# Mechanical Properties and Weibull Scaling Laws of Unknown Spider Silks

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

_{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.

## 2. Results

_{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.

_{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.

_{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.

_{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}.

## 3. Discussion

## 4. Material and Methods

#### 4.1. Spiders and Silk

#### 4.2. Tensile Tests

#### 4.3. Weibull Statistics

^{®}, The MathWorks, Natick, MA, USA). Moreover, for each data set we used the maximum likelihood principle to investigate if the Weibull’s parameters were consistent with respect to the number of tested samples (following Peterlick et al. [36] and by means of Mathemathica

^{®}software) (Wolfram Research, Inc., Champain, IL, USA).

#### 4.4. ANOVA Analysis

_{g}is the number of tests of the same sample, m

_{u}is the mean value of all the data, m

_{g}is the mean value within the group (i.e., sample), and x is the single quantity value. These sums of squares were used to compute the T value, as:

_{F}, with a significance level of 5%. If T > F

_{F}, we reject the null hypothesis and thus we can consider the difference among the data set as significant. In this case, the null hypothesis is that the differences among the mean values of five groups of the independent variable (sample length or strain rate) are consequences of the internal variance in the groups, and thus they are not due to an intrinsic difference. The test function T and the two-tailed p-value were computed with the support of Matlab

^{®}.

#### 4.5. Effect Size (ES)

_{1}and n

_{2}are the dimensions of the two groups compared, and s

_{1}and s

_{2}their standard deviations. With s, Cohen defined the following parameter

_{1}and m

_{2}are respectively the means of the two groups. Based on Cohen [54] and Sawiloski [44], d

_{c}helps to define qualitatively the magnitude of the difference of the means as very small (d

_{c}≥ 0.01, i.e., circa 100% distributions’ overlap), small (d

_{c}≥ 0.20, i.e., circa 85% distributions’ overlap), medium (d

_{c}≥ 0.50, i.e., circa 67% distributions’ overlap), large (d

_{c}≥ 0.8, i.e., circa 53% distributions’ overlap), very large (d

_{c}≥ 1.20, i.e., circa 40% distributions’ overlap), and huge (d

_{c}≥ 2.0, i.e., circa 19% distributions’ overlap).

## 5. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Sample Availability: Samples of the compounds are available from the authors. |

**Figure 1.**Mechanical properties of the spider silk of Cupiennius salei at different strain rates: (

**a**) strain at break, (

**b**) strength, (

**c**) Young’s modulus, and (

**d**) toughness modulus.

**Figure 2.**(

**a**) The diameter of the dragline vs the length of the sample. (

**b**) Strength of the dragline vs the length of the sample. (

**c**) Linear regression plot of the strength data set used to compute the Weibull parameters. (

**d**) Weibull’s probability density distribution of the strength at different lengths. (

**e**) Shape parameter vs length of the sample. (

**f**) Plot of Equation (6) with linear regression to compute the Weibull fractal dimension.

Species | Nr. Samples | Strain at Break (mm/mm) | Strength (MPa) | Young’s Modulus (GPa) | Toughness Modulus (MJ/m^{3}) | Scale Parameter b (MPa) | Shape Parameter (c) |
---|---|---|---|---|---|---|---|

Araneus diadematus | 16 | 0.25 ± 0.09 | 655 ± 286 | 8.5 ± 4.9 | 267 ± 164 | 949 | 2.56 |

Ancylometes bogotensis | 15 | 0.24 ± 0.08 | 897 ± 441 | 21.8 ± 9.9 | 191 ± 175 | 1112 | 2.01 |

Ceratogyrus marshalli | 15 | 0.17 ± 0.15 | 163 ± 158 | 2.6 ± 2.1 | 14 ± 11 | 236 | 1.16 |

Cupiennius salei | 36 | 0.29 ± 0.15 | 253 ± 217 | 3.5 ± 2.8 | 37 ± 35 | 859 | 2.35 |

Grammostola rosea | 15 | 0.17 ± 0.15 | 13 ± 9 | 3.0 ± 2.2 | 26 ± 16 | 42 | 1.58 |

