1 Introduction

Polyethylene is one of the most common polymers and is used in a vast array of consumer goods. In fact, polyethylene and polypropylene represent about 92% of the synthetic plastics produced [23]. The mechanical behaviour of this polymer is therefore of great practical interest. In the present study, stress relaxation in a high-density polyethylene (HDPE) and under finite strain conditions is investigated both experimentally and theoretically.

Polyethylene (PE) that solidifies at rest forms a spherulitic microstructure, which is essentially isotropic from a mechanical point of view [28]. PE is a very ductile polymer and can be deformed up to 100% of straining and beyond. At large strains, this means that the crystallites are disrupted, and the polymer chains align in the direction of deformation. The microstructural processes associated with large straining of HDPE and other semi-crystalline polymers have been documented in a number of studies (e.g. [3, 4, 10, 21, 28, 38, 39]). With regard to the macroscopic mechanical response, PE shows a strong strain rate dependence, and increasing strain-rate generally causes loss of ductility [19, 50, 57].

Due to its wide usage and industrial importance, polyethylene is one of the most investigated polymers. Different aspects of the mechanical behaviour of HDPE have been investigated in previous studies, including the response during monotonic loading [13, 43, 47, 66] as well as during creep and relaxation loading [17, 27, 42, 62].

In applications, polyethylene often undergo large deformations, and constitutive models based on a theory for finite strains are therefore needed. Many modelling approaches are based on the multiplicative decomposition of the deformation gradient into elastic and inelastic parts [11, 30, 37]. Some models are derived on the basis of microstructural considerations, and mixture theory is often used to account for the contributions from the amorphous and crystalline phases, e.g. Parks and Ahzi [46], Lee et al. [36], van Dommelen et al. [16], Nikolov et al. [43], Dusunceli and Colak [18], Selles et al. [56], and Sreejith et al. [63]. Most models, however, are more phenomenologically oriented, and many models have been proposed over the last years, e.g. Ayoub et al. [6], Zeng et al. [66], Balieu et al. [7], Pouriayevali et al. [48], Shojaei and Li [58], Uchida and Tada [65], Krairi and Doghri [29], Abdul-Hameed et al. [1], Garcia-Gonzalez et al. [22], Ayadi et al. [5], Barriere et al. [9], Qi et al. [49], Barba et al. [8], Felder et al. [20], Cundiff et al. [14], Hao et al. [25], Satouri et al. [55], Song et al. [61], Makki et al. [40], Reuvers et al. [51], Zeng et al. [67].

The models listed in the previous paragraph can all be said to be Lagrangian in the sense that they depend on measures of total and inelastic deformation. The model proposed here is Eulerian in the sense that all state variables are defined in the current state of the material (cf. [44]), and the evolution of elastic deformations is prescribed directly. The basic framework of the model has been established before (e.g. [2, 32,33,34,35, 52,53,54]). In these previous works, the same basic Eulerian framework has been applied for modelling residual stresses in metals during melting and solidification [32], general directional hardening [33], residual stresses at welds [35], volumetric inelasticity in semi-crystalline polymers [34], and stress relaxation and damage in semi-crystalline polymers [2]. In the present work, the ability to model stress relaxation at multiple time scales is added. The proposed model is calibrated by use of uniaxial tensile tests, both monotonic loading tests and stress relaxation tests. Relaxation at different initial stress levels and for different time periods is considered. The material model is also implemented in Abaqus as a user subroutine (UMAT), i.e. a fully implicit numerical implementation, including an analytic tangent modulus. The mechanical response of a 3D geometry is also tested experimentally, and these results are used to validate the material model.

The paper is organised as follows: Sect. 2 describes the experimental setups. The constitutive model is presented in Sect. 3. The new features of the proposed material model are demonstrated in Sect. 4. In Sect. 5, the material model is compared to and calibrated by use of uniaxial tensile test results. In Sect. 6, the UMAT implementation is validated by applying it to compression testing of a 3D structure. Section 7 contains a discussion and some concluding remarks.

2 Experiments

2.1 Uniaxial tensile tests

The uniaxial tensile stress relaxation tests used in the present work have been published previously in a master thesis [24]. These tests were performed on high-density polyethylene (HDPE), which is a semi-crystalline polymer. The HDPE used had a melt flow index of 26 g/10 min (190 \(^\circ \)C, 2.16 kg) and a density of 0.953 g/cm\(^3\). The test specimen used followed the standard ISO 527-2-1BA and is shown in Fig. 1. The specimen was flat with a rectangular cross-section, and the specimens were punched out from test plates with a thickness of \(T_0=0.60\) mm. The test plates were manufactured through injection-moulding (ISO 294-5). The width of the specimen in the test region was \(W_0=5.0\) mm and the initial distance between the grips in the testing machine was \(L_0=60\) mm. The resulting initial cross-sectional area in the test region was \(A_0=W_0T_0=3.0\) mm\(^2\).

Fig. 1
figure 1

Test specimen used in tensile tests

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During the tests, force versus time was recorded using a load cell and a computer. The raw data from the tests consisted of force versus time data, F(t), and load-line displacement versus time data, \(\delta (t)\).

The information above is enough to establish the nominal stress vs. engineering strain relation for the different load cases and also the evolution of the engineering stress versus time at different constant engineering strain levels.

For all the tests, the grip separation rate was \(\dot{\delta }=30\) mm/min. This resulted in an engineering strain rate in the test specimens of \(\dot{\delta }/L_0=0.00833\)/s.

