A Constitutive Model for Large Multiaxial Deformationsof Solid Polypropylene at High Temperature

John Sweeney, Robert Spares, Mike WoodheadSchool of Engineering, Design, and Technology/IRC in Polymer Science and Technology, University ofBradford, Bradford BD7 1DP, UK

Experiments on a blow-molding grade of polypropylenehave been performed at 1358C using a biaxial testingmachine. Both simultaneous and sequential equibiaxialtests were performed at strain rates relevant to solidphase processing regimes. A constitutive model hasbeen developed that includes a single Eyring processand two Edwards-Vilgis networks. The effectiveness ofthis model for predicting the observed stress-strainbehavior is explored. Predictions of simultaneousstretching and the first stretch in sequential experi-ments are excellent. The second stretch in sequentialexperiments is less well predicted, but the model’s per-formance is useful overall. The model is incorporatedinto a commercial finite element code and its practical-ity is demonstrated. POLYM. ENG. SCI., 49:1902–1908,2009. ª 2009 Society of Plastics Engineers

INTRODUCTION

There are a number of important polymer processes in

which products are formed by deformation to large strains

at elevated temperatures below melting point. These proc-

esses include thermoforming, blow molding, and stretch

blow molding. There is also an increasing need for effi-

ciency in polymer processing, both in terms of energy

consumption and material usage. This latter factor has led

to a greater interest in process modeling, which in turn

demands a fundamental understanding of large polymer

deformations under the appropriate conditions. At the

heart of this understanding is the material constitutive

equation, which is the subject of this study.

We present both a constitutive model and its experi-

mental verification. For a complex, large deformation

model, uniaxial testing cannot yield sufficient information

to fully define the material parameters, as demonstrated

by Sweeney et al. [1], and this makes a necessity for

biaxial experiments. Furthermore, when deciding what

experiments are required to verify a constitutive theory,

the program of testing should encompass the combina-

tions of strain along the different axes—the strain paths—

that relate to relevant applications. Given the impossibility

of including all possible strain paths, it makes sense to

define the material parameters using those that are partic-

ularly relevant.

In this article, we present a constitutive model that

includes the essential features of the material behavior:

large deformations, strain rate dependence of stress, and

yielding. This theory is applied to a particular grade of

polypropylene. A program of large strain biaxial testing

provides verification at conditions relevant to processing

regimes. Both simultaneous equibiaxial and sequential

biaxial strain paths are modeled. The model is imple-

mented as a user-defined subroutine in the finite element

package ABAQUS.

MATERIALS AND TESTING

A commercial polypropylene material, designed for

injection stretch blow molding (TOTAL Petrochemicals

PPR 7225), supplied in the form of granules was formed

into sheet using a film line that extrudes through a die

onto a chill roll. According to the manufacturer’s data,

this material has a melt temperature of 1468C and is

specially designed for injection stretch blow molding. The

film line incorporated a Betol BK38 extruder connected

to a Davis Standard lab3 Roll ‘‘Hi-Press’’ 3 roll stack.

The four extruder barrel zones were set at 145, 165, 175,

and 1808C respectively with the clamp, adapter, and die

maintained at 1808C. These temperature settings were

arrived at after consideration of the low melting point of

the material in comparison with typical polypropylenes,

and they are each lower by 158C than that which would

be appropriate according to our experience for these more

conventional materials with melting temperature of

around 1658C. The water cooling temperature for the chill

roll was 138C, the extruder screw speed was set at 75

rpm, and the line speed of the stack set to 35 mm/s. By

operating with a die that has a width of 150 mm and gap

of 0.7 mm, the resulting material was 0.65 mm thick.

Correspondence to: J. Sweeney; e-mail: j.sweeney@bradford.ac.uk

Contract grant sponsor: The European Commission under the Framework

6 Program via the Apt-Pack project.

DOI 10.1002/pen.21426

Published online in Wiley InterScience (www.interscience.wiley.com).

