The influence of process and material parameters on
shrinkage and shape deviations of injection moulded
thermoplastics
Citation for published version (APA):
Caspers, L. W. (1989). The influence of process and material parameters on shrinkage and shape deviations of injection moulded thermoplastics. (DCT rapporten; Vol. 1989.023). Technische Universiteit Eindhoven.
Document status and date: Published: 01/01/1989
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L.W. Caspers
bush-1
PREFACE
This report is a result
of a
three month's term of probationat
the Philips Research Laboratories Eindhoven, which took placeas
part of the examination for the'Ir.'
degreemechanical engineering. The first month was spent studying some literature on the subject and mastering the software. The second month and the first half of the last month was used to carry out the calculations. The remaining two weeks were needed to write this report.
Although many people helped me
a
lot during these three months, I particularly wish to thank my attendant Dr. Ir. F.P.T. Baaijens for his good advice and for his patience when answering my many questions.CONTENTS PREFACE 1. INTRODUCTION 2. GOVERNING EQUATIONS 2.1. Conservation laws 2.2. Constitutive equations 3.1. Time discretization 3.2. Spatial discretization 4.1. Problem defenition 4.2. Analytical solution 4.3. Numerical solution 3. DISCRETIZATION 4. TEST PROBLEM
5. THE ACTUAL PROBLEM 5.1. Problem defenition 5.2. Material parameters 6. CALCULATIONS OF THE SHAPE
6.1. A first calculation
6.2. Influence of the mesh coarseness 6.3. Influence of the compressibility
6.4. Influence of the coefficient of expansion 6.5. Intermezzo: reconsidering the Tait equation 6.6. Influence of the length of the time steps 6.7. Influence of neglecting the convective terms 6.8. Influence of the rate of temperature decrease 6.9. Influence of the initial temperature
6.10. Influence of the heat conduction coefficient 6.11. Influence of the mould temperature
6.12. Influence of the initial pressure 7. FURTHER INQUIRIES
% 7.1. Extending the model with a mould
7.2. The Taylor Hood element 7.3. The whole object
8. CONCLUSIONS 9. LIST O F SYMBOLS 10. LITERATURE Page 1 3 5 5 5 8 8 8 10 10 10 13 14 14 16 18 18 15 15 19 20 20 21 21 21 22 22 22 23 23 23 23 24 25 27
bush-3 . '
1.
INTRODUCTIONWhen manufacturing injection moulded plastic products it may be necessary to
accomplish
a
very high precision where the shape of the product andits
dimensions are concerned. For lenses of compact disc players for instance the required precision is abouta few pm per em. Factors that determine the errors in shape and dimensions of tlze
product are in summary sometimes called the five M's: Machine, Mould, Materia.1, Mail
and Method. By method we mean the process parameters such
as
temperaturesof
mould and polymer and the cavity pressure.The injection moulding process of thermoplastics can be subdivided into five stages: 1. Plastification stage.
By supplying thermal and mechanical energy to the polymer the temperature of the material rises above the so-called glass transition temperature such that the plastic melts and can be processed.
2.
Filling stage.The mould cavity is filled with the molten plastic (temperature about 300 OC) at great speed,
to
prevent the cavity from being blocked b y solidified plastic. While filling the cavity, pressure rises becauseof
the cavity getting narrower and the polymer becoming more viscous, but most because of the filled part of the cavity getting longer.3. Packing stage.
The polymer is put under high pressure
to
compensate for thermal shrinkage.4. Cooling stage.
The polymer is cooling down and solidifies. To neutralise the shrinkage, some fluid polymer is supplied, while the high pressure is maintained.
4 5 . Ejection stage.
The product is released from the mould.
Stage 3 and 4 are often called the post-filling stage. During the process stresses arise in the plastic. We can subdivide these into two categories:
Thermally induced stresses
After the part of the plastic close
to
the cavity wall is cooled down, the hot plastic in theinner part of the cavity cools down causing shrinkage of this part. Thus stresses arise in the material.
