Rheologica Acta

, Volume 44, Issue 5, pp 513–520

Recoil from elongation using general network models

Authors

    • School of Aerospace, Mechanical and Mechatronic EngineeringUniversity of Sydney
  • Anthony M. Zdilar
    • School of Aerospace, Mechanical and Mechatronic EngineeringUniversity of Sydney
  • Simin Nasseri
    • School of Aerospace, Mechanical and Mechatronic EngineeringUniversity of Sydney
Original Contribution

DOI: 10.1007/s00397-005-0433-8

Cite this article as:
Tanner, R.I., Zdilar, A.M. & Nasseri, S. Rheol Acta (2005) 44: 513. doi:10.1007/s00397-005-0433-8

Abstract

In this paper, we use two new models and the irreversible KBKZ model of Wagner (Rheol Acta 18:681–692, 1979) to describe the famous experiments of Meissner (Rheol Acta 10:230–242, 1971) on the recoil of polyethylene. The new models are based both on network and reptation-type ideas. One of the new models (PTT-X) is a member of the PTT family and shows good agreement with polyethylene data in shearing, elongation, and recoil from elongation.

Keywords

PolyethyleneRecoilPTT modelElongation rheology

Introduction

This paper discusses the behaviour of some microstructure-based models in inertialess recoil from a stress state induced by an elongational flow history. Here the word ‘model’ is used to indicate a set of constitutive equations which describe the polymer rheology. The test situation to be investigated is the following:
  1. 1.

    Starting from a rest state at some initial time, to, a purely elongational flow history, not necessarily at a constant elongation rate, is imposed up to time zero (t=0).

     
  2. 2.

    At t=0 the stress state is made zero, and an instantaneous recoil occurs.

     
  3. 3.

    For t>0, further recoil may occur slowly; the stresses remain at zero.

     
This type of history is shown in Fig. 1.
Fig. 1

Strain history of recoil test. There is no stress in the sample for t>0; inertia is ignored

The reason for exploring this flow is that it was shown by Wagner (1979) that some plausible constitutive models of the KBKZ type failed to describe such recoil (they were too ‘elastic‘), although otherwise they were useful. Wagner (1979) then used his “irreversible” idea in a KBKZ framework to describe recoil quite successfully. Langouche and Debbaut (1999) investigated the recoil of high-density polyethylene (HDPE) experimentally and used a multi-mode exponential Phan-Thien-Tanner (expPTT) model (Tanner 2000) to fit the elongational recoil data. The maximum Hencky strains used were three or four, and strain rates of 0.01, 0.1 and 1 s−1 were used. Some overestimation of the recovery data at large initial strains was noted—again, the model showed excessive elasticity.

Joshi and Denn (2004) have surveyed failure and recovery in elongational flow. They concluded that the results for recovery were model-dependent, and cited the work of Wagner (1979). While the Wagner irreversible model appears to be successful, we believe the irreversibility device is somewhat mysterious, and we prefer to explore other models.

Many new models have been proposed, and often they describe steady and some transient shear and transient elongational flows quite well. However, there is little information about their elongational recoil behaviour. In double-step shear, Chodankar et al. (2003) have done some careful work, but the differences in behaviour between models are often not as marked in shear as in elongational flow, and in many ways elongation is simpler. Therefore, we shall concentrate on it here.

The constitutive relations to be explored are all of the general network theory type ; for discussions of the assumptions involved see the books of Larson (1988), Tanner (2000) and Huilgol and Phan-Thien (1997). Special cases of this model include the Lodge rubber-like model (1964) and various PTT models (Tanner and Nasseri 2003). We also consider the XPP model of Verbeeten et al. (2002). In this model, which is said to be derived using reptation theory ideas and is a development of the pom–pom model (Blackwell et al. 2000), we note that if the Giesekus-type term (1982) in it is set to zero, the model falls into the general network class (Tanner and Nasseri 2003). (Note that Lodge (1989) has insisted that the Doi-Edwards tube model gives zero recoil; this problem does not arise with the models discussed here). In the present paper, we concentrate on models that are expressible in differential forms (these are easier to incorporate in large-scale computer programs) and so the more recent models of Wagner,for example,are not included.

