INJECT-3, a simulation code for the filling stage of the injection moulding process of thermoplastics

INJECT-3, a simulation code for the filling stage of the

injection moulding process of thermoplastics

Citation for published version (APA):

Werf, van der, J. J., & Boshouwers, A. H. M. (1988). INJECT-3, a simulation code for the filling stage of the injection moulding process of thermoplastics. Technische Universiteit Eindhoven.

https://doi.org/10.6100/IR288542

DOI:

10.6100/IR288542

Document status and date: Published: 01/01/1988

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a simulation code for the filling stage of

the injection moulding process of

thermoplastics

Guus Boshouwers

Johan van der Werf

INJECT-3,

a simulation code for the filling stage of

the injection moulding process of

thermoplastics

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de rector magnificus, prof. dr. F.N. Hooge, voor een commissie aangewezen door het college van dekanen in het openbaar te verdedigen op dinsdag 20 mei 1988 te 14.00 en 15.30 uur

door

Johannes Joseph van der Werf

geboren te Reeuwijk

en

Augustinus Hubert Maria Boshouwers

geboren te Roermond

en

Prof. dr. ir. J.D. Janssen Co-promotor:

Dr. ir. J.F. Dijksman

Beide promovendi zijn verantwoordelijk voor het gehele proefschrift. In het bijzonder verantwoordelijk is

Johan van der Werf voor de hoofdstukken 2 en 3 en Guus Boshouwers voor de hoofdstukken 4 en 5

aan Annelies

The technology of mass manufacturing of plastic and metal parts is the main research area of department 1, 'Mechanica! Production Technology', of the Philips' Centre For manufacturing Technology (CFT). The goal of this department is not only to develop technological knowledge about the manufacturing of parts, but also to assist with the transfer of this knowledge within Philips. If this information is available to the product designers and themakers of manufacturing tools (such as moulds and dies), their designs can he completed in less time, with better results.

In the 'Plastics Moulding Technology' group, of the 'Mechanica! Produc-tion Technology' department, wedevelopeda computer program, known as INJECT- 3, which simulates the stage of the injection moutding of a part, in which it is filled with a thermoplastic polymer. In 1984,it was decided to develop such a program, since the software available on the market did not come up to expectations. The program serves as a design tooi for engineers of plastic parts and moulds and enables them to check the consequences of their design decisions, during the moulding process. In this way, the design can he adapted at a very early stage, resulting in a shorter lead time for product innovation and better design quality.

Although much time has been spent on the implementation of the pro-gram code, this thesis mainly discusses the physical-mathematical model (chapters 2 and 3) and the numerical procedures (chapter 4), that are used in INJECT-3, to simulate the injection moulding process. We have tried to build a model, with the strongest possible theoretica! and empirica! base. However, the main requirement for INJECT-3 is the ability to solve the every-day problems of plastic parts engineers. This sometimes involves taking a rather pragmatic approach, in order to overcome certain problems.

The results obtained with INJECT-3 and comparison with practical ex-periences and literature, show the program's ability to make predictions, and have thus helped to justify this type of approach.

Preface Table of contents List of figures List of tables List of Symbols 1 Introduetion

1.1 The injection moulding process . . . . 1.2 Physical phenomena in the mould cavity . . . . 1.3 Computer simulation as a problem solving tooi 1.4 Requirements for the INJECT-3 program . 1.5 Literature review . . . .

2 Conservation laws applied to the injection moulding

pro-i iii vii x

x

i 1 1 3 5 9 13 cess 16 2.1 General . . . 16 2.2 Equations of change . . . 17 2.2.1 Conservation of mass . 17 2.2.2 Conservation of momenturn 18 2.2.3 Conservation of energy . . . 19

2.3 Equations of change for a Generalised Newtonian Fluid 22 2.4 Reformulation of the conservation laws in terms of the

quan-tities pressure and temperature . . . 26

2.4.1 Replacement of the term ~~ in the equation of energy 27 2.4.2 Replacement of the term ~ in the equation of

con-tinuity . . . 28 2.5 Application of the equations of change to a slit flow 30 2.5.1 Equation of continuity 30 2.5.2 Equation of motion . . . 33 2.5.3 Equation of energy . . . 37 2.6 Towards two equations with two unknowns 41 2.6.1 Integration of the equation of motion. 41 2.6.2 Integration of the equation of continuity 45 2.6.3 Symmetry . . . 4 7

2. 7 Boundary and initia! value conditions; progression of the flow front . . . 50

3 Material behaviour 53

3.1 General . . . 53 3.2 The rheological behaviour of thermoplastics . . . 54 3.2.1 Models for the shear rate dependenee of the viscosity 56 3.2.2 The choice of a model for the shear rate dependenee

of the viscosity . . . 58 3.2.3 The temperature dependenee of the viscosity

3.2.4 The pressure dependenee of the viscosity 3.3 p VT-relationships . . . .

3.4 Heat capacity and heat conduction

4 Numerical solution procedures

4.1 General . . . . 59 61 62 67 71 71 4.2 The geometrical representation of the flow domain . . . . 72 4.2.1 The finite element model of the product geometry 72 4.2.2 Locating the flow front . 74 4.3 The system of equations . . . 78 4.4 The pressure equation . . . 80

4.4.1 The triangle element equations for completely filled elements . . . 81 4.4.2 The triangle element equations for partly filled elements 89 4.4.3 The right-hand side vector. . . 95

4.5 The magnitude of the pressure gradients and the veloeities . 101

4.6 The progression of the flow front 104

4. 7 The temperature equation . . . 108 4.7.1 The convection . . . 113

4.7.2 Newly reached nodal points 115

4.7.3 The dissipation . . . 116

4. 7.4 Stability considerations . . 116

4. 7.5 The solidified layer thickness 118 5 Sirnulation results

5.1 General . . . .

5.2 Case presentation . . . . 5.2.1 Case 1: Di:'!monstration part . 5.2.2 Case 2: Deflection unit . . . . 5.2.3 Case 3: Strip . . . . 5.2.4 Case 4: the quadrangle part . 5.2.5 Case 5: Test mould . . . .