Flow induced stresses
As the molten polymer is a viscoelastic material, stresses caused by large deformations when filling the cavity, get "frozen-in" as soon as the polymer solidifies. Because these kind of stresses are relatively small and very difficult to calculate, we will direct our
attention only
to
the thermally induced stresses.These stresses will lead to shape deviations. Our goal is to predict these. For this the finite element method is used. The bush illustrated in fig. i.i is the object considered. To verify the results, some tests were done at the Plastics & Metalware Factories (PMF). Shape and dimensions of the bush were determined.
bush-5
2. GOVERNING EQUATIONS 2. i . Conservation laws
Continuity equation
The thermally induced stresses arise because of a varying density. Therefore the continuity equation to be used is:
tr(Q) = 0 Y I/
--
- (2.1.1) Equation of equilibriumAccelerations and bodyforces are neglected, hence: (2.1 .2)
Equation of angular momentum
(2.1.3)
Energg equation
(2.1.4)
2.2. Constitutive equations
Specijc volume
As
the thermally induced stresses arise because ofa
varying density it should bemodelled pressure and temperature dependend. For this the so-called Tait equation [Zoller 19821 is used:
(2.2.1) Y (p,T) = vT(O,T).(l - 0.0894.h (1
+
i))
vT(O,T) = a0+
a1.TThe values of
ao, al,
Bo
and BI depend on the temperature.glass transition temperature
(2.2.2)
St resses
While the temperature is above T, we will model the stresses like those of a
compressible viscous fluid:
(2.2.3) O = - p a l
+
2 . 7 (T,p)*Q-
-
- -
and below
T,
we wil! consider the material being linear elastic:(2.2.4)
Viscosity
(2.2.5)
t
=t,
when T =T,
Internal energy
[Sitters 19881 has shown that the following equation is valid:
a
(2.2.6)
bush-7 .
(2.2.7)
:
=c,.T
Heat
fiux
3. DISCRETIZATION 3.1. Time discretization
We will now briefly discuss the time discretization as implemented.
The first term of (2.1.1) can be written as:
E =
a.T
+
&.i;
U (3.1.1)
where
a
and K can be derived from(2.2.1).
The Picard-iteration scheme to handle the non linearity in the continuity equation is as follows:(3.1.2)
(3.1.4)
From (2.1.4) and
(2.2.12),
(2.2.13) one can derive:(3.1.7)
3.2. Spatial discretization
This is done with two elements, one for the temperature, and one for velocities and
pressure. The first will be a six nodes quadratic element, the latter a PZ-P,
Crouzeix-Raviart, which can be replaced by
a
P 2-PI
Taylor-Hood element. This leadsto two sets of equations, one of which has to be solved iterative (see time
bush-9
discretization). First the temperatures will be solved, then the velocities and the
pressure are calculated, from which we can derive the stresses that will cause the errors in shape and dimensions.
4. TEST PROBLEM
4.1. Problem definition
To test the elements a problem is generated, which can be solved analytically as well. Consider a cylinder, height h = 1, radius r = R
(see
fig. 4.1.1). The uniform temperature obeys the following functionof
time:To be able to solve the problem analytically it is assumed that: (4.1.2)
TO
>
"min>
T g(4.1.3) v = ~0*(1
+
K - P+
a * T )(4.1.4) X
>>
1 (uniform temperature distribution) Boundary condit ions :(4.1 -5) (4.1.6) (4.1.7) (4.1.8)
r = O, O
5
z5
h: u = O, v = O, h n = O (no heat transport in z direct ion)r =
R,
O5
z5
h: or = O z = O, O5
r5
R: v = Oz = h, O
5
r5 R:
v = O4.2. Analvtical solution
The second term in (2.1.1) can be written as:
-+
U ( ) * @ - Si
= U 0 . K . P+
v0.a.T N UO'K.A t
(4.2.3)
No motion in the Mirection is allowed:
(4.2.4) Hence:
(4.2.5)
h e
= u g = o
Examination of the problem under consideration leads to the conclusion:
(4.2.6)
E=O
ihr
which leads to:i a
up =
- * (
r c X
4- - - r a*S)*At 4- PO(4.2.7)
The rate of deformation tensor looks like:
(4.2.8)
O 0
which is based on symmetry considerations and the specific problem. Hence:
O
-.