Hence, the present paper will address the behaviour of general network theory models as its principal concern. Inertia and compressibility will be ignored—it is believed they are of minor importance in the cases described here.

The general network model

We may follow the notation of Huilgol and Phan-Thien (1997) in presenting the model as (in single-mode form to begin with)
$$ \lambda \frac{{\Delta {\varvec{\tau }}}} {{\Delta t}} + H{\varvec{\tau }} = G_1 {\mathbf{I}} $$
(1)
where λ is a relaxation time, Δ τt is the upper-convected time derivative of the extra stress tensor τ, H is a network destruction function, G1 is a network creation function, and I is the unit tensor. We assume affine deformation (Huilgol and Phan-Thien 1997) and so
$$ \frac{{\Delta {\varvec{\tau }}}} {{\Delta t}} = \frac{{\partial {\varvec{\tau }}}} {{\partial t}} + {\varvec{\nu }} \cdot \nabla {\varvec{\tau }} - {\varvec{\tau L}}^{\text{T}} - {varvec{L\tau }} $$
(2)
where v is the velocity field and L is the velocity gradient tensor, so
$$ {\mathbf{L}}^{\text{T}} \equiv \nabla {\varvec{\nu }}\left( {\left( {{\mathbf{L}}_{ij} \equiv \partial {\varvec{\nu }}_i /\partial x_j } \right)} \right) $$
(3)
The total stress σ is related to τ and the pressure p by
$$ {\varvec{\sigma }} = - p{\mathbf{I}} + {\varvec{\tau }} $$
(4)
The functions H and G1 are actually functions of the configurations of the network strand vectors h (see Tanner 2000) in particular the average h2. In the usual form of the general network model a closure approximation is made so that H and G1 are considered as functions of tr τ. This closure is obtained as follows. In (Gaussian) network theory it is known that (Tanner 2000)
$$ {\varvec{\tau }} = {\text{constant}} \cdot \int {{\mathbf{h}}\;{\mathbf{h}}\;\varphi {\text{d}}} h^3 $$
(5)
where ϕ is the probability distribution function for h, the network end-to-end vector. Taking the trace of Eq. 5 gives
$$ {\text{tr}}\,{\varvec{\tau }} = {\text{constant}} < h^2 > $$
(6)

Hence a dependence on the average square of h can be replaced by a dependence on tr τ.

In addition, we can write the constitutive relation Eq. 1 in the integral form: (Huilgol and Phan-Thien (1997))
$$ {\mathbf{\tau }} = {\int\limits_{ - \infty }^t {\frac{1} {\lambda }} }G_{1} ({t}\ifmmode{'}\else$'$\fi)\exp {\left( { - \frac{1} {\lambda }{\int\limits_{{t}\ifmmode{'}\else$'$\fi}^t {H({t}\ifmmode{''}\else$''$\fi)\,{\text{d}}{t}\ifmmode{''}\else$''$\fi} }} \right)}{\mathbf{C}}^{{ - 1}} ({t}\ifmmode{'}\else$'$\fi)\,{\text{d}}{t}\ifmmode{'}\else$'$\fi $$
(7)
We assume no slip in Eq. 7, so C−1 (t′) is the Finger strain tensor at time t′ relative to the configuration at time t. When H=1, G1o/λ, Eq. 7 reduces to the Lodge (1964) rubber-like model:
$${\varvec{\tau }} = \int\limits_{ - \infty }^t {\frac{{\eta _{\text{o}} }} {{\lambda ^2 }}} \exp \left\{ { - (t - t')/\lambda } \right\}{\mathbf{C}}^{ - 1} (t')\,{\text{d}}t' $$
(8)
In addition, we may define a new stress S
$$ {\varvec{\tau }} = \left( {\eta _{\text{o}} /\lambda } \right){\mathbf{I}} + {\mathbf{S}} $$
(9)
where ηo is the zero-shear rate viscosity.
Substitution in Eq. 1 gives the result
$$ \lambda \frac{{\Delta {\mathbf{S}}}} {{\Delta t}} + H{\mathbf{S}} + \left[ {\left( {\eta _{\text{o}} /\lambda } \right)H - G_1 } \right]{\mathbf{I}} = 2\eta _{\text{o}} {\mathbf{d}} $$
(10)
where 2d=L+LT, and H and G1 are now regarded as functions of tr S. The form Eq. 10 is the XPP model of Verbeeten et al. (2002) when the Giesekus term in that model is set to zero (α=0 in their notation) and affine motion is assumed. Thus, the general network model covers a number of cases. For example, when H=1, G1o/λ, then Eq. 10 becomes the familiar upper-convected Maxwell (UCM) model (Tanner 2000); when G1=Hηo/λ, the third term vanishes, and the resulting model is a member of the general PTT family.