119 119 121 121 125 129 135 139 6 Condusion 145 6.1 Discussion . 145 6.2 Recommendations 146 Bibliography 148

A Derivation of the equation of change for kinetic energy 153 B The equations of change with substantial derivatives 155 C The cornponents of the pressure gradient and the rnean

velocity 157

D Line elernents D.l Cylinder element D .2 Reetangolar element

E Reforrnulation of the equation of energy

V

159 159 161 163

Abstract Samenvatting Curricula vitae Nawoord VI 167 168 169 170

1.1 Cross section of an injection moutding machine . . . 2 1.2 Cross section of flow, showing solidification near the mould

metal . . . 4 1.3 Weid lines and air enclosures . . . 6 1.4 Four-fold mould cavity, giving rise to balancing problems 7 1.5 Sink marks on product, caused by thermal shrinkage . . . 8 1.6 Warpage . . . 8 1.7 Information exchange between user, CAD system and INJECT-3 11 2.1 Volume element V with boundary S . . . 17 2.2 General slit geometry . . . 30 2.3 The fountain effect, occuring at the flow front 36 2.4 Definition of liquid and solid area . . . 42 2.5 A mould cavity being filled, forming a domain with boundaries 50 3.1 Viscosity versus shear rate at different temperatures of a

LDPE, [Bird 76] . . . 55 3.2 pVT-relationship of an HDPE, [Vdma 79] . . . 63 3.3 p VT -relationship of a PC, [V dma 79] . . . 64 3.4 y;; as a function of p and Tof an HDPE, [Vdma 79J 65 3.5 p VT-relationship of a PC with pressure on the absciss, [Vdma 79] 66 3.6 p VT -relationship with a independent of p . . . . 67 3.7 Temperature dependenee of heat capacity of a PC, [Ober 80] 68 3.8 Temperature dependenee of heat capacity of an HDPE, [Ober 80] 69 3.9 Temperature dependenee of heat conductivity of an HDPE,

[Ober 80J . . . 69 4.1 Surface representation of an arbitrary thin-walled mould cavity 73

4.2 Finite element model of the product of figure 4.1 . . . 73 4.3 Triangular element and the finite difference grid for the

tem-perature . . . 7 4 4.4 The three elements for the pressure solution procedure . . 75 4.5 Application of cylindrical and rectangular line elements 76 4.6 Finite element mesh with side definitions and flow fronts . 77 4.7 The iteration scheme for the total solution . 79 4.8 Triangulation of 0 . . . 84 4.9 Basis function 4>i . . . • . . • • . . • • • • . 84 4.10 Linear basis function for FEM, within an element . 86 4.11 Definition of side veetors . . . 88 4.12 Definition of "wet" and "dry" nocles . . . . 89 4.13 The two possible intersections of a triangle 90 4.14 Quantities for intersection 2 . . . 91 4.15 Quantities for intersection 1 . . . 92 4.16 All occurring intersection situations 94

4.17 Transition when two flowsmeet . . 94

4.18 Flow rate into boundary element b 96

4.19 Two injection points . . . 97 4.20 lteration scheme to solve non-linear pressure equation 99 4.21 Iteration scheme for simplified flow problem 100 4.22 A moving flow front . . . 105 4.23 Determination of rate of fill factor . . . 105 4.24 Update of the fill factors . . . 106 4.25 Initia! fill factors and flow conduction coefficient S 107 4.26 Differential molecule for the discretisation of the

tempera-ture equation . . . 109 4.27 Elements with velocity vectors, surrounding the noclal point

of interest . . . 113 4.28 The initia! temperatures at the flow front . . . 115 4.29 Determination of the solid fraction of the flow 118 5.1 Geometry for case 1 . . . 121

5.2 Finite element mesh for case 1 123

5.3 Flow front progression (case 1) 123

5.4 Pressures when 60% full (case 1) 124

5.5 Core temperatures, just before filling is complete (case 1) 124 viii

5.8 Pressures in deflection unit, just before filling is complete (case 2) . . . 127 5.9 Temperatures in deflection unit, just before filling is

com-plete (case 2) . . . 128 5.10 The 'strip' mould cavity (case 3) . . . 129 5.11 Finite element mesh for case 3 . . . 129 5.12 Flow front progression inthestrip (case 3) 131 5.13 Pressures, just before filling is complete (case 3) 131 5.14 Core temperatures, just before filling is complete (case 3) 132 5.15 Frozen layer thickness, just before filling is complete (case 3) 132 5.16 Pressures along the mould axis, just before filling is complete

(case 3) . . . 133 5.17 Frozen layer thickness along the mould axis (case 3), at

t=l.O, 2.0 and 2.5 s . . . 133 5.18 Temperature profiles, just before filling is complete (case 3) 134 5.19 Geometry and fini te element mesh of quadrangle part (case 4) 135 5.20 Flow fronts for case 4 . . . 137 5.21 Pressure distribution at t=0.79 s. for case 4 .

5.22 Core temperatures at t=O. 79 s. for case 4 . . 5.23 Solid layer thicknesses at t=0.79 s. for case 4 5.24 The mould cavity for measurement (case 5) 5.25 The element distribution (case 5)

5.26 Cakulated pressures (case 5) . 5.27 Calculated flow fronts (case 5) 5.28 Measured pressures (case 5) . 5.29 Measured flow fronts (case 5) .

137 138 138 140 142 142 143 143 144 C.1 Determination of the intersection point for the convection 157

List of Tables

2.1 General equations of change . . . 21 2.2 Equations of change for a Generalised Newtonian fluid . . 25 2.3 Equations of change in termsof pressure and temperature 29 2.4 General equations for a slit geometry . . . 40 2.5 Integrated equations of continuity and motion . 47 2.6 Integrated equations for the symmetrie case 49 3.1 Material properties and their dependenee . 70 4.1 Definitions of sicles and registration of tili-state of geometry

of tigure 4.6 77

5.1 Overview of cases . 120

5.2 Material properties and process conditions for case 1 122 5.3 Material properties and process conditions, used to mould

the deflection unit (case 2) . . . 126 5.4 Material properties and process conditions for case 3 130 5.5 Material properties and process conditions for case 4 136 5.6 Material properties and process conditions for case 5 141

General

Conventions about the notations have been adopted from [Bird 60] and an overview is given below:

Multiplication signs may he interpreted as follows

1

rn~'t!~!~c~!~~l"l si~

11

L~r-~~r o~ r~s~~~~--~

. - - - - ---~ - --- .. ---~ -- --none

x

E

E-1

E 2 E -4

Quantities are given narnes as follows:

• Scalars are printed as lightface italics or Greek symbols, for example p, T or p.

