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Let us continue our earlier theme about fiber orientation. We derived a set of equation
probability and density and so on. Number of fibers for fiber structures for for planar
planar types of of fiber structures, this distribution of directions is very important
for lot of properties at most for mechanical properties of fibers assemblies.
I want in this lecture to present one case, how is possible to apply our earlier knowledge
to the mechanical behavior of fibers assembly, but I must say in our lecture, it will be
an easiest case. So easy, that it is on the border of unreality, each real structure is
much more complicated. So that, but I want to present it to you because it can show,
how is the style of our work, when we when we when we must to derive some model for mechanical
property often fiber assembly with with with some some distribution significant distribution
of fiber direction because this methodological sense, I want to present you this easiest
case. On the final sentences, I will in short comment what is possible to do more and go
more to the real means more complicated structure. So, that is easiest case of mechanical behavior
of planar fiber assembly. Let us let us accept six assumptions, which which make our problem
easy. First, our model is planar fiber orientation is the planar fiber, second each fiber is
straight, it means no crimped also cotton have cotton fiber have some small crimp, but
we assume that that the each fiber is straight then third each fiber is clamped by both jaws
of tensile machine or breaking machine. And we neglect the effect of the margins of jaws.
You know, when we have something like non-woven some some warp or something so. In in couple
of jaws by in breaking machine, we must cut this this structure, where the ends of jaws
know. So, that some fibers are like this here, we do not want to calculate these fibers,
you can imagine that the the jaws are very very very long and this age between this this
this value between between between couple of jaws is very small fourth because I said
easiest model, let us imagine linear force-strain relation same for each fiber. So, between
the fiber force F l and the fiber elongation epsilon l is their relation strained by breaking
strained times epsilon, this is the constant for fiber times epsilon l and 0 of course,
after breaking the this next next assumption small deformations so small that that no one
fiber will rupture it destroy it through our process. And the last of our assumption is
that the fibers are deformed mutually independently, we do not calculate the friction fiber to
fiber friction in in our structure and so on.
Well, first step let us do derive do derive the function of of pair one fiber one general
fiber between jaws. The, we have some couple of of jaws to jaws in breaking machine
is is clumped between these two, two between these two, two, two jaws. Lengths of this
fiber is l, starting angle we we will need, it is enough to know non oriented angle theta
is is is shown on the on the picture here. To vertical vertical axis is our earlier y
axis. Well, so, h is vision after first intuitively
after elongation after after jaw displacement, the jaw B is changed the position to the new
position B dash. Therefore, the fiber is fiber is now this here, plus one dash higher than
length. So, dash epsilon, we call as a relative jaw displacement and it is from h lengths
and h dash defined, the traditional here. Jaw displacement in opposite to them, the
strain in fiber strain in fiber work is l length is. So, that it is l dash l minus l
by l evident here. Starting angle theta for the starting angle theta is specific to write
that it is cosine of this is h by l it is shown from the figure.
The final angle theta is the angle after elongation, which is similarly, from the picture here
h dash by l dash and because h dash is h times 1 plus epsilon and l is 1 dash is l times
1 plus epsilon l fiber strain. Then we can write that the cosine is theta dash is cosine
is theta starting angle times this ratio 1 plus epsilon by 1 plus epsilon l.
You know, the Pythagorean Theorem is not it? It is very known theorem in whole world. Therefore,
we want to use it two times, first time from this triangle from this yellow triangle and
what we obtain from Pythagorean Theorem x square is l square minus h square is not it?
Well, from the second right green triangle, we obtain x square is l dash square minus
h square because this length x is same for yellow triangle assess for for for white green
triangle. Why? Because our jaws are on are from some metal deformable moving of the jaw
B.
So, that it must be valid, this is equal to this expression is here, it is here. Now some
small rearranging, which of we divide this equation by l. So, that we obtain this here,
sorry l square of course, by l square we divide using our symbols our our equations now from
here, we can write 1 minus cosine is square theta is 1 plus epsilon l square minus h times
1 plus epsilon by l square. So that, it is it is this one because h by l is cosine. So,
I have this here from this equation, this is possible to write in this to this form
trivially and epsilon one is this here or epsilon l we obtain epsilon or epsilon l is
possible to explain using this expression. That is well, that is good. It is shown that
the strain of fiber is not the same than jaw displacement epsilon.