For the XPP model, G1o/λ. In general, a single relaxation time λ is not sufficient to describe realistically the response of a typical polymer, and usually several λ-values will be needed. This will be considered next.

Model responses in simple flows

Before looking at recoil, we mention the capabilities of various forms of the general network model in steady and transient shear and elongational flows. Tanner and Nasseri (2003) have discussed four-mode models of the so-called XPP (Verbeeten et al. 2002) and PTT-X models, as well as the usual exponential PTT model (Tanner 2000). (In Tanner and Nasseri (2003), what we refer to here as the PTT-X model is called the PTT-XPP model, which is too long a name for comfort).

Comparison of XPP and PTT-X models for steady and transient shear flow of a low-density polyethylene (LDPE) show they give fairly similar predictions. These are, at least for the steady shear flow PTT-X model, very close to experimental data. In both of these models, the form used for H is as follows:
$$ H = \Lambda ^{- 2} + 2r(1 - \Lambda ^{- 1} )\exp \left[ {v(\Lambda - 1)} \right] $$
(11)
where \(\Lambda = \left( {1 + ({\lambda \mathord{\left/ {\vphantom {\lambda 3}} \right. \kern-\nulldelimiterspace} 3})({{{\text{tr}}\,{\mathbf{S}}} \mathord{\left/ {\vphantom {{{\text{tr}}\,{\mathbf{S}}} {\eta _{\text{o}} }}} \right. \kern-\nulldelimiterspace} {\eta _{\text{o}} }})} \right)^{1/2} ;\)r and ν are parameters. (In a multi-mode model, each mode has its own value of r, ν, λ and ηo).
We have shown (Tanner and Nasseri 2003) that the transient (and steady) elongational behaviour of both single-mode models is very similar, and the same is true for multi-mode systems. (They also showed that the α-parameter of Verbeeten et al. (2002) played an insignificant role in the response: here it is set to zero). We compare with experimental data for an LDPE (Verbeeten et al. 2002) using the parameters of Table 1 in a four-model model. The close fit of the experimental data by both models suggests that they may be useful in elongational recoil calculations (Fig. 2a, b).
Table 1

Linear and non-linear parameters for fitting of the DSM Stamylan LD 2008 XC43 LDPE melt

Mode

Maxwell parameters

XPP and PTT-X parameters

i

Gi (Pa)

λi (s)

qi=2/νi

rib,is,i

1

7.2006×104

3.8946×10−3

1

7.0

2

1.5770×104

5.1390×10−2

1

5.0

3

3.3340×103

5.0349×10−1

2

3.0

4

3.0080×102

4.5911×100

10

1.1

The parameters are obtained from Veerbeeten et al. (2002). Temperature 170 °C

Fig. 2

Quasi-steady state (a) and transient uniaxial viscosity (b); showing a comparison between models and experimental data. The PTT-X and XPP models are used to describe the response of DSM Stamylan LD2008 XC403 low-density polyethylene (LDPE) at 170 °C. Parameters used in the models show in Table 1. Symbols show in experimental data

