Three-dimensional viscoplastic FEM simulation of a stretch blow molding process

Advances in Polymer Technology, Vol. 17, No. 3, 189–202, 1998Q 1998 by John Wiley & Sons, Inc. CCC 0730-6679/98/030189-14

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Three-Dimensional ViscoplasticFEM Simulation of a StretchBlow Molding Process

SONG WANG and AKITAKE MAKINOUCHIMaterials Fabrication Laboratory, Institute of Physical and Chemical Research,2-1 Hirosawa, Wako-city, Saitama 351-01, Japan

TAKEO NAKAGAWADepartment of Precision Engineering, University of Tokyo, Tokyo, Japan

Received: June 17, 1997Accepted: January 27, 1998

ABSTRACT: In this study, a viscoplastic FEM simulation of the stretch blowmolding process is presented. The material’s behavior at elevated temperature isassumed to be strain-rate-dependent, and a non-Newtonian creeping materialmodel is employed to specify the strain-rate-sensitive characteristic of thematerial. A degenerated quadrilateral shell element, which enables properevaluation of the bending effect, is employed. The mold surface is described bytriangular surfaces, and the contact phenomenon is modeled as a node-to-meshcontact. In using the code developed, the stretch blow molding process is takenas an example and discussed qualitatively. First, the process under threeboundary conditions is simulated and examined, and then the condition leadingto the wrinkle phenomenon, which is encountered in the simulation, isdiscussed. q 1998 John Wiley & Sons, Inc. Adv Polym Sci Techn 17: 189–202,1998

Introduction

I n the last decade, blow molding has undergonevery rapid growth. Thousands of hollow plastic

products are produced by the blow molding pro-cess, such as soft drink bottles, large containers, au-tomobile gasoline tanks, etc. There are three main

forming methods: injection blow molding; stretchblow molding; and extrusion blow molding.1 Oneof the main difficulties in blow molding is the op-timization of the thickness distribution. Because theextension ratio is .10, it is very difficult to estimatethe final thickness distribution. Until now, most ofthe work has relied on the trial-and-error process.This is both expensive and time-consuming. An ef-

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THREE-DIMENSIONAL VISCOPLASTIC FEM SIMULATION

ficient and economic way should be found to solvethis problem.

Much research has been conducted on numericalsimulation of the blow molding process. The simu-lation of thermoforming, which is also a kind ofblow molding process, has been studied by manygroups. Most of these works employed the hy-2–8

perelastic material model, assuming rubberlike de-formation behavior.

Compared with thermoforming simulation,fewer works have been done with simulation of thestretch blow molding process. Among these,Schmidt et al.,9 for example, employed Newtonianflow to approximate the deformation behavior andobtained a reasonable thickness distribution, but thecalculated plunger force could not fit the measuredresults. Haessly and Ryan carried out detailed10,11

experimental research on the injection blow mold-ing process, and employed the neo-Hookean con-stitutive equation for its simulation. However, theyfailed to give a precise prediction of the materialsdeformation shape. Hartwig et al. also performed12

stretch blow molding simulation using the com-mercial FEM program ABAQUS/Standard. All ofthese are based on the two-dimensional (2D)FEManalysis.

The viscoplastic constitutive equation waswidely employed in the simulation of superplasticforming, the forging process, and sheet metal13 14

forming. Recently, attempts have been made to ap-15

ply it to the blow molding process. The visco-16–19

plastic formulation is more promising than the hy-perelastic one. Two main advantages can be notedfirst, because of the introduction of viscosity, thereal forming time can be evaluated; and second, theanisotropic constitutive model can be implemented.

In this article, a three-dimensional (3D) visco-plastic FEM code developed by the authors is intro-duced. The material behavior is assumed to be non-Newtonian under flow, where viscosity is anonlinear function of the strain rate. The quadrilat-eral degenerated shell element is implemented,treating the bending effects properly. The mold sur-face is described by triangular surfaces. Contact be-tween the parison and tool is formulated as a node-to-mesh contact, namely, a node of material gettinginto contact with a triangular surface of the mold.The no-penetration condition is the basis of this con-tact search. Experimental observation through atransparent mold, by Imamura et al. showed that20

the slipping phenomenon between the polymer andthe mold could be omitted; therefore, the stickingcontact assumption is employed.

Constitutive Equations

A generalized viscoplastic approach is em-21,22

ployed to determine the constitutive equation. Us-ing this approach, the rigid viscoplastic, Binghamflow, creeping material, and Newtonian flow couldbe obtained as special cases. In this study non-New-tonian creeping material behavior is employed.

