Simulation of compression molding of long-glass-fiber-reinforced thermoplastic sheets

Simulation of Compression Molding of Long-Glass-Fiber-Reinforced Thermoplastic Sheets

R. DUCLOUX, M. VINCENT, and <J. F. AGASSANT

Centre de Mise en Forrne des Materiaux Unite Associee au CNRS no 1374

Ecole Nationale Superieure des Mines dc Paris Sophia-Antipolis, 06560 Valbonne, France

and

A. POITOU

Laboratoire de Mecanique et Technologie ENS d e Cachan, Universite Paris Vl, CNRS

94230 Cachan. France

An isothermal two-dimensional shear-thinning model of the compression molding of long-glass-fiber-reinforced thermoplastic is presented. A transverse isotropic behavior is introduced by considering two different pairs of power-law coefficients. First ( K 1 , rn,) is linked to the shear deformations through the thickness and to the compression deformation: second ( K Y , rn2) is associated to the expansion motions in the plane. KY is significantly greater than K , because of the influence of the glass mat structure, which resists stretching in the plane of the part. These coefficients are determined by fitting the computed and experimental mold closing forces in a circular isothermal compression-molding test. The average velocity through the thickness is obtained by a variational method. Incom- pressibility is ensured by penalization. A finite element method is used, with a convected mesh and remeshing technique. The agreement between this model and experimental results (shape of the sample. mold closing force, pressure) performed on a rectangular mold is good.

INTRODUCTION

ompression molding of mat fiber-reinforced poly- C mers is a process that is widely used with ther- moset matrices. There is a n increasing interest in the compression molding of reinforced thermoplas- tics because of their greater manufacturing flexibility and properties. Reinforced thermoplastic sheets are obtained either by coating of several glass fiber mat layers by thermoplastic sheets (essentially polypro- pylene or polyamide), or by a paperlike technique that consists of mixing short fibers and polymer solutions, laying out on a porous conveyor belt, and finally eliminating the solvent. During the process itself, several reinforced sheets are heated in a n oven and pressed in a temperature controlled mold. Two goals must be reached for a n optimized process: first, filling the mold in the shortest period of time with a limited mold closing force, and second, obtaining a final part with optimal mechanical properties that is without fiber degradation and fiber-matrix separation and with a controlled fiber orientation.

During the last ten years, several research studies have been conducted in this field but mainly on sheet molding compound (SMC). In 1978 Owen, et al. (1) proposed an empirical method based on mass con- servation to predict the flow front evolution as filling proceeds. In 1980, Silva-Nieto, et al. (2) solved the generalized Newtonian Hele-Shaw equations for compression molding between infinite parallel plates. They assumed at each time step a local para- bolic velocity profile throughout the thickness. Menges and Derek (3) extended this approach in 198 1 by introducing a power-law viscosity. In 1983 Tucker and Folgar (4) proposed a finite element resolution of the generalized Hele-Shaw equations. They considered at each time step a zero-pressure condition at the flow front, and slipping conditions when the material comes into contact with the lateral mold boundaries. They compared the computed and experimental shapes of the sample at successive time steps. When the initial sample thickness is thin, the agreement is fairly nice, but there is a significant discrepancy for thick initial charges. This model has

30 POLYMER COMPOSITES, FEBRUARY 1992, Vol. 13, No. 1

K . Ducloux, A. Poitou, M . Vincent, and J . F. Agassant

been improved by taking into account successively shear thinning effects (5)- and temperature effects (6). but the disagreement between computations and experiments did not disappear. Barone and Caulk (7) developed in 1986 a quite different model. They introduced sliding conditions along the upper and lower walls so that the gapwise velocity profile is supposed to be flat. They introduced shear components in thin layers located along the upper and lower walls, and an elongational behavior in the remaining part of the charge. The resulting model takes into account the influence of the initial thickness of the sample, and the computed shapes are in qualitative agreement with the experimental ones. Recently, Davis and Tucker (8) determined the rheological parameters of the Rarone and Caulk theory by fitting their previous experimental results. Hojo, et al. (9) implemented also the generalized Hele-Shaw equations, but they introduced two shear viscosity coefficients, which are functions of the shear rate, through a classical power law, and also of fiber orientation and fiber- matrix separation. The agreement between experiments and calculations is fairly good.

