A constitutive model for large multiaxial deformations of solid polypropylene at high temperature.

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 processes include thermoforming, blow molding, and stretch blow molding. There is also an increasing need for efficiency 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 experimental 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 combinations 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 particularly 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 implemented 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 146[degrees]C 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 180[degrees]C respectively with the clamp, adapter, and die maintained at 180[degrees]C. 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 15[degrees]C than that which would be appropriate according to our experience for these more conventional materials with melting temperature of around 165[degrees]C. The water cooling temperature for the chill roll was 13[degrees]C, 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.

Ideally, the product from this process would be isotropic, 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 (corresponding to extrusion and transverse directions) were indistinguishable 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 X 55 mm were stretched in tension using a biaxial testing machine in air at a temperature of 135[degrees]C. This corresponds to an appropriate temperature 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 manufactured 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 written 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 piston 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.

[FIGURE 1 OMITTED]

The modes of stretching were simultaneous equibiaxial and sequential equibiaxial. The latter comprises two planar 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, corresponding to initial strain rates of 0.37 and 2.3 [s.sup.-1] 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 processing, strain rates are often much higher. However, even if the much higher strain rates were achievable experimentally, the problems of anisothermal conditions associated with adiabatic heating would further complicate the experimental interpretation. We believe that this experimental regime can establish the principles of the constitutive behavior that should be applicable at higher strain rates.

Strains were calculated from the grip displacements [delta] and the gauge length L, the distance between opposite grips. The extension ratios [lambda] are thus given by

[lambda] = 1 + [delta]/L (1)

and the true strains e by

e = ln ([lambda]) (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 [sigma] calculated as

[sigma] = [lambda]F/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 constitutive model by means of an Eyring process in which the scalar plastic strain rate [e.sub.p] is given by

[e.sub.p]=A exp([V.sub.p][bar.[sigma]])sinh([V.sub.s][tau]) (4)

where A is a temperature dependent term that includes activation enthalpy, and [V.sub.p] and [V.sub.s] 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 [bar.[sigma]] and the octahedral shear stress [tau], to be defined below.

The other major component in the model represents large elastic deformations. Edwards and Vilgis (4) derived the following expression for the change in strain energy per unit volume W, as a result of principal extension rations [[lambda].sub.I], [[lambda].sub.II], and [[lambda].sub.III]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here, k is Boltzmann's constant and T, the absolute temperature. [N.sub.c] and [N.sub.s] are crosslink and sliplink densities respectively, and [eta] is a 'slipperiness factor' that characterizes the degree of mobility of the sliplink; [eta][greater than or equal to] 0 where [eta] = 0 corresponds to no mobility. The model includes the phenomenon of finite strain extensibility, which is characterized by the parameter [alpha] and corresponds to a singularity in the strain energy as the denominator 1 - [[alpha].sup.2][I.sub.1]approaches zero. [I.sub.1] is the first strain invariant, equal to

[I.sub.1] = [[lambda].sub.I.sup.2] + [[lambda].sub.[II].sup.2] + [[lambda].sub.[III].sup.2] (6)

The material is assumed incompressible with [[lambda].sub.I][[lambda].sub.II][[lambda.sub.III] = 1. As a result, principal stresses are derived from Eq. 5 by

[[sigma].sub.i] = [[lambda].sub.i][[partial derivative]W/[partial derivative][[lambda].sub.i]] - p (i = I, II, III) (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

[[sigma].sub.i] = [[lambda].sub.i][[partial derivative]W/[partial derivative][[lambda].sub.i]] - [[lambda].sub.III][[partial derivative]W/[partial derivative][[lambda].sub.III]] (i = I, II) (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 precedents 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 deformation D and rigid body rotation R (via the use of the Cauchy-Green strain measure) to give

G = DR (9)

[FIGURE 2 OMITTED]

In the X arm, the strain is split into elastic and plastic components, [D.sup.ex] and [D.sup.p], associated with stress-strain laws Eqs. 4, 5, respectively:

G = [D.sup.eX][D.sup.p]R (10)

The principal values of [D.sup.eX], [[lambda].sub.I.sup.eX], and [[lambda].sub.II.sup.eX], define the stress in the X arm via Eq. 8, with parameters [N.sub.c.sup.X], [N.sub.s.sup.X], [[eta].sup.X] and [[alpha].sup.X] . An incremental approach is employed, with the current plastic stretch related to the plastic strain [D.sub.0.sup.p] at the end of the previous time increment and the increment of plastic strain [[DELTA][D.sup.p], developed during the current increment, related by

G = [D.sup.eX][DELTA][D.sup.p][D.sub.0.sup.p]R (11)

For a given total deformation, G, [[DELTA]D.sup.eX], and [[DELTA][D.sup.p] in Eq. II 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 relative proportions of the components of [[DELTA]D.sup.p] 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 [[SIGMA].sup.X].