And that is the function of jaw of angle theta, it based on the orientation of our fiber segment,
our fiber. Well, we said our model will be the easiest as possible. Therefore, let us
imagine small deformations, very small deformations then epsilon is very small and then the the
approximate value, approximate equations are valid. You know that we can construct different
approximate formulas using Taylor series is not it? It is known from mathematics. And
using such, but exist also the some content of different often used approximated, approximation
equations. The one say, 2 times epsilon plus epsilon square is roughly for very small epsilon
2 times epsilon because this square is extremely small then we can neglect it, square root
of 1 plus 2 times epsilon cosine square theta if epsilon is small, if this part is very
small is possible to write approximated s 1 plus 1 half of this.
The third is nothing this 2 is in the moment enough then our epsilon epsilon, l was this
here using these approximation equations, we obtained that it is. This is this is approximately
this? So, that epsilon one is epsilon times cosine square theta for small deformations.
In this easy equation, you can see you can see that the function of angle theta, when
we have when we have fiber is higher angle theta then cosine is is smaller than 1, cosine
square is much more smaller than than 1. So, that the epsilon l is smaller than epsilon,
the highest value of strain of fiber is, when the fiber is parallel to jaw axis a higher
this angle theta from the direction of jaw axis. So, smaller is fiber elongation.
Is there is out of this equation? It was something about the fiber, about fiber strain, now about
fiber force forces. We said that the force in fiber follows the linear function. So,
that the force in fiber F l is proportioned to P by a fiber strain by fiber breaking strain
times fiber strain across fiber strain epsilon l. How is the force, what I can say vertical
force? This force F l have 2 components and we measure in our breaking machine, we measure
this vertical force. It can be the spacious speech about this horizontal force, it helps
together this this moment frictional moment in yarn, but it is no in yarn, in each structure
specially in yarn is the plays interesting role.
But we spoke about our vertical force emulation to our picture. So that so that, how is the
vertical force no to force F l times cosine of our angle theta dash, from our picture
using our equations, we obtained this here. Using our we we assume this small deformation
than epsilon l is given by such equation after rearranging, we obtained this equation. And
because small deformation, we can also write that 1 by 1 plus something small is roughly
1 minus something small, over some approximation formula known formula.
And we can also write that epsilon plus epsilon square time sine square minus epsilon power,
epsilon cube times cosine square and so on, is roughly equal to epsilon because epsilon
square is small and epsilon power periphery is much more smaller is not it? So, we can
write epsilon using this approximation, we can this function rearrange as follows, I
think I need to command this rearranging this rearranging on the level of your high school.
On the end we we obtain this this expression and because this is approximately epsilon,
we obtained that the vertical force proportional to epsilon is imaginable and for cosine is
of angle theta power to 3.
Now, it was one fiber, we derived a vertical force per one fiber. Now, how is the total
force on the breaking machine by jaw displacement epsilon? I say the, we assume that the vertical
axis, the axis of jaws is y our earlier, y axis the distribution of angles theta corresponds
to our probability density function u star theta, which we derived in in our earlier
lecture. Because clump line of jaw is the same as the section line evidently, it is
not the same, but in the model it is the same. We derived earlier, the number of sectioned
fibers per unit length. It was g by t times k n, where k n was also this integral and
what was G? G was mass weight of our our planar textile per mass unit, t is fiber fineness,
you know it mostly in decitex or something so is not it? And what is k? And we discussed
long time in last in the last lecture. Now, we will speak about fibers, in short
I say having the direction theta. What I mean, I mean that the in more precisely let us imagine,
the group elemental group of fibers, which have angles from some value theta to theta
plus d theta, some elemental angular class, class interval. So, that in this class interval,
the relative frequency of fibers is u star theta times u star theta times differential
quantity d theta. So, that the number of fibers in this interval per unit length of jaw is
what? Total number of fibers per unit length of jaw, it was in our earlier lecture nu,
the times probability density in earlier lecture we remember that this is evident and this
is valid. And then the vertical force due these fibers is d R, which is force per one
fiber times number of fibers times, number of all fibers times relative frequency of
these 2 2 numbers together means number of fibers having angle theta. It is a number
of all fibers per unit length of jaw times relative frequency of fibers having angle
theta. Using these expressions, we know the equation
for each each of these quantities; we obtain d R in such. It is not well may be I have
here, I have here one one one mistake, sorry nobody is perfect. It must be, no no no no
no no no all back, all all back, all is well all is well, it is not new equation, it is
the same equation, it is continued after rearranging.
