A material model for multiaxial stretching and stress relaxation of polypropylene under process conditions

A material model for multiaxial stretching and stress relaxation of polypropylene under process conditions

Mechanics of Materials 54 (2012) 55–69 Contents lists available at SciVerse ScienceDirect Mechanics of Materials journal homepage: www.elsevier.com/...

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Mechanics of Materials 54 (2012) 55–69

Contents lists available at SciVerse ScienceDirect

Mechanics of Materials journal homepage: www.elsevier.com/locate/mechmat

A material model for multiaxial stretching and stress relaxation of polypropylene under process conditions J. Sweeney a,⇑, C.P.J. O’Connor b, P.E. Spencer a, H. Pua a, P. Caton-Rose a, P.J. Martin b a b

School of Engineering, Design and Technology/IRC in Polymer Science and Technology, University of Bradford, Bradford, BD7 1DP, UK School of Mechanical and Aerospace Engineering, Queen’s University Belfast, Ashby Building, Stranmillis Road, Belfast, BT9 5AH, Northern Ireland, UK

a r t i c l e

i n f o

Article history: Received 1 November 2011 Received in revised form 9 June 2012 Available online 28 June 2012 Keywords: Polypropylene Large deformations Constitutive equation

a b s t r a c t Polypropylene sheets have been stretched at 160 °C to a state of large biaxial strain of extension ratio 3, and the stresses then allowed to relax at constant strain. The state of strain is reached via a path consisting of two sequential planar extensions, the second perpendicular to the first, under plane stress conditions with zero stress acting normal to the sheet. This strain path is highly relevant to solid phase deformation processes such as stretch blow moulding and thermoforming, and also reveals fundamental aspects of the flow rule required in the constitutive behaviour of the material. The rate of decay of stress is rapid, and such as to be highly significant in the modelling of processes that include stages of constant strain. A constitutive equation is developed that includes Eyring processes to model both the stress relaxation and strain rate dependence of the stress. The axial and transverse stresses observed during loading show that the use of a conventional Levy–Mises flow rule is ineffective, and instead a flow rule is used that takes account of the anisotropic state of the material via a power law function of the principal extension ratios. Finally the constitutive model is demonstrated to give quantitatively useful representation of the stresses both in loading and in stress relaxation. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The mechanical testing of polymers to large deformations under multiaxial conditions is both of technological relevance and an essential tool for the evaluation of material constitutive equations. A number of solid phase deformation processes for polymers – for example, film blowing, thermoforming, blow moulding and stretch blow moulding – involve multiaxial stretching operations to large strains. For modelling these processes, material constitutive equations are required that need to be verified experimentally. Multiaxial testing is essential for establishing well-defined parameters relating to the material model. It is also desirable because it can replicate the deformation conditions ⇑ Corresponding author. Address: University of Bradford, School of Engineering, Design and Technology, Richmond Road, Bradford, W Yorkshire, BD7 1DP, United Kingdom. Tel.: +44 1274235456. E-mail address: [email protected] (J. Sweeney). 0167-6636/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.mechmat.2012.06.003

found in solid phase processes, and thus verify the material model under particularly relevant conditions. In many of the processes concerned the multiaxial stretching processes take place in stages, which makes it essential for models to also include the effects of stress relaxation. We shall describe a programme of work in which sheets of polypropylene have been stretched sequentially along perpendicular axes to achieve states of equibiaxial strain, and then allowed to stress relax at constant strain, forces being monitored throughout the process. The study of stress relaxation under large biaxial strains is an area that has received little or no attention hitherto. In practice, however, a stress relaxation stage will occur whenever a solid phase processing operation is either partially or fully complete and the component has either achieved a preform stage or its final shape. The sequential strain path employed to achieve the equibiaxial state, i.e. planar extension followed by a second planar extension perpendicular to the first, is revealing about the constitutive behaviour, and also

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which for polypropylene is close to 0 °C, is largely irrelevant to the solid phase processing temperature. In the principal experiments, straining was carried out biaxially in a two stage sequential operation – an initial planar extension to ratios of 1  3, followed by a second perpendicular planar extension to 3  3. While the second stretch was programmed to follow immediately on completion of the first step, the extension measurements showed that there was a delay of 20–90 ms caused by the response time of the testing machine. The length of delay depended on the testing speed and was measured during each test. The extensions were at constant speeds corresponding to initial nominal strain rates of either 0.39, 1.24, or 2.17 s1. Once the state of equibiaxial strain had been attained it was followed by stress relaxation for a period of 120 s. Other subsidiary biaxial experiments, both simultaneous biaxial and planar extension but not followed by stress relaxation, were carried out at four strain rates (0.3, 0.8, 1.2 and 2.2 s1 in order to fully define the strain rate dependence of stress. Additionally, for the same purpose, uniaxial experiments were carried out in the same machine at the same set of rates on rectangular specimens of dimensions 35  55 mm, which were gripped on the shorter sides only and stretched along the 55 mm axis. The biaxial testing machine was manufactured originally by T M Long, and currently is fitted with PC-based displacement and temperature control, and data capture. Draw rates, ratios and starting time of each axis can be controlled independently of one another. The specimens are held by six pneumatic grips on each of their four edges, with force transducers incorporated into one of the grips on two perpendicular sides; this arrangement has been shown to give accurate estimates of the total forces on the specimen sides for stress calculations (Sweeney et al., 2009a). The specimens and stretching mechanism were contained in an environmental chamber, heated by a com-

approximates to the strain path in the stretch blow moulding and plug-assisted thermoforming processes. In this paper a single constitutive theory is used to model the loading and relaxation processes, with Eyring mechanisms representing strain rate dependence, yielding and stress relaxation phenomena. We have found it necessary to include two Eyring processes, one broadly corresponding to the loading phase of the experiments and the other to stress relaxation. Flow rules incorporating straininduced anisotropy are found to be necessary to represent the two-dimensional stress fields observed during planar extension loading. The Eyring mechanisms are combined with entropic elastic network models. The time scales involved are 2–10 s in loading and 120 s in stress relaxation, with testing temperature 160 °C.

