Elsevier

Engineering Fracture Mechanics

Volume 110, September 2013, Pages 330-349
Engineering Fracture Mechanics

Influence of plastic slip localization on grain boundary stress fields and microcrack nucleation

https://doi.org/10.1016/j.engfracmech.2013.04.019Get rights and content

Highlights

  • Slip localization occurs frequently during plastic deformation of metallic polycrystals.

  • It is known to trigger grain boundary (GB) brittle fracture.

  • Based on FEs computations, analytical formulae are proposed for decribing GB stress fields.

  • The predicted remote stress to GB microcracking is lower than predicted by pile-up theories.

  • It is closer to experimental data due to the finite thickness of slip bands.

Abstract

Slip localization is widely observed in metallic polycrystals after tensile deformation, cyclic deformation (persistent slip bands) or pre-irradiation followed by tensile deformation (channels). Such strong deformation localized in thin slip bands induces local stress concentrations in the quasi-elastic matrix around, at the intersections between slip bands and grain boundaries where microcracks are often observed.

Since the work of Stroh, such stress fields have been modeled using the dislocation pile-up theory which leads to stress singularities similar to the LEFM ones. The Griffith criterion has then been widely applied, leading usually to strong underestimations of the macroscopic stress for microcrack nucleation.

In fact, slip band thickness is finite: 50–1000 nm depending on material, temperature and loading conditions. Then, many slip planes are plastically activated through the thickness. Stress fields have probably been overestimated using the pile-up theory which assumes that all dislocations are located on the same atomic plane. To evaluate more realistic stress fields, crystalline finite element (FE) computations are carried out using microstructure inputs (slip band aspect ratio and spacing). Slip bands (low critical resolved shear stress) are embedded in an elastic matrix. The following results are obtained concerning grain boundary normal stress fields:

  • strong influence of slip band thickness close to the slip band corner, which is not accounted for by the pile-up theory. But far away, the thickness has a negligible effect and the predicted stress fields are close to the one predicted by the pile-up theory,

  • analytical formulae are deduced from the numerous FE computation results which allows the prediction of surface/bulk slips as well as grain boundary stress fields. Slip band plasticity parameters, slip band length and thickness, Schmid factor and remote stress are taken into account. The dependence with respect to the various parameters can be understood in the framework of matching expansions usually applied to cracks with V notches of finite thickness,

  • as the exponent of the GB stress close-field is lower than the pile-up or crack one, that is 0.5, the Griffith criterion may not be used for GB microcrack prediction in case of finite thickness. That is why finite crack fracture mechanics is used together with both energy and stress criteria,

  • the pile-up theory leads to large underestimation of the critical remote stress leading to GB microcrack nucleation measured in the case of pre-irradiated austenitic stainless steels subjected to tensile loading in inert environment, probably because of the overestimation of the local GB stress field. And the critical remote stress computed using the proposed modeling of slip bands of finite thickness is much closer to the experimental values.

Introduction

Slip localization occurring at the grain scale has been extensively observed, particularly in Faced-Centred Cubic (FCC) metals and alloys subjected to either post-irradiation tensile tests [1], [2], [3], [4], [5], [6] (proton or neutron irradiation with high dose), cyclic loadings [7], [8], [9], [10], [11], [12], [13], [14], or even simply tensile loading [15]. Plastic slip is localized in thin slip bands. The thickness is lower than 1 μm but higher than a few ten nm. Usually, slip bands cross all the grains from one grain boundary to the opposite one. Therefore the slip band length is approximately equal to the grain size which usually varies from a few ten microns to a few hundred microns depending on the material. The degree of localization seems to be high. It could be evaluated using the ratio between the slip band and macroscopic axial plastic strains. The case of channels (or clear bands) produced in pre-irradiated materials by the removing of irradiation defects by mobile dislocations. Following the AFM (Atomic Force Microscopy) measurements of Was et al. [5], this ratio is equal to about 10 for austenitic steels subjected to post-irradiation tensile loading (macroscopic axial strain of 0.03). The TEM (Transmission Electronic Microscopy) observations of Sharp [1] and Edwards et al. [4] concerning either single crystals or polycrystals of copper subjected to post-irradiation tensile loading permitted these authors to evaluate the clear band plastic slips. The obtained localization degrees seem to be about ten too. Similar results have been obtained in irradiated austenitic stainless steels [16]. Plastic strain is highly localized in slip bands induced by cyclic loadings as well. Following the Winter model [9], the plastic strain is essentially localized in persistent slip bands (PSBs). For very small macroscopic plastic strains (about 10−4), the localization degree could be as high as 100. The recent AFM measurements of Wejdemann and Pedersen [17] gave more precise evaluations: the PSB’s plastic strain is about fifty times larger than the macroscopic plastic strain.