The Y arm consists of an Edwards-Vilgis network with parameters [N.sub.c.sup.Y], [N.sub.s.sup.Y], [[eta].sup.Y] and [[alpha].sup.Y]. The principal extension ratios are those of D, and Eq. 8 defines the stress. From this, we derive the stress tensor in global axes [[SIGMA].sup.Y]. The total stress [SIGMA] is then given by

[SIGMA] = [[SIGMA].sup.X] + [[SIGMA].sup.Y] (12)

The stress quantities driving the Eyring process of Eq. 4 relate to the stress [[SIGMA].sup.X]. The stress deviator [tau] is given by

[tau] = [[SIGMA].sup.X] - [bar.[sigma]]I (13)

where the mean stress is given by

[bar.[sigma]] = [1/3]tr([[SIGMA].sup.X]) (14)

The scalar octahedral shear stress [tau] used in (Eq. 4) is then given in terms of the stress deviator tensor [tau] by

[tau] = [square root of ([1/3][tau]:[tau])] (15)

where the colon denotes the double contraction.

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 [D.sup.P] with zero associated plastic spin. The scalar strain rate used in (Eq. 4) is derived from the rate of plastic deformation [Q.sup.P]

[Q.sup.p] = [D.sup.p][D.sup.[p-1]] (16)

by the relation

[e.sub.p] = [square root of ([1/3][Q.sup.p]:[Q.sup.p])] (17)

The Levy-Mises flow rule can then be expressed as

[[Q.sup.p]/[e.sup.p]] = [[tau]/[tau]] (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.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

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 simultaneous 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 envisage 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

[e.sub.p] = A exp([V.sub.p][bar.tau] + [V.sub.s][tau]) (19)

[FIGURE 7 OMITTED]

Typically, [V.sub.p] [much less than] [V.sub.s] and the first term in the argument of the exponential may be neglected. Then (Eq. 19) can be rearranged to give

[tau] = [1/[V.sub.s]]ln([e.sub.p]/A) (20)

which implies that the gradient of the plot in Fig. 7 is equal to 1/[V.sub.s]. On this basis, the gradient of the line fitted through all the results in Fig. 7 corresponds to [V.sub.s] = 6.9 MPa (1). 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 [V.sub.s] determined, the value of the pressure activation parameter [V.sub.p] is then derived on the basis that it is a fixed proportion of [V.sub.p] = 0.06 [V.sub.s], in line with the conclusions of other workers on polymers (7-9). With the activation volumes established, the Eyring constant A 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 crosslink term (the first term involving [N.sub.c] in Eq. 5) with no finite strain extensibility, so that we are assuming a Gaussian network. This is defined by the single parameter [N.sub.c.sup.X] 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.

TABLE 1. Parameters used for the constitutive model.

[N.sub.c.sup.X]/MPa  [N.sub.s.sup.X]/  [[eta].sup.X]  [[alpha].sup.X]
                            MPa

        4.37                 0               -                0

[N.sub.c.sup.Y]/MPa  [N.sub.s.sup.Y]/  [[eta].sup.Y]  [[alpha].sup.Y]
                           MPa

         0.36             0.085              0.2            0.08

[V.sub.s]/      [V.sub.P]/[MPa.sup.-1]    [10.sup.3] [A/s.sup.-1]
[MPa.sup.-1]

     6.9                 0.4                        8.0

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 determined by the observed stress levels and strain rate dependence. 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 transverse 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 quarter model is used, stretched simultaneously along its two perpendicular axes at an overall strain rate of 2.3 [s.sup.-1] to a state of true strain of 1.4 X 1.4, as shown in Fig. 8b. The circular unstrained regions near the outer boundaries represent 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 specimen 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 X 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 calculating specimen stress-strain data produces little error.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

DISCUSSION

There have been some previous studies of polypropylene under conditions that, like those used here, are relevant to the solid phase processing regime. Briatico-Vangosa et al. (10) performed high temperature tests at rates of 0.5-2 [s.sup.-1] 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 [s.sup.-1] in simultaneous equibiaxial, planar, and uniaxial modes. Their observed strain rate dependence of yield stress, reported for simultaneous equibiaxial stretching at 150[degrees]C (14[degrees]C below the melting point), which is of a similar magnitude to that reported here (at 11[degrees]C 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 inclusion 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 material 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 capable of predicting the consequences of this, such as localized wall thinning. The second issue also applies to process 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 discrepancy 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 network. In particular, the stress predicted for the transverse direction during the second stretch is consistently low. This could reflect some increase in crystallinity or network entanglements during the first stretch.