2. Materials and experimentation The polypropylene material is a commercial homopolymer manufactured Basell Moplen with designated grade HP540J and density 900 kg/m3. It was obtained as granules and formed into sheet using a 38 mm diameter Killion single screw extruder with a sheet die of width of 610 mm and die gap of 1.8 mm operating at a screw speed of 75 rpm. The melt temperature was 250 °C, melt pressure 8 MPa and haul-off speed 0.42 m/min. The sheet was output onto a chill roll at 90 °C. For the biaxial experiments, square samples 55  55 mm and 1.04 mm thick were stretched in tension using a biaxial testing machine after heating to 160 °C. This corresponds to an appropriate temperature for solid phase processing of the material, close to the melting point which is less than 170 °C, and falls within the processing window for this material estimated by O’Connor et al. (2010). For semicrystalline polymers, the glass transition temperature,

5

Stress / MPa

4

3

1-axis

2

2-axis

1

0 1

2

Extension ratio Fig. 1. Simultaneous equibiaxial stretch at 0.39 s1.

3

J. Sweeney et al. / Mechanics of Materials 54 (2012) 55–69

bination of electric heating plates and blown air. Extension and force signals were logged at 300 Hz. Simultaneous equibiaxial testing at 160 °C has shown that the sheet material is essentially isotropic in the plane, as illustrated in Fig. 1. Here the 1 axis refers to the extrusion direction. There is a noticeable drop in the stress in the 2 axis at yield which is absent in the 1 axis stress, but thereafter the stresses converge and are essentially indistinguishable. The observed difference is a consistent effect and is probably due to preorientation arising from the extrusion process. However, it is not detectable at the strains of interest here. 3. Analysis of loading and stress relaxation In these experiments strains were applied sequentially in the two perpendicular (1 and 2) directions and then held constant, following the strain histories shown in Fig. 2 in terms of extension ratios k1 and k2 . Strains are applied at constant speeds so that, with t denoting time, dkdt1 or dkdt2 were constant, corresponding to the initial true strain rate. Figs. 3 and 4 show the loading and stress relaxation results for the three strain rates. The essential element of the constitutive equation is a Maxwell-type model in which the dashpot takes the form of an Eyring process and the series elastic element is hyperelastic. The latter takes the form in this case of the neo-Hookean or Gaussian model. The model is illustrated in Fig. 5. In polyropylene the finite strain limit is far in excess of the extensions applied here, and so it is appropriate that we use an elastic model that does not include finite strain extensibility. The Gaussian network has this property and is the simplest such model, with a single fitting parameter that is adequate for the present purposes. In the Eyring process the scalar plastic strain rate e_ p is given by (Ward and Sweeney, 2004; Buckley and Jones, 1995)

 Þ sinhðV s sÞ; e_ p ¼ a expðV p r

Extension ratio

ð1Þ

where a is a constant pre-exponential factor, Vp and Vs are proportional to the pressure and shear activation volumes vp and vs respectively. These quantities are given by v v s where k is Boltzmann’s constant and T V p ¼ kTp ; V s ¼ kT  is the mean stress and s an the absolute temperature. r appropriate driving stress, customarily the octahedral shear stress. The scalar plastic strain rate is related to the plastic strain rate tensor Lp by

e_ p ¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 p p L :L 3

ð2Þ

and in general the values of the individual tensor components are arrived at by the use of a flow rule, to be specified below. The plastic strain is assumed to be incompressible. In the experiments that we are modelling, the principal directions of both strain and stress do not change and can be associated with the global 1–2–3 system. The plastic extension ratios kp1 ; kp2 and kp3 and their time derivatives k_ p1 ; k_ p2 and k_ p3 then form the plastic strain rate tensor

0 _p

k1

p B k1

B B Lp ¼ B 0 B @ 0

0 p k_ 2 p k2

0

0

1

C C C 0 C: C p A k_

ð3Þ

3 p

k3

The elastic element of the model is subject to the same stress as the Eyring element, and is defined by a neo-Hookean strain energy function. The strain energy U per unit volume is given by



 1  e 2 G ðk1 Þ þ ðke2 Þ2 þ ðke3 Þ2  3 ; 2

ð4Þ

where G is a material constant and ke1 ; ke2 and ke3 are the elastic extension ratios. These are related to the total extension ratios and plastic extension ratios by multiplicative decomposition:

3

First stretch

2

1

57

Second stretch

Time Fig. 2. Strain history for sequential biaxial tests.