VVC 2009 Society of Plastics Engineers

POLYMER ENGINEERING AND SCIENCE—-2009

Ideally, the product from this process would be iso-

tropic, but a small haul-off force was exerted on the sheet

during processing, inducing some molecular orientation

along the extrusion direction, and this was observable in

the equibiaxial stretching experiments reported below. The

stretching forces along the two experimental axes (corre-

sponding to extrusion and transverse directions) were indis-

tinguishable up to strains of 0.2, but at strains greater than

this became measurably different, with the force along the

extrusion direction exceeding that along the transverse

direction by 12% at strains of 0.5. The results presented

here are for the stress averaged over the two directions.

Square samples of 55 3 55 mm were stretched in

tension using a biaxial testing machine in air at a temper-

ature of 1358C. This corresponds to an appropriate tem-

perature for solid phase processing of the material. The

specimens were held by six pneumatic grips per side, with

force transducers incorporated into two of the grips as

shown in Fig. 1. The biaxial testing machine was manu-

factured originally by T M Long, and as a result of recent

modifications, it is fitted with PC-based displacement,

temperature control, and data capture. Software was writ-

ten using National Instruments Labview 8.0 to interface

with two NI PCI-6221 cards housed in a single PC. One

card is responsible for monitoring the position of the

hydraulic pistons and set the required speed via two

Parker VRD350 hydraulic motion controllers. The loads

are also monitored to prevent overload and damage to the

machine. Draw rates, ratios, and starting time of each pis-

ton can be controlled independently of one another. The

second card monitors the load and position, with the data

captured in block mode to ensure precise time increments

of the data for subsequent analysis.

The modes of stretching were simultaneous equibiaxial

and sequential equibiaxial. The latter comprises two pla-

nar extension (constant width) steps, the first always along

the sheet extrusion direction, followed immediately by a

second perpendicular stretch, to achieve a final equibiaxial

stretch. Tests were performed at two constant speeds, cor-

responding to initial strain rates of 0.37 and 2.3 s21 up to

extension ratios of 4.75 (true strains of 1.56). Loads were

monitored at a sampling rate of 400 Hz. The specimens

were observed to deform uniformly.

The higher value of strain rate is the greatest practicable

rate for the present stretching apparatus. In industrial proc-

essing, strain rates are often much higher. However, even

if the much higher strain rates were achievable experimen-

tally, the problems of anisothermal conditions associated

with adiabatic heating would further complicate the experi-

mental interpretation. We believe that this experimental re-

gime can establish the principles of the constitutive behav-

ior that should be applicable at higher strain rates.

Strains were calculated from the grip displacements dand the gauge length L, the distance between opposite

grips. The extension ratios k are thus given by

l ¼ 1þ dL

(1)

and the true strains e by

e ¼ lnðlÞ (2)

The stresses were calculated by assuming that the total

force on each specimen side was equal to six times the

force measured in one of the six transducers. The validity

of this procedure is explored below by modeling. For the

calculation of true stresses, the material is assumed to be

incompressible, with the true stress s calculated as

s ¼ lF=a (3)

where F is the total force on the specimen side, and a the

initial cross-sectional area.

The results presented are averages of three tests.

THEORY

Time and rate dependence are included in the constitu-

tive model by means of an Eyring process in which the

scalar plastic strain rate ep is given by

ep ¼ A expðVpsÞ sinhðVstÞ (4)

where A is a temperature dependent term that includes

activation enthalpy, and Vp and Vs are material constants

being (at constant temperature) proportional to pressure

and shear activation volumes respectively (following, for

example, Buckley and Jones [2] or Spathis and Kontou

[3]). The process is driven by the mean stress s and the

octahedral shear stress t, to be defined below.

The other major component in the model represents

large elastic deformations. Edwards and Vilgis [4] derivedFIG. 1. Arrangement of grips and transducers around biaxial specimen.

