# On the evolution of cyclic deformation microstructure during relaxation test in austenitic stainless steel at 823K

Xavier Feaugas, S. Catalao, Philippe Pilvin, Marie-Thérèse Cabrillat

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Xavier Feaugas, S. Catalao, Philippe Pilvin, Marie-Thérèse Cabrillat. On the evolution of cyclic deformation microstructure during relaxation test in austenitic stainless steel at 823K. Materials Science and Engineering: A, 2008, 483-484, pp.422-425. 10.1016/j.msea.2006.12.167. hal-00400617

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# On the evolution of cyclic deformation microstructure during relaxation test in austenitic stainless steel at 823 K

X. Feaugas<sup>a</sup>, S. Catalao<sup>b</sup>, P. Pilvin<sup>c</sup>, M.-Th. Cabrillat<sup>b</sup>

The evolution of cyclic patterns during relaxation time has been studied in terms of long-range internal stresses and effective stress evolution as a function of time at 823 K. The main features are the decrease of intra-granular back stress in relation to dipole annihilation promoted by self-diffusion and a decrease of inter-granular back stress which results in a homogenization of dislocation distribution in all the grains. Moreover, the static aging leads to an increase of the athermal component of the effective stress in association with the hardening effect of solution segregation on dislocation junctions.

*Keywords:* Recovery process; Dipole annihilation; Dislocation patterns; Cyclic deformation

## 1. Introduction

In the past, the creep-fatigue interaction has been extensively studied in austenitic stainless steel at high temperatures [1–5]. Generally, a decrease of stress amplitude $\Sigma_a$ is observed as a function of the hold time, $t_m$ [1–3,5]. However, some time a previous hardening was reported before the softening stage [5]. The origin of this hardening stays unclear. However, the “time softening” has been correlated with recovery of dislocation structures. It was suggested that phenomena arise in association with dipole annihilation in dislocation walls [4,5]. The competition between dipole annihilation during the hold time and the formation of this one during cyclic loop conduces to an impossibility to identify the kinetic parameters associated with recovery process. Consequently, in present study we propose an alternative experimental procedure to evaluate recovery of dislocation structure arising to cyclic loading. Moreover, this experimental approach allows discussing the origin of creep-fatigue behaviour at high temperature (823 K).

## 2. Experimental procedures

Experiments were performed on polycrystalline austenitic stainless steel AISI 316L. This alloy presents an average grain size of 53 µm and a dislocation density below $10^{10} m^{-2}$ [6,7]. Cylindrical specimens with a 8 mm diameter and a 10 mm gauge length were machined for tension-compression tests. All tests were performed at 823 K with a constant strain rate of $3 \times 10^{-3} s^{-1}$ and a true plastic strain control. In a first step, a cyclic loading is conduced to stress saturated stage with a constant plastic strain equal to $4 \times 10^{-3}$. In a second step, relaxation tests are performed following by an unloading sequence for different time. The unloading sequence offers a measure of back stress X and effective stress $\Sigma_{ef}$. The analysis proposed by Dickinson et al. [6–9] was used to provide stress partition, taking into account the thermal part of flow stress $\Sigma^*$ (Fig. 1a). The reverse yield stress $\Sigma_r$ is obtained on unloading sequence with a plastic strain offset equal to $5 \times 10^{-5}$. The notions of back stress and effective stress have been extensively discussed in the past [6–9]. The back stress which is the directional component of hardening is linked to the local straining process that introduces long-range interactions with mobile dislocations. The effective stress is an isotropic component which present a thermal component $\Sigma^*$ and an athermal component $\Sigma_\mu$ (Fig. 1a). $\Sigma_{ef}$ represents

<footnote><sup>a</sup>Laboratoire d'Etude des Matériaux en Milieux Agressifs (LEMMA), EA3167, Université de la Rochelle, La Rochelle, France</footnote>

<footnote><sup>b</sup>CEA/DEN/DER/SESI/LCSI Laboratoire de Conception des Systèmes Innovants CEA-Cadarache, France</footnote>

<footnote><sup>c</sup>Laboratoire Génie Mécanique et Matériaux (LG2M), Centre de Recherche Sciences de l'Ingénieur Université de Bretagne Sud, Lorient, France</footnote>

