Research Article Finite Element Analysis of PMMA Stretch Blow Moldingdownloads.hindawi.com/archive/2014/175743.pdf · 2019-07-31 · InternationalJournalof ManufacturingEngineering

Research ArticleFinite Element Analysis of PMMA Stretch Blow Molding

Afef Bougharriou Mohieddine Jeridi Mohamed Hdiji Anoir Boughrira and Kacem Sa

UGPMM National Engineering School of Sfax Route Soukra Km 25 BP 1173 Sfax Tunisia

Correspondence should be addressed to Kacem Saı kacemsaiyahoofr

Received 30 May 2014 Revised 14 August 2014 Accepted 15 August 2014 Published 27 August 2014

Academic Editor Hailiang Yu

Copyright copy 2014 Afef Bougharriou et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The electric bubbles are a useful product made of PMMA material They are produced by the stretch blow molding processThickness which reflects the quality of the electric bubble is a crucial parameter that deserves special attention for the moldingprocess In this work finite element simulations of the stretch blow molding process are performed aiming at the determination ofthe preform geometry to ensure homogeneous thickness of the finished product The geometrical parameters of the preform areoptimized allowing a better homogeneity thickness compared to existing dataThe predicted parameters allow the improvement ofthe thickness distribution The standard deviation of the thickness is reduced to about 95 compared to the existing bubble

1 Introduction

Stretch blow molding is one of the most important man-ufacturing processes of polymer via injection process [1ndash3]or extrusion process [4 5] to produce hollow plastic partsThe majority of clear bottles are produced using injectionstretch blow molding process Accordingly several workswere devoted to the simulation of the stretch blow moldingin order to optimise the process and improve the qualityof these containers Chung [6] proposed a finite element(FE) model for injection stretch blow molding of PET bottlesusing the software ABAQUS These simulations focused onthe optimisation of the side wall thickness of the bottle byadjusting the variations of histories of plungermovement andgas pressure Cosson et al [7] used the so-called constrainednatural elementsmethod to simulate and optimize the stretchblowing of a PET preform by assuming an axisymmetricaland viscoplastic preform model under an internal pressureThe thickness of the bottle is linked to the variation ofthe stretch rod speed and time where the blow pressure isapplied Cosson et al [8] simulated the stretch blow moldingprocess of PET bottles at the usual process temperature Thesimulation results modeled the crystalline microstructureevolution and predicted the elastic behaviour of the finalbottle It also provides Youngrsquos modulus distributions inthe bottles Tan et al [9] developed a 2D isothermal FEsimulation of the injection stretch blow molding process for

PET containers To inflate the PET preform two approacheswere used in the simulation (i) a direct pressure input and(ii) a constant mass flow rate of air The results show that thesimulation with the fluid flow method gives good predictionof the volume versus time curve and the preform shapeevolution Tan et al [9] concluded that applying the pressurevia a mass flow rate of air is the most appropriate method tomodel the pressure inside the preform Salomeia et al [10]simulated the stretch blow molding process for PET bottleby the means of thermodynamic model implemented in anISBM The simulations provide the evolution of the pressureand the temperature profiles on both inside and outsidethe surfaces Yang et al [11] proposed a FE model for theinjection stretch blowmolding process of PET bottles aimingat optimizing the process conditions (ie the bottle wallthickness the distributions of the material the temperatureand the strain) Gupta et al [12] simulated the blow moldingprocess for PET bottle in order to study the thickness andthe stress distribution Zimmer and Stommel [13] presentedFE simulations for stretch blow molding with free blowtrials of a PET bottle The model provides the temperaturedistribution and the stress-strain fields The poly(methylmethacrylate) (PMMA) material of this study is an eco-nomical versatile general purpose material Various types ofPMMA presenting high mechanical properties are used indifferent fields and applications Numerous studies weredevoted to the modeling of the mechanical behavior of this

Hindawi Publishing CorporationInternational Journal of Manufacturing EngineeringVolume 2014 Article ID 175743 6 pageshttpdxdoiorg1011552014175743

2 International Journal of Manufacturing Engineering

R

r

H

H998400

De

K

120572

2∘

2R2

2R1

(a) Geometrical parameterization (b) Typical mesh (zoom)

