# Re-straining induced triggering of the Portevin-Le Chatelier effect in recrystallised 2198 T3 and T8 aluminium alloy: Experimental observations and FE simulations for smooth and notched samples

S.C. Ren, Thilo F. Morgeneyer, G. Rousselier, M. Mazière, S. Forest

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S.C. Ren, Thilo F. Morgeneyer, G. Rousselier, M. Mazière, S. Forest. Re-straining induced triggering of the Portevin-Le Chatelier effect in recrystallised 2198 T3 and T8 aluminium alloy: Experimental observations and FE simulations for smooth and notched samples. International Journal of Solids and Structures, 2025, 323, pp.113636. 10.1016/j.ijsolstr.2025.113636. hal-05315416

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# Re-straining induced triggering of the Portevin-Le Chatelier effect in recrystallised 2198 T3 and T8 aluminium alloy: Experimental observations and FE simulations for smooth and notched samples

S.C. Ren<sup>a,b</sup>, T.F. Morgeneyer<sup>a,b,*</sup>, G. Rousselier<sup>a</sup>, M. Mazière<sup>a,b</sup>, S. Forest<sup>a,b</sup>

## ARTICLE INFO

### Keywords:

Al-Cu-Li aluminium alloy

Digital image correlation

Relaxation

Portevin-le Chatelier effect

Strain localisation

DSA

Serrated yielding

## ABSTRACT

The susceptibility of 2198 Al-Cu-Li alloys to the Portevin-Le Chatelier (PLC) effect was investigated under various testing conditions, including constant strain rate, relaxation, and strain rate jump tests on tensile specimens. The localisation behaviour in the naturally aged state (T3R) was compared with that in the artificially aged state (T8R) using digital image correlation (DIC). Additionally, the triggering of the PLC effect following a change in the strain path was observed. The T8R grade, which contains various precipitates, exhibits no localisation across temperatures from -50 to 80 °C. In contrast, localisations are observed in the T3R grade, which lacks precipitates, suggesting that dynamic strain ageing (DSA) remains the dominant mechanism responsible for serrated yielding, rather than precipitate shearing. A McCormick-type macroscopic elastoviscoplastic model, implemented in a finite element solver, was calibrated with experimental data to simulate the effects of different loading conditions on tensile specimens. The triggering effect and spatio-temporal patterns were well captured compared to DIC results for both T3R and T8R grades. The model was also employed to simulate the early plasticity during tearing tests of flat compact-tension-like samples in order to make a direct comparison with laminography-DVC (Digital Volume Correlation) experiments. Slant localisation patterns ahead of the notch are predicted by the model but are more mobile than those observed in laminography-DVC experiments of previous studies. The differences between constant strain rate and interruptive loading conditions are evidenced. Interestingly, even for the artificially aged T8 material, strain bands were predicted ahead of the notch after restraining.

## 1. Introduction

Since the 1930s, aluminium alloys have been widely used as the primary structural material for aircraft due to their favourable physical and mechanical properties. Although composite materials have gained increasing popularity in fuselage applications in recent years, aluminium alloys remain attractive in airframe construction. Compared to composite materials, aluminium alloys still offer several advantages, including lower certification and production costs, relatively high resistance to impact, stable mechanical behaviour under changing environmental conditions (moisture/temperature), mature assembly techniques, repair and maintenance procedures, and ease of recycling at the end of the aircraft's life (Dursun and Soutis, 2014). Among aluminium alloys, the 2xxx series alloys that contain copper as the principal alloying element are particularly popular for airframe structural applications due to their superior damage tolerance. The addition of copper results in higher strength through the precipitation of Al₂Cu and

Al₂CuMg phases. Currently, Al-Cu-Li alloys are considered potential replacements for Al-Cu alloys in the aeronautic industry. With lithium as the lightest metal, the addition of lithium to aluminium alloys results in significant weight reduction (3% for each 1% Li added), making it an effective means of lowering the structural weight of aircraft to increase fuel efficiency. The latest generation of Al-Cu-Li alloys, such as the third generation 2198 Al-Cu-Li alloys, are increasingly utilised for their high specific strength, excellent damage tolerance, corrosion resistance, and formability.

It has been demonstrated that the 2xxx series alloys can exhibit the Portevin-Le Chatelier (PLC) effect under certain experimental conditions, as reported by Ren et al. (2019). Investigations on the PLC effect have been conducted to improve the surface finishing quality of materials by avoiding strain localisation bands caused by PLC effects. Various studies have also been dedicated to investigating the impact

<footnote><sup>a</sup> MINES Paris, PSL University, MAT-Centre des Matériaux, CNRS UMR 7633, 78000 Versailles, France</footnote>

<footnote><sup>b</sup> VTT Technical Research Centre of Finland, P.O. Box 1000, FI-02044, VTT Espoo, Finland</footnote>

<footnote>* Corresponding author.
E-mail address: thilo.morgeneyer@mines-paristech.fr (T.F. Morgeneyer).</footnote>
of the PLC effect on mechanical properties and showed a correlation between PLC effect and the degradation of ductility, fracture toughness and fatigue resistance (Brindley, 1970; Wang et al., 2012; Ren et al., 2021; Rowlands et al., 2023). However, the specific mechanism behind this relationship remains unclear. The degradation may occur due to the increased strengthening from PLC, which reduces plasticity at the crack tip. Abnormal degradation under PLC conditions may also be attributed to the strain localisation effects induced by PLC. Within these localised bands, voids can easily link up, further contributing to the observed drop in ductility and fracture toughness. The modelling of these interactions remains challenging (Ren et al., 2021). It may be crucial to consider the potential impact of the PLC effect on material and component design.

