Viscoelastic simulation of PET stretch/blow molding process

Jonr~ofNou-Newtonian ELSEVIER J. Non-Newtonian Fluid Mech., 64 (1996) 19-42 Fluid Mechanics Viscoelastic simulation of PET stretch/blow molding process F.M. Schmidt a'*, J.F. Agassant a, M. Bellet a, L. Desoutter b aEcole des mines de Paris, CEMEF-URA C N R S no. 1374, 06904 Sophia-Antipolis, France bSIDEL Corporation, 76600 Le Havre, France Received 4 August 1995; in revised form 4 December 1995 Abstract In the stretch/blow molding process of poly(ethylene terephthalate) (PET) bottles, various parameters such as displacement of the stretch rod, inflation pressure, and polymer temperature distribution, have to be adjusted in order to improve the process. An axisymmetric numerical simulation code has been developed using a volumic approach. The numerical model is based on an updated-Lagrangian finite element method together with a penalty treatment of mass conservation. An automatic remeshing technique has been used. In addition, a decoupled technique has been developed in order to compute the viscoelastic constitutive equation. Successful stretch/blow molding simulations have been performed and compared to experiments. Keywords: Finite element method; Splitting technique; Stretch/blow molding; Viscoelastic fluid 1. Introduction 1.1. Description of the stretch~blow molding process An amorphous injected molded tube-shaped preform of poly(ethylene terephthalate) (PET) is heated in an infrared oven above the glass transition temperature (T ~ 100°C), transferred inside a mold and then inflated with stretch rod assistance in order to obtain the desired bottle shape (Fig. 1). The performance of the produced bottle (wall thickness distribution, transparency, mechanical properties...) is determined both by the material properties and the operating conditions: the initial preform shape, the initial preform temperature and the balance between stretching and blowing rate. *Corresponding author. 0377-0257/96/$15.00 © 1996 - Elsevier Science B.V. All rights reserved S S D I 0377-0257(95)01420-9 20 F.M. Schmidt et al. / J. Non-Newtonian FluM Mech. 64 (1996) /9 42 1.2. Literature on blow molding simulations Numerical simulations of the blow or stretch/blow molding processes have been extensively developed during the last decade. Most of the models assume a thin shell description of the parison. Warby and Whiteman [1] as well as De Lorenzi and coworkers [2,3] propose isothermal finite element calculations. These models, first developed for thermoforming processes, have since been applied to the blow molding process. The rheological behavior is given by a nonlinear-elastic constitutive equation derived from the rubber-like theory. Kouba and Vlachopoulos [4] have extended the previous model to the blow molding of a viscoelastic fluid (KBKZ constitutive equation). Several models use a volumic finite element approach. In 1986, Cesar de Sa [5] simulated the blowing process of glass parisons assuming Arrhenius temperature dependent Newtonian behavior. Chung [6] has carried out simulations of PET stretch/blow molding using the code ABAQUS ®. The model assumes elasto-visco-plastic behavior and thermal effects are neglected. Poslinski and Tsamopoulos [7] have introduced nonisothermal parison inflation in a simplified geometry. In order to take into account the phase change, the latent heat of solidification has been included in the heat capacity of the material. Recently, Debbaut et al. [8] have also performed viscoelastic blow molding simulations with a Giesekus constitutive equation. They introduce thermal effects but present numerical results only in the case of a Newtonian fluid. In blow molding simulations, numerical models have to take into account large biaxial deformations of the material, the evolving contact between tools (mold and stretch rod) and polymer, and temperature gradients. In the stretch/blow molding process, the contact between the stretch rod and the bottom of the preform induces localized deformations which need volumic approaches in order to obtain an accurate description. 