Figure 1 Geometry of the preform

material see for instance [14ndash17] The FE studies related tothe simulations of stretch blow molding process for PMMAmaterial are scarce Dong et al [17] presented a FE simulationof thermoforming process of rectangular bubble inflationThe obtained critical material parameters of PMMA havebeen used in the simulation of free inflation of a bubbleprofileThe light bubbles are highly demanded in theTunisianmarket The Tunisian market imports these bubbles or themanufactured molds from abroad IMR a Tunisian companyspecialized in design andmanufacture of injection tools aimsat producing these bubbles at lower cost and good qualityIMR needs to optimize the preform geometry leading toquasihomogeneous distribution of the thickness along thebubbles In this work FE simulations of the stretch blowmolding process are performed aimed at the determinationof the preform geometry to ensure homogeneous thickness ofthe finished product The paper is organized in the followingmanner in Section 2 the preform geometry (ie mesh)the boundaries conditions the loading and the materialbehavior are detailed Section 3 is devoted to the discussionof the FE results in terms of stress and bubble thicknessdistributions Finally recommendations are given on thegeometric design of the preform

2 Finite Element Modeling

The simulations are performed using the FE softwareABAQUSThematerial behavior may affect in a considerableway the obtained results To this end the FE model of

Table 1 Number of elements in the FE model

120572119890 36 38 4 4221 42211872 40051776 38251696 3609160023 mdash 44011952 41491840 3879172025 mdash mdash mdash 43291920

the molding process is investigated taking into account thehyperelastic behavior of the PMMA at 150∘C by the means ofMooney-Rivlin model

21 Geometry and Meshing The design parameters of thepreform are shown in Figure 1(a) The geometry is controlledby the three independent parameters 119890 120572 and 119867 Assumethat the deformation of the preform occuring under constantvolume (119881 = 119881

0) reduces the number of parameters from 3 to

2 as will be shown below The FE analysis is performed usingsix-noded axisymmetrical elements with a full integrationschemeThe preform is designed according to the pairs (120572 119890)Table 1Themesh size is chosen in such a way that subsequentrefinement does not affect the stress-strain behavior much Atypical FE mesh is shown in Figure 1(b)

The volume of the preform is calculated as the sum of asemisphere and a conical cylinder as shown in Figure 1 Aftersome algebraic manipulations the expression of the volumeof the preform is given by the three terms 119860 119861 and 119878

119881 = 119860 + 119878 minus 119861 (1)

International Journal of Manufacturing Engineering 3

with

119860 =

120587

3

1198671015840

(1198772

1

+ 1198772

+ 1198771119877)

119861 =

120587

3

1198671015840

(1198772

2

+ 1199032

+ 1198772119903)

119878 =

4120587

6

(1198773

minus 1199033

)

(2)

119860 119861 and 119878 depend on the geometrical characteristics of thepreform

119877 = 119863 minus (1198671015840

minus 119870) tan120572

119903 = (119863 minus 119890) minus (1198671015840

minus 119870) tan 2

1198771= (119863 + 119870 tan120572)

1198772= (119863 minus 119890 + 119870 tan 2)

119867 = 119877 + 1198671015840

(3)

It is assumed that the deformation of the preform occursunder constant volume during the mold flow process (119881 =

1198810) This leads to a third degree equation in 119890

119881 = 1198810997904rArr 1198903

+ 1198861198902

+ 119887119890 + 119888 = 0 (4)

where

119886 = minus3 (119863 minus (1198671015840

minus 119870) tan 2) minus 3

2

1198671015840

119887 = 3(119863 minus (1198671015840

minus 119870) tan 2)2

+

3

2

1198671015840

(2119863 + (2119870 minus 1198671015840

) tan 2)

119888 =

3

2120587

(119860 minus 1198810) +

1

120587

(1198773

minus (119863 minus (1198671015840

minus 119870) tan 2)3

)

minus

1

2

((119863 + 119870 tan 2)2 + (119863 minus (1198671015840

minus 119870) tan 2)2

+ (119863 + 119870 tan 2) (119863 minus (1198671015840

minus 119870) tan 2) )

