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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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(2)

L.W. Caspers

(3)

bush-1

PREFACE

This report is a result

of a

three month's term of probation

at

the Philips Research Laboratories Eindhoven, which took place

as

part of the examination for the

'Ir.'

degree

mechanical 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.

(4)

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

(5)

bush-3 . '

1.

INTRODUCTION

When manufacturing injection moulded plastic products it may be necessary to

accomplish

a

very high precision where the shape of the product and

its

dimensions are concerned. For lenses of compact disc players for instance the required precision is about

a 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

temperatures

of

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 because

of

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:

(6)

Thermally induced stresses

After the part of the plastic close

to

the cavity wall is cooled down, the hot plastic in the

inner 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.

(7)

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 equilibrium

Accelerations 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 of

a

varying density it should be

modelled 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.T

(8)

The 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)

(9)

bush-7 .

(2.2.7)

:

=

c,.T

Heat

fiux

(10)

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 leads

to two sets of equations, one of which has to be solved iterative (see time

(11)

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.

(12)

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 function

of

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

z

5

h: u = O, v = O, h n = O (no heat transport in z direct ion)

r =

R,

O

5

z

5

h: or = O z = O, O

5

r

5

R: v = O

z = h, O

5

r

5 R:

v = O

4.2. Analvtical solution

The second term in (2.1.1) can be written as:

(13)

-+

U ( ) * @ - S

i

= 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

u

p =

- * (

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

(14)

As

g r = - p

+

2

-

q.Dd11 it

can

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)

(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 was

specified in the program. When the Taylor-Hood element is used, the numerical solution approximates the analytical one equally

well.

(16)

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 find

the 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 pressure

at

the entrance of the cavity wall is not. The

(17)

bush-15

boundaries of the bush are allowed to move in both directions, for except

at

the entrance

of 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 5.0

5

r

5

7.0, z = 27.0: u = v = O dynamic thermal r = 5.0, r = 7.0, 0.0

5

z _< 27.0: T = T s i n k (t) 5.0

5

r _< 7.0, z = O:

h

=

0

where Tsink

(t)

= max (To

+

T"t,Tmin)

where To

>

Tmin,

T

<

O

Problem

2:

kinematic

5.0

5

r

5

7.0, z = 27.0: v = O

dynamic

(18)

thermal

r = 5.0, r = 7.0, 0.0

5

z

5

27.0:

T

= Tmin

5.2. 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/cm2

BI

= 2.609 * 10-3 a0 = 0.8302 cm3/g a1 = 2.20 + 10-4 cm3/gOC

The parameters designated with a asterisk were not determined by Zoller but derived from the formula:

(5.2.1) In u (0,T) = bi

+

b2.T3l2

of which Zoller determined the parameters bi and b2. Other material parameters are:

70 = 1.698.10-2 MPas

A = 4.217

(19)

B = 94.95 OC G = 894.7 N/mm2 X = 0.26'10-3 J/smmK cP = 1.5 J/gK

(20)

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 (uniform

temperature 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 in

fig. 6.1.3 and 6.1.4. The calculations,'presented in this report were all performed using

the

C.R.

element. There was

no

time

to

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 further

calculations.

6.3. Influence of the compressibility

The compressibility can be written as follows: (6.3.1)

U

1

T

B + p ' T = - 0.0894 *

(21)

bush-19

P

(6.3.2)

P

(ao+al.Tg).(l - 0 . 0 8 9 4 . l n ( l

+

Bo.e-Bl.Sg

1

We will change the parameter Bo so that we will not introduce new temperature effects.

Hence:

(6.3.4) Bo=--

*

Bo 1.1

Qualitative prediction

of the

solution.

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

(22)

(6.4.2)

As

a0

is only

a

constant factor in

a

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, about

21

%.

The shape errors are not influenced by the change of

a

(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 identical

at 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 results

see

fig 6.5.1, 6.5.2. Compared

to

the measurements (fig. 6.1.1, 6.1.2) shape deviations are underestimated and the shrinkage

at r = 5.0 is overestimated about

10

pm. This solution will be our new standard, of

which 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 on

its

own history. Therefore it is

step. Consider a cylinder

(see

fig. 4.1.1) with boundary conditions:

(23)

bush-21 kinematic r =

R:

O

5

z

5

0.5.h: u = v = O (symmetrical) O

5

r

5

R, z = O : u = v = O O

5

r

5

R, z

= 0.5.h: v = O (symmetry) dynamic

initial pressure: 80 MPa

thermal

The coefficient

of

heat conduction X is

set

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 rate

of

temperature decrease (from

100

to

90 OC/s) has hardly any

effect

on the global shrinkage, which reduces

2%

(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 OC

(24)

6.10. 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 = 60

oc).

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 is

a

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 has

not

been modelled yet.

(25)

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 lead

to

the model,

shown

in fig. 7.1.1. Although a start has been made, there was

no

time left

to

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 left

to

find the exact cause of the errors.

(26)

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 varied

to 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.

(27)

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 radius

mat 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

(28)

70

x

IE v v T v0 a - - sqmscripts 1

[-I

subscripts n

[-I

reference viscosity (2. i .5) compressibility

heat 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

(29)

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,

(30)
(31)

fig. 1.1

v

ll E/////,,///.//,

,y

fig 4.1.1

(32)

fig. 6.1.2

O

(33)

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 .

(34)

27.0-I I Y Y I iR-r.0 Y I I --e

fig. 6.2.2

- 0 . 0 2 ' ~

Y

(35)

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 - I

(36)

i

0 . 0 2.7 5 . 4 8.1

5

2 7 . 0 - 2 I W I IR-ï.0 Y y i -C

(37)

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

(38)

- 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

(39)

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 - 0.15

(40)

fig. 6.6.3 t 1 b t < 4 b 4 c fig. 6.6.4

(41)

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

(42)

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

(43)

- 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

(44)

- 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

(45)

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

-0.031 4 &V - 0 . 0 3 6 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 i 7 . t - i 11.1 iii-5.0 1 1 1 +

(46)

fig. 6.11.3 -0.04 1 - *

:::::]/I/

-0.04

ii

- 0 . O S I 4 4 . . ; . . . 0 . 0 2.1 5 . 4 11.1 10.8 13.5 16.2 111.9 21.6 24.3 21.0 27.0-2 11.11 IR-7.0 W l + fig. 6.11.4 0 . 0 2 . 1 5 . 4 6.1 10.8 13.5 16.2 18.9 21.6 24.3 27.0 27.0-2 I W I 11-5.0 WI +

(47)

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