1
00:00:24,609 --> 00:00:31,609
Let us continue our earlier theme about fiber
orientation. We derived a set of equation
2
00:00:37,550 --> 00:00:44,550
probability and density and so on. Number
of fibers for fiber structures for for planar
3
00:00:50,149 --> 00:00:57,149
planar types of of fiber structures, this
distribution of directions is very important
4
00:00:59,619 --> 00:01:06,619
for lot of properties at most for mechanical
properties of fibers assemblies.
5
00:01:11,520 --> 00:01:18,520
I want in this lecture to present one case,
how is possible to apply our earlier knowledge
6
00:01:22,719 --> 00:01:29,719
to the mechanical behavior of fibers assembly,
but I must say in our lecture, it will be
7
00:01:33,420 --> 00:01:40,420
an easiest case. So easy, that it is on the
border of unreality, each real structure is
8
00:01:43,750 --> 00:01:50,750
much more complicated. So that, but I want
to present it to you because it can show,
9
00:01:53,360 --> 00:02:00,360
how is the style of our work, when we when
we when we must to derive some model for mechanical
10
00:02:06,880 --> 00:02:13,880
property often fiber assembly with with with
some some distribution significant distribution
11
00:02:18,260 --> 00:02:25,260
of fiber direction because this methodological
sense, I want to present you this easiest
12
00:02:26,709 --> 00:02:33,709
case. On the final sentences, I will in short
comment what is possible to do more and go
13
00:02:35,500 --> 00:02:42,500
more to the real means more complicated structure.
So, that is easiest case of mechanical behavior
14
00:02:44,550 --> 00:02:51,550
of planar fiber assembly. Let us let us accept
six assumptions, which which make our problem
15
00:02:57,170 --> 00:03:04,170
easy. First, our model is planar fiber orientation
is the planar fiber, second each fiber is
16
00:03:09,650 --> 00:03:16,650
straight, it means no crimped also cotton
have cotton fiber have some small crimp, but
17
00:03:18,140 --> 00:03:25,140
we assume that that the each fiber is straight
then third each fiber is clamped by both jaws
18
00:03:28,220 --> 00:03:35,220
of tensile machine or breaking machine. And
we neglect the effect of the margins of jaws.
19
00:03:38,019 --> 00:03:44,099
You know, when we have something like non-woven
some some warp or something so. In in couple
20
00:03:44,099 --> 00:03:51,099
of jaws by in breaking machine, we must cut
this this structure, where the ends of jaws
21
00:03:53,080 --> 00:04:00,080
know. So, that some fibers are like this here,
we do not want to calculate these fibers,
22
00:04:06,450 --> 00:04:13,450
you can imagine that the the jaws are very
very very long and this age between this this
23
00:04:18,250 --> 00:04:25,250
this value between between between couple
of jaws is very small fourth because I said
24
00:04:27,470 --> 00:04:34,470
easiest model, let us imagine linear force-strain
relation same for each fiber. So, between
25
00:04:38,360 --> 00:04:45,360
the fiber force F l and the fiber elongation
epsilon l is their relation strained by breaking
26
00:04:49,879 --> 00:04:56,879
strained times epsilon, this is the constant
for fiber times epsilon l and 0 of course,
27
00:04:58,499 --> 00:05:05,499
after breaking the this next next assumption
small deformations so small that that no one
28
00:05:12,650 --> 00:05:19,650
fiber will rupture it destroy it through our
process. And the last of our assumption is
29
00:05:22,039 --> 00:05:28,919
that the fibers are deformed mutually independently,
we do not calculate the friction fiber to
30
00:05:28,919 --> 00:05:32,490
fiber friction in in our structure and so
on.
31
00:05:32,490 --> 00:05:39,490
Well, first step let us do derive do derive
the function of of pair one fiber one general
32
00:05:45,710 --> 00:05:52,710
fiber between jaws. The, we have some couple
of of jaws to jaws in breaking machine
33
00:06:08,219 --> 00:06:15,219
is is clumped between these two, two between
these two, two, two jaws. Lengths of this
34
00:06:21,999 --> 00:06:28,999
fiber is l, starting angle we we will need,
it is enough to know non oriented angle theta
35
00:06:33,279 --> 00:06:40,279
is is is shown on the on the picture here.