• Veetors are printed as boldface italics, for instanee v and g

• Second order tensors are printed as underlined Greek symbols, for instanee Q. and !.·

The results of special vector and tensor multiplications, such as the 'sin-gle dot' (·),'double dot'(:) or 'cross multiplication' (x) are usually enclosed in parentheses. The following conventions are used for these parentheses:

• ( ) Ordinary parentheses indicate that the result of the multiplication between them is a scalar

• [ ] Square brackets indicate that the result of the multipikation be-tween them is a vector

• { } Curly braces indicate that the result of the multiplication between them is a second order tensor

Symbols

a vector function that describes the position of the flow front a side vector of partly filled element

a:r;, a₁₁ x- and y-component of a

A element area

A', A1 , A2 element area of partly filled element A cc A 00 A cO A Oe

'

'

'

Submatrices of system matrix

b sealing factor for the temperature differential dT

b' sealing factor for the temperature T

b Boundary condition vector for temperature equation b Boundary condition vector for pressure equation b Side vector of partly filled element

B Conduction matrix for temperature equation

BC Boundary conditions Br Brinkman number

d differential operator

d,, dj, dk element side veetors

D coordinate transformation matrix

DPt. substantial derivative Dt P = .!!. öt

+

v zax JL

+

v llöy JL

+

v ZlJz JL

E

number of triangular elements

é,

el k'th and l'th finite element

e1, ej, ek Si de vector of partly lilled element

f

{Source) function

f right-hand side vector for thermal expansion in pressure equation

f*,

/1,

fz functions

[; component of f

fconv convection vector in temperature equation fdis8 dissipation vector in temperature equation

/ij fill factor between node i and j at node i /ij fill rate factor

Fi

vector element of right hand side vector of pressure equation

Fe, F0 _{sub veetors of right hand si de vector of pressure equation}

g body force on fluid (e.g. gravity)

g.:,g₁₁,gz x-, y- and z-component of g

Gz Graetz number

h half height of liquid portion of (symmetrical) thin slit xiii

h +, h- height of up per and lower liquid half of thin slit, respectively

H half height of slit

'

A

1

the unit tensor I unit matrix

[!!.., II!!.., [[[!!.. first, second and third invariant of second order tensor ([ k time increment number

L sealing factor for dimensions in x- and y-direction of mould cavity L heat of fusion

L0 _{lengthof boundary element b}

m viscosity constant of Power Law model M mass matrix

Me mass matrix of element e Mij components of mass matrix

n shear ra te dependenee of Power Law, Truncated Power Law and Car-reau model

n normal unit vector

n coordinate in direction of n n1 number of iteration cycles

N, N* number of basis functions for pressure problem Nupstr number of upstream elements

Ng number of gridpoints for temperature problem

0 ()

Order of magnitude of ()

p Pi

Pm

Pold p* Po Po Po p q Q r r Tmin' Tmaz R 8

s

.s

approximate solution of pressure

value of approximate solution of pressure in noclal point i

mechanica! pressure

pressure value of previous iteration cycle dimensionless pressure

sealing factor for pressure prescribed pressure

..!... pressure depence of viscosity

Pu

pressure vector

subveetors of pressure vector heat flux

x-,y- and z-component of q flow rate into mould cavity curve parameter

right hand side vector of pressure equation matrix element of R

minimum and maximum value of curve parameter r heat capacity matrix for temperature equation

coordinate in the convection direction volume conduction coefficient

surface

Sald Volume conduction coefficient of previous iteration cycle

S

entropy per unit mass

S volume conduction coefficient obtained by relaxation scheme t time

t₀ start time

t0 sealing factor for time

tinj injection time T temperature vector t• dimensionless time

T

temperature

()T

transposition of a tensor or a matrix T• dimensionless temperature

Tg

glass transition temperature

Ti

temperature of inflowing fluid Tm melt temper at ure

T mld mould temperature

Tnf no-flow temperature

T0 reference temperature for temperature dependenee of viscosity and

shear rate

u arbitrary function of z-coordinate Û internat energy per unit mass

v magnitude of velocity vector

v

magnitude of mean velocity vector

v velocity

v

mean velocity

v• climensionless velocity

V volume

V sealing factor for velocity

V, V N, V N· function space

V

volume per unit mass

v8 mean velocity component in s-clirection v", v_{11 , Vz} x-, y- ancl z-component of velocity v", v11 x- ancl y-component of mean velocity

v;, v;, v;

x-,

y- ancl z-component of climensionless velocity w, w* test function

x, y, z x-, y- ancl z-coorclinate of rectangular coorclinate system

x*, y*, z* climensionless x-, y- ancl z-coorclinate of rectangular coorclinate system

xi position vector of i'th noclal point

x i, yi,

i

coorclinates of i 'th noclal point

z'

auxiliary variabie for integration

Zo z-coorclinate where transition from Newtonian to Power Law be-haviour occurs

Zo,,," zo of previous iteration cycle

a volume expansion coefficient xvii

{j temperature dependenee of vicosity

r;•

temperature dependenee of

"to

j_ rate-of-deformation tensor

j_1

deviatoric rate-of-deformation tensor

'1

shear rate

"fo shear rate of transition of Carreau or Truncated Power Law model

'16

shear rate of transition at reference temperature T0

"too shear rate of transition for Truncated Power Law fiuid at T =

ooc

r

boundary of flow domain

a

partial derivative

~ finite increment

f. quotient H / L

71 {Generalised-Newtonian) viscosity tJ* dimensionless viscosity

7]1 reference viscosity for pressure dependenee model

7Jo sealing factor for viscosity

7Jo viscosity at low shear rate of Carreau or Truncated Power Law model

7]00 viscosity at very high shear rate of Carreau fiuid

t16

viscosity at low shear rate at reference temperature

T0

tJoo viscosity at low shear rate of Truncated Power Law model at T

=

ooc

K isothermal compressibility coefficient

>.,

À8 , Àz heat conduction coefficient (solid,liquid)

A magnitude of pressure gradient

Az,A₁₁,Az x-, y- and z-component of pressure gradient

Áotd magnitude of pressure gradient of previous iteration cycle JL Newtonian viscosity

p, Pa, Pt mass density (solid, liquid)

Q stress tensor of Cauchy Ut; components of Q

r deviatoric stress tensor (shear stress tensor)

<Pt basis function, betonging to nodal point

i

n

two-dimensional field

\7 nabla operator

Chapter

1

Introduetion

1.1

The injection moulding process

Plastic parts are used extensively for the professional and consumer mar-kets, especially in the car and electronica industries. Their mass manufac-ture is very often acheived using the injection moulding process.

The shape of the product or part is determined by the geometry of the

mould cavity. One of the major benefits of injection moulding is the great

freedom of product shapes it offers. Many plastic products are thin-walled, which means that the wall thickness is small, compared with the other dimensions.

