Temperature dependent deformation localization in irradiated tungsten
Introduction
Tungsten has many outstanding physical properties, and is considered as the plasma-facing material in the current and future high-temperature energy conversion systems, such as the divertor of nuclear fusion device and the cathode of plasma thruster (Bohnert et al., 2016; Norajitra et al., 2008). In these applications, tungsten has to withstand the severe temperature condition, ranging from about 2000 K to room temperature or even lower. At the same time, tungsten components are subjected to a mixed spectrum of irradiation energy, which induces a high density of irradiation defect clusters. The formation of dislocation channels and deformation localization are widely observed in numerous irradiated materials, which is believed to further induce irradiation embrittlement (Das et al., 2020; Muroga et al., 2002; Odette, 1983; Odette and Lucas, 2001) and increase the susceptibility to stress corrosion cracking initiation (McMurtrey et al., 2015). Therefore, the effective prediction of localized deformation is crucial to guarantee the safe operation of fusion systems. However, it is still far from well understood whether and how dislocation channels form and the deformation localization occurs in irradiated tungsten.
Numerous efforts have been devoted to understanding the microstructure feature and mechanical response of irradiated tungsten. From the experimental perspective, ion irradiation and neutron irradiation tests are usually carried out. The ion irradiation has lower energy and leads to a limited penetration depth of about a few micrometers. The corresponding experiments are performed quickly and mainly focus on the change of surface topography (Kajita et al., 2009; Li et al., 2020; Lindig et al., 2009; Zheng et al., 2019), mechanical properties of the material near the surface (Armstrong et al., 2013; Chen et al., 2018; Das et al., 2019a; Terentyev et al., 2016) and the ion trap effect (Alimov et al., 2009; Ogorodnikova et al., 2003; Piaggi et al., 2015; Zheng and Han, 2020). The irradiation defects are mainly dislocation loops and tangles (Ferroni et al., 2015; Guo et al., 2020). On the contrary, neutron or proton irradiations with much higher energy can cause the apparent change of the whole testing sample, which is similar to the service conditions in nuclear power plants. However, these experiments take a few months or years to carry out. The experimental data of the neutron-irradiated tungsten are relatively limited. The main focus is on the relationship between the neutron doses and the microstructure of irradiation defects (Abernethy et al., 2019; Fukuda et al., 2016; Koyanagi et al., 2017), and the overall mechanical properties, such as the irradiation hardening and embrittlement (Abernethy et al., 2019; Gorynin et al., 1992; Maloy et al., 2005; Sommer, 1995; Steichen, 1976). As summarized in Table A1 in Appendix A, the neutron-induced irradiation defects in tungsten include many types, such as vacancies, interstitials, voids, precipitates and dislocation loops, depending on the irradiation dose and temperature (Abernethy et al., 2019; Fukuda et al., 2016; Koyanagi et al., 2017). When the irradiation temperature is lower than about 500 K, dislocation loops are the primary type of irradiation defects. For lower irradiation doses and higher temperatures (>500 K), dislocation loops and voids are the two main kinds of defects, and precipitates appear for higher irradiation doses.
These irradiation defects serve as the obstacle for the dislocation motion, and induce the irradiation hardening and embrittlement. Steichen et al.'s experimental studies (Steichen, 1976) show that the yield strength of neutron-irradiated tungsten is much higher than that of unirradiated ones and exhibits significant dependence on the temperature and strain rate. Maloy et al. (Maloy et al., 2005) studied the effect of proton irradiations on the ductility of tungsten. It is found that the irradiation defects could dramatically decrease the ductility and quickly induce fracture at room temperature. The decrease of the ductility is also observed in other experiments. For example, fission neutron-irradiated tungsten specimens directly fracture in the elastic regime at the testing temperature of 300 °C (Gorynin et al., 1992), and proton-irradiated tungsten beams exhibited zero ductility at 150 °C (Sommer, 1995). These studies suggest that irradiation leads to the apparent increase of ductile to brittle transition temperature (DBTT) of tungsten (Abernethy et al., 2019; Steichen, 1976). On the other hand, experimental results of Maloy et al. (Maloy et al., 2005) show that the localized heavy deformation (plastic instability) first appears and then turns into the fracture zone at a higher temperature of 475 °C. The failure model is characterized by the faceted fracture surface, which is different from the quasi-cleavage surface at room temperature. These observations indicate that the failure mechanisms of irradiated tungsten are different at different temperatures.
