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A study of the die-swell behaviour of rubber compounds during short-die extrusion
,lore'halof
Materials Processing Technology Journal of Materials Processing Technology 59 (1996) 268-271
A study of the die-swell behaviour of rubber compounds during short-die extrusion J.Z. Liang Department of Chemical Machinery, South China University of Technology, Guangzhou 510641, People's Republic of Chh~a Received 15 February 1995; accepted 14 August 1995
Industrial summary The theological properties of a rubber unvulcanizate in capillary extrusion are investigated using a Monsanto processability tester (MPT). It is found that the die-swell ratio (B) increases with increasing shear rate @w) and the diameter ratio (Dp/D) of the reservoir to the die, at the particular test temperatures employed. A mathematical model for describing the die swell in short-die extrusion of polymer melts is presented, i.e., B = (! + 2KSR + K2S~t) I/4
where K is the entry convergence flow parameter and SR is the recoverable shear strain. Good agreement is found between predicted and measured values of B. Keywords: Die-swell behaviour; Rubber compounds; Short-die extrusion
1. Introduction
2. Modeling
The die-swell behavior is an important elasicity characteristic of polymer melts in processing opelations, such as extrusion and injection. Short-die extrusion is one of the common methods of polymer product shaping, being used particularly in the rubber industry. There are many papers in the literature which discuss the mechanism of extrudate swell for viscoelastic fluids during channel flow, and there are available some semi-theoretical and theoretical models for estimating the values of the die-swell ratio [1-3]. However, for short-die extrusion, research advances in the die-swell behaviour of polymer melts have not been made in theory, except for Huang and White [4] who presented an extrudate-swell equation for a spray nozzle with an acute entrance angle. Recently, Liang and co-workers [5-9] have been researching this field, especially with regard to rubber-compound extrusion. In this paper, the factors influencing the dieswell behaviour of polymer melts in short-die flow are discussed. To simplify conditions, the author attempts to establish a new extrudate-swell equation for short dies, which is suitable for engineering applications.
When a polymer melt flows into a small cross-section channel from a large cross-section channel, it is compressed and sheared intensely to form an entra,-,~e converging flow. In general, the entry flow consists, elongational flow and shear flow. Correspondingly, both elongational and shear deformation are produced, resulting in a large pressure drop at the channel inlet (APen). If the elastic part of the deformation of the melt in flow has not been completely relaxed before the melt leaves the channel, elastic recovery of the strain in the extrudate along the flow direction will occur due to the elastic-memory effect of polymer materials. Therefore, die swell is produced. Usually, the degree of swell of the extrudate is expressed using the die-swell ratio (B). For a circular die, B = DJD, where De and D are the extrudate diameter and the die diameter, respectively. For a short die, of which the length-to-diameter ratio (L/D) is very small, it can be said that the elastic recovery of the elongational deformation and the shear deformation produced in the die entrance flow is the main factor which results in the extrudate swelling of polymer melts [5]. In view of the above discussion, the theoretical analysis for short-die extrusion in this study is based on
0924-0136/96/$15.00 © 1996 Elsevier Science S.A, All rights reserved SSDi 0924,0136(95)02153-D
J,Z. Liang / Journal of Materials Process#tg Technology 59 (1996) 268-271
269
the following simplifying hypotheses: (i) the melt flow is isothermal and incompressible; (ii) the relationship between the elastic strain and stress of the melt in flow obeys Hooke's law; and (iii) the influence of inertia, gravity, and surface tension on the melt extrusion flow can be neglected. A constitutive equation proposed by Lodge [9] is used in this paper, which is based on the 'network' theory for rubbery fluids:
where Dp is the reservoir diameter, and ¢ is the coefficient related to the melt coherence. In Eq. (4), SR is the recoverable shear strain, and is defined as [9]
o"+ p l =
SR -
;'
(1)
m ( t - t')C' dt'
where o- is the stress tensor, P is the hydrostatic pressure, ! is the unit tensor, C' is the strain tensor at time t', and m ( t - t') is a memory function of the form m(t - t') =
G; exp i=i
(2)
Ti
where G~is me shear modulus and r~is the relaxation time const0nt. Suppose the melt is a viscoelastic fluid with a single relaxation time constant then m(t-
t') = _G exp
(,t)
~
(3)
T
According to the above hypotheses, and the application of tensor analysis method, polymer-melt element analysis for flat entry flow, as shown in Fig. 1, is carried out, and a mathematical model is derived for estimating the extrudate swell of polymer melts emerging fi'om short dies using Eqs. (1) and (3) giving the equation B = (1 -F 2KSR + K2S2) I/4
(4)
where K is the entrance converging flow parameter of polymer melts, given by [9]: K = 0.5tg~o
(5)
where % is the half-angle of the natural convergence for the melt in the entry flow, being a function of the flow behavior index (n), the Bagley correction factor (e), and the die geometry parameters for flat inlet flow [5,9], i.e., t g % =4[1 - ( D / D p ) l'SO'+ l)] 3e(n + 1)
2 ~ [ ( D p / D ) l . 5 ( , , - I j _ 1] +
3e(n - 1)
(6)
AP U
i
Dp
D©
2~.~"-
Fig. 1. Diagram for the analysis of short-die extrusion flow.