<footnote>* Corresponding author. Tel.: +33 05 46 45 82 11.
E-mail address: xfeaugas@univ-lr.fr (X. Feaugas).</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1. Definition (a) and evolution (b) of long-range internal stress X, thermal Σ' and athermal Σ_μ parts of effective stress Σ_{ef} as a function of time during relaxation sequence.</figcaption></figure>

the stress locally required for a dislocation to move (short-range interactions). The evolution of these stresses versus relaxation time is correlated to dislocation density and distribution investigated by transmission electronic microscopy (TEM). Using a JEOL JEM 2011 apparatus operating at 200 kV, TEM investigations were conducted on thin foils cut parallel to the stress axis. TEM sample preparation was described in detail elsewhere [6,7]. For each test condition, dislocation features are studied and quantitatively measured over 30 grains. The principal stress direction is reported in the cubic crystallographic system for each grain (Fig. 2c). The spatial arrangement of dislocations in each grain is determined in terms of dislocation densities in soft phase (ρ_s) (channels) and hard phases (ρ_w) (walls) and

the fraction of hard phase (f_w) is calculated. The mean value of intra-granular long-range internal stress (X_intra) which arises in grain in relation with dislocation patterning has been evaluated using the composite model initially proposed by Mughrabi [10]. Details of this calculation are given in former publications [6,9,10]. (X_intra) is a mean value of long-range internal stresses experienced by mobile dislocations in inter-walls spacing. The crude approach which does not take into account the geometrical aspect of dislocation pattern allows an evaluation of (X_intra) as follows:

$$ \langle X_{\text{intra}} \rangle = M \langle f_w \rangle \alpha \mu b (\sqrt{\langle \rho_w \rangle} - \sqrt{\langle \rho_s \rangle}) \quad (1) $$

<figure><img src="image_3.png" /><figcaption>Fig. 2. Evolution of dislocation patterns during relaxation sequence: (a) dipolar walls (cyclic loading), (b) partial destruction of dipolar walls (total relaxation), (c) inverse pole figure and (d) statistical analysis of dislocation pattern in terms of fraction of grain which present one kind of dislocation microstructure.</figcaption></figure>
<table><caption>Table 1
Evolution of dislocation densities $\langle\rho_s\rangle$, $\langle\rho_w\rangle$ and distribution $\langle f_w \rangle$ as a function of time during relaxation sequence</caption><thead><tr><th>Test</th><th>Times (s)</th><th>⟨f<sub>w</sub>⟩ (%)</th><th>⟨ρ<sub>s</sub>⟩ (10<sup>14</sup> m<sup>-2</sup>)</th><th>⟨ρ<sub>w</sub>⟩ (10<sup>14</sup> m<sup>-2</sup>)</th><th>α</th></tr></thead><tbody><tr><td>Cyclic loading</td><td>2 × 10<sup>3</sup></td><td>29.5</td><td>0.94</td><td>6.4</td><td>0.2</td></tr><tr><td>Partial relaxation</td><td>7 × 10<sup>3</sup></td><td>16.9</td><td>2.1</td><td>5.4</td><td>0.22</td></tr><tr><td>Total relaxation</td><td>7.6 × 10<sup>4</sup></td><td>4.8</td><td>1.3</td><td>2</td><td>0.29</td></tr></tbody></table>

where $\alpha$ is a constant depending on the elastic interaction of dislocations and $M$ is a Taylor factor. Plastic strain incompatibilities between grains induce long-range internal stresses namely inter-granular back stress $\langle X_{\text{inter}} \rangle$. The evolution of this component can be deduced from $X$ and $\langle X_{\text{intra}} \rangle$ assuming the inter-granular and intra-granular components of back stress additivity [6,9]:

$$ \langle X_{\text{inter}} \rangle = X - \langle X_{\text{intra}} \rangle. \qquad (2) $$