1.3. Objectives of the present approach In a previous paper [9], we pointed out that the computed stretching force using a Newtonian volumic model was very far from the experimental one. In the present work, an isothermal Stretch rod Stretched& II I III / air;;;;ureI ~~ J Fig. l. Description of the stretch/blow molding step. so. (1) where ~ is the velocity field and fl is the domain occupied by the parison.__. In addition. / J. 64 (1996) 19 42 21 RADIAL SYMMETRY BOTTLE MOLD Z ~ PLUG j AXIAL / I i . 2. finite element volumic calculation of the PET stretch/blow molding of a viscoelastic fluid is presented.\ PREFORM J Fig. Schmidt et al.F. 2..__~. SYMMETRY I I I I . Non-Newtonian Fluid Mech. Basic equations and boundary conditions The material is assumed to be incompressible. the continuity equation may be expressed as V.~=0 one.M. the weak form of the dynamic equilibrium can be written over the whole domain fl at any time t and for any velocity field g*: . B o u n d a r y c o n d i t i o n s . Thickness and stress profiles in the bottle will be discussed. The improvement in terms of force prediction will be shown. r/s and r/v are the viscous part and the viscoelastic part respectively of the total viscosity r/ (r/= qs + ~7v). The boundary F of the domain fl is decomposed as F = F v w F p L9 F f. T + T ' d). the extra-stress tensor which is related to 4 by a nonlinear partial differential equation: DT T + 2 ~ = 2~/. 4" =½(V~*+W~*). a t [ . In the present approach.d. but this will not be considered here): a • r/= ~ (7) Internal free surface 1-'Pnt. the liquid-like viscoelastic constitutive equation of JohnsonSegalman type [10] with additional solvent viscosity is used: a=-p'l+2qs4+T onfl. Non-Newtonian Fluid Mech. p. (6) where F V is the part of the boundary F where the velocity is prescribed (bottom of the rod).M. 2.1 . (8) . F p the part of the boundary F where a pressure is applied and F f the part of the boundary F contacting the tools. For the stability of the simulations. A zero pressure condition is assumed. D/Dt is the Gordon-Schowalter convective time derivative: DT Dt ST + ( ~ " V ) T + T" ~ . the rate of strain tensor associated with ~*. we took r/~ > r/v/8 according to the criteria of Crochet et al. serving as a reference for pressure values (this could just as well be the atmospheric pressure or even an evolving pressure resulting from the balance between air compression between the preform and the mold and air leakage flow through the vents of the mold. the identity tensor. T. the acceleration. +1] is the "slip" parameter which determines the type of convective derivative. [11]. d. 64 (1996) 19-42 a is the Cauchy stress tensor. /.i = -aP. External free surface 1-'Pxt . r/ and 2 are constant in the case of an isothermal computation. the acceleration due to gravity. The initial geometry of the preform and the boundary conditions are presented in Fig. / J. For a = 1 (upper-convected). the unit outward normal vector to the boundary of the domain F. A differential inflation pressure AP(t) is applied which can evolve during the successive blowing stages: • . ~.