To vertical vertical axis is our earlier y
36
00:06:45,589 --> 00:06:52,279
axis.
Well, so, h is vision after first intuitively
37
00:06:52,279 --> 00:06:59,279
after elongation after after jaw displacement,
the jaw B is changed the position to the new
38
00:07:05,119 --> 00:07:12,119
position B dash. Therefore, the fiber is fiber
is now this here, plus one dash higher than
39
00:07:14,169 --> 00:07:21,169
length. So, dash epsilon, we call as a relative
jaw displacement and it is from h lengths
40
00:07:29,080 --> 00:07:36,080
and h dash defined, the traditional here.
Jaw displacement in opposite to them, the
41
00:07:38,459 --> 00:07:45,459
strain in fiber strain in fiber work is l
length is. So, that it is l dash l minus l
42
00:07:47,099 --> 00:07:54,099
by l evident here. Starting angle theta for
the starting angle theta is specific to write
43
00:07:55,830 --> 00:08:02,719
that it is cosine of this is h by l it is
shown from the figure.
44
00:08:02,719 --> 00:08:09,719
The final angle theta is the angle after elongation,
which is similarly, from the picture here
45
00:08:12,219 --> 00:08:19,219
h dash by l dash and because h dash is h times
1 plus epsilon and l is 1 dash is l times
46
00:08:25,339 --> 00:08:32,339
1 plus epsilon l fiber strain. Then we can
write that the cosine is theta dash is cosine
47
00:08:33,349 --> 00:08:40,200
is theta starting angle times this ratio 1
plus epsilon by 1 plus epsilon l.
48
00:08:40,200 --> 00:08:47,200
You know, the Pythagorean Theorem is not it?
It is very known theorem in whole world. Therefore,
49
00:08:49,640 --> 00:08:56,640
we want to use it two times, first time from
this triangle from this yellow triangle and
50
00:08:57,300 --> 00:09:04,300
what we obtain from Pythagorean Theorem x
square is l square minus h square is not it?
51
00:09:07,230 --> 00:09:14,230
Well, from the second right green triangle,
we obtain x square is l dash square minus
52
00:09:15,360 --> 00:09:22,360
h square because this length x is same for
yellow triangle assess for for for white green
53
00:09:26,089 --> 00:09:33,089
triangle. Why? Because our jaws are on are
from some metal deformable moving of the jaw
54
00:09:36,670 --> 00:09:38,740
B.
55
00:09:38,740 --> 00:09:45,740
So, that it must be valid, this is equal to
this expression is here, it is here. Now some
56
00:09:50,230 --> 00:09:57,190
small rearranging, which of we divide this
equation by l. So, that we obtain this here,
57
00:09:57,190 --> 00:10:04,190
sorry l square of course, by l square we divide
using our symbols our our equations now from
58
00:10:08,089 --> 00:10:15,089
here, we can write 1 minus cosine is square
theta is 1 plus epsilon l square minus h times
59
00:10:16,790 --> 00:10:23,790
1 plus epsilon by l square. So that, it is
it is this one because h by l is cosine. So,
60
00:10:31,310 --> 00:10:37,670
I have this here from this equation, this
is possible to write in this to this form
61
00:10:37,670 --> 00:10:44,670
trivially and epsilon one is this here or
epsilon l we obtain epsilon or epsilon l is
62
00:10:50,560 --> 00:10:57,560
possible to explain using this expression.
That is well, that is good. It is shown that
63
00:10:59,029 --> 00:11:06,029
the strain of fiber is not the same than jaw
displacement epsilon.