(5)

Analytical solution can easily be obtained for the third degree(4) Two cases are distinguished depending on the value ofthe determinant Δ = 119902

2

+ 41199013

27 (119902 = 21198863

27 minus 1198861198873 + 119888 and119901 = 119887minus119886

2

3) For the studied problem Δ is negative so that 3real zeros verify (4) Each solution 119896 (119896 = 0 119896 = 1 and 119896 = 3)is given by the means of polar coordinates

119890 = 211990313 cos(120579

3

+

2119896120587

3

) (6)

with 119903 = 05radic1199022

minus Δ and 120579 = arctan(minusradicminusΔ119902) Only onesolution lies on the allowed range The diagram in Figure 2can be divided into regions The divisions between theseregions are bounded by the curve with the equation 119867 =

119867max The feasible region is above this curve whereas theinfeasible region below the curve leads to preform geometryin which the height119867 is greater than the height of the mold

36

38

4

42

21 22 23 24 25

120572

e (mm)

H = Hmax

Figure 2 Limit curve

22 Boundary Conditions and Loading The following bound-ary conditions are considered for the FE analysis

(i) The ldquocontainerrdquo is fully constrained and the preformcould move only along the axial direction

(ii) A displacement along the 119911-axis (Figure 3) to thebottom of the mold is applied first to the preformThis displacement is preconized by the manufacturerAccordingly the preform moves down to be in con-tact with the mold

(iii) A pressure of 03MPa was applied in the second stepto the inside of the preform with a gradual increase

(iv) A friction law corresponding to Coulomb law witha constant friction coefficient 120583 = 01 is consideredbetween the preform and the ldquocontainerrdquo It is worthnoting that some simulations (see Section 3) in which120583 was taken in the range of 01 to 05 have shown thatthe resultswere not affected by the value of the frictioncoefficient

23 Material Behavior Law To describe the mechanicalbehavior of the PMMA hyperelastic and viscoelastic modelsare the two commonly used approaches The emphasis ofcurrent study is a phenomenological hyperelastic approachdue to its better suitability for the PMMA which is isotropichighly elastic and incompressible for the forming conditions[17] It is already established that the Mooney-Rivlin modelis accurate in simulating polymer sheet forming see forinstance [17 18] It should be noted that this approach failsto represent the speed influence The material behavior isdescribed with a specific free energy density 119882 which is afunction of strain and temperature For isotropic behavior119882 depends on the invariants of the transformation tensor 119865which is given by

119889119909 = 119865119889119883

119861 = 119865119879

119865

(7)

where119889119883 is the elementary vector in the initial configuration119889119909 is the transformation of 119889119883 in the actual configuration


Z

RP

RP RT

Figure 3 Loading

119865 is the transformation gradient tensor and 119861 is the Cauchy-Green Gauche tensor The stresses can be obtained as thepartial derivatives of the strain energy density function withrespect to invariants 119868

1 1198682 and 119868

3 For isotropic materials

119882 is a linear combination of the invariants 1198681 1198682 and 119868

3

of the strain tensor 119882 = 119882(1198681 1198682 1198683) where 119868

1= trace119861

1198682= (12)((trace1198612) + (trace119861)2) and 119868

3= det(119861) These

invariants can be expressed with the elongations (120582119894 119894 = 1 2

and 3) as follows

1198681= 1205822

1

+ 1205822

2

+ 1205822

3

1198682= 1205822

1

1205822

2

+ 1205822

1

1205822

3

+ 1205822

2

1205822

3

1198683= 1205822

1

1205822

2

1205822

3

(8)

0

05

1

15

2

1 15 2 25 3

ExperimentSimulation

120582

120590(M

Pa)

Figure 4 Stress strain curve

The principal stretch ratio 120582 is the ratio of the gauge lengthsafter and before the test (120582 = 119871119871

0) The hyperelastic

theory (Mooney-Rivlin model) is adopted in this study forthe simulation of the nonlinear material behavior The strainenergy density 119882 is expressed by Mooney-Rivlin model asfollows