The polymers used for the moulding process may he of the thermoset-ting or thermoplastic type. Thermoplastics have the property of softening

at higher temperatures (typically 200-350°0) and this allows them to flow through a mould cavity, at high pressures (typically 50-200 MPa). Ther-mosets, once heated, become irreversibly solid [Rodr 83].

This thesis concentratea on the injection moutding of thermoplastics. Thermoplastics account for about 80% of the raw material used for injec-tion moulding. There is a wide variety of thermoplastic polymers of many different chemical compositions on the market. Each polymer offers its own particular properties and possibilities. The properties that influence

the flow behaviour are discussed in chapter 3.

c: cylinder e: ejector pius he: heater ho: hopper i: iujection chamber m: mould ma: material me: mould cavit.y r: runner system s: screw v: check valve a. r m

- ...

,.ijjiijjiijjiijji,.,.~,.,.,._

•

..._ii~I" rnc b. c.

Figure 1.1: Cross section of injection moulding machine

a: plastificatiou and ejectiou stage b: the filling stage

c: the cooling and packiug stage

The thermoplastic raw material is mostly supplied in granular form. In order to transform this raw material into the desired product, the following cyclic processing takes place, in a reciprocating screw injection moulding machine (see figure 1.1):

1. Plastijication stage. The material is heated, homogenised and com-pressed. This process takes place in the cylinder (c). The material enters the cylinder from the hopper (ho) and is transported by the rotating screw (s) to the injection chamber (i). The screw threads its way back into the cylinder, until enough material to fill a product is in front of it in the ÏI1jection chamber. The heating of the material takes place, both by friction of the screw rotation and heaters (he) (see figure 1.1 a).

1. Introduetion - - - _ _ _ _ _ _ _ _ _ _ 3

moves the screw in the direction of the injection eh amber. A check valve (v) prevents the plastic from flowing backwards along the screw. The material is injected at high pressure through the runner system (r) filling the mould cavity (me) with liquid material (see figure 1.1

b).

3. Packing stage. As the mould is cool (50-120°C), the material starts

to cool down as soon as it enters the mould cavity. When the cavity has been filled, and the filling stage has passed, the cooling process goes on. In order to compensate for thermal shrinkage, which might cause sink marks on the product, pressure is kept on the material in the cavity, allowing more material to enter the mould cavity (post filling) (see figure 1.1 c).

4. Cooling stage. Even after the material has cooled down to the stage

that no more material can enter the cavity, one still has to wait until the part is firm enough to be ejected (see tigure 1.1 c).

5. Ejection. The rnould is opened now and the part is ejected, usually

by means of ejector pins (e) (see figure 1.1 a).

The filling, packing and cooling stages take place in the mould cavity.

If the cavity has a cornplicated shape, two or three stages may take place simultaneously, in different sections: while one section of the cavity is still being filled, another section may be already full and in the packing or even cooling stage. Typkal values for the total cycle time of the process range from about 1 to 100 seconds. The filling stage takes roughly 10 to 50% of this time. The rest of the cycle time is needed for packing and cooling.

Problems arising in the filling, packing or cooling stages seem to have a strong conneetion with the geometry of the product and mould.

1.2

Physical phenomena in the mould cavity

The computer simulation performed by INJECT-3, is limited to the filling stage, at least as far as this thesis is concerned. The physical process of the mould filling can bedescribed as a non-isothermal flow of a non-Newtonian fiut·d, through a narrow gap with a moving boundary. The following char-acteristics apply to a polymer melt flowing into a mould cavity:

Figure 1.2: Cross section of flow, showing solidification near the mould me tal

• Pressure and velocity of the Huid depend not only on the geometry and the location of the flow front, but also on the viscosity of the Huid. The viscosity of polymer roelts is high ( 17 :::::i 104 Pa·s whereas 17 :::::i 10 3 Pa·s for water) and strongly dependent on temperature and

shear rate.

INJECT-3 takes into account viscous behaviour only, although, in principle, viscous as well as elastic forces play a role in the flow of polymer me lts ( visco-elast ie behaviour). The viscous behaviour is assumed to he dominant.

• Heat transport in the Huid takes place by convection and conduction.

Although thermoplastics are relatively bad heat conductors, all heat has to he discharged by heat conduction to the mould. For this reason, plastic parts are very often designed to be thin-walled.

Heat generation in the Huid takes place by viscous dt'ssipation.

• The mould is cooled at a relatively low temperature -in any case, bclow the solidification temperature of the thermoplastic. As a con-sequence, the Huid wil! solidify near the mould wal!, immediately after entering the cavity. The thickness of the solidtfied layer determines the remairring gap thickness, and thus influences the local flow resis-tance (see fig 1.2).

1. Introduetion - - - · · _---5

• The process takes place at very high pressures and relatively high changes in temperature are involved. Thus, thermal expansion and

fiuid compressibihty may play a role.

• The filling behaviour, i.e. the position of the flow front as a function of time, is affected by the phenomena mentioned above: viscosity, temperatures, solidified layer thickness, pressures and velocities. All the quantities mentioned above and the relationships between them have to he taken into account, in order to carry out a complete simulation of the filling stage.

1.3

Computer simulation as a problem

solv-ing tool

Computer simulation can he used to predict, analyse and solve problems that can arise in the injection moutding process, before the mould is actually built. A number of possible problems are listed below:

1. Filling problems. The mould cavity cannot he filled, because the

pressure needed to fill it bec01nes too high or the material cools down too soon.

2. Unwanted filling behaviour. The way the flow front moves through

the mould cavity is unexpected or even unwanted:

• Weid lines occur at unwanted locations. They may affect the visual appearance and the mechanica! strength of the part.

• Air enclosures occur at a location where air is trapped by the flow in the mould cavity (see figure 1.3). This aften leads to a large increase in air temperature, giving rise to material degra-dation and filling problems.

3. Hes~·tation. Especially in narrow sections of the mould cavity, the flow may hesitate, i.e. stop temporarily, because the flow resistance in other sections is much lower. Until the sections with low flow resistance have been filled, the pressure may not increase sufficiently

flow front

~/;air

enelosure

/ / ₁weldline

I

I

Figure 1.3: Weid lines and air enclosures

to overcome the flow resistance in the narrow section. Meanwhile, however, because of the absence of convection and dissipation, the temperatures in this section may have decreased to such an extent, that flow has become impossible. Badly filled or unfilled portions may he the final effect of this. Hesitation usually results in poor product quality.