2 d u l u
3 -dT - 3 . r
As
g r = - p+
2-
q.Dd11 itcan
be derived(T
>
Tg):(4.2.1 O)
The solution of this differential equation can be written as: 1
u
(r) = c1+
c2.r+
c3.-r
(4.2.11)
From examination of the problem we obtain: (4.2.12) Hence: (4.2: 13) r = O: u = O r = R : a r = O c1 = c3 =
o
a'c2+
b-c2+
c = O - c CL =aSb
and from this we obtain:
(4.2.14)
Hence the stresses will be: (4.2.15)
bush-13
(4.2.17) a - b
a+b
az = c*
4.3. Numerical solution
We will now briefly discuss the results of the tests of the elements using the Crouzeix-Etaviart element. Equation (2.2. i> was replaced by
(4.3.1) v = ~ 0 * ( 1
+
K - P+
CYST)in the program. All the parameters po, To,
??,
Tmin, Q, K, VO, 7, A, cp, and A t were varied, but each time the numerical approximation was perfect within the error that wasspecified in the program. When the Taylor-Hood element is used, the numerical solution approximates the analytical one equally
well.
5. THE ACTUAL PROBLEM
5.1. Problem definition
At PMF an object has been injection moulded of which the dimensions and shape were measured. Our goal is
to
calculate the deviation from the nominal dimensions, and findthe parameters of great influence on this quantity. The object, a bush, is pictured in fig. 1.1. In this report only the purely cylindrical part of the bush is considered (fig.
5.1.1).
This problem can be split into two sequential problems:Problem
1
The bush is forced to cool down to a (almost) uniform temperature Tmin by prescribing
the tem-perature along the boundaries in contact with the cavity wall âs â function of time:
To is the initial temperature,
T
the rate of temperature decrease. On the remaining boundary h, = O is used as boundary condition.The pressure is forced to demìnish by prescribing the pressure
at
the entrance of the cavity (cylindrical part) as a funtion of time:&ink
(t)
= max (po+
p.t,o>which is done this way because measurements
at
PMF showed the validity of this equation.No motion is allowed in any direction along the cavity wall. This way stresses that cause
the shape errors can be evaluated. Problem 2
To calculate the shape errors caused by the stresses, calculated in problem
1
the boundary conditions must be changed. The temperature along the cavity wall is still prescribed (T = Tmin). The pressureat
the entrance of the cavity wall is not. Thebush-15
boundaries of the bush are allowed to move in both directions, for except
at
the entranceof the cavity in the z direction (z = 27.0). The velocities calculated along the boundaries
are the displacements of these boundaries when the length of the (only) time step is chosen
1.0
s.
In summary the boundary conditions are:Problem
1:
kinematic r = 5.0, r = 7.0,0.0
5 z
5
27.0: u = v = O 5.05
r5
7.0, z = 27.0: u = v = O dynamic thermal r = 5.0, r = 7.0, 0.05
z _< 27.0: T = T s i n k (t) 5.05
r _< 7.0, z = O:h
=0
where Tsink
(t)
= max (To+
T"t,Tmin)where To
>
Tmin,T
<
OProblem
2:
kinematic5.0
5
r5
7.0, z = 27.0: v = Odynamic
thermal
r = 5.0, r = 7.0, 0.0
5
z5
27.0:T
= Tmin5.2. Material properties
[Zoller 19S2] has measured the parameters applying to the material used at PILIF, i.e. polycarbonate (Lexan 101 of General Electric), which are used to solve the problem:
Tg (O) = 150.4 OC s = 0.52 oC/MPa
T
>
T, (p<
1000 kg/cm2)Bo
= 3161 kg/cm2 B~ = 4.07s. 10-3*
a0 = 0.7752 cm3/g a1*
= 5.57.10-4 cmJ/goC T I T,Bo
= 3954 kg/cm2BI
= 2.609 * 10-3 a0 = 0.8302 cm3/g a1 = 2.20 + 10-4 cm3/gOCThe parameters designated with a asterisk were not determined by Zoller but derived from the formula:
(5.2.1) In u (0,T) = bi
+
b2.T3l2of which Zoller determined the parameters bi and b2. Other material parameters are:
70 = 1.698.10-2 MPas
A = 4.217
B = 94.95 OC G = 894.7 N/mm2 X = 0.26'10-3 J/smmK cP = 1.5 J/gK
6. CALCULATIONS OF THE SHAPE 6.1. A first calculation
After the injection had taken place, the bush was cooled down in two phases during the tests at PMF: first it remained in the mould for a long time cooling down t o the mould temperature, then it was ejected from the mould and cooled down in the open air for a long time, before its dimensions were taken. To immitate this process numerically, the bush is cooled down first to 80 OC (for the time being) and then to
20
OC (uniformtemperature distributions). The initial pressure is set to 80 MPa (for the time being), p to
-10
MPa/s. The test results are shown in fig, 6.1.1 and 6.1.2, the numerical results infig. 6.1.3 and 6.1.4. The calculations,'presented in this report were all performed using
the
C.R.