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Fig. 3. Loading as a function of extension ratio at the three strain rates.

ki ¼ kei kpi

ði ¼ 1; 2; 3Þ:

ð5Þ

3.1. Stress relaxation

e_ p ¼

Guiu and Pratt (1964) proposed a method of analysis of stress relaxation curves in terms of an Eyring process using a Maxwell-type model as described above. Their analysis concerned uniaxial conditions, whereas our approach, for plane stress conditions in the 1–2 plane, nevertheless gives the same time dependence for the stress decay at constant strain. As with the experiments, the stress relaxation is assumed to take place under conditions of equibiaxial strain with k1 ¼ k2 . For simplicity we also assume that the in-plane principal stresses are equal with r1 ¼ r2 . Since the strain is attained after a process of sequential loading in the 1 and 2 directions (see Fig. 2), this is a significant assumption, but the results of Fig. 4 show that it is true to a good approximation. We can therefore be confident that the analysis will be fit for its purpose, which is to derive values for the Eyring parameters a, Vs and Vp of Eq. (1). We adopt the conventional approach and equate the driving stress s of Eq. (1) with the octahedral shear stress soct , given by (Ward and Sweeney, 2004)

soct

1=2 1 ðr1  r2 Þ2 þ ðr2  r3 Þ2 þ ðr3  r1 Þ2 ¼ : 3

With

ð6Þ

r ¼ r1 ¼ r2 and r3 ¼ 0, the driving stress becomes

s ¼ soct

pffiffiffi 2 ¼ r: 3

ð7Þ

Also, under these conditions

1 3

by an exponential function such that sin hðxÞ ¼ 12 expðxÞ. Then, using Eqs. (7) and (8), Eq. (1) becomes

2 3

r ¼ ðr1 þ r2 þ r3 Þ ¼ r:

ð8Þ

We now follow Guiu and Pratt and assume that the hyperbolic sine function of Eq. (1) can be approximated

a 2

" exp

pffiffiffi # ! 2 2 Vp þ Vs r ; 3 3

ð9Þ

which we can simplify on introducing the quantity V,



pffiffiffi 2 2 Vp þ V s; 3 3

ð10Þ

as

e_ p ¼

a 2

expðV rÞ:

ð11Þ

In terms of true strain rates, the incompressibility condition for the plastic flow is

k_ p1 k_ p2 k_ p3 þ þ ¼ 0: kp1 kp2 kp3

ð12Þ

Then, with kp1 ¼ kp2 ¼ kp ,

k_ p3 k_ p p ¼ 2 p k k3 and via the use of Eqs. (2) and (3) we obtain

  pffiffiffik_ p  e_ p ¼ 2 p : k  For positive plastic strain rates Eq. (11) becomes

k_ p a ¼ pffiffiffi expðV rÞ: kp 2 2

ð13Þ

We now relate this to the elastic strain using Eq. (5), which after differentiation produces the result p k_ i k_ i k_ ei ¼ pþ e ki ki ki

ði ¼ 1; 2; 3Þ;

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a

7

6

First stretch 5

Stress / MPa

Second stretch 4

3

2

1

0 0

20

40

60

80

100

120

80

100

120

Time / s

b

8

7

First stretch

6

Stress / MPa

Second Stretch 5

4

3

2

1

0 0

20

40

60

Time / s Fig. 4. Loading and stress relaxation for strain rates in loading of (a) 0.39 s1, (b) 1.24 s1 and (c) 2.17 s1.

which for equibiaxial conditions in the 1–2 plane we may write as

k_ k_ p k_ e ¼ þ ; k kp ke

k_ e a ¼ pffiffiffi expðV rÞ: ke 2 2







ði ¼ 1; 2Þ; ð16Þ

ð14Þ

where ke1 ¼ ke2 ¼ ke . Under conditions of stress relaxation, k_ ¼ 0 and, using Eq. (14) in Eq. (12), we derive





ri ¼ G ðkei Þ2  ðke3 Þ2 ¼ G ðkei Þ2  1=ðke1 ke2 Þ2

ð15Þ

For incompressible deformation and with r3 ¼ 0, standard arguments (Ogden, 1997) produce for the stresses in the plane the result

which for the equibiaxial stress state becomes





r ¼ G ðke Þ2  ðke Þ4 : After differentiating with respect to time, rearrangement of this expression returns

k_ e r_ : ¼  ke 2G ðke Þ2 þ 2ðke Þ4

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c

8

7

First stretch

6

Second stretch Stress / MPa

5

4

3

2

1

0 0

20

40

60

80

100

120

Time / s Fig. 4 (continued)

> 0. As shown in Figs. 6–8, for the complete relaxation curves (t0 = 0) the fit is poor. However, the fit is markedly improved for values of t0 as small as 2 s. This suggests that

Fig. 5. Single Eyring process model.

On substitution of this expression into Eq. (15) we obtain the differential equation

r_ a   þ pffiffiffi expðV rÞ ¼ 0; e 2 e 4 2 G ðk Þ þ 2ðk Þ

ð17Þ

which has the same form as that derived for the uniaxial case by Guiu and Pratt. The solution at time t is

rðt0 Þ  rðtÞ ¼

1 lnð1 þ ðt  t 0 Þ=cÞ V

Fig. 6. Guiu–Pratt fits for stress relaxation loaded at 0.39 s1.

ð18Þ

where 0 6 t0 < t and c is defined by



pffiffiffi 2 exp ðV rð0ÞÞ  : aG ðke Þ2 þ 2ðke Þ4

ð19Þ

We have fitted Eq. (18) to the stress relaxation curves of Fig. 4 using a least-squares procedure to derive values for V and c and, via Eq. (19), a. The quality of the fits provides an indication of the appropriateness of the use of the single process model of Fig. 5. The equation is fitted to the stress arising from the second stretch, r2 . For all three loading rates, the equation is fitted to the entirety of the curve–i.e. from the point at which the constant strain is attained such that t0 = 0 – and also to sections of the curves beginning at times t0

Fig. 7. Guiu–Pratt fits for stress relaxation loaded at 1.24 s1.

