# Influence of Test Specimen Geometry on Probability of Failure of Composites Based on Weibull Weakest Link Theory

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## Abstract

**:**

## 1. Introduction

## 2. Motivation

## 3. The Weibull Model for the Probability of Failure

#### 3.1. Test Specimens

- Straight-sided specimen, based on standard ISO 527-5:2009 [21]; thickness $t=2\mathrm{mm}$, cross-sectional area $A=30{\mathrm{mm}}^{2}$, tested volume $V=4800{\mathrm{mm}}^{3}$.
- Straight-sided specimen, based on standard ISO 527-5:2009 [21]; thickness $t=5\mathrm{mm}$, cross-sectional area $A=75{\mathrm{mm}}^{2}$, tested volume $V=12,000{\mathrm{mm}}^{3}$.
- Non-straight-sided specimen, butterfly-shaped, designed at DTU Wind Energy [14]; thickness $t=2\mathrm{mm}$, cross-sectional area $A$ in the range $30\u201355{\mathrm{mm}}^{2}$, tested constant stress volume $V=1800{\mathrm{mm}}^{3}$, tested non-constant stress volume $V=9800{\mathrm{mm}}^{3}$.

#### 3.2. Comparison between Constant Stress Specimens

#### 3.3. Comparison between Constant vs. Non-Constant Stress Specimens

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Bazant, Z.P.; Chen, E.P. Scaling of structural failure. Appl. Mech. Rev.
**1997**, 50, 593–627. [Google Scholar] [CrossRef] - Hu, X.; Duan, K. Size effect and quasi-brittle fracture: The role of FPZ. Int. J. Fract.
**2008**, 154, 3–14. [Google Scholar] [CrossRef] - Gao, X.; Koval, G.; Chazallon, C. Energetical formulation of size effect law for quasi-brittle fracture. Eng. Fract. Mech.
**2017**, 175, 279–292. [Google Scholar] [CrossRef] - Hu, X.; Guan, J.; Wang, Y.; Keating, A.; Yang, S. Comparison of boundary and size effect models based on new developments. Eng. Fract. Mech.
**2017**, 175, 146–167. [Google Scholar] [CrossRef] - Wisnom, M.R. Relationship between strength variability and size effect in unidirectional carbon fibre/epoxy. Composites
**1991**, 22, 47–52. [Google Scholar] [CrossRef] - Sørensen, B.F.; Toftegaard, H.; Linderoth, S.; Lundberg, M.; Feih, S. Strength and failure modes of ceramic multilayers. J. Eur. Ceram. Soc.
**2012**, 32, 4165–4176. [Google Scholar] [CrossRef] - Lei, W.-S. A generalized weakest-link model for size effect on strength of quasi-brittle materials. J. Mater. Sci.
**2018**, 53, 1227–1245. [Google Scholar] [CrossRef] - Lei, W.S.; Qian, G.; Yu, Z.; Berto, F. Statistical size scaling of compressive strength of quasi-brittle materials incorporating specimen length-to-diameter ratio effect. Theor. Appl. Fract. Mech.
**2019**, 104, 102345. [Google Scholar] [CrossRef] - Weibull, W.A. Statistical Theory of the Strength of Materials; Royal Swedish Academy of Engineering Sciences: Stockholm, Sweden, 1939. [Google Scholar]
- Bullock, R.E. Strength Ratios of Composite Materials in Flexure and in Tension. J. Compos. Mater.
**1974**, 8, 200–206. [Google Scholar] [CrossRef] - Wisnom, M.R.; Khan, B.; Hallett, S.R. Size effects in unnotched tensile strength of unidirectional and quasi-isotropic carbon/epoxy composites. Compos. Struct.
**2008**, 84, 21–28. [Google Scholar] - Maheri, M.R. An improved method for testing unidirectional FRP composites in tension. Compos. Struct.
**1995**, 33, 27–34. [Google Scholar] [CrossRef] - Matsuo, T.; Hojo, M.; Kageyama, K. Influence of gripping method on tensile properties of unidirectional thermoplastic CFRP—Round-robin activity for international standardization in Japan. J. Compos. Mater.
**2019**, 53, 4161–4171. [Google Scholar] [CrossRef] - Korkiakoski, S.; Brøndsted, P.; Sarlin, E.; Saarela, O. Influence of specimen type and reinforcement on measured tension-tension fatigue life of unidirectional GFRP laminates. Int. J. Fatigue
**2016**, 85, 114–129. [Google Scholar] [CrossRef][Green Version] - De Baere, I.; Van Paepegem, W.; Hochard, C.; Degrieck, J. On the tension–tension fatigue behaviour of a carbon reinforced thermoplastic part II: Evaluation of a dumbbell-shaped specimen. Polym. Test.
**2011**, 30, 663–672. [Google Scholar] [CrossRef][Green Version] - Kumar, R.; Mikkelsen, L.P.; Lilholt, H.; Madsen, B. Experimental Method for Tensile Testing of Unidirectional Carbon Fibre Composites Using Improved Specimen Type and Data Analysis. Materials
**2021**, 14, 3939. [Google Scholar] [CrossRef] [PubMed] - Ueki, Y.; Lilholt, H.; Madsen, B. Fatigue behaviour of uni-directional flax fibre/epoxy composites. In Proceedings of the 20th International Conference on Composite Materials ICCM20 Secretariat, Copenhagen, Denmark, 19–24 July 2015. [Google Scholar]
- Wisnom, M.R. Size effects in the testing of fibre-composite materials. Compos. Sci. Technol.
**1999**, 59, 1937–1957. [Google Scholar] [CrossRef] - Bunsell, A.R.; Renard, J. Fundamental of Fibre Reinforced Composite Materials; Institute of Physics Series in Materials Science and Engineering: Bristol, UK; Institute of Physics Publishing: Philadelphia, PA, USA, 2005. [Google Scholar]
- Hademenos, G.J.; Spiegel, M.R. Mathematical Handbook of Formulas and Tables: Based on Schaum’s Outline of Mathematical Handbook of Formulas and Tables; McGraw-Hill: New York, NY, USA, 2001. [Google Scholar]
- ISO 527-5:2009; Plastics—Determination of Tensile Properties—Part 5: Test Conditions for Unidirectional Fibre-Reinforced Plastic Composites. ISO: Geneva, Switzerland, 2009.
- Mikkelsen, L.P.; Kumar, R. Probability of failure in composites: Influence of test specimen design [Software]. Zenodo
**2021**. [Google Scholar] [CrossRef] - Giannadakis, K.; Mannberg, P.; Joffe, R.; Varna, J. The sources of inelastic behavior of Glass Fibre/Vinylester non-crimp fabric [±45]s laminates. J. Reinf. Plast. Compos.
**2011**, 30, 1015–1028. [Google Scholar] [CrossRef]