SR -
-
o'll -- 6~-, -- 2(O'12) w
-
NI
-
2(0"12) w
(7)
or: (O"12) w
(8)
G
where al~ and 022 are the normal stresses, (a.2)w is the wall shear stress, and N~ is the first normal stress difference.
3. Experimental The sample material used in this study was an industrial rubber unvulcanizate, which included natural rubber (NR), styrene-butadiene rubber (SBR), sulfur, s~earic acid, carbon black, and some other ingredients. The major test apparatus was a Monsanto processability testor (MPT) manufactured by the Monsanto Company in the US, a constant-rate type of capillary rheometer. A pressure transducer was installed at the entrance of the capillary die to measure the total pressure drop of the die in the extrusion of the melts. A set of capillary dies with various length-to-diameter ratios ( L / D ) and diameters was used. Extrusion rheology experiments for the sample were carried out at temperatures of 80-120 °C, the apparent shear rates varying from 50 to 1000 s ~. The die-swell ratio (B) of the sample was measured by using both a constant-length extrudate weighing method, and a laserscanning device installed in the MPT.
4. Results and discussion Eq. (4) describes the relationship between B and the rheological parameter of polymer melts, such as K and SR, for short-die extrusion. In other words, B depends, equally, on the material, the channel geometry, and the extrusion conditions, such as extrusion rate and temperature. In general, for polymer melts, d~ in Eq. (6) can be taken to be approximately equal to 0.8. Therefore, when D o is much greater than D, Eq. (6) can be simplified as follows: 4 t g % - 3e(n + l )
+
2( 3e(l-n)
(9)
According to the present experimental data, ~o is estimated using Eq. (9) with the values of the sample varying from 27 to 38° under the present test conditions, the flow-behavior index (n) of the sample being about 0.18-0.62. In this paper, the Bagley correction
270
J.Z. Liang/ Journal q/' Materials Processing Technology59 (1996)268-271 die swell ratio
die swell ratio 1.4~ - - 9 0 "C, 500 1Is 135 "C, 800 1/s 1.35 I
1.3 I
.-x- 105" C, 500 1/s
1.3
1.25 1.2-
""- 90"C, 50 1Is
1.25
1.15
1.2
1.1 1 .O5
0
'
I
i
L,
I
I
1
I
1(~0 200 300 400 500 600 700 800
1.15
6
shear rate(I/s)
~ 12
J 14
~6 1
' 18
' 20
Fig. 3. Dependenceof B o n Dp/Dfor the sample, where L/D = 16.
factor (e) is determined using Bagley's plotting method, with the values of the sample varying from 2.3 to 4.4. The values of K are, thus about 0.25-0.39. Fig. 2 shows the dependence of B of the sample on the apparent shear rates at the wall (~w) and LID. It is found that B increases with increasing ~w, but decreases with increasing LID at the test temperature employed. Generally, the extension deformation, and the shear deformation of the melt in extrusion, increase with increasing ~w under given conditions. On the other hand, the residence time (tR) of the melt in the die decreases with increasing ~w and decreasing L/D. Thus, the Deborah number (Nd) increases with increasing shear rate for a given die and temperature. The elastic deformation of the melt in extrusion, therefore, cannot be completely relaxed in a short die, which results in increased extrudate swelling. Zheng [7] and Tan [8] carried out extrusion experiments on some natural rubber compounds using several short dies (L/D varied from 0,1 to 7.5), obtaining similar results. It can be seen from Eqs. (4)-(6), that B is related closely to Dp/D for a given material and operational conditions. Fig. 3 shows that effect of Dp/D on the value of B of the sample, It is found that B increases with increasing Dp/D under the present experimental conditions: this is because the entry converging flow is intensified and the elastic energy stored in the melt increases with increasing value of Dp/D, within limits. It is believed generally that the phenomenon of extrudate swelling of viscoelastic fluids can be attributed mainly to the normal stress effect, several researchers relating B to the first normal stress difference (N~) [11]. Tanner [1] investigated the swelling behavior of polymeric fluids flowing through a long tube and presented an equation for evaluating B as follows: o \ ~ / l
1
diameter ratio
Fig. 2. B vs. ;;,~ for the sample tested at 105 °C.
