A question on a recent test asked me to determine what the stress
inside a uniaxial composite was at a given strain value. The given

strain value was below the failure strain for the fibers and matrix
material, but above the yield strain for the matrix. I was docked
points for assuming that the matrix material would still be bearing
load once it was beyond yielding, and my professor says that once the
matrix has begun yeilding, it won't carry any of the load. Why is this
so? I'm still not 100% sure about my professor's explanation.
Thanks in advance,
Dave

Are you sure that you accurately described the professor's remarks?
Was there any mention made of the type of material in the fiber and the
matrix? Ductileness or brittleness - were these specified?
Was the reinforcement continuous or discontinuous?
Was the loading tensile, not compressive?
The most common assumption is that the strain will distribute more or
less uniformly between the fiber and the matrix if the fibers are
continuous and parallel to the loading direction. If not, then the
fibers would likely be slipping in the matrix. In other words, the
total length of the fibers would be different from the total length of
the matrix, as obtained by the integration of the axial strain along the
specimen length.
If the matrix is carrying no load, then it doesn't elastically stretch
and there must be slippage between the fiber and the matrix, again by
integrating the strain along the axis of the fiber. Of course, a
material with zero post yield load bearing capability would, by
definition satisfy such a criterion, but with the exception of quite
brittle materials, such a constitutive equation is unexpected. The
sketchy description you gave would imply that the matrix isn't brittle
in this sense.
Hit the library and look in other introductory textbooks. You should
find the printed answer there, and you can take that back to your
professor for further discussion. The old text by R M Jones may have it.

the
this
Yes, his solution only accounted for the fiber's loadbearing, not the
matrix (which he said did not carry any load under yielding).

the
The values given to us were:
1. ultimate tensile stress of the fiber and matrix
2. ultimate tensile strain of the fiber and matrix
3. yield stress and strain of the matrix material
4. volume fraction
The question was along the lines of "what is the elastic modulus of the
composite (a) prior to matrix yielding and (b) after matrix yielding?"
For (a), he included the loadbearing of the matrix into the solution,
but not for (b) as he said it was no longer bearing load...???

Continuous
Yes, tensile.

of
the
Yes, I'm assuming that under yeilding, strain(fiber)=strain(matrix).

stretch
brittle
it.
Well, I (and some of my fellow classmates) are having trouble with
this. I'm not too sure the professor hasn't made an incorrect
assumption about matrix yielding.
Thanks for your input, and any additional advise would be appreciated!
Dave

You might make discrete inquiries as to any formal classes in continuum
mechanics that may be in the professor's background, or ABSENT from it.
Without a decent background in solid mechanics (elasticity, continuum
mechanics, plasticity....) it is easy to make absurd statements on loads
and stresses and strains.
Being ignorant of mechanics is not a forbidding variable to claim
competence in composite materials. Unfortunately, understanding just
mechanics is sometimes considered adequate to be an expert in composite
materials.
Conventional Chemistry professors would distain solid mechanics because
it was so "elementary" (just the same old Newton's Laws). These
chemistry profs knew all about stress and strain, except that they
couldn't actually accurately distinguish between stress and strain when
asked.
Frankly, assuming the correctness of what you describe, your professor
is using mechanics as taught on Venus or some other planet. So, if the
professor is a Venusian, then it would make sense to allow him or her to
conform to the physics of his or her native planet.

given
matrix
docked
bearing
is
If the matrix does not work harden, ie. is elastic-perfectly plastic,
the stress it supports above its yield point is constant. Since a
perfectly plastic material's resistance to deformation does not change
with stress, it does not contribute to modulus even though it does
provide a constant resistance to deformation. (modulus is the *change*
in stress with deformation.)
Elastic-perfectly plastic is not a bad description for many cases; it
just means that the work hardening slope is much, much less than the
pre-yield modulus.
Are you sure your professor said "once the matrix has begun yeilding,
it won't carry any of the load?" Might s/he have said that "above the
yield point the matrix contribution to the modulus is very small."
The point is "it won't carry any of the load" is not the same as "it
doesn't contribute to the modulus".
It might have been a poorly posed question about a subtle point.
Dave

Ok, I have my test in front of me. The actual question was:
"Titanium alloys are reinforced with longitudinal continuous silicon
carbide fiber. The following are material properties for both matrices
and fiber:
Ef@0Gpa
Ultimate stress (fiber)440Mpa
Volume fraction (fiber/total)=.36
Em0Gpa
Yield stress (matrix)0Mpa
Ultimate stress (matrix)00Mpa
Ultimate strain (matrix)=1.0%
a) Calculate the longitudinal modulus of the lamina (with matrix 1)
before and after matrix 1 yielding."
Any insight?
Thanks,
Dave
snipped-for-privacy@newarts.com wrote:

You said...
" Em0Gpa {I take this to refer to "matrix 1" in the problem}
Yield stress (matrix)0Mpa
Ultimate stress (matrix)00Mpa
Ultimate strain (matrix)=1.0% "
The matrix yield point was .00818 strain = 900mpa/110,000mpa
The average effective modulus of the matrix above its yield point is
(1000-900)mpa/(.01-.00818)T.945 gpa.
While not exactly zero it is much smaller than the modulus of the
fibers.
A simple parallel model for the composite above the yield point has an
effective modulus of 0.36x400+0.64x54.99 gpa
If the matrix had zero modulus the effective modulus would be
0.36*400+0.04 gpa
Dave

is
Well, he may be correct above a total strain greater than 1% (ie. *if*
the matrix truly ruptures) but he's incorrect between 0.818% and 1%
because the problem statement implies work hardening plastic behavior
from 0.818-1.00% and continuity requires the matrix' strain to increase
in that range. The work to cause that strain must come from somewhere.
You'll have to decide if it is in your best interests to argue about a
range of 0.182%.
Also be sure he's refering to the first strain excursion above the
yield point. While subsequent lower strain modulii may the same as in
the original case, the effects of residual stresses and strains make it
harder to understand.
If you are going to pursue the matter, you'd better think through the
residual strain case so you can defend your position.
Dave

The question doesn't specifiy which region above yeilding, leading me
to think that if I can proove that the matrix's modulus in the area
immediately beyond yeilding is not zero, I can make a good case for my
solution. On the other hand, getting on the professor's bad side could
proove more harmful than taking the 4 point hit.
Dave

my
could
Be sure you say "the matrix' effective modulus", ie. the slope of the
workhardening curve, for the first excursion.
And you better understand why.
Dave

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