~ " St T-a(d. Schmidt et al. we recover the Oldroyd B model. r/. (3) where p' is an arbitrary pressure. (4) 2 is the relaxation time. (5) where f~ ~-~-l(~7~--TV~) is the rotation tensor.22 F. the specific mass. the unit tangential vector. Explicit time-marching algorithm The whole process is divided into time intervals Ati so that the current time tn may be written t. (9) where ~f is the friction coefficient (~f= 0 results in a perfectly sliding contact). assuming that any contacting node remains fixed until the end of the process. 23 J. which can be defined as (a. Numerical resolution 3.M. the non-penetration condition is written Aft. Fixed-point algorithm. the velocity difference along the tools interface. Non-Newtonian Fluid Mech.4 2 Regions in contact with the tools F f.~)/'= ._-)fix-1 ---) u = u n __~fix ] --) u n Tn I fix = fix + 1 UTfix-T =rl = fix-1 II n IITnll > ~ a n d fix < F i x m a x Fig. (12) i=1 I fix-1 fix = 1 T = n =T =n-I fix-1 Use T= n to solve P S P I Use u n to solve [14] --~ .f t o o l s (11) " /~" 3. (10) Regions where the velocity of the nodes is prescribed F v. A perfectly sticking contact can also be considered. Schmidt et al. 64 (1996) 1 9 . fi < O. /. = ~ At. Af. f " /~ -~.F. ] n-I . A Newtonian friction law is assumed.1. 3. In addition. (n ~> 1).~fr/Aa{. +._I has been calculated on the domain f~. ) = 2qv~.._~ at the previous time step. (2) can be written at each time t. Non-Newtonian Fluid Mech. we use a Newmark type [12] integration rule: 1 (ft. • ~. is the coordinate vector at time t. (15) and (16) together with the incompressibility condition (I). / J. deals with an incompressible Newtonian fluid flow. the mechanical equations (see Sections 3.~_1 ( 1 .+~ using the second order explicit Euler rule: At] + 1 . ' ~ . ' T . called the Perturbed Stokes Problem (PSP). . a splitting technique is presented.. ' T .._ ) (13) where 0 is the arbitrary implicit parameter.:.~ . (15) where T. which belongs to [0. In this paper.. For a complete review of these techniques.3) are solved on the deformed configuration f~. . + T . which split the global set of equations into two sub-systems.. an iterative procedure based on a fixed-point method is used. Schmidt et al.. ~.F. 3.+2[ T" -At.M. . = X.. the geometry is updated from ~ . and the extra-stress tensor Tn are computed.. see Keunings [13].T"-1 + T . 3. the pressure p'.0 ) . Eq.g ) " u* d r = 0 .. Splitting technique Using Eqs. Time descretization of the constitutive equation The time differential constitutive equation (4) is approximated by an implicit Euler's scheme over the time increment At..+. J(.a ( i . Such an implicit algorithm is well known for its non-conditional stability. Then. 64 (1996) 19 42 24 At each time step tn.2. Jn np (16) In order to solve Eqs. Basombrio [14] and Baaijens [15]. : fo n n + I dr r ~fq(AJ)" n ... In order to compute the acceleration field ~. The first sub-problem. which are successively solved. two families of computation methods are available: the coupled methods which solve the complete discretized equations and the decoupled methods. to f~.v+fr {'~* d S + 6 P ( ~ . the current values of the velocity vector ~. + At.1]. ' ~ . perturbed by a known extra-stress tensor computed at the previous fixed-point iteration (fix-l).. At each time step. This fully-implicit algorithm leads to ] 1". + ~ ~.2 and 3. (3) and (5) and the boundary conditions (7)-(11). (14) where )(.3.:... The second sub- . It is to be noted that the penalty method is equivalent to the resolution using a discontinuous pressure. Finite element approximation for PSP The reference domain f~n is approximated by a set of 6-node isoparametric triangles (P2 element). pp the penalty coefficient. (18) The incompressibility constraint (1) is prescribed in a penalized form as ! V'a. Schmidt et al. . the components of the current extra-stress tensor T. 3. Non-Newtonian Fluid Mech. Consequently.F. it reduces to a (4 x 4) linear algebraic system. 4. 