Governing Equations

An Alternative Model for the Stress Build-Up during Cooling of Injection Moulded Thermoplastics

5.3 Governing Equations

Chapter 5: An Alternative Model for Cooling Stress 48

This force stretches the solid polymer laying on the mould walls, but as long as the mould remains closed, neither the length nor width of the sample can increase above its initial values. This pressure build-up also causes densification of the layers at the instant of their solidification. This density increase, counteracts the tendency to shrink that was caused by the temperature decrease. At each instant, the stress . distribution in the solid shell laying on the mould walls must be in equilibrium with the external forces acting on it. These forces are related to the melt pressure that squeezes the solid polymer against the mould, and the interaction between the solid polymer and the mould.

Chapter 5:AnAlternative Model for Cooling Stress

Defining one third of the trace of the stress tensor:

. and substituting equation (5.5) into (5.4) gives:

E= =-{(l+vp= -3vcr}

1 E

49

(5.5)

(5.6) Consider the sample when solidification is taking place atYs as shown in Figure 5.6.

Referring to the solid layer at the solidification temperature T_s, the strain along a direction x normal to y can be written as

Ex =-.!.-{(l+vPx -3vcr}+E

_T (5.7) E

where ET is the linear thermal contraction evaluated from the solidification temperature. For an amorphous polymer, ETcan be written as

ET =

r

^{,a(T)dT (5.8)}

where a is the linear thermal expansion coefficient. From Figure 5.6, we consider normal stress along width and length directions to be equal; that is, the normal stress along any direction z normal to both x and y is equal to

cr

x :

As long as pressure in the melt remains positive, the solid layer is held against the mould walls. Under these conditions

cry

does not depend on y. Hence:

cry ⁼ cr_{normal -} P

where p = pressure in the melt, and since we are neglecting normal stress, from equation (5.5):

(5.9) Substitute equation (5.9) into (5.7), so the strain at the solidification layer becomes:

Chapter 5:AnAlternative Model for Cooling Stress

Ex =E1{ax +vax -2vax +VP}+ET

Ex =-{ax -vax +VP}+E1 ^T E

Ex

=~{(l-v

^{Px +Vp}+ET}

E

50

(5.10) At the solid-melt interface, solidification takes place under hydrostatic pressure, so that all three stress components are equal to the hydrostatic pressure (-p) at this point.

fory=ys (5.11)

Mould

This layer solidified at timet'

Deformation (8) at surface

rr-- .._-r--+----Solidification layer

y. y

p

_ _ _- - - - '_ _----L. Midplane

Figure 5.8 Polymer sample undergoes a continuous length shrinkage due to thennal contraction of the layers.

The stress distribution will change with time, and is not dependant on y. This means that the change in deformation of each solid layer between two different instants t and t' does not depend on y. However, the changein strainin the layers is equal to the changeinshrinkage at the mould surface, giving:

Ex

(t, y)-

^Ex

(1', y) =

6(t)-

6(1')

where 6(t) is the shrinkage with respect to the mould, and t' is the instant when solidification takes place at y. Further manipulation gives

Sx(t, y) = Ex

(t',

y)+ 6(t)- 6(t')

=

S5(Y)+

6(t)-6₅

(y)

where bs(Y) is the shrinkage of the layer y when it solidified, and Es(Y) is the strain of that layer at the instant of solidification.

Chapter 5: An Alternative Model for Cooling Stress

Grouping Es(Y) and bs(y) into l1(Y), the above equation becomes

Substituting equation (5.12) into (5.10) gives

51

(5.12)

(5.13) This equation holds for each layer of y in the solid polymer shelL Since ^ET only becomes applicable once the layer is solidified, at the solid-melt interface

and at the surface layer

at t=0

(5:14)

(5.15) Although friction between the two solid polymer shells and the mould walls gives rise to a change of the stress distribution along the x and z planes it is neglected. Only the interaction between the solid polymer and mould edges is accounted for, since it prevents the sample from exceeding its initial dimensions. The melt pressure stretches the solid shell through the force acting on the sample edges.

At any solidification layer

py+interactions with mould edges= internal forces

=

f

^{O"x(y)dy (5.16)}

py=

f

^O"x(y)dy when 8(t)<O (5.17) 5.4 Solution Procedure

The solution to the final stress distribution will now be discussed.

Mould

b

Solidified att= t'

Solidifiedatt= t'+M

..

y,

=tt

_ _ _..L..-_ _~ ^{- ' - -} Midplane

Figure 5.9 Polymer cross section.

Chapter 5: An Alternative Model for Cooling Stress

Integrating equation (5.13) over the solidified polymer layers

where 1=b Y ="L.. and Y =1.

b' ' b b

and substituting equation (5.17):

52

-.!.-(I-v)PY, +-.!.-vp(l-

Y,)+ r

^{l:TdY =}

r

T\(Y}lY +8(1- Y,)

E E • •

E.[(I-v)Y, +v(l-

Y,)]+ r

^{l:TdY =}

r

T\(Y}lY +8(1- Y,)

E • •

E.(y, -2vY, +v)+

r

^{l l:TdY =}

r

^l T\(Y}lY + 8(1- Y,) (5.18)

E

_{J~ ~}

Assume that the function T\(Y) is known from Y down toY'" whereY', refers to the layer that solidified at time t'. At time t = t' + /It, solidification takes place atY,. 8 can be numerically calculated from equation (5.81) once the actual pressure and temperature fields are known. Within first order numerical approximation, the integral of T\ can be written as

and substituting equation (5.14):

Equation (5.15) is the starting point to the numerical procedure, when solidification takes place on the surface layer. Equation (5.19) is substituted into equation (5.18) in order to determine 8(t). If the value obtained for 8 is positive, the interaction between the sample and the mould cavity keeps it zero. Equation (5.15) is used only to start the procedure at the surface layer. For subsequent layers, the value of 8(t), provided it is negative, as obtained in equation (5.18) will be used in equation (5.14), to calculate TJ. Once 8(t) andTJ are kno\V"Il, the stress can be calculated from equation (5.13). This procedure is then repeated in order to solve the shrinkage throughout melt solidification.

Chapter 5: An Alternative Model for Cooling Stress

Mould Wall

~ }-Ll

Yo YI I

110 =-{2v-l}po E

Midplane

T

G

{2v-I}PI -110 XYo

~

^Y1)+

~

(Y1-2Y1v+v)PI +u(To,u - TJyo- Y1) Su

(Yo - YJ+(Yo

~

^Y1)

.Jl..

111 =-{2v-I}PI- SuI E

~

Gu

E[( 111 -

~I

- u(To,u - T1))]

I-v

Mould Wall

~ }-Ll

.. }-L2

Yo

Y, Midplane

53

G

{2v-I}P2 -11IY1

; Y2

)-110(Yo- YJ+

~

(Y2-2Y2v+V)P2 +u(To,L2 -T2Xy^{O -} Y2) SL2

(Yo- y 2)+(Y1; Y2)

~

112 = E {2v-I}P2 -SL2I

~

GL2

E(-Su +112 -

~2

^{-U(TO,L2 -T,))}

I-v

Figure 5.10 Algorithm for the solution of shrinkage during melt solidification,

Chapter 5:AnAlternative Model for Cooling Stress