64
00:11:09,420 --> 00:11:16,420
And that is the function of jaw of angle theta,
it based on the orientation of our fiber segment,
65
00:11:19,120 --> 00:11:26,120
our fiber. Well, we said our model will be
the easiest as possible. Therefore, let us
66
00:11:32,860 --> 00:11:39,860
imagine small deformations, very small deformations
then epsilon is very small and then the the
67
00:11:44,519 --> 00:11:51,519
approximate value, approximate equations are
valid. You know that we can construct different
68
00:11:52,829 --> 00:11:59,829
approximate formulas using Taylor series is
not it? It is known from mathematics. And
69
00:12:02,040 --> 00:12:09,040
using such, but exist also the some content
of different often used approximated, approximation
70
00:12:11,480 --> 00:12:18,480
equations. The one say, 2 times epsilon plus
epsilon square is roughly for very small epsilon
71
00:12:23,620 --> 00:12:30,620
2 times epsilon because this square is extremely
small then we can neglect it, square root
72
00:12:31,449 --> 00:12:37,269
of 1 plus 2 times epsilon cosine square theta
if epsilon is small, if this part is very
73
00:12:37,269 --> 00:12:44,269
small is possible to write approximated s
1 plus 1 half of this.
74
00:12:44,660 --> 00:12:51,660
The third is nothing this 2 is in the moment
enough then our epsilon epsilon, l was this
75
00:12:56,670 --> 00:13:03,670
here using these approximation equations,
we obtained that it is. This is this is approximately
76
00:13:06,829 --> 00:13:13,829
this? So, that epsilon one is epsilon times
cosine square theta for small deformations.
77
00:13:18,060 --> 00:13:25,060
In this easy equation, you can see you can
see that the function of angle theta, when
78
00:13:27,730 --> 00:13:34,730
we have when we have fiber is higher angle
theta then cosine is is smaller than 1, cosine
79
00:13:37,449 --> 00:13:44,449
square is much more smaller than than 1. So,
that the epsilon l is smaller than epsilon,
80
00:13:46,759 --> 00:13:53,759
the highest value of strain of fiber is, when
the fiber is parallel to jaw axis a higher
81
00:13:56,480 --> 00:14:03,480
this angle theta from the direction of jaw
axis. So, smaller is fiber elongation.
82
00:14:06,529 --> 00:14:13,529
Is there is out of this equation? It was something
about the fiber, about fiber strain, now about
83
00:14:19,649 --> 00:14:26,649
fiber force forces. We said that the force
in fiber follows the linear function. So,
84
00:14:32,459 --> 00:14:39,459
that the force in fiber F l is proportioned
to P by a fiber strain by fiber breaking strain
85
00:14:41,589 --> 00:14:48,589
times fiber strain across fiber strain epsilon
l. How is the force, what I can say vertical
86
00:14:55,420 --> 00:15:02,420
force? This force F l have 2 components and
we measure in our breaking machine, we measure
87
00:15:03,189 --> 00:15:10,189
this vertical force. It can be the spacious
speech about this horizontal force, it helps
88
00:15:11,480 --> 00:15:18,480
together this this moment frictional moment
in yarn, but it is no in yarn, in each structure
89
00:15:20,339 --> 00:15:25,050
specially in yarn is the plays interesting
role.
90
00:15:25,050 --> 00:15:32,050
But we spoke about our vertical force emulation
to our picture. So that so that, how is the
91
00:15:33,740 --> 00:15:40,740
vertical force no to force F l times cosine
of our angle theta dash, from our picture
92
00:15:43,660 --> 00:15:50,660
using our equations, we obtained this here.
Using our we we assume this small deformation
93
00:15:55,009 --> 00:16:02,009
than epsilon l is given by such equation after
rearranging, we obtained this equation. And
94
00:16:03,740 --> 00:16:10,740
because small deformation, we can also write
that 1 by 1 plus something small is roughly
95
00:16:12,279 --> 00:16:19,279
1 minus something small, over some approximation
formula known formula.
96
00:16:20,740 --> 00:16:27,740
And we can also write that epsilon plus epsilon
square time sine square minus epsilon power,
97
00:16:28,589 --> 00:16:35,589
epsilon cube times cosine square and so on,
is roughly equal to epsilon because epsilon
98
00:16:36,970 --> 00:16:43,740
square is small and epsilon power periphery
is much more smaller is not it? So, we can
99
00:16:43,740 --> 00:16:50,740
write epsilon using this approximation, we
can this function rearrange as follows, I
100
00:16:56,279 --> 00:17:03,279
think I need to command this rearranging this
rearranging on the level of your high school.