4. Balancz"ng problems. More than one part may he moulded in a single process cycle. In that case, the mould has been constructed with multiple part cavities, fed by a runner system (see figure 1.4). If the system is assymmetrical with respect to the flow, it may he difficult to balance the runner system, such that all parts are tilled at the same time and under the same conditions.

5. Overpacking. If too much material is forced into the mould cavity, ejection becomes difficult or even impossible. Furthermore, this may give rise toflashes and unbalanced mechanicalloading of the mould. Overpacking is often caused by unbalanced filling of the mould. 6. Extreme temperatures. Temperatures may increase so much by

dissi-pation, that degradation of the material occurs. Low temperatures may give rise to hesitation.

7. Bink marks. If the post filling in the packing stage is not accomplished adequately, the effects of thermal shrinkage may become visible on the product ( see figure 1.5).

1. Introduetion

-Figure 1.4: Four-fold mould cavity, giving rise to balancing problems

8. Warpage. Residual stresses, frozen in the product (for instanee by

irregular cooling of the mould) may cause distortion of the product shape (see figure 1.6).

9. Long cycle times. lf the parts cooling time becomes too long, this leads to long cycle times and poor manufacturing efficiency, but usu-ally results in a higher product quality.

Problems 1, 2, 3, 4, 5 and 6 may occur in the filling stage; problem 7 is typical of the packing stage and 8 and 9 are mainly caused by the cooling stage.

A plastics engineer may solve these problems and avoid their actual occurrence, in the following way:

• He may make adjustments to the part geometry. For instance, the wall thicknesses may be changed, within the margins allowed by con-siderations of mechanica) strength and raw material usage.

• He may choose a different materiaL • He may alter the process conditions:

Figure 1.5: Sink marks on product, caused by thermal shrinkage

Figure 1.6: Warpage

The mould temperature. This may he influenced by the cooling system of the mould.

The temperature of the injected materiaL The flow-rate or the pressure at the sprue.

Changing these conditions may sametimes have far from obvious conse-quences for the filling behaviour. For this reason, there is a certain mystique about injection moulding knowledge, which is often basedon experience and empiricism. This may result in a large amount of trial and error, with the mould being repeatedly adjusted. However, the demands fora shorter lead time in product innovation and a higher product quality, and the appear-ance of new materials on the market, for which no experience exists, make the "trial-and-error" approach no longer acceptable.

1. Introduetion ~---9

If a computer simulation prediets the process reliably, the design may be checked for possible errors, and adapted at an early point in the de-sign process. This would speed up product innovation and reduce costs considerably.

The resulting program must he able of simulating the filling behaviour of arbitrary, 3-dimensional, thin-walled mould cavities. This means that know-how, which was previously only available in laboratories, becomes at the disposal of part and mould designers.

1.4

Requirements for the

INJECT-

3 program

A number of conditions must be satisfied, in order to supply product and mould designers with an effective analysis tooi. This subparagraph dis-cusses the requirements and the constraints for the simulation program INJECT-3.

The constraints for INJECT-3 have been mentioned already in the pre-vions sections:

• The injection moulded part must be thin-walled

• Only the filling stage of the injection moulding process is simulated • The material is a thermoplastic and is considered to be purely viscous The program must predict the problems, that may arise during the Hlling stage, detailed in section 1.3: filling problems, unwanted filling be-haviour, hesitation, balancing probierus and extreme temperatures. More-over, the program should enable the user to analyse the causes of possible problems. lf he makes a change in the geometry, process conditions or ma-terial selection, the program should be able to predict the consequences of this change.

In genera), the calculation time required to perfarm a simulation is con-siderable. The program must, therefore, be able to carry out the complete calculation in a batch job, without the intervention of the user. However, since computers are becoming faster and faster, there must be an option to run it interactively and even to steer the calculation intermediately. Fur-thermore, the program must be robust enough to give either meaningful results or error messages, if the input data is meaningless or inconsistent.

The program must he flexible and portable, enabling it to integrate with the hardware and software of the user's design environment. The flexibility of the interfacing software is particularly important: INJEOT- 3 has been interfaced with several CAD systems, and it must he possible to interface with other finite element modeHers and CAD systems, if required.

The program's requirements will now he discussed in more detail, m terms of the input required and output produced.

Input for INJEOT-3

1. The geometry of the moulded part. The geometry of the moulded part,

which is equivalent to the geometry of the mould cavity, is supplied to INJEOT- 3 in the form of a finite element mesh. Because the parts are thin-walled, a shell element with a specified thickness is used. The use of triangular elements offers much freedom for the modeHing of complex three-dimensional parts. Special bar-type elements are available for modeHing runner systems, or other pipe-like sections of the product. The selection of element type is easy: there are only three types and selection is governed by geometrical criteria only. Subpara 4.2.1 discusses the geometrie model and its finite element representation in more detail.

An important advantage of the use of finite elements for geometrie modelling is its straightforwardness. The user does nothave to create two-dimensional fold-outs of a complex three-dimensional part, nor to make laborious specifications of flow elements, based on the user's anticipation of the flow behaviour.

Nowadays, CAD systems are used extensively for the design of part.s and moulds. Because CAD systems incorporate finite element mod-ellers, an interface with IN JEOT- 3 seems appropriate. Basic geomet-rical information about the design can he transformed into a finite element mesh, by means of the finite element module of the CAD system. Next, the mesh is passed to INJEOT-3, via a formatted file (ANSYS Prep7 format) (see figure 1.7).

The graphical display of INJEOT-3 results on the product geometry is also clone with the help of the CAD system. So the user interacts

1. Introduetion - - - 1 1

USER

Figure 1.7: Information exchange between user, CAD system and INJECT-3

mainly with the CAD system, while IN JECT- 3 perfarms it simulation in background.

Although the problem domain changes with time, because of the pro-gression of the flow front, we do notwant INJECT- 3 to remesh it with each progression. Calculations have to he carried out on a fixed finite element mesh, for two reasons: firstly because the remeshing is calcu-latively intensive and, secondly, because the results of the calculation must fit on the original finite element mesh, which is stored in the database of the CAD system.

2. Materials data. When the user selects a material for calculation with

IN JECT- 3, material parameters are retrieved from a database, w hich contains all commonly used plastics. If necessary, he may alter them, before the calculation is started. Easy retrieval of material data con-tributes greatly to convenient employment of the program.

3. Process conditions. This data is related to the injection moulding machine and its settings. It consists of:

• The location of the gates. This is specified as a set of noclal point numbers of the finite element mesh. If single points are specified, the program simulates a point gate. If two or more adjacent points are specified, the program simulates a slit gate. • The flow rate at the gate. A simple way to specify the flow rate

is to give an injection time. The program then uses the flow rate required to fiJI the volume of the parts in the specified amount of time.