Neglecting elastic deformation, Perzyna de-23

fined the strain rate, by the general form:.« ,ij

dQ.« 5 g^f(F)& (1)ij dsij

where is a yield surface, and is a plasticF 5 0 Q 5 0potential surface. Assuming that both the yield andplastic potential surfaces are identified, and can beexpressed by the Von Mises yield surface, we have:

F 5 Q 5 3J 2 s (2)p 2 y

where the second invariant of the deviatoric stress

tensor, is and is the uniaxial yield1

9 9s9 J 5 s s , sij 2 ij ij y2stress. From eqs. (1) and (2) we obtain:

93 s. ij« 5 g^f(F)& (3)ij 2 s

where is the equivalent stress. Squar-9 9s 5 3/2s sp ij ij

ing both sides of eq. (3) we obtain:

9 9s s. . ij ij2« « 5 L9 (4)ij ij 2s

where3

L9 5 g^f(F)&.2

Multiplying both sides by 2/3, and taking theirsquare root, we have:

«̄̇ 5 2L9/3 (5)

where is the equivalent strain.. .

«̄̇ 5 2/3 « «p ij ij

From eqs. (5) and (3), we have:

2s .9s 5 « (6)ij ij˙3«̄

Equation (6) is equivalent to the isotropic viscousflow:

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.9s 5 m « (7)ij ij

where m is a viscosity.Substituting eq. (7) in eq. (3) we have:

1 3 15 g^f(F)& (8)

m 2 s

Assuming we can obtain:nf(F) 5 ( 3J 2 s )p 2 y

1/n˙2(s 1 («̄/g) )ym 5 (9)˙3«̄

As shown in Table I, for a certain condition of eq.(9), the rigid viscoplastic material, Bingham flow,creeping material, and Newtonian flow could bespecified.22,24,25

Materials Behavior

The deformation of polymer at elevated temper-ature is approximated by non-Newtonian flow. Theequation has often been derived from the uniaxialtension test, where the uniaxial tensile stress, s, isrelated to the tensile strain rate,

.«, by:

. ms 5 k «

It can be generalized into the 3D case:

m˙s 5 k«̄ (10)

Equation (10) is identical with the creeping materialmodel shown in Table I with m 5 1/n.

Geometric and Kinematic Fieldof Shell Element

In this section, the formulation of the geometricand kinematic field of the degenerated quadrilateralshell element employed are presented.26,27

As illustrated in Figure 1, the degenerated quad-rilateral shell element has four nodes in the middlelayer, and each node is coupled with a nodal fiberthat is assumed to be inextensible within one incre-

mental time step. Boundaries of the element are j,h, in the natural coordinate system.z 5 61

The geometric field can be expressed by:

za a a 2ax 5 N (j,h) x 1 h ei S i 3i D2

(a 5 1,4 and i 5 1,3) (11)

where is the coordinate of the node a, repre-a ax hi

sents the length of the nodal fiber, and denotesae3

the unit vector of the nodal fiber.is the shape function taking a value of unityaN

at node a and zero at all other nodes:

(1 1 j j)(1 1 h h)a aaN 5 (12)4

in which:

j 5 21 h 5 21 j 5 1 h 5 211 1 2 2

j 5 1 h 5 1 j 5 21 h 5 13 3 4 4

The nodal coordinate system is defined as:

a ae 5 e 3 i1 3

or:

a a ae 5 e 3 j (if e is parallel to i)1 3 3

a a ae 5 e 3 e2 3 1

where i, j, and k are the base vectors of the globalcoordinate system. The velocity field inside thevi

element can be found by:

azha a a a a av 5 N v 1 (e v 2 e v )i H i 1i 1 2i 2 J2

(a 5 1,4 and i 5 1,3) (13)

where and represent two rotation velocitiesa av v1 2

with respect to and local coordinate bases, re-a ae e2 1

spectively, and represents the velocity of node a.avi

Equation (13) can be rewritten in a compact form:

a av 5 N (j,h)v̂ (14)i ri r

where:

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FIGURE 1. Degenerated quadrilaterial shell element.

av r 5 1,3ra av̂ 5 v r 5 4 (15)r 1H

av r 5 52

and:

aN d r 5 1,3ri

za a a aN (j,h) 5 N e h r 5 4 (16)ri 1i2H z

a a a2N e h r 5 52i2

FEM Formulation

The constitutive eq. (6) can be rewritten as:

2 s.9s 5 a« , where a 5 (17)ij ij ˙3 «̄

Because the plane stress condition is assumed forshell elements, is imposed in eq. (17), ands 5 033

we obtain:

.s 5 aD « (18)

where:

