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Stress intensiﬁcation at grain boundary free ends in anisotropic materials - Application to austenitic stainless steel Intergranular Stress Corrosion Cracking susceptibility G. Meric de Bellefon a, *, J.C. van Duysen b, c, d a

University of Wisconsin-Madison, United States University of Tennessee-Knoxville, United States Unit e Mat eriaux et Transformation (UMET) CNRS, Universit e de Lille, France d Ecole Nationale Sup erieure de Chimie de Lille, France b c

a r t i c l e i n f o

a b s t r a c t

Article history: Received 23 February 2017 Received in revised form 30 April 2017 Accepted 21 May 2017 Available online 23 May 2017

Most studies on Intergranular Stress Corrosion Cracking (IGSCC) in Light Water Reactor environment consider the macroscopic applied stress as the relevant mechanical parameter to characterize crack initiation. Progress in understanding, modeling, and forecasting IGSCC could be made by correlating crack initiation susceptibility to the stress ﬁeld at crack initiation sites. Through ﬁnite element method (FEM) calculations, the present article assesses this stress ﬁeld in polycrystalline 316 stainless steel and identiﬁes key parameters controlling local stress intensiﬁcation. It is in particular shown that the auxetic behavior (i.e., negative Poisson's ratio) of 316 single crystals in some orientations plays an important role. FEM calculation results are used to set up a preliminary expression that can be used to rank IGSCC initiation sensitivity of components made of the same fcc alloy (i.e., austenitic stainless steels, Ni-based alloys) from their surface texture. Further work is in progress to integrate inﬂuence of grain boundary triple junctions and validate the whole approach against experimental SCC data. © 2017 Elsevier B.V. All rights reserved.

1. Introduction Intergranular Stress Corrosion Cracking (IGSCC) is a recurring issue in Light Water Reactors (LWR). Since the early 1960s (e.g. [1]), IGSCC was successively reported on: sensitized 304 and 316 austenitic stainless steels in Boiling Water Reactors (BWR), nickelbased alloys (alloys 600 then X750) in primary or secondary water of Pressurized Water Reactors (PWR), irradiated stainless steels as well as cold-worked non-sensitized stainless steels in PWR and BWR. Such degradation entailed large maintenance costs, e.g., to replace steam generators, reactor vessel heads, or bafﬂe plate bolts in PWR. Many research programs have been carried out worldwide [2] to ﬁnd out solutions to mitigate or suppress IGSCC in LWR. Through corrosion tests, those programs mostly aimed to understand and quantify the role of stress, water chemistry, electrochemical

* Corresponding author. Engineering Physics, University of Wisconsin-Madison, 1500 Engineering Drive, Madison, WI 53706, United States. E-mail address: [email protected] (G. Meric de Bellefon). http://dx.doi.org/10.1016/j.jnucmat.2017.05.028 0022-3115/© 2017 Elsevier B.V. All rights reserved.

potential, irradiation, creep, hydrogen charging, material deformation history, and surface machining (e.g. [3e11]). Experimental studies were also carried out to characterize phases, chemistry, crack morphology, and plasticity at crack tips (e.g. [12e14]). This experimental endeavor uncovered some of the key parameters contributing to SCC (e.g. [15]) and enabled the development of predictive models (e.g., see review for cold-worked austenitic stainless steels in Ref. [16]). It also supports the current effort on multi-scale modeling of SCC (e.g. [17]). This outcome was highly useful to select materials and operational conditions that help to limit IGSCC degradation in LWR. However, it must be admitted that a full understanding of the mechanisms controlling this degradation has not been established yet, and further work is still needed. It is well known that SCC phenomena require a proper combination of stress, environment, and material conditions. They are in general described in two main stages: crack initiation and crack propagation. From a mechanical standpoint, crack propagation rate is in general correlated with the stress ﬁeld at crack tip. This stress ﬁeld is evaluated through classical approaches used in Fracture Mechanics (e.g. [18]) and captures local intensiﬁcation of the