3.5. Each point )(n of the elementary domain fl~ is located by means of the vector of nodal position )(~ and the matrix N of shape functions )(n = N)(]. This algorithm is enforced by a dichotomic procedure on the time step in the case of non-convergence of the algorithm after a few iterations (Fixmax 10). P2-P0 element. For each time step t~. = NK~. are only needed at the Gaussian points of each element. (15) are evaluated by the Gauss-Legendre point integration rule. 3. pp" (19) in which. problem consists in determining the components of the extra-stress tensor for a known velocity vector by solving the time-discretized constitutive equation (15). 64 (1996) 19-42 25 • Pressure e Velocity Fig. is expressed in terms of the nodal velocity vectors ~ with the same shape functions: K. is a large number (typical value is 107). (17) The current value of the velocity field t~. the tensorial equation (15) is solved at a local level. constant per element (see Fig. Numerical resolution of the viscoelastic equation As the different integrals in Eq. 4).4. the complete procedure is summarized in Fig. The procedure is repeated until convergence.- P.M. / Y. 0 ) .[" /~. N dS b.e . dng The gravity forces are expressed by (24) f ~ = P g I" NdvL . J 0 - WN. lI. . are given by PF~en-1 Fe" = -d L a t . 5. (P~ + f~r + Fg) + ~ (fb + rb). the momentum equation (16) becomes C. Thus. Schmidt et al. (23) The viscoelastic forces ff~.26 F. " D N dv e. ~.which come from the 4-component extra-stress vector /~. where the vector /7 is the assembly of the nodal velocity components: Are 17= y' fig. (21) e=l P is the vector of the applied forces: Ne Nb P = Y'. Non-Newtonian Fluid Mech.M. Position of the nodes after remeshing. we use the same approximation as for the velocity field (Galerkin method).Ndv e. l? = (20) F.)ng (25) The forces associated with the application of inflation pressure are Fpb = . . Orb (26) ~nA' unknown at t h e ) ew nodes Newmesh (E~t(Old m e s h ) ) Fig. at current time tn are set by f~r = . / J. + (1 . 64 (1996) 19-42 For the virtual velocity ~*. e=l (22) b=l The inertia forces F~.{" APt/. t) ( Radial vc ~¢ity ) .p c o r r e s p o n d s to the i n c o m p r e s s i b i l i t y r e q u i r e m e n t : C/ep = pp [ TV" N" (V.C. / J. b (27) C is the m a t r i x d e f i n e d b y Ne C = Z (C~. Schmidt et al. (30) an g Table 1 Physical data and process parameters So (m) R o (m) Lo (m) vo (m s 1) AP (Pa) p (kg m 3) .13 0.. 64 (1996) 19-42 27 Slip contact 8 R U(r. Non-Newtonian Fluid Mech.09275 0.fV ~¢l~Afit" N d S b. Simultaneous inflation and extension of a tube. e=l (28) T h e e l e m e n t a r y m a s s m a t r i x C~.J ?W(z. 6. "1.1 2.M.125 0.F. is e_ Cp P .C~s ).10 × 105 .p -{.M' Imposed pressure - .i (s) r/V (Pa" s) 0.I• OAth _ ~N-Ndv% (29) C~. N) dv e. T h e f r i c t i o n f o r c e s are Fb = -. t) ( Axial velocity ) Fig.4 10 6 1380 0. 3.. Thickness of the tube vs. . 64 (1996) 19-42 i I I i i 4L 3. ~x.. allows one to determine the future trajectory of each node....5 "\ 0 0 [ I I I I 0.Ts= 2t/s fo g TDN" D N dv% (31) where D N expresses in vectorial form the strain rate tensor in terms of nodal velocities.. The bounded set of linear algebraic equations (20) is solved by a direct Crout decomposition. 2 : S A .5' 2 1. .. when large deformations occur... Schmidt et al. 0. .2 0. .2 Fig. An automatic remeshing procedure is used [16]. 2 :w %.~x \~...I. At time step t. curvature of boundary e d g e s .M. Non-Newtonian Fluid Mech.8 1 1. the procedure consists in the following steps: (a) Check the distortion of the elements and the accuracy of the mesh boundary (penetration of the boundary nodes into the tools. 7.+1. One can then compute the intersection of each trajectory with the tools. . 2.. 