2 1 0 0 01 2 0 0 0

D 5 0 0 1/2 0 0 (19)F G0 0 0 1/2 00 0 0 0 1/2

and the components of vectors.« and s are;

Ts 5 (s s s s s )11 22 12 23 31

. . . . . .T« 5 ( « « 2« 2« 2« )(20)

11 22 12 23 31

Assuming a viscoplastic body under surfacetraction p, the viscoplastic functional can be givenas:

f 5 EdV 2 p v dS (21)E E i iv St

where and v is the velocity of thee .ijE 5 s d«E ij ij

0

material on the surface traction boundary. Invokingthe stationary of the functional f (i.e., ), anddf 5 0applying the standard finite-element discretizationprocedure, we obtain:

.T Ts d «dV 2 p Ndv̂dS 5 0 (22)E Ee eV St

Note that strain rate.« can be written as:

.« 5 Bv̂ (23)

Substituting eqs. (18) and (23) into eq. (22), we have:

. T Ta« DBdV 5 p NdS dv̂ 5 0 (24)E ES De eV St

therefore:

TaKvdV 2 N pdS 5 0 (25)E Ee eV St

with:

TK 5 B DB (26)

To linearize eq. (25), the Newton–Raphson methodis applied. The final linearized form for the kth it-eration step is:

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FIGURE 2. Stretch blow molding process.

FIGURE 3. Initial thickness distribution of the preform.

2Tk b KdV 1 c H H dV Dv̂E ES (k21) (k21) (k21) (k21) D k

e e3 V V

T5 N pdS 2 a H dV (27)E E (k21) (k21)e eSt V

where:

2(m 2 1)m21 (m23)˙ ˙b 5 «̄ , c 5 «̄ ,(k21) k21 (k21) (k21)3

TH 5 Kv̂ , K 5 B DB(k21) (k21)

Calculating from eq. (27), the velocity of the kthDvk

iteration step is obtained:

v̂ 5 v̂ 1 aDv̂ (28)k k21 k

where a is a stabilizing parameter with a value be-tween 0 and 1.

The convergence criterion for the iterative solu-tion is:

||Dv̂ ||k# d (29)

||v̂ ||k

where d is a prescribed small value (we use).25d 5 10

Applications

The stretch blow molding process of a PET (poly-ethylene terephthalate) bottle was simulated. Thegeneral stretch blow molding procedure of the PETbottle is shown in Figure 2. In the first step, the pre-form of the PET bottle is produced by injectionmolding, where the space between the metal coreand the injection mold is filled with melted PET.After the cooling and heat adjustment process, thepreform is enclosed inside the blow mold and thenstretched with a metal rod (step 2). In the third step,PET is deformed, by the combination of stretchingload and air pressure, to the shape of the mold. Fi-nally, when the bottle is sufficiently cooled insidethe mold, it is ejected (step 4).

Simulation of the stretching and blowing stages(step 3 and step 4) is the main objective of the de-veloped code. The shape and the prescribed thick-ness of the preform employed in the simulation areshown in Figure 3. The radius of the preform tapersdown from the top to the bottom, and the thickness

of is uniform throughout the sidewall, but4.5 mmit decreases gradually to at the bottom. The1.7 mmshape of the bottom of the preform is designed tofit the shape of the head of the stretching rod, andtherefore almost no deformation occurs. The shapeof the mold (Fig. 4) is constructed of the intercon-nected triangular surfaces. Because the mold isplane symmetric, only a quarter of the model wassimulated. To describe the preform shape of PET,1220 quadrilateral degenerated shell elements were



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FIGURE 4. Mold employed in the simulation.

FIGURE 5. Blowing without stretching (case 1).

employed. Inflation pressure was set at and0.5 MPathe material parameters were chosen to be m 50.53789, k 5 292529.

The simulations were performed for three differ-ent cases. In the first case, the preform was blownwithout stretching. In the second case, the preformwas initially stretched by a rod to the bottom of themold and, after finishing this process, air pressurewas applied. In the third case, the blowing pressurewas applied before the stretching rod reached thebottom, and the material was inflated in the surfacenormal direction while being stretched in the lon-gitudinal direction. These different forming proce-dures resulted in different material deformationmodes. The simulated deformation shapes of thedifferent cases are shown in Figures 5–7. For eachcase, the intermediate profiles are chosen so that acontinuous deformation process could be observed(the time increment between each profile is not con-stant).

In case 1, during the early stage of the process,the preform tended not to extend in the longitudinaldirection, but rather inflated in the radial direction.It deformed like a balloon and, due to the constraintof the surrounding mold, most of the material gotinto contact with the tool in this early stage. Morethan 70% of the material contacted 40% of the totalarea of the mold surface. Because the node of theparison was assumed to stick to the mold surfaceafter contact, the remaining 30% of material wouldgovern the further deformation and be distributedto the rest of the mold surface. This resulted in se-rious thinning.