In the stress relaxation tests, the test specimen was first loaded up to a prescribed force, \(F_0\). After that, the grip separation in the testing machine was kept constant, and the evolution of force, F(t), was registered. Results from 5 different initial force levels are presented: \(F_0=30\) N, 40 N, 60 N, 65 N, and 69 N. In general, three tests for each load-case were performed.

2.2 Compression testing of 3D geometry

A 3D geometry was also manufactured using injection-moulding and using the same material as in the uniaxial tensile testing, see Fig. 2. Figure 2a shows an example of the physical specimen, both the whole specimen and the cross-section of the specimen. Figure 2b shows the setup for the compression test. The 3D geometry was loaded in compression between two rigid blocks, as indicated in Fig. 2. The applied force, F(t), and the load-line displacement, \(\delta (t)\), were registered. The tests were performed in displacement control. The 3D geometry was loaded in compression up to a prescribed force \(F_0\) and using the displacement rate \(\dot{\delta }=0.2\) mm/s. Two target forces were used: \(F_0=150\) N and 200 N. Once the target force had been attained, the displacement was kept constant for a time of 3 h, allowing the geometry to relax, corresponding to a decreasing force F(t). After 3 h, the geometry was unloaded by moving the cross-head of the testing machine back to \(\delta =0\) using the same displacement rate as during loading, i.e. \(\dot{\delta }=-0.2\) mm/s.

Fig. 2
figure 2

3D geometry tested in compression

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3 Constitutive model

3.1 Kinematics

The position of a material point in the deformed configuration is denoted by the vector \({\textbf {x}}\), and the velocity is \({\textbf {v}}=\dot{{\textbf {x}}}\), where \(\dot{\overline{(\bullet )}}\) denotes differentiation with respect to time. The velocity gradient \({\textbf {L}}\) and the rate of deformation tensor \({\textbf {D}}\) are defined by

$$\begin{aligned} {\textbf {L}}= \frac{\partial {\textbf {v}}}{\partial {\textbf {x}}} \,,\quad {\textbf {D}}= \frac{1}{2}({\textbf {L}}+{\textbf {L}}^T) \,. \end{aligned}$$
(1)

The deformation state of the material is decomposed into its volumetric and distortional parts. The elastic distortional deformation is modelled by the symmetric, positive definite, unimodular tensor \(\bar{{\textbf {B}}}_\textrm{e}\) (i.e. \(\textrm{det}\bar{{\textbf {B}}}_\textrm{e}\equiv 1\)). The entity \(\bar{{\textbf {B}}}_\textrm{e}\) can be interpreted as the elastic left Cauchy–Green deformation tensor in the Lagrangian theory. The evolution of this tensor is given by

$$\begin{aligned} \dot{\bar{{\textbf {B}}}}_\textrm{e} = {\textbf {L}}\bar{{\textbf {B}}}_\textrm{e} + \bar{{\textbf {B}}}_\textrm{e}{\textbf {L}}^\textrm{T} - \frac{2}{3}({\textbf {D}}:{\textbf {I}})\bar{{\textbf {B}}}_\textrm{e} - \varGamma \bar{{\textbf {A}}}_\textrm{p}, \quad \varGamma \ge 0, \end{aligned}$$
(2)

where \({\textbf {Q}}:{\textbf {R}}= \textrm{tr}({\textbf {Q}}{\textbf {R}}^\textrm{T})\) (\({\textbf {Q}}\) and \({\textbf {R}}\) are just two general second-order tensors), and \({\textbf {I}}\) is the second order unit tensor. The first two terms in (2) on the right hand side can be seen as being related to objectivity. The third term ensures that \(\bar{{\textbf {B}}}_\textrm{e}\) remains unimodular, i.e. \(\textrm{det}\bar{{\textbf {B}}}_\textrm{e}=1\). Finally, the fourth term is where the actual constitutive behaviour comes into play. A detailed description of how the present Eulerian formulation of kinematics relates to the more standard Lagrangian formulations is provided in “Appendix A”.

The rate of (distortional) inelastic deformations is defined by \(\varGamma \ge 0\), which plays the role of the plastic multiplier in standard Lagrangian formulations of plasticity. The direction of inelastic deformations is determined by the tensor \(\bar{{\textbf {A}}}_\textrm{p}\), which takes the form

$$\begin{aligned} \bar{{\textbf {A}}}_\textrm{p} = \bar{{\textbf {B}}}_\textrm{e} - \left( \frac{3}{\bar{{\textbf {B}}}_\textrm{e}^{-1}:\textbf{I}}\right) \textbf{I}, \end{aligned}$$
(3)

where

$$\begin{aligned} \bar{{\textbf {A}}}_\textrm{p}: \bar{{\textbf {B}}}_\textrm{e}^{-1} = 0. \end{aligned}$$
(4)

This choice of \(\bar{{\textbf {A}}}_\textrm{p}\) ensures that \(\bar{{\textbf {B}}}_\textrm{e}\) remains unimodular. Furthermore, in standard formulations of plasticity, the directions of plastic deformations are usually derived from a plastic potential (the yield criterion in the case of associative formulations). In the present formulation, the direction of plastic/inelastic deformations, \(\bar{{\textbf {A}}}_\textrm{p}\), is not related to any plastic potential but is simply prescribed directly according to (3).