3 : F E . / J... For each time interval. The smallest of these intersection times is then retained as the value to be used for the next time step At. ) in order to decide if remeshing must be started according to prescribed tolerances.5 i :FE I : S A . 2.6. Automatic remeshing With an updated Lagrangian formulation.4 0... the nodes of the mesh following the kinematic evolution of the material points. time. Time step control associated with tool contact monitoring The geometry of the tools (stretch rod and mold) is defined by a piecewise linear approximation. ..... . the velocity field ~. .7. 3:sA .6 T I M E (s) 0. The viscous elementary matrix C er/s takes the following form: Ce.k 3 .28 F... %k"...5 x'x 1 "'. This method may result in excessively distorted elements. 3... L L' z' -- (34) (35) . The rate of strain tensor as well as stress tensor are diagonal at each time step.A ) 2 d v ' e ) = 1 e (32) e = element Applications 4. L) and (r'. 6). One can take advantage of the axisymmetric tube growth and use the Lagrangian coordinate transformation (r. 64 (1996) 1 9 . Validation test The simultaneous inflation and stretching of a tube limited by two planes has been considered (Fig. Non-Newtonian Fluid Mech. Schmidt et al. Thus. L') are respectively the radial coordinate. Consequently. the axial coordinate. (c) triangulation using Delaunay's algorithm [17] with addition of internal nodes in order to get triangle elements with the best possible shape. A differential inflation pressure AP is applied to the inner surface of the tube. A constant elongation velocity v0 is prescribed on the lower plane and the upper one has no displacement in the vertical direction. Z): X = zc(r2 . \e fn ( A ' . the global least squares method is employed in order to minimize the quadratic error function between the unknown variable A' at new nodes of the domain fgt and the known variable A at old nodes of the domain f2t (see Fig. There is a perfectly sliding contact between the tube and the two planes which means that the part will always remain a tube. R. we have Min(~ VA' 4.F.M. R'.R'z)-~-+R 2.R'2)L '. A quasi-analytical model may be considered. (e) numerical interpolation of the variables (especially stress variables) from the old mesh f~t to the new o n e ~'~tt . / J.4 2 29 (b) addition of nodes on the current boundary (overdiscretization) and elimination of some of these nodes in order to generate an appropriate set of boundary nodes which must be compatible with the old mesh boundary and satisfy non-penetration conditions and the curvature condition. 5). the inner radius and the length of the tube at time t and t'. (d) improvement of the shape of the elements by changing the diagonal of two adjacent neighboring triangles. z. Its results will be compared to the numerical model. and regularization of the mesh by moving the internal nodes toward the barycenter of adjacent nodes. Z Z-L Zt L' (33) where (r.1. As regards this last step of the procedure. z = R = Ro L = Lo. z) ~ (X. z'.R2)L = rc(r'2 . we have Vt>>_t'>O Vt < 0 r= r = r0 ~/ L' (r '2 . . N'~%.12 1. .... % n~ o 250 r. The components of the extra-stress tensor can be more easily determined via the integral form of the Johnson-Segalman constitutive model which reduces in the case of an elongational flow to g~retohtnii tore..0 II o t> 315..00 0.. z 200 £5. 9.9 Fig.I l .. . S t r e t c h i n g force o f the t u b e vs. Stretching force o f the t u b e vs. time... 8... Non-Newtonian Fluid Mech..~e (..~ ] 0. 0~oMe-~ =On~n~...7 0. .0 605.F..i 0. .0 0. [ t9 . . / J..] k = 0. r~ o~ 150 i00 50 0 I I I [ 0..... 64 (1996) 19-42 30 400 I I [ I i I I I l ' I:FE I:SA 2:FE 2:SA 3:FE 3:SA 350 300 • ... Schmidt et al.. ... time.0 1'7'0.. ...4 [ I 0.56 ] I 0.. .5 0..0 ° ~-11 =021~u l l a x ~ H .) Fig..84- 1.6 TIME (s) I I I 0..8 0.j 460. I 25...0 ' ~ - . .4 TL. ( N ) 750.M..3 0..2 0... The orthoradial coordinates remain constant.