The PLC effect is typically characterised by a negative strain rate sensitivity (nSRS) range, serrated yielding, and propagating localisation bands at the macroscopic scale (Penning, 1972; McCormick, 1988; Kubin and Estrin, 1990; Xu et al., 2022). At the micro-scale, the influence of Li atoms and precipitates on the PLC effect in Al–Cu–Li alloys at the micro-scale is still controversial. Two explanations for serrated flow in Al–Li system can be found in the literature, with some studies suggesting that Li-based dynamic strain ageing (DSA), involving repeated pinning and unpinning of mobile dislocations by diffusive solute atoms, is responsible for the serrated flow, as proposed by Gregson et al. (1988), Kubin et al. (1992), Kumar et al. (1996), Chmelík et al. (1998) and Wu et al. (2022). On the other hand, some studies propose that the effect is directly caused by the shearing of precipitates such as $\delta'$ (Al$_3$Li) by mobile dislocations, as suggested by Kumar and Pink (1995), Brechet and Estrin (1995), Behnood and Evans (1989), Zambo and Wert (1993) and Pink et al. (2000).

In contrast to AA2024 or 5xxx series aluminium alloys, the occurrence of the PLC effect in AA2198 alloy has not been widely reported. Deschamps et al. (2013) did not observe the PLC effect in AA2198 alloy in the T351 temper. However, they did observe serrations on the tensile curves after heating the material at 155 °C for less than 2 h, which were attributed to the dissolution of atom clusters. These serrations disappeared after a longer ageing time (>8 h). Interestingly, the serrated flow was observed in a very similar alloy, 2196, in the T351 temper, which contains more $\delta'$ phase and higher Li content (1.4–2.1 wt.%) compared to AA2198 (0.8–1.1 wt%). It was also observed that plastic instabilities persisted until 9 h of artificial ageing. After 24 h, serrations were suppressed with the nucleation of the $T_1$ phase. The higher Li content in 2196 makes the alloy more sensitive to PLC effects, while serrations were suppressed earlier in AA2198 due to its lower Li content. This suggests that the PLC effect in AA2198 can still be explained by the dynamic strain ageing (DSA) theory related to Li atoms. Later, Ovri and Lilleodden (2015) observed the PLC phenomenon in AA2198 alloy for an over-aged (OA) temper at a higher temperature (370 °C). Through transmission electron microscopy (TEM), the authors showed that the OA AA2198 material contains less Li solute in the matrix due to high-density $T_2$ and $T_B$ precipitates. They also found that no plastic instability was observed in the naturally aged (NA) temper, which has a higher concentration of solute atoms. The authors concluded that the DSA theory alone could not account for the occurrence of plastic instability in AA2198 alloy. Instead, they proposed a diffusion-controlled mechanism involving a pseudo-locking mechanism that accompanies order hardening. However, more recently, Nayan et al. (2019) reported the manifestation of the PLC effect in AA2198 in the solution-treated condition. It was discovered that this alloy contains no $\delta'$ precipitates. The authors indicated no evidence of precipitate shearing in the alloy. In addition, DSA-induced strain localisation in an AA2198-T3-like state has been reported based on crystal plasticity simulations (Gupta et al., 2019).

To confirm whether the PLC effect exists in this alloy under the most beneficial heat treatment conditions for industrial applications (T3 and T8), additional in-depth investigations are required, along with the implementation of digital image correlation (DIC) measurements.

Previous experimental investigations have predominantly relied on constant strain rate tensile tests as a means to characterise the PLC phenomenon. However, recent studies have demonstrated that factors such as pre-straining history and ageing time can influence the critical strain at which the PLC effect initiates, as emphasised in previous research. For instance, Balík et al. (2000) observed the modification of critical strains for the onset of PLC in an Al–Mg alloy induced by pre-straining. Furthermore, Allain et al. (2011) conducted research on an FeMnC steel and reported an enhancing effect on the static strain ageing process during relaxation tests, subsequently triggering the dynamic strain ageing mechanism in a systematic manner. In another study, Ren et al. (2019) analysed the PLC effect triggered by various loading paths in an Al–Cu aluminium alloy (2139) using DIC. They reproduced these effects through numerical simulations and tests, which were conducted exclusively on smooth tensile specimens. In a recent study (Ge et al., 2022), the authors demonstrated the remarkable enhancing effect of pre-straining on the negative strain-rate sensitivity in a 5xxx series Al–Mg alloy. Furthermore, some alloys may exhibit weak serrations on the stress–strain curve, making it difficult to detect the manifestation of the PLC effect without digital image correlation (DIC) measurements. Hence, the primary objective of this study is to investigate the potential occurrence of the PLC phenomenon in AA2198 alloy using a combination of constant strain rate tests, relaxation tests, and strain rate jump tests, complemented with DIC measurements.

The second objective of this study is to investigate the PLC effect on the strain localisation ahead of the notch of a compact tension (CT)-like specimen, aiming to draw a comparison with the in-situ synchrotron scanning technique involving non-constant strain rate and relaxations. Recent studies have reported strain localisation and intermittent plastic activities in the notched area of thin CT-like specimens fabricated from AA2198-T851 (Morgeneyer et al., 2014; Buljac et al., 2016) and 2198-T351 materials (Buljac et al., 2018), utilising a combination of in-situ laminography and digital volume correlation (DVC). However, the origin of these strain localisation patterns remains uncertain. To shed light on this phenomenon, macroscopic dynamic strain ageing (DSA) based models is employed for simulating the CT tearing test. The resulting localisation patterns are then compared and analysed in this study.

The paper is structured as follows: Section 2 provides an overview of the materials and experimental studies, detailing the studied materials in Section 2.1 and the mechanical testing methods in Section 2.2. The results of constant strain rate tensile tests are presented in Section 2.3, followed by an analysis of the triggering of the PLC effect after relaxation in Section 2.4. Section 2.5 discusses the strain rate jump tests. Section 3 describes the McCormick type elastoviscoplastic strain ageing model used in this study. Section 4 focuses on the FEM simulations for the AA2198 alloy. Section 4.1 covers the FEM simulation of the triggering effect of relaxation, while Section 4.2 discusses the thin sheet compact tension (CT) specimen. Section 4.3 includes a discussion of the findings. Finally, Section 5 presents the conclusions of the study.

## 2. PLC effect in AA2198 aluminium alloy

### 2.1. *Studied material*

AA2198 alloy belongs to the third generation of low-density Al–Cu–Li alloy developed in the past decade, which is starting to be applied for the latest aircraft (Warner, 2006). The addition of Lithium (Li) provides higher strength and damage tolerance along with a reduction of density. Li is the major alloying element in terms of atomic percentage (see Table 1). Li participates in the formation of the $T_1$ (Al$_2$CuLi) precipitate which is the main hardening phase in the current material.