In this paper, a transverse isotropic behavior is introduced. This means that the structure of the fiber mat resists stretching in the plane of the sheet, but does not resist cross-plane shearing. A s a conse- quence, two sets of viscosity coefficients are introduced for these two kinds of deformations. The first set is linked to in-plane isotropic deformations, and the second one to shear and compression deformations throughout the thickness. Flow-induced fiber- matrix separation and fiber orientation are neglected so that the shear and in-plane expansion viscosity coefficients are homogeneous and constant. In this paper, a n isothermal hypothesis is assumed. A two- dimensional finite element model is proposed, first considering a Newtonian behavior and then a more realistic shear-thinning one. Then the different rheological parameters of the model are measured using an axisymmetric isothermal compression test. Finally, the computation results are compared with several experiments performed with a GMT Symalit40 reinforced thermoplastic on a well-instrumented mold.

MECHANICAL MODEL

Geometry and Flow Kinematics Description

At each time step t , the geometry of the stamped material R is supposed to be plane with a varying gap h(x, y) . dR is the boundary of the domain at time t (Fig. 1 ) . The material kinematics is described by four velocities: U is the average velocity through the thickness, U1 and U1 are the material velocities tangent to the plane of the upper and lower walls. If a sticking contact is assumed, U1 and U1 are equal to the in- plane velocities of both parts of the mold. These two velocities are equal to zero in the plane compression- molding conditions with sticking contact that are considered here. This is not the case for more gen-

eral geometries or boundary conditions, and U , and U, will be considered in the model. Notice that ( l / h ) (U - ( U , + U 2 ) / 2 ) is an estimate of the shear rate through the thickness. The fourth material velocity is h = d h / d t , which is usually (but not neces- sarily) equal to the mold closing speed (Fig. 2). The continuity equation for a n incompressible materia leads to E q 1.

(1 div(hU) + h - - u1 -I- u2 grad h = 0. 2

Virtual Power Principle

The virtual power principle is used neglecting in- ertia; for each virtual velocity field (U*, U:, U 2 , h*) the dissipated energy in the material P, is equal to the energy dissipated by the external forces (-Pc). The internal stress dissipation is:

- a:grad U’h d S

where A , A,, A2, A are forces and c is the in-plane stress tensor. They are functions of both velocities and velocity gradients.

Fig. 1 . Definition of the domain.

l h

uz +

Fig. 2. Definition qf the velocities.


Simulation of Compression Molding

Considering simple rigid motions and pure rota- tions leads to a simplified form of Eq 2:

,. ,.

where k is the two-dimensional rate of strain tensor in the plane (x, y). The dissipated energy by the external forces is:

P,. = FU' ds + s F , U f ds + i F,Uf ds

+ S K h ' ds + TU*h dZ. <I!)

(4)

F is a mass force, F1 and F, are forces associated with the upper and lower walls. K is the classical mold closing force and T represents a stress that may be exerted along the border dQ. The pressure p is introduced as a Lagrangian multiplier associated with the incompressibility condition (Eq 1 ) :

J A (U' - U 7)h ds + s A 2 (U' - Uc)h ds

-SAh*hds - $u:i(U*)hds

+ l F U * ds + l F l U? d s + $., U t d s (5)

+ A,Kh* ds + J TU*h dz d!!

gradh ds=O + lp[div[hU*) + h* - ~ 1 u:+ U," 2

The equilibrium equations are the local expressions of Eq 5. which are obtained after integration by part. Thus, we obtain on the domain Q :

(6.1) - h grad p + hA1 + hAn + h div cr + F = 0

grad h - hA1 +F1 - p ~ = o

(6.3)

and on the boundary dQ:

- hU.n + hp + hT= 0

where n is the normal of an. (7)

Linear Newtonian Behavior In a first step, linear relations between A1, Az, g,

and A and the velocities and velocity gradients are assumed. Several considerations of invariance lead

to

u = 2u&(U)

where a , b, U, and X are four rheological parameters. The internal stress dissipation can now be written as