Recoil with a single relaxation time

In the case of a single relaxation time, we know, from the results of Lodge et al. (1965) and Zdilar and Tanner (1992), that sudden unloading does not engender post-jump recoil, so that only the instantaneous jump (Fig. 1) has to be found. It is convenient to use the integral form Eq. 7 and split the deformation into two parts—the continuous flow part (Fc) and the strain jump, defined by Fo. The deformation gradient F relating the configuration at time t′ to the final shape is then (Tanner 2000, p. 433)
$$ {\mathbf{F}} = {\mathbf{F}}_{\text{c}} {\mathbf{F}}_{\text{o}} $$
where Fo occurs at time zero. Explicitly, in terms of the final coordinate x (at t=0+) and pre-jump coordinate x* (at t=0−) Fo is given by
$$ {\mathbf{F}}_{\text{o}} = \frac{{\partial {\mathbf{x}}^ * }} {{\partial {\mathbf{x}}}} $$
(12)
The component Fc is given as
$$ {\mathbf{F}}_{\text{c}} = \frac{{\partial {\mathbf{r}}}} {{\partial {\mathbf{x}}^ * }} $$
(13)
where r is the (particle) coordinate at time t′. Hence, since (Tanner 2000) C(t′)=FTF, we find
$$ \begin{aligned} {\mathbf{C}}^{ - 1} (t') = & {\mathbf{F}}_{\text{o}}^{ - 1} {\mathbf{F}}_{\text{c}}^{ - 1} {\mathbf{F}}_{\text{c}}^{ - {\text{T}}} {\mathbf{F}}_{\text{o}}^{ - {\text{T}}} \\ = & {\mathbf{F}}{}_{\text{o}}^{ - 1} {\mathbf{C}}_{\text{c}}^{ - 1} {\mathbf{F}}_{\text{o}}^{ - {\text{T}}} \\ \end{aligned} $$
(14)
where Cc−1 is a continuous strain history relative to x* at t=0−. Substitution of Eq. 14 into Eq. 7 and noting that Fo is not a function of t′, we get
$$ {\varvec{\sigma }} + p{\mathbf{I}} = {\varvec{\tau }} = {\mathbf{F}}_{\text{o}}^{ - 1} \left[ {\int\limits_{ - \infty }^t {\frac{1} {\lambda }G_1 \exp \left( { - \frac{1} {\lambda }\int\limits_{t'}^t {H\,{\text{d}}t''} } \right){\mathbf{C}}_{\text{c}}^{ - 1} (t')\,{\text{d}}t'} } \right]{\mathbf{F}}_{\text{o}}^{ - {\text{T}}} $$
(15)
For t=0+, the total stress σ is zero, and hence, following Lodge (1964), we have
$$ p{\mathbf{F}}_{\text{o}} {\mathbf{F}}_{\text{o}}^{\text{T}} = {\varvec{\tau }}^ - $$
(16)
where τ is the stress just before the jump. Taking the determinant of Eq. 16 (and noting det Fo=1), we have
$$ p^3 = \det {\varvec{\tau }}^ - $$
(17)
or
$$ {\mathbf{F}}_{\text{o}} {\mathbf{F}}_{\text{o}}^{\text{T}} = \frac{{{\varvec{\tau }}^ - }} {{\left( {\det {\text{ }}{\varvec{\tau }}^ - } \right)^{1/3} }} $$
(18)
If we now concentrate on axisymmetric extensions, so that
$$ {\mathbf{F}}_{\text{o}}^{\text{T}} = {\mathbf{F}}_{\text{o}} = {\text{diag}}\left( {\varepsilon ^{ - 1} ,\varepsilon ^{1/2} ,\varepsilon ^{1/2} } \right),{\varvec{\tau }}^ - = {\text{diag}}(\tau _1 ,\tau _2 ,\tau _3 ) $$
(19)
then we find
$$ \varepsilon ^{- 1} = \left( {\tau _1 /\tau _2} \right)^{1/3} $$
(20)

Here ε−1 is the ratio of the length just before recoil to the length after recoil. Thus the prediction of the instantaneous recoil can be done using only the results from the continuous history.

In the case of the Wagner (1979) relation, where strain measures are included in the kernel, no simple resolution of the equations occurs, and the recoil must be found by a numerical iterative process.