At least two main mechanisms explain the slip band formation. The first mechanism is holds for post-irradiation tensile tests carried out on FCC metals and alloys as well as alloys containing shearable precipitates. It is based on interactions between mobile dislocations and irradiation-induced defects (proton or neutron irradiation) [18]. Cycling of precipitation-hardened alloys is known to induce the formation of thin slip bands as well, if the precipitates are shearable [19]. Following the 3D Dislocation Dynamics computations of Shin et al. [20], interactions between mobile dislocations and precipitates could explain the formation of the slip bands even if the precise interaction mechanisms differ from the previous ones. The second mechanism corresponds to the case of FCC (ductile) metals and alloys subjected to cyclic loadings. Slip localization seems to be essentially due to dislocation glide and interactions. Slip band formation has been investigated either by dislocation dynamics [21], [22] or by energy minimizing [23]. Cross-slip seems to be a necessary mechanism for PSB formation of finite thickness instead of pile-ups [21], [22]. Recent 3D Dislocation Dynamics computations, taking into account cross-slip, permitted Déprés et al. to reproduce numerically the formation of slip bands in an austenite crystal [24]. It should be noticed that similar observations of slip localization have been often reported in either body-centred cubic (BCC) or hexagonal compact (HCP) metals and alloys and is therefore very commonly observed in metals and alloys whatever their crystallographic structure.

Several computations were carried out for evaluating the plastic slips inside slip bands, particularly in the framework of cycling of ductile metals. Authors first modelled slip bands as elongated inclusions embedded in a matrix which mimics the whole polycrystal [25]. This permitted them to use the analytical solution given by Eshelby for bulk inclusion [26]. Then, Finite Element computations using crystalline plasticity permitted the investigation of surface effects [27], [28]. In the case of type B slip bands, both slip magnitude and heterogeneity are considerably enhanced by surface effects [29], which explains partially the preferential surface fatigue crack initiation.

Clear bands and slip bands impinge to grain boundaries. This induces stress or plastic strain concentrations as shown by TEM observations on copper polycrystal deformed after neutron irradiation [4]. Edwards et al. observed indeed either local lattice rotations corresponding to high elastic strain concentrations or considerable amount of (plastic) shearing at the grain boundary if another channel has been nucleated on the opposite side of the grain boundary [4]. Such propagation of a channel in the neighbouring grain was observed [5], [30], but almost only in the case of singular grain boundaries such as twin boundaries. Because of these interactions with grain boundaries, clear bands or slip bands are often considered as triggering grain boundary crack initiation and propagation. The corresponding crack initiation mechanism has been investigated experimentally for copper [30] and nickel [31] subjected to cyclic loadings. Similarly, slip localization in clear bands has been considered as promoting intergranular crack initiation in case of irradiation assisted stress corrosion cracking (IASCC) [32], [33], [4]. Impinging deformation-induced twin bands have been found to trigger intergranular crack initiation as well [33], [34], [35]. In general, whatever the environment, clear bands are considered as promoting intergranular crack initiation. Finally, previous cold-work deformation of austenitic steels leads to earlier GB crack initiation during SCC tests, which is considered to be due to slip localization during cold-work [36]. Many experimental studies concerning the influence of grain boundary characteristics on intergranular crack initiation have been published. Those studies are based on EBSD measurements which permit the evaluation of the grain crystallographic orientations and then of grain boundary misorientations. Concerning grain boundaries, two extreme cases can be considered [37], [38]. On the one hand, general grain boundaries display mostly very high Σ values. That value is defined as the inverse of the fraction of coincident atoms between the two crystallographic networks. Therefore, these grain boundaries present quite no periodicity along the grain boundary. Their grain boundary energies as well as their diffusion coefficients are very high [37], [38]. On the other hand, the special boundaries have low Σ values and present generally a periodicity along the grain boundary. Their grain boundary energies as well as their diffusion coefficients are low [37], [38]. The Σ3 twin boundary is a well-known example of special grain boundary. Based on microscopic observations, the authors of the different studies could evaluate which grain boundaries are the most prone to crack initiation and which ones are the less prone to crack initiation. All the authors concluded that the special boundaries, and particularly the Σ3 twin boundaries, are the less prone to stress corrosion cracking (SCC) initiation even if some of them could crack [38], [39], [40]. It should be noticed that the same result was obtained in copper [30] or nickel [31] subjected to cycling in air or either environment condition [6]. Their low GB energy values lead to high fracture energy and the observed slip band transmission through special GBs decreases GB stress concentrations which is not the case for general GBs.