Two grades of AA2198 alloy are used for the experimental study, namely T351 (naturally aged) and T851 (artificially aged). In addition, these material sheets are in the recrystallised state thus also referred to as T3R and T8R respectively. The grain structure of AA2198-T3R
<table><caption>**Table 1**

Chemical composition in weight percent (wt.%) and atomic percent (at.%) of the AA2198 alloy (Chen, 2011).</caption><thead><tr><th></th><th>Cu</th><th>Li</th><th>Zn</th><th>Mn</th><th>Mg</th><th>Zr</th><th>Si</th><th>Ag</th><th>Fe</th></tr></thead><tbody><tr><th>wt.%</th><td>2.9–3.5</td><td>0.8–1.1</td><td>≤0.35</td><td>≤0.5</td><td>0.25–0.8</td><td>0.04–0.18</td><td>≤0.08</td><td>0.1–0.5</td><td>≤0.01</td></tr><tr><th>at.%</th><td>1.23–1.48</td><td>3.11–4.26</td><td>≤0.14</td><td>≤0.25</td><td>0.28–0.88</td><td>0.01–0.05</td><td>≤0.08</td><td>0.02–0.06</td><td>≤0.05</td></tr></tbody></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** Geometry of the smooth tensile specimen used for experiments. The strain measured along the line Y will be used for creating the spatio-temporal diagram. EBSD inverse pole figure showing the grain structures and orientations of the 2198-T8R material in TS, LS, LT planes (L: rolling direction, T: long transverse direction and S: short transverse direction). EBSD data is from Kong et al. (2022).</figcaption></figure>

is shown in Fig. 1. Flat grains were observed in the L-T plane. The grain size was measured by Kong et al. (2022) using a mean linear intercept method: 82 µm along the rolling direction (L) and 80 µm along the transverse direction (T), 20 µm in the short transverse direction (S). No difference in grain structure was observed between T3R and T8R.

In addition, TEM investigations reveal significant differences between the two temper conditions. In the naturally aged T3R condition, only dispersoids such as Al₂₀Cu₂Mn₃ and Al₃Zr are observed, while in the artificially aged T8R condition, plate-shaped T₁(Al₂CuLi) precipitates and θ' (Al₂Cu) are clearly visible (see Chen, 2011; Decreus et al., 2013). In addition, the volume fraction of intermetallic particles and void content in 2198 alloy is very low (only 0.34%).

## 2.2. Mechanical response

Tensile tests were performed on flat tensile specimens of both T3R and T8R grades under uniaxial loading conditions. All the specimens were prepared in the L-T plane with the axis along the *T* direction, as shown in Fig. 1. The thickness of the sheets was 2 mm. Machine stiffness is known to influence the manifestation and characteristics of serrated flow during DSA, as highlighted in recent experimental and numerical studies on superalloys (Guillermin et al., 2023). All experiments were conducted on an electromechanical testing machine with a stiff load frame (Instron 5582, 100 kN capacity), minimising such effects. Careful attention was given to maintain constant displacement rate control, which minimises external dynamic feedback that could alter localisation dynamics. Unless otherwise specified, tests were conducted at room temperature.

Three types of tensile tests were carried out:

1. Usual monotonic constant strain rate tests with prescribed strain rates of $\dot{E} = 10^{-5} \text{ s}^{-1}$, $10^{-4} \text{ s}^{-1}$, $10^{-3} \text{ s}^{-1}$ and $10^{-2} \text{ s}^{-1}$ (CSR).

2. Tests involving single or multiple relaxations.

3. Strain rate jump (SRJ) tests: increasing or decreasing strain rate at different strain levels.

## 2.3. Constant strain rate tensile tests

Fig. 2 displays the nominal stress-strain curves of T3R specimens subjected to different applied strain rates ranging from $\dot{E} = 10^{-2} \text{ s}^{-1}$ to $10^{-5} \text{ s}^{-1}$. The strain rate sensitivity between two applied strain rates is calculated continuously during the tensile tests up to the ultimate strength point using the formula $m(\dot{E}_i, \dot{E}_j) = \log(\sigma_i/\sigma_j)/\log(\dot{E}_i/\dot{E}_j)$, where $\sigma_i$ and $\sigma_j$ are the stress values measured in tests conducted at overall strain rates $\dot{E}_i$ and $\dot{E}_j$, respectively. As shown on the right y-axis, a slight negative strain rate sensitivity is observed, limited to a specific strain range.

Similarly, Fig. 3 illustrates the nominal stress-strain curves of T8R specimens tested at different strain rates. It can be observed that T8R material exhibits higher yield and ultimate strength compared to T3R, but lower ductility. Moreover, T3R shows a higher negative strain rate sensitivity (the maximum observed negative value: $m \approx -0.005$) compared to T8R (the maximum observed negative value: $m \approx -0.001$). Type A serrations are observed at $\dot{E} = 10^{-2} \text{ s}^{-1}$ and $\dot{E} = 10^{-3} \text{ s}^{-1}$ in T3R specimens, and these tests have been repeated multiple times with good reproducibility.

In contrast to T3R, the stress-strain curves of T8R are smoother. The tensile curves of T8R specimens tested at various temperatures ranging from -50 °C to +80 °C are shown in Fig. 4, and no marked serrations are observed. The material's tensile properties are significantly influenced by temperature. As the temperature decreases, the material exhibits an increase in both yield and ultimate tensile strength. However, ductility is reduced at lower temperatures.

## 2.4. Triggering of PLC effect after relaxations

In addition to the constant strain rate tensile tests, relaxation and strain rate jump tests are also performed for both T3R and T8R grades of AA2198 alloys. The results of the digital image correlation (DIC) measurements from the constant strain rate tensile tests will be presented alongside those from the relaxation tests for direct comparison.