- (Ul - Uz)'h d s - 2vk(U),h ds (9) s - 1s h'h ds

I t is interesting to notice that Eq 9 includes several kinematic invariants of the compression-molding process:

1. Il = ( l /h2)(U - (U , + U,)/Z)' represents an invariant related to the shear throughout the sample thickness.

2. I2 = (1/h2)(U1 - U 2 ) * represents a shear invariant between the upper and the lower walls.

3. I3 = ( h2/h') represents a compression invariant. 4. I? = &(U)" represents the second invariant of the

Considering the general three-dimensional rate of strain tensor 9 associated with the compression- molding process:

rate of strain tensor in the plane R.

(10)

it can be noticed that II and 1' are associated with the shear components throughout the thickness i..yz and I:3 is associated with the compressive term i.zz, and I.? with the in-plane expansion rates of deformation i.x*, Y r y , yy,. In the classical Hele-Shaw ap- proximation, only II and I, are taken into account. and I3 and I4 are neglected.

To determine the four coefficients, a, b. u , and X, it is useful to consider two simple flow situations that admit analytical solutions. The first one is the one- dimensional flow of a Newtonian fluid submitted to a pressure gradient between two infinite parallel plates moving in translation. This leads to a = 67, b = -27. The second one is the case of the perfectly lubricated squeeze flow which leads to X = 27 and u = 7.

Because of their great length, all the fibers of the reinforced thermoplastic belong to the plane of the part. This will promote the anisotropic behavior of

' I [Y,, ?XI, Yxz

Y = Y A ! J YV<i Y!,. Y!lL YZL


R. Ducloux, A. Poitou, M . Vincent, and J. F. Agassant

the material. This has been taken into account by introducing two different viscosity coefficients: q l is associated with the shear deformation through the thickness (a = 6771, b = - 2 ~ ~ ) and with the compression deformation (A = 2q1); q2 is associated with the in-plane deformation (v = q2) . According to the mat structure, 4, will be much more important than q,. The internal stress dissipation is now written as

,. - Ji, $ (u, - U,),h ds

- 19 h2h ds - I 2 ~ ~ i ' ( U ) ~ h ds

and the stresses are

= 2q,E(U)

ti A = 2771 7 h

Interestingly, if the reinforced thermoplastic has an isotropic behavior, q1 = q2 = q and the internal stress dissipation is

PI = - qq2h ds with I Notice that even if the expansion invariant I4 (in

which elongational terms are important) is small rel- ative to the shear invariants II and 12, the associated stress dissipation in Eq 1 1 may be of the same order of magnitude because of the value of the corresponding rheological coefficients (respectively q2 and q1 ].

Shear-thinning Behavior

In the following, and especially for comparison with experiments, a more realistic shear-thinning behavior will be considered. For an isotropic shear thinning fluid of consistency Ki and power-law index m, it is possible to write the internal stress dissipation as

P , = -1 Kiqm+'h ds with 4

To take into account the nonisotropic behavior of the material, two power-law functions, q l and q2, have been defined with consistencies K1 and K 2 and

power-law indexes m, and m2:

771 = Kly;"'-' (15)

with

and

with

The expressions of A,, A2, a, and A are easily obtained by replacing q l and q2 in Eq 12 by their new values. As indicated by E q s 15 and 17, ql is a function of the invariants associated with the gapwise shearing deformations and with the compression deformation: q2 is a function of I, associated with in- plane deformations. The internal stress dissipation is now

P, = - ( r < , + ; " l + ' + K2qT2")h ds. (19) i Numerical Resolution

At each time step, the flow domain Q has been discretized using a finite element method. The planar velocity field U is obtained by minimizing a functional associated with the virtual power principle (Eq 5). The incompressibility (Eq 1 ) is imposed by penalization. In the Newtonian case, the functional is

+ Sq2C(U)'hds - 1 F U d s - l F I U l ds (20)

r. r.