element. There wasno
timeto
employ the T.H. element.6.2. Influence of the mesh coarseness
Qualitavely one can state that the coarser the mesh, the more the overall shrinkage will
be exaggerated. The errors in shape are not much influenced by the mesh coarseness. For
results see fig. 6.2.1, 6.2.2, 6.2.3 and 6.2.4. The
21
x 8 mesh will serve our furthercalculations.
6.3. Influence of the compressibility
The compressibility can be written as follows: (6.3.1)
U
1
TB + p ' T = - 0.0894 *
bush-19
P
(6.3.2)
P
(ao+al.Tg).(l - 0 . 0 8 9 4 . l n ( l
+
Bo.e-Bl.Sg1
We will change the parameter Bo so that we will not introduce new temperature effects.
Hence:
(6.3.4) Bo=--
*
Bo 1.1Qualitative prediction
of thesolution.
Less material is packed initially which will cause
a
lower shrinkage.Results
The global shrinkage declined 7 pm, which is about 14%. There is
no
difference between the shape errors, nor between the residual stresses (see fig. 6.3.1, 6.3.2).6.4. Influence of the coefficient of exDansion For this coefficient we can write:
(6.4.1) Q =
3.m
1 d u(6.4.2)
As
a0
is onlya
constant factor ina
we will change al. In general al.Tg<<
$0 so an
increase of ai of 10% equals an increase of Q
of
the same percentage.Qualitative prediction
B y increasing Q the material will be more sensitive to changes in temperature, so the
global shrinkage will be higher.
Results
The global shrinkage has increased
10
pm, about21
%.
The shape errors are not influenced by the change ofa
(see
fig. 6.4.1, 6.4.2).6.5. Intermezzo: reconsidering the Tait eauation
When computing certain special cases sometimes problems occured evaluating the Tai t
equation. Cause of this was the extremely
low
pressure at which the specific volume was to be evaluated: the argument of the logarithm became negative. To solve this problem the rather plausible assumption was made that the behaviour of the material is identicalat both positive and negative pressure. The calculated shrinkage approximates the measurements better when the new Tait equation is used, but this is merely a coincidence: the material will probably never experience this low pressure in the injection moulding process
at
all. For the resultssee
fig 6.5.1, 6.5.2. Comparedto
the measurements (fig. 6.1.1, 6.1.2) shape deviations are underestimated and the shrinkageat r = 5.0 is overestimated about
10
pm. This solution will be our new standard, ofwhich we will vary some more parameters in the next paragraphs. 6.6. Influence of the length of the time steDs
The density is modelled in such
a
way that it depends onits
own history. Therefore it isstep. Consider a cylinder
(see
fig. 4.1.1) with boundary conditions:bush-21 kinematic r =
R:
O5
z5
0.5.h: u = v = O (symmetrical) O5
r5
R, z = O : u = v = O O5
r5
R, z
= 0.5.h: v = O (symmetry) dynamicinitial pressure: 80 MPa
thermal
The coefficient
of
heat conduction X isset
to 1 0 8 Jm/soC then YT =0. Both pressure
and density were evaluated for different time steps and two different initial temperatures
(see fig. 6.6.1 to 6.6.4). It appears that a temperature drop of
10
OC per time step is small enough when T> T,,
20
OC when T<
T,.6.7. Influence of neglecting the convective terms
When discretizing the governing equations, we neglected the convective terms in both energy equation and equation of equilibrium. Implementation into the element appeared
to
be relative simple matter, but the computed shrinkage and dimensions did not change a pm (compare fig. 6.7.1, 6.7.2 with 6.5.1, 6.5.2).6.8. Influence of the temperature decrease per second
A change
of
10%
on the rateof
temperature decrease (from100
to
90 OC/s) has hardly anyeffect
on the global shrinkage, which reduces2%
(see
fig. 6.8.1, 6.8.2).6.9. Influence of the initial temperature
This
effect
is more significant:a
change of initial temperature from 300 to 270 OC6.10. Influence of the heat conduction coefficient
There is hardly any influence of
a
change of X(10%)
on
the solution(2%).