### 3. Experimental results

After cyclic loading sequence the saturation stress reaches a value equal to 370 MPa. This stress is partially reduced during relaxation sequence to reach a value near 260 MPa after 4 days. The evolution of the different components ($X$, $\Sigma_{\mu}$, $\Sigma^*$) of stress during relaxation stage is presented in Fig. 1b. Back stress $X$ and thermal part of effective stress $\Sigma^*$ decrease as a function of time. However, athermal part of effective stress $\Sigma_{\mu}$ increases as a function of time. These results can be correlated with TEM observations. Dislocation patterns formed after cyclic loading (veins, ladders and cells) are partially vanished during relaxation sequence (Fig. 2a and b). After 4 days some grains show tangles but the deformation is mainly homogeneous in grains (Fig. 2d). The consequence of this redistribution of dislocations on stress evolution as a function of time can be evaluated using mean values of the parameters which characterize dislocation patterns ($\langle\rho_s\rangle$, $\langle\rho_w\rangle$, $\langle f_w \rangle$). Inter-granular $\langle X_{\text{inter}} \rangle$ and intra-granular $\langle X_{\text{intra}} \rangle$ long-range internal stresses have been evaluated using data reported in Table 1, Eqs. (1) and (2). Both stresses decrease as a function of relaxation time (Fig. 3). Stress relaxation is mainly associated with back stress softening in relation with the recovery of dislocation pattern (decrease of $\langle X_{\text{intra}} \rangle$) and a reduction of inter-granular plastic strain incompatibilities $\langle X_{\text{inter}} \rangle$. The origin of the reduction of inter-granular plastic strain incompatibilities is well illustrated (Fig. 2d) by the fact that the diversity of dislocation patterns observed in the early stage of relaxation sequence vanishes at the end of relaxation time (homogeneous deformation in all the grain). The vanishing of walls is mainly linked with the annihilation of edge dipoles and the dislocations junctions seems to be less affected by recovery process.

### 4. Discussion

According with a large number of works on cyclic behaviour of fcc metals, there exist not only forest dislocations but also debris such as long dipole loops and point defects in fatigue deformation. Point defects like vacancies intersected by mobile dislocations give rise to much larger temperature and strain rate sensitivity than for tensile deformation. Under a relaxation

sequence vacancies can be also contributed to dipole annihilation. Recently, TEM investigations of dipole height in cyclically deformed AISI 316L in the temperature range 300–873 K have demonstrated that “spontaneous” dipole annihilation proceeds by edge climb process promoted by pipe-diffusion [11]. In present case, the original distances of dipoles that decompose are probably much farther than minimum value of height dipole observed under cyclic loading. Moreover, the decomposition process observed under relaxation sequence is far being considered “spontaneous”. As proposed by Argon and Moffatt [12] the climb of dislocation in fcc metals can be controlled by vacancy emission from extend jog. The rate of decrease of dipole density in wall is given by [12]:

$$ \begin{gathered} d\rho_{\text{dip}} = -B_0\rho_{\text{dip}}^2 dt \quad \text{with} \\ B_0 = \frac{4}{\pi(1-\nu)} \frac{A\mu\Omega Dn}{k_B T} c_j \left[ \frac{\gamma_{\text{sfe}}}{\mu b} \right]^2 \end{gathered} \qquad (3) $$

with $A \sim 1050$ is a numerical factor which depend on elastic constant, $n=12$ is the coordination number of neighbour sites, $c_j b = 10^{-2}$ is the concentration of jogs, $\Omega = 4.65 \times 10^{-29} \text{ m}^3$ is the atomic volume, $\mu = 60 \text{ GPa}$, $\nu = 0.3$, $b = 0.295 \text{ nm}$, $T = 823 \text{ K}$, $k_B = 8.61 \times 10^{-5} \text{ eV K}^{-1}$, $\gamma_{\text{sfe}}/\mu b = 1.2 \times 10^{-3}$ and $D$ is the diffusion constant (self-diffusion or pipe-diffusion). The decrease