#' . Schmidt et al. . ------ 600 a = 0. . .. 10... ff is the dimensionless volume of the circular tube. .J_ oc where C-a(t')l~= l is the Finger strain tensor..2 0. .v L t=7~' ~ £--Co' R ~_ x =Ro' g_ --~C0R~' o (So~ 2 ~LoRg \goo/ 1. / J.5 . The dimensionless components of ir are 800 700 a = l(Oldr°Yd0U)a 5 =~'.2 1. . (36) a . I I I I I I t 0. 11.F. 64 (1996) 19-42 T= 1 | m(t - t')c-a(t ') dr' 31 (Va ¢ 0). "~.. The components of C-i(t ') are deduced from Eq.4 1. (34) (see Chung and Stevenson [18]). -'. .6 Fig... . .~ eq ..005 (Jaumann) . . G e o m e t r y o f t h e b o t t l e m o l d a n d initial p r e f o r m m e s h .6 0. Vs > _ 0 is the so-called memory function defined as re(s) rn ( s ) .4 0.r _.M. . I n f l u e n c e o f t h e slip p a r a m e t e r o n t h e s t r e t c h i n g force. 500 400 300 2O0 100 i 0 0 / .. Non-Newtonian Fluid Mech. Dimensionless variables are introduced: t=. . . a = -1(De Witt) .. .8 t(s) 1 1. / Fig. a =-0. 00 i a i l i 0 (Bars) \/ .~) \/ \ / i I L.lo16 i i ~).2288 l a ~.500 i 60.O0 (Bars) i i i 0 i i t =..t o l l l l l t = .00 I (s) P = 2.2002 i o t-...o i~) .L~ i t .oo ($) P = 2.5oo i 6o.O000E+O0 (s) P • .0o (z) P = Z S o o (Oars) (!~.O000Et. on (s) P . 2.3151 i ( i i i 60.00 (s) p = 2. = t-..3801 0 t =.oo (8) P = 2.500 i 60. 12. Intermediate bottle shapes.00 (a) P .~ s o o oo.soo Fig.500 (Bern) (Bans) \ / i 0 | l .6641 o l | ~.L o i * i t . il :! E i li E t .3¢s~ (Oar~) (ear=) \/ \/ J 4~ . M..0 43 x 103 0. /72~ .0"00 AP + R r dr = PTr dr. 1Iv Re . / J.. this gives (r/. The first calculation has been achieved using a constitutive equation of Oldroyd B type . Non-Newtonian Fluid Mech. De + \r/v) ©t ~ + 1 + 5 J0 )7' +/7.5. At each time step. the dimensionless radius /~ is determined from Eq. Classical values for the rheology of PET and for the blow molding parameters [20] (see Table 1) lead to the following characteristic values for Reynolds and Deborah numbers: Re = 6 x 10 . Simpson's first rule was used to evaluate the integrals. This indicates that the contribution of inertia effects will be much smaller than viscous and elastic effects. De = 0.7()7a/~" + £ 2)o { ()7+ 1)-" j0 j d? . for example.7 ( ['reTdt'~ iP. Schmidt et al.s) 2 (s) r/v (Pa" s) 1. using a quasi-Newton iterative procedure. (41) precludes any simple analytical solution for /~ except for some limiting cases.= e 1+ ~a do L'2U]" (38) (39) Using the boundary condition given in Section 2.pR2 2Vv (42) The nonlinear form of Eq.~ __AP.a [/2()7--+--~/72)1~ ()7 + 1)~ + fieF[/2'()7 +/2'/~'Z)]a d~}.. Then.4 .~ ~ I l n ( 0 )J 1 ~'gTrr~ ~()(! ig~2d)7. 64 (1996) 19-42 Table 2 Rheological parameters for the blow molding process a t/~ (Pa. (40) R Using the previous dimensionless variables. the components of the stress tensor and the stretching force are deduced.34 F. when inertia effects are strongly dominant [19] (Re >> 1) or for a Newtonian tube with inertia terms neglected (Re << 1). The iterative scheme was stopped when successive values of /? differed by less than 10-3%. Lo = e./~ 2 d)7' = Re J0 2LF (41) where Re is the Reynolds number and De the Deborah number which are defined as: De = .7 35 x 104 eT { (37) iP~r-. (41). the integrated stress balance equation in the r-direction is O'rr . 5 g °ii 0 \ I I 2O 40 I I I I 60 80 100 120 Longitudinal coordinate (mm) Fig. 7 and 8 show the comparison between the finite element calculation (FE) and the semi-analytic solution (SA) for the thickness and the stretching force vs. The agreement is fair. time (i.F. then reaches a m a x i m u m and decreases continuously. .0 . Fig. 1). 