Figs. 5, 6, and 7 display the spatio-temporal patterns obtained through DIC for the CSR tests and single relaxation tests conducted on T3R specimens at different strain rates of $\dot{E} = 10^{-2} \text{ s}^{-1}$, $10^{-3} \text{ s}^{-1}$, and $10^{-4} \text{ s}^{-1}$. These patterns provide information about the strain rate localisation bands' location at various stages of the experiment, as described in previous works (Chmelík et al., 2002; Nogueira de Codes and Bennallal, 2011). The incremental axial strain over the entire deformation process is measured along a line positioned in the gauge area (referred to as line Y in Fig. 1). The colour contour in the figures represents the ratio between the measured strain rate and the applied macroscopic strain rate, denoted as $\dot{\epsilon}_{yy}/\dot{E}$, where $\dot{\epsilon}_{yy}$ is calculated as the change in $\epsilon_{yy}$ over a constant time interval $\Delta t$ through image correlation. It should be noted that the choice of $\Delta t$ affects the visual appearance of the spatio-temporal maps. Wave-like features occasionally observed are artefacts introduced during DIC post-processing, particularly when using smaller time steps, which increase noise levels. These features do not correspond to real mechanical instabilities.

In order to investigate the triggering effect in the AA2198-T3R material, a comparison was made between the spatio-temporal patterns obtained from constant strain rate tests and tests involving a single relaxation at a strain level of $\epsilon = \Delta L/L_0 = 0.02$. Fig. 5 presents the results for specimens tested at a strain rate $\dot{E} = 10^{-2} \text{ s}^{-1}$. In the Conventional CSR test, the PLC effect initiates at $\epsilon = 0.125$, whereas in the relaxation test, localisation bands start to propagate right after the re-straining at $\epsilon = 0.02$. Fig. 6 gives the results at a lower strain rate of
<figure><img src="image_5.png" /><figcaption>Fig. 2. Constant strain rate tensile tests of AA2198-T3R at different strain rates. The strain rate sensitivity indicator *m* is superposed (right y-axis).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 3. Results of constant strain rate tensile tests of AA2198-T8R at different strain rates. The strain rate sensitivity indicator *m* is superposed (right y-axis).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 4. Tensile tests at different temperatures of AA2198-T8R specimens tested at $\dot{E} = 10^{-3}$ s$^{-1}$.</figcaption></figure>

$\dot{E} = 10^{-3}$ s$^{-1}$. The PLC effect initiates at $\varepsilon = 0.14$ under CSR conditions, and the spatio-temporal pattern for the relaxation test at this strain rate is similar to that at $\dot{E} = 10^{-2}$ s$^{-1}$. It should be noted that serrations on the stress-strain curve are subtle and can often go unnoticed, potentially overlooking the true extent of the material's deformation. This showcases the indispensable role of DIC in uncovering the intricate details of the material's deformation behaviour.

Fig. 7 presents the spatio-temporal patterns of CSR and single relaxation tests at the strain rate $\dot{E} = 10^{-4}$ s$^{-1}$. In the CSR test, there

is no signal of the PLC effect. In contrast to the tests at $\dot{E} = 10^{-2}$ s$^{-1}$ or $10^{-3}$ s$^{-1}$, strain rate localisation bands do not appear right after re-straining at $10^{-4}$ s$^{-1}$. The critical strain for the onset of the PLC effect after re-straining is around $\varepsilon = 0.1$.

The experimental results, comparing the effects of relaxation at various strain levels, tested at a strain rate of $10^{-2}$ s$^{-1}$ for both T3R and T8R materials, are presented in Fig. 8. In both tests, four relaxations were applied at strains $\varepsilon = 0.02, 0.04, 0.06$, and $0.08$, followed by a displacement hold time of $15$ s for each relaxation. For the T3R specimen, localisation bands appeared after the first re-straining around the shoulder and propagated through the gauge area. The second re-straining triggered only a single band, while the bands after the third and fourth re-straining started directly from the location where the previous localisation bands disappeared. This observation highlights the influence of localisation history on the triggering effect, which has not been reported previously. In contrast, unlike the T3R specimen, there were no significant localisation bands observed after re-straining in the T8R specimen. However, slight localisation behaviour was still observed around the shoulder. The spatio-temporal patterns of T8R specimens under constant strain rate (CSR) conditions are presented in Fig. A.20. Notably, no PLC effect was observed in T8R specimens under CSR conditions.

## 2.5. *Strain rate jump tests*

Fig. 9(a) shows the nominal stress-strain curves of the strain rate jump tests for AA2198-T3R specimens with 2 different jump sequences. The increasing strain rate test (blue curve) started with strain rate $10^{-5}$ s$^{-1}$. At $\varepsilon = 0.02$, the applied strain rate was increased to $10^{-4}$ s$^{-1}$. At $\varepsilon = 0.04$, strain rate was increased from $10^{-4}$ s$^{-1}$ to $10^{-3}$ s$^{-1}$. After $\varepsilon = 0.06$, applied strain rate was increased to $10^{-2}$ s$^{-1}$ until fracture. The
<figure><img src="image_8.png" /><figcaption>Fig. 5. Comparison of the experimental spatio-temporal diagrams between the constant strain rate test and the test with relaxation at a strain $ε = 0.02$ for AA2198-T3R, tested at an applied strain rate $\dot{E} = 10^{-2}$ s$^{-1}$.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 6. Comparison of the experimental spatio-temporal diagrams between the constant strain rate test and the test with relaxation at a strain $ε = 0.02$ for AA2198-T3R, tested at an applied strain rate $\dot{E} = 10^{-3}$ s$^{-1}$.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 7. Comparison of the experimental spatio-temporal diagrams between the constant strain rate test and the test with relaxation at a strain of $ε = 0.02$ for AA2198-T3R, tested at an applied strain rate of $\dot{E} = 10^{-4}$ s$^{-1}$.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Fig. 8. Tensile tests involving 4 relaxations. Left: AA2198-T8R at $\dot{E}=10^{-3}$ s$^{-1}$. Right: AA2198-T8R at $\dot{E}=10^{-2}$ s$^{-1}$.</figcaption></figure>

decreasing strain rate test (red curve) was conducted in the opposite direction. The first strain rate jump is set at $\epsilon = 0.04$ where the applied strain rate decreases from $10^{-2}$ s$^{-1}$ to $10^{-3}$ s$^{-1}$. The second jump is set at $\epsilon = 0.09$ along with a decrease of applied strain rate from $10^{-3}$ s$^{-1}$ to $10^{-4}$ s$^{-1}$. Fig. 9(b) and (c) show the spatio-temporal diagrams of these two strain rate jump tests. In the increasing strain rate test, a single localisation band started after the second strain rate jump ($\epsilon = 0.04$) which is earlier than the CSR test at $\dot{E}=10^{-3}$ s$^{-1}$. In the decreasing strain rate jump test, the triggering effect is less pronounced. We can observe slight bands after the second strain rate change at $\epsilon = 0.04$. Localisation bands showed different behaviour at the same strain rate.