119882 = 11986210(1198681minus 3) + 119862

20(1198681minus 3) (9)

where 11986210and 119862

20are two material parameters For the case

of incompressible material under uniaxial elongation (1205821=

120582 1205822= 1205823= 1radic120582) the true Cauchy stress differences can

be calculated as

12059011minus 12059033

= 211986210(1205822

minus

1

120582

) + 211986220(120582 minus

1

1205822

)

12059022minus 12059033

= 0

(10)

In the case of simple tension 12059022

= 12059033

= 0 the uniaxialbehavior law writes

12059011

= (211986210+

211986220

120582

)(1205822

minus

1

120582

) (11)

TheMooney-Rivlin parameters of PMMAmaterial are takenfrom the literature One element test subjected to monotonictensile loading is carried out in order to check the accuracyof the material parameters in simulating the experimentalstresselongation curve (Figure 4) It can be seen that theexperimental data are well described by the Mooney-Rivlinmodel for the parameters values of 119862

10= minus00303MPa and

11986220

= 04503MPa at the temperature of 150∘C

3 Results and Discussions

In this section the FE results are discussed in terms ofthickness distribution Since the friction behaviour neededfor the simulation of the stretch blow molding process isunknown parametric study related to the effect of frictioncoefficient between themold and the preform is preliminarily


1

12

14

16

18

2

0 50 100 150 200 250 300 350

f = 05

f = 01

f = 0

s (mm)

e(m

m)

Figure 5 Effect of the friction coefficient

conductedThe prediction using three different friction coef-ficients namely 00 (frictionless) 01 and 05 is investigatedFigure 5 shows that the variation of the friction coefficientdoes not affect the bubble thickness Further simulationswill be carried out with friction coefficients of 01 Figure 6shows the distribution of the final product thickness versusthe ordinate-abscissa for several geometries of the preformEach geometry is characterized by a given pair (120572 119890) (givenin Table 1) The thicknesses are in agreement with the resultsobtained by Cosson et al [7] and Yang et al [11] Thesimulation results are superimposed to the measured data ofan existing bubble Figure 6 shows how the scattering param-eters (119890 and 120572) can greatly improve the thickness uniformityThe standard deviation of the thickness is reduced to about95 compared to the existing bubble For a better evaluationof the effect of the geometrical parameters 119890 and 120572 of thepreform on the final results a contour plot of the standarddeviation of the thickness is shown in Figure 7 Classicallyan appropriate cost function is mathematically formulatedin which the objective function depends on the standarddeviation of the computed thicknesses In our work thenumber of driving parameters is reduced to 2 (ie 119890 and 120572)It is then possible thanks to the contour plot of the standarddeviation to optimize manually the thickness distribution Itcan be seen that the thickness is widely spread for high valuesof 119890 and low values of 120572 Low standard deviation showingthat the thickness is clustered closely around a mean value isnoticed for high values of 119890 and low values of 120572 The increaseof the preform thickness improves the thickness distributionof the bubble and decreases the height Accordingly thestretching increases and improves the mechanical propertiesof material of the finished product as demonstrated byCosson et al [8] The ldquooptimalrdquo distribution of the thicknessshould be obtained for 120572 ≃ 21

∘C and 119890 ≃ 42mm It givesa light bubble with a uniform distribution of the thickness inaddition to a better mechanical quality [7 8]

05

1

15

2

25

3

0 50 100 150 200 250 300 350

Stan

dard

dev

iatio

nof

the t

hick

ness

Ordinate-abscissa (mm)

Measured profilee = 25 120572 = 22

e = 38 120572 = 23

e = 42 120572 = 21

e = 36 120572 = 21

e = 40 120572 = 21e = 42 120572 = 23

e = 38 120572 = 21

e = 40 120572 = 23

Figure 6 Bubble thickness evolution ordinate-abscissa

42

41

4

39

38

37

3621 215 22 225 23 235 24 245 25

01

015

02

025

03

035

Figure 7 Contour plot of the standard deviation of the thickness (120572versus 119890)

4 Conclusion

In this study a finite element simulation is carried out to sim-ulate the stretch blow molding The geometrical parametersof the preform leading to a homogeneous thickness distri-bution of the electric bubbles are predicted The simulationresults show an improvement compared to an existing bubblein terms of thickness distribution andmechanical propertiesSeveral possible extensions can be performed in the nearfuture