9 compares the stretching force as a function of time for two viscoelastic constitutive equations (upper-convected Maxwell.05 s (no. 5 . and 2 = 0. Fig.2 s (no.M. The curve for the stretching force starts from zero. 13.2. 64 (1996) 19-42 35 (a = 1 in Eq.1 to +1. 3). Non-Newtonian Fluid Mech. + 1) and for 2 = 0. the thickness of the tube decreases more rapidly and the initial slope of the stretching force decreases. Three different relaxation times have been considered: 2 = 0. 2 = 0. we present a blow molding process (no stretch rod).5.5 x 105 Pa is 1.005. Oldroyd B) and a Newtonian fluid with a viscosity ~IN = r/s + r/v.1 s (no.J. The blowing time decreases as the slip parameter increases from .1. the force exerted on the moving plane which is related to the stress in the z-direction). Schmidt et al. Figs.1 s. / J. +0. 2). 0. Thickness distribution of the bottle. The geometry of the preform and the bottle has been furnished by Professor R.e. (5)]. A constant internal pressure AP of 2. Set up of real stretch/blow molding examples Once the numerical model has been validated by comparing with a semi-analytic solution. We note that when the relaxation time increases. 4. 14o . 10 compares the stretching force as a function of time for different values of the slip parameter a (a = . We notice that the shape of the curves remains identical. First. The difference between the curves clearly indicates that a purely viscous model does not represent the increasing part of the curve which is directly related to the elastic response of the material. we now investigate some more realistic cases. Crawford from the Queen's University of Belfast [20]. 12 presents intermediate bottle shapes from the beginning of the process to the end. Non-Newtonian Fluid Mech. 14) shows the stress distribution at the end of the process. injected under the same conditions as the tube shaped preform. 11 shows the geometry of the bottle mold and the initial mesh of the preform. 14.36 F. The rheological parameters of the PET are given in Table 2.38 ~ 8.13 Maxi = 24. 13 presents the thickness distribution vs. A zoom of the neck and the bottom of the bottle (Fig. They have been determined by fitting to the traction force on an amorphous PET sample.00<g<5. Schmidt et al.69<o<8.5 Mini= 0.M. Fig.06<o<13.75 Bottom of the bottle 1375 o 1644 ~ 16. / J. 64 (1996) 19-42 m •< 3.44<o<19. prescribed. the bottom of the bottle is submitted to high stresses.06 ~ 11.69 ~ 5. . Generalized stress distribution at the end of the process. Fig.00 ~ 3. Fig.38<o<11.31 Unit : MPa Neck of the bottle Fig. In cleary indicates that at the end of the process. longitudinal coordinate at the end of the process. e.M. Fig. Non-Newtonian Fluid Mech. Pps is the m a x i m u m pre-blowing pressure (low pressure) imposed during t ~ [0. stretch/blow using preblow and blow). The prescribed pressure is a time-dependent function (i. and P~ the m a x i m u m blowing pressure (high pressure) imposed during t ~ [tps. Table 3 Dimensions of the bottle mold and the preform Material Length (mm) Inner radius (mm) External radius (mm) Preform Bottle mold 125 310 9. / J. The dimensions of the bottle mold and the preform are given in Table 3. Vo is the velocity of the stretch rod which is applied as long as the preform contacts the bott o m of the mold. Schmidt et al. The process parameters are given in Table 4. Geometry of the bottle mold and initial preform mesh.3 13.275 44. 64 (1996) 19-42 37 Fig. 15 shows the geometry of the bottle m o l d and the initial mesh of the preform.025 44. We study now a stretch/blow molding operation. tps + ts].F.3 . tps]. 15. The rheological parameters are the same as those given in Table 2. 0 100 ii 100 (s) H = 819..0000E+00 (s) H .3 t J I ! (ram) (mm) gt 1 1 .3301 =l. 900... 