Moreover, the volumetric part of the deformation, the dilatation J, evolves according to

$$\begin{aligned} \frac{\dot{J}}{J} = {\textbf {D}}:{\textbf {I}}. \end{aligned}$$
(5)

In the present model, the yield criterion is formulated in strain space. The equivalent elastic distortional deformation, corresponding to the von Mises stress in stress-based formulations, is given by

$$\begin{aligned} \gamma _\textrm{e} = \frac{1}{2} \sqrt{\frac{3}{2}\bar{{\textbf {B}}}'_\textrm{e}:\bar{{\textbf {B}}}'_\textrm{e}}, \end{aligned}$$
(6)

where

$$\begin{aligned} \bar{{\textbf {B}}}'_\textrm{e} = \bar{{\textbf {B}}}_\textrm{e} - \frac{1}{3}(\bar{{\textbf {B}}}_\textrm{e}:{\textbf {I}}){\textbf {I}} \end{aligned}$$
(7)

is the deviatoric part of \(\bar{{\textbf {B}}}_\textrm{e}\). Below, \((\bullet )'\) is used to denote the deviatoric part of \((\bullet )\), in accordance with the definition in (7). Also, the equivalent rate of total distortional deformation is defined as

$$\begin{aligned} \dot{\varepsilon } = \sqrt{\frac{2}{3}{\textbf {D}}':{\textbf {D}}'}, \end{aligned}$$
(8)

where \({\textbf {D}}'\) is the deviatoric part of \({\textbf {D}}\).

3.2 Strain energy and stress

The strain energy of the material, \(\varSigma \), defined per unit mass, is taken to be

$$\begin{aligned} \rho _\textrm{0}\varSigma = \frac{\mu }{2}(\alpha _1 - 3) + \frac{K}{2}(J - 1)^2, \end{aligned}$$
(9)

i.e. a standard neo-Hookean material is adopted. Above, \(\rho _\textrm{0}\) is the density of the material in the reference configuration, \(\alpha _1=\bar{{\textbf {B}}}_\textrm{e}:{\textbf {I}}\) is the first invariant of \(\bar{{\textbf {B}}}_\textrm{e}\), and \(\mu \) and K are the positive shear and bulk moduli of the material. The dilatation, J, is taken to be purely elastic, i.e. inelastic deformations are taken to be incompressible.

Assuming isothermal conditions, the rate of material dissipation requires that

$$\begin{aligned} {\textbf {T}}:{\textbf {D}}- & {} \rho \dot{\varSigma } = \ldots = \left\{ {\textbf {T}}- \frac{\rho }{\rho _\textrm{0}} \left( \mu \bar{{\textbf {B}}}'_\textrm{e} + K(J - 1)J{\textbf {I}}\right) \right\} :{\textbf {D}}\nonumber \\+ & {} \frac{\rho \mu }{2\rho _\textrm{0}} \varGamma (\bar{{\textbf {A}}}_\textrm{p}:{\textbf {I}}) \ge 0, \end{aligned}$$
(10)

where \({\textbf {T}}\) is the symmetric Cauchy stress, and \(\rho =\rho _0/J\) is the current density of the material. The Cauchy stress is taken to be

$$\begin{aligned} {\textbf {T}}= \frac{\mu }{J}\bar{{\textbf {B}}}'_\textrm{e} + K(J - 1){\textbf {I}}, \end{aligned}$$
(11)

and the von Mises stress \(\sigma _\textrm{e}\) is defined by

$$\begin{aligned} \sigma _\textrm{e} = \sqrt{\frac{3}{2}{\textbf {T}}':{\textbf {T}}'}. \end{aligned}$$
(12)

In the last term of (10), \(\mu \), \(\rho \), \(\rho _0\), and \(\varGamma \) are all positive constants or variables. Also, since \(\bar{{\textbf {B}}}_\textrm{e}\) is unimodular, \(\bar{{\textbf {A}}}_\textrm{p}:{\textbf {I}}\ge 0\) holds. Taken together, these properties ensure that the dissipation in (10) is always non-negative.

For the numerical implementation of the model, to be discussed in a later section, the Kirchhoff stress, \(\varvec{\tau }\), is also introduced as

$$\begin{aligned} \varvec{\tau }= J{\textbf {T}}, \end{aligned}$$
(13)

and when comparing the model to experiments, the first Piola–Kirchhoff stress is used, i.e.

$$\begin{aligned} {\textbf {P}}= \varvec{\tau }{\textbf {F}}^\mathrm{-T}, \end{aligned}$$
(14)

where \({\textbf {F}}\) is the deformation gradient.

3.3 Inelastic deformations

The rate of inelastic deformations, \(\varGamma \), is expressed as

$$\begin{aligned} \varGamma = \varGamma _\textrm{vp} + \varGamma _\textrm{r}, \end{aligned}$$
(15)

where \(\varGamma _\textrm{vp}\) accounts for short-term viscoplasticity, and \(\varGamma _\textrm{r}\) accounts for long-term stress relaxation. The specific forms for \(\varGamma _\textrm{vp}\) and \(\varGamma _\textrm{r}\) are chosen according to

$$\begin{aligned} \varGamma _\textrm{vp}= & {} (a_0 + b_0\dot{\varepsilon }) \langle \gamma _\textrm{e} - \kappa \rangle , \end{aligned}$$
(16)
$$\begin{aligned} \varGamma _\textrm{r}= & {} \sum _{i=1}^n c_i \langle \gamma _\textrm{e} - \zeta _i \rangle , \end{aligned}$$
(17)

where \(\kappa \) and \(\zeta _i\) are hardening variables, and \(a_0\), \(b_0\), and \(c_i\) are material constants. Specifically, \(a_0\) and \(b_0\) govern the rate-dependent and rate-independent short-term inelastic response of the material, respectively, whereas \(c_i\) can be seen at time constants for the relaxation behaviour of the material. The model is taken to consist of n relaxing terms/units, which enables modelling of stress-relaxation at different time-scales. The hardening variable \(\kappa \) models isotropic hardening in strain space. The hardening variables \(\zeta _i\) are more abstract, but model a type of hardening associated with the relaxation in the material.

