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Let us start today’s lecture, which is of
mechanics of Parallel fiber bundles. You know
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that the fiber bundles are very important
type of textile’s textures. Bundles are
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basis of all linear textiles, but also for
different other types. We will speak today,
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about the ideal bundle with parallel fibers.
To solve some model in this direction is either
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easy or very difficult. We plan to show you
one model, which is relatively very easy and
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this is known as the Hamburger’s model.
Then, we will introduce also some probabilistic
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model which is a little more complicated.
So, let us start our first idea, how to model
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fiber bundle mechanics. Let us use a general
assumption which are they. We assume that
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our bundle is created from great number of
fibers. Each fiber is straight, is linear.
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Each fiber is gripped in by both jaws. Fibers
are mutually parallel, so parallel fiber bundle.
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Fibers are mechanically independent to each
other. It means something like friction amount.
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Fiber’s is not used in our model or this
model of Mr. Hamburger terminologically. We
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were speaking about the strength of fiber
and from strength of fiber we understand their
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maximum tensile force in a fiber and breaking
strain of fiber which is strain by fiber strength
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point ok.
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In this case, we will speak about the variance,
but at first, some terms, some symbols. Let
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us imagine one easiest bundle having only
one fiber, so that one fiber between two jaws
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of breaking machine. The gauge length, we
call h and strain or relative elongation,
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we call epsilon. Then, we will speak about
number of fibers in bundle in this case, which
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is here, number of fibers. Red fiber is one,
then tensile force. Tensile force is s. We
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will speak about force strain relation of
fiber. So, the force s is the function of
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epsilon. Isn’t it, some function? Next term
is strength. What is strength? Strength is
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the maximum value of force s. Isn’t it?
Finally, we will speak about the breaking
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strain and breaking strain quote a. It is
a special value of epsilon of strain of fiber
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in the point in which the force is equal to
strength P. So, the P is the function S in
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the point a. This is a case with 1 fiber.
The second is fiber assembly having more fibers.
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So, then the number of fibers is in fiber
bundles schematically here. Lot of red fibers
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is here. Number of fibers is n. We call it
n.
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Tensile force is s sigma, capital sigma as
memo technique symbol subscript for summation
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all forces together. So, force is s sigma.
This force is function of epsilon, the bundle
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of forces. Therefore, a sigma must be a sigma
epsilon function. Strength of bundle maximum
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force of bundle is P sigma which is maximum
of a sigma. Breaking strain is called a sigma.
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Breaking strains of bundle, of course a sigma,
so that if P sigma is equal to the function
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S sigma in the point a sigma. It is evident.
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We will speak about three cases. In this theory,
case 1 is trivial. Case 2 is very easy and
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case 3 is not too easy for you.
Case 1 the trivial case. Let us assume that
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all fibers have same force strain curve, the
same strength P and same breaking strain a,
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at each fiber is same as each other fiber.
All fibers are same. How is then it is evident?
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It is really trivial case. It is evident that
the force in fiber bundle, what is the force
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in fiber bundle. Now, it is force per 1 fiber
time’s number of fibers. So, that n times
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s epsilon. Strength, what is the strength?
How the loading curve of such bundle is longer
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in one moment ping and all fibers are destroyed
in one moment?
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So, that the strength is evidently strength
of bundle is evidently n times strength of
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fiber. Of course, strain that the breaking
strain, the breaking strain of bundle is same
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than the breaking strain of each fiber. It
is as this case, trivial.
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Case 2 is solved in 1949 year by Hamburger
and it is known as Hamburger’s linear theory.
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Let us imagine a bundle from 2 types of fibers.
Here on outer scheme, the fibers are red and
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green yeah. Bundle from 2 types of fibers.
All fibers of 1 type have same force strain
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curve, same strength P and same breaking strain
a. Let us imagine for example, in reality
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the bundle from viscose fibers and polyester
fibers.
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All viscose fibers, you mean that all viscose
fibers have same properties. Also, how polyester
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fibers have same properties, but between viscose
fiber and polyester fiber are very high different
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properties, have very fine, very significant
differences.
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Well, let
us formulate one convention. Now, fiber of
1 type having smaller value of breaking strain
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is denoted as number 1. In the brand viscose
polyester fiber, evidently viscose fiber having
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small value of breaking strain, isn’t it?