Strain rate jump tests were also performed for T8R specimens. As shown in Fig. 10, the increasing strain rate test (blue curve) involves 3 strain rate jumps: from $10^{-5}$ s$^{-1}$ to $10^{-4}$ s$^{-1}$ at $\epsilon = 0.02$; $10^{-4}$ s$^{-1}$ to $10^{-3}$ s$^{-1}$ at $\epsilon = 0.04$; $10^{-3}$ s$^{-1}$ to $10^{-2}$ s$^{-1}$ at $\epsilon = 0.06$. The decreasing strain rate test (red curve) involves 2 strain rate jumps: from $10^{-2}$ s$^{-1}$ to $10^{-3}$ s$^{-1}$ at $\epsilon = 0.04$ deformation; $10^{-3}$ s$^{-1}$ to $10^{-4}$ s$^{-1}$ at $\epsilon = 0.65$ deformation. Unlike T3R specimens, moving localisation bands are not observed in these strain rate jump tests of T8R specimens. However, for the increasing strain rate test, slight localisation bands could be observed initiating from the two shoulders of specimens. The location of the final necking seems to be a result of competition between these two early localisation spots.

### 3. The McCormick type elastoviscoplastic strain ageing model

The McCormick type model (McCormick, 1988; Graff et al., 2004, 2005; Mazière et al., 2010) can be presented within the finite strain framework using the concept of local objective frames following Besson et al. (2009). Second and fourth-order tensors are defined by a single tilde $\tilde{Z}$ and a double tilde $\tilde{Z}$ respectively. Observer invariant stress and strain rate measures $\sigma$ and $\dot{\epsilon}$ are defined by the transformation of the Cauchy stress tensor $\tilde{T}$ and the Eulerian strain rate tensor $\tilde{D}$ into the corotational frame characterised by the rotation $\tilde{Q}(\tilde{x}, t)$ at each material point:

$$ \left\{ \begin{array}{l} \tilde{\sigma} = \tilde{Q} \cdot \tilde{T} \cdot \tilde{Q}^T \\ \dot{\tilde{\epsilon}} = \tilde{Q} \cdot \tilde{D} \cdot \tilde{Q}^T \\ \tilde{Q} \quad \text{such as} \quad \dot{\tilde{Q}}^T \cdot \tilde{Q} = \tilde{Q} \quad (\text{corotational}) \end{array} \right. $$

where $\tilde{D}$ and $\tilde{Q}$ respectively are the symmetric and skew-symmetric parts of the velocity gradient.

The strain rate is then decomposed into elastic $\dot{\epsilon}^e$ and plastic $\dot{\epsilon}^p$ parts:

$$ \dot{\epsilon} = \dot{\epsilon}^e + \dot{\epsilon}^p, \quad \sigma = \tilde{C} : \epsilon^e \quad (1) $$

where $\tilde{C}$ is the Hooke tensor of elasticity. The plastic flow is described by the normality rule

$$ \dot{\epsilon}^p = \dot{p} \underline{n}, \quad \underline{n} = \frac{\partial F}{\partial \underline{\sigma}}, \quad (2) $$

where $p$ is the cumulative plastic strain and the yield function, $F$, is taken in the form

$$ F(\underline{\sigma}) = \sigma_{eq}(\underline{\sigma}) - R(p) - R_a(t_a) \quad (3) $$

$$ R(p) = R_0 + Q_1[1 - e^{-b_1 p}] + Q_2[1 - e^{-b_2 p}] \quad (4) $$

$$ \dot{p} = \dot{\epsilon}_0 \sinh \left( \frac{\max(0, F)}{\sigma_0} \right) \quad (5) $$

where $\sigma_{eq}$ can be any of the well-known equivalent stress measures associated with classical yield criteria (von Mises, Hosford, Tresca etc.). $R(p)$ is one of the conventional hardening functions. $R_0$, $Q_1$, $Q_2$, $b_1$ and $b_2$ are hardening parameters. The ageing hardening term $R_a$ reads

$$ R_a(t_a) = P_1 \left[ 1 - e^{-\left(\frac{t_a}{t_0(p)}\right)^n} \right], \quad i_a = 1 - \frac{t_a}{w}, \quad t_0(p) = (1/P_2 p^\alpha)^{1/n} \quad (6) $$

where $P_1$ is the maximum possible serration amplitude; $t_0$ is characteristic time of diffusion processes; $n = 0.33$ is usually used for aluminium. The ageing time $t_a$ grows like time in the absence of plastic flow. Its evolution law depends on the plastic strain rate $\dot{p}$ and $\omega$ represents the strain increment produced when all arrested dislocations overcome local obstacles and move forward to the next pinned configuration. The influence of cumulated plastic strain on the ageing process is introduced by the power $\alpha$.

The elastic modulus and Poisson's ratio are kept the same for the T3R and T8R states. Parameters $\sigma_0$ and $\dot{\epsilon}_0$ related to viscosity were determined based on the relaxation test and CSR tests at different strain rates. These values are kept unchanged for T3R and T8R states. The value of $\dot{\epsilon}_0$ ($1.27 \times 10^{-6}$) used here is smaller than that ($\dot{\epsilon}_0 = 3.5 \times 10^{-5}$) used for the 2024 Al-Cu alloy (Böhlke et al., 2009).