**Figure 1.**Stress–strain curves of straight-sided (black curves) and butterfly (red curves) test specimens. The crosses (+) at the end of the curves represent standard deviations of failure stress and strain to failure. The inset shows pictures of the test specimens before and after the test.

**Figure 2.**Geometry and dimensions of analyzed test specimens: (I) straight-sided, $t=2\mathrm{mm}$; (II) straight-sided, $t=5\mathrm{mm}$; (III) non-straight-sided, butterfly-shaped, $t=2\mathrm{mm}$. For all three test specimens, the width in the gauge section is $15\mathrm{mm}$. Arrows indicate the two comparisons of the stress ratio for the same probability of failure made in the present study. The analyzed part of the specimens is the non-hatched part between the grips.

**Figure 3.**The ratio of stress for the same probability of failure as a function of the Weibull modulus. Shown are the two analyzed comparisons between test Specimens I and II (blue curve) and test Specimens I and III (red curve).

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**MDPI and ACS Style**

Kumar, R.; Madsen, B.; Lilholt, H.; Mikkelsen, L.P. Influence of Test Specimen Geometry on Probability of Failure of Composites Based on Weibull Weakest Link Theory. *Materials* **2022**, *15*, 3911.
https://doi.org/10.3390/ma15113911

**AMA Style**

Kumar R, Madsen B, Lilholt H, Mikkelsen LP. Influence of Test Specimen Geometry on Probability of Failure of Composites Based on Weibull Weakest Link Theory. *Materials*. 2022; 15(11):3911.
https://doi.org/10.3390/ma15113911

**Chicago/Turabian Style**

Kumar, Rajnish, Bo Madsen, Hans Lilholt, and Lars P. Mikkelsen. 2022. "Influence of Test Specimen Geometry on Probability of Failure of Composites Based on Weibull Weakest Link Theory" *Materials* 15, no. 11: 3911.
https://doi.org/10.3390/ma15113911