B
~0
8
(1 O)
Substituting Eq. (7) into Eq. (4): KNI B=
1+ ~
K2N 2 ~1/4 -:1-4(0. 2)2 )
(11)
It can be seen by comparing Eq. (10) with Eq. (11), that channel geometry, such as Dp/D and the inlet shape of the die, are important factors relating to B for short-die extrusion. The experimental data (measured at T = 105 °C, LID-- 16) in this study, and the prediction values calculated using Eq. (4) or Eq. (11), are drawn in a coordinate system of B - S a, as shown in Fig. 4. It can be found by comparing the measured data in this study, and the results obtained by Zheng [7] and Tan [8], that the results fall on the theoretical curve, even though the data were gained under different test
Die swell ratio
2,2
,f
2 1.8 1,6 "" t h e o r e t i c a l
1,4
"" experim.(L/D=
1,2 1
0
1,25)[8]
"~" e x p e r i m . ( L / D = 0 . 2 ) [ 7 ] • "" t h i s e x p e r i m e n t a l
1
2
t
J
I
I
I
I
I
3
4
5
6
7
8
9
Recoverable shear strain
Fig. 4. Comparison between experimental data and theoretically predicted values of B.
J.Z. Littng /Journal of Materials Processing Technology 59 (1996) 268-271
conditions. The sample used in Refs. [7,8] was a cispolybutadiene rubber/NR compound; the test temperatures were 90 and 100 °C, respectively. It was found in previous work that the value of 2~0 for a rubber compound was about 75° [6].
271
agreement between the predicted and measured values of B from the present experiments on the sample and those reported earlier in the literature.
References 5. Conclusions
For a short die, the elastic recovery of the deformation produced in die entrance flow, and the normalstress effect in the extrusion of polymer melts, are the main factors which lead to die swell, from the viewpoint of the phenomenological theory of polymer rheology. The die-swell ratio (B) depends, therefore, to a significant degree, on the die geometry and the operational conditions, such as extrusion rates and temperatures for a given material. It is found that the value of B of the sample increases with increasing shear rates and D p / D at the test temperatures employed. Eq. (4) or Eq. (11) describes the relationship between the rheological parameters of polymer melts, the channel geometry, the extrusion conditions, and the value of B during short-die extrusion. The results show good
[1] [2] [3] [4] [5] [6] [7]
[8]
[9] [10] [11]
R.I. Tanner, J. Appl. Polym. Sci., A-2(8) (1970) 2067. W.Y. Chiu and G.D. Shyu, J. Appl. Polym. Sci., 35 (1988) 847. M.B. Bush, J. Non-Newtonian Fluid Mech., 34 (1990) 15. D. Huang and J.L. White, Polym. Eng. Sci., 20 (1980) 182. J.Z. Liang and G.J. Tang, China Synth. Rubber Indust., 13 (1990) 195. J.Z. Liang, Y.Q. Huang, G.J. Tang and J.N. Ness, Plast. Rubber Compos. Appi., 18 (1992) 311. R. Zheng, A Irnenomenologicai study of the viscoelastic behaviour of polymer melts in entrance and exit of capillaries, Master Thesis, South China University of Technology, People's Republic of China, 1984. Z.M. Tan, A study of the relationship between the parameters and the rheaiogicai properties of rubber during capillary extrusion, Master Thesis, South China University of Technology, People's Republic of China, 1984. J.Z. Liang, Acta Mechanica Sinica, 22 (1990) 79. A.S. Lodge, Elastic Liquids, Academic Press, New York, 1964. C.D. Han, Rheology in Polymer Processing, Academic Press, New York, 1976.