DOI 10.1002/pen POLYMER ENGINEERING AND SCIENCE—-2009 1903

the following expression for the change in strain energy

per unit volume W, as a result of principal extension

rations kI, kII, and kIII

W

kT¼ 1

2Nc

ð1� a2ÞI11� a2I1

þ lnð1� a2I1Þ� �

þ 1

2Ns

Xi

ð1þ ZÞð1� a2Þl2ið1þ Zl2i Þð1� a2I1Þ

þ lnð1þ Zl2i Þ8>>>:

9>>>;"

þ lnð1� a2I1Þ#

ði¼ I; II; IIIÞ ð5Þ

Here, k is Boltzmann’s constant and T, the absolute tem-

perature. Nc and Ns are crosslink and sliplink densities

respectively, and Z is a ‘slipperiness factor’ that charac-

terizes the degree of mobility of the sliplink; Z ‡ 0 where

Z ¼ 0 corresponds to no mobility. The model includes

the phenomenon of finite strain extensibility, which is char-

acterized by the parameter a and corresponds to a

singularity in the strain energy as the denominator 1 2 a2I1approaches zero. I1 is the first strain invariant, equal to

I1 ¼ l2I þ l2II þ l2III (6)

The material is assumed incompressible with lIlIIlIII ¼1. As a result, principal stresses are derived from Eq. 5 by

si ¼ liqWqli

� p ði ¼ I; II; IIIÞ (7)

where p is an unknown hydrostatic pressure. In the present

analysis, we assume plane stress conditions with zero stress

along the III direction, so that on elimination of p the

stresses in the I-II plane are

si ¼ liqWqli

� lIIIqWqlIII

ði ¼ I; IIÞ (8)

The model consists of two ‘arms’ arranged in parallel.

One arm consists of an Edwards-Vilgis network in series

with an Eyring process (the X arm), and the other entirely

of an Edwards-Vilgis network (the Y arm), the general

arrangement being shown in Fig. 2. There are many prece-

dents for this configuration, perhaps the earliest being due

to Haward and Thackray [5]. The formulation is suitable

for implementation as a user-defined ‘UMAT’ subroutine

within the finite element package ABAQUS. The total

strain is the same as the strain in each arm and is defined

by the deformation gradient G. G is split into pure defor-

mation D and rigid body rotation R (via the use of the

Cauchy-Green strain measure) to give

G ¼ DR (9)

In the X arm, the strain is split into elastic and plastic

components, DeX and Dp, associated with stress-strain laws

Eqs. 4, 5, respectively:

G ¼ DeXDpR (10)

The principal values of DeX, leXI, and leXII, define the

stress in the X arm via Eq. 8, with parameters NXc , N

Xs , Z

X

and aX. An incremental approach is employed, with the

current plastic stretch related to the plastic strain Dp0 at the

end of the previous time increment and the increment of

plastic strain DDp, developed during the current increment,

related by

G ¼ DeXDDpDp0R (11)

For a given total deformation, G, DeX, and DDp in

Eq. 11 are derived via an iterative process, involving

Eqs. 4, 5, to impose the condition that the stresses in

the network and the Eyring process are equal. The rela-

tive proportions of the components of DDp are fixed

by the use of a Levy-Mises flow rule, defined in the

relation (Eq. 18) below. The resulting true stress is

then transformed to global directions to give the stress

tensor SX.

The Y arm consists of an Edwards-Vilgis network with

parameters NYc , N

Ys , g

Y and aY. The principal extension

ratios are those of D, and Eq. 8 defines the stress. From

this, we derive the stress tensor in global axes SY. The

total stress S is then given by

� ¼ �X þ�Y (12)

The stress quantities driving the Eyring process of Eq.4 relate to the stress SX. The stress deviator t is given by

t ¼ �X � sI (13)

where the mean stress is given by

s ¼ 1

3tr �X� �

(14)

The scalar octahedral shear stress t used in (Eq. 4) is

then given in terms of the stress deviator tensor s by

t ¼ffiffiffiffiffiffiffiffiffiffiffiffi1

3s : s

r(15)

where the colon denotes the double contraction.

FIG. 2. Schematic arrangement of constitutive model.