A more advanced specification of the flow rate divides the filling time into equal time intervals and prescribes the flow rate for each time interval.

• The pressure Hmz"t. If this pressure value is exceeded at the gate, the program switches from prescribed flow rate to prescribed pressure at the gate.

• The temperature of the infiowing plastic. A temperature advice range is given by the plastics database for the processing of the selected plastic. The program assumes a constant, uniform tem-perature profile for the fluid at the gate.

• The mould temperature. The database also provides an advice range for the mould temperature, specific to the selected plastic. In principle, it is possible to have a time and place dependent mould temperature. Generally, a constant temperature is as-sumed for the entire mould.

The results generated by INJECT-3

They consist of the following quantities: pressure, flow fronts, core temper-ature, solidified layer thickness, shear rate, shear stress and cooling times. Flow fronts are given in the form of contours of a time value: for each point, the time elapsed since it has been reached by the flow front is calculated.

Because the simulation is time dependent, the program must be able to give the above-mentioned quantities, at any point requested by the user during the filling stage. The quantities are generally presented as contours on the product geometry.

Besides these contours, the following results can also be requested: • Graphs of core temperatures, pressures and solid layer thicknesses, as

a function of time at specified points.

• Graphs of pressure and flow rate at the gate, as a function of time. • Graphs of temperature and velocity profiles at specified noclal points

1. Introduetion _ _ __ _________________________ 13

• A plot of velocity veetors on the product geometry, at requested points in time.

As well as the graphical information, a report is generated which sum-marises the course of events during the filling process. lt reports events such as hesitation and solidification and logs the occurrences of maximum and minimum values of pressure, temperatures, soldified layer, shear rate and shear stress.

1.5

Literature review

The first studies in the area of cavity filling were reported by SPENCER and GILMORE [Spen 51] in the early 1950's. They made a visual study of the filling of the mould and derived an empirica! equation for determining the filling time. Other early important flow visualisation and tracer studies include the contributions of BALLMAN, et al. [Bali 59] and RIELLY and PRICE [Riel 71]. In the early seventies, some authors began to analyse the mould-filling process as a problem of combined transport of momenturn and energy. The transport phenomena approach has been applied to this transient and non-isothermal flow problem by many researchers.

HARRY and PARROTT [Harr 70] coupled one-dimensional flow analysis with a heat balance equation for a rectangular cavity, in one of the first transport phenomena models. KAMAL and KENIG [Kama 72], [Keni 72] worked on the filling of a disc mould, using the assumption of a radial creeping flow. They also presented an experimental test and showed good agreement with calculation results.

The work of RICHARDSON [Rich 72], suggesting that the lubrication theory may he applied in many mould cavities, has been used extensively. WILLIAMS and LORD [Will 75], [Lord 75] measured and simulated the flow in a rectangular plaque mould. VAN WIJNGAARDEN et al. [Wijn 82] im-proved the method by taking into account the heat transport by convection perpendicular to the wal!. Their results are discussed in more detail in chapter 5.

Later on, compressibility and crystallisation was considered by, among others, KAMAL and DIETZ [Kama 72], [Karna 82], [Kama 84], [Diet 78]. The packing and cooling stages can be simulated fairly accurately, with these models. Visco-elasticity and stress relaxation were still neglected at

that time, but these days visco-elasticity is being investigated by many researchers [ Grme 84].

HIEBER and SHEN [Hieb 80] introduced the finite element method to the field of injection moulding. As a result, moulds with complex geome-tries can he studied. New problems arise in descrihing the progression of the flow front and hesitation effect. Also, instability problems and long calculation times appear. Little attention is payed in literature to specific problems in complex geometries. Few experimental results are available in literature. KRUEGER and TADMOR [Tadm 80] published some results, but their conclusions about the inftuence of rheological properties and temper-ature are rather doubtful.

In parallel with our work, a study of the injection moutding process was performed by SITTERS [Sitt 88] at the Technica) University in Eindhoven. This work appeared just a few months ago, so no reference will he made to it. In the investigation period, we exchanged views very satisfactorily.

The workof HIEBER and SHEN is implemented in a commercial software package, known as C-ftow. We evaluated this package and it gave good results in a number of cases. However, sometimes the temperature and the shear rate distribution showed some problems. We expect that this is caused by the calculation of the heat convection.

The theories used in other commercial packages such as Procop ( Cisi-graph, France), Cadmould (IKV, Germany) and Moldftow (Moldftow, Aus-tralia) were not published. Evaluation of these packages showed us accept-able results in a qualitative sense. However, the value of output quantities, such as pressures, temperatures etc., may deviate considerably (up to 100

%) from results presented in literature, obtained by means of accurate cal-culations and measurements.

In 1984 at the C.F .T. a package called INJECT- 2 was released for Philips internal use. This program is still in use and has already been of benifit in a nurnber of applications. Evaluation of INJECT-2 showed that it could not always corne up to the expectations of the user.

1. Introduetion _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ 15

The main problems occur in: • the temperature equation

• the calculation of the flow front progression • modelling of complex parts

• the conneetion with CAD systems

Conservation laws applied to

the injection moulding process

2.1

General

In this chapter, a mathematica! model is derived that describes the filling stage of the injection moulding process of thermoplastics. First, the equa-tions of change from the continua mecham'cs are introduced. These will he transformed into equations descrihing the non-isothermal flow of a viscous thermoplastic fluid, in mould cavities. They form the starting point for the numerical solution procedures.

In order to transform these equations into their ultimate form, state-ments about the fluid and the geometry of the flow domain will he made. In chapters 2 and 3 these statements are classified into assumptions and

decisions. Assumptions are essential for the transformation of the equa-tions. They are mostly based on empirica! experiences. The consequences of deviations from these assumptions fall outside the scope of this thesis.

Decisions are mostly made for simplification purposes only. The conse-quences of deviations from decisions are not essential to the course of the argumentation.

When the equations of change (which express the conservalion laws) are introduced for the first time, they are given in a coordinate-free notation, in terms of veetors and second order tensors. A coordinate system is not introduced until subparagraph 2.5. For a concise treatment of the tensor

2. Ganservation laws applied to the injection moulding process --···-17

V _n

V

Figure 2.1: Volume element V with boundary S

calculus, the reader is referred to [Bird 60, appendix A]. Some conventions about the notations have been adopted from this work and an overview is given in the list of symbols.