In case 2, in the stretching stage, obvious shrink-age was observed in the radial direction and, whenstretching was finished, the total length of the par-ison became more than two times the initial length.

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FIGURE 6. Blowing after stretching rod reaches thebottom (case 2).

FIGURE 7. Blowing before stretching rod reaches thebottom (case 3).

It is interesting to see that, in the blowing stage,bulging started at the top and bottom ends and thetwo inflated ends joined in the final stage of form-ing. In an actual forming process, under certain con-ditions, a folded line can be observed on the prod-uct. This deformation mode might be the reason forthe generation of that folded line.

In case 3, the material was first stretched for acertain length and, subsequently, air pressure wasapplied. Inflation was observed initially at the topof the preform, and gradually extended to the lowerpart. The bottom of the preform was also enlarged,but the bulging speed was slower than thatof the upper part, so that the two balloons formedin case 2 were not generated. As a result, the de-formation started from the top and ended at the bot-tom; a very smooth deformation mode was ob-served.

The final thickness distribution is plotted for thefinal deformation shape in the three cases (Figs. 8–10). In Figure 8 (case 1), due to the aforementioned

reason, the mesh at the middle part is severely dis-torted where the thickness decreased to 0.05 mm.Because no fracture criterion was used in thepresent FEM model, calculation could be continued.However, this thickness value indicates that frac-ture may have already occurred in this area. In Fig-ure 9, the deformed mesh shape becomes smoother,and the mesh at the joint between the two balloonsformed in case 2 appears to be less deformed thanthe surrounding parts. Consequently the thicknessat this area becomes greater than that of the sur-rounding area. In Figure 10, except for the bottompart, the deformed mesh shape is very regular, andsmooth thickness distribution can also be observedfrom the top to the bottom. Because the calculationwas performed under the assumption of a uniformtemperature distribution, the thickness at the bot-tom is less than in the actual case where the tem-perature is always lower at the bottom. The effectof temperature on the thickness distribution will notbe discussed in this article.



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FIGURE 8. Final thickness obtained in case 1.


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Analysis of WrinklingPhenomenon

Besides enabling the prediction of the thicknessdistribution, the simulation is also expected to pre-dict the defects of the product. The wrinkling en-countered in the simulation (case 2) is one of thedefective phenomena; therefore, the conditions un-der which it occurs are discussed qualitatively inthis section.

From the stretched profile and the thickness dis-tribution obtained in first stage of case 2 (Fig. 11),we can see that, from the top to the bottom, thethickness gradually decreases. Also, the radius atthe area near the bottom becomes the smallest. Thiscan be observed from the two rings cut from the topand the bottom of the preform, as illustrated in Fig-ure 12. The stress normal to the upper and the lowersurface of the ring is assumed to be the same and isdenoted by and for the two rings. Also, for thes sa b

two rings, the inner radii are denoted as andRa

and the thickness of the walls of the rings de-R ,b

noted and and can be simply derived as:T T ; s sa b a b

F Fs 5 5 (30a)a A p(2R T )Ta a 1 aa

F Fs 5 5 (30b)b A p(2R T )Tb b 1 bb

Due to the initial tapered shape of the parison, theradius of the top cross-section is larger than that ofthe bottom cross section (i.e., ). Additionally,R . Ra b

because the normal forces (F) acting on the two ringsare the same and from eqs. (30a) and (30b),T 5 T ,a b

we have Under the assumption of uniforms , s .a b

temperature distribution, the material parametersare the same for the two rings; therefore, ring b ismore easily deformed than ring a. Furthermore, dueto the volume conservation assumption,

.« 51

2.«

.« extension in the longitudinal direction2 ,2 3

causes a decrease in the thickness as well as in theradius. Therefore, after stretching, the thickness andradius become smallest near the bottom.

In the blowing stage of case 2, wrinkling oc-curred. Two reasons are possible: the thickness dis-tribution after stretching and the shape of the pari-son after stretching. To discuss the two factorsseparately, two additional calculations (case 4 andcase 5) in the blowing stage of case 2 were carriedout. The differences lie in the thickness and theshape of the parison before blowing. In case 4, thestretched shape was kept the same as that calculatedin the first stage, whereas the thickness was initial-ized to be with uniform distribution. The cal-3 mmculated blown deformation shapes are shown inFigure 13, from which we can see that the defor-mation mode is similar to that in case 2, and infla-tion in the bottom part could not be avoided, even



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FIGURE 11. Thickness distribution after stretching.

