Micromechanics- Multimodal, Single Fibre and Bundle Testing Flashcards
Are short or long fibres stronger and why?
Shorter fibres are stronger. There is a smaller chance of having catastrophic flaw in a shorter length
Effect of doing Weibull analysis on fewer data points
As number of data points N reduces the averages for m and σ* become less accurate and error bars increase. The graph of double log things vs ln(σ) fits the regression line poorer for less data
What happens when there is more than 1 flaw type?
Multimodal distributions are needed
Bimodal case
A combined analysis fits the data. Long equation on page 11 lecture 2 used. Need two different m and σ* values
Problem of combining distributions
When compared with the original strength data there is poor fit at the low (more important) strengths
Solution to combining distributions problem
For bimodal, there are two lines of best fit used on the double log vs ln(σ) graph which intersect. Split the two regions (of data that follow one line or the other) and treat them as separate distributions doing two analyses. The combined fit then is better at the lower strengths. In reality more complex analyses are required
How does two different fibre lengths affect arithmetic mean?
Get two different arithmetic means one for each fibre length.
σbar1=(σ0/L1^1/m)Γ(1+1/m), same for 2
So relation between two arithmetic means is
σbar1/σbar2=(L2/L1)^1/m (databook)
Assumes m doesn’t vary with length
Using relation between two arithmetic means for a graph
σbar1/σbar2=(L2/L1)^1/m start
In principle ln(σbar)=-1/m ln(L)+C
Plot graph ln(σ) vs ln(L) and gradient is -1/m
Still assumes m doesn’t vary with L
So if have mean strength at one L then can use first equation to find mean strength at other L
How to test a single fibre
Test in tension. Fibres selected from two (bundle). Individually fix to a card of paper frame. Load frame in tensometer. Cut frame along horizontal dotted lines (see diagram page 16 lecture 2). Apply load. Practically very challenging
What happens in fibre bundles if one fibre fails?
If a single fibre fails, the load it was bearing is redistributed through load sharing to other fibres and the matrix. Ignoring matrix gives lower limit. Subsequent failure possibilities are that adjacent fibres near initial failure if redistributed load raises above their strength, other wise fibres well away from initial failure site
What is a bundle?
Multiple fibres collected together. Aka a tow
Force applied to bundle formula
F=σAN
N is number of fibres that survive at that load
A is individual CSA
Can you use bundles to determine m?
Yes. Plot scaled applied load against scaled bundle stresses. Slope of line joining origin to maximum of curve is exp(-1/m). So can use to find m. But gauge length is also an issue. Compatible results with single filaments of the same gauge length.
Bundle strength formula
σsubB=Fmax/N0A
Nsub0 is original number of fibres
Eventually get to relation of σB/σbar in databook
Bundle strength compared to mean fibre strength
Bundle strength lower than mean fibre strength (fails at weakest part) but at higher m values bundle strength tends towards mean fibre strength. Unlikely to get m above 30