See for results fig. 6.10.1, 6.10.2).6.11. Influence of the mould temDerature
The calculations were also done at a mould temperature of
100
OC and 60 OC. Results of these calculations with this varied process parameter are shown in fig. 6.11.1, 6.11.2(Tmould =
100
OC) and 6.11.3, 6.11.4 (Tmould = 60oc).
Conclusion is that the mould temperature has only little effect on the shrinkage. Alas we had some convergence problems: the deviation of the new solution from the previous iteration remained constant after a few iterations. This isa
problem that is not solved yet.6.12. Influence of the initial pressure
The pressure at the entrance of the cavity was varied too. Numerical solution resulted in
pure nonsense, because negative pressures arose in the fluid polymer. Therefore the polymer has
to
be able to come loose from the cavity wall. This hasnot
been modelled yet.bush-23
7. FURTHER INQUIRIES
7.1. Extending the model with a mould
The people from PMF wanted to know the usefulness of using a piece of ceramic in the mould, especially when the product has
a
bevelled edge. This leadto
the model,shown
in fig. 7.1.1. Although a start has been made, there wasno
time leftto
finish these (numerical) inquiries.7.2. The P$-Pi Taylor Hood element
This element has not been used for the calculations yet. There is no saying what profits there might be using this other element
7.3. The whole object
A start was made
to
calculate the shape when the whole bush is taken into account. Alas there were some Fortran or Sepran problems, or bugs in our own software. There was no time leftto
find the exact cause of the errors.8. CONCLUSIONS
The main results of this term of probation
are
the effects on the global shrinkage and the shape deviations of process and material parameters. The process parameters were variedto compare the numerical results with the measurements. The material parameters were
changed now and then because the value of these parameters are not precisely known. Especially the values of the compressibility and the coefficient of expansion have a 1a.rge influence on the global shrinkage, less on the shape of the bush. The initial temperature appeared to be of much greater influence on the shrinkage then the rate of temperature decrease.
Another result is that these calculations suffer quite easily from numerical problems, which are not always easy to solve.
bush-25 9. LIST OF SYMBOLS Y material parameter in (2.2.5) material parameter in (2.2.1) material parameter in (2.2.1)
m-aterial param-eter delved frcm (5.2.1) material parameter derived from (5.2.1) material parameter in (2.2.5)
material parameter in (2.2.1) material pararneter in (2.2.1) heat capacity at constant pressure rate of deformation tensor
deviatoric part
of
0
shear modulus heat flux unity tensor pressure reference pressure (4.2.3) first cylindrical coordinate radiusmat er ia1 parameter (2.2.2) temperature
glass transit ion temperat ure see (4.1.1)
see (4.1.1) see (4.1.1) time
length of time step
velocity first cylindrical coordinate
analytical calculated velocity first cylindrical coordinate velocity third cylindrical coordinate
third cylindrical coordinate coefficient of expansion
-
rate of heat production viscosity
70
x
IE v v T v0 a - - sqmscripts 1[-I
subscripts n[-I
reference viscosity (2. i .5) compressibilityheat conduction coefficient specific volume
see (2.2.1)
reference value specific volume Cauchy stress tensor
conjugated Cauchy stress tensor first main stress component second main stress component third main stress component
iteration number
bush-27
10. LITERATURE [Sitters 19881:
[Zoller 19821 :
Sitters C.W.M.: "Numerical simulation of injection moulding", thesis Eindhoven University of Technology
Zoller P.:
"A
study of Pressure-Volume-Temperature Relationships of Four Related Amorphous Polymers: Polycarbonate, Polyarylate, Phenoxy, and Polysulphone",
Journal of Polymer Science: Polymer Physics Edition, Vol. 20,fig. 1.1
v
ll E/////,,///.//,,y
fig 4.1.1
fig. 6.1.2
O
fig. 6.1.3 . . I . . . ' : . . . : . . . . ; . . . . : . . . ~ . . . . _ _ . . . _ _ . _ _ . _ _ _ _ 0.0 2 . 7 5 . 4 8.1 10.8 13.5 16.2 18.9 21.6 2 4 . 3 27.0 fig. 6.1.4 \ - 0 . 0 3 8 . : . . . ' ~ ~ ~ : ' . . ; ' ' ' . ; . . . . ; . . . ~ : ~ . : - . . - i 0.0 2 . 7 5 . 4 8.1 10.8 1 3 . 5 1 6 . 2 18.9 2 1 . 6 2 4 . 3 2 7 . 0 b ~ s h p l o t ~ - 3 .