Linothele fallax | 15 | 0.27 ± 0.16 | 110 ± 86 | 5.0 ± 4.3 | 21 ± 20 | 127 | 1.22 |

Nuctenea umbratica | 15 | 0.21 ± 0.06 | 1199 ± 725 | 10.2 ± 4.2 | 138 ± 81 | 1693 | 2.35 |

Phoneutria fera | 15 | 0.32 ± 0.19 | 936 ± 544 | 27.2 ± 13 | 202 ± 141 | 1191 | 3.14 |

Zygiella x-notata | 15 | 0.19 ± 0.07 | 283 ± 137 | 5.0 ± 2.6 | 36 ± 25 | 597 | 2.19 |

Strain Rate (mm/s) | Nr. Samples | Diameter (μm) | Strain at Break (mm/mm) | Strength (MPa) | Young’s Modulus (GPa) | Toughness Modulus (MJ/m^{3}) |
---|---|---|---|---|---|---|

0.08 | 33 | 3.5 ± 1.5 | 0.21 ± 0.15 | 288 ± 241 | 6.6 ± 3.9 | 36 ± 41 |

0.10 | 37 | 4.0 ± 1.2 | 0.21 ± 0.12 | 289 ± 218 | 8.8 ± 6.5 | 45 ± 46 |

0.11 | 36 | 3.3 ± 0.9 | 0.29 ± 0.15 | 253 ± 217 | 3.5 ± 2.8 | 37 ± 35 |

0.15 | 31 | 3.0 ± 1.6 | 0.23 ± 0.17 | 510 ± 311 | 13.5 ± 6.7 | 76 ± 63 |

0.17 | 35 | 3.9 ± 1.5 | 0.20 ± 0.11 | 259 ± 168 | 8.6 ± 5.4 | 37 ± 36 |

**Table 3.**Mechanical properties and Weibull parameters of the dragline silk at different lengths. In italics and between brackets are the Weibull parameters obtained through the maximum likelihood method, following Peterlik [36].

Length (cm) | Nr. Samples | Diameter (μm) | Strain at Break (mm/mm) | Young’s Modulus (GPa) | Toughness Modulus (MJ/m^{3}) | Strength (MPa) | Shape Parameter c | Scale Parameter b (MPa) |
---|---|---|---|---|---|---|---|---|

0.55 | 29 | 5.5 ± 2.9 | 0.23 ± 0.11 | 8.0 ± 4.1 | 39 ± 40 | 932 ± 345 | 2.7 (3.10) | 1054 (1044) |

0.75 | 29 | 4.1 ± 1.6 | 0.20 ± 0.15 | 7.6 ± 5.3 | 45 ± 41 | 805 ± 371 | 2.4 (2.4) | 909 (910) |

1.0 | 27 | 3.2 ± 1.2 | 0.25 ± 0.14 | 6.1 ± 3.8 | 60 ± 43 | 754 ± 315 | 2.4 (2.7) | 860 (849) |

1.25 | 28 | 3.3 ± 1.9 | 0.22 ± 0.10 | 7.1 ± 4.2 | 51 ± 45 | 790 ± 317 | 2.6 (2.8) | 894 (889) |

1.5 | 33 | 3.4 ± 1.4 | 0.21 ± 0.17 | 7.5 ± 2.5 | 64 ± 39 | 515 ± 260 | 2.3 (2.1) | 579 (583) |

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**MDPI and ACS Style**

Greco, G.; Pugno, N.M.
Mechanical Properties and Weibull Scaling Laws of Unknown Spider Silks. *Molecules* **2020**, *25*, 2938.
https://doi.org/10.3390/molecules25122938

**AMA Style**

Greco G, Pugno NM.
Mechanical Properties and Weibull Scaling Laws of Unknown Spider Silks. *Molecules*. 2020; 25(12):2938.
https://doi.org/10.3390/molecules25122938

**Chicago/Turabian Style**

Greco, Gabriele, and Nicola M. Pugno.
2020. "Mechanical Properties and Weibull Scaling Laws of Unknown Spider Silks" *Molecules* 25, no. 12: 2938.
https://doi.org/10.3390/molecules25122938