The ageing hardening term $R_a$ involves 5 distinct parameters: $P_1$, $P_2$, $\alpha$, $n$, $w$. As shown in Table 2, $P_2$, $n$, $w$ have the same values for T3R and T8R states. $P_1$ and $\alpha$ have been optimised to better fit the experimental results. A typical value of $w = 1 \times 10^{-4}$ for aluminium alloy is used (Benallal et al., 2008) which is smaller than that ($6.81 \times 10^{-4} \sim 7.4 \times 10^{-4}$) proposed by Böhlke et al. (2009). The parameter $n$ is fixed at 0.33 as used by Dierke et al. (2007), Benallal et al. (2008) and Böhlke et al. (2009). The optimised value of $P_1$ is consistent with that obtained by Graff et al. (2008) and Böhlke et al. (2009) (19–50 MPa). The resultant $t_0$ varies from 1.4 to 5 s ($t_0 = 0.125$ s in Böhlke et al., 2009, $t_0 = 10$ s in Gupta et al., 2017).
<figure><img src="image_12.png" /><figcaption>Fig. 9. AA2198-T3R strain rate jump tests. (a) stress–strain curves; The spatio-temporal diagram of strain rate jump tests: (b) increasing strain rate test; (c) decreasing strain rate test.</figcaption></figure>

The hardening parameters $Q_1$, $Q_2$, $b_1$ and $b_2$ together with $P_1$ and $\alpha$ are determined by an optimisation procedure. The Levenberg–Marquardt algorithm is used to minimise the deviation between data from material point simulations and that of the smoothed experimental stress–strain curves during the optimisation procedure. This strategy avoids time-consuming FEM simulations on full-dimension specimens (Marais et al., 2012).

## 4. FEM simulations for AA2198 alloy

### 4.1. FEM simulation of the triggering effect of relaxation

Fig. 11 shows the 2D and 3D meshes used for the simulations of CSR and relaxation tensile tests. The 2DST6 mesh contains 3369 nodes and 1056 C2D8R 2D quadratic elements with reduced integration. The 3DST6_x1 mesh is completed with 7895 nodes and 1056 C3D20 3D quadratic brick elements. 3DST6_x4 mesh has 21473 nodes and 4 times more elements than 3DST6_x1 due to the partition in the thickness plane. The simulated PLC band at $\dot{E} = 10^{-2} s^{-1}$ using these 3 meshes are presented in Fig. 12. The band on the 2D mesh looks

<table><caption>Table 2
Parameters of the elastoviscoplastic model identified for T3R and T8R.</caption><thead><tr><th colspan="7">AA2198-T3R</th></tr><tr><th>E(GPa)</th><th>ν</th><th>P<sub>1</sub> (MPa)</th><th>n</th><th>ω</th><th>P<sub>2</sub> (s<sup>-n</sup>)</th><th>α</th></tr></thead><tbody><tr><td>71</td><td>0.3</td><td>43</td><td>0.33</td><td>10<sup>-4</sup></td><td>3.49</td><td>0.29</td></tr><tr><td>Q<sub>1</sub> (MPa)</td><td>b<sub>1</sub></td><td>Q<sub>2</sub> (MPa)</td><td>b<sub>2</sub></td><td>R<sub>0</sub> (MPa)</td><td>ε<sub>0</sub> (s<sup>-1</sup>)</td><td>σ<sub>0</sub> (MPa)</td></tr><tr><td>50</td><td>456</td><td>205</td><td>10</td><td>196</td><td>1.27 × 10<sup>-6</sup></td><td>3.0</td></tr></tbody><tbody><tr><th colspan="7">AA2198-T8R</th></tr><tr><th>E(GPa)</th><th>ν</th><th>P<sub>1</sub> (MPa)</th><th>n</th><th>ω</th><th>P<sub>2</sub> (s<sup>-n</sup>)</th><th>α</th></tr><tr><td>71</td><td>0.3</td><td>30</td><td>0.33</td><td>10<sup>-4</sup></td><td>3.49</td><td>0.22</td></tr><tr><td>Q<sub>1</sub> (MPa)</td><td>b<sub>1</sub></td><td>Q<sub>2</sub> (MPa)</td><td>b<sub>2</sub></td><td>R<sub>0</sub> (MPa)</td><td>ε<sub>0</sub> (s<sup>-1</sup>)</td><td>σ<sub>0</sub> (MPa)</td></tr><tr><td>70</td><td>544</td><td>94</td><td>20</td><td>358</td><td>1.27 × 10<sup>-6</sup></td><td>3.0</td></tr></tbody></table>

different from that on 3D mesh. The band morphology simulated using the mesh with 4 elements in the thickness plane is similar to that using a single element in the thickness plane. It is well-known that 3D simulations are preferable to 2D plane stress elements to study localisation problems (Besson et al., 2009).
<figure><img src="image_13.png" /><figcaption>**Fig. 10.** AA2198-T8R strain rate jump tests. (a) stress–strain curves; The spatio-temporal diagram of strain rate jump tests: (b) increasing strain rate test; (c) decreasing strain rate test.</figcaption></figure>

Fig. 13 shows the simulated macroscopic stress–strain curves for both T8R and T3R states at $\dot{E}=10^{-2} s^{-1}$. The numerical results are in good agreement with experiments.

Fig. 14(a) and (b) shows the numerical spatio-temporal pattern of the CSR test and single relaxation test using 2D mesh of T3R material. Fig. 14(c) and (d) present the result on the 3D mesh (3DST6_x1). The triggering effect after relaxation is reproduced successfully. The 3D simulation is closer to DIC measurements presented in Fig. 5. Fig. 15 shows the stress–strain curve at strain rate $10^{-2} s^{-1}$ with one relaxation applied at $\epsilon = 0.02$ (corresponds to Fig. 14(d)). The band morphology is also presented.

Fig. 16 gives the numerical spatio-temporal patterns of CSR and multiple-relaxation tests of T8R material. The model does not predict any localisation band for the CSR conditions. However, slight trace of localisation bands initiated after re-straining. These results are very similar to the DIC measurements presented in Fig. 8(b).