FIGURE 12. Analytical model of the stretchedthickness distribution.

by increasing of the thickness. Additionally, becausethe thickness at the small-radius part increased, thebending effect induced by the inflated part also in-creased. This further limited the inflation of thesmall-radius part, and therefore wrinkling becamemore serious than that in case 2. In case 5, the lengthof the parison was kept the same as the stretchedparison obtained in the first stage of case 2, but theshape was designed in such a way that the top andbottom of the stretched parison were connected bya straight wall. The thickness was plotted such that,in the longitudinal direction, the thickness of the

manipulated parison corresponded to the thicknessof the stretched result. This initial shape and thick-ness distribution is shown in Figure 14. The de-formed shape after blowing is shown in the Figure15. The deformation started from almost the middleof the parison. Inflation of the part near the bottomwas always slower than that of the middle region.Therefore, a smooth deformation mode was ob-tained. From this we can see that the deformationmode was completely changed from that in case 2.The initial shape of the parison seems to be a deter-minative factor of the further deformation mode.

FIGURE 13. Deformation shape obtained in case 4.

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FIGURE 15. Deformation shape obtained in case 5.

FIGURE 14. Manipulated intermediate shape of the parison.

To clarify the situation, the analytical modelshown in Figure 16 was considered. Half of two sec-tions, which represent the parts of the parison withlarge inner radius and small inner radius be-R Ra b

fore blowing, were drawn. The thickness of the tworings is denoted as and and the stress actingT T ,a b

on the section of the two rings is denoted as andsa

Both rings undergo the same pressure, p, froms .b

the inside. We compare the increment of the radiiof the two rings.

From the equivalent of the inner stress with thesurface traction, p, we write:

R p R pa bs 5 and s 5 (31)a bT Ta b

Strain increment induced by the stress in the tworings for a strain rate sensitive material can be ex-pressed as:

1/m1De 5 s Dt, anda S aDk

1/m1De 5 s Dt (32)b S bDk

where Dt denotes the time increment.The radius deformed to and can be written:9 9R Ra b

2pR 1 De · 2pRa a a9R 5 5 R 1 R De (33a)a a a a2p

2pR 1 De · 2pRb b b9R 5 5 R 1 R De (33b)b b b b2p

Therefore, the increments of the two radii, andDRa

are:DR ,b

DR 5 R De (34a)a a a

DR 5 R De (34b)b b b

From eqs. (28), (29), (31a), and (31b) we have:



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FIGURE 16. Analytical model of wrinkling.

1/m (111/m) 21/mDR 5 {(p/k) Dt}R T (35a)a a a

1/m (111/m) 21/mDR 5 {(p/k) Dt}R T (35b)b b b

From eqs. (35a) and (35b), we have, if

then the deformation of the(111/m) 1/mR Ta a. ,S D S DR Tb b

large ring will be faster than that of the small ring.Clearly, the initial radius is a more important factorthan the initial thickness.

From this we can see that the initial shape of theparison plays a key role in the deformation mode,and the wrinkle phenomenon is caused by the dif-ferences in the radius of the parison before blowing.

However, the analytical model can only explainthe initial deformation speed of the two rings, butthe situation may differ due to the strain-rate-sen-sitive characteristic of the material. Because strainrate hardening effect will slow down the speed ofdeformation for the large radius part, a full analysis,based on the deformation history, is necessary.Based on this viewpoint, the FEM simulation is ex-pected to be a powerful tool.

Conclusions

Using the viscoplastic blow molding simulationcode, the material deformation behavior underthree boundary conditions was simulated and dis-cussed qualitatively.

Based on the results of the simulations we canconclude that, first, it is not possible to apply case 1to produce a real product; second, in case 2, espe-cially when the stretching ratio is large, wrinklingthat should be avoided might occur and remain onthe final product; third, because the most stable andsmooth deformation is achieved in case 3, it isstrongly suggested to apply this method in the realprocess; fourth, the shape of the parison beforeblowing is a determining factor in the further de-formation mode in which wrinkling can be induced

due to the difference in the radius of the parison.The trends of the deformation behavior of the ma-terial in the blow molding process have been clari-fied, but further experimental investigation is nec-essary to obtain more detailed information. Inparticular, the strain hardening effect, which wasnot discussed in the present study, will be investi-gated in the near future.

Acknowledgments

The authors express their special thanks to Mr.Ibe, Mr. Onuma, Mr. Tamura, and Mr. Maejima ofthe Aoki Co. for helping us to understand the realprocess.

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Three-dimensional viscoplastic FEM simulation of a stretch blow molding process

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