The Macaulay brackets \(\langle \bullet \rangle \) are defined by

$$\begin{aligned} \langle x \rangle = \text {max}(x,0) . \end{aligned}$$
(18)

The evolution of the hardening variables \(\kappa \) and \(\zeta _i\) are governed by the functions

$$\begin{aligned} \dot{\kappa }= & {} \xi \varGamma _\textrm{vp}, \end{aligned}$$
(19)
$$\begin{aligned} \dot{\xi }= & {} m\varGamma _\textrm{vp}(\xi _\textrm{s} - \xi ), \end{aligned}$$
(20)
$$\begin{aligned} \dot{\zeta }_i= & {} n_i\dot{\varepsilon }(\zeta _{\textrm{s},i} - \zeta _i) \cdot \varTheta (\gamma _\textrm{e} - \zeta _i), \end{aligned}$$
(21)

where \(m\ge 0\) and \(n_i\ge 0\) are hardening rates, \(\zeta _{\textrm{s},i}\) are the saturated values of the hardening variables, and \(\varTheta (\bullet )\) is the Heaviside step function. The rate of the evolution of \(\kappa \) is governed by the additional state variable \(\xi \), which evolves to the saturated value \(\xi _\textrm{s}\). This is a convenient way to model the hardening of the present material. The initial values of the respective hardening variables are denoted by \(\kappa _{0}\), \(\xi _0\), and \(\zeta _{i0}\).

In principle, (20) can be solved analytically, such that \(\xi \) is expressed as a function of the time integral of \(\varGamma _\textrm{vp}\). But in the present implementation (as will be shown below), the differential relation in (20) is used, since history variables are avoided in the present model formulation.

3.4 Summary of material model

The model contains the following state variables and non-standard kinematic entities:

  • \(\bar{{\textbf {B}}}_\textrm{e}\): elastic distortional deformation

  • \(\alpha _1\): first invariant of \(\bar{{\textbf {B}}}_\textrm{e}\)

  • J: elastic dilatational deformation

  • \({\textbf {A}}_\textrm{p}\): direction of inelastic deformation

  • \(\varGamma \): rate of inelastic deformation

  • \(\varGamma _\textrm{vp}\): rate of short-term viscoplasticity

  • \(\varGamma _\textrm{r}\): rate of long-term stress relaxation

  • \(\kappa \): isotropic hardening

  • \(\xi \): state variable controlling the rate of isotropic hardening

  • \(\zeta _i\): relaxation hardening

  • \(\gamma _\textrm{e}\): equivalent distortional elastic deformation

  • \(\dot{\varepsilon }\): equivalent rate of total distortional deformation

In addition, the model contains the following model parameters:

  • \(\mu \): shear modulus

  • K: bulk modulus

  • \(a_0\): controls the rate of rate-dependent viscoplasticity

  • \(b_0\): controls the rate of rate-independent plasticity

  • m: controls the rate of isotropic hardening

  • \(c_i\): controls the rate of relaxation

  • \(n_i\): controls the rate of relaxation hardening

  • \(\xi _\textrm{s}\): maximum rate of isotropic hardening

  • \(\zeta _{\textrm{s,}i}\): saturated value of relaxation hardening

4 Model behaviour in simple load cases

The behaviour of the proposed model is illustrated for some simple load cases, i.e. uniaxial tension, biaxial tension, and simple shear. The numerical implementation of these simple load cases is described in “Appendix B”. Uniaxial and biaxial loading are applied in terms of a prescribed strain \(\varepsilon _{11}\) (\(\varepsilon _{22}=\varepsilon _{11}\) for biaxial loading), and for simple shear, the shear angle \(\gamma \) is instead prescribed. The shear modulus and Poisson’s ratio were set to \(\mu =300\) MPa and \(\nu =0.4\), respectively. This gives a bulk modulus of \(K=2\mu (1+\nu )/3(1-2\nu )=1400\) MPa. These elastic constants are listed in Table 1.

Table 1 Parameters for elastic behaviour
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By use of \(\varGamma _\textrm{vp}\), rate-independent as well as rate-dependent inelasticity can be modelled. Yielding starts when \(\gamma _\textrm{e}\) exceeds \(\kappa _0\). The material hardening is modelled by \(\kappa \), and the hardening rate is governed by m and the state variable \(\xi \). The parameter \(a_0\) models the viscoplasticity, and \(b_0\) governs the rate-independent plasticity. The set of viscoplastic model parameters that is used in this study of the model behaviour is listed in Table 2.

Table 2 Parameters for viscoplastic behaviour
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The ability to model stress relaxation at different time scales is a new aspect of the present model. The contribution of \(\varGamma _\textrm{r}\) is therefore demonstrated. Figure 3 shows the stress response when the strain is increased quickly and then held at a constant level. The elasticity constants in Table 1 and the viscoplasticity parameters in Table 2 are applied. For the stress relaxation behaviour, only one relaxation term is used (\(n=1\)), and \(\zeta _{\textrm{s},1}=0.06\).