## 4.2. Thin sheet compact tension (CT) specimen

The previous simulation results reproduced the major experimental observations in tensile tests which allows us to apply the model to other

specimen geometries. As reported by Morgeneyer et al. (2014), Buljac (2017), multiple localisation bands have been observed in the notch area of CT specimens made by AA2198-T3R and T8R alloys. The final fracture is found to develop following these localisation bands. However, the origin of these bands is still not fully clear. To investigate the localisation patterns due to strain ageing and compare them with that observed in CT specimens by laminography-DVC (Morgeneyer et al., 2014), the early plasticity during the tearing test of CT specimens was predicted with the models presented in the previous parts. Particularly, the relaxation effect in the CT specimen is investigated as the stepwise loading mode during the in-situ laminography test is closer to the current relaxation test. During the in situ experiment, the specimen was scanned after each loading by maintaining the notch opening displacement (Morgeneyer et al., 2014; Buljac et al., 2018).

The objective of finite element simulations in this part is to investigate the localisation behaviour around the notch tip of a thin sheet CT specimen. The strain rate pattern after re-straining in a CT specimen is compared with that under constant strain rate loading. We use the same specimen geometry (60 × 70×1 mm) as that in Morgeneyer et al. (2014), Rousselier et al. (2017) in order to make a direct comparison with previous laminography and simulation results. Fig. 17 illustrates
<figure><img src="image_14.png" /><figcaption>Fig. 11. Finite element meshes used for the simulations of CSR and relaxation tensile tests.</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 12. PLC band morphologies at $10^{-2} s^{-1}$ using different meshes.</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>Fig. 13. Simulated constant strain rate tensile curves at $\dot{E} = 10^{-2} s^{-1}$ for T3R and T8R with the macroscopic model.</figcaption></figure>

the detailed mesh structure. The mesh comprises 29360 elements and 128649 nodes in total. The design utilises C3D20R hexahedral elements and C3D15R prismatic elements. The element size in the refined notch region is $0.125 \times 0.1 \times 0.125 \text{ mm}^3$. Displacements at a constant velocity of $10^{-3} \text{ mm s}^{-1}$ were applied as loading conditions through two “rigid” triangular blocks. The positions of these two blocks are determined according to experimental conditions (Morgeneyer et al., 2014). To fix

the rigid body motion, the point at (x = 60 mm, y= 0 mm, z= 0 mm, the origin of the coordinate system is located at the mid-plane of left side) is fixed ($U_x = 0, U_y = 0, U_z = 0$). The displacement $U_z$ is set to be 0 at the mid-plane z = 0 mm of the area surrounding the critical notch zone.

The simulation results for the AA2198-T3R CT specimen are shown in Fig. 18. The section located at 1 mm to the notch tip ($\Delta x = 1$ mm) is extracted for observation. Five snapshots are selected for both the CSR and stepwise loading tests whose corresponding time instants are marked on the force-time curves. The “waffle-like” strain rate pattern composed of multiple parallel or orthogonal slant bands is observed for both CSR and stepwise loading conditions. The pattern for the CSR condition exhibits more intermittent behaviour. The pattern is less symmetric compared with that after relaxation. A “X” type pattern is always found after the restraining which is very stable. The strain rate inside these bands is visibly higher than that during CSR loading. These “X” type bands split into multiple bands after a very short time and seem to try to escape from the notch area. Fig. 18 shows the patterns after the first (t= 42 s) and second (t= 83 s) re-straining.

CSR and stepwise loading conditions were also performed on the CT specimen using the parameters of T8R material as shown in Fig. 19. For this material, there is no PLC effect for the CSR loading which is confirmed by both DIC measurement and simulations on tensile specimens. For the relaxation tests, only a very slight tendency for localisation band formation can be seen right after re-straining. Compared to T3R specimen, these shallow bands can easily be neglected as shown in Figs. 8 and 16. Only two snapshots of the strain rate pattern for CSR loading are presented in Fig. 19. There is no multiple band formation under CSR condition which has been checked during the whole deformation process. For the stepwise loading, “X” type localisation bands are observed again as in the T3R material. These bands evolved into multiple bands later. Compared to T3R, there are fewer localisation bands in T8R and the pattern is more homogeneous. Once the bands propagate towards the region far from the notch tip, no more bands could be seen.

### 4.3. Discussion

Previous studies have provided insights into the behaviour of the AA2198 alloy, specifically regarding the Portevin-Le Chatelier (PLC) effect under various ageing conditions. Deschamps et al. (2013) did not find PLC effect in the naturally aged AA2198 alloy (T351). Ovri and Lilleodden (2015) also reported the absence of plastic instability in the naturally aged AA2198 alloy in which higher concentration of solute atoms is found. Meanwhile, the PLC phenomenon was observed for an overaged temper treatment at a higher temperature (370 °C) different from Deschamps et al. (2013). The authors made the conclusion that DSA is not sufficient to account for the plastic instability in AA2198 alloy. A diffusion-controlled mechanism which involves the operation of a pseudo-locking mechanism that accompanies order hardening was proposed.

Contrary to Deschamps et al. (2013) or Ovri and Lilleodden (2015), we showed that the naturally aged AA2198 alloy is indeed very sensitive to PLC effect. It is important to note that we cannot identify the PLC effect based solely on tensile curves. DIC is necessary to clearly show the associated localisation events. This effect might have been present in other studies but went unnoticed because DIC analysis was not performed. This is consistent with the work of Alexopoulos et al. (2013), though the authors did not pay attention to the plastic instabilities showed on the T351 curve. The artificially aged temper (T8R) is less sensitive to PLC effect. This result suggests that DSA is still the main mechanism for triggering plastic instabilities in AA2198-T3R alloy due to higher Li concentration in this naturally aged state. The T8R state is less sensitive because of the depletion of Li due to precipitation.

To some extent, the triggering effect after relaxation observed here is very similar to that observed in an FeMnC alloy (Allain et al., 2011). There is always a Lüders like plateau after re-straining followed by
<figure><img src="image_17.png" /><figcaption>**Fig. 14.** Comparison of the spatio-temporal patterns of CSR and relaxation tests at $10^{-2}$ s⁻¹ for AA2198-T3R: (a) 2D FEM simulation of CSR test; (b) 2D FEM simulation involving a single relaxation at $\epsilon = 0.02$; (c) 3D FEM simulation (3DST6_x1) of CSR test; (d) 3D FEM simulation (3DST6_x1) involving a single relaxation at $\epsilon = 0.02$. These simulated patterns are in good agreement with DIC results shown in Fig. 5.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>**Fig. 15.** Strain rate bands in 3D tensile specimen (3DST6_x1) at strain rate $10^{-2}$ s⁻¹ with one relaxation at $\epsilon = 0.02$.</figcaption></figure>

type A serrations. Allain et al. (2011) attributed this phenomenon in the FeMnC alloy to the triggering of twinning promoted by dislocation pinning during relaxation, whilst this explanation cannot be applied to the current aluminium alloy.