0 No: 0 100 100 No: 45 t = ...2 (s) H = 768..|.. .J=....im.0 No: 30 t = .0 No: 15 t.~ H-871.7197E-01 (s) i I i .2018 IH.= . 0 (ram) (ram) %. 0 100 No: 69 t = .l.3991 Fig. H = 736.3glg (s) (s) (ram) I | (ram) 4 M 100 Ii: 0 . 0 No: 55 t = .9 (s) (mm) (ram) .4 (s) H = 727.~ I 4~ < 2- c~ . Intermediate bottle shapes.2 !©.4094 ...ll.100 .i Jl.i. 100 No: 75 t = ..4302 0 No: 60 t = .J=. 16. H = 743..2 H = 740. - 500 400 et 300 I .J:... Non-Newtonian Fluid Mech. 64 (1996) 19-42 Table 4 Rheological parameters for the stretch/blow molding process Vo (m s-l) Pp~ (Pa) tps (s) P~ (Pa) t~ (s) 0. 17. Stretching force vs. mm/s .3 t(s) 0. time are plotted in Fig...\ .. The comparison between the computed thickness and the experimental data is shown in Fig. lJ 0. 5.2 20 x 105 0.4 8 x 105 0.... . reaches a maximum and then decreases. The measured and computed stretching forces on the stretch rod vs.8 Fig.... 18.4 m s-~. effects of the plug velocity.'V 0~" 0 I I I I ! 0.:? . 17). A Newtonian analysis would lead to a continuously decreasing stretching force.. We note that even if the calculated and experimental results of stretching differ (Fig../ '.~ I P 200 J" " // j7 ""' ~ ". \ '.4 0..: 100 !2" • \ /ii. mrrVs . Comparisons with experimental measurements have been done.. Conclusion Successful numerical simulations of the stretch/blow as well as blow molding processes have 600 I i Computed Data Computed Data - Vo Vo Vo Vo = = 400 400 200 200 mrrVs mrn/s -~--." ?'. 0... We note that there is a qualitative agreement between computation and measurement: the stretching force starts from zero (or from a very low value). / J.2 0.. •'" iss] .40 F.M.2 and 0.... the agreement between the computed thickness and the experimental data is fair.. Schmidt et al. i t .. 17 for two velocities of the stretch rod..1 0.6 .. \ '...5 Fig. * . time.°* / s ... 16 presents intermediate bottle shapes from the beginning of the process to the end. 64 (1996) 19-42 3.M. Schmidt et al. / J.Vo . However. The volumic mechanical computations using the finite element method have allowed us to predict the thickness distribution. Thickness distribution at the end of the process. . Fig. . Acknowledgments This research was supported by the Sidel Company and the French Minist~re de la recherche (MRT no. this raises the problem of the identification of the constitutive equation parameters for PET at high strain rates and evolving temperature. been performed using viscoelastic constitutive equations. a coupled thermomechanical formulation should be developed in order to account for the temperature gradients that affect the preform during the process. taking into account the transient heat transfer in the preform [21]. In the future. 2.5 A E 2 E v 1. 90A 136).Vo = 200 mrn/s • Computed . . which still remains an open issue. A preliminary development has been carried out in that sense. 18. .F.200 mrrVs .5 " 0 0 I 5O I 100 [ I 150 200 Longitudinal coordinate ( mm ) I I 250 300 350 [.5 i I- f ! Il 0.5 J 41 i Data . Non-Newtonian Fluid Mech. the contact kinetic and the stress distribution. and more generally. the problem of coupling between microstructural evolution and the thermomechanical history. . [2] H. [7] A. Finite element simulation of thermoforming and blow molding. Warby and J. Mech. Comput. 1991. J. Methods Appl. H. Numerical modelling of glass forming processes. Debbaut. Ecole des Mines de Paris. Schmidt et al. J.J. Mech. theoretical and applied rheology. B.. [8] B. [19] R. [14] F. Comput. Chung. September 1992. J. Simulation of viscoelastic flow. Non-Newtonian Fluid Mech. Modeling of thermoforming and blow molding. [9] F. De Lorenzi and H.. Polym.). 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