101
00:17:05,540 --> 00:17:11,740
On the end we we obtain this this expression
and because this is approximately epsilon,
102
00:17:11,740 --> 00:17:18,740
we obtained that the vertical force proportional
to epsilon is imaginable and for cosine is
103
00:17:31,270 --> 00:17:38,270
of angle theta power to 3.
104
00:17:38,649 --> 00:17:45,649
Now, it was one fiber, we derived a vertical
force per one fiber. Now, how is the total
105
00:17:54,799 --> 00:18:01,799
force on the breaking machine by jaw displacement
epsilon? I say the, we assume that the vertical
106
00:18:07,809 --> 00:18:14,809
axis, the axis of jaws is y our earlier, y
axis the distribution of angles theta corresponds
107
00:18:20,059 --> 00:18:27,059
to our probability density function u star
theta, which we derived in in our earlier
108
00:18:28,179 --> 00:18:35,179
lecture. Because clump line of jaw is the
same as the section line evidently, it is
109
00:18:41,990 --> 00:18:48,990
not the same, but in the model it is the same.
We derived earlier, the number of sectioned
110
00:18:49,529 --> 00:18:56,529
fibers per unit length. It was g by t times
k n, where k n was also this integral and
111
00:19:01,610 --> 00:19:08,610
what was G? G was mass weight of our our planar
textile per mass unit, t is fiber fineness,
112
00:19:14,799 --> 00:19:21,720
you know it mostly in decitex or something
so is not it? And what is k? And we discussed
113
00:19:21,720 --> 00:19:28,720
long time in last in the last lecture.
Now, we will speak about fibers, in short
114
00:19:32,049 --> 00:19:39,049
I say having the direction theta. What I mean,
I mean that the in more precisely let us imagine,
115
00:19:41,779 --> 00:19:48,779
the group elemental group of fibers, which
have angles from some value theta to theta
116
00:19:49,200 --> 00:19:56,200
plus d theta, some elemental angular class,
class interval. So, that in this class interval,
117
00:20:03,549 --> 00:20:10,549
the relative frequency of fibers is u star
theta times u star theta times differential
118
00:20:13,990 --> 00:20:20,990
quantity d theta. So, that the number of fibers
in this interval per unit length of jaw is
119
00:20:28,080 --> 00:20:35,080
what? Total number of fibers per unit length
of jaw, it was in our earlier lecture nu,
120
00:20:35,390 --> 00:20:42,390
the times probability density in earlier lecture
we remember that this is evident and this
121
00:20:45,559 --> 00:20:52,559
is valid. And then the vertical force due
these fibers is d R, which is force per one
122
00:20:56,200 --> 00:21:03,200
fiber times number of fibers times, number
of all fibers times relative frequency of
123
00:21:11,980 --> 00:21:18,980
these 2 2 numbers together means number of
fibers having angle theta. It is a number
124
00:21:25,710 --> 00:21:32,460
of all fibers per unit length of jaw times
relative frequency of fibers having angle
125
00:21:32,460 --> 00:21:39,250
theta.
Using these expressions, we know the equation
126
00:21:39,250 --> 00:21:46,250
for each each of these quantities; we obtain
d R in such. It is not well may be I have
127
00:21:55,419 --> 00:22:02,419
here, I have here one one one mistake, sorry
nobody is perfect. It must be, no no no no
128
00:22:06,460 --> 00:22:13,460
no no no all back, all all back, all is well
all is well, it is not new equation, it is
129
00:22:14,730 --> 00:22:18,590
the same equation, it is continued after rearranging.
130
00:22:18,590 --> 00:22:25,590
Well, and now how is the total force? Total
force is not the force only from the fibers
131
00:22:26,610 --> 00:22:33,610
having our angle theta, but for all fibers.
So, the force per unit lengths of jaw are
132
00:22:36,010 --> 00:22:43,010
must be an integral, must be an integral from
d R, is not it? Over all angles theta, it
133
00:22:45,760 --> 00:22:52,760
is un-oriented and obtain for theta equal
0 to theta 90 degree, theta pi by 2. I must
134
00:22:53,909 --> 00:23:00,909
remember that in theoretical works every times,
we we degrees, we can say degrees because
135
00:23:02,960 --> 00:23:09,090
it is better for our imagination, but we must
work this radiance.