27.0-I I Y Y I iR-r.0 Y I I --e
fig. 6.2.2
- 0 . 0 2 ' ~
Y
fig. 6.2.3 - 0 . 0 4 6 $
/i
- 0 . 0 5 4 -0.056 -0.057 -o.osede . - . . . . 0 . 0 2.7 5 . 4 0.1 10.8 fig. 6.2.4 t I I a - 0 . 3 4 7 7 0 . i - 0 . 3 5 6 7 5 - I - 0 . 3 6 5 8 0 - - 0 . 3 7 4 6 5 - i - 0 . 3 8 3 9 0 - I - 0 . 3 9 2 9 5 - I - 0 . 3 4 7 7 0 . i - 0 . 3 5 6 7 5 - I - 0 . 3 6 5 8 0 - - 0 . 3 7 4 6 5 - i - 0 . 3 8 3 9 0 - I - 0 . 3 9 2 9 5 - Ii
0 . 0 2.7 5 . 4 8.1
5
2 7 . 0 - 2 I W I IR-ï.0 Y y i -C
fig. 6.4.1 -0.0567 O -0.058 o -0.059 - __.. fig. 6.4.2 -0.034
!
Y -0.048 0 . 0 1.1 10.8 13.5 16.2 18.9 21.6 2 4 . 3 21.0 2 7 . 0 - i ISM) 11-5.0 Y : -D- 0 . 0 5 4 1 - 0 . 0 5 5 . ' - 0 . 0 5 7 8 . . . , , - ~. . . . . . . . . , . , . , , 2 . 7 5 . 4 8 . 1 10.8 13.5 16.2 18.9 21.6 2 4 . 3 27.0 27 0-2 I Y Y I r R - I . 0 w l , + fig. 6.5.2 f
fig. 6.6.1 + 4. t + t c t t t + t t + c + O . c C J fig. 6.6.2 -700 -1000 ~ I F r n li -2- -270. -yoc 4 4 + * 4 r + t & * + ++$ A t = 0.rs ... A t
-
0 . 2 5_ _ _
b k - 0.15fig. 6.6.3 t 1 b t < 4 b 4 c fig. 6.6.4
t
fig. 6.7.1
0.0 2 . 7 5 . 4 8.1 10.8 13.5 16.2 18.9 21.6 2 4 . 3 2 7 . 0
27.0-2 IWI I R - 7 . 0 W I -e
fig. 6.8.2 -0.027 -0.028 -0.029 -0.030 -0.032 2 ï . ö - Z I Y Y I CR-5.0 Y Y I d
- 0 . 4 7 9 O E L : ~ ' i . 0 . 0 2 . 7 5.4 8 . 1 10.8 13.5 16.2 18.9 2 1 . 6 2 4 . 3 2 7 . 0 2r.a-2 rni 111-7.0 ni d fig. 6.9.2 0.0 2 . 7 5 . 4 8 . 1 10.8 1 3 . 5 16.2 18.9 2 1 . 6 2 4 . 3 2 7 . 0 2 7 . 0 - 1 I Y Y I ( I - S . 0 W i d
- 0 . 0 5 4 5 6 8 ' . . ; . . . - ;
0 . 0 2 . 7 5 . 4 8.1 10.1) 15.5 16.2 18.9 21.6 24.5 2 1 . 0
2I.O-I I Y Y I IR-7.0 U I +
fig. 6.10.2
b ~ s h ~ l o t ~ - I . 5 . fig. 6.11.1 - 0 . 0 4 4 9 - 0 . 0 4 -0.04 - 0 . 0 4 -0.04 - 0 . 0 5 - 0 . 0 5 - 0 . 0 5 - O . O S & - . . ; . . , 0.0 2 . 7 5 . 4 8 . 1 10.8 13.5 1 6 . 2 18.9 2 1 . 6 2 4 . 3 2 7 . 0
t
27.0-2 I y Y 1 I R - 7 . 0 uli + fig. 6.11.2-0.030111
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