). On the other hand, although the differences between the diameters of the samples at 0.75 cm and those at 1.25 cm and 1.5 cm were significative, their size effects were small. Respectively, we measured 5.5 ± 2.9 μm, 4.1 ± 1.6 μm, 3.2 ± 1.2 μm, 3.3 ± 1.9 μm and 3.4 ± 1.4 μm. This is in agreement with a recent work on the same species [

35], which has highlighted that the dragline near the attachment disc (where the thread is attached) is composed of more fibers than the dragline far (in cm) from the attachment point, which thus has a smaller diameter.

Figure S2 shows optical and SEM images that highlight this feature, and could explain the observed difference. In fact, in spiders that fell greater distances (longer samples), it was unlikely that the dragline collected was near the attachment disc, i.e., had more fibers.

The strength of the silk at different lengths is reported in

Figure 2b (for the

p-values and the Cohen’s coefficient, see

Table S6). Respectively, the values measured were 931 ± 345 MPa, 805 ± 371 MPa, 754 ± 315 MPa, 789 ± 317 MPa and 515 ± 260 MPa. We observed, thus, a slightly significant decrease in strength for the longer fibers. In particular, the fibers 0.55 cm long were significantly stronger with respect to those 1.0 cm long (with a medium ES) and those 1.5 cm long (with a very large ES). Moreover, the fibers 1.5 cm long were significantly weaker than those 0.75 cm, 1.0 cm and 1.25 cm long (with a large ES).

In

Figure 2c we plotted the linear regression method [Equation (3), see materials and methods] used to compute the Weibull parameters. All the data sets were acceptable to be dealt by means of Weibull statistics under the Kolmogorov–Smirnov test, and the obtained parameters were comparable with those obtained through the maximum likelihood method [

36], confirming the applicability of Weibull statistics for our samples. In

Table 3, these Weibull parameters are reported. Respectively, we obtained the following scale (

b) and shape (

c) parameters (in the brackets we inserted the parameters obtained through the maximum likelihood method): 1054 MPa and 2.7 (1044 MPa and 3.10), 909 MPa and 2.4 (910 MPa and 2.4), 860 MPa and 2.4 (849 MPa and 2.7), 894 MPa an 2.6 (889 MPa and 2.8) and 579 MPa and 2.3 (583 MPa and 2.1). The Weibull parameters that were obtained with the support of the linear regression method were used to plot the density probability function of the fracture strengths of the five different lengths (

Figure 2d). These were similar, and no major differences appeared, indicating the similarity in statistical fracture behavior.

Figure 2e represents the shape parameters obtained with the support of the linear regression method. No major differences appeared, indicating that Weibull statistics could be applied to this spider silk (i.e., the shape parameter can be considered constant).

Figure 2f shows the decrease of the scale parameter [as predicted from Equation (6), see materials and methods, and the Weibull statistics] as a function of the length. Although the decrease is not big, we estimated the Weibull fractal dimension, using Equation (6), directly from the slope of the plot (i.e., −0.4584). We calculated

d = 1.1, which is consistent with our prediction (

d close to 1, due to the unidimensional nature of the silk fiber).

## 3. Discussion

The silks of nine species of spider were analyzed by means of mechanical tests. We found that the two distinct typologies of silk behaved differently from a statistical and mechanical point of view. One belongs to the infraorder of Mygalomorphae (Dipluridae and Theraphosidae families) and presented low shape parameters (between 1 and 2), which means that the fracture probability is more constant [

8,

30,

31]. On the other hand, the Araneomorphae presented silks that were better in terms of mechanical performance, and had Weibull shape parameters higher than 2 (commonly found in other polymeric fibers, silk included [

30,

37]).

The silk of the species

Cupiennius salei was tested firstly at different strain rates. It being a polymeric viscoelastic material, the strain rate affects the capability of the silk to relax, and thus its mechanical properties [

19,

20,

38,

39]. In particular, we noticed a difference in terms of the mechanical properties, although the range of variation in the strain rates was relatively small, compared to that analyzed in a recent work [

25]. Slightly higher values of Young’s modulus, strength and toughness modulus were found at 0.15 mm/s, suggesting that such a value is somewhat optimal, and is a figure compatible with natural conditions [

19,

40]. The improvement of such properties with increasing strain rates is in agreement with what has been found in the literature, here derived from a broader range of strain rates [

25,

41].

Together with the strain rates, we investigated how the length of the sample affects the mechanical properties of the silk (analysis performed on

Cupiennius salei silk at five different lengths). We noticed a slightly significant decrease in the diameter for longer samples. This could be explained by the different numbers of fibers that compose the collected threads. It has been shown for the same species that draglines have a higher number of fibers when they are near the attachment discs [

35], i.e., they have a greater diameter. This observation is relevant given the procedure of sample collection we employed; each sample being a consequence of a single fall, for spiders that fell greater distances it was unlikely that the dragline collected was near the attachment disc, i.e., had more fibers. At the same time, in the same work [

35], it was also shown that these two distal portions of the same dragline have the same mechanical properties at a fixed length. Moreover, we found a slightly significant decrease in strength for longer threads, but not a significant difference in the other mechanical properties. This is due to the fact that the constitutive laws are highly nonlinear (

Figure S1), e.g., the toughness modulus is not proportional to strength.

As we expected, the consistency of the Weibull shape parameters for spider dragline silk at different lengths suggests that Weibull statistics could be applied to the strength of silk, and thus used to obtain its scaling laws. With the support of Equation (6), we plotted the Weibull scale parameters at different lengths, and computed the Weibull fractal dimension [

31]. We obtained

d = 1.1, which confirms the fact that defects in spider silk are proportional to the fiber length. This is in agreement with what was found through simulations and numerical works, where the failures of the silk occurred in the aligned crystalline regions along the pulling direction (i.e., traction tests) [

42,

43,

44].