G. Meric de Bellefon, J.C. van Duysen / Journal of Nuclear Materials 493 (2017) 294e302

macroscopic applied stress. For the initiation stage, most modeling studies and analysis of experimental data consider the macroscopic applied stress as the relevant mechanical parameter [19]. Under static loading and LWR conditions, SCC initiation sites for austenitic stainless steels and nickel-based alloys are the emergences of grain boundaries on the free surface. These emergences will be named grain boundary free-ends in the rest of this document. Owing to anisotropy of fcc single crystals (see Appendix), grain boundaries in fcc alloys have to be considered as interfaces between materials with different elastic responses when stressed. The stress ﬁeld at such interfaces has been the subject of many studies, in particular in the area of adhesive joints for structural applications. As shown in Fig. 1, under a tensile stress in air at room temperature, it is experimentally observed (e.g. [20]) that the onset of fracture on these joints mainly happens at the emergences of the interfaces between the involved materials (e.g., a metal and an adhesive such as epoxy) on the specimen surface (interface freeends). This preferential location is attributed to a local stress intensiﬁcation resulting from both the presence of the free surfaces and the elastic mismatch between the materials [25]. A stress intensiﬁcation can also been expected at grain boundary free-ends in austenitic stainless steels and nickel-based alloys, while being not high enough to initiate a crack without a proper combination of environment and other material conditions (for such alloys, SCCtype tests carried out in inert atmosphere do not lead to any crack). Several models have been developed to evaluate the stress intensiﬁcation at interfaces between isotropic or anisotropic materials with different elastic properties (e.g., [21e27]. For isotropic materials [25,28,29], the maximum stress appears to be located several degrees away from the interface in the stiffer material. Several ﬁnite element method (FEM) studies conﬁrmed that the applied stress might be intensiﬁed at grain boundary free-ends in austenitic stainless steels and nickel-based alloys [30,31]. Progress in understanding, modeling, and forecasting IGSCC in LWR could be made by identifying the key parameters controlling this stress intensiﬁcation, and correlating crack initiation times and sites with those parameters. This should also help to reduce SCC susceptibility of components by optimizing their manufacturing process or the chemical composition of their alloy (chemical composition impacts the texture thus the nature of grain boundaries). The present work relies on FEM calculations to i) assess stress intensiﬁcation at grain boundary free-ends in 316 stainless steel under constant elastic loading, and ii) identify the key parameters controlling this stress intensiﬁcation. It is part of a broader effort to tailor austenitic

Fig. 1. Fracture at the interfaces between two materials with different elastic properties. Cracks start at interface free-ends A and C, grow along the interface and deﬂect into the adhesive [20].

295

stainless steels to LWR applications [32]. 2. Methods 2.1. FEM model We consider two 316 grains (Grain 1 and Grain 2) at the surface of a tensile specimen under a constant macroscopic applied stress s. The specimen is solicited in its elastic domain (i.e., s is lower than the ﬂow stress). This latter condition fulﬁlls conventional structural design rules, which in general limit in-service mechanical loading to 60% of yield stress (80% for severe loading). The grains are anisotropic, which implies that their elastic behavior depends on their orientation with respect to the tensile direction. The calculation of the anisotropic elastic constants is detailed in the Appendix. The presence of an oxide layer on the specimen surface is not taken into account. 2.2. FEM calculations The stress ﬁeld near grain boundary free-ends was determined through linear elastic FEM calculations run with the commercial LISA program. The simulated model specimen is shown in Fig. 2, along with the considered laboratory basis (e1 , e2 , e3 ). It has two anisotropic 316 grains (Grain 1 and Grain 2) embedded in an isotropic 316 matrix. The grain boundary between the two grains is perpendicular to the applied stress, based on results of [30] that showed that this conﬁguration leads to the highest stress intensiﬁcation. As mentioned in section 5, other orientations are currently under study. The grain boundary is limited to a line, which is a purely theoretical case since real grain boundaries have a non-zero thickness. Thickness, width, and height of the model specimen are 40, 300, and 300 mm, respectively. The height of both grains is 37.5 mm. Boundary conditions are the following (see Fig. 2a): displacement along e2 is null on face D, macroscopic stress is applied on face D’. in order to simulate the behavior of two grains embedded in a test specimen and having only one free surface, displacement is null along e3 on surfaces B and B’ (i.e., plane strain conditions), and null along e1 on face A’. With those constraints, the model specimen behaves as if it was inserted into a larger test specimen stressed along e2 and having a free surface comprising face A. The meshes are parallelepipedshaped elements that get smaller closer to the grain boundary (see Fig. 2b). As the effect of temperature on the elastic constants of stainless steels is poorly known, the calculations were run at room temperature (RT). As discussed in section 4, the obtained results are expected to be meaningful in the LWR in-service temperature range (z280e370 C). Several teams have experimentally measured the RT elastic properties of annealed Fe-Cr-Ni alloy single crystals (see review in Ref. [33]). We used stiffness values proposed in Ref. [34] for 316 single crystals: c11 ¼ 2.06 1011 N/m2, c12 ¼ 1.33 1011 N/m2 and c44 ¼ 1.19 1011 N/m2. We note Ei the Young's modulus along direction i, Gij the shear modulus along j in plan i (ratio of the shear stress to the shear strain along direction j in the plan perpendicular to i, Gij ¼ Gji), and nij the Poisson's ratio along j for a stretching along i (ratio of length evolution along direction j to that along the stretching direction i, nij may differ from nji). For the isotropic 316 matrix, we used the values proposed in Ref. [35]: E ¼ 1.95 1011 N/m2 and n ¼ 0.29. Calculations were run with an applied stress of 300 MPa, which is a meaningful value considering the possible range of RT ﬂow