2.2

Equations of change

For the sake of completeness, the equations of change are given in three forms [Bird 60]. First, the equations of continuity, motion and energy, in integral form, are given for a volume element, fixed in space (Euler's de-scription). Next, we come to the equations of changefora volume element, fixed to the material (Lagrange's description). The relationship between the two descriptions will be derived, by means of the substantial derivative. The third form of the equations of change is the differential formulation. Using Du Bois Reymond's lemma [Cuve 86] of the calculus of variations, the transition from integral to differential formulation will take place.

2.2.1

Conservation of mass

Consider an arbitrary volume element, V, which has been fixed in space (see figure 2.1). The volume is enclosed by a surface, S. In each point of S we define a vector, n, perpendicular to S and pointing outwards.

If we equalize the rate of addition of mass of the medium crossing the surface of V, with the rate of increase of mass within V, then we get the law of conservation of mass in integral formulation.

(2.1)

where p is the mass density and v the velocity of the materiaL

Sirree the volume element, V, is fixed, we may bring the time derivative inside the integral. Using the Gauss-Ostrogradskii divergence theorem as well [Bird 60] we get

fv

~dV

=

fv

('\7 ·

pv)dV

=>

(2.2)

(2.3)

Because of the arbitrariness of the volume, the integrand may he set to zero (according to Du Bois Reymond's lemma [Cuve 86]). Thus we obtain the equation of continuity in differential formulation.

op

at

- (\7.

pv)

2.2.2

Conservation of momenturn

(2.4)

For momenturn too, we can formulate the law of conservation in the integral formulation for the same volume, V.

!

fv

pvdV ' - - - v - - - - ' tot al Wh ere

Is

[n ·

p {vv }] dS

+Is

[n ·

q])dS

+

fv

pgdV ~~

flow ~tres$ body

total: rate of increase of momenturn of the medium within V. flow: rate of addition of momenturn across S by bulk flow.

(2.5)

stress: rate of addition of momenturn across S by molecular motions and interactions.

2. Conservation laws applied to the injection moulding process 19

The tensor, Q:, known as the stress tensor of Cauchy, is a symmetrie

tensor of the second orderand vector g is the gravity force. Applying the Gauss-Ostrogradskii theorem and bringing dj dt inside the integral, we get

f

apv dV = -

f

[\7.

p { vv}] dV

+ {

[\7 ·

2:]

dV

+ {

pgdV

(2.6)

lv

at

lv

lv

lv

from which the equation of motion in differential formulation can he de-rived.

apv

at

= -

[V .

P { vv}

l

+

[V . 2:]

+

pg

(2.7)

2.2.3

Conservation of energy

The energy within the arbitrary volume V consists of kinetic energy and

internal energy. The law of conservation of energy in integral formulation

can he written as follows.

!!_ {

(~pv

2

+

pÛ)

dV dt

lv

2 tot al flow - fs(n·q)dS+

fs(n·[q_·v])dS

+

~ heat stress [ (v · pg) dV ~ body

Here Û is the internal energy per unit mass.

total: rate of increase of kinetic and internal energy.

(2.8)

flow: rate of addition of kinetic and internal energy across S, by bulk flow.

heat: rate of addition of energy across S by molecular motions.

stress: rate at which medium outside V works against medium inside V. body: rate at which gravity or other body forces works on the medium.

In differential formulation, the equation of energy reads:

_q_

(1

pv2

+

pÛ)

at

2

- (V'·

(~pv

2

+

pÛ) v) - (V'·

q)

+

(V'·

[Q ·

v])

+

(v · pg)

(2.9)

From this equation, we can subtract the equation of change for kinetic energy (2.10). We can derive this equation by forming the dot product of v with the equation of motion (2. 7) and using the equation of continuity (2.4). In appendix A, the derivation of this equation can he found.

1

apv

2

(

1 2 )

"i at=-

V'·

₂

pv v

+

(v ·[V'·

g])

+

(v · pg)

(2.10)

where

(2.11)

This gives the internal energy equation.

apû

(

A )

at

= -

V'·

pUv

-(V'·

q)

+

(ç:

V'v)

(2.12)

Here, we used the following identity, which is valid for a symmetrie tensor Q [Bird 76].

(Q:

V'v)

=

(V'·

[Q ·

v]) - (v · [V'·

Q]) (2.13)

Table 2.1 summarizes the equations of change. They are now also given in the form, where the substantial derivatz"ve is used. This is the time derivative of a property of a partiele of the materiaL In other words, the coordinate system is thought to he fixed to the materiaL The substantial derivative can he written as

2. Ganservation laws applied to the injection moulding process 21

D

a

Dt

=

Bt

+

(v. V) (2.14)

In appendix B, the transition to the formulation in substantial derivative is presented.

Continuity '!e _àt -V ·pv (2.15a)

Qe _{-p(V·v) (2.15b)}

Dt

Motion - [V · pvv]

+

[V · Q]

+

pg (2.15c)

( 2.15)

piJyT = [V . Q]

+

pg (2.15d)

Energy à

_-§i-

û _{= -} ( _{V· pUv} ~ ) _{-(V· q)}

₊

_{(Q: Vv) (2.15e)}

p~~

=-(V· q)

+

(Q: Vv) (2.15!)

2.3

Equations of change for a Generalised

Newtonian Fluid

The general equations of change, listed in table 2.1, apply to any continuous medium, without the necessity for making additional assumptions. In order to obtain a relationship between the stress tensor, CJ, and the velocity, v, we have to make an assumption about the thermoplastic fluid. This leads to the so-called constitutive equations.

The following assumption is made about the fluid:

Assumption 2.1 The thermoplastic medium can be classified as a

Gener-a[,·sed Newtonian fiuid.

This assumption implies that only viscous forces are present in the fluid. There are no elastic forces.

As a first step to obtaining the equations of change for viscous fluids, we wiJl introduce an expression for the stress tensor, CJ, of Newtonian fluids. For more information on this subject, the reader is directed to [Bird 60] and [Bird 76]. Two criteria are imposed upon the fluid to obtain this expression. Criterion 2.1 In a state of rest, all shear stresses are 0; i.e. the

compo-nents, CJ;j, for i :j::. j of the stress tensor, Q:., are 0. Furthermore, the normal

stresses are equal to each other; i.e. CJii

=

CJjj·

as:

Thus, it becomes obvious that the mechanica! pressure should be defined 1

Pm

=

3 LCJ;; I

(2.16)

The stress tensor,

u,

is thought to consist of a hydrostatic stress tensor,

Pml, and a deviatoric stress tensor, r_. A deviatoric tensor describes the deviation from the hydrostatic state.