(i) the determination of the blow injection mold geome-try allowing the realization of the electric bubbles

(ii) the choice of the mold cavity disposition the alimen-tation system and the cooling mechanism

(iii) the simulation of the process using the designedmoldto optimise the implementation parameters such ascooling time filling time and closing force


Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] A Bendada F Erchiqui and A Kipping ldquoUnderstanding heattransfer mechanisms during the cooling phase of blowmoldingusing infrared thermographyrdquoNDT and E International vol 38no 6 pp 433ndash441 2005

[2] F Daver and B Demirel ldquoA simulation study of the effectof preform cooling time in injection stretch blow moldingrdquoJournal of Materials Processing Technology vol 212 no 11 pp2400ndash2405 2012

[3] G H Menary C G Armstrong R J Crawford and J PMcEvoy ldquoModelling of poly(ethylene terephthalate) in injectionstretch-blow mouldingrdquo Plastics Rubber and Composites Pro-cessing and Applications vol 29 no 7 pp 360ndash370 2000

[4] K Peplinski and A Mozer ldquoAnsys-polyflow software use toselect the parison diameter and its thicknes distribution inblowing extrusionrdquo Journal of Polish CIMAC 2010

[5] K Peplinski and A Mozer ldquoComparison of bottle wall thick-ness distribution obtain in real manufacturing conditions andin ansys polyflow simulation environmentrdquo Journal of PolishCIMAC vol 7 pp 231ndash235 2012

[6] K Chung ldquoFinite element simulation of pet stretchblowmold-ing processrdquo Journal of Materials Shaping Technology vol 7 no4 pp 229ndash239 1989

[7] B Cosson L Chevalier and J Yvonnet ldquoOptimization by theC-NEM method of the stretch-blow molding process of a petbottle near Tgrdquo International Journal of Material Forming vol1 no 1 pp 707ndash710 2008

[8] B Cosson L Chevalier and G Regnier ldquoSimulation of thestretch blow moulding process from the modelling of themicrostructure evolution to the end-use elastic properties ofpolyethylene terephthalate bottlesrdquo International Journal ofMaterial Forming vol 5 no 1 pp 39ndash53 2012

[9] C W Tan G H Menary Y Salomeia et al ldquoModelling ofthe injection stretch blow moulding of PET containers viaa Pressure-Volume-time (PV-t) thermodynamic relationshiprdquoInternational Journal of Material Forming vol 1 no 1 pp 799ndash802 2008

[10] Y M Salomeia G H Menary and C G Armstrong ldquoInstru-mentation andmodelling of the stretch blowmoulding processrdquoInternational Journal of Material Forming vol 3 no 1 pp 591ndash594 2010

[11] Z J Yang EHarkin-Jones GHMenary andCG ArmstrongldquoA non-isothermal finite element model for injection stretch-blow molding of PET bottles with parametric studiesrdquo PolymerEngineering and Science vol 44 no 7 pp 1379ndash1390 2004

[12] S Gupta V Uday A S Raghuwanshi S Chowkshey S NDas and S Suresh ldquoSimulation of blow molding using ansyspolyflowrdquo APCBEE Procedia vol 5 pp 468ndash473 2013

[13] J Zimmer and M Stommel ldquoMethod for the evaluation ofstretch blow molding simulations with free blow trialsrdquo IOPConference Series Materials Science and Engineering vol 48Article ID 012004 2013

[14] N Saad-Gouider R Estevez C Olagnon and R Seguela ldquoCal-ibration of a viscoplastic cohesive zone for crazing in PMMArdquoEngineering Fracture Mechanics vol 73 no 16 pp 2503ndash25222006

[15] Z H Stachurski ldquoStrength and deformation of rigid poly-mersthe stress-strain curve in amorphous PMMArdquo Polymervol 44 no 19 pp 6067ndash6076 2003

[16] M Nasraoui P Forquin L Siad and A Rusinek ldquoInfluence ofstrain rate temperature and adiabatic heating on the mechan-ical behaviour of poly-methyl-methacrylate experimental andmodelling analysesrdquoMaterials and Design vol 37 pp 500ndash5092012