Materials Processing Technology Journal of Materials Processing Technology 59 (1996) 268-271
A study of the die-swell behaviour of rubber compounds during short-die extrusion J.Z. Liang Department of Chemical Machinery, South China University of Technology, Guangzhou 510641, People's Republic of Chh~a Received 15 February 1995; accepted 14 August 1995
Industrial summary The theological properties of a rubber unvulcanizate in capillary extrusion are investigated using a Monsanto processability tester (MPT). It is found that the die-swell ratio (B) increases with increasing shear rate @w) and the diameter ratio (Dp/D) of the reservoir to the die, at the particular test temperatures employed. A mathematical model for describing the die swell in short-die extrusion of polymer melts is presented, i.e., B = (! + 2KSR + K2S~t) I/4
where K is the entry convergence flow parameter and SR is the recoverable shear strain. Good agreement is found between predicted and measured values of B. Keywords: Die-swell behaviour; Rubber compounds; Short-die extrusion
1. Introduction
2. Modeling
The die-swell behavior is an important elasicity characteristic of polymer melts in processing opelations, such as extrusion and injection. Short-die extrusion is one of the common methods of polymer product shaping, being used particularly in the rubber industry. There are many papers in the literature which discuss the mechanism of extrudate swell for viscoelastic fluids during channel flow, and there are available some semi-theoretical and theoretical models for estimating the values of the die-swell ratio [1-3]. However, for short-die extrusion, research advances in the die-swell behaviour of polymer melts have not been made in theory, except for Huang and White [4] who presented an extrudate-swell equation for a spray nozzle with an acute entrance angle. Recently, Liang and co-workers [5-9] have been researching this field, especially with regard to rubber-compound extrusion. In this paper, the factors influencing the dieswell behaviour of polymer melts in short-die flow are discussed. To simplify conditions, the author attempts to establish a new extrudate-swell equation for short dies, which is suitable for engineering applications.
When a polymer melt flows into a small cross-section channel from a large cross-section channel, it is compressed and sheared intensely to form an entra,-,~e converging flow. In general, the entry flow consists, elongational flow and shear flow. Correspondingly, both elongational and shear deformation are produced, resulting in a large pressure drop at the channel inlet (APen). If the elastic part of the deformation of the melt in flow has not been completely relaxed before the melt leaves the channel, elastic recovery of the strain in the extrudate along the flow direction will occur due to the elastic-memory effect of polymer materials. Therefore, die swell is produced. Usually, the degree of swell of the extrudate is expressed using the die-swell ratio (B). For a circular die, B = DJD, where De and D are the extrudate diameter and the die diameter, respectively. For a short die, of which the length-to-diameter ratio (L/D) is very small, it can be said that the elastic recovery of the elongational deformation and the shear deformation produced in the die entrance flow is the main factor which results in the extrudate swelling of polymer melts [5]. In view of the above discussion, the theoretical analysis for short-die extrusion in this study is based on
0924-0136/96/$15.00 © 1996 Elsevier Science S.A, All rights reserved SSDi 0924,0136(95)02153-D
J,Z. Liang / Journal of Materials Process#tg Technology 59 (1996) 268-271
269
the following simplifying hypotheses: (i) the melt flow is isothermal and incompressible; (ii) the relationship between the elastic strain and stress of the melt in flow obeys Hooke's law; and (iii) the influence of inertia, gravity, and surface tension on the melt extrusion flow can be neglected. A constitutive equation proposed by Lodge [9] is used in this paper, which is based on the 'network' theory for rubbery fluids:
where Dp is the reservoir diameter, and ¢ is the coefficient related to the melt coherence. In Eq. (4), SR is the recoverable shear strain, and is defined as [9]
o"+ p l =
SR -
;'
(1)
m ( t - t')C' dt'
where o- is the stress tensor, P is the hydrostatic pressure, ! is the unit tensor, C' is the strain tensor at time t', and m ( t - t') is a memory function of the form m(t - t') =
G; exp i=i
(2)
Ti
where G~is me shear modulus and r~is the relaxation time const0nt. Suppose the melt is a viscoelastic fluid with a single relaxation time constant then m(t-
t') = _G exp
(,t)
~
(3)
T
According to the above hypotheses, and the application of tensor analysis method, polymer-melt element analysis for flat entry flow, as shown in Fig. 1, is carried out, and a mathematical model is derived for estimating the extrudate swell of polymer melts emerging fi'om short dies using Eqs. (1) and (3) giving the equation B = (1 -F 2KSR + K2S2) I/4
(4)
where K is the entrance converging flow parameter of polymer melts, given by [9]: K = 0.5tg~o
(5)
where % is the half-angle of the natural convergence for the melt in the entry flow, being a function of the flow behavior index (n), the Bagley correction factor (e), and the die geometry parameters for flat inlet flow [5,9], i.e., t g % =4[1 - ( D / D p ) l'SO'+ l)] 3e(n + 1)
2 ~ [ ( D p / D ) l . 5 ( , , - I j _ 1] +
3e(n - 1)
(6)
AP U
i
Dp
D©
2~.~"-
Fig. 1. Diagram for the analysis of short-die extrusion flow.