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Fig. 2. a) model specimen used for the FEM calculations and its orientation with respect to the laboratory basis (e1 , e2 , e3 ), and b) part of the mesh on face A in the vicinity of the grain boundary between Grain 1 and Grain 2, the smallest mesh spacing along e1 et e2 in the ﬁgure is 0.29 mm.

stress for 316 steel: about 250, 550 and more than 800 MPa in annealed, 20% cold-worked, and a few dpa-irradiated at about 370 C conditions, respectively. Stress intensiﬁcation next to grain boundary free-ends was characterized by the ratio of the highest value of the local stress in the e2 direction on the free surface ðsSe2 Þ to that of the macroscopic applied stress s. Preliminary results, not presented here, showed that the highest value of sSe2 increases almost linearly with that of the applied stress. Thus, this latter value has little effect on stress intensiﬁcation. An analysis of the inﬂuence of the mesh spacing on the calculated stress intensiﬁcation was performed by considering that Grain 1 and Grain 2 are 316 single crystals having ([111] and [100]) along the tensile direction e2 , repectively. The calculations were run with the following mesh spacings along the e2 direction in the vicinity of the grain boundary free-end: 4.68, 2.34, 1.17, 0.58, 0.29, 0.15 and 0.07 mm. Results are not shown here but it was observed that convergence was reached for spacings below 0.29 mm. This latter value was used for all the results given in this article. Stress intensiﬁcation at grain boundary free-ends was assessed by considering that Grain 1 and Grains 2 have a high rotational symmetry axis (½100; ½110; or ½111) parallel to the tensile direction e2 . For the [110] case, two orientations, named [110]a and [110]b, that can be distinguished by a rotation around e2 , were taken into account: for [110]a, the [110] direction is parallel to e1 , and for [110]b, the [001] direction is parallel to e1 . As shown in Appendix (see Fig. 9), these two orientations account for two extreme values of the Poisson's ratio along the e1 direction: n½110;110 ¼ - 0.17 and n½110;001 ¼ 0.76. It can be noted that the Poisson's ratio may have values higher than 0.5 and lower than 0 in 316 steel single crystal (n½110;110 ¼ - 0.17, means that the crystal dilates along ½110 when stretched along ½110). Such properties are generally believed to be rare. In contrast to this belief, many

materials exhibit them [36,37,49]. Materials with negative Poisson's ratios are said to be auxetic. The elastic constants for the considered grain orientations are given in Table 1.

3. Results 3.1. Stress intensiﬁcation at grain boundary free-end An example of result of a calculation run on the model specimen presented in Fig. 2 is given in Fig. 3 for Grain 1 and Grain 2 having [111] and [100] parallel to e2 , respectively. As expected from the values of the elastic constants given for both orientations in Table 1, the two grains stretch in the e2 direction and contract along e1 . However, Fig. 3a shows that the contraction of Grain 2 along the e1 direction is higher than that of Grain 1, owing to the difference in the values of the Poisson's ratio (0.39 and 0.2). The contraction difference entails the creation of a zone (in red) on the free surface of Grain 1 where the stress in the e2 direction sSe2 is intensiﬁed with respect to the applied stress. Fig. 3b maps more precisely the distribution of sSe2 near the grain boundary free-end. It can be noted that i) this distribution is uniform along e3 , which is in conformity with the imposed FEM model and boundary conditions, ii) the zone of stress intensiﬁcation on the surface is not exactly located on the grain boundary free-end but at about 0.3 mm from it, and iii) the stress intensiﬁcation is about 1.49 (sSe2 max ¼ 446 MPa, for 300 MPa applied). The stress intensiﬁcation proﬁle extends on about 1 mm along e1 . It can also be noted that on the free surface of Grain 2, the stress in the e2 direction is lower than the applied stress (about 200 MPa for an applied stress of 300 MPa). The ability of Grain 1 and Grain 2 to contract or expand along e1 (depending if the Poisson's ratio is positive or negative) under a macroscopic tensile stress s applied in the e2 direction can be

Table 1 Stress intensiﬁcation at grain boundary free-end according to grain orientation. Grain orientation and elastic constants for grains stressed along the e2 direction Grain 1

[100]// e2 Ee2 ¼ 102 GPa Ge2 ;e3 ¼ 119 GPa ne2 ;e1 ¼ 0.39

Grain 2

[110]b// e2 Ee2 ¼ 196 GPa Ge2 ;e3 ¼ 37 GPa ne2 ;e1 ¼ 0.76

Stress intensiﬁcation

1

a

[110] //e2 means [110]//e2 and [11 0]//e1 . [110] b//e2 means [110]//e2 and [001]//e1 .