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ably consistent with findings for a range of polymers, both semicrystalline and amorphous. Examples of values derived for Vp =Vs are 0.05–0.072 (Bauwens-Crowet and Bauwens, 1972) and 0.075 (Nazarenko et al., 1994) for polycarbonate; 0.15 for polypropylene (Joseph and Duckett, 1978); 0.13 for ultra-high molecular weight polyethylene (Naz et al., 2010). The values of Vp and Vs are reasonably consistent over the three strain rates. The large variation of a arises from the fact that it is multiplied by an exponential function of stress in Eq. (17); the logarithmic values show a variation similar to that of Vp and Vs . 3.2. Incorporation of loading steps Fig. 8. Guiu–Pratt fits for stress relaxation loaded at 2.17 s1.

The Guiu–Pratt analysis is based on the assumption that the state of constant strain and associated stress that is being analysed is attainable. A complete model of the stress relaxation behaviour must include the process whereby the state of strain was reached. This can be achieved readily by the use of a numerical analysis of the model of Fig. 5, (the Eyring–Gaussian model) when subjected to a strain history consisting of an interval of positive strain rate followed by a constant strain. For this analysis, plane stress conditions are assumed with r3 ¼ 0. Eqs. 1,2,3,4,5,6 and (16) remain valid. We need to introduce explicitly a flow rule, which for this initial analysis is that of Levy–Mises, taking the form

the short term stress relaxation behaviour may need to be represented by a separate mechanism, such as another parallel Maxwell-type arm with an Eyring process having a faster rate of stress relaxation. Model parameters derived using the fitting process for t0 = 8 s are given in Table 1 for the three strain rates applied in the stress relaxation experiments. The value of G was derived from the initial slopes of the loading curves and was found to be 2.8 MPa. The value of a was derived from the fitted value of c via Eq. (19). The quantity V was obtained from the fitting process, and separated into values of Vp and Vs using Eq. (10) and the assumption that Vp ¼ 0:1Vs . This assumption is reason-

0 _p

k1 p

B k1

Table 1 Model parameters from Guiu–Pratt fitting. Initial strain rate/ s1

Vp / MPa1

Vs / MPa1

a/s1

G/ MPa

0.39

2.92

29.2

4.39  1023

2.8

1.24

2.94

29.4

2.93  1019

2.8

2.17

3.64

36.4

2.97  1024

2.8

1B B B0 _ep B @ 0

0 p k_ 2 p k2

0

0

1

0 1 C  0 0 C 1 r1  r C B C r2  r 0 0 C ¼ @0 A C s 0 0 r3  r _kp A 3

kp3

under the constraint of Eq. (12) and the plane stress condition r3 ¼ 0. A time-stepping approach is adopted. The stress in the Eyring process is evaluated using Eqs. 1,2,3

2.9

2.7

Stress / MPa

2.5

Eyring-Gauss

Guiu-Pratt

2.3

2.1

1.9

1.7

1.5 0

20

40

ð20Þ

60

80

100

120

Time / s Fig. 9. Comparison of stress relaxation curves from Guiu–Pratt equation and from numerical implementation of model illustrated in Fig. 6.

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and (20), and the stress in the Gaussian model is evaluated using Eq. (16). For given total extension ratios k1 and k2 , stress equilibrium between the elastic and Eyring process is enforced to arrive at the values of the elastic and plastic extension ratios kei and kpi ði ¼ 1; 2Þ, while subject to the condition 5. A more complete account of the numerical framework used for a more general case is given by Naz et al. (2010). It is instructive to simulate the complete process of loading and stress relaxation using this model and the parameter values of Table 1. We do this first for simultaneous equibiaxial loading to compare with that obtained with the Guiu–Pratt analysis outlined above. The applied strain rate is 1.24 s1 up to a state of strain of k1 ¼ k2 ¼ 3. The analyses are compared in Fig. 9. The agreement is excellent from around 20 s. The initial discrepancy arises because the stress drop depends on the level of stress; the initial stress in the Guiu–Pratt fit is from the experimental

a

observation, whereas in the Eyring–Gaussian model it depends on the yield characteristics of the Eyring process. The important observation here is that yielding takes place during loading of the Eyring–Gauss model, so that the initial stress obtained is lower than observed in Fig. 3. This again emphasises the need for a process or processes to be added to the Eyring–Gauss model of Fig. 5 to represent the initial transient effects. The same model has been subject to sequential straining as defined in Fig. 2, again using the strain rate 1.24 s1. The results are shown in Fig. 10. Yielding is at a similar level to that for simultaneous loading. Towards the end of the first stretch (t  1.5 s in Fig. 10(a)), r1  2r2 . At this stage, both stresses are approximately constant with time, so that the Gaussian network is not being extended and the plastic strain rate is equal to the total strain rate. Since k_ p2 ¼ 0 it fol , i.e. r1 ¼ 2r2 consislows from the flow rule 20 that r2 ¼ r tent with the numerical result. Similarly, during the second

3.5

3

Stress / MPa

2.5

2

1.5

First stretch

1

Second stretch

0.5

0 0

0.5

1

1.5

2

2.5

3

3.5

Time / s

b 3.5 3

Stress / MPa

2.5

2

1.5

First stretch

1

Second stretch

0.5

0 0

20

40

60

80

100

120

Time / s Fig. 10. Predictions for the model of Fig. 6 when using the Levy–Mises flow rule. Loading at 1.24 s1 (a) and complete loading and stress relaxation (b).