[17] Y Dong R J T Lin and D Bhattacharyya ldquoDetermination ofcritical material parameters for numerical simulation of acrylicsheet formingrdquo Journal of Materials Science vol 40 no 2 pp399ndash410 2005

[18] G Marckmann E Verron and B Peseux ldquoFinite elementanalysis of blow molding and thermoforming using a dynamicexplicit procedurerdquoPolymer Engineering and Science vol 41 no3 pp 426ndash439 2001

International Journal of

AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of



RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal ofEngineeringVolume 2014

Submit your manuscripts athttpwwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



R

r

H

H998400

De

K

120572

2∘

2R2

2R1

(a) Geometrical parameterization (b) Typical mesh (zoom)

Figure 1 Geometry of the preform

material see for instance [14ndash17] The FE studies related tothe simulations of stretch blow molding process for PMMAmaterial are scarce Dong et al [17] presented a FE simulationof thermoforming process of rectangular bubble inflationThe obtained critical material parameters of PMMA havebeen used in the simulation of free inflation of a bubbleprofileThe light bubbles are highly demanded in theTunisianmarket The Tunisian market imports these bubbles or themanufactured molds from abroad IMR a Tunisian companyspecialized in design andmanufacture of injection tools aimsat producing these bubbles at lower cost and good qualityIMR needs to optimize the preform geometry leading toquasihomogeneous distribution of the thickness along thebubbles In this work FE simulations of the stretch blowmolding process are performed aimed at the determinationof the preform geometry to ensure homogeneous thickness ofthe finished product The paper is organized in the followingmanner in Section 2 the preform geometry (ie mesh)the boundaries conditions the loading and the materialbehavior are detailed Section 3 is devoted to the discussionof the FE results in terms of stress and bubble thicknessdistributions Finally recommendations are given on thegeometric design of the preform

2 Finite Element Modeling

The simulations are performed using the FE softwareABAQUSThematerial behavior may affect in a considerableway the obtained results To this end the FE model of

Table 1 Number of elements in the FE model

120572119890 36 38 4 4221 42211872 40051776 38251696 3609160023 mdash 44011952 41491840 3879172025 mdash mdash mdash 43291920

the molding process is investigated taking into account thehyperelastic behavior of the PMMA at 150∘C by the means ofMooney-Rivlin model

21 Geometry and Meshing The design parameters of thepreform are shown in Figure 1(a) The geometry is controlledby the three independent parameters 119890 120572 and 119867 Assumethat the deformation of the preform occuring under constantvolume (119881 = 119881

0) reduces the number of parameters from 3 to

2 as will be shown below The FE analysis is performed usingsix-noded axisymmetrical elements with a full integrationschemeThe preform is designed according to the pairs (120572 119890)Table 1Themesh size is chosen in such a way that subsequentrefinement does not affect the stress-strain behavior much Atypical FE mesh is shown in Figure 1(b)

The volume of the preform is calculated as the sum of asemisphere and a conical cylinder as shown in Figure 1 Aftersome algebraic manipulations the expression of the volumeof the preform is given by the three terms 119860 119861 and 119878

119881 = 119860 + 119878 minus 119861 (1)


with

119860 =

120587

3

1198671015840

(1198772

1

+ 1198772

+ 1198771119877)

119861 =

120587

3

1198671015840

(1198772

2

+ 1199032

+ 1198772119903)

119878 =

4120587

6

(1198773

minus 1199033

)

(2)


119877 = 119863 minus (1198671015840

minus 119870) tan120572

119903 = (119863 minus 119890) minus (1198671015840

minus 119870) tan 2

1198771= (119863 + 119870 tan120572)

1198772= (119863 minus 119890 + 119870 tan 2)

119867 = 119877 + 1198671015840

(3)



119881 = 1198810997904rArr 1198903

+ 1198861198902

+ 119887119890 + 119888 = 0 (4)

where

119886 = minus3 (119863 minus (1198671015840


2

1198671015840

119887 = 3(119863 minus (1198671015840

minus 119870) tan 2)2

+

3

2

1198671015840

(2119863 + (2119870 minus 1198671015840

) tan 2)

