Tensile testing of Fiber Bundles, Weibull Fracture Probability

Résumé du document

Glass fibers are often used as reinforcement for composite materials, which makes it necessary for an engineer to know their strength. However, testing a single fiber is technically difficult if not impossible, and we have to use statistical models to describe the strengths and weaknesses of these fibers. To do so, tensile tests are generally done on fiber bundles. The objective of the present Lab work was to deeply understand the failure model of brittle glass fibers, by realizing tensile tests on several bundles to determine Weibull parameters of glass fibers. In this report the following points will be explained: a theory summery about Weibull failure model and the associated parameters; the experimental protocol and data; the results coming from these experiments and the discussion we can have upon those.

Extraits

[...] After that, we had to determine the single fiber strength distribution, by using two different methods: - Method by plotting force versus strain, we can measure the slope S0 of the curve for each sample, then calculate tensile load and measure the applied tensile force in order to determine the probability of survival Ps of each sample. Then, by using the double logarithm of the Weibull distribution function, we can find Weibull distribution parameters β and ε0. the following equations were used: 6 ln( p s = ln( + β ln(ε f ) + β ln(ε 0 ) where ln( + β ln(ε 0 ) = a. Where a is a constant. This constant a will be the value where the linearized curve to the ln( p s ))vsβ ln(ε f ) . [...]

[...] We calculated an average of all the values and found that the diameter of one fibre corresponded to 76,1725 pixel, which corresponds to a diameter d = 24,414 µm. Then we could calculate the area of one fibre: A = π.r2 = mm2 = 4,68. 10-10 m2 and the volume of a fibre of one meter: Vf1m= 4,68. 10-10 m3. Thanks to that we could have found the Vb1m number of fibres in the bundle: Nf = Vf = 1875 fibres. 1m Finally the area of the bundle was: Abundle = A b u n d le = m m 2 Then, we began the real experiments. [...]

[...] And the slope of the curve will be β . This gives us ε 0 = exp((ln(L) a ) / β ) Method with S0, the maximum load Fmax and the maximum strain ε m known from experiments, we determine directly the shape and the scale parameters from given relationships β= ε S ln( m 0 ) Fmax ε0 = ( εm 1 β ) Lβ Theses calculations can be found on the 250mm.xls” and “Pure 250.xls” which are attach in the email together with this file. [...]

[...] Unfortunately for us, the lab equipment only gives us the load F vs. elongation : Figure 1 - Curve in theory Figure 2 - Experimental Curve σ F E modulus Inverse of Compliance 1/Ca ε But, in order to use the curve, we have to consider the fact that in is also contained the testing system elongation, which means the system compliance. Therefore, we will have to realise a secondary study to calculate that system compliance CS This can be done very simply actually: by testing numerous bundles of the same length we can apply Weibull theory, and by testing bundles of different lengths we can find the correction factor CS: Figure 1 Compliance versus initial length Ca L0 In this plot each group of point correspond to the results obtained by the group testing the L0-length bundles. [...]

[...] The excel file with all the calculations of the Ca is in appendix. We obtain the curves of Ca vs length for pure fibres and fibres soaked in oil: 0,016 Indicated compliance Ca y = 3E-05x + - Linear (Ca average (pure)) Linear (Ca average Ca average (oil) Ca average (pure) y = 4E-05x - 0,0002 Length of fibres Figure Indicated compliance vs length of fibers The compliance value for 0mm gage length represents the system compliance Cs. Therefore in these curves, the ordinate when is the value of Cs. [...]