1904 POLYMER ENGINEERING AND SCIENCE—-2009 DOI 10.1002/pen

At each time increment, any rigid body rotations are

stripped from the plastic strain tensor using the Cauchy-

Green process and polar decomposition, to give a pure

stretch Dp with zero associated plastic spin. The scalar

strain rate used in (Eq. 4) is derived from the rate of plas-

tic deformation Qp

Qp ¼ DpDp�1 (16)

by the relation

ep ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

3Qp : Qp

r(17)

The Levy-Mises flow rule can then be expressed as

Qp

ep¼ s

t(18)

The above analysis has been programmed as a user-

defined ‘UMAT’ subroutine in the finite element package

ABAQUS. For convenience, the calculations of stress for

the polymer matrix in uniform states of strain, as used in

Figs. 3–6, and are obtained by runs of four-element

square models in ABAQUS.

RESULTS AND MODELING

In Fig. 7, we plot yield stress in the form of octahedral

stress (defined in Eq. 15) against the natural logarithm of

octahedral strain rate (defined in Eq. 17) for both simulta-

neous and sequential stretching. Yield is observed at

strains of 0.1–0.2, and is taken as the stress corresponding

to when the gradient in the stress-strain curve stops

decreasing (this is observable in Figs. 3–6). If we envis-

age the action of the two-arm model, this stress would

correspond to the stress in the X-arm becoming constant

(Fig. 2). At this point, the rate of strain in the Eyring

model matches that of the total applied strain, and the

network in the X-arm ceases to stretch. Conventionally

[6], the rate dependence of the Eyring process is derived

by approximating the hyperbolic sine function of Eq. 4

with an exponential, which then becomes

ep ¼ A expðVpsþ VstÞ (19)

Typically, Vp � Vs and the first term in the argument

of the exponential may be neglected. Then (Eq. 19) can

be rearranged to give

t ¼ 1

Vs

lnepA

8>: 9>; (20)

which implies that the gradient of the plot in Fig. 7 is

equal to 1/Vs. On this basis, the gradient of the line fitted

through all the results in Fig. 7 corresponds to Vs ¼ 6.9

MPa1. The results presented in Fig. 7 suggest that the rate

dependence of the planar extension (sequential biaxial)

results differs from that obtained for the simultaneous

equibiaxial results. However, we shall proceed on the

assumption that there is one average rate dependence for

the two stretching modes in accordance with the theory

presented above, and accept that this may introduce an

approximation.

With the value of Vs determined, the value of the pres-

sure activation parameter Vp is then derived on the basis

that it is a fixed proportion of Vp ¼ 0.06 Vs, in line with

the conclusions of other workers on polymers [7–9]. With

the activation volumes established, the Eyring constant A

FIG. 3. Simultaneous equibiaxial drawing for strain rate 0.37 s21.

FIG. 4. Sequential drawing for strain rate 0.37 s21. Axial strain

increases from 0 to 1.56, and is then constant. Transverse strain is zero

until axial strain reaches 1.56, and then is increased to 1.56.

FIG. 5. Simultaneous equibiaxial drawing for strain rate 2.3 s21.

DOI 10.1002/pen POLYMER ENGINEERING AND SCIENCE—-2009 1905

determines the stress at yield, and is fixed on that basis

with respect to the experimental curves of Figs. 3–6.

There remain two Edwards-Vilgis networks to be

defined. The X-arm network ceases to extend, once the

Eyring process begins to flow, so is subject only to strains

up to around 0.1. The chief aspects of this network model

are not invoked, and therefore, we include only the cross-

link term (the first term involving Nc in Eq. 5) with no

finite strain extensibility, so that we are assuming a Gaus-

sian network. This is defined by the single parameter NXc

which is derived from the initial slopes of the stress-strain

curves. The other network is defined by shape of the

stress-strain curve up to maximum strain. The parameters

for this network are determined by trial and error, using

both kinds of experiment and, in the case of sequential

experiments, both axial and transverse stresses. The set of

parameters for the whole model is given in Table 1.