Generally, the Hencky (Tanner and Tanner 2003) strain is used to report recoil, so if we denote the recoverable strain as εr, then
$$ \varepsilon _{\text{r}} = {\text{ln}}(\varepsilon ^{ - 1} ) = \frac{1} {3}\ln \left( {\frac{{\tau _1 }} {{\tau _2 }}} \right) $$
(21)
where τ1 and τ2 are stresses in the continuous history before the jump.

Some examples of instantaneous recoil

We now assume a single relaxation time model of the type given in Eq. 1. We can make this equation dimensionless by letting t~t/λ, LL, G1~G1/G (where G is a modulus equal to ηo/λ), τ ~ τ/G; H is already dimensionless.

Since the recoil, from Eq. 20, only depends on the stresses just before the jump, we shall assume a state of steady elongation before recoil. When the elongation rate ( \(\dot \varepsilon \)) is high, we can follow earlier results (Tanner and Nasseri 2003) to see that the pre-jump stresses obey, for arbitrary H and G1, the equations (recall \(\dot \varepsilon\) is now dimensionless and constant).
$$ - 2\dot \varepsilon \tau _1 + \tau _1 H = G_1 ;\quad \dot \varepsilon \tau _2 + \tau _2 H = G_1 ;\quad {\text{tr}}\,\tau = \tau _1 + 2\tau _2 $$
(22)
where τ1 is in the axial direction; τ23, and \(\dot \varepsilon \) is the dimensionless elongation rate.
Hence,
$$ \tau _1 = \frac{{G_1 }} {{(H - 2\dot \varepsilon )}};\quad \tau _2 = \frac{{G_1 }} {{(H + \dot \varepsilon )}} $$
(23)
For large \(\dot \varepsilon \), τ1 is large and must remain positive, so H~2\(\dot \varepsilon \), and tr τ~τ1, so we may find τ1. For all general network models, we may write H=2\(\dot \varepsilon \)+G11 and for large \(\dot \varepsilon \), one has
$$ \tau _2 = \frac{{G_1 }} {{H + \dot \varepsilon }} = \frac{{G_1 }} {{3\dot \varepsilon + G_1 /\tau _1 }} \approx \frac{{G_1 }} {{3\dot \varepsilon }} $$
(24)
Also, for all general network models with the same H function, the result for τ1 is the same at high enough extension rates (Tanner and Nasseri 2003). So, we can write
$$ \frac{{\tau _1}} {{\tau _2}} = \frac{{3\dot \varepsilon \tau _1}} {{G_1}} $$
(25)
For the XPP model G1=1 (made dimensionless with ηo/λ and for all PTT models, G1=H~2 \(\dot \varepsilon\) at high elongation rates. Hence, the results are
$$ \left( {\frac{{\tau _1 }} {{\tau _2 }}} \right)_{{\text{XPP}}} = 3\dot \varepsilon \tau _1 ;\quad \left( {\frac{{\tau _1 }} {{\tau _2 }}} \right)_{{\text{PTT}}} = \frac{{3\tau _1 }} {2} $$
(26)

The asymptotic results for τ1 have been given for several specific H-forms in our previous paper (Tanner and Nasseri 2003).

For the exponential PTT, where H=exp(ε*(tr τ−3)), τ1 has the form ε*-1ln \(\dot \varepsilon \) for large \(\dot \varepsilon \), and so the (Hencky) recoil strain is
$$ \varepsilon _{\text{r}} = \frac{1} {3}\ln \left[ {\left( {\frac{3} {{2\varepsilon ^* }}} \right)\ln \dot \varepsilon } \right] $$
(27)
which increases very slowly with \(\dot \varepsilon \).
For the XPP and PTT-X models where H has a more complex form, we find the results in Table 2. Here the PTT-X and XPP models both have the same H-function:
$$ H = \Lambda ^{- 2} + 2r(1 - \Lambda ^{- 1} )\exp \left[ {\nu (\Lambda - 1)} \right] $$
(28)
where \(\Lambda = (1 + (1/3){\text{tr}}\,S)^{1/2} ,\) and r and ν are parameters.
Table 2

εr as function of dimensionless \(\dot \varepsilon \)\(\dot \varepsilon \))

 