[111]// e2 Ee2 ¼ 285 GPa Ge2 ;e3 ¼ 47 GPa ne2 ;e1 ¼ 0.20

1.38

[110]a// e2 Ee2 ¼ 196 GPa Ge2 ;e3 ¼ 37 GPa ne2 ;e1 ¼ - 0.17

[111]// e2 Ee2 ¼ 285 GPa Ge2 ;e3 ¼ 47 GPa ne2 ;e1 ¼ 0.20

[100]// e2 Ee2 ¼ 102 GPa Ge2 ;e3 ¼ 119 GPa ne2 ;e1 ¼ 0.39

[110]b// e2 Ee2 ¼ 196 GPa Ge2 ;e3 ¼ 119 GPa ne2 ;e1 ¼ 0.76

[111]// e2 Ee2 ¼ 285 GPa Ge2 ;e3 ¼ 47 GPa ne2 ;e1 ¼ 0.20

[100]// e2 Ee2 ¼ 102 GPa Ge2 ;e3 ¼ 119 GPa ne2 ;e1 ¼ 0.39

1.56

1.39

1.18

1.49

G. Meric de Bellefon, J.C. van Duysen / Journal of Nuclear Materials 493 (2017) 294e302

297

Fig. 3. Results of an FEM calculation run to study the behavior of the test specimen shown in Fig. 2 under a stress of 300 MPa (mesh spacing along e2 = 0.29 mm). a) Deformed test specimen with outlines of Grain 1 and Grain 2 - edeformations in all directions have been multiplied by 200. b) Stress distribution on the free surface in the vicinity of the grain boundary free-end - (e2, e3) plane.

nie2 ;e1

characterized by the term si21 ¼

Eei 2

where si21 is a term of the

compliance matrix of grain i in the laboratory basis (see Appendix), εie

i ¼ 1 or 2 according to the grain. Indeed: sz Eei 2 εie2 and nie2; e1 z εi 1 , e2

which leads to

εie1 z si21

s where

εie1

and

εie2

are the strain of Grain i

along e1 and e2 , respectively. The higher the value of si21 (i.e., the smaller the contraction or the larger the dilation along e1 , according to the sign), the stiffer is the grain. In this reasoning, we used the sign z instead of ¼ , because the latter would be correct only for uniform tensile conditions, which is not the case near the grain boundary free-end. Fig. 4 gives the evolution of s21 for 316 single crystals in the stereographic triangle according to the direction of tension. As ne2 ;e1 may vary according to grain rotation around the tensile direction, Fig. 4 a) and b) give the lowest and highest possible values of s21 ¼

ne2 ;e1 Ee2

, respectively. Negative (positive)

values mean that the crystal contracts (dilates) in the e1 direction. It appears that positive values are possible when the crystal is stretched along directions next to 〈110〉.

The absolute value of the difference of contraction along the e1 direction between the two grains can be assessed with Equation (1), where 1 and 2 refer to Grain 1 and Grain 2

2 εe1 ε1e1 z s221 s121 s

(1)

On the example shown in Fig. 3, the grain stretched along [111] may be considered as the stiffer grain, since its Young's modulus and Poisson's ratio are respectively higher and lower than those of the grain stretched along [100] (see Table 3). Thus, as foreseen in Ref. [25], the stress intensiﬁcation occurs in the stiffer material and is not exactly located on the grain boundary. A series of FEM calculations was carried out with different mesh spacings for the specimen shown in Fig. 3. It was noticed that as the mesh spacing decreases: i) the highest value of the stress in the e2 direction on the free surface ðsSe2 Þ increases and tends to inﬁnity when the spacing approaches 0, and ii) the distance between the location of the highest stress value and the grain boundary linearly decreases and tends to 0 when the spacing approaches 0. It can be concluded that there is a stress singularity at sharp (limited to a

Fig. 4. Value of s21 plotted in the stereographic triangle according to some tensile directions. The higher s21 , the stiffer is the grain. As n may vary according to grain rotation around the tensile direction, a and b give the lowest and highest possible values of s21 for each tensile direction, respectively. Negative (positive) values mean that the crystal contracts (dilates) in the e1 direction.

G. Meric de Bellefon, J.C. van Duysen / Journal of Nuclear Materials 493 (2017) 294e302

line) grain boundary free-ends, as foreseen in previous modeling work (e.g. [25]). The performed Linear Elastic FEM calculations provide an asymptotic approximation of the “true solution” of this theoretical case. In practice, 0-thickness grain boundaries do not exist, and the highest stress value at grain boundary free-ends should be limited to about the yield stress (as aforementioned at RT: about 250 or 550 for 316 steel in annealed or 20% cold-worked conditions). In the rest of the article, the asymptotic approximation provided by FEM calculations (with a mesh spacing along e2 ¼ 0.29 mm) is used to identify grain conﬁgurations and elastic characteristics susceptible to lead to the highest stress intensiﬁcation on real components or test specimens.

M – Calculated intensification factor

298

2,5 2,0 1,5 1,0 0,5

Hypothetical materials

0,0 0,00

0,01

316

0,02

0,03

P

3.2. Effect of 316 grain orientation on stress intensiﬁcation

Fig.