+ lp[div(hU) + - grad h ds 2 I'

where p is a penalty coefficient. In this paper, plane parts are considered. U1 and U2 are equal to zero, the mass forces, F, are neglected, and no stress is exerted along the boundary dQ. Thus, Eq 20 is simplified as follows:

I$ I = 1 h2 h ds + 6 2 h2h ds

+ 6 p[div(hU) + ti]' ds.

POLYMER COMPOSITES, FEBRUARY 1992, Vol. 13, No. 1 33

Simulation of Compression Molding

The boundary conditions are the following:

-A mold closing speed, h, or a mold closing force, K , is imposed:

-As indicated previously, no stress is exerted along the free boundaries. When the reinforced thermoplastic comes in contact with the lateral mold walls, sticking or slipping conditions may be imposed.

The deformation of the domain Q between two successive time steps is obtained by an explicit integration of the velocity field. The problem is to adjust the time increment according to two constraints. First, the deformation of the domain must be small during the time increment to obtain small velocity variations. Second, as each node of the mesh is convected according to the material velocity, the time increment has to be decreased if a node crosses the mold wall, especially a t the flow front. Remeshing techniques are used when the mesh is too distorted.

The pressure field is determined using E q s 6.1 and 7 , which can be written now as

- grad p - - 12" U + 2q2div(E(U)] h2

with the following condition along the border:

- ha.n + hp = 0. (23) The pressure p is obtained by minimizing D defined

in E q 24 with the boundary condition ( E q 23) on the finite element mesh:

D =

12" U + 2 q2h div[E(U)] h

For the shear-thinning behavior, we use the same E q s 6.1 and 7 , but with the corresponding formula- tion of A , , Az, CJ, and A. The mold closing force may be obtained either by integration on the whole domain of the normal stress ( - p + Ah) deduced from E q 6.4 or directly using the internal dissipated energy.

RHEOLOGY OF THE REINFORCED THERMOPLASTIC

Experiments

Classical rheometers are not usable for reinforced thermoplastic sheets. For example, in a cone-and- plate rheometer, the composite material slips along the plate and the cone. This is the reason why most rheological studies on long-fiber-reinforced mate- rials, mainly SMC, are performed with axisymmetric isothermal compression experiments (8, 10).

We measured the rheology on a mold designed by FIAT car company with dimensions (0.5 m in diameter) very near industrial ones. This permits handling of samples in which the orientation and distribution of the fibers can be considered as homogeneous, and the performance of large deformations. The mold is

mounted on a 5000 kN press. The mold temperature can be varied from room temperature up to the compression temperature. The initial temperature of the charge is controlled with a thermocouple. Since the initial charge temperature is identical to the mold temperature, the experiment is assumed to be nearly isothermal. The sample thickness, the mold closing force, and the pressure on four different transducers are recorded on a fast data acquisition system.

A 40 wt.% mat glass-fiber-reinforced polypropyl- ene has been studied (GMT Symalit 40). The initial charge was 0.2 m in diameter and 0.015 m in thickness. The final thickness was 0.003 m. The material and mold temperatures were 200°C. The nominal mold closing speed was h = 0.005 m/s. In fact, the actual velocity was 0.005 m/s at the beginning of the process, and it decreased to 0.002 m/s at the end. The actual velocity is obtained from the thickness versus time records (Fig. 3).

Results and Discussion

The values of the four rheological parameters of the transverse isotropic shear thinning model (K1, K s , rn, , m,) are obtained by minimizing the difference between the experimental and computed mold closing forces (Fig. 4). It is valuable to determine rnl and

0.016

h 0,012 E v

v1

% 0,008 E

E .- 5 t+ 0,004

0.OOO 0 1 2 3 4

Time (s) Fig. 3. Experimental and nominal thickness versus time.

10 1 0 1 2 3 4

Time (s)

Fig. 4. Experimental (Exp) and calculated (Cal) moId clos- ingforce versus time.