The current relaxation tests and simulations show that the DSA mechanism could explain this premature triggering effect. As shown in Fig. 7, the PLC effect did not appear right after the re-straining. Due to solute diffusion, the ageing during relaxation and pre-strain history modified local Li distribution which might result in a modification of the critical strain for PLC. In addition, the strain rate heterogeneity due to the Lüders effect after reloading could also allow an early triggering of PLC bands.

In the broader context of strain localisation phenomena, it is worth noting that the observed behaviour can also be interpreted within the framework of autowave theories of plastic deformation. The experimental and modelling approach by Zuev and Khon (2025) was applied to Lüders and PLC effects in Danilov et al. (2018) for steels and duralumin. The autowave approach was recently described in Zuev et al. (2025). The proposed model represents an alternative approach to the Estrin-Kubin-McCormick class of models. It is based on reaction-diffusion types of constitutive equations involving second spatial gradients of strain and stress. In contrast to Zuev's very detailed analysis of one-dimensional jerky flow and patterning, the present work is based on three-dimensional finite element analyses of tensile and notched specimens which requires solving equilibrium and constitutive equations at each time step. This is particularly important at stress concentrators such as sample fillets and notches. In that respect, the EKMC model which includes a single additional internal variable controlled by a time differential equation is particularly efficient for structural applications, with systematic comparison with strain field measurements.
<figure><img src="image_19.png" /><figcaption>**Fig. 16.** The FEM simulation (3DST6_x1) involving relaxation is compared with the constant strain rate (CSR) curve at the same applied strain rate (10⁻² s⁻¹) for AA2198-T8R. These results can be compared with the experimental results of Fig. 8(b).</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>**Fig. 17.** Mesh of the thin sheet CT-like specimen.</figcaption></figure>

**5. Conclusions**

In conclusion, this study investigated the strain ageing effect in two grades of AA2198 alloy by tensile tests, DIC measurements and finite element simulations of smooth and deep-notched samples. The results revealed notable differences in the localisation behaviour of AA2198-T3R and AA2198-T8R specimens.

The PLC effect was observed in AA2198-T3R specimens under constant strain rate loading conditions by DIC, whilst serrations were weak. Notably, relaxation and sudden strain rate changes were found to trigger localisation bands, leading to an earlier onset of the PLC effect in the T3R material. In contrast, the PLC effect was not observed in AA2198-T8R specimens. Only shallow localisation bands were observed after re-straining, and these bands failed to propagate.

An elastoviscoplastic model incorporating strain ageing has been identified through tensile experiments conducted at various strain rates, enabling the simulation of strain ageing effects in T3R and T8R specimens. The model effectively captures the spatio-temporal evolution of localisation in the tensile specimens. It reproduces the triggering of localisation bands following relaxation and strain rate jump-induced

re-straining in T3R specimens, as well as the formation of immobile bands in T8R specimens.

Three-dimensional FE simulations of CT-like specimens demonstrate that the triggering of the PLC effect after re-straining observed in tensile tests can also be noticed in CT specimens. For T3R material, localisation bands appear under both constant strain rate and stepwise loading conditions. For the less sensitive T8R material, multiple bands also appear after re-straining, whereas the constant strain rate loading does not produce multiple localisation bands. The multiple strain bands near the notch tip predicted by the simulations have similar patterns compared to those observed in Tomography-DVC experiments by Morgeneyer et al. (2014, 2016) where stepwise loading was applied during in-situ laminography tests. However, unlike experiments, the simulated bands using the strain ageing model show high mobility. Microstructure-induced localisation leads to a more stable form of localisation, as detailed in Morgeneyer et al. (2021) and Gille et al. (2024), which more closely resembles the localisation events observed through Tomography-DVC.
<figure><img src="image_21.png" /><figcaption>**Fig. 18.** Force-time curves of the CSR and the stepwise loading of AA2198-T3R CT specimen. Strain rate patterns on the surface at $\Delta x = 1$ mm to the notch tip. The time instant of each snapshot is given and marked on the force-time curves. Simulations are performed with the macroscopic strain ageing model.</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>**Fig. 19.** Multiple strain rate localisation bands triggered after re-straining in AA2198-T8R compared with constant strain rate loading condition. The corresponding positions of these images are marked on the force-time curve. Simulations are performed with the macroscopic strain ageing model.</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>Fig. A.20. Spatio-temporal patterns of CSR tests at room temperature and different strain rates for AA2198-T8R.</figcaption></figure>

## CRediT authorship contribution statement

**S.C. Ren:** Visualization, Software, Investigation, Conceptualization, Writing – original draft, Validation, Methodology, Data curation. **T.F. Morgeneyer:** Writing – review & editing, Visualization, Resources, Methodology, Funding acquisition, Conceptualization, Writing – original draft, Supervision, Project administration, Investigation, Data curation. **G. Rousselier:** Supervision, Investigation, Validation, Project administration, Conceptualization. **M. Mazière:** Visualization, Supervision, Investigation, Conceptualization, Writing – review & editing, Validation, Software, Funding acquisition. **S. Forest:** Visualization, Supervision, Resources, Methodology, Funding acquisition, Conceptualization, Writing – review & editing, Validation, Software, Project administration, Investigation, Formal analysis.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

Agence National de la recherche (ANR) is gratefully acknowledged for partial financial support on projects (LAMBDA: ANR17-CE08-0051). Constellium C-TEC is thanked for materials supply. Institut Carnot M.I.N.E.S is also acknowledged for funding.

## Appendix. DIC measurements of CSR tests on AA2198-T8R alloy

Fig. A.20 shows the DIC spatio-temporal patterns for AA2198-T8R CSR tests at different strain rates.

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