In the following, grains with 〈hkl〉 parallel to the tensile axis will be named 〈hkl〉 grains. Stress intensiﬁcation at grain boundary freeend has been assessed for Grain 1 and Grain 2 having a high symmetry axis parallel to e2 (at mesh spacing 0.29 mm). Results are given in Table 1. It can be noticed that stress intensiﬁcation varies signiﬁcantly among the considered conﬁgurations, the highest values being for grain boundaries between 〈110〉a and 〈100〉 grains (about 1.6) as well as between 〈111〉 and 〈100〉 grains (about 1.5), and to a lower degree for grain boundaries between 〈111〉 and 〈110〉b grains as well as between 〈110〉a and 〈110〉b grains (about 1.4). It is worth mentioning that the auxetic behavior (negative Poisson's ratio) of 〈110〉a grains has a key impact since it leads to the highest stress intensiﬁcation (z1.6) - 〈110〉a grain tends to dilate while the grain on the other side of the boundary tends to contract along e1 . 3.3. Structural parameters controlling stress intensiﬁcation To identify parameters that control stress intensiﬁcation at grain boundary free-ends, FEM calculations were carried out by assuming that Grain 1 and Grain 2 were made up with hypothetical fcc materials. The parametric study was performed by varying grain orientation as well as the Young's modulus Ee2 between 100 and 300 GPa, the shear moduli Gei ;ej between 50 and 500 GPa, and Poisson's ratio nei ;ej between - 0.2 and 0.7, with i and j ¼ 1, 2 or 3 (i sj). Analysis of the simulation results led the correlation given in Fig. 5 (diamonds). It appears that the stress intensiﬁcation is mainly controlled by the Young's modulus Ee2 and Poisson's ratio ne2 ;e1 of both grains, and to a lower extent by the shear modulus of the stiffer grain. Regression analysis on the calculation results gives the following approximation:

M ¼ 18:11 P þ 1 where: M ¼ stress intensiﬁcation, P ¼ !0:1 ¼

1 sa66

sb22 sa22

(2) 0:1 Ea e2 Gae2 ;e1 Eb

nae2; e1 Eea2

e2

nbe2; e1

!

Eeb2

sa21 sb21 , where subscripts a and b refer to the

stiffer and more compliant grains, respectively ðsa21 > sb21 Þ, and G and E are expressed in GPa. In the expression of P,

nae2; e1 Eea2

nbe2; e1 Eeb2

charac-

terizes the difference of contraction of the two grains in the e1 direction (i.e., the origin of the stress intensiﬁcation), as shown in Eea

section 3.1, Eb2 reﬂects the ability of the more compliant grain (Grain e2

b) to transfer the load to the stiffer one (Grain a), and the ability of the stiffer grain to increase the stress for a given strain. Expression

P¼

5. Correlation a

Ee ðGae2 ;e1 Þ0:1 Eb2 e 2

yae2; e1 Eea2

between !

ybe2; e1 Eeb2

the ¼

1 sa66

stress !0:1 sb22 sa22

intensiﬁcation

ðsa21

sb21 Þ,

and

the

parameter

where subscripts a and b refer

to the stiffer and more compliant grains, respectively ðsa 21 > sb21 Þ, et G and E are expressed in GPa. The squares correspond to 316 stainless steel for different grain orientations.

(2) should be applicable for any fcc materials and, for a given material, it can allow to determine the surface texture leading to the highest SCC initiation susceptibility. In Fig. 5 the results are plotted along with the results for 316 single crystals given in Table 1 (squares).

4. Discussion Occurrence of SCC requires a proper combination of stress, environmental, and material conditions. The approach developed in this work focuses on the sole role of stress, and aims to rank material textures according to their effect on SCC initiation susceptibility, by considering all other parameters constant. The role of the stress on IGSCC initiation is still unclear, and could be explained through several models (e.g., see review for cold-worked austenitic stainless steels in Ref. [16]). For instance, it has been alleged that local stress can break the passive oxide ﬁlm, modify the chemical composition and thickness of this ﬁlm [38], entail plasticity mechanism (e.g. [39]), or induce diffusion of species such as hydrogen or oxygen (e.g. [17,39]). For all models, it is well accepted that a higher stress at initiation sites eases crack initiation. From the results presented in this article, it can be concluded that for 316 stainless steel: i) the stress at grain boundary free-ends (i.e., IGSCC initiation sites) can be much higher than the macroscopic applied stress, (stress intensiﬁcation z 1.6 for 〈100〉/〈110〉 grains, z 1.5 for 〈100〉/〈111〉 grains, z 1.4 for 〈111〉/〈110〉) and 〈110〉/ 〈110〉 grains), and ii) differences of stress intensiﬁcation can be large enough to entail signiﬁcant variations of SCC initiation susceptibility among grain boundary types. An effect of grain orientation on the IGSCC initiation susceptibility has already been revealed by several experimental studies (e.g. [40]), which led to grain boundary engineering studies. It is worth mentioning that those studies often linked IGSCC sensitivity with grain boundary energy, while in the approach proposed in this article IGSCC is associated with the elastic mismatch between grains. The FEM-based approach developed here is based on stiffness values measured at room temperature, while SCC phenomena in LWR occur at higher temperature (at most about 330 C and 370 C in non-irradiated and highly irradiated areas, respectively). As aforementioned, we made this choice because the effect of temperature on stiffness values of 316 steel single crystals above RT is badly known (this effect is much better known at temperature lower than RT [41]). The impact on the overall conclusion is

