1 Introduction

Faced with increasing market competitiveness as well as consumption, manufacturers must guarantee a long service life of their components. Elastomers are among these components. They are used in various fields and different industrial applications [1]. In high safety demanding applications (e.g. hydrogen infrastructure [2] or containers for radioactive materials [3]) in addition to long-term effects, the safety assessment also raises the question of lifetime predictions and failure behaviour under extreme conditions. Probably the most encountered conditions to which the elastomer seal is exposed to are mechanical loading, temperature, and oxygen [4]. The combination of elevated temperature and oxygen leads to thermo-oxidative ageing which degrades the material properties over exposure time. The degradation is irreversible and microstructural changes (chain scission and crosslinking) occur during thermo-oxidative ageing [5,6,7]. Since accelerated ageing experiments are sometimes very long and therefore costly, numerical simulations have gained further importance. According to the two-network theory, continuous stress relaxation is a measure of chain scission, while continuous stress relaxation subtracted from intermittent stress relaxation at each ageing time is a measure of crosslinking [8, 9]. Inspired by the two-network theory very interesting modelling approaches were developed [10,11,12,13].

On the other hand, experimental compression stress relaxation (SR) and compression set (CS) present key indicators of elastomer seal’s long-term performance and are often used for lifetime predictions. However, care must be taken when analyzing the data as diffusion-limited oxidation effects (DLO) may appear [14]. DLO effects occur when the oxygen in the interior of the sample is consumed faster than it can diffuse in from the surrounding air. This can lead to oxygen shortage in the core of the sample and thus to less ageing in the core region. This heterogeneous ageing may distort the ageing data and can result in misleading lifetime predictions.

Moreover, during SR physical (reversible) and chemical (irreversible microstructural changes) processes take place [4]. To separate physical and chemical processes during SR, most experiments were performed either in inert atmosphere or first at low temperature or short time and then the resulting relaxation is subtracted at high temperature or longer time by using the time-temperature superposition technique [15,16,17]. In this paper, a new approach to separate physical and chemical processes that occur during stress relaxation is proposed through the analysis of compression set measurements under various test conditions. Characteristic times with regard to the degradation processes are then determined by examining CS measurements and compression SR measurements. These times are then used in the identification of material parameters in the simulation section based on the model of Shaw et al. [13] that was motivated by the two-network theory of Tobolsky [8, 9].

2 Experimental

2.1 Material

Commercial ethylene-propylene-diene rubber (EPDM) O-rings with a cord diameter of 10 mm and an inner diameter of 190 mm were used. The peroxide cured EPDM is based on a polymer containing 48 wt% ethylene, 4.1 wt% ethylidene norbornene (ENB) and was compounded with 90 phr of carbon black filler, and no plasticizer.

2.2 IRHD

IRHD measurements were taken on O-ring cross sections as shown in Fig. 1 across the sample profile using a linear table according to DIN ISO 48-2:2021-02 with the method M. As the O-rings cross sections are thick (10 mm), ageing can be influenced by diffusion-limited oxidation (DLO) effects. Slices of O-rings with a thickness of 3–4 mm were cut from O-rings that had been aged at 125 \(^{\circ }\,\hbox {C}\) and 150 \(^{\circ }\,\hbox {C}\) compressed by 25%, respectively. The measuring points on the cross section are illustrated in Fig. 1. Only x direction will be considered as the outsides on the left- and the right-hand side were in direct contact with oxygen during ageing in the compressed state.

Fig. 1
figure 1

IRHD microhardness measuring positions across O-ring cross section

Full size image

2.3 Compression set

CS was measured at room temperature on EPDM O-ring segments aged compressed by 25% at \(125\,^{\circ }\,\hbox {C}\) and \(150\,^{\circ }\hbox {C}\) for different ageing times ranging from 1 to 185 d as shown in Fig. 2. The measurements were taken 30 min after disassembly according to standards ASTM D395-18 and DIN ISO 815-1:2016-09 and for different times after disassembly for up to 1 year. This showed that the recovery proceeds even after several days. Therefore to accelerate the recovery towards equilibrium [18], the O-ring segments were placed in an oven for 1 day at \(100\,^{\circ }\,\hbox {C}\) and the recovered height after thermal conditioning was measured. For each type of CS measurement, the recovered height of O-ring segments was measured at 7–10 positions using a calliper and CS was calculated from the average values.