(2.17)

The tensor, r, is known as the shear stress tensor and I is the unit tensor. The components of the shear stress tensor, r_, are 0 in the case where the fluid is in a rest state, because of this criterion.

2. Ganservation laws applied to the injection moulding process 23

Criterion 2.2 The relationship between the shear stress tensor, r_, and the devz.atoric rate-of-deformation tensor,

j',

is given by a scalar, JL. The scalar,

JL, represents the viscosity of the fluid.

In formulae this criterion can be expressed as:

(2.18) The rate-of-deformation tensor,

i,

and the deviatoric rate-of-deformation

tensor,

j_',

are defined by: -·

j'

i

HV

·v)I

(2.19)

The tensor VvT is the transposition of Vv. Note that the rate-of-deformation tensor

i

is symmetrical as well as the shear stress tensor r_, and that the compoiients of the deviatoric rate-of-deformation tensor ]' are 0 when the fiuid is in a rest state. So, criterion 2.1 is not violated.

In the case of a simple fluid flow in the x-direction, where the velocity only depends on the y-coordinate, equation (2.18) reduces to the viscosity

law of Newton:

(2.20) For the class offluids we are interested in, namely thermoplastic polymer fiuids, the relationship given by equation (2.18) with the viscosity, JL, a constant, is too rough an approximation. It has been found [Bird 76] that, under such circumstances, the viscosity may vary by a factor 10 to 1000.

For this reason, we introduce a viscosity, 'fl, that may depend on the pressure, p, the temperature, T, and on the components of the deviatoric rate-of deformation tensor,

i'.

Fluids whose viscosity cannot be considered constant are called Generalised Newtonian fluz'ds. We will now pay some attention to the dependenee of the viscosity, 'fl, of the components of the rate-of-deformation tensor,

j_'.

Of course, the viscosity, '1'}, may not depend on the choice of the co-ordinate system, to which the components of

j_'

are related. This implies that '11 must depend on the so called invariants of the tensor. A second

order tensor has three independent invariants. For the rate-of-deformation tensor,

j_',

these invariants are:

lrst invariant: J_1, = tr(:t)

2::''

1;; (2.21) 2nd invariant : flr ₁ = (i' :i') _{- -}

LL•I•I

1ij1j; (2.22)

j

3rd invariant : _{I lij_'}

₌

L L L

_i:ji;k i~; _(2.23)

j k

The first invariant, L'r', of the deviatoric rate-of-deformation tensor,

j_',

is 0, by definition. The third invariant, IIL,,, can he shown to he 0 for shear flows (see [Bird 60] and [Bird 76]). In subpara 2.5, the assumption is made that the geometry is thin walled, which implies that we have an approximate shear flow. Anticipating this, we therefore consider the viscosity, '1'}, to depend on the second invariant, IL₁•

=

(i' :

j').

More specifically: the viscosity, '1'}, depends on the shear rate -=r which is defined as:

.

~('''')

1= -1=1 _{2 - -} (2.24)

In summary, the following applies to the viscosity of Generalised Newtonian fluids:

'11

rJ(p,T,'=t)

(2.25)

with

'1

2 frac12(j_' :

j')

('Vv: 'Vv)

+

('Vv : 'VvT) 2/3(\7 · v) 2 (2.26) Besides pressure, temperature and shear rate, one could consider the viscosity, '1'}, todependon other quantities, for instanee on the components of the shear stress tensor, r_ [Bird 76]. However, we wil\ limit ourselves to the dependenee given by equation (2.25). In chapter 3, we will deal with

2. Conservation laws applied to the injection moulding process 25

a number of material models, which give the value of the viscosity, 17, at a given pressure, temperature and shear rate.

With the aid of equation {2.17), we can derive the following identities.

(~: Y'v) [V'· { ~vmi

+dl

Y'pm

+

[v ·

111']

[V' ·

Pmi]

+

[V' ·

rJ

=

({

~vmi

+

L}: Y'v) ( -pml: Y'v)

+

(r: Y'v) -pm {V'· v)

+

(111':

Y'v)

-Pm

(V'· v)

+

(11 {

Y'v

+

V'vT -pm (V'· v)

+

11'l

{2.27)

2/3(Y' ·v)I}: Y'v) = (2.28) With the help of these identities, the equations of change, as listed in table 2.1, can he transformed into the equations of changefora Generalised Newtonian Huid, listed in table (2.2). Equation (2.29b) is often called the

Navz'er-Stokes equation.

Mass ~i = - p (V' .

V)

(2.29a)

Motion

P~'[

= -Y'pm

+

[v

·111]

+

pg (2.29b)

(2.29)

Energy PDÛ _ _{Dt -} (V'· q)- Pm (V'· v)

+

17-)2 (2.29c)

Shear rate

') =

_{y2_ -}

lf(~~~~ (2.29d)

2.4 Reformulation of the conservation laws

in terms of the quantities pressure and

temperature

In most cases, it is more convenient to have the conservation laws of ta-bie (2.2) in terms of the thermadynamie state variables, pressure, p, and temperature, T. Some principles from thermodynamics, as wellas the law of Fourier for heat conduction are needed, in order to reformulate these equations. In particular, the internat energy, Û, and the heat flux, q, wil! he replaced by expressions in terms of pressure, p, and temperature, T.

For a more thorough treatment of thermodynamics, the reader is di-rected to [Fast 48] and [Erin 80]. In fact, thermodynamics is not well devel-oped as a science for the treatment of dynamic and dissipative phenomena. Introducing the conceptsof entropy and temperature as primitive quantities seems to he the safest approach with the present state of thermodynarn-ics. In this subpara, the derivations are made using thermodynamthermodynarn-ics. A more detailed and precise derivation, lead to the same results, is given in appendix E, in accordance with [Wijn 88] among others. We start with the following assumption.

Assumption 2.2 The mechanica/ pressure, Pm, and the thermadynamie

pressure, p, are equal.

This assumption implies that P=, in equation (2.29b) and (2.29c), can he replaced by p, which represents the thermodynamic pressure. In the case of the injection moutding process, where no shockwaves are involved, this assumption comes very close to reality.

Furthermore we state:

Assumption 2.3 The moulded material is a homogeneaus substance or mixture, which can always be cansidered to be in local thermadynamie equi-librium.

Sorne processes that may play a role in injection moulding -ageing and undercooling for instance- are excluded from the mathematica! model, be-cause of this assumption.