G. Meric de Bellefon, J.C. van Duysen / Journal of Nuclear Materials 493 (2017) 294e302

expected to be small, because: i) temperature has a relatively small effect on elastic constants of austenitic stainless steels between RT and 370 C (e.g., for isotropic 316 steel, E ¼ 1.95 1011 N/m2 at RT and 1.71 1011 N/m2 at 370 C [42]), ii) the most inﬂuential elastic constants in Equation (2) intervene in ratios, thus the temperature effects on the numerator and denominator may offset each other. To conﬁrm, several FEM calculations have been run at 370 C by using the temperature dependence of c11 ; c12 ; c44 proposed in Ref. [43]. It was noticed that stress intensiﬁcation is only few percent higher at 370 C than at RT. However, the proposed approach suffers from several weaknesses, in particular: i) it provides only an asymptotic approximation of the real stress ﬁeld, ii) the impact of the anisotropy of the grains surrounding the two considered grains is not taken into account, and iii) the presence of the passive layer that forms on the surface of stainless steels is neglected. Work is in progress to address the two latter points (see Fig. 6). The obtained results are expected to be applicable to other austenitic stainless steels (e.g., 304). More generally, the expression (Equation (2)) to assess the stress intensiﬁcation at grain boundary free-ends according elastic constants of grains should be applicable to any fcc alloys (e.g., nickel based-alloys) in annealed, cold-worked or irradiated states. If it is considered that susceptibility to IGSCC initiation rises as the number of grain boundary free-ends with high stress intensiﬁcation increases, Equation (2) allows ranking IGSCC resistance of fcc alloys from their texture. It should also allow optimizing alloy texture, through optimization of the fabrication process and chemical composition, to reduce IGSCC susceptibility. Some experimental results conﬁrm a role of the deformation path on SCC susceptibility of austenitic stainless steels (e.g. [44e46]). Stacking fault energy (thus chemical composition) has also to be considered since it impacts grain rotation during cold work operation, in particular in compression [47]. 5. Conclusion and future work The present work evaluates stress intensiﬁcation entailed by the elastic mismatch between grains at the emergence of grain boundaries on the surface of components or test specimens made up of 316 stainless steel. Through FEM calculations it was shown that i) the stress at grain boundary emergences (i.e., IGSCC initiation sites) can be much higher than the macroscopic applied stress, (stress intensiﬁcation z 1.6 for 〈100〉/〈110〉 grains, z 1.5 for 〈100〉/

299

〈111〉 grains, z 1.4 for 〈111〉/〈110〉) and for 〈110〉/〈110〉 grains), ii) differences of stress intensiﬁcation can be large enough to entail signiﬁcant variations of SCC initiation susceptibility among grain boundary types. Key parameters controlling the intensiﬁcation were also identiﬁed. It was in particular noticed that the auxetic behavior (i.e., negative Poisson's ratio) of 316 single crystals in some directions plays an important role. Results were used to set up an analytical expression that can be used to rank IGSCC initiation susceptibility of components made of the same fcc alloy (i.e., austenitic stainless steels, Ni-based alloys) from their surface texture. This expression should be applicable for annealed, cold-worked, or irradiated states. To complement the results presented in this article, work is currently in progress to assess the effect of i) oxide layer, ii) grain boundary orientation, iii) grain boundary triple junctions (see type of used model test specimen in Fig. 6), and the role played by strain at grain boundary free-end. The validation of the whole approach against experimental data is also in progress. The set of current and future results might be useful to propose ways of reducing IGSCC susceptibility of LWR components though appropriate manufacturing operation and chemical composition. Acknowledgments The ﬁrst author was funded under a DOE Integrated University Program Graduate Fellowship. Appendix. Anisotropy of 316 steel single crystals Performing linear elasticity calculations in anisotropic materials requires deﬁning the used coordinate basis. In this document, we considered two direct orthonormal bases as shown in Fig. 7: the (e1 ; e2 ; e3 ) laboratory basis and the ([100], [010], [001]) natural basis of cubic single crystals (in which elastic constants are generally given). The orientation matrix A ¼ (aij) between the two bases is deﬁned by a1i ¼ ½100:ei ; a2i ¼ ½010:ei, and a3i ¼ ½001:ei (i ¼ 1, 2 or 3). It is such that for a vector expressed as x in the crystal cubic cell basis and as x0 in the laboratory basis, x ¼ Ax0 . Using linear elasticity theory, stress (s) and strain (ε) secondorder tensors are given by s ¼ C ε or ε ¼ S s, where C ðcefkl Þ and S ðsefkl Þ are the fourth-order stiffness and compliance tensors, respectively. If C and S are known in the crystal cubic cell basis, they can be expressed in the laboratory basis by using the following relationships: crystal

clab mnuv ¼ aem aen aku alv cefkl

and

crystal

slab mnuv ¼ aem afn aku alv sefkl

(3) Voigt's notation takes advantage of the symmetry of the tensors to express s and ε as 6 1 vectors by applying s11/s1, s22/s2,

Fig. 6. Type of model test specimen used to assess stress intensiﬁcation at grain boundary triple junctions.