It is generally believed that the deformation localization, induced by the formation of dislocation channels, is one of the main origins of irradiation embrittlement (Byun and Hashimoto, 2006; Was et al., 2012; Xiao, 2019). In the single crystal, the dislocation channel will lead to significant shear bands. For polycrystalline metals, the grain boundary breaks down the continuity of dislocation channels. The pile-up of dislocations around the grain boundary induces higher local stresses, and further increases susceptibility to stress corrosion cracking initiation (McMurtrey et al., 2015). This conclusion has been further proved by the experiments (McMurtrey et al., 2014) and atomistic simulations (McMurtrey et al., 2011). Dislocation channel formations were widely observed and studied in irradiated body center cubic (BCC) metals, such as iron (Dai et al., 2001; Lee et al., 2001), molybdenum (Mastel et al., 1963) and niobium (TUCKER et al., 1969). However, concerning tungsten, the investigations about the dislocation channel formation is very limited. There are only a few ion-irradiated experiments to support the existence of the dislocation channeling in tungsten. Das et al. (Das et al., 2018; 2019a; 2019b) carried out nanoindentation experiments on the helium-irradiated tungsten single crystal to study the orientation-dependence of plastic deformation for the surface region. There is a pronounced irradiation hardening process due to the intense obstacles of helium-defects, which is followed by the formation of dislocation channels and even “channel failure” (Chopra and Rao, 2011). On the other hand, there are some studies focusing on the shear localization of non-irradiated polycrystalline tungsten (Chen et al., 2021; Wei et al., 2005; Yu et al., 2020), but the corresponding mechanism is different from that of irradiated tungsten. Till now, there is still lack of in-depth studies on the dislocation channel formation of neutron-irradiated tungsten under different conditions of irradiation, temperature, and loading.
As a typical BCC metal, the plastic behavior of tungsten is found to exhibit several unique features.
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Significant temperature dependence (Beardmore and Hull, 1965; Brunner, 2010). Due to the non-planar core structure of screw dislocations in BCC, the Peierls barrier is high (Cereceda et al., 2016; Terentyev et al., 2015). The glide of screw dislocations is controlled by the thermal-activated nucleation and migration of kink-pairs, which is different from the phonon drag mechanism of edge dislocations (Ito, 2001; Ji et al., 2020; Srivastava et al., 2013; Vitek, 2004).
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Non-Schmid effect. Generally, the priority of dislocation motions is determined by the magnitude of the resolved shear stress (RSS) at a given slip system. Other components of the stress tensor than shear in the slip plane along the slip direction play no role in driving the glide of dislocations. However, in BCC, its non-planar core structure of screw dislocations breaks down the standard geometric projection rule of the RSS from the total stress tensor (known as Schmid law). The non-Schmid effect appears (Gröger et al., 2008; Knezevic et al., 2014; Lim et al., 2015a, 2013; Weinberger et al., 2012). It means that the non-glide components of the applied stress tensor also affect the dislocation behavior by modifying the dislocation core spreading pattern and further giving rise to a significant change in the critical shear stress for the onset of dislocation slip (Cho et al., 2018; Ito, 2001). Two typical magnifications are the tension/compression asymmetry and strong loading orientation dependence. In addition, the BCC lattice is lack of mirror symmetry with respect to planes orthogonal to the dominant 〈111〉 slip direction, leading to the twining/anti-twinning asymmetry (Po et al., 2016).
Till now, a variety of crystal plasticity models were proposed to describe the mechanical behavior of tungsten, which fall roughly into two kinds. The first kind describes the plastic strain rate for a given slip system as a function of RSS and a critical stress based on the power law model, where the critical stress can well reflect the additive nature of strengthening mechanics, such as the resistance stress induced by forest dislocations or other obstacles (Chen et al., 2014; Lee et al., 1999; Terentyev et al., 2015; Vrielink et al., 2020; Xiao, 2019; Xiao et al., 2019; Yao and You, 2017). These models are not directly associated with the special features of screw dislocations, thus is difficult to capture the plastic characteristics of tungsten listed above. Some studies changed the expression of RSS, which to some extent can consider the non-Schmid effect (Gröger et al., 2008; Knezevic et al., 2014; Lim et al., 2015a; Weinberger et al., 2012). The second kind of models are based on the Orowan equation (Arsenlis and Parks, 2002; Cereceda et al., 2016, 2015; Yu et al., 2021). The plastic strain rate is directly associated with the dislocation mobility law through the dislocation velocity, and the multiplication and annihilation process of dislocations through the dislocation density. Therefore, these models can better reflect the physical picture that dislocations are the main carrier of plasticity, and the dislocation mobility controls the plasticity of tungsten. The related parameters have clear physical meanings, and can be obtained through the atomic simulation or experiments (Po et al., 2016), which makes these models be more broadly applicable and is therefore used in the current work.