Fig. 3
figure 3

Parametric study for the relaxation behaviour (parametres in Tables 1 and 2 applied): \(n_1=1\) (blue lines) and \(n_1=10\) (red lines); \(c_1=1\)/s (solid lines), \(c_1=3\)/s (dashed lines), \(c_1=10\)/s (dotted lines) (Color figure online)

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The \(\varGamma _\textrm{r}\)-term causes stress relaxation. The relaxation behaviour for three different choices of \(c_1\) is shown, i.e. \(c_1=1\)/s (solid lines), 3/s (dashed lines), and 10/s (dotted lines). Two values of the hardening rate \(n_1\) are explored, i.e. \(n_1=1\) (blue lines) and 10 (red lines). It is clear from Fig. 3, that increasing \(c_1\) causes the stress to relax faster to the equilibrium stress level. Also, increasing \(n_1\) causes the equilibrium stress level to increase.

More relaxation units are now activated, and the three relaxation units listed in Table 3 are added to the model parameters. The elastic constants in Table 1 together with the inelasticity parameters in Tables 2 and 3 together constitute the set of model parameters that provided the best fit of the model to the experimental data, as will be shown in the next section. For now, the proposed model is explored a bit further for different types of loading.

Table 3 Parameters for relaxation
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Figure 4 shows the response of the model for some different simple load-cases. Figure 4a shows the response of the model in uniaxial tension, biaxial tension, and simple shear during cyclic loading. The material is first loaded well into the plastic regime, then unloaded and reloaded in compression or the opposite shearing direction in the case of simple shear. The material is then unloaded and reloaded in tension/positive shearing again. As expected, the curve for biaxial loading (red line) produces the highest stresses, whereas simple shear produces the lowest stresses (green line).

Fig. 4
figure 4

Model responses using the parameters in Tables 1, 2 and 3: a responses in uniaxial tension (blue line), biaxial tension (red line), and simple shear (green line) (loading rate: \(\dot{\varepsilon }_{11}=\dot{\gamma }=0.01\)/s); b uniaxial tension response during repeated cyclic loading (\(\dot{\varepsilon }_{11}=0.1\)/s); c uniaxial tension response for loading rates \(\dot{\varepsilon }_{11}=0.01\)/s (blue line), \(2.3\times 10^{-5}\)/s (red line), and \(7.7\times 10^{-7}\)/s (green line) (Color figure online)

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Figure 4b demonstrates the model response in the case of repeted loading in uniaxial tension. The material is first loaded up to a strain of \(\varepsilon _{11}=0.2\), then unloaded to 0.18, reloaded to 0.4, unloaded to 0.38, etc.

Figure 4c again shows a case of cyclic loading in uniaxial tension but in this case for three different loading rates. The first loading rate (\(\dot{\varepsilon }_{11}=0.01\)/s, blue line) is a moderate loading rate, where short-term viscoplasticity primarily governs the material response. The other two lines (red and green) show the model response at lower loading rates where stress relaxation comes into play. The red and green lines represent the cases \(\dot{\varepsilon }_{11}=2.3\times 10^{-5}\)/s and \(7.7\times 10^{-7}\)/s, respectively, which correspond to completing the load cycle in 24 h and in 30 days, respectively. It is clear from Fig. 4c that stress relaxation generally decreases the magnitude of the stress levels during the load cycle.

5 Model calibration

5.1 Prerequisites

The proposed constitutive model was calibrated using experimental uniaxial tensile tests. Two types of tests were used: monotonic loading at constant strain rate and relaxation testing at fixed test specimen length. These test data came from two sources: the master thesis by Görtz [24] that was described in Sect. 2, and another experimental study by Abelen and Kroon [2] on the same material. Görtz [24] performed testing both in terms of monotonic loading at constant strain rate as well as stress relaxation testing. The output from this study was nominal/engineering stress versus either strain or time. In the study by Abelen and Kroon [2], digital image correlation (DIC) was used to establish the true stress in the specimen, and this study produced data in terms of true (Cauchy) stress versus strain. The data from Görtz [24] are therefore represented by the first Piola–Kirchhoff stress, \(P_{11}\), whereas the data from Görtz [2] come in terms of the Cauchy stress, \(T_{11}\). These data are compared to the corresponding stress measures from the model.

5.2 Comparison with uniaxial tensile tests

The model is first compared to the monotonic stress–strain tests and also the relaxation tests. As indicated above, the model parameters that provided the best fit to the experimental data are listed in Tables 1, 2 and 3. Three terms in the expression for \(\varGamma _\textrm{r}\) were needed to be able to fit all of the relaxation data (\(n=3\)). The optimisation of the model parameters was performed manually. When plotting the results, the engineering strain, \(\varepsilon _{11}=F_{11}-1\), is again used. Stresses are represented both by the Cauchy stress, \(T_{11}\), and the nominal/engineering stress, \(P_{11}\), as indicated above.