119888 =

3

2120587

(119860 minus 1198810) +

1

120587

(1198773

minus (119863 minus (1198671015840

minus 119870) tan 2)3

)

minus

1

2

((119863 + 119870 tan 2)2 + (119863 minus (1198671015840

minus 119870) tan 2)2

+ (119863 + 119870 tan 2) (119863 minus (1198671015840

minus 119870) tan 2) )

(5)


2

+ 41199013

27 (119902 = 21198863

27 minus 1198861198873 + 119888 and119901 = 119887minus119886

2


119890 = 211990313 cos(120579

3

+

2119896120587

3

) (6)

with 119903 = 05radic1199022



36

38

4

42

21 22 23 24 25

120572

e (mm)

H = Hmax








119889119909 = 119865119889119883

119861 = 119865119879

119865

(7)



Z

RP

RP RT

Figure 3 Loading


1 1198682 and 119868



3


1= trace119861

1198682= (12)((trace1198612) + (trace119861)2) and 119868



and 3) as follows

1198681= 1205822

1

+ 1205822

2

+ 1205822

3

1198682= 1205822

1

1205822

2

+ 1205822

1

1205822

3

+ 1205822

2

1205822

3

1198683= 1205822

1

1205822

2

1205822

3

(8)

0

05

1

15

2

1 15 2 25 3


120582

120590(M

Pa)



0) The hyperelastic


119882 = 11986210(1198681minus 3) + 119862

20(1198681minus 3) (9)

where 11986210and 119862




be calculated as

12059011minus 12059033

= 211986210(1205822

minus

1

120582

) + 211986220(120582 minus

1

1205822

)

12059022minus 12059033

= 0

(10)


= 12059033


12059011

= (211986210+

211986220

120582

)(1205822

minus

1

120582

) (11)



11986220





1

12

14

16

18

2

0 50 100 150 200 250 300 350

f = 05

f = 01

f = 0

s (mm)

e(m

m)




05

1

15

2

25

3

0 50 100 150 200 250 300 350

Stan

dard

dev

iatio

nof

the t

hick

ness



e = 38 120572 = 23

e = 42 120572 = 21

e = 36 120572 = 21

e = 40 120572 = 21e = 42 120572 = 23

e = 38 120572 = 21

e = 40 120572 = 23


42

41

4

39

38

37

3621 215 22 225 23 235 24 245 25

01

015

02

025

03

035


4 Conclusion








References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










with

119860 =

120587

3

1198671015840

(1198772

1

+ 1198772

+ 1198771119877)

119861 =

120587

3

1198671015840

(1198772

2

+ 1199032

+ 1198772119903)

119878 =

4120587

6

(1198773

minus 1199033

)

(2)


119877 = 119863 minus (1198671015840

minus 119870) tan120572

119903 = (119863 minus 119890) minus (1198671015840

minus 119870) tan 2

1198771= (119863 + 119870 tan120572)

1198772= (119863 minus 119890 + 119870 tan 2)

119867 = 119877 + 1198671015840

(3)



119881 = 1198810997904rArr 1198903

+ 1198861198902

+ 119887119890 + 119888 = 0 (4)

where

119886 = minus3 (119863 minus (1198671015840


2

1198671015840

119887 = 3(119863 minus (1198671015840

minus 119870) tan 2)2

+

3

2

1198671015840

(2119863 + (2119870 minus 1198671015840

) tan 2)

119888 =

3

2120587

(119860 minus 1198810) +

1

120587

(1198773

minus (119863 minus (1198671015840

minus 119870) tan 2)3

)

minus

1

2

((119863 + 119870 tan 2)2 + (119863 minus (1198671015840

minus 119870) tan 2)2

+ (119863 + 119870 tan 2) (119863 minus (1198671015840

minus 119870) tan 2) )

(5)


2

+ 41199013

27 (119902 = 21198863

27 minus 1198861198873 + 119888 and119901 = 119887minus119886

2


119890 = 211990313 cos(120579

3

+

2119896120587

3

) (6)

with 119903 = 05radic1199022



36

38

4

42

21 22 23 24 25

120572

e (mm)

H = Hmax








119889119909 = 119865119889119883

119861 = 119865119879

119865

(7)