Concerning modelling in the continuous framework, several studies have been dedicated to the evaluation of grain boundary stress concentrations. Neumann showed that crystalline elasticity induces stress concentration at grain boundaries [41]. This partially explains that Margolin and co-workers deduced from their optical observations of slip traces that stresses are higher in the vicinity of grain boundaries [42], [43]. Recently, Diard et al. used large-scale Finite Element computations for evaluating stress gradients in the vicinity of grain boundaries, induced by plastic deformation incompatibilities between neighbour grains [44]. All these studies highlighted stress concentrations which could promote intergranular crack initiation. Concerning specifically the influence of slip band impingement, GB stress fields have been evaluated analytically using the theory of discrete or continuous dislocation pile-ups. This approach is based on the well-known Stroh model [45]. The stress singularity induced by an edge or screw pile-up of length Lpile-up is the same as the one of a crack in the framework of linear elastic fracture mechanics (LEFM) [45], [46]. Thanks to the similarity with the LEFM crack problem, the energy release rate, G, may be computed on a straightforward way. Following the pioneering work of Griffith, an energy balance criterion has been applied by Smith and Barnby for predicting allowing GB microcrack nucleation [46]. As the stress singularity exponent is ½, the application of the Griffith criterion leads to possible microcrack nucleation which should not be true for lower stress exponent value [47], [48]. The Griffith criterion is based on the equality between the energy release rate, G, and the GB fracture energy, γfract. This means that only an energy criterion is used and the crack increment is assumed to be infinitesimal. This modelling has been applied to the prediction of GB microcrack nucleation, either in copper polycrystals subjected to cyclic loading [30] or pre-irradiated austenitic stainless steels subjected to tensile loading [49]. Applying such modelling to inter-granular crack initiation at the free surface of copper subjected to cyclic loading, Liu and co-workers found that the predicted critical remote stress was generally reached when GB microcracks were observed [30]. A similar approach was followed by Tanaka and Mura [50]. But Kim and Laird [51] showed that GB microcracks appear only when the steps between adjacent grains increasing cycle by cycle are high enough. Generally many cycles are required which is in contradiction with the conclusions of Liu et al. which predicted instantaneous microcrack nucleation provided slip bands exist and stress saturation is reached. Using the pile-up theory as well, Evrard and Sauzay predicted critical remote tensile stresses much lower than the observed ones, whatever the environment [49]. Therefore, the pile-up theory seems to lead to overestimations of the critical remote stresses when compared to experimental data.