Fig. 5
figure 5

Stress–strain response in monotonic uniaxial tensile tests. Strain-rates: \(\dot{\varepsilon }_{11}=0.0004/\)s (experiments: blue lines, model prediction: green solid line), and \(\dot{\varepsilon }_{11}=0.0083/\)s (experiments: red line, model prediction: green dashed and dotted lines) (Color figure online)

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In Fig. 5, the experimental results for the case of monotonic loading at two different loading rates are shown together with the model response. The result for the higher loading rate is from Görtz [24] and the result for the lower loading rate is from Abelen and Kroon [2]. In the tests with the lower loading rate, 8 specimens were tested, and DIC (digital image correlation) and manual thickness measurements of the specimens during the test enabled determination of the true stress–strain response. The dispersion in these results (blue lines) is very low, as can be seen in Fig. 5. The uniaxial test carried out in Görtz [24] only included measurement of the evolution of the force, and together with the initial cross-sectional area, the nominal stress could be established. As can be seen in Fig. 5, the true stress (solid green line) increases monotonically with the strain, whereas the nominal stress (green dashed line) reaches a peak and then starts to decrease (both in experiments and model prediction). The calibrated model is able to reproduce the stress–strain responses in the experiments very well at both loading rates.

The data from Görtz [24] (red line) are only shown up to the point of necking, since beyond necking, the engineering stress–strain results are more or less meaningless (unless the necking behaviour is analysed in more detail). The associated model prediction (dashed green line) reaches a peak and then decreases. It is emphasised, that this decrease in the model prediction is related to the fact that cross-sectional area changes are not accounted for in the engineering stress, and hence, this decrease in the numerical prediction has nothing to do with necking. (We are just simulating one material element and not the whole test specimen.) Furthermore, the green dotted line is the predicted true stress response at the higher loading rate, illustrating that the (predicted) true stress increases monotonically also at the higher loading rate.

In the relaxation tests, the material was loaded at a specified deformation rate up to a prescribed force, and then the grips in the testing machine were kept fixed. The force relaxation was registered. Figure 6a–d demonstrate the outcome from these tests together with the fitted model response. Results are shown in terms of nominal stress, \(P_{11}\), versus time. In most of these tests, 3 tests were performed for each load-case, except for the long-term test in Fig. 6d, where only two tests were performed. The experimental data points in Fig. 6a–d represent the average results from the tests. The dispersion in these tests was very low.

Fig. 6
figure 6

Comparison between experiments (symbols) and model predictions (solid lines) for relaxation tests: a short-term response for \(F_0=30\) N (blue), 40 N (red), 60 N (green), and 69 N (magenta); b close-up of the first 100 s of the short-term response in (a); c medium-long response for \(F_0=40\) N (red), 60 N (green), and 69 N (magenta); d long-term response for \(F_0=65\) N (orange) (Color figure online)

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The short-term stress relaxation is shown in Fig. 6a, i.e. stress relaxation during the first 350 s. The agreement between experiments and model is excellent for the two lower load-cases. For the two higher load-cases, the model overpredicts the initial relaxation somewhat. Figure 6b shows a close-up of the response of the material during the first 100 s. Overall, the model does a good job of predicting the initial relaxation observed in the tests.

The medium-long stress relaxation is shown in Fig. 6c. In this case, there are only three load cases. The agreement between model and experiments is generally good, even though the long-term stress level is slightly overestimated by the model in the lowest load case.

In Fig. 6d, the long-term relaxation behaviour is shown. Due to the length of this experiment, it was only carried out for one force level, i.e. \(F_0=65\) N. As can be seen from Fig. 6d, the agreement between model and experiments is good.

In Fig. 6a, c, and d reference curves (dashed lines) are also included. These curves show what the predicted response would be if the relaxation is turned off. The dashed lines in Fig. 6a shows the outcome for the respective load levels if \(c_1=c_2=c_3=0\) is applied. There is some relaxation present anyway, which is associated with viscoplasticity and the parameter \(a_0\). However, the continued stress relaxation cannot be captured if \(c_1\), \(c_2\), and \(c_3\) are turned off.

The dashed lines in Fig. 6c show the outcome if \(c_1\) is set according to Table 3, but \(c_2=c_3=0\) is applied. In this case, the mid-range relaxation behaviour cannot be properly captured. Finally, the orange dashed line in Fig. 6d illustrates the outcome for the case \(F_0=65\) N when \(c_1\) and \(c_2\) are chosen according to Table 3 but \(c_3\) is turned off. The long-term relaxation is not captured by the model then.

6 Model validation using 3D geometry

6.1 Preliminaries

The material model was implemented in Abaqus [26] as a user subroutine (UMAT). The numerical implementation of the stress update algorithm and the consistent (algorithmic) tangent stiffness is described in detail in “Appendix C”.

6.2 Problem formulation

The implemented model is applied to the problem illustrated in Fig. 7, i.e. the compression test of a 3D geometry is simulated. The FE representation of the 3D geometry is shown in Fig. 7a, and the FE problem to be solved is shown in Fig. 7b. The geometry is placed between to rigid blocks. The contact between the HDPE geometry and the rigid blocks is taken to be friction-less. The whole geometry was discretized using about 89,000 8-node linear elements, and the analyses were carried out as quasi-static, implicit simulations. The simulations of the compression tests took about 4 h on a standard computer (no parallelization), and on the average about 60 load increments were used to simulate the loading-unloading procedure.

Fig. 7
figure 7

a Mesh of 3D geometry; b FE problem

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6.3 Results

The compression tests described in Sect. 2 were simulated. Figure 8 shows contours of the von Mises stress, \(\sigma _\textrm{e}\), and the hardening variable, \(\kappa \), at different stages of the simulation. A cross-section of the geometry is shown, i.e. the interior of the geometry is seen. Just as in the experiments, the displacement rate during loading and unloading was \(\dot{\delta }=0.2\) mm/s.