Fig. 2
figure 2

O-ring segments between flanges for CS measurements

Full size image

2.4 Continuous stress relaxation

Continuous SR measurements were taken at \(125\,^{\circ }\,\hbox {C}\) and \(150\,^{\circ }\,\hbox {C}\) on O-ring segments with a length of approximatively 40 mm. The O-ring segments were thermally and mechanically conditioned according to the standard DIN ISO 3384:2008. Three O-ring segments were tested using the SR rigs EB 02 in a cell oven EB 22 from Elastocon. The median curve of three measurements will be presented and is expressed as the ratio \({R}_{\mathrm{c}}\) of the continuously measured force \({F}_{\mathrm{c}}\) to the initial force \({F}_{0}\). \({F}_{0}\) is the measured force 30 min after the full compression by 25% was achieved according to the DIN ISO 3384:2008.

3 Modelling

Within the framework of large deformation of hyper-elastic quasi-incompressible materials, the strain energy function is split into a deviatoric and volumetric part as expressed in Eq. (1).

$$\begin{aligned} {\Psi }\left( {\, }{\varvec{B}} \right) =\overline{\Psi } \left( {\varvec{B}} \right) +{\Psi }^{\mathrm{vol}}\left( J \right) \end{aligned}$$
(1)

\({\varvec{B}}\) is the left Cauchy green tensor and J is the transformation Jacobian. \({\varvec{B}}\) is expressed in terms of the multiplicative decomposition as:

$$\begin{aligned} {\varvec{B}}=\bar{{\varvec{B}}}.{\varvec{B}}^{\mathrm{vol}} \end{aligned}$$
(2)

Since the applied strains are not large (25%), the simplified Neo–Hook potential is adopted. On the one hand, the model has a physical meaning as it is based on the statistical thermodynamics of crosslinked polymer chains. On the other hand, the Neo-Hook potential has already proven its accuracy to model the material behaviour during thermo-oxidative ageing [19].

$$\begin{aligned} {\Psi }\left( {\, }{\varvec{B}} \right) =C.\left( \mathrm{tr}\left( \bar{{\varvec{B}}} \right) {-3} \right) +\frac{1}{D}{{\, }\left( J-1 \right) }^{2} \end{aligned}$$
(3)

where \(C{=}\frac{1}{2}n\left( t \right) {\cdot }\,k{\cdot }T\).

\(n\left( t \right) \) is the effective number of crosslinks as a function of exposure time t, k is Boltzmann’s constant, T is the absolute temperature and \(D=2/K_{0}\) where \(K_{0}\) is the initial bulk modulus.

Physical relaxation is a reversible process and is due to chain rearrangement under compression load. In a real system, chemical degradation and physical relaxation occur simultaneously during ageing, but physical relaxation is dominant at lower ageing times and chemical relaxation on the other hand is preponderant at longer ageing times [20]. In modelling, it is possible either to separate the variables (physical and chemical) [13] or to couple them. The coupling can be used when the experimental basis required to identify the coupling parameters is available (e.g. intermittent ageing in oxygen and inert atmosphere). In this paper, the stress relaxation is presented as an additive contribution of the physical and chemical relaxation.

The short-term physical relaxation in a standard static analysis and, under isothermal conditions, corresponds to an elastic solution based on instantaneous elastic moduli (Eq. 4). Here the relaxation coefficients are applied to the constant C that define the energy function in Eq. (3).

$$\begin{aligned} C^{R}\left( t \right) =C^{0}\cdot (1-\sum \limits _{k=1}^n {\bar{g}_{k}^{p}{\cdot }\left( 1-\exp \left( {-t}/\tau _{k} \right) \right) } \end{aligned}$$
(4)

where \(C^{0}=\frac{1}{2}n\left( 0 \right) {\cdot }k{\cdot }T\) and \(\bar{g}_{k}^{p}\) and \(\tau _{k}\) are the Prony parameters.