Well, and now how is the total force? Total force is not the force only from the fibers
having our angle theta, but for all fibers. So, the force per unit lengths of jaw are
must be an integral, must be an integral from d R, is not it? Over all angles theta, it
is un-oriented and obtain for theta equal 0 to theta 90 degree, theta pi by 2. I must
remember that in theoretical works every times, we we degrees, we can say degrees because
it is better for our imagination, but we must work this radiance.
Well, using this, we obtain such equation as a resulting equation for a force, which
we need because realize the displacement between jaws equal epsilon. So, it based not only
to our epsilon, it is based also to distribution of orientation of fiber segments in our in
our structure. It is evident that if G is higher, if mass of unit area mass per square
meter; for example, is higher than we have more mass and the force will be higher and
so on and so on. It is not too important important. Therefore,
let us calculate a special case and then something like it will became some utilization coefficient,
which can say utilization of mechanic utilization fibers material. In opposite to earlier case,
in which all fibers in between jaws have its own, each fiber have its own special angle
theta. Let us imagine another structure in which all fibers be parallel to jaw axis,
this structure is shown here, all green both at as far as came to the line are fibers here.
So, let us imagine the situation in which between our jaws, our fibers are parallel
to jaw axis. Let us imagine that we take each fiber and we rotate each fiber to the position
to be to be parallel to to to jaw axis A, then we obtain such structure. Same h length
unit and jaw, the number of fibers in the lengths unit of jaw in this case is evidently
maximum. Mass area is mass by mass area in this in this rectangle one times h is each
G is mass here, of this fibers times area, one times h, but the mass, what is the mass?
The mass is the mass is t times nu max times h, nu max is number of fibers here, in lengths
unit of jaw, each have fiber have length h. So that so that l times nu max, its total
length of fibers in our rectangle, one times h total length and nu times and from the definition
of from the definition of fineness, we can write that the mass is 3 times nu max times
h, using it we obtain this here t times nu max. So, that number of fibers nu max is mass
mass per R R unit by fiber by fiber fineness. Well, this is the number of such fibers.
How is the force, the strain of fiber generally at rest epsilon l is now equal epsilon because
all fibers are parallel is equal to jaw displacement epsilon. So, that I can write that one fiber,
it is very easy, one fiber on one fiber is the force f max, which is our linear equation
for a force strain relation by fiber, but times epsilon epsilon l is no needed, it now
because epsilon l is equal to epsilon.
So, pair 1 fiber we obtain the first f max and then the total vertical force per unit
length of the jaw clamp line, what is it? It is number of fibers times force per one
fiber, using it we obtain this easy expression. We have 2 forces, one is for our, let us imagine
real structure having the orientation and second is for the structure from same fibers
with same mass areal mass, but orient it parallel to to jaw axis. One is R, second is R max,
we can construct the ratio R by R max. It can say us, how is how is the the mechanical
utilization of fibers during the effect of fiber orientation, is not it? Here we have
real include include affect of orientation and in denominator, it is without the affect
of the orientation and using this you can see that this blue card before integral here
is the same that R max. So, that we can write that the mechanical utilization in our easiest
case, which we solved is given by integral from theta from 0 to pi by 2 from cosine is
power to 4, where it is strong very hard effect of cosine times u theta d theta.