R. Ducloux, A . Poitou, M . Vincent, a n d J. F. Agassan t

rnz, which represent the sensitivity to the rate of strain on a single experiment conducted at nearly constant velocity, because the values of the rate of strain tensor components change drastically throughout the sample and during mold filling. The following values were obtained:

KI = 2.73 lo4 Pa.sml K 2 = 130 K ,

rnl = 0.3 rn2 = 0.7

When varying one of these rheological parameters, the distance between the experimental and computed curves increases. For example, on Fig. 5, rnl is varied between 0.2 and 0.5, and on Fig. 6 KS is varied between 80 K 1 and 200 K1. We can notice the important value of the ratio between K z and K , , which reflects the material anisotropy. Lee, e t al. (10) obtained a similar ratio for a SMC material. Figure 7 shows that the agreement between experiments and calculation is less correct when considering an isotropic behavior with the best set of consistency and power-law index ( K , = K 2 = 5.6 lo4 Pa.sm, rn = rnl =

1500 I ? I

0 1 2 3 4

Time (s) Fig. 5. Experimental (0) and calculated mold closing force: (A) optimum rheological parameters, (X) same except m, = 0.2; (0) same except m, = 0.5.

0 1 2 3 4

Time (s) Fig. 6. Experimental (0) and calculated mold closing force: (A) optimum rheologicalparameters; (X) same except K 2 = 200 K1 : (0) same except K 2 = 80 K1 .

rnz = 0.3). When considering now K, = K z = 2.73 lo4 Pa.s" and rn = rn, = rnz = 0.3, thedifference between computation and experiment will be markedly en- hanced.

COMPARISON BETWEEN STAMPING EXPERIMENTS AND COMPUTATION RESULTS

Experiments

The same GMT Symalit 40 material has been molded in a rectangular plaque mold built by Peugeot S.A. car manufacturer (Fig. 8). It is equipped with four pressure transducers. Some moldings were performed with a thermocouple, the end of which was located in the charge. Unfortunately, it broke because of the high viscosity of the material and high stresses during mold filling. The mold temperature was 80°C; the initial material temperature was 200°C. The initial charge shape was rectangular (0.150 by 0.396 m'), with a thickness of 0.015 m. At the end of the test, the mold was nearly full and the material thickness was 0.003 m. Three nominal mold closing velocities have been used (0.005, 0.01. and 0.02 m/s); as previously, the actual velocities were recorded.

Results and Discussion

Figure 9 shows the measured pressure on transducers 1 and 2 as a function of time for the three mold closing velocities. Pressure on transducer 1 is greater than pressure on transudcer 2, and pressure increases with mold closing speed.

Figure 10 and 1 1 compare the computed and ex-

isotmpic 1500

non-isotropic CalCUlatiOn

0 1 2 3 4 Time (s)

Fig. 7. Comparison between measured and calculated (nonisotropic and isotropic models) mold closing forces.

O.115m 0.15 m

k 0 . 1 5 m a pz P3 w- 0.3m - a + 0.025m -- I----- * i- 0.45 m

Fig. 8. Rectangular plaque mold (only a quarter is repre- sented) with four pressure transducers. PI, Pp. PSI Pa.


Simulation of Compress ion Molding

0 1 2 3 Time (s)

Fig. 9. Measured pressures versus time on transducers PI and P2 for three mold closing Velocities (0.02; 0.01; 0.005 m/s) .

Exp.0.005m/s Num.0.005m/s Exp.0.01 m/s Num.O.Olm/s

v Exp. O.OZm/s

0 I 2 3

Time (s) Fig. 10. Comparison between experimental (Exp) and computed (Numlpressure on transducer PI for three mold closing velocities (0.02, 0.01 I 0.005 m/s) .

perimental pressures on transducers 1 and 2 for the three velocities. The agreement is fair. The difference between computations and experiments is greatest at the end of filling. This may be due to the influence of the contact between the material and the lateral boundaries of the mold, which may induce a change of the fiber structure and consequently a change in the material behavior. This point has to be studied more precisely in future work. The agreement is better at high compression velocities than at low ones. This can be explained by thermal effects and more precisely by the balance between viscous dissipation and heat transfer between the mold and the material. The heat transfer is proportional to the filling time. In slow compression-molding conditions, the dissipation is not large enough to balance the conduction. When the mold closing speed increases, the dissipation increases too, and the heat transfer decreases. The experiment tends to be isothermal like the calculation.