Therefore, viscose fiber will be fiber number
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1 in our schemes. Let us assume that they
are red fibers. The second fibers, green fibers
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will have a number 2. We will use subscripts
1 and 2 for first and second type of fibers
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material. Symbols for the fineness t 1 or
t 2, there are symbols for material number
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1. There are symbols for material number 2.
So, t 1 and t 2. For strain relation is parallel
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fiber is S 1 epsilon and are S 2 epsilon.
For material number 2, a breaking strain is
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a 1 and a 2 and based on our convention, a1
is smaller than a2 yeah. Then, fiber strength
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is P 1 or P 2. Number of fibers in our bundle
is n 1 and n 2. Total number of fibers in
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bundle is n which is n 1 plus n 2.
Mass of fibers in our bundle, all is related
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to our bundle among the couple of jaws. Mass
of fiber is m 1 and m 2. Total mass is m sum
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of both. Bundle fineness is bundle count is
capital T which is total mass of our bundle
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by lengths of our bundle. Length of our bundle
is h lengths. Mass portion, we have spoken
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of first lessons about the mass portions.
Mass portion of first material is m1 mass
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of first fibers by total mass. Similarly,
g 2 is m 2 by m. Let us remember that g 1
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plus g 2 must be equal 1.
You know in the industry we used Parasynchuk
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values, so that we in our theoretical like
g 1, g 2 can be 0.4, 0.6 something between
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0 and 1 in Parasynchuk, sometimes 40 percentages,
then 60 percentages here. So, in theoretical
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way if we speak about a dimensionless quantity
from interval 0 to 1, this is g 1 and g 2.
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Well, how it is this number of fibers in our
fiber bundle? You know that m1 mass of fibers
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from first material is g 1 times m. It is
going cut from definition of g from mass portion.
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Isn’t it? Then, also it is t 1 fineness
is mass by lengths. How is the length of fibers
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in our bundle? Number of fiber times lengths
of each 1 n times h 1 h and times h. From
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the second equation we obtained 1 is and 1
by t 1 h, but n 1 from here, from this equation
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and 1 is g 1 times m, g 1 times m, but the
ratio m by h, it was the fineness, the linear
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density of our bundle capital T.
Then, we can write n 1 is g 1 time capital
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T by t 1 evidently. Similarly, we can derive
n 2, n 2 is g 2 times T by t 2. This equation
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we will use for number of fibers from first
and second materials in our blended bundle.
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Now, let us think about our scheme which this
one here. On the other side, this is the scale
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of epsilon, strain fiber strain and though
on the ordinate our forces. Schematically,
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let us imagine that the red curve is force
strain relation of fiber number 1 from first
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material red curve. The green curve is similar.
Similarly, the force strain relation of our
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fiber number 2 from second material.
The first curve is increasing from 0 to some
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end point which represents the break of fiber.
This end point has 2 coordinates. Epsilon
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is equal a1 breaking strain of red fiber and
the force is p 1. So, that it is strength
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of our red fiber number 1. Similarly, green
fiber is increased have another force strain
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relation. End point has the coordinate a 2
breaking strain of green fiber and p 2 strength
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of green fiber.
On this, this scheme is well because a1 is
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more than a 2. Our convention is valid. On
the red, sorry on the green function, we have
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1 white point is here which shall be important.
What is it? Which of point is it? The green
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fiber in this moment have the strain epsilon
equal a1 as a breaking strain of red fiber,
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but for green fiber, it is not breaking strain.
It is only some general strain. In this moment,
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a green fiber has some force S 2 because S
2 is whole this green function in the point
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epsilon equal to, a 1 S 2 a 1. Let us now
divide this scheme to three parts.
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First part is the part from 0 to epsilon equal
a 1 breaking strain of red fiber. It is a
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little area in my picture. The second interval
is from a 1 to a 2. It is green color and
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the third is over a 2. It is white. Do your
study separately, the forces in these three
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intervals.
In the first interval from 0 to a 1, in which
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point from this interval is the total force
in bundle, the maximum force in which point?
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You see each fiber is epsilon take higher
and higher force red as well as green. So,
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the highest force must be in the point epsilon
equal a1. Isn’t it? Is it logically clear?
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Well. So, which of force bundle is when epsilon
is equal a 1, it is shown here. It is force
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S sigma and the point epsilon equal, a 1.