Fig. 7. Used direct orthonormal bases - (e1 ; e2 ; e3 ) laboratory basis and ([100], [010], [001]) crystal cubic cell basis.

300

G. Meric de Bellefon, J.C. van Duysen / Journal of Nuclear Materials 493 (2017) 294e302

Table 2 Relationships between compliance and stiffness matrice coefﬁcients, and expressions of elastic constants along arbitrary orthogonal directions for cubic crystals (cij and sij are the coefﬁcients of the matrices in the crystal cubic cell basis). þ c12 1 12 s11 ¼ ðc11 cc1211Þðc s12 ¼ ðc11 c12c Þðc11 þ2c12 Þs44 ¼ c44 c11 ¼ 11 þ2c12 Þ ! 1 1 h2 k2 þh2 l2 þk2 l2 2 2 2 2 Ehkl ¼ s11 2 s11 s12 2S44

Compliance and stiffness coefﬁcients Young's modulus Ehkl along [hkl] Shear modulus Ghkl,h’k’l’ ratio of the shear stress to the shear strain along the direction [hkl] in the plane {h’k’l’} Poisson's ratio nhkl,h’k’l’: ratio of length evolution along direction [h’k’l’] to that along the orthogonal stretching direction [hkl]

Ghkl;h’ k’ l’

1s 2 44

¼ s44 þ 4 s11 s12 s12 þ

nhkl;h’ k’ l’ ¼ With m ¼

s12 ½ðs11 s12 Þðs11 þ2s12 Þc44

¼ s144

þ

s11 s12 12s44

k2

þ

ðh2 h’2 þ k2 k’2 þ l2 l’2 Þ ðh2 þ k2 þl2 Þðh’2 þ k’2 þl’2 Þ

ðm2 m’2 þ n2 n’2 þ p2 p’2 Þ

s11 s12 12s44

s11 2

h/ðh2

¼

ðh þk þl Þ

1

s11 þs12 ½ðs11 s12 Þðs11 þ2s12 Þc12

ðm2 n2 þn2 p2 þ p2 m2 Þ

1 l2 Þ2

1

n ¼ k/ðh2 þ k2 þ l2 Þ2 1

p ¼ l/ðh2 þ k2 þ l2 Þ2 (and likewise by replacing {h,k,l,m,n,p} by {h’,k’,l’,m’,n’,p’}).

s33/s3, s12 ¼ s21 / s6, s13 ¼ s31 / s5, s23 ¼ s32 / s4, and likewise for εef with (e, f) ≡ (1,2,3). C and S become 6 6 matrices under the Voigt's notation. Elastic constants in any (i,j,k) basis are then [48,49]: Young's modulus in direction i is given by Ei ¼ 1/sii, shear modulus Gij (ratio of the shear stress to the shear strain along direction j in the plan perpendicular to i, Gij ¼ Gji) is given by G23 ¼ 1/s44, G31 ¼ 1/s55, G12 ¼ 1/s66, and Poisson ratio nij (ratio of length evolution along direction j to that along the stretching direction i), is given by nij ¼ -sij/sii (i s j), nij may differ from nji). In cubic crystals, C and S are given by Equation (4) in the crystal cubic cell basis. 2

3 C11 C12 C12 0 0 0 6 7 6 C12 C11 C12 0 0 0 7 6 7 6C 0 0 0 7 6 12 C12 C11 7 Ccrystal ¼ 6 7 6 0 0 0 C44 0 0 7 6 7 6 0 0 7 0 0 0 C44 4 5 0 0 0 0 0 C44 2 3 S11 S12 S12 0 0 0 6 7 6 S12 S11 S12 0 0 0 7 6 7 6S 0 0 0 7 6 12 S12 S11 7 crystal ¼ 6 7 S 6 0 7 0 0 S 0 0 44 6 7 6 0 7 0 0 0 0 S 4 5 44 0 0 0 0 0 S44

Fig. 8. Spatial dependence of the Young's modulus of 316 single crystals.

and

(4)

Table 2 provides the relationships between coefﬁcients of C and S, as well as derivations of elastic constants according to arbitrary orthogonal directions [hkl] and [h’k’l’] in the cubic cell basis, which can be derived from (3) [48,49]. Using the stiffness values proposed in Ref. [34] for 316 single crystals (c11 ¼ 2.06 1011 N/m2, c12 ¼ 1.33 1011 N/m2 and c44 ¼ 1.19 1011 N/m2) and the set of equations given in Table 2, the spatial evolution of Ehkl was calculated; Ghkl;k’ k’ l’ and nhkl;k’ k’ l’ were also calculated for a few high symmetry orientations. Results are shown

Fig. 9. Dependence of G110;k0 k0 l0 in 316 single crystals with 〈h’k’l’〉 direction in {110} plane.