Fig. 8
figure 8

Contours of \(\sigma _\textrm{e}\) and \(\kappa \) for a load cycle with \(F_0=200\) N

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At \(t=0\), the whole geometry is stress-free and no hardening has taken place, i.e. \(\kappa =\kappa _0=0.005\). At time \(t=6.4\) s, the target force, \(F_0=200\) N, has been reached. This is the point when the structure experiences the highest stresses, and the peak stress reaches a value of about 25 MPa. Some inelastic deformations have taken place, and some hardening has taken place. For \(t>6.4\) s, the applied displacement, \(\delta \), is kept constant, and the structure experiences stress relaxation. At the end of the relaxation phase, at \(t=3\) h, the applied force has decreased to a value of about 77 N, which is less than half of \(F_0\). During the relaxation phase, inelastic deformations take place, which results in material hardening. The peak value of \(\kappa \) is about 0.025 at the end of the relaxation phase. The last stage of the loading cycle consists of unloading of the geometry. Some parts of the structure have undergone significant inelastic deformations and associated material hardening. However, this does not result in any significant residual stresses after the external load has been removed, as can be seen in the lowest figure in Fig. 8.

Figure 9 shows comparisons between the simulations and experiments in terms of the force response during tests with \(F_0=150\) N and 200 N. Three tests were performed for each target stress \(F_0\). The experimental results are indicated by solid blue lines (\(F_0=150\) N) and solid red lines (\(F_0=200\) N). Model predictions are indicated by magenta dashed lines (\(F_0=150\) N) and green dashed lines (\(F_0=200\) N).

Figure 9a shows the evolution of the force, F, as a function of time, t. Time goes from 0 to 3 h, which means that the initial loading phase is hardly visible. For the case \(F_0=200\) N, the force starts at 200 N and decreases down to about 77 N after 3 h. In other words, the material experiences significant stress relaxation. The three experimental curves (solid red lines) are well collected. The model overestimates the initial relaxation compared to the experiments, but at the end of the relaxation process, the model prediction is quite close to the experimental stress level. The same conclusions hold for the case \(F_0=150\) N, i.e. the experimental results show a low dispersion. The force decreases from 150 N to about 60 N at the end of the relaxation process. The model again overestimates the initial relaxation, but at the end of the relaxation phase, the model agrees very well with the experimental force level.

Fig. 9
figure 9

Force relaxation of 3D geometry for two different initial loads, \(F_0\). \(F_0=150\) N: tests (solid blue lines) and model prediction (dashed magenta line), \(F_0=200\) N: tests (solid red lines) and model prediction (dashed green line). a Force response versus time; b force response versus logarithm of time; c Force response versus load-line displacement (Color figure online)

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Figure 9b essentially shows the same information as in Fig. 9a, but in Fig. 9b, the horizontal axis shows the logarithm of time instead. In Fig. 9b, it can be seen that in the very beginning of the relaxation process, the model actually underestimates the relaxation, but after about 100 s the model instead starts to overestimate the relaxation. It should also be noted that the experimental curves have been translated in the horizontal direction (t-dimension) so that they reach the peak force at the same time as the simulations.

In Fig. 9c, force versus displacement results are shown. In this diagram, it can be seen even more clearly that the experimental curves exhibit an initial non-linearity. In the experiments, the specimens always undergo some initial adjustments, which causes this initial non-linearity. Again, in order to help the comparison between model and experiments, the experimental curves were shifted in the displacement dimension so that the peak force in the experiments appears at the same displacement as in the simulations. In the force-displacement diagram, the experimental curves show some dispersion. The overall impression is that the model seems to somewhat underestimate the initial stiffness of the structure. As noted earlier, the total amount of relaxation is very well predicted by the model. From Fig. 9c it is also evident that the final unloading path is relatively well predicted by the model. The experiments show a non-linear unloading path, whereas the model response is virtually linear. But the value of \(\delta \) that results in zero force during unloading is well predicted by the model and occurs at about \(\delta =0.5\) mm and 0.8 mm for \(F_0=150\) N and 200 N, respectively.

7 Discussion

The proposed model is intended to model the rate- and history-dependent response of semi-crystalline polymers, and this includes stress relaxation. In general, stress relaxation in polymers is caused by time-dependent rearrangements of the long, entangled polymer chains, where the polymer network reorganises itself in order to minimise the elastic energy. But the details of this process are more complicated. Doi and Edwards [15] suggested three basic mechanisms for stress relaxation in the amorphous phase, i.e. (a) relaxation between entanglements in order to re-establish a constant chain density, (b) chain retraction inside a deformed (and imagined) ‘tube’ in the material, and (c) chain disengagement from the ‘tube’ by a reptation process. These three steps would then be characterised by three different relaxation times. For semi-crystalline polymers, the situation is even more complex. Studies suggest that stress relaxation can be associated with the formation of microvoids in the amorphous phase in between the crystallites in the microstructure [12, 56]. Furthermore, partial crystallisation of the amorphous phase is another mechanism that could cause stress relaxation [31, 41, 64].