So, a sigma a 1, isn’t it? What is it? Logically,
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how many fibers, red fibers are in our bundle
n 1? Each fiber takes the force which is its
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strength force p 1. So, n times p 1, it is
the part from red fiber, same bundle. Which
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force take in the green fibers? Each fiber
has or takes the force S 2 a 1. Is it so?
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So, total force is n times p 1 plus n 2 times
S 2 a 1 yeah. After using this couple of equations
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here, we obtained this, this, this, this,
this expression. So, we know that in interval
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from 0 to a 1, the highest force in bundle
which is in moment epsilon equal a 1 is given
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by this formula, by this equation.
Now, let us solve the second interval for
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a 1 to a 2. How it is here? If epsilon is
higher than a1, evidently all red fibers are
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broken. Only green fibers are working. Nevertheless,
with increasing of epsilon, the force in each
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green fiber is increasing too and their maximum
of force in each green fiber is in the point
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epsilon equal a 2. Clear?
So, how is the total force in this moment
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epsilon equal a 2 in our bundle? How it is?
Where are the forces in our bundle? A number
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of red fibers times for 0, all are broken
plus number of green fibers times the maximum
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possible force, it is strength of fiber. So,
we can write n 1 times 0 plus n 2 times P
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2 using n 2. From this equation, we obtain
s sigma a 2 is t times, g 2 P 2 by t 2 and
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for completeness, if epsilon is higher than
a 2, all fibers are broken. So, it is evident
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that the force in the bundle is equal 0. Clear?
Now, let us solve the problem. What is the
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strength of bundle? We said that strength
is the maximum force. It must be one of our
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earlier two forces. It can be this one or
this one. May be this, may be this. In the
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moment nobody knows.
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So, if we must write that strength of bundle
is maximum from 2 values S sum a 1 and S sum
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a 2. Which was right using expressions derived?
We can write that strength of bundle is T
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because we can give before brackets, before
the operator of maximum T times, maximum of
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these two expressions. Well, what we have
here, it is the first thing. Here is a ratio
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P 1 by t 1. What is P 1? It is strength of
fiber by fiber linear density by fiber fineness.
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It is tenacity. It is known as a tenacity
of fiber, for example or something so on.
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Similarly, what is it P 2 by t 2? It is a
tenacity of fiber number 2, green fiber. When
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we get the t on the left hand side of our
equation, we obtain ratio P sigma by T. What
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is this? It is evidently tenacity of our bundle.
So, that we can write our equations in such
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form and we can say that the bundle tenacity,
it is given by such expression is maximum
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of these 2 values in which we have the breaking
tenacity of first fiber, tenacity of second
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fiber and also ratio S 2 a 1 by t 2 force
in our earlier white point on the green curve
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by linear density by fineness and it is called
as a specific stress.
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You know, from earlier lecture that the quantities
for linear density is equal to stress by rho
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by specific mass and it is called generally
in the theory of mechanics is the specific
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stress. So, our tenacity is also something,
is also specific stress, but in an end point
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of force strain curve. Now, let us solve the
breaking strain of bundle. If the first member
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here is higher than the second, then from
compacts, from logical contact is evident
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that the breaking strain of bundle will be
same as the breaking strain of red fiber.
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It is a1 and similarly, if the second number
is higher than first, then it will be a 2.
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Now, let us solve the graphically interpretation
of our equation. We write it here. It is the
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same as this one. Well, we want to create
some graphical interpretation of the equation.
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The g 1 and g 2 are the mass portions of first
and second material. Let us give on their
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quantity g 2 from left to right, from 0 to
10 percentage of fiber number 2 green fibers.
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Therefore, green arrow 2, 1, 100 percentages
yeah.
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G 1 must go from right hand side to left,
also from 0 to 1. Isn’t it? I do not know
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if g 2 is 70 percentages, 0.7 for example.
Here, then g 2 must be at the percentage 0.3,
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then is clear on or the right we will have
tenacities.
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Let us study now this expression. The first
member here, have 2 numbers. Let us study
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the first of this g 1 time g 1 times P 1 by
t 1, P 1 by t 1. It is tenacity of fiber number
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1. It is a given value here by P 1 by t 1
and this value is multiplied by g 1, so that
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if g 1 is equal 0 that this member is equal
0. If g 1 is maximum, is equal 1. This member
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is equal P 1 by t 1 tenacity and linear relation,
so that this member Linear A increased with
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g 1 from 0 to 1.