J. Sweeney et al. / Mechanics of Materials 54 (2012) 55–69

stretch, k_ p1 ¼ 0 and a similar argument gives r2 ¼ 2r1 , as observed at t  3 s in Fig. 10(a). Thus, the qualitative pattern of stress development during loading is a direct consequence of the flow rule. The same pattern will result in a model consisting of any number of arms like that in Fig. 5 acting in parallel, provided that all the Eyring dashpots are substantially yielded and obey the same flow rule. This stress development is also completely different from that observed experimentally in Fig. 3. The model as currently constituted is thus a qualitative failure. To devise a model with the correct response to sequential loading, there are two possible approaches: to adapt the present approach with a different flow rule; or to introduce an entirely unrelated mechanism that acts to give a realistic overall response. The latter approach was used by Sweeney et al. (2009a), who introduced a hyperelastic model acting in parallel with an Eyring-hyperelastic Maxwell-type model to predict the biaxial loading behaviour of polypropylene. In the present work, there is a limit to the stress introduced by such an elastic component arising from the stress level observed during relaxation. A second consideration is the relative size of the stress maxima observed at the end of the first and second stretches. They are equal in Fig. 10(a) for the model of Fig. 5, and so if a hyperelastic arm were added in parallel the second peak would be higher than the first. In contrast, in Fig. 3, the second peak is lower than the first. This arises from the fundamental properties of the Levy–Mises flow rule. From this second consideration alone, we are obliged to explore the application of a different flow rule. We also add an elastic mechanism to represent the stress at long times. The stresses due to the Eyring components then must decay ultimately to zero, allowing the total stresses to decay to a common value, approximately as observed. 4. Constitutive model In a previous publication Sweeney et al. (2009b) observed that, for room temperature testing of ultra-high molecular weight polyethylene under conditions of plane strain compression, the ratio of axial stress to lateral stress was seriously in contradiction to that predicted by the Levy–Mises flow rule. To model this behaviour they introduced a flow rule in which anisotropy was developed as a function of strain. This work was extended to tensile behaviour (Sweeney et al., 2011). As described above, the present work has presented a similar problem, and we shall explore the application of the same flow rule here. Hill’s flow rule (Hill, 1985) is used. We note that it has been applied to oriented polymer previously by Van Erp et al. (2009), for polypropylene tapes with fixed levels of orientation tested in tension at room temperature. In their work the different levels of yield stress at a range of angles to the axis of orientation were used to derive the Hill coefficients, defined below. Our problem is to model the orientation as it develops from the initially isotropic state, and so we use a power law function of the extension ratios to model the coefficients. Hill’s flow rule (Hill, 1985) can be written, for an arbitrary 1–2–3 axis set, as

Lp11 ¼ e_ p ½Hðr11  r22 Þ þ Gðr11  r33 Þ=3s; Lp22 ¼ e_ p ½Fðr22  r33 Þ þ Hðr22  r11 Þ=3s; Lp33 ¼ e_ p ½Gðr33  r11 Þ þ Fðr33  r22 Þ=3s; Lp23 ¼ e_ p Lr23 =3s; Lp13 ¼ e_ p M r13 =3s; Lp12 ¼ e_ p Nr12 =3s;

63

ð21Þ

where F, G, H, L, M and N are the Hill coefficients, the Lpij ði; j ¼ 1; 2; 3Þ are the components of the plastic strain rate tensor Lp, with the scalar plastic strain rate e_ p defined as above in Eq. (2). For our present purposes the principal directions of strain are fixed and remain coincident with the principal directions of stress, and we assign them to global directions 1, 2 and 3. The dependence of the anisotropy on strain is introduced by defining F, G and H as power law functions of the total extension ratios k1 ; k2 and k3 . In principal directions the flow rule is then

  k_ p1 ¼ e_ p ðk3 Þm ðr1  r2 Þ þ ðk2 Þm ðr1  r3 Þ =3s; kp1   k_ p2 ¼ e_ p ðk1 Þm ðr2  r3 Þ þ ðk3 Þm ðr2  r1 Þ =3s; kp2   k_ p3 ¼ e_ p ðk2 Þm ðr3  r1 Þ þ ðk1 Þm ðr3  r2 Þ =3s; kp3

ð22Þ

where the exponent m controls the anisotropy; for m = 0, the isotropic Levy–Mises flow rule is returned. This equation replaces Eq. (20) in this revised analysis, with the driving stress now redefined in principal directions as:

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 3 u 2 m m u u 6 ðk3 Þ ðr1  r2 Þ þ ðk2 Þ ðr1  r3 Þ 2 7 1 u1 6  s¼ u þ ðk Þm ðr2  r3 Þ þ ðk3 Þm ðr2  r1 Þ 7 5: 3 t3 4  1 2 m m þ ðk2 Þ ðr3  r1 Þ þ ðk1 Þ ðr3  r2 Þ

ð23Þ

Inspection of Eq. (23) for the case of plane stress with

r33 ¼ 0 shows that, for the end of the first stretch when k1 =3.0 and k2 =1.0 and it is assumed that k_ p2 ¼ 0, the observed stress ratio r11 =r22  7 can be achieved with m  0.8. However, at the end of the second stretch when k1 ¼ k2 ¼ 3:0, a similar argument with this value of m and assuming that k_ p1 ¼ 0 gives the result r11  0:07r22 . This is much smaller than observed in Fig. 3, where the two stresses are approximately equal. However, the complete constitutive model includes a hyperelastic component, representing the long-term relaxed response, which has the effect of reducing the percentage difference between these two components. For the reasons discussed so far, our constitutive model consists of three parallel components. Two of these are Maxwell-type models as illustrated in Fig. 5: one (the X arm) represents the long-term stress relaxation response quantified above in Table 1; the other (the Y-arm) represents the transient stresses to contribute to the initial stress response on loading. Finally, the Z arm is a Gaussian hyperelastic process that provides the long-term relaxed stress. The model is illustrated in Fig. 11. In the following Eqs. (24)–(30), superscripted symbols are notational except in the case of mX and mY , which rep-