SR -
-
o'll -- 6~-, -- 2(O'12) w
-
NI
-
2(0"12) w
(7)
or: (O"12) w
(8)
G
where al~ and 022 are the normal stresses, (a.2)w is the wall shear stress, and N~ is the first normal stress difference.
3. Experimental The sample material used in this study was an industrial rubber unvulcanizate, which included natural rubber (NR), styrene-butadiene rubber (SBR), sulfur, s~earic acid, carbon black, and some other ingredients. The major test apparatus was a Monsanto processability testor (MPT) manufactured by the Monsanto Company in the US, a constant-rate type of capillary rheometer. A pressure transducer was installed at the entrance of the capillary die to measure the total pressure drop of the die in the extrusion of the melts. A set of capillary dies with various length-to-diameter ratios ( L / D ) and diameters was used. Extrusion rheology experiments for the sample were carried out at temperatures of 80-120 °C, the apparent shear rates varying from 50 to 1000 s ~. The die-swell ratio (B) of the sample was measured by using both a constant-length extrudate weighing method, and a laserscanning device installed in the MPT.
4. Results and discussion Eq. (4) describes the relationship between B and the rheological parameter of polymer melts, such as K and SR, for short-die extrusion. In other words, B depends, equally, on the material, the channel geometry, and the extrusion conditions, such as extrusion rate and temperature. In general, for polymer melts, d~ in Eq. (6) can be taken to be approximately equal to 0.8. Therefore, when D o is much greater than D, Eq. (6) can be simplified as follows: 4 t g % - 3e(n + l )
+
2( 3e(l-n)
(9)
According to the present experimental data, ~o is estimated using Eq. (9) with the values of the sample varying from 27 to 38° under the present test conditions, the flow-behavior index (n) of the sample being about 0.18-0.62. In this paper, the Bagley correction
270
J.Z. Liang/ Journal q/' Materials Processing Technology59 (1996)268-271 die swell ratio
die swell ratio 1.4~ - - 9 0 "C, 500 1Is 135 "C, 800 1/s 1.35 I
1.3 I
.-x- 105" C, 500 1/s
1.3
1.25 1.2-
""- 90"C, 50 1Is
1.25
1.15
1.2
1.1 1 .O5
0
'
I
i
L,
I
I
1
I
1(~0 200 300 400 500 600 700 800
1.15
6
shear rate(I/s)
~ 12
J 14
~6 1
' 18
' 20
Fig. 3. Dependenceof B o n Dp/Dfor the sample, where L/D = 16.
factor (e) is determined using Bagley's plotting method, with the values of the sample varying from 2.3 to 4.4. The values of K are, thus about 0.25-0.39. Fig. 2 shows the dependence of B of the sample on the apparent shear rates at the wall (~w) and LID. It is found that B increases with increasing ~w, but decreases with increasing LID at the test temperature employed. Generally, the extension deformation, and the shear deformation of the melt in extrusion, increase with increasing ~w under given conditions. On the other hand, the residence time (tR) of the melt in the die decreases with increasing ~w and decreasing L/D. Thus, the Deborah number (Nd) increases with increasing shear rate for a given die and temperature. The elastic deformation of the melt in extrusion, therefore, cannot be completely relaxed in a short die, which results in increased extrudate swelling. Zheng [7] and Tan [8] carried out extrusion experiments on some natural rubber compounds using several short dies (L/D varied from 0,1 to 7.5), obtaining similar results. It can be seen from Eqs. (4)-(6), that B is related closely to Dp/D for a given material and operational conditions. Fig. 3 shows that effect of Dp/D on the value of B of the sample, It is found that B increases with increasing Dp/D under the present experimental conditions: this is because the entry converging flow is intensified and the elastic energy stored in the melt increases with increasing value of Dp/D, within limits. It is believed generally that the phenomenon of extrudate swelling of viscoelastic fluids can be attributed mainly to the normal stress effect, several researchers relating B to the first normal stress difference (N~) [11]. Tanner [1] investigated the swelling behavior of polymeric fluids flowing through a long tube and presented an equation for evaluating B as follows: o \ ~ / l
1
diameter ratio
Fig. 2. B vs. ;;,~ for the sample tested at 105 °C.