The Eyring parameter [V.sub.s] can be interpreted as an activation volume v, via the relation

v = [V.sub.s]kT (21)

where k is a Boltzmann's constant and T, the absolute temperature. For the testing temperature 135[degrees]C, the current value of 6.9 [MPa.sup.-1] corresponds to v = 39 [nm.sup.3]. This can be compared with values obtained by others for polypropylene, invariably obtained at lower test temperatures. Working on the oriented polypropylene fibers at room temperature, Duxbury and Ward (12) identified two activation volumes, with the larger one in the range 0.5-0.6 [nm.sup.3]. Teoh et al. (13), working in tension at room temperature on a commercial isotropic polypropylene, identified a single Eyring process with an activation volume of 6.13 [nm.sup.3]. 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 activation volume of around 3.0 [nm.sup.3] at 25[degrees]C and for annealed material around 7.7 [nm.sup.3] at 60[degrees]C. Dasari et al. (15) have used room temperature testing of polypropylene to determine an activation volume of 2.77 [nm.sup.3]. The general indication is that, for polypropylene up to 60[degrees]C, there is an activation volume of 3-8 [nm.sup.3], 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 polypropylene at 135[degrees]C, at true strains of up to 1.56. Both simultaneous 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 second 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
[DELTA] D      Stretch increment tensor
[e.sub.p]      Scalar plastic strain rate
Superscript e  elastic
F              Force
G              Deformation gradient tensor
[I.sub.1]      First strain invariant
k              Boltzmann's constant
L              Gauge length
[N.sub.c]      Crosslink density
[N.sub.s]      Sliplink density
P              Hydrostatic pressure
Superscript p  plastic
[Q.sup.p]      Rate of plastic strain tensor
R              Rotation tensor
T              Absolute temperature
[upsilon] p    Activation volume
[V.sub.p]      Parameter proportional to pressure activation volume
[V.sub.s]      Parameter proportional to shear activation volume
w              Strain energy density
Superscript X  X-arm of model
Superscript Y  Y-arm of model

Greek Symbols
[alpha]        Finite strain extensibility factor
[delta]        Grip displacement
[eta]          Sliplink slipperiness factor
[lambda]       Extension ratio
[sigma]        Component of normal stress
[bar.[sigma]]  Mean stress
[tau]          Scalar octahedral shear stress
[tau]          Stress deviator tensor
[SIGMA]        Stress tensor

REFERENCES

(1.) J. Sweeney, T.L.D. Collins, P.D. Coates, and I.M. Ward, Polymer, 38, 5991 (1997).

(2.) C.P. Buckley and D.C. Jones, Polymer, 36, 3301 (1995).

(3.) G. Spathis and E. Kontou, Polym. Eng. Sci., 41, 1337 (2001).

(4.) S.F. Edwards and T.A. Vilgis, Polymer, 27, 483 (1986).

(5.) R.N. Haward and G. Thackray, Proc. Roy. Soc A., 302, 453 (1968).

(6.) I.M. Ward and J. Sweeney, An Introduction to the Mechanical Properties of Solid Polymers, John Wiley & Sons Ltd, Chichester, UK, 233 (2004).

(7.) S. Nazarenko, S. Bensason, A. Hiltner, and E. Baer, Polymer, 35, 3883 (1994).

(8.) C. Bauwens-Crowet and J.-C. Bauwens, J. Mater. Sci., 7, 176 (1972).

(9.) L.E. Govaert, P.H.M. Timmermans, and W.A.M. Brekelmans, J. Eng. Mater. Technol., 122, 177 (2000).

(10.) F. Briatico-Vangosa, M. Rink, F. D'Oria, and A. Verzelli, Polym. Eng. Sci., 40, 1553 (2000).

(11). L. Capt, S. Rettenberger, H. Munstedt, and M.R. Kamal, Polym. Eng. Sci., 43, 1428 (2003).

(12.) J. Duxbury and I.M. Ward, J. Mater. Sci., 22, 1215 (1987).

(13.) S.H. Teoh, A.N. Poo, and G.B. Ong, J. Mater. Sci., 29, 4918 (1994).

(14.) R. Seguela, E. Staniek, B. Escaig, and B. Fillon, J. Appl. Polym. Sci., 71, 1873 (1999).

(15.) A. Dasari, S. Sarang, and R.D.K. Misra, Mater. Sci. Eng., A368, 191 (2004).

John Sweeney, Robert Spares, Mike Woodhead

School of Engineering, Design, and Technology/IRC in Polymer Science and Technology, University of Bradford, Bradford BD7 1DP, UK

Correspondence to: J. Sweeney; e-mail: [e-mail]

Contract grant sponsor: The European Commission under the Framework 6 Program via the Apt-Pack project.

DOI 10.1002/pen.21426