This one is this straight line which represents
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this member here. It starts from 0 and this
g1 increasing from 0 to 1 is increasing to
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the value P 1 by t 1. Clear? Similarly, the
second member is here, if g 2 is equal 0 into
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0, if g 2 is 1, then it is S 2 a 1 by t 2
and it is linear function. So, that picture
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of this part of this member is increasing
to this g 2 from 0 to S 2 a 1 by t 2. It is
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increasing from this point to this here to
the value S 2 a 1 by t 2, but our first member
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is sum of both. Sum of these two lines, evidently
this thick black line.
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So, is the picture of our first member in
relation to g 1 g 2 proportions? How it is
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with second member? Second member is g 2 times
P to by t 2 p 2 times. T 2 is tenacity of
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second fiber given value for a, this of that
fiber which we use from the place of our green
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fibers. This member value of this member is
increasing with g 2 from 0 to P 2 by t 2.
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So, we can have the line from 0 g 2 0 with
g 2 increased it to P 2 by t 2.
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Now, what is the bundle tenacity P sigma by
capital T? It is maximum of these two black
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thick lines on our picture when we are in
this region. What is higher, this thick line
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or this thick line? Higher value is evidently
on this thick line. Yes. So, in this region,
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this line is tenacity of bundle to this yellow
point here. How it is on the right hand side?
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This region which this couple of thick black
line has the higher position, evidently this,
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so it is from this yellow point to the right
end. This line represents the tenacity of
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bundle. Isn’t it? Altogether, the tenacity
of bundle is given by such blue curve sign
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which is a break shape. Isn’t it?
It is interesting. Why? See, let us imagine
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we start this 100 percentage of fibers number
1, then the tenacity is P 1 by t 1. Then,
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on the price of red fibers, we give some portion
of green fibers on the price of I do not know
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viscose fibers. We give some fibers from polyester.
Polyester is of higher value of its stand.
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For example, we use this g 1 and g 2. What
do we obtain? We obtain value which is smaller
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than earlier. Starting bundle is not, it is
not right when somebody is meaning that when
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on the price of one fiber, we use some fibers
which have higher strength. Then, the bundle
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the blend together will increase in strength.
You can see that it can be also a situation
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in which it is decreased than the tenacity
is smaller. So, this is when we create in
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spinning mill. For example, the blanks because
when we choose no good portions, mass portions
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of material, our final yarn is not ideal fiber
bundle, but similarly can have smaller tenacity
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than earlier result blending.
This blue curve, break curve is typical for
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blending, but not every time. It is possible
also to obtain such picture in this case really
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direct point is the minimum point. Let us
study, now how is the minimum bundle tenacity?
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It is the point in which mechanical properties
the highest is not.
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What is minimum of bundle tenacity? Minimum
of bundle tenacity can be or if this structure
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is at 12 or in our red point, if the result
is this 1 or in our yellow point, usually
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in our yellow point. So, let us calculate
this in this points the quantities which in
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the red point. It is very easy in the red
point. Every time it is P1 by t1 tenacity
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of first fiber’s material in yellow point.
What is this yellow point? It is section of
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two lines. One line, equation of line is given
by this expression. Equation for the second
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is given by this expression.
In yellow point, our yellow point here both
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must be valid because it is section of two
lines. So, that is valid that the first member
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g 1 P 1 by t 1 plus g 2 S 2 a 1 by t 1 must
be equal to g 2 P 2 by t 2. No. Well, of three
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arranging of this using g 1 is 1 minus g 2
because g 1 plus g 2 equal 1 and after rearranging,
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we obtain the mass portion for second material.
Our green material in our lecture as shown
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in our equation here, it is to be rearranging.
We know g 2 in this position g 1 is 1 minus
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g 2 evidently.
Using this value, we can calculate the minimum
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tenacity of bundle for which, for this line,
but also from this line because yellow point
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is section of both. I recommend you to use
this line because mechanically, it is easier
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you need not. So, long write by numerical
calculation. Therefore, this tenacity is g
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1 times P 2 by t 2. It is the minimum tenacity.
No, precisely minimum tenacity is minimum
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from two values or this one from red points
or this one from our yellow points
of tenacity of fiber bundle. I said after
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addition of fibers having higher tenacity,
the tenacity of resulting bundle can decrease.
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Yes, this story can be applied for rough estimation
of blended staple yarns too. Of course, staple
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yarn is not ideal parallel fiber bundle, but
the preferential direction in yarn is longitudinal.