Table 3 Values of elastic constants of 316 single crystals for some high symmetry orientations. 〈hkl〉 parallel to the tensile axis Ehkl (GPa) Ghkl;k0 k0 l0 (GPa)

〈100〉 102 119

nhkl;h’ k’ l’

0.39

〈110〉 196 G110;001 ¼ 119 G110;110 ¼ 37

n110;001 ¼ 0.76 n110;110 ¼ 0.17

〈111〉 285 47 0.20 Fig. 10. Dependence n110;h’ k’ l’ in 316 single crystals with 〈h’k’l’〉 direction in {110} plane.

G. Meric de Bellefon, J.C. van Duysen / Journal of Nuclear Materials 493 (2017) 294e302

301

Fig. 11. Effect of the auxetic behavior of 316 steel single crystals shown by FEM calculation of a test specimen made up of two single crystals (length: 5 mm, section: 1 1 mm) under a tensile stress of 300 MPa (colors help comparing the behavior of the different segments of the specimen according to the considered face) e measures are in mm and the deformations are highly ampliﬁed. The tensile stress is in the direction of e2 . a) Test specimen in the laboratory basis (e1 , e2 , e3 ) and cubic cell orientation of the two single crystals, before applying stress. b) View of the strained specimen in the (e1 , e2 ) plane. The upper crystal is dilated along its ½110 direction due to the negative Poisson's ratio, the lower crystal is contracted along its [100] direction, c) View of the strained specimen in the (e2 , e3 ) plane. Both crystals are contracted along their [100] direction.

in Fig. 8, Fig. 9 and Fig. 10, and values are given in Table 3. Fig. 8 shows that the Young's modulus is strongly dependent on crystal orientation. Its highest (2.85 1011 Pa) and lowest (1.02 1011 Pa) values are for 〈111〉 and 〈100〉 crystals, respectively. Very similar anisotropies have been calculated for other fcc metals: Ni [50], Cu, Ag, Au and Pb [51]. According to [51], anisotropy of Al is much weaker. The shear moduli G100;k0 k0 l0 and G111;k0 k0 l0 are independent of the 〈h’k’l’〉 direction in the {100} and {111} planes, respectively: G100;k0 k0 l0 ¼ 119 GPa and G111;k0 k0 l0 ¼ 47 GPa. On the contrary, Fig. 9 shows that G110;k0 k0 l0 strongly depends on the 〈h’k’l’〉 direction in the {110} plane. Its highest (119 GPa) and lowest (47 GPa) values are for 〈001〉 and < 110 > directions, respectively. The Poisson's ratios n100;h’ k’ l’ and n111;h’ k’ l’ are independent of the 〈h’k’l’〉 direction in the {100} and {111} planes, respectively: n100;h’ k’ l’ ¼ 0.393 and n111;h’ k’ l’ ¼ 0.198. On the contrary, Fig. 10 shows that n110;h’ k’ l’ strongly depends on the 〈h’k’l’〉 direction in the {110} plane. As aforementioned, it can be noted that the Poisson's ratio may be higher than 0.5 and have negative values. A negative value means that the crystal dilates along 〈h’k’l’〉 when stretched along 〈110〉. Materials with negative Poisson's ratio are said to be auxetic. The effect of grain anisotropy in 316 steel, and in particular of negative Poisson ratios, can be visualized by running FEM calculations of the behavior of a stressed test specimen made up of two 316 single crystals with different orientations. Result of such a calculation is given in Fig. 11. The length of the test specimen is along e2 in the laboratory basis, and the two single crystals have [100] and [110] parallels to e2 (see elastic constants of both grains in Table 3), respectively. It can be noted in Fig. 11 that i) the shape of the strained specimen is very different according to the observed face, and ii) the crystal stressed in the [110] direction exhibits a dilatation along ½110 due to the negative Poisson's ration. References [1] NRC report, BNL-NUREG-77111e2006, Expert Panel Report on Proactive Materials Degradation Assessment, Brookhaven National Laboratory, February 2007. [2] V.S. Raja, T. Shoji (Eds.), Stress Corrosion Cracking - Theory and Practice, Woodhead Publishing, 2011. [3] T.M. Angeliu, D.J. Paraventi, G.S. Was, Corros. Sci. 51 (11) (1995) 837. [4] G.S. Was, P.L. Andresen, Corrosion 3 (Issue 1) (2007) 19e45.

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