Pile-up theories assume that slip is localized on one atomic plane only. But, many experiments and observations show that for many materials and loading conditions, a non-negligible fraction of the slip occurs inside the fatigue slip bands (cf interferometry measurements [8], in situ TEM observations [48] and AFM measurements [5], [16], [52]). Concerning 316L austenitic stainless steel deformed after pre-irradiation, Byun et al. [6] concluded that shear strain is uniformly distributed through the thickness of channels (clear bands). Similar conclusions were drawn by Jiao et al. [5] and Sauzay et al. [16]. As plastic slip is indeed much more homogeneously distributed than assumed by pile-up theories, these last ones may overestimate the local GB normal stress fields as well as energy release rate values which may lead to the underestimation of the critical remote stress mentioned previously. Taking into account not only the slip band length, L (or pile-up length often assumed to be about one-half of the grain size), but also its thickness, t, may improve the predictions. As mentioned previously, the thickness varies between a few ten nm in pre-irradiated polycrystals to one micron in polycrystals subjected to cyclic loadings. Slip bands produced in alloys containing shearable precipitates seem like channels in pre-irradiated metallic materials after deformation. The finite element (FE) method is used in the framework of crystalline elastoplasticity because of the non-linear behaviour of slip bands. Slip bands of various thickness and lengths are embedded at the free surface of an elastic matrix. The effect of crystalline plasticity parameters, slip band thickness and length as well as remote tensile stress is studied in details thanks to the results of numerous FE computations. Analytical formulae describing the GB stress singularities induced by slip bands are deduced with a large range of validity. The dependence with respect to the various considered parameters is similar to the one found by the theory of matching expansions with considers elastic problems involving a crack of length l with a V notch crack tip of finite thickness, t  l [47], [53]. The direct use of the Griffith criterion would not be useful because the stress singularity exponent is found to be indeed lower than 0.5 contrarily to the pile-up stress stress singularity one [45], [46]. In the framework of LEFM theory, this low exponent value leads to a value of G equal to zero if no GB microcrack increment is added. In order to predict GB microcrack formation, the G (or rather J) integral is computed by the FE method using meshes in which short GB cracks of various lengths, a, have been introduced. Then finite fracture mechanics [47], [48], [53] is used replacing the infinitesimal crack increment by a finite one. This is called quantized fracture mechanics as well [54]. Not only the energy criterion but also a local stress criterion should be applied in order to predict both critical remote stress, Σc, and critical crack length, ac. Finally the GB normal stress fields and critical remote tensile stress predicted by pile-up theories are compared to the ones taking into account slip bands of finite thickness, t > 0. An application to the prediction of GB microcrack nucleation in pre-irradiated austenitic stainless steels subjected to tensile loading is proposed. The predictions obtained for either channels of finite thickness or pile-ups are compared to experimental data.

Section snippets

Modeled microstructures

To study the influence of the dislocation channel thickness and length on GB stress fields, a first microstructure, named M1, (Fig. 1a) has been generated using the FE software Cast3m [23]. A dislocation channel, defined by the thickness, t, the length, L, inclined to 45° with respect the tensile axis and surrounded by a grain, has been generated. Grain and dislocation channels are surrounded by a matrix. The vector perpendicular to the GB is inclined by an angle α = 35° with respect the tensile

Grain boundary stress fields

First of all, the dependence of the grain boundary normal stress field with respect to the slip band crystalline parameter values is studied. As mentioned earlier, because of the small considered scale, these ones are not always exactly known and such parameter study is needed. Additionally, this will help later for proposing analytical formulae taking into account the crystalline plasticity parameter as well as the remote tensile stress, Σ0. Then, the influence of both slip band length, L, and

Results of the theory of matching asymptotic fields

The theory of matching expansions [53], [47] applied to the case of a crack with a V-noch tip in an elastic matrix predicts the dependence of the stress close field with respect to the crack length, L, the notch thickness, t, and the stress exponent characterizing the V notch singularity, 1  λ:σij(r,θ)=AΣ0Lttr1-λpij(θ)+Σ0,ij

With A′ a geometry factor which is usually computed by the FE method [47]. Polar coordinates are used as plane strain problems are considered here. This theory is based on

Finite fracture mechanics

As mentioned in part 1, the weak stress singularity (1  λ  0.32 < 0.5) does not allow us to use the Griffith criterion which corresponds to an energy balance equation for an infinitesimal crack nucleus. On the contrary, the use of finite fracture mechanics [47], [48] (or quantized fracture mechanics [54]) involving a finite crack nucleus, ac > 0 allows us to predict GB microcrack nucleation. An additional equation is therefore needed because two unknown values should be computed, on one hand the

Application to GB microcrack nucleation in pre-irradiated austenitic stainless steels

Nishioka et al. [64] and Fukuya et al. [65] tested in argon environment austenitic stainless steels previously pre-irradiated up to a neutron dose equivalent to 35 displacements per atom (dpa) at a temperature of about 320 °C. The tensile tests have been carried out at very slow strain rate (≈10−8 s−1) and at a temperature of about 320 °C. In these experimental conditions, dislocation channels are observed at the surface of the specimens as well as close to the GBs. Both authors measured an

Conclusions

Deformation localization in thin slip bands induces local stress concentrations in the quasi-elastic matrix around, at the intersections between slip bands and grain boundaries where microcracks are often observed. Such stress fields have been modeled using the dislocation pile-up theory which leads to stress singularities similar to the LEFM ones. The Griffith criterion has then been widely applied in the literature. In fact, slip band thickness is finite: t = 50–1000 nm depending on material,

Acknowledgments

The authors are grateful to CEA (DEN-RSTB, projects RACOC and MASOL) and the European project PERFORM60 for financial and scientific support.

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