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So, as it is little similar to our ideal bundle.
Therefore, our result can be roughly used
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also for evaluation of yarn tenacity of blended
yarn. How it is applied?
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On the place of P 1 by t 1, earlier to this
moment, it was tenacity of fiber number 1.
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We use the tenacity of single yarn from 100
percentage of material number 1 on the place
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P 2 by t 2. Now, we use the mean tenacity
of single yarn from second, only from second
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material. On the place of value S 2 a 1 by
t 2, our earlier white point which means a
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specific stress of the single yarn, now from
100 percentage of material number 2. When
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the strain is equal to the breaking strain
of the single yarn from 100 percentage of
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material number 1, it is epsilon is a 1 or
for example, in N/tex.
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So, similarly only on the position of earlier
tenacities and breaking corrugation of fibers,
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we use analogical quantities from yarns. We
make on my university some experiments with
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blended yarns. Of course, we did not obtain
so idealize break graph, but really such curve
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experimentally measured have such, usually
it is so that our curve is going but it
have half minimum, roughly near to our yellow
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point, this expression is often used. For
our work in industry is Hamburger’s theory,
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bring one important result desired possibility
to calculate it numerically.
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It is shown that when we prepare some blend,
the tenacity of such blend may be yarn tenacity
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of such blend can be higher than earlier tenacity
of 100 percentage of yarn from 1 component.
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When you will prepare some blend in your brain
must start some red light carefully that the
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strength of your yarn will not be enough well
for following application.
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This you must prove it and check it and be
sure that your idea of this or that blend
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is fully useful also from the point of your
mechanical properties. Therefore, this theoretical
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concept is very useful for industry. You can
calculate when you have the starting values.
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You can calculate it and say it quantitatively,
but in your brain, when you will be some technologies
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in industry must start by blending some, I
said red light in your brain be carefully
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re-strength is strength means tenacity of
yarn. Well, it is about the Hamburger’s
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theory. is original work of Hamburger 1949.
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The fact, the case 3 which in short, we want
to start now and in our other lecture, we
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will continue with this. It is not so easy.
That is very trivial but useful Hamburger’s
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theoretical model. Some intuitive introduction
to case 3. We spoke in Hamburger’s model
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about two components. It was red and green
fibers, yeah only blend from two components.
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Similarly, it is possible to derive in corresponding
equations for three components. Similar logical
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way for five components, for ten components,
for thousand components, for million components
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theoretical, isn’t it? Now, let us see by
cotton fiber material, each fiber from its
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natural fiber. Each fiber has another value
of tenacity and other value of breaking strain.
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Isn’t it? Let us make from the fiber theoretically.
Not practically. It is too difficult. Let
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us make a separate of fibers. The groups,
the fiber is having same tenacity and same
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breaking strain.
You may have, may be thousand different groups.
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Then, make blend. It is our original material.
So, the material having variable tenacity
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and variable breaking strains of fibers is
some sink like Hamburger’s case, but with
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no tools and then, thousands very much components
intuitively. Is it intuitively clear, this
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idea?
So, this similar effect by Hamburger must
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be also in the case when we use fibers having
the distribution of tenacity and distribution
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of breaking strain. Breaking points of fibers
are random usual, breaking points I mean these
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couples force breaking strain at this case.
In this graph on the is a breaking strains
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of fibers, on the ordinate is force strength
of fibers and each fiber of another end point
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by break. So, that altogether we obtain such
set of red point as the symbolic set of all
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couple’s strength breaking strain. Symbols
which we will use, P is fiber strength, a
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is fiber breaking strain. Let us imagine that
P is from some interval from P min to P max
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and a is from some interval from a min to
a max. No, because write it in shorter form
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this domain we will call under the symbol
omega.
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Omega, it means P from interval P min to P
max, a from interval a min to a max. This
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00:48:23,569 --> 00:48:30,569
distribution, the distribution of all such
points here, of all points strength breaking
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elongation of fibers have some joint probability
density function of this couples. This probability
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00:48:47,210 --> 00:48:54,210
density function, joint probability density
function we call UPA. It is probability density
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function U of two parameter random variables.
First random variable is P fiber strength.
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Second random variable is a fiber breaking
strain.
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Well, I think this introduction to our third
case in this lecture is finished. In following
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lecture, we will continue with the relation
of solving of this problem. Well, thank you
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for your attention.