Re: stress vs. strain determines crack limit

seferiad wrote:
> Thanks for the detailed responses.  I'll pose a more specific question.
> Let's say I have a bunch of fibers that perfectly obey Weibull statistics
> when  I do dynamic pull testing to destruction for the fibers. The
> distribution when plotted on Weibull gives a slope = m (it doesn't matter
> what m is), but the fiber population is sufficiently strong so that a
> significant fraction of the fiber will break at sufficiently high pull force
> so that the non-linear stress-strain relationship of glass will be apparent.
>
> I have two "ideal" Instron machines. One machine pulls with a constant
> stress rate and the other machine pulls with a constant strain rate.
>
> If I now plot these two Weibull distributions:  %Failure vs. Stress (or
> Strain), I would expect that for both graphs at low stress (or strain) that
> the plots should be perfectly linear (based on the original problem
> statement of having ideal fiber).  However, at higher Stress (or Strain) one
> of the graphs will have a change in slope.
>
> I'm trying to figure out conceptually, which is the plot that is more likely
> to be straight -- at least with regards to the equations set forth by
> Griffith (for the moment, nevermind that the theory is incomplete I'm trying
> to make sure that I am applying that theory correctly).  I realize that I
> have oversimplified the problem, but this is my simple-minded way of trying
> to explain the problem.
>
> Thanks,
> Jay

>
In general,  you should think very carefully about what you think is 
happening in a specimen and what is actually happening.

Imagine three fibers having identical strengths. Imagine first that each 
fiber is exactly the same length as the others and is perfectly 
gripped... repeat, perfectly gripped .... in an Instron.

As the specimen is loaded, all three fibers load in exactly the same way 
and will fail at the exact same time.... because of the exact identical 
strength assumption.

Now imagine these very same fibers. Imagine imperfect gripping of the 
specimens so that the length between the perfect grips for each fiber is 
slightly different.  Imagine looking at a piece of yarn after you have 
untwisted is and you wlll see that some fibers are more slack (and 
curver) than others).

Now when you load this array of three fibers in an Instron tensile 
tester, because of the difference in lengths (Slack Lengths) the 
shortest fiber loads first, and then the next and then the next. So, the 
load initiation of the fibers reflects the gauge length statistics.

Eventually, the shortest fiber reaches it's failure strength and it 
fails. There is a load drop from load redistrubution. Then as the 
machine is further run and the specimen continues to elongate, the 
remaining fibers will subsequently fail.

Remember the assumptin of a perfect Instron with infinite stiffness. 
Because of this pure displacement loading condition, the fibers won't 
actually undergo loading increases as the individual fibers fail. For an 
Instron represented by a finite spring constant, the second and third 
fiber WILL  experience a loading increase as a result of the failure of 
the first fiber.

But, the result is that you gain evident fiber failure statistics 
(incorrect inference) even though the fibers were assumed to have no 
failure stress variability.... What you are seeing are geometric 
statistics that relate to when fibers begin to be loaded in the perfect 
Instron with perfect gripping.

Slack Length statistics.

These slack length statistics will destroy the correlation you are 
attempting to draw.

And this work dates to the 1970's.

Even old researchers could and did think.

The basic lesson is that when you are testing fibers, think it through 
and understand what the mechanics details really are.

It was a common problem with carbon/graphite fibers that the bare yarn 
strengths were typically only about 1/3 of the average fiber strength.

If you understood that the bare yarn test was a lousy way to attempt to 
measure filament strengths, then it made sense. Otherwise, folks would 
just look at the results and contemplate the mysteries of composites.

I published a paper on this in the "Extended Abstracts of the American 
Carbon Society" in about 1975 and discovered some related excellent work 
of a similar nature (including twist effects) by an author I can't 
remember.  There probably have been several prior dissertations on the 
detailed effects of similar considerations.

Your best goal to equalizing the load among the fibers is to include a 
matrix to transfer load between fibers. However, here you need to 
remember to account for the load redistribution.... in two key ways.
1) load redistribution locally from fiber to fiber.
2) load redistribution along the length of the fiber so that an 
individual fiber can sustain more than one fracture along its length.

The penalty for the loading equalization of the matrix is to ruin the 
assmuption of independent mechanical fracture of the individual fibers.

You can't have your lunch and eat it too.

Remember that the advantage of the matrix is precisely this form of load 
redistribution that tends to make each fiber equally share the load.

It helps to think the details through carefully, and even sometimes draw 
simple cartoons of what one presupposes the mechanics are.... and then 
to actually look at a test sample to guess if your assumptions were a 
match to the actual test.

Jim