To take into account chemical ageing, the effective number of crosslinks n is chosen as a decreasing function of t according to the proposed function of Shaw et al. [13] so that the Cauchy stress is given in Eq. (5). Here, a number of first-order chemical relaxation processes take place simultaneously, each one of which is Maxwellian. Each process (i) has a characteristic time \(\tau _{i}\) which is evolved with a portion of the network interpreted as a phase fraction \(A_{i}\).

$$\begin{aligned} \sigma ^{c}\left( t \right) =\frac{n_{0}.\sum _{i=1}^n {A_{i}\mathrm {exp}(-t/\tau _{i})} k.T}{J}\left( \bar{{\varvec{B}}}-\frac{1}{3}\, tr\left( \bar{{\varvec{B}}} \right) \mathbf {.1} \right) +\frac{2}{D}\left( J-1 \right) .\mathbf{1} \end{aligned}$$
(5)

where \(A_{i}\ge 0\) and \(\sum \nolimits _{i=1}^n A_{i} =1\).

From Eqs. (4) and (5), the total Cauchy stress \(\sigma (t)\) is expressed as:

$$\begin{aligned} \sigma \left( t \right) =\frac{n_{0}.k.T.(1-\sum _{k=1}^n {\bar{g}_{k}^{p}{\cdot }\left( 1-\exp \left( -t/\tau _{k}\right) \right) } +\sum _{i=1}^n A_{i}\mathrm {exp}(-t/ \tau _{i}))}{J}\left( \bar{{\varvec{B}}}-\frac{1}{3}\, tr\left( \bar{{\varvec{B}}} \right) \mathbf {.1} \right) +\frac{2}{D}\left( J-1 \right) .\mathbf {1} \nonumber \\ \end{aligned}$$
(6)

4 Results

IRHD microhardness measurements were taken over the cross section of compressed O-rings aged at \(125\,^{\circ }\,\hbox {C}\) and \(150\,^{\circ }\,\hbox {C}\) according to DIN ISO 48-2:2021-02 with the method M. The development over time is shown in Fig. 3. The microhardness profiles show no changes for samples aged up to 909 d at \(125\, ^{\circ }\, \hbox {C}\) and 75 d at \(150\, ^{\circ }\, \hbox {C}\). For samples aged at \(125^{\circ }\, \hbox {C}\) for 1093 d and those aged at \(150\,^{\circ }\,\hbox {C}\) for 101 d, slight DLO effects are detected. This is clearly emphasized in Fig. 3 as the measured values of microhardness near the surface are higher than those measured in the interior of the O-ring cross section. The microhardness profiles of EPDM aged at \(125\,^{\circ }\, \hbox {C}\) present slight decrease of maximum 2 Shore D Hardness in the core region between 909 and 1093 d which might be due to anaerobic chain scission reactions.

Fig. 3
figure 3

IRHD microhardness measurements at \(125\,^{\circ }\hbox {C}\) and \(150\,^{\circ }\hbox {C}\) at different ageing times

Full size image

CS was measured 30 min after disassembly according to ASTM D395-18 and DIN ISO 815-1:2010 and after 0.5 d, 1 d, 5 d, 10 d, 18 d, 50 d and 365 d. An exemplary result of the CS measurement of EPDM aged at \(125\,^{\circ }\,\hbox {C}\) for 10 d is presented in Fig. 4. It shows the evolution of the recovery over time and its corresponding fit (\(\hbox {CS} = \hbox {CS}_{\infty }+A_{d}\cdot e^{-t/\tau d})\). The calculated errors of measurements are within the size of the used symbols. The measured CS after thermal conditioning reflects only the irreversible chemical reactions (chain scission and crosslinking), and its value is set as \(\hbox {CS}_{\infty }\). According to our investigation, it corresponds to about 1-year recovery time as the difference between the measured CS after 1 year and the measurement after tempering is 0.1%. The term \(A_{d} \cdot e^{-t/\tau d}\) accounts for reversible physical processes. \( A_{d}\) and \(\tau _{d} \) are the retardation strength and the characteristic time at a given ageing time, respectively. For 10 d ageing at \(125\,^{\circ }\,\hbox {C}\), \(A_{d\, }= 10.8\) and \(\tau _{d} = 24.7\, \hbox {d}\) with a good correlation fit (\(R^{2\, }= 0.98\)). It is to mention that two representative samples were previously tempered again [21] in order to confirm that the CS after the first tempering corresponds to the equilibrium value \(\hbox {CS}_{\infty \,}\) and the results after first and second tempering were almost the same. The results of CS at different times after disassembly that are presented in Fig. 4 call into question the test method according to the standard (ASTM D395-18 - DIN ISO 815-1:2010) as the recovery proceeds further after 30 min after disassembly. A common error of the measurements is about \(\pm \,3{\%}\) [22]. From the results of Fig. 4, the difference in percent between a measurement after 30 min and \(\hbox {CS}_{\infty \, }\) is about 59%. These differences should be considered when comparing tempered CS results to those measured with the standard procedure.