136
00:23:09,090 --> 00:23:16,090
Well, using this, we obtain such equation
as a resulting equation for a force, which
137
00:23:16,980 --> 00:23:23,980
we need because realize the displacement between
jaws equal epsilon. So, it based not only
138
00:23:26,929 --> 00:23:33,929
to our epsilon, it is based also to distribution
of orientation of fiber segments in our in
139
00:23:35,890 --> 00:23:42,890
our structure. It is evident that if G is
higher, if mass of unit area mass per square
140
00:23:47,669 --> 00:23:53,260
meter; for example, is higher than we have
more mass and the force will be higher and
141
00:23:53,260 --> 00:23:58,700
so on and so on.
It is not too important important. Therefore,
142
00:23:58,700 --> 00:24:05,700
let us calculate a special case and then something
like it will became some utilization coefficient,
143
00:24:08,580 --> 00:24:15,210
which can say utilization of mechanic utilization
fibers material. In opposite to earlier case,
144
00:24:15,210 --> 00:24:22,210
in which all fibers in between jaws have its
own, each fiber have its own special angle
145
00:24:24,600 --> 00:24:31,600
theta. Let us imagine another structure in
which all fibers be parallel to jaw axis,
146
00:24:36,529 --> 00:24:43,529
this structure is shown here, all green both
at as far as came to the line are fibers here.
147
00:24:48,240 --> 00:24:54,980
So, let us imagine the situation in which
between our jaws, our fibers are parallel
148
00:24:54,980 --> 00:25:01,980
to jaw axis. Let us imagine that we take each
fiber and we rotate each fiber to the position
149
00:25:03,679 --> 00:25:10,679
to be to be parallel to to to jaw axis A,
then we obtain such structure. Same h length
150
00:25:15,110 --> 00:25:22,110
unit and jaw, the number of fibers in the
lengths unit of jaw in this case is evidently
151
00:25:25,750 --> 00:25:32,750
maximum. Mass area is mass by mass area in
this in this rectangle one times h is each
152
00:25:45,919 --> 00:25:52,919
G is mass here, of this fibers times area,
one times h, but the mass, what is the mass?
153
00:25:58,600 --> 00:26:05,600
The mass is the mass is t times nu max times
h, nu max is number of fibers here, in lengths
154
00:26:10,710 --> 00:26:17,710
unit of jaw, each have fiber have length h.
So that so that l times nu max, its total
155
00:26:23,210 --> 00:26:30,210
length of fibers in our rectangle, one times
h total length and nu times and from the definition
156
00:26:42,440 --> 00:26:49,000
of from the definition of fineness, we can
write that the mass is 3 times nu max times
157
00:26:49,000 --> 00:26:56,000
h, using it we obtain this here t times nu
max. So, that number of fibers nu max is mass
158
00:27:05,210 --> 00:27:12,210
mass per R R unit by fiber by fiber fineness.
Well, this is the number of such fibers.
159
00:27:22,519 --> 00:27:29,519
How is the force, the strain of fiber generally
at rest epsilon l is now equal epsilon because
160
00:27:34,679 --> 00:27:41,679
all fibers are parallel is equal to jaw displacement
epsilon. So, that I can write that one fiber,
161
00:27:42,760 --> 00:27:49,760
it is very easy, one fiber on one fiber is
the force f max, which is our linear equation
162
00:27:51,620 --> 00:27:58,620
for a force strain relation by fiber, but
times epsilon epsilon l is no needed, it now
163
00:27:59,610 --> 00:28:02,090
because epsilon l is equal to epsilon.
164
00:28:02,090 --> 00:28:09,090
So, pair 1 fiber we obtain the first f max
and then the total vertical force per unit
165
00:28:10,049 --> 00:28:17,049
length of the jaw clamp line, what is it?
It is number of fibers times force per one
166
00:28:20,710 --> 00:28:27,710
fiber, using it we obtain this easy expression.