Exp-PTT

PTT-X

XPP

\({\dot \varepsilon}\)

ε*=0.25

ν=1, r=3

ν=1, r=3

10

0.875

1.145

2.14

100

1.106

1.526

3.29

1,000

1.241

1.789

3.55

In Table 2, we have chosen the moderate values ν=1, r=3 for illustrative purposes (compare with Table 1). Surprisingly, there is a tremendous range of recoil shown by these very similar models. The XPP model recoils about two to three times as much as the other models. These results may be confirmed by calculation; and we will return to the presentation of Wagner (1979) in the following figures. The exp PTT model, if compared at the same stress to the PTT-X model, will give the same recoil (Eq. 26) for high strain rates. Thus one should not compare the two PTT models directly in Table 2.

For the Lodge model, τ12=(1+ \(\dot \varepsilon \))/(1−2 \(\dot \varepsilon \)) which rises indefinitely as \(\dot \varepsilon \to 0.5\); it can only be considered for transient (start-up from −to) flows. We now consider the effect of multiple relaxation times.

Multiple relaxation times

In this case, the analysis for the jump strain may be repeated and it is clear that Eq. 18 holds also for multiple relaxation times. The pre-jump stress state (τ) is in this case the sum over the multiple modes:
$${\varvec{\tau }}^ - = \sum\limits_{i - 1}^n {{\varvec{\tau }}_i^ - } $$
(29)
where i is the number of modes. Each mode may have different λ, H and G functions. We find that Eqs. 17, 20 and 21 still hold, and
$$ \varepsilon _{\text{r}} = \frac{1} {3}\ln \left( {\frac{{\sum {(\tau _i )_1 } }} {{\sum {(\tau _i )_2 } }}} \right) $$
(30)
For the linear viscoelastic case, we can illustrate the effect of multiple modes explicitly. In this case (Tanner 2000),
$$ {\varvec{\sigma }} = - p{\mathbf{I}} + 2\int\limits_{ - \infty }^t {G(t - t'){\mathbf{d}}(t')\,{\text{d}}t'} $$
(31)
Let the stretching direction be 1, the transverse direction 2. Then σ22=0 at all times, so p is determined, and hence the axial stress σ (σ11) is given by (since d22= \( - \dot \varepsilon /2\))
$$\sigma = 3\int\limits_{- \infty}^t {G(t - t')\dot \varepsilon (t'){\text{d}}t'} $$
(32)
where \(\dot \varepsilon (t)\) is the extension rate in the axial direction at time t.
Suppose the stress at t=0, just before any jump, is σ. Then
$$\sigma (0^ -) = 3\int\limits_{- \infty}^0 {G(t - t')\dot \varepsilon (t'){\text{d}}t'} $$
(33)
Let us split G(t) into modes:
$$G(t) = \sum\limits_i {\frac{{\eta _i}}{{\lambda _i}}} \exp (- t/\lambda _i)$$
(34)
Then σ=Σ σi say, at t=0, and for t≥0, since σ=0,
$$ 0 = \sigma = \sum\limits_i {\sigma _i^{\text{o}} } \exp ( - t/\lambda _i ) + 3\int\limits_{\text{o}}^t {G(t - t')\dot \varepsilon (t')\,{\text{d}}t'} $$
(35)
The initial jump at t=0 can be found to be
$$ \varepsilon (0^ + ) \sim - \frac{{\frac{{\sigma ^0 }} {3}}} {{\left( {\sum {g_i } } \right)}} $$
(36)
which is physically understandable as the recoil against the combined elasticity or “springs” of all the modes, ∑gi.
Similarly, the total recoil as t → ∞ can be found to be:
$$ \varepsilon (\infty ) = - \frac{{\sum {\sigma _i^{\text{o}} \lambda _i } }} {{3\sum {\eta _i } }} $$
(37)
where ηi=giλi.
As an example, if \( \dot {{\varvec{\epsilon}}} \) is constant, equal to \( \dot {{\varvec{\epsilon}}}_0 \) for t<0, then we find
$$\frac{{\varepsilon (0^ +)}}{{\varepsilon (\infty)}} = \frac{{\left({\sum {\eta _i}} \right)^2}}{{\sum {(\eta _i /\lambda _i)\sum {(\eta _i \lambda _i)}}}}.$$
(38)

This ratio is clearly less than one, except for a single mode. Thus, generally the sudden jump decreases as a percentage of the total recoil when multiple modes are assumed.