Fig. 4
figure 4

Evolution of the recovery of EPDM aged at \(125\,^{\circ }\,\hbox {C}\) for 10 d over time and its corresponding fit

Full size image

Figure 5a, b shows the evolution of CS measurements at \(125\,^{\circ }\,\hbox {C}\) and \(150\,^{\circ }\,\hbox {C}\) after 30 min and after thermal conditioning over ageing time, respectively, which illustrates again their different values. In Fig. 5b, the values of CS after ageing at 101 d which are marked with a blue frame will not be taken into account for further analysis as these samples exhibited slight DLO effects according to the IRHD microhardness profiles shown in Fig 3. In Fig. 5a, b, the normalised relaxation \(R_{\mathrm{c}}\) over ageing time is also presented at \(125\,^{\circ }\,\hbox {C}\) and \(150\,^{\circ }\,\hbox {C}\). \(R_{\mathrm{c}}\) exhibits two regions that could be approached asymptotically [23]. For the first region, the line was obtained by fitting a tangent to the linear region starting from the first recorded value of \(R_{\mathrm{c}}\). The second line was obtained by fitting a tangent to the half-time relaxation at 50% force loss \(t_{50\% }\) ((*) in Fig. 5a, b). The first region correlates with physical relaxation [23]. Its duration \(t_{\mathrm{physical}}\) ((\(+)\) in Fig. 5a, b) is defined as the time at which the measurement data deviate by 1% from the straight line. The second phase is associated with chain scission reactions [23]. The induction time \(t_{\mathrm{induction}}\) which represents the onset of the dominant chemical reactions can be estimated from the intersection of the two linear regions [22]. It is noteworthy to clarify that the induction time (\(t_{\mathrm{induction}})\) should not be confused either with the induction period in chemistry which is an initial slow stage of chemical reaction or the oxidative induction time (OIT) which is determined by means of Fourier transform infrared spectroscopy (FTIR), chemiluminescence (CL) or differential scanning calorimetry (DSC). The latter is described in ASTM D3895-07. The induction time here represents the threshold from which chain scission during the continuous SR test is noticeable. The threshold was previously confirmed for EPDM aged at \(125\,^{\circ }\,\hbox {C}\) by comparing the continuous and intermittent SR measurements [22] that represent the evolution of chain scission and crosslinking according to the two-network theory of Tobolsky [8, 9]. It has been shown for EPDM that the influence of chain scission reactions competes with that of crosslinking at longer ageing times. Moreover, chain scission and crosslinking were occurring with approximately equal impact until 10 d for EPDM aged at \(125\,^{\circ }\,\hbox {C}\) [22], which is the determined threshold in Fig. 5a. At a macromolecular scale, the induction time represents the time at which the concentration of the oxidative products is sufficiently high to accelerate the decomposition. In fact, during thermo-oxidative ageing, macromolecules are mostly cleaved by the \(\upbeta \)-scission process to fragments with ketone and vinyl end-groups. In the early reaction stages, these oxidation products are only formed to a certain extent. However, at a certain ageing time (induction time) when the concentration of these products is increased enough, their more energy-efficient bimolecular decomposition begins [24].