We have 2 forces, one is for our, let us imagine
167
00:28:37,789 --> 00:28:44,690
real structure having the orientation and
second is for the structure from same fibers
168
00:28:44,690 --> 00:28:51,690
with same mass areal mass, but orient it parallel
to to jaw axis. One is R, second is R max,
169
00:29:01,049 --> 00:29:08,049
we can construct the ratio R by R max. It
can say us, how is how is the the mechanical
170
00:29:11,299 --> 00:29:18,299
utilization of fibers during the effect of
fiber orientation, is not it? Here we have
171
00:29:24,529 --> 00:29:31,529
real include include affect of orientation
and in denominator, it is without the affect
172
00:29:32,809 --> 00:29:39,809
of the orientation and using this you can
see that this blue card before integral here
173
00:29:44,370 --> 00:29:51,370
is the same that R max. So, that we can write
that the mechanical utilization in our easiest
174
00:29:53,159 --> 00:30:00,159
case, which we solved is given by integral
from theta from 0 to pi by 2 from cosine is
175
00:30:02,539 --> 00:30:09,539
power to 4, where it is strong very hard effect
of cosine times u theta d theta.
176
00:30:15,429 --> 00:30:22,429
So, you know, you can see that this effect
is utilization is can be very can be sometime
177
00:30:26,440 --> 00:30:33,440
very small, is going from 0 to 1, is not it?
But sometimes it can be very small, when we
178
00:30:35,269 --> 00:30:42,269
use on the place of u theta, our u theta from
our model imaginary flexible belt and so on.
179
00:30:48,419 --> 00:30:54,630
So, that you use this u theta according this
this expression.
180
00:30:54,630 --> 00:31:01,630
We can to to calculate it and you obtained
based on C value following following curves.
181
00:31:08,740 --> 00:31:15,740
This is alpha angle from minus 90 degree to
0 to plus means, let us imagine something
182
00:31:17,470 --> 00:31:24,019
like web or from web some freeze or I do not
know what? May be all those slides today we
183
00:31:24,019 --> 00:31:30,120
discussed is Professor Ishteyak about the
possibility to apply such equation to and
184
00:31:30,120 --> 00:31:37,120
so on.
Let us imagine some structure and now, when
185
00:31:39,570 --> 00:31:46,570
it is planar structure, you can take your
experimental part to breaking machine in different
186
00:31:50,840 --> 00:31:57,840
angles. You can in jaw with to longitudinal
direction or right or left, it is small angle,
187
00:31:57,980 --> 00:32:04,590
higher angle, much more higher angle, clamp
it in the. There is also different based on
188
00:32:04,590 --> 00:32:11,590
this angle alpha, why because the angle alpha
is, it is here, the alpha is here here we
189
00:32:14,509 --> 00:32:21,480
integrate, we are integrating over theta,
but angles steady here is a constant is a
190
00:32:21,480 --> 00:32:28,480
parameter of orientation of our web; for example,
in relation to jaw axis. So, we obtain this
191
00:32:31,740 --> 00:32:38,740
curves based for different way of C.
For example, for evidently for C equal 1,
192
00:32:39,179 --> 00:32:46,179
what it means isotropic structure, you can
rotate this structure how you want. The utilization
193
00:32:46,659 --> 00:32:53,659
will be permanently same, constant. When you
use web then maybe you obtain C, C equal to
194
00:33:00,539 --> 00:33:07,539
roughly, you show that the typical web have
C roughly equal to 2 then you your work is
195
00:33:11,549 --> 00:33:18,549
this function. The the the value here by the
the utilization value is roughly 2 times higher
196
00:33:23,509 --> 00:33:30,509
than here. The utilization value value by
longitudinal direction is roughly 2 times
197
00:33:33,600 --> 00:33:40,600
higher than by cross direction.
It is often used in in textile industry, the
198
00:33:45,539 --> 00:33:52,539
people do not have time and instruments for
deeper study of similar problems, but they
199
00:33:55,080 --> 00:34:02,080
have breaking machines and they have the products
which they produce. In a non-woven often used,
200
00:34:06,580 --> 00:34:13,580
the strength in longitudinal direction and
strength in cross direction and ratio constructed
201
00:34:18,629 --> 00:34:25,629
from these 2 values. Longitudinal to to cross
cross direction, it can show it is the very
202
00:34:30,800 --> 00:34:37,800
easy the practical the practical way, how
to characterize the the to say intuitive and
203
00:34:42,180 --> 00:34:49,050
the degree of orientation, here the intensity
of orientation which was used.