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J. Sweeney et al. / Mechanics of Materials 54 (2012) 55–69

sX

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2 3 u 2n mX X mX X u X u 6 ðk3 Þ r1  r2 þ ðk2 Þ r1 7 u 6 o2 7 1u 16 n 7 mX X mX X u ¼ u 6 þ ðk1 Þ r2 þ ðk3 Þ r2  rX1 7: 7 3 u3 6 5 o t 4 n 2 mX X mX X þ ðk2 Þ r1 þ ðk1 Þ r2

ð25Þ

For the neo-Hookean mechanism of Eq. (4), the incomeX eX pressibility condition keX 1 k2 k3 ¼ 1 leads to the stress– strain law



2



eX rXi ¼ GX keX  1= keX i 1 k2

2 

ði ¼ 1; 2Þ:

ð26Þ

The elastic and plastic extension ratios are linked to the total extension ratios applied to the model by the decomposition of Eq. (5) pX ki ¼ keX i ki

ði ¼ 1; 2Þ:

ð27Þ

For the Y arm, a similar set of equations apply Fig. 11. Proposed constitutive model.

k_ pY 1

resent powers. For a plane stress analysis in the 1–2 plane, the flow rule for the X arm is given by 20:

kpY 1 k_ pY 2 kpY 2

k_ pX 1 kpX 1 k_ pX 2 kpX 2

h i X

X ¼ e_ pX ðk3 Þm rX1  rX2 þ ðk2 Þm rX1 =3sX ð24Þ

h i X X

¼ e_ pX ðk1 Þm rX2 þ ðk3 Þm rX2  rX1 =3sX

sY

and the driving stress by

h i Y

Y ¼ e_ pY ðk3 Þm rY1  rY2 þ ðk2 Þm rY1 =3sY ð28Þ

h i Y Y

¼ e_ pY ðk1 Þm rY2 þ ðk3 Þm rY2  rY1 =3sY ;

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi o2 3 u 2n mY Y mY Y u Y u 6 ðk3 Þ r1  r2 þ ðk2 Þ r1 7 u o2 7 1 u1 6 7 6 n mY Y mY Y Y ¼ u 7; 6 þ ðk1 Þ r2 þ ðk3 Þ r2  r1 7 3u u3 6 5 o t 4 n 2 mY Y mY Y þ ðk2 Þ r1 þ ðk1 Þ r2

ð29Þ

Table 2 Model parameters. V Xp MPa1

Analytical

2.94

Optimised

0.675

aX s1

mX

GX MPa

V Yp MPa1

V Ys MPa1

aY s1

mY

GY MPa GZ MPa

29.4

2.12  104

0.8

2.8

0.1

1.0

0.05

0.8

0.5

0.27

13.5

2.0  105

0.8

2.9

0.05

1.0

0.05

0.8

0.4

0.28

V Xs MPa1

6

5

4

τ / MPa

Method

3 Uniaxial Equibiaxial Planar extension

2

1

0 -2

-1.5

-1

-0.5

0

0.5

1

1.5

Ln (octahedral strain rate / s-1) Fig. 12. Stresses on attainment of axial extension ratio 3 for the three stretching modes.

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J. Sweeney et al. / Mechanics of Materials 54 (2012) 55–69



2



eY rYi ¼ GY keY  1= keY i 1 k2

2 

ði ¼ 1; 2Þ

This scheme has been programmed in a more general form as a user-defined UMAT subroutine in the finite element package ABAQUS/Standard 6.10 (Simulia, Providence RI USA) and also implemented in FORTRAN in this specific form.

ð30Þ

and pY ki ¼ keY i ki

ði ¼ 1; 2Þ:

ð31Þ

For the Z arm the stress–strain law is



rZi ¼ GZ ðki Þ2  1=ðk1 k2 Þ2



5. Model parameters

ði ¼ 1; 2Þ:

ð32Þ There are eleven parameters in this model, summarised in Table 2. We have used two methods to derive values for them. In one-the analytical approach-we use conventional methods of deriving Eyring parameters, namely Guiu–Pratt fitting and analysis of the strain rate dependence of stress, together with estimation of the powers mX and mY from the relative values of r1 and r2 , and combined with some degree of trial-and-error. In the other,-the optimisation approach-we use a numerical optimisation technique. Both are described in turn.

Incompressibility conditions apply for all processes and for the overall model: pX pX pX pY pY pY eX eX eY eY eY keX 1 k2 k3 ¼ k1 k2 k3 ¼ k1 k2 k3 ¼ k1 k2 k3

¼ k1 k2 k3 ¼ 1:

ð33Þ

Finally the total stresses are given by

ri ¼ rXi þ rYi þ rZi ði ¼ 1; 2Þ:

a

ð34Þ

7

6

Stress / MPa

5

4

3

First stretch

Second stretch

2

First stretch model 1 Second stretch model 0 0

2

4

6

8

10

12

Time / s

b

7 First stretch 6 Second stretch 5

Stress / MPa

First stretch model 4

Second stretch model

3

2

1

0 0

20

40

60

80

100

120

Time / s Fig. 13. Observations and model for loading at 0.39 s1 (a) and complete loading and stress relaxation (b).