To compare CS and \(R_{\mathrm{c}}\), a correction factor is used [25] since CS measurements were taken at room temperature (\(\hbox {RT}=23\,^{\circ }\,\hbox {C}\)), but \(R_{\mathrm{c}}\) measurements at \(125\,^{\circ }\,\hbox {C}\) and \(150\,^{\circ }\,\hbox {C}\). A linear expansion coefficient of \(2.10^{-4}/\hbox {K}\) was previously determined by the Dynamic Mechanic Thermal Analysis (DMTA) between room temperature and the test temperature [22]. For a temperature difference of \(102\,^{\circ }\,\hbox {C}\) and \(127^{\circ } \hbox {C}\) which are the differences between the ageing temperature and the RT, the correction factors are 0.944 and 0.99, respectively. By using the correction approach, the \(R_{\mathrm{c}}\) at \(125\,^{\circ }\,\hbox {C}\) and \(150\,^{\circ }\,\hbox {C}\) can be compared to the corrected equilibrium compression set and the corrected CS after 30 min at each ageing time. Figure 5c, d illustrates the evolution of \(R_{\mathrm{c}}\) versus the corrected CS after 30 min and CS after tempering for EPDM O-rings aged at \(125\,^{\circ }\,\hbox {C}\) and \(150\,^{\circ }\,\hbox {C}\), respectively. The data represented in a blue frame in Fig. 5d will not be analysed due to DLO effects as stated before. The evolution (Fig. 5c, d) is linear, and the contribution of physical and chemical processes is determined through the analysis of the corrected CS after tempering. The intercept of the linear fit of \(R_{\mathrm{c}}\) versus CS after tempering with \(R_{\mathrm{c}}\) ((X) in Fig. 5c, d) corresponds to the shift factor to which the \(R_{\mathrm{c}}\) curve should be moved if only irreversible chemical relaxation is to be considered. This intercept represents the value at which irreversible chemical degradation begins. The difference between the shifted curve and the real one (\(\sim \,10{\%}\) for the ageing temperature of \(125\,^{\circ }\,\hbox {C}\)) accounts for the physical processes that occur during ageing. The ageing time corresponding to \(R_{\mathrm{c}}\) intercept represents the initiation time \(t_{\mathrm{initiation}}\) of the chemical degradation and is determined from Fig. 5a, b. Again, the induction period (\(t_{\mathrm{induction}})\) is very long and lasts 10 d and 2 d for EPDM aged at \(125^{\circ }\, \hbox {C}\) and \(150^{\circ }\, \hbox {C}\), respectively. On the other hand, the initiation time starts earlier. The determined characteristic times (\(t_{\mathrm{physical}}=0.06\,\hbox {d}\), \(t_{\mathrm{initiation}}=1\,\hbox {d}\), \(t_{\mathrm{induction}}=10\,\hbox {d}\) and \(t_{50\% }=69\,\hbox {d}\)) are intrinsic only for the investigated rubbers (geometry and compounds) at the ageing temperature of \(125\,^{\circ }\,\hbox {C}\). For \(150^{\circ }\, \hbox {C}\) the characteristic times are \(t_{\mathrm{physical}}=0.05\,\hbox {d}\), \(t_{\mathrm{initiation}}=0.13\,\hbox {d}, t_{\mathrm{induction}}=2\,\hbox {d}\) and \(t_{50\% }=25\,\hbox {d}\).

Fig. 5
figure 5

Normalised compression stress relaxation \(R_{\mathrm{c}}\) and compression set CS measurements after 30 min and after tempering. a for EPDM aged at \(125\,^{\circ }\,\hbox {C}\) and b for EPDM aged at \(150\,^{\circ }\,\hbox {C}\). Contribution of physical relaxation to ageing c for EPDM aged at \(125\,^{\circ }\,\hbox {C}\) and d for EPDM aged at \(150\,^{\circ }\,\hbox {C}\) Data in the blue square are slightly affected by DLO and therefore excluded from the analysis

Full size image

Since the determined \(t_{\mathrm{initiation}\, }\) is representative for the initiation time of the chemical degradation, the experimental results of SR shown in Fig. 5a, b until 1 d and 0.13 d relative to the ageing temperatures of \(125\,^{\circ }\,\hbox {C}\) and \(150\,^{\circ }\,\hbox {C}\) are introduced in the ABAQUS\(\circledR \) simulation software in order to determine the parameters of physical relaxation. An internal calibration tool identifies the Prony parameters according to Eq. (4) until the given initiation time.