A second important comparison between experiments and calculations concerns the evolution of the flow front during mold filling. Figure 12 shows the sample shape at 50% deformation. The agreement is

h

Num. 0.01 m/s Exp. 0.02 m/s E N m . 0.02 m/s

k

0 1 2 3

Time (s) Fig. 1 1. Comparison between experimental IExp) and computed “urn) pressure on transducer P p for three mold closing velocities (0.02, 0.01, 0.005 m/s).

0,12 isotropic calculation I

I

0,08

0,06

0,04

Non 1 \ 1 isotropic ~, 2 l c u l a t i o n

i I

Opo2! 0.00 0 , o 0,1 0,2

x (m)

Fig. 12. Comparison between experimental and computed sample shape at 50% deformation.

really better for our nonisotropic calculation than for an isotropic one.

CONCLUSIONS

This study shows that the material has a significant transverse isotropic behavior: The viscosity associated with in-plane expansion is about a hundred times greater than the viscosity associated with shearing through the thickness. The introduction of a transverse isotropic shear-thinning behavior permits one to calculate precisely the mold closing force and the pressure field and to explain some specific flow front shapes that have been observed by several authors. The comparisons with experiments performed on well-instrumented molds show fair agreement.. We have now to take into account thermal transfer phenomena by developing a nonisothermal model. For further development, fiber-matrix separation and fiber orientation phenomena will be studied.

ACKNOWLEDGMENTS

We are grateful to Fiat, Peugeot S.A. , Renault, and Volvo companies for experimental and financial sup- port.


R. Ducloux, A. Poitou, M . Vincent, and J . F . Agassant . .

NOMENCLATURE 71, 72 = anisotropic invariants of the rate of strain tensor. A. A], AP, A = internal forces.

a, b, v, F = external mass force.

P = penalty coefficient. R

l3R = boundary of R. &

= in-plane stress tensor. = intermediate rheological coefficients.

= external forces along the upper and

= mean surface sample and calculation IJ

domain.

= in-plane rate of strain tensor. FI, Fz lower walls.

h = sample thickness. ti = mold closing speed. I ] , 12, I4 = kinematic invariants.

= mold closing force. = isotropic shear thinning consistency. = anisotropic shear-thinning consisten-

= isotropic power-law index. = anisotropic power-law indexes. = internal dissipated energy. = dissipated energy by external forces. = pressure. = external stress exerted along dsL. = mean velocity vector through the thick-

= material velocities along the upper and

= isotropic Newtonian viscosity. = anisotropic Newtonian viscosities. = three-dimensional rate of strain ten-

cies.

ness.

lower parts of the mold.

sor.

REFERENCES

1 . M. J. Owen. I>. H. Thomas, and M. S. Found. Proc. 33rd Ann. Tech. Conf.. SPI RP/C Div.. Sect. 20-B (1978).

2. R. J. Silva-Nieto, B. C. Fisher. and A. W. Birley, Polym. Compos. 1, 14 (1980).

3. G. Menges and H. Derek, Proc. 36th Ann. Tech. Conf., SPI RP/C Div., Sect. 23-C (1981).

4. C. L. Tucker and F. Folgar, Polym. Eng . Sci.. 23, 69 (1 983).

5. C. C. Lee. F. Folgar, and C. L. Tucker, J . Eng. Industry, 106, 114 (1984).

6. C. C. Lee and C. L. Tucker, J . Non-Newt. Fluid Mech.. 24, 245 (1987).

7. M. R. Barone and D. A. Caulk. J . Appl. Mech.. 53, 361 (1 986).

8. S. M. Davis and C. L. Tucker. SPE ANTEC Tech. Pa- pers, 34, 524 (1988).

9. H. Hojo, H. Yaguchi, T. Onodera, a n d E . G. Kim, Intern. Polym. Proc. rr i , 1, 54 (1988).

10. L. J. Lee, L. Marker, and R. M. Griffith, Polym. Com- pos., 2. 209 (1981).


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Simulation of compression molding of long-glass-fiber-reinforced thermoplastic sheets