The quality of the model predictions at the two strain

rates is apparent from Figs. 3–6. For the simultaneous

stretches of Figs. 3 and 5, the initial yielding behavior is

well modeled. At higher strains, there is negligible error

up to a strain of 1.0 at the lower speed in Fig. 3, with the

maximum error around 10% as the maximum strain is

approached. At the higher speed in Fig. 5, the maximum

error is around 7%. For the sequential tests of Figs. 4 and

6, the predictions of the first stretches are of a quality

similar to those of the simultaneous experiments. For the

second stretches, there are significant errors. For the axial

stresses at both testing speeds, there is a sharp drop in the

observed stress after the end of the first stretch that is not

well captured. The rate of stress relaxation observed is

much higher than that associated with the single Eyring

process of the model, the parameters of which are deter-

mined by the observed stress levels and strain rate de-

pendence. We must conclude that there are in reality

other flow processes in operation at shorter time scales

that are not featured in this simple model. The other

major discrepancy is in the underprediction of the trans-

verse stress during the second stretch, which is apparently

an inherent feature of the Edwards-Vilgis network model;

this issue is developed in the discussion below.

A finite element model of the experimental specimen

has been made, using the mesh shown in Fig. 8a. A quar-

ter model is used, stretched simultaneously along its two

perpendicular axes at an overall strain rate of 2.3 s21 to a

state of true strain of 1.4 3 1.4, as shown in Fig. 8b. The

circular unstrained regions near the outer boundaries rep-

resent the grips, which are in the form of cylindrical

pneumatic pistons. We calculate the stress with the

method used to interpret the experiments, using only the

reaction force at the grips nearest the centre of the speci-

men sides (corresponding to two circular regions in Fig.

8a and b, one at lowest right and the other at highest left)

to calculate the total force, and making use of Eqs. 1–3 to

calculate the stress-strain curve. This is compared with

the stress-strain curve at the position corresponding to the

specimen center (the 3 3 3 array of elements at the lower

left corner of the model). The comparison is shown in

Fig. 9. This plot suggests that the simple method of calcu-

lating specimen stress-strain data produces little error.

DISCUSSION

There have been some previous studies of polypropyl-

ene under conditions that, like those used here, are rele-

vant to the solid phase processing regime. Briatico-

Vangosa et al. [10] performed high temperature tests at

rates of 0.5–2 s21 under uniaxial tensile conditions, and

modeled their results using elastic theories. Capt et al.

[11] performed high temperature stretches at strain rates

in the range 0.24–1.43 s21 in simultaneous equibiaxial,

planar, and uniaxial modes. Their observed strain rate de-

pendence of yield stress, reported for simultaneous equi-FIG. 7. Rate dependence of yield stress.

FIG. 6. Sequential drawing for strain rate 2.3 s21. Axial strain

increases from 0 to 1.56, and is then constant. Transverse strain is zero

until axial strain reaches 1.56, and then is increased to 1.56.

TABLE 1. Parameters used for the constitutive model.

NXc /

MPa

NXs /

MPa gX aXNYc /

MPa

NYs /

MPa gY aYVs/

MPa21

Vp/

MPa21

103

A/s21

4.37 0 – 0 0.36 0.085 0.2 0.08 6.9 0.4 8.0

1906 POLYMER ENGINEERING AND SCIENCE—-2009 DOI 10.1002/pen

biaxial stretching at 1508C (148C below the melting

point), which is of a similar magnitude to that reported

here (at 118C below the melting point). There have also

been observations of inhomogeneous multiaxial stretching

of polypropylene, modeled using an elastic theory [1].

This work advances the field in two ways: the inclu-

sion of strain rate dependence into the constitutive model;

and the use of both simultaneous and sequential biaxial

stretching experiments to verify the model. The first issue

is important because the strain rate sensitivity of a mate-

rial determines whether or not strain inhomogeneities,

such as necks, will form. When used in a process model,

it is highly desirable that the constitutive model be capa-

ble of predicting the consequences of this, such as local-

ized wall thinning. The second issue also applies to pro-

cess modeling. Any constitutive equation will provide at

best an approximation to the real material behavior, and

will represent some strain states better than others. It is

therefore desirable to examine its performance for strain

paths that are relevant to processing. Sequential biaxial

drawing is a strain path that is particularly relevant to

stretch blow molding and representative of the complexity

that can be encountered in practice. Examination of a

model’s performance under these conditions provides a

more realistic assessment of its practical value. For the

present model, we may conclude that the quality of its

predictions are good to excellent for the simultaneous

stretching and the first stretch in the sequential stretching

and less good but still useful for the second sequential

stretch.