For chemical stress relaxation, two first-order Maxwellian chemical relaxation processes are applied in Eq. (5). The material parameters used in the simulation of physical and chemical relaxation are summarised in Table 1. The physical relaxation parameters (\(\bar{g}_{k}^{p}\) and \(\tau _{k})\) for samples aged at \(125\,^{\circ }\,\hbox {C}\) and \(150\,^{\circ }\,\hbox {C}\) are the same even when the initiation of the chemical degradation happens at different times (\(t_{\mathrm{initiation}}=1\,\hbox {d}\) for \(125\,^{\circ }\,\hbox {C}\) and \(t_{\mathrm{initiation}}=0.13\,\hbox {d}\) for \(150^{\circ }\, \hbox {C}\)). On the other hand, the chemical relaxation times \(\tau _{1}\) and \(\tau _{2}\) (that represent early and later degradation, respectively) differ due to the different reaction rates at \(125\,^{\circ }\,\hbox {C}\) and \(150\,^{\circ }\,\hbox {C}\). The shift factor between \(125\,^{\circ }\,\text{ C }\) and \(150\,^{\circ }\,\hbox {C}\) is 5.9 for \(\tau _{1}\) and 7.9 for \(\tau _{2}\). Moreover, when we examine the SR experimental results shown in Fig. 5a, b, the \(150\,^{\circ }\,\hbox {C}\) curve can be shifted to the \(125\,^{\circ }\,\hbox {C}\) curve until superposition with a shift factor of 6.9 (considering the whole relaxation process). This represents the macroscopic average of the superposition factors of low and high chemical relaxation times. The simulation results of the normalised continuous stress relaxation of EPDM aged at \(125\,^{\circ }\,\hbox {C}\) and \(150\,^{\circ }\,\hbox {C}\) with physical relaxation and chemical ageing are shown in Fig. 6 and exhibit very good agreement with the experimental data.

Table 1 Set of parameters used in the simulation
Full size table
Fig. 6
figure 6

Experimental and numerical normalised compression stress relaxation results

Full size image

The experimental SR results were performed at \(125\,^{\circ }\,\hbox {C}\) and \(150\,^{\circ }\,\hbox {C}\) until a degradation stage of \(R_{\mathrm{c}}=0.47\) (ageing time \(=173\,\hbox {d}\)) and \(R_{\mathrm{c}}=0.04\) (ageing time \(=120\,\hbox {d}\)), respectively. At this degradation stage no DLO effects were detected for samples aged at \(125\,^{\circ }\,\hbox {C}\) as shown in Fig. 3. Since the proposed model and its determined parameters are based on DLO free measurements, the delivered results are also DLO free. From Fig. 3 no DLO effects are detected for samples aged at 909 d and slight DLO effects are noticed after 1093 d. At the ageing interval between 909 and 1093 d where probably DLO effects occur, the Rc values according to our simulation are between 0.03 and 0.004, respectively. So that at \(125\,^{\circ }\,\hbox {C}\) ageing temperature, the EPDM ages homogeneously during the SR experiment since the relaxation is almost completed. On the other hand, samples aged at \(150\,^{\circ }\,\hbox {C}\) present slight DLO effects after 101 d as shown in Fig. 3. However, as the measured value is very close to the DLO-free simulation, one can suppose that the slight detected DLO effects at 101 d do not significantly affect the measured SR. However, for lower ageing temperatures much longer relaxation times (\(t_{\mathrm{physical}}\), \(t_{\mathrm{initiation}}\), \(t_{\mathrm{induction}}\) and \(t_{50\% })\) are to be expected. Taking the example of EPDM aged at \(75\,^{\circ }\,\hbox {C}\), even after 168 d of ageing, the experimental results showed an \(\hbox {Rc} = 0.88\) [26] which is almost within the physical relaxation regime. Similarly, the CS after tempering [21] at this ageing stage was quite low with about 6%. When elastomer components are operated at moderate temperature ranges, physical relaxation should not be neglected. Even at high temperature where chemical processes are dominant, physical processes still contribute to over 10% force drop. With the separation of physical and chemical processes, more reliable lifetime predictions, especially for lower ageing temperatures where samples show slower degradation, can be performed [26] as these predictions are principally governed by the chemical processes.

5 Conclusion

An experimental method to separate physical and chemical relaxation was developed and applied in the present study through the analysis of stress relaxation (SR) and compression set (CS) measurements. An identification strategy for the material parameters based on the relationship between SR and equilibrium CS was proposed. Physical and chemical relaxations were modelled with the Prony series and two decay functions, respectively. This modelling is only valid for homogeneously aged samples which was controlled by the IRHD analysis at different ageing times. However, an extension of the model to take into account DLO effects too is under progress which is applied to another material which degrades faster and experiences DLO effects already during the performed SR measurements (at about \(t_{50\% }\)).