B
~0
8
(1 O)
Substituting Eq. (7) into Eq. (4): KNI B=
1+ ~
K2N 2 ~1/4 -:1-4(0. 2)2 )
(11)
It can be seen by comparing Eq. (10) with Eq. (11), that channel geometry, such as Dp/D and the inlet shape of the die, are important factors relating to B for short-die extrusion. The experimental data (measured at T = 105 °C, LID-- 16) in this study, and the prediction values calculated using Eq. (4) or Eq. (11), are drawn in a coordinate system of B - S a, as shown in Fig. 4. It can be found by comparing the measured data in this study, and the results obtained by Zheng [7] and Tan [8], that the results fall on the theoretical curve, even though the data were gained under different test
Die swell ratio
2,2
,f
2 1.8 1,6 "" t h e o r e t i c a l
1,4
"" experim.(L/D=
1,2 1
0
1,25)[8]
"~" e x p e r i m . ( L / D = 0 . 2 ) [ 7 ] • "" t h i s e x p e r i m e n t a l
1
2
t
J
I
I
I
I
I
3
4
5
6
7
8
9
Recoverable shear strain
Fig. 4. Comparison between experimental data and theoretically predicted values of B.
J.Z. Littng /Journal of Materials Processing Technology 59 (1996) 268-271
conditions. The sample used in Refs. [7,8] was a cispolybutadiene rubber/NR compound; the test temperatures were 90 and 100 °C, respectively. It was found in previous work that the value of 2~0 for a rubber compound was about 75° [6].
271
agreement between the predicted and measured values of B from the present experiments on the sample and those reported earlier in the literature.
References 5. Conclusions
For a short die, the elastic recovery of the deformation produced in die entrance flow, and the normalstress effect in the extrusion of polymer melts, are the main factors which lead to die swell, from the viewpoint of the phenomenological theory of polymer rheology. The die-swell ratio (B) depends, therefore, to a significant degree, on the die geometry and the operational conditions, such as extrusion rates and temperatures for a given material. It is found that the value of B of the sample increases with increasing shear rates and D p / D at the test temperatures employed. Eq. (4) or Eq. (11) describes the relationship between the rheological parameters of polymer melts, the channel geometry, the extrusion conditions, and the value of B during short-die extrusion. The results show good
[1] [2] [3] [4] [5] [6] [7]
[8]
[9] [10] [11]
R.I. Tanner, J. Appl. Polym. Sci., A-2(8) (1970) 2067. W.Y. Chiu and G.D. Shyu, J. Appl. Polym. Sci., 35 (1988) 847. M.B. Bush, J. Non-Newtonian Fluid Mech., 34 (1990) 15. D. Huang and J.L. White, Polym. Eng. Sci., 20 (1980) 182. J.Z. Liang and G.J. Tang, China Synth. Rubber Indust., 13 (1990) 195. J.Z. Liang, Y.Q. Huang, G.J. Tang and J.N. Ness, Plast. Rubber Compos. Appi., 18 (1992) 311. R. Zheng, A Irnenomenologicai study of the viscoelastic behaviour of polymer melts in entrance and exit of capillaries, Master Thesis, South China University of Technology, People's Republic of China, 1984. Z.M. Tan, A study of the relationship between the parameters and the rheaiogicai properties of rubber during capillary extrusion, Master Thesis, South China University of Technology, People's Republic of China, 1984. J.Z. Liang, Acta Mechanica Sinica, 22 (1990) 79. A.S. Lodge, Elastic Liquids, Academic Press, New York, 1964. C.D. Han, Rheology in Polymer Processing, Academic Press, New York, 1976.