66

J. Sweeney et al. / Mechanics of Materials 54 (2012) 55–69

ues of activation volume. Here it is necessary to equate the plastic strain rate with the total strain rate, an assumption that is only true at a yield point. In the present work we plot the stress at a fixed axial strain on the basis that the plot gives a useful indication when significant yielding has occurred. The results are plotted in Fig. 12 in terms of scalar strain rate (Eq. (2)) and driving stress s (Eq. (23), calculated with m = 0.8–see below), where the three testing modes are fitted to a common slope that suggests the value V Ys =0.5 MPa1. This was used as a basis for a trial and error process in which the numerically generated stress–strain curves were compared with observations to arrive at the value of 1.0 MPa1. We note that, if the octahedral shear stress soct of Eq. (6) is used in this plot rather than s, the slope suggests a value V Ys ¼ 0:3 MPa1 and there is more scatter (R2 = 0.44 rather than R2 = 0.61). The parameter aY was fitted to produce appropriate levels of

The analytical approach is implemented as follows. For the X arm, the Eyring parameters are obtained by fitting the Guiu–Pratt Eq. (18) for t0 = 8 s to the transient proportion of the total stress, r2  rZ2 for the second stretch. This produces Vp and Vs values identical to those in Table 1 since they depend only on the stress drop, but the a value is different as it depends on the stress level. For the Y arm, we examine the rate dependence of the stress during the initial loading for uniaxial, simultaneous equibiaxial and planar extension at the fixed extension ratio of k1 ¼ 3. Adopting the exponential approximation of Eq. (1) gives a linear relation between logarithm of strain rate and stress. This relation has been exploited by a number of workers (for instance: Bauwens-Crowet et al., 1969; Teoh et al., 1994; Seguela et al., 1999; Govaert et al., 2000; Klompen et al., 2005; Van Erp et al., 2009) to produce plots of yield stress against log strain rate and thus estimate val-

a

8

7

6

Stress / MPa

5

4

3 First stretch 2

Second stretch First stretch model

1

Second stretch model 0 0

0.5

1

1.5

2

2.5

3.5

3

Time / s

b

8

7 First stretch

6

Stress / MPa

Second stretch 5 First stretch model 4

Second stretch model

3

2

1

0 0

20

40

60

80

100

120

Time / s Fig. 14. Observations and model for loading at 1.24 s1 (a) and complete loading and stress relaxation (b).

67

J. Sweeney et al. / Mechanics of Materials 54 (2012) 55–69

stress. For both Eyring processes the power law exponents mX and mY were assumed to be equal at 0.8, derived from the ratio of stresses during loading in the sequential tests as discussed above. Finally, the Z arm parameter GZ was derived from the average level of ultimate stress at t > 120 s. The parameter values are summarised in Table 2. For the optimisation approach, we have combined the use of the ABAQUS finite element program (ABAQUS/Standard, Simulia, Providence RI USA) with HyperStudy optimisation software (Altair Engineering Ltd, Warwickshire, UK). HyperStudy is ‘wrap-around’ software that can be used in conjunction with any finite element solver software. The experiments of Figs. 3 and 4 were simulated with ABAQUS using four-element square geometries in plane stress. HyperStudy was used to derive parameter values that minimised the square error function E, given by

a



X

ri1;exp 1 i r1;cal

i

!2 þ

X i

ri2;exp 1 i r2;cal

!2 ;

where i runs through the total number of experimental data points for the three strain rates, and the subscripts exp and cal refer to the experimental and calculated results respectively. The optimised parameters are given in Table 2. The Y arm parameter V Ys corresponds to an activation vY

volume v Ys , such that V Ys ¼ kTs , with k Boltzmann’s constant and T the absolute temperature, giving a shear activation volume of 6 nm3. This is very similar to the values found by Zhou and Mallick (2002) for polypropylene, which ranged from 4.7 nm3 at 21.5°C to 6.1 nm3 at 100 °C, and were obtained from the analysis of the strain rate dependence of yield stress. Similarly for polypropylene, Teoh et

8

7

Stress / MPa

6

5

4

3 First stretch 2 Second stretch First stretch model

1

Second stretch model 0 0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Time / s

b

8

7 First stretch

6

Stress / MPa

Second stretch 5

First stretch model Second stretch model

4

3

2

1

0 0

20

40

60

80

100

120

Time / s Fig. 15. Observations and model for loading at 2.17 s1 (a) and complete loading and stress relaxation (b).

2

68

J. Sweeney et al. / Mechanics of Materials 54 (2012) 55–69

al. (1994) derived a value of 6.1 nm3 at room temperature and Seguela et al. (1999) found a value of 7.7 nm3 at 60 °C. Our activation volume is thus similar to many of those arrived at by other workers. For the X arm, the relatively large value of V Xs -corresponding to an activation volume v Xs of 78 nm3 for the optimised case-is by contrast quite dissimilar from other reported values. However, we have not seen activation volumes reported at temperatures this close to the melting point, other than by one of the present authors (Sweeney et al., 2009a). In that study of a different grade of polypropylene stress relaxation was not included, and a single activation volume of 39 nm3 was derived from the rate dependence of stress. 6. Model performance The predicted loading and stress relaxation curves, obtained using the optimised parameters in Table 2, are compared with observations in Figs. 13–15 for the three loading rates. The strain history inputs were as observed experimentally and included the small time lags between the end of the application of the first stretch and the beginning of the application of the second. For loading (Figs. 13(a), 14(a) and 15(a)) the model satisfactorily predicts that the stress peak on completing the second stretch is less than that associated with the first. The driving stresses sY of Eq. (30) are, for the same principal stresses, higher for the state of equibiaxial strain at the end of the second stretch than for the state of planar extension at the end of the first. This results in a higher plastic strain rate and a lower level of stress for the second peak, as observed experimentally. The present model gives a good quantitative fit for the peak values, in contrast to that which would be obtained with the conventional Levy–Mises flow rule, for which the first and second peaks would be of equal magnitude. For stress relaxation, model predictions using the optimised parameters are significantly in error, particularly during the first 40 s. Better agreement is obtained using the analytical parameter values, since the Guiu–Pratt parameters that relate specifically to stress relaxation are used directly. However, this is at the expense of poorer fits during loading, where the initial yield points are predicted with greater error. Quantitatively the predictions are at a useful level of accuracy, and certainly far superior to those obtainable with a conventional flow rule. 7. Conclusions After fast loading at processing temperatures, the stress in polypropylene relaxes to less than half its initial level within 20 s. This has significant implications for process modelling. Eyring processes possess inherent capabilities that enable them to readily represent rate dependence of stress and stress relaxation, so that effective models can be created using only a small number of elements. The Guiu–Pratt equation is useful for acquiring Eyring parameters for the longer term processes governing stress relaxa-