Tremendous progress has been achieved to understand the deformation localization in irradiated materials, such as iron, steel, zircaloy, through the crystal plasticity model at the continuum scale (Barton et al., 2013; Erinosho and Dunne, 2015; Guan et al., 2017; Isavand and Assempour, 2021; Patra and McDowell, 2013, 2016; Tasan et al., 2014), dislocation dynamic at the micro-scale (Arsenlis et al., 2012) and coupled dislocation dynamics-continuum model (Cui et al., 2018a; Das et al., 2020). However, how these unique plasticity features of tungsten contribute to the dislocation channel formation and the deformation localization under neutron irradiations is still mostly unknown.
On the other hand, under the current experimental conditions, tungsten samples are generally irradiated firstly and then adopted for the mechanical tests at room temperature to study the deformation feature. However, these tests on post-irradiated samples at room temperature may not truly reflect the formation process of dislocation channels at high temperature, due to the strong temperature dependence of tungsten plasticity. Therefore, it is urgently needed to establish the theoretical model and computation methods to reveal the underlying mechanism of the localized deformation for tungsten and the related temperature effect.
Based on the analysis above, this work aims to develop a continuum multi-scale theoretical model, which considers the thermal activation feature of microstructure evolutions at the micro-scale and the mechanical properties under the combined effects of temperature, mechanical loading, and irradiation at the macro-scale. This model will be used to demonstrate the role of the characteristics of tungsten plasticity in the deformation localization and study the temperature effect on the mechanical behavior of neutron-irradiated tungsten. The corresponding mechanism is hoped to provide the theoretical foundation to predict the response during the process of coupled thermal condition and mixed energy irradiations in future energy devices and design new tungsten structures with a higher deformation localization resistance under irradiation conditions.
Section snippets
Theoretical model
A continuum multi-scale model is developed as schematically described in Fig. 1 to explore the coupled effect of temperature, irradiation condition and mechanical loading on the deformation localization of tungsten. At the micro-scale, there are three models to represent the evolution of microstructures. The first one is the hardening model to calculate the resistance stress induced by dislocations and irradiation defects. The second one represents the mobility laws of edge and screw
Theoretical model verification
To verify the accuracy and effectiveness of the developed model, the predicted mechanical response of both unirradiated and irradiated tungsten single crystals are compared with the available experimental data.
The role of the unique plasticity features of tungsten
After the careful verification above, the theoretical model is further used to unveil the mystery of whether or how deformation localization occurs under the combined effect of mechanical loading, thermal condition, and irradiation. The role of the temperature dependent plasticity and the non-Schmid effect of tungsten will be discussed.
Critical condition of deformation localization
To predict the critical condition of deformation localization, a simplified model is proposed in this section.
Based on the analysis in Section 4.4, can be adopted as the criterion for the formation of dislocation channel (Barton et al., 2013; Cui et al., 2018b; Patra and McDowell, 2016). According to Section 2.4, the required applied shear stress and its derivation to the shear strain can be calculated as follows,
Conclusion
In summary, investigating material response under the controllable irradiation-mechanical-thermal conditions remains inaccessible experimentally, which calls for the further development of theoretical and simulation methods. As a material for plasma-facing components, tungsten attracted lots of interest. However, how its unique plasticity features influence its response under the coupled irradiation, mechanical loading and temperature is far from well understood. To solve this problem, a
CRediT authorship contribution statement
Zhijie Li: Investigation, Formal analysis, Writing – original draft. Zhanli Liu: Writing – review & editing. Zhuo Zhuang: Writing – review & editing. Yinan Cui: Conceptualization, Supervision, Formal analysis, Writing – original draft, Writing – review & editing.
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.
Acknowledgement
This material is based upon work supported by the National Natural Science Foundation of China under Grant No. 11972208, 11921002, Science Challenge Project, No. TZ2018001 and Tsinghua University Initiative Scientific Research Program.
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