204
00:34:49,050 --> 00:34:56,050
So, we can see on this on this picture, how
it is by C equal to and so on and so on. So,
205
00:35:00,660 --> 00:35:07,660
fiber orientations have significant role or
play significant role to mechanical properties
206
00:35:10,300 --> 00:35:17,300
of fibers assemblies, different fibers assemblies.
Now to our easiest case and reality some last
207
00:35:20,350 --> 00:35:27,350
words. Normally, we often work with stepper
fibers. So, that some of fibers are not clamp
208
00:35:31,570 --> 00:35:38,570
into in both in both jaws.
So, you need to formulate this problem mathematically
209
00:35:39,620 --> 00:35:46,620
are reduced the number of fibers in jaw only
to fibers, which are clump by both jaws. Second
210
00:35:48,040 --> 00:35:55,040
on the end of jaws, on the end of jaws some
fibers on the end of jaws here, on the other
211
00:36:03,940 --> 00:36:10,940
side too, some fibers are cut it. Therefore,
they are not in both jaws because was section
212
00:36:15,720 --> 00:36:22,720
that your preparation of your experiment material,
it is possible to calculate how how many fibers
213
00:36:22,740 --> 00:36:28,360
and so on and give it in our model 2, the
second influence.
214
00:36:28,360 --> 00:36:35,360
Third, you said fibers are straight, no fibers
are not straight; usually, fibers have some
215
00:36:36,880 --> 00:36:43,880
crimping may be very important, may be small,
but every time I am saying, it is. Then the
216
00:36:47,150 --> 00:36:54,150
fiber for strain curve is modified through
this scrimping, through this scrimp also in
217
00:36:56,350 --> 00:37:03,350
parallel fiber bundle about which we will
speak later. When the fibers have some distribution
218
00:37:03,530 --> 00:37:10,530
of such scrimp then the mechanical effects
are significant. Doctor does help me prepare
219
00:37:12,810 --> 00:37:19,810
a special publication to to this problem.
Well, this is the, it is I do not know third,
220
00:37:21,460 --> 00:37:26,190
may be third influence which we do not to
use.
221
00:37:26,190 --> 00:37:33,190
Linear force strain relation, it is it is
only on easiest theoretical example, but in
222
00:37:40,560 --> 00:37:47,560
reality the force strain curve of fiber is
fiber to fiber different. So, that you need
223
00:37:48,240 --> 00:37:55,240
to use similar way, but use inside of our
equations non-linear non-linear curve of for
224
00:37:58,440 --> 00:38:05,440
strain relation related to specially your
fibers. Well, all is this all is possible
225
00:38:14,280 --> 00:38:21,280
may be good may be not so good, but to formulate
using some theoretical tools and make some
226
00:38:24,100 --> 00:38:31,100
modified model, which is much more which is
much more complicated, but nearer to the reality.
227
00:38:33,290 --> 00:38:40,290
The last assumption, the fibers deformed mutually,
it is it is very difficult to to to to give
228
00:38:48,220 --> 00:38:55,220
out because to this time, I personally, I
do not know and neither good model, how to
229
00:39:04,860 --> 00:39:11,860
input to our model of friction phenomenon
between fibers. I know only that the traditional
230
00:39:13,180 --> 00:39:20,180
equations like a coulomb equation forces friction
forces proportional to normal force. Or the
231
00:39:26,290 --> 00:39:33,290
the the other equation is based on the coulomb
idea means no other friction and so on are
232
00:39:37,310 --> 00:39:43,850
not enough well for textile structures, but
what is well from point of view of friction,
233
00:39:43,850 --> 00:39:50,850
it is quite open question, which is waiting
for you may be. Some of you will be researchers,
234
00:39:53,740 --> 00:40:00,740
you will be scientist in future and then you
must solve the problems, which we all generation
235
00:40:03,960 --> 00:40:09,800
did it to solve.
Well, I think this is for this theme. All
236
00:40:09,800 --> 00:40:16,800
thank very much for your attention. Be happy
and good bye.