tion. The examination of strain rate dependence of stress is useful for the shorter term processes. A constitutive model has been developed that gives a good representation of multiaxial loading followed by stress relaxation. Rate dependent plastic flow is represented by separate Eyring mechanisms for long and short term processes. In conjunction with the Eyring elements, an essential component of this model is a flow rule that takes account of the anisotropic state of the polymer. This is based on the Hill criterion with coefficients derived from a power law function of the principal total extension ratios. The use of sequential loading to an equibiaxial state of strain is particularly revealing of the performance of the flow rule used in any constitutive model. In this case it demonstrated the inappropriateness of the Levy–Mises flow rule, which gave qualitatively incorrect predictions. The alternative flow rule was found to give a good performance. For the Levy–Mises flow rule, plastic flow depends only on the stress field and takes no account of material anisotropy. In contrast, the flow rule introduced here takes account of anisotropy as it develops during deformation. The model has been implemented within a finite element package, and has the capability for application in process simulations.

References Bauwens-Crowet, C., Bauwens, J.C., Homes, G.J., 1969. Tensile yield-stress behavior of poly(vinyl chloride) and polycarbonate in the glass transition region. J. Polymer Sci. Part A-2: Polymer Phys. 7, 1745– 1754. Bauwens-Crowet, C., Bauwens, J.-C., 1972. The temperature dependence of yield of polycarbonate in uniaxial compression and tensile tests. J. Math. Sci. 7, 176–183. Buckley, C.P., Jones, D.C., 1995. Glass–rubber constitutive model for amorphous polymers near the glass transition. Polymer 36, 3301– 3312. Govaert, L.E., Timmermans, P.H.M., Brekelmans, W.A.M., 2000. The influence of intrinsic strain softening on strain localization in polycarbonate: modelling and experimental validation. J. Eng. Mater. – T ASME 122, 177–185. Guiu, F., Pratt, P.L., 1964. Stress relaxation and the plastic deformation of solids. Phys. Status Solidi. 6, 111–120. Hill, R., 1985. The Mathematical Theory of Plasticity. Oxford University Press, Oxford 1985. Joseph, S.H., Duckett, R.A., 1978. Effects of pressure on the non-linear viscoelastic behaviour of polymers:1 Polypropylene. Polymer 19, 837–849. Klompen, E.T.J., Engels, T.A.P., van Breeman, L.C.A., Schreurs, P.J.G., Govaert, L.E., Meijer, H.E.H., 2005. Quantitative prediction of longterm failure of polycarbonate. Macromolecules 38, 7009–7017. Naz, S., Sweeney, J., Coates, P.D., 2010. Analysis of the essential work of fracture method as applied to UHMWPE. J. Mater. Sci. 45, 448–459, 201. Nazarenko, S., Bensason, S., Hiltner, A., Baer, E., 1994. The effect of pressure on the necking of polycarbonate. Polymer 35, 3883–3892. O’Connor, C.P.J., Martin, P.J., Menary, G., 2010. Viscoelastic material models of polypropylene for thermoforming applications. Int. J. Mater. Form. 3, 599–602. Ogden, R.W., 1997. Non-linear Elastic Deformations. Dover Publications, Inc., New York, Chapter 4. Seguela, R., Staniel, E., Escaig, B., Fillon, B., 1999. Plastic deformation of polypropylene in relation to crystalline structure. J. Appl. Polymer Sci. 71, 1873–1885. Sweeney, J., Naz, S., Coates, P.D., 2009b. Viscoplastic constitutive modeling of polymers—Flow rules and the plane strain response. J. Appl. Polymer Sci. 111, 1190–1198. Sweeney, J., Naz, S., Coates, P.D., 2011. Modeling the tensile behavior of UHMWPE with a novel flow rule. J. Appl. Polymer Sci. 121, 2936– 2944.

J. Sweeney et al. / Mechanics of Materials 54 (2012) 55–69 Sweeney, J., Spares, R., Woodhead, M.A., 2009a. Constitutive model for large multiaxial deformations of solid polypropylene at high temperature. Polymer Eng. Sci. 49, 1902–1908. Teoh, S.H., Poo, A.N., Ong, G.B., 1994. Effective and recovery stresses in deformation studies of polyvinyl chloride and polypropylene using the modified strain transient dip test. J. Mater. Sci. 29, 4918–4926. Van Erp, T.B., Reynolds, C.T., Peijs, T., Van Dommelen, J.A.W., Govaert, L.E., 2009. Prediction of yield and long-term failure of oriented

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polypropylene: kinetics and anisotropy. J. Polymer Sci. Part B: Polymer Phys. 47, 2026–2035. Ward, I.M., Sweeney, J., 2004. An Introduction to the Mechanical Properties of Solid Polymers, second ed. Wiley, Chichester. Zhou, Y., Mallick, P.K., 2002. of temperature and strain rate on the tensile behaviour of unfilled and talc-filled polypropylene Part I: Experiments. Polymer Eng. Sci. 42, 2449–2460.