In these experiments, it is clear that, once a significant

strain has been realized and the Eyring process activated,

the greater part of the stress is due to the network in the

Y arm of the model. On the basis of this model, the dis-

crepancy in the predictions of stress for the second stretch

in the sequential experiments must to a large extent be

accounted for in the response of the Edwards-Vilgis net-

work. In particular, the stress predicted for the transverse

direction during the second stretch is consistently low.

This could reflect some increase in crystallinity or net-

work entanglements during the first stretch.

The Eyring parameter Vs can be interpreted as an acti-

vation volume v, via the relation

v ¼ VskT (21)

where k is a Boltzmann’s constant and T, the absolute

temperature. For the testing temperature 1358C, the cur-

rent value of 6.9 MPa–1 corresponds to v ¼ 39 nm3. This

can be compared with values obtained by others for poly-

propylene, invariably obtained at lower test temperatures.

Working on the oriented polypropylene fibers at room

temperature, Duxbury and Ward [12] identified two acti-

vation volumes, with the larger one in the range 0.5–0.6

nm3. Teoh et al. [13], working in tension at room temper-

ature on a commercial isotropic polypropylene, identified

a single Eyring process with an activation volume of

6.13 nm3. Seguela et al. [14] made measurements in

tension on both quenched and annealed polypropylene at

various temperatures, and found for quenched material

(conditions similar to the production of our sheet) an acti-

FIG. 9. Comparison of model stresses deduced from stretching force

with those calculated for specimen center.

FIG. 8. (a) Mesh for quarter model of biaxial specimen. (b) Deformed

model, stretched to true strains of 1.4 3 1.4.

DOI 10.1002/pen POLYMER ENGINEERING AND SCIENCE—-2009 1907

vation volume of around 3.0 nm3 at 258C and for annealed

material around 7.7 nm3 at 608C. Dasari et al. [15] haveused room temperature testing of polypropylene to deter-

mine an activation volume of 2.77 nm3. The general indi-

cation is that, for polypropylene up to 608C, there is an

activation volume of 3–8 nm3, with a trend of increasing

activation volume with temperature. When viewed in this

context, our value is of a reasonable order of magnitude.

CONCLUSION

When operating at large strains, biaxial experiments

are an essential tool for the determination of the material

parameters that define complex constitutive equations. It

is desirable that the strain paths used in the experiments

reflect those that occur in the applications for which the

constitutive equation is to be used.

A constitutive equation, in which yielding and strain

rate dependence are controlled by an Eyring process and

large strain behavior by an Edwards-Vilgis network, has

been shown to give an effective representation of the

multiaxial stretching of a blow-molding grade of polypro-

pylene at 1358C, at true strains of up to 1.56. Both simul-

taneous and sequential equibiaxial strain paths were used

in the experiments. The latter provided a severe test of

the constitutive model, particularly in the case of the sec-

ond stretch. For the simultaneous stretching and for the

first stretch in the sequential experiment, i.e., in planar

extension, the model provided excellent predictions.

NOMENCLATURE

A Pre-exponential factor in Eyring

equation

a Cross-sectional area

D Pure stretch tensor

DD Stretch increment tensor

ep Scalar plastic strain rate

Superscript e elastic

F Force

G Deformation gradient tensor

I1 First strain invariant

k Boltzmann’s constant

L Gauge length

Nc Crosslink density

Ns Sliplink density

p Hydrostatic pressure

Superscript p plastic

Qp Rate of plastic strain tensor

R Rotation tensor

T Absolute temperature

v Activation volume

Vp Parameter proportional to pressure

activation volume

Vs Parameter proportional to shear

activation volume

w Strain energy density

Superscript X X-arm of model

Superscript Y Y-arm of model

Greek Symbols

a Finite strain extensibility factor

d Grip displacement

g Sliplink slipperiness factor

k Extension ratio

r Component of normal stress

r Mean stress

s Scalar octahedral shear stress

s Stress deviator tensor

S Stress tensor

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