We can compare the total recoil of the multi-mode result to an equivalent single-mode model; clearly the effective average relaxation time here is \(\left({\sum {\eta _i \lambda _i} /\sum {\eta _i}} \right),\) which is heavily biased toward the longer relaxation times.

Unfortunately, the experiments quoted by Wagner (1979) lie far outside the linear theory, and a numerical approach to calculating long-term recoil is needed. In order to compare with the IUPAC low-density polyethylene experiments of Meissner (1971), we use 9-mode XPP and PTT-X models, using H as in Eq. 28. In Eq. 10, if Go/λ, then we have the XPP model, and if G1λ/ηo=H then the PTT-X model is found (Tanner and Nasseri 2003). The material properties of the IUPAC low-density polyethylene have been described by Blackwell et al. (2000) and have been used here, see Table 3. The linear spectrum is from Laun (1986). The results were computed using an iterative procedure described by Zdilar and Tanner (1992), with the addition of an adaptive time stepping scheme. Time stepping started at 10−6 s and went up to a maximum of 10−2 s. Convergence was set at 500 s.
Table 3

The mode parameters used for the fitting of IUPAC A/LDPE

Mode no., i

Modulus, gi (Pa)

λi (s)

qi

ri

1

1.520×105

1.0×10−3

1.0

2.0

2

4.005×104

5.0×10−3

1.0

2.0

3

3.326×104

2.8×10−2

2.0

2.0

4

1.659×104

1.4×10−1

2.0

2.5

5

8.690×103

7.0×10−1

4.0

2.0

6

3.151×103

3.8×100

7.0

2.0

7

8.596×102

2.0×101

8.0

1.5

8

1.283×102

1.0×102

12.0

1.0

9

1.8495×100

5.0×102

30.0

1.0

The values for qi and ri are from Blackwell et al. (2000) and the linear spectrum is from Laun (1986). The νi=2/qi in Eq. 28

The results are shown in Fig. 3. At a strain-rate of 0.01 s−1, both models fit the experimental data very well. At 0.1 s−1 the models begin to diverge and a fairly large difference is evident at \(\dot \varepsilon = 1.0\,{\text{s}}^{ - 1} \). A reasonable agreement with experiment is shown by the PTT-X model overall, with slightly too much predicted recoil for larger strains. The XPP model is clearly too elastic in its recoil behaviour, as one would expect from Table 2.
Fig. 3

Recoverable strain as a function of total strain and strain rate for the IUPAC LDPE at 150 °C. Solid lines XPP and PTT-X models. Dashed lines purely elastic response (recoverable strain =  \(\dot \varepsilon _{\text{o}} t\)) at top; Wagner calculations (1979) below. Squares experimental data

The results of Wagner’s irreversible hypothesis (1979) are also plotted in Fig. 3 (dashed lines). These lie somewhat closer to the experimental data than the PTT-X model results, but the differences are small, especially for large total strain and large recoverable strain.

Figure 4 shows the complete recovery curves from a total strain \(\left( {\dot \varepsilon _{\text{o}} t} \right)\) of 4. The recovery is very slow.
Fig. 4

Time-dependence recovery for a total strain of four for two models

Conclusion

The above-mentioned results confirm the value of the PTT-X model, at least for low-density polyethylene modelling. While it has previously been shown to give good results in shearing, we show here that these extend to unsteady elongation and recoil flows.

Its obvious drawback from the user’s point of view is the larger number of parameters involved—as opposed to the exponential PTT model, say. However, since it is based on both a molecular model and network theory, it is probably suitable for both branched and unbranched polymers, but further work will be needed to confirm this.

Acknowledgements

We thank the Australian Research Council, the Polymer Cooperative Research Centre and the University of Sydney for supporting this investigation.

Copyright information

© Springer-Verlag 2005