There are other models available in the literature that essentially set out to accomplish the same thing as the present model, and some of these were mentioned briefly in Sect. 1. An alternative Eulerian approach was proposed by Simo [60]. However, in Simo’s model, the yield criterion is formulated in stress space. Furthermore, the evolution of inelastic deformations is derived from a plastic potential and the principle of maximum plastic dissipation is applied in Simo’s work, which is not the case in the present work. With regard to Lagrangian models, van Dommelen et al. [16] proposed a micromechanically motivated constitutive model for semi-crystalline polymers. van Dommelen et al. [16] make use of the multiplicative decomposition of the deformation gradient into elastic and inelastic parts, i.e. \({\textbf {F}}={\textbf {F}}_\textrm{e}{\textbf {F}}_\textrm{i}\). They use mixture theory to represent the contribution to the total stress from the two phases (crystalline and amorphous), and crystal plasticity is used for modelling the inelastic deformations of the crystalline phase. Ayoub et al. [6] also proposed a model for semi-crystalline polymers that is valid for finite strains and based on the multiplicative split of the deformation gradient. Ayoub et al. [6] propose a three-phase material where the total stress is produced by a ‘molecular network resistance’, an ‘amorphous intermolecular resistance’, and a ‘crystalline intermolecular resistance’. This translates into three spring-dashpot units carrying the load in parallel, which among other things enables modelling of stress relaxation at different time scales. Other models exist that use more or less the same basic theoretical frameworks as the two models mentioned above. Both of these models are Lagrangian in the sense that their constitutive framework depends on the reference state through state variables like the total deformation gradient and the plastic/inelastic part of the deformation gradient. In contrast, the model proposed here is Eulerian in the sense that it only depends on state variables that are defined in the current state of the material. One attractive feature of the Eulerian framework is that the state variables are—at least in principle—measurable in the current state of the material. For example, the elastic strains (i.e. \(\bar{{\textbf {B}}}_\textrm{e}\) and J) can be measured directly by experimental techniques, such as diffraction methods. In contrast, total strains or plastic strains (in Lagrangian models) cannot be directly measured from the current state of the material. Furthermore, inelastic deformations are often taken to be incompressible, and another difference between the frameworks can be found in how this incompressibility is treated numerically. In numerical implementations of Lagrangian formulations of inelasticity, special numerical schemes must be used for ensuring that the plastic deformations remain incompressible (\(\textrm{det}{\textbf {F}}_\textrm{i}=1\)). In the present Eulerian formulation, this issue simply never arises. The evolution laws in the present Eulerian formulation are also formulated in strain space, in contrast to the two Lagrangian formulations above, which are formulated in stress space. This also means that the numerical implementation of inelasticity becomes somewhat simpler.

The experimental results presented here can to some extent be compared to other studies. Drozdov and Christiansen [17] investigate the monotonic stress–strain response and also the stress relaxation of high-density polyethylene. At the strain rate 0.01/s, Drozdov and Christiansen [17] get an engineering stress peak of about 20 MPa. This stress peak is slightly lower than what was seen in the present study (23 MPa) at a slightly lower strain rate (0.0083/s). Drozdov and Christiansen [17] also perform stress relaxation tests, but they consider stress relaxation starting from an engineering stress of about 20 MPa, which is lower than in the present study. When starting a relaxation test at 20 MPa and then keeping the applied strain constant, the stress has relaxed to below 10 MPa after about 3000 s. The results from Drozdov and Christiansen [17] are consistent with the results in the present study.

The rate aspect of the relaxation behaviour in the model is primarily governed by the \(c_i\)-constants. For the present set of experiments, three relaxation terms (\(n=3\)) were needed to describe the relaxation behaviour for all time ranges. The optimal values of these parameters were \(c_1=1\)/s, \(c_2=0.05\)/s, and \(c_3=0.00003\)/s. Inversion of these values gives an estimate of the time constants of the microstructural relaxation processes involved. Hence, the inverted values are \(1/c_1=1\) s, \(1/c_2=20\) s, and \(1/c_3\approx 33{,}000\) s\(\approx 9.3\) h\(\approx 0.4\) days. It is difficult to say in detail what the microstructural processes are, but this observation illustrates the complicated rate-dependence that these materials exhibit.

In the simulations of the 3D geometry, the simulations seemed to underestimate the initial stiffness of the structure. This could be due to the fact that the contact between the geometry and the rigid plates was taken to be friction-less. Introducing friction in the contact will make the simulated structure stiffer. Another reason why the simulation underestimated the structural stiffness could be that the inelasticity in the proposed model is pressure-independent. The hydrostatic stress usually has an impact on the inelastic flow in polymers, meaning that a positive hydrostatic stress facilitates inelastic flow whereas a negative hydrostatic stress has the opposite effect. Previous studies have also confirmed that there is such an effect in polyethylene [42, 62]. In the present study, the material model was calibrated by use of tensile tests, whereas the validation was performed by exposing the 3D geometry to compression loading. It could be expected that the model would then overestimate the relaxation of the 3D geometry compared to experiments. An overestimation of the relaxation was indeed observed but only after some time of relaxation. Overall, the present results suggest that the relaxation properties of the present material in tension and compression are not too different, at least not within the stress and strain regimes considered here.

In the simulations, linear 8-node elements with incompatible modes were used. A limited mesh study was also carried out, where the element size in the regions with the largest deformations was decreased by a factor 2. This did not have any significant influence on the force response of the structure. In some regions of the 3D structure, the material undergoes large plastic deformations, and the material response is therefore close to near incompressible. In the element formulation in Abaqus, selectively reduced integration is applied, i.e. reduced integration is applied to the volumetric terms. This is done in order to prevent volumetric locking. Taken together, significant mesh-dependence should not be an issue in the present simulations.

In summary, the stress relaxation behaviour of HDPE has been studied both experimentally and theoretically. An Eulerian constitutive model for isotropic polymers was proposed. The model was able to describe the uniaxial tension results well, and it was also able to predict the behaviour of a 3D structure exposed to compression.