So, you know, you can see that this effect is utilization is can be very can be sometime
very small, is going from 0 to 1, is not it? But sometimes it can be very small, when we
use on the place of u theta, our u theta from our model imaginary flexible belt and so on.
So, that you use this u theta according this this expression.
We can to to calculate it and you obtained based on C value following following curves.
This is alpha angle from minus 90 degree to 0 to plus means, let us imagine something
like web or from web some freeze or I do not know what? May be all those slides today we
discussed is Professor Ishteyak about the possibility to apply such equation to and
so on. Let us imagine some structure and now, when
it is planar structure, you can take your experimental part to breaking machine in different
angles. You can in jaw with to longitudinal direction or right or left, it is small angle,
higher angle, much more higher angle, clamp it in the. There is also different based on
this angle alpha, why because the angle alpha is, it is here, the alpha is here here we
integrate, we are integrating over theta, but angles steady here is a constant is a
parameter of orientation of our web; for example, in relation to jaw axis. So, we obtain this
curves based for different way of C. For example, for evidently for C equal 1,
what it means isotropic structure, you can rotate this structure how you want. The utilization
will be permanently same, constant. When you use web then maybe you obtain C, C equal to
roughly, you show that the typical web have C roughly equal to 2 then you your work is
this function. The the the value here by the the utilization value is roughly 2 times higher
than here. The utilization value value by longitudinal direction is roughly 2 times
higher than by cross direction. It is often used in in textile industry, the
people do not have time and instruments for deeper study of similar problems, but they
have breaking machines and they have the products which they produce. In a non-woven often used,
the strength in longitudinal direction and strength in cross direction and ratio constructed
from these 2 values. Longitudinal to to cross cross direction, it can show it is the very
easy the practical the practical way, how to characterize the the to say intuitive and
the degree of orientation, here the intensity of orientation which was used.
So, we can see on this on this picture, how it is by C equal to and so on and so on. So,
fiber orientations have significant role or play significant role to mechanical properties
of fibers assemblies, different fibers assemblies. Now to our easiest case and reality some last
words. Normally, we often work with stepper fibers. So, that some of fibers are not clamp
into in both in both jaws. So, you need to formulate this problem mathematically
are reduced the number of fibers in jaw only to fibers, which are clump by both jaws. Second
on the end of jaws, on the end of jaws some fibers on the end of jaws here, on the other
side too, some fibers are cut it. Therefore, they are not in both jaws because was section
that your preparation of your experiment material, it is possible to calculate how how many fibers
and so on and give it in our model 2, the second influence.
Third, you said fibers are straight, no fibers are not straight; usually, fibers have some
crimping may be very important, may be small, but every time I am saying, it is. Then the
fiber for strain curve is modified through this scrimping, through this scrimp also in
parallel fiber bundle about which we will speak later. When the fibers have some distribution
of such scrimp then the mechanical effects are significant. Doctor does help me prepare
a special publication to to this problem. Well, this is the, it is I do not know third,
may be third influence which we do not to use.
Linear force strain relation, it is it is only on easiest theoretical example, but in
reality the force strain curve of fiber is fiber to fiber different. So, that you need
to use similar way, but use inside of our equations non-linear non-linear curve of for
strain relation related to specially your fibers. Well, all is this all is possible
may be good may be not so good, but to formulate using some theoretical tools and make some
modified model, which is much more which is much more complicated, but nearer to the reality.
The last assumption, the fibers deformed mutually, it is it is very difficult to to to to give
out because to this time, I personally, I do not know and neither good model, how to
input to our model of friction phenomenon between fibers. I know only that the traditional
equations like a coulomb equation forces friction forces proportional to normal force. Or the
the the other equation is based on the coulomb idea means no other friction and so on are
not enough well for textile structures, but what is well from point of view of friction,
it is quite open question, which is waiting for you may be. Some of you will be researchers,
you will be scientist in future and then you must solve the problems, which we all generation
did it to solve. Well, I think this is for this theme. All
thank very much for your attention. Be happy and good bye.