Z

RP

RP RT

Figure 3 Loading


1 1198682 and 119868



3


1= trace119861

1198682= (12)((trace1198612) + (trace119861)2) and 119868



and 3) as follows

1198681= 1205822

1

+ 1205822

2

+ 1205822

3

1198682= 1205822

1

1205822

2

+ 1205822

1

1205822

3

+ 1205822

2

1205822

3

1198683= 1205822

1

1205822

2

1205822

3

(8)

0

05

1

15

2

1 15 2 25 3


120582

120590(M

Pa)



0) The hyperelastic


119882 = 11986210(1198681minus 3) + 119862

20(1198681minus 3) (9)

where 11986210and 119862




be calculated as

12059011minus 12059033

= 211986210(1205822

minus

1

120582

) + 211986220(120582 minus

1

1205822

)

12059022minus 12059033

= 0

(10)


= 12059033


12059011

= (211986210+

211986220

120582

)(1205822

minus

1

120582

) (11)



11986220





1

12

14

16

18

2

0 50 100 150 200 250 300 350

f = 05

f = 01

f = 0

s (mm)

e(m

m)




05

1

15

2

25

3

0 50 100 150 200 250 300 350

Stan

dard

dev

iatio

nof

the t

hick

ness



e = 38 120572 = 23

e = 42 120572 = 21

e = 36 120572 = 21

e = 40 120572 = 21e = 42 120572 = 23

e = 38 120572 = 21

e = 40 120572 = 23


42

41

4

39

38

37

3621 215 22 225 23 235 24 245 25

01

015

02

025

03

035


4 Conclusion








References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Z

RP

RP RT

Figure 3 Loading


1 1198682 and 119868



3


1= trace119861

1198682= (12)((trace1198612) + (trace119861)2) and 119868



and 3) as follows

1198681= 1205822

1

+ 1205822

2

+ 1205822

3

1198682= 1205822

1

1205822

2

+ 1205822

1

1205822

3

+ 1205822

2

1205822

3

1198683= 1205822

1

1205822

2

1205822

3

(8)

0

05

1

15

2

1 15 2 25 3


120582

120590(M

Pa)



0) The hyperelastic


119882 = 11986210(1198681minus 3) + 119862

20(1198681minus 3) (9)

where 11986210and 119862




be calculated as

12059011minus 12059033

= 211986210(1205822

minus

1

120582

) + 211986220(120582 minus

1

1205822

)

12059022minus 12059033

= 0

(10)


= 12059033


12059011

= (211986210+

211986220

120582

)(1205822

minus

1

120582

) (11)



11986220





1

12

14

16

18

2

0 50 100 150 200 250 300 350

f = 05

f = 01

f = 0

s (mm)

e(m

m)




05

1

15

2

25

3

0 50 100 150 200 250 300 350

Stan

dard

dev

iatio

nof

the t

hick

ness



e = 38 120572 = 23

e = 42 120572 = 21

e = 36 120572 = 21

e = 40 120572 = 21e = 42 120572 = 23

e = 38 120572 = 21

e = 40 120572 = 23


42

41

4

39

38

37

3621 215 22 225 23 235 24 245 25

01

015

02

025

03

035


4 Conclusion








References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










1

12

14

16

18

2

0 50 100 150 200 250 300 350

f = 05

f = 01

f = 0

s (mm)

e(m

m)




05

1

15

2

25

3

0 50 100 150 200 250 300 350

Stan

dard

dev

iatio

nof

the t

hick

ness



e = 38 120572 = 23

e = 42 120572 = 21

e = 36 120572 = 21

e = 40 120572 = 21e = 42 120572 = 23

e = 38 120572 = 21

e = 40 120572 = 23


42

41

4

39

38

37

3621 215 22 225 23 235 24 245 25

01

015

02

025

03

035


4 Conclusion








References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












References





















RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











RoboticsJournal of





Journal of



RotatingMachinery





VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Documents

Research Article Finite Element Analysis of PMMA Stretch Blow Moldingdownloads.hindawi.com/archive/2014/175743.pdf · 2019-07-31 · InternationalJournalof ManufacturingEngineering