<figure><img src="image_4.png" /><figcaption>Fig. 3. Evolution of $X$, $\langle X_{\text{inter}} \rangle$ and $\langle X_{\text{intra}} \rangle$ as a function of time during relaxation sequence.</figcaption></figure>
of dislocation density in the wall $\rho_w$ is interpreted as a decrease of dipole density in the walls. Consequently, the time decrease of dislocation density in the wall can be evaluated as follow: $d\rho_w = d\rho_{dip} \approx -B_0\rho_w^2 dt$. A good description of the decrease of $\rho_w$ as a function of time (Table 1) is obtained using this last equation with $B_0 = 4.5 \times 10^{-20} \text{ m}^2 \text{ s}^{-1}$. Consequently, the value of $B_0$, obtained experimentally, supports the fact that recovery process associated with climb is controlled by vacancy emission from extending jogs. This result seems in agreement with a previous work in copper which finds a similar value of $B_0$ [13]. Eq. (3) provides a fair account of experimental data using the value $D = 2.64 \times 10^{-18} \text{ m}^2 \text{ s}^{-1}$. Taking the pre-exponential factor $D_0 = 10^{-4} \text{ m}^2 \text{ s}^{-1}$ for self-diffusion [19] we evaluate the activation energy of vacancy diffusion. The value of 2.2 eV does not correspond to the activation for pipe-diffusion (0.72 eV [11]) but it is of the order of the energy for self-diffusion (2.4 eV [19]). The dipole annihilation process contributes to intra-granular back stress softening in relation with the recovery of dipolar dislocation pattern (dipolar walls).

The athermal part of effective stress ($\Sigma_\mu$) is mainly linked with forest dislocations like dislocation junction. The increase of this stress component as a function of relaxation time is in accordance with the occurrence of a static strain aging (SSA). Chrome diffusion and segregation on dislocation junction has been extensively reported to explain static and dynamic strain aging (DSA) under tensile loading for austenitic stainless steel [15,16]. In present work, the hardening effect of solution segregation on dislocation junctions can be experimentally evaluated. The athermal effective stress evolution is associated with short-range interactions on mobile dislocations in the “soft phase” (inter-walls spacing) [6,9]. Consequently, the relation between $\Sigma_\mu$ and $\langle\rho_s\rangle$ can be expressed using Crowan by passing relation: $\Sigma_\mu = Ma\beta\mu\sqrt{\langle\rho_s\rangle}$. Using the values of $\Sigma_\mu$ and $\langle\rho_s\rangle$ obtained from different relaxation time the value of $\alpha$ can be obtained (Table 1):

$$ \alpha = \frac{\Sigma_{\mu}}{[Mb\mu\sqrt{\langle\rho_{s}\rangle}]} \quad (4) $$

An increase of 45% of $\alpha$ is observed as a function of the relaxation time (Table 1). In other term the hold time favours solute segregation which increases the elastic interactions of dislocations. The modelling of this effect on athermal stress can be easily described using Estrin-Kubin-McCormick approach (EKMC) [17,18]. This aspect will be developed in a further publication.

To resume two physical mechanisms occur during relaxation sequence: the annihilation of dipole and the solute segregation. The first one favours the dissolution of dipolar walls (ladders, veins) and the second one promotes the formation of incidental dislocation boundaries as recently observed [14]. Consequently, the evolution of intra-granular long-range internal

stress observed as a function of time is the results of a competition between these both phenomenons. Finally, creep-fatigue behaviour recently reported on AISI 316L stainless steel [5,14] can be partly explained using these results. At 823 K, an increase of cyclic saturation stress followed by a decrease has been reported as a function of the hold time [5,14]. The hardening observed for low values of hold time can be associated with static and dynamic strain aging and the softening results with dipoles annihilation process. If the physical parameters associated with SSA and DSA can be the aim of the future work, the physical mechanism of dipole annihilation has been identified in present work.

## 5. Conclusion

The physical origin of the time relaxation of cyclic saturation stress has been identified in austenitic stainless steel at 823 K. Two mechanisms occur during relaxation sequence: the dipole annihilation promoted by self-diffusion and static aging in relation with chrome segregation on dislocation junctions.

## Acknowledgments

The authors thank the Centre Commun d'Analyse, Université de La Rochelle for electron microscopy facilities and the Roberval Laboratory, Université de Technologie de Compiègne for fatigue testing facilities. Financial support from Commissariat à l'Energie Atomique is gratefully acknowledged.

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