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Article
Calculated Shoulder to Gauge Ratio of Fatigue Specimens in
PWR Environment
Igor Simonovski 1, *, Alec Mclennan 2, Kevin Mottershead 2, Peter Gill 2, Norman Platts 2, Matthias Bruchhausen 1,
Joshua L. Waters 2, Marc Vankeerberghen 3, Germán Barrera Moreno 4, Sergio Arrieta Gomez 5
and Radek Novotny 1
Citation: Simonovski, I.; Mclennan,
A.; Mottershead, K.; Gill, P.; Platts, N.;
Bruchhausen, M.; Waters, J.L.;
Vankeerberghen, M.; Moreno, G.B.;
Gomez, S.A.; et al. Calculated
Shoulder to Gauge Ratio of Fatigue
Specimens in PWR Environment.
Metals 2021,11, 376. https://
doi.org/10.3390/met11030376
Academic Editors: Vincenzo Crupi
and Janice Barton
Received: 15 December 2020
Accepted: 18 February 2021
Published: 24 February 2021
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Copyright: © 2021 by the authors.
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Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1European Commission, Joint Research Centre, Westerduinweg 3, 1755 ZG Petten, The Netherlands;
Matthias.Bruchhausen@ec.europa.eu (M.B.); Radek.Novotny@ec.europa.eu (R.N.)
2Jacobs, Walton House 404 Faraday Street, Warrington WA3 6GA, UK; Alec.Mclennan@jacobs.com (A.M.);
Kevin.Mottershead@jacobs.com (K.M.); Peter.Gill@jacobs.com (P.G.); Norman.Platts2@jacobs.com (N.P.);
Joshua.L.Waters@jacobs.com (J.L.W.)
3Nuclear Materials Science Institute, Studiecentrum voor Kernenergie, Centre d’Étude de l’Énergie
Nucléaire (SCK CEN), Boeretang 200, 2400 Mol, Belgium; marc.vankeerberghen@sckcen.be
4Centro de Investigaciones Energéticas, Medioambientales y Tecnológicas (CIEMAT),
Avenida Complutense 40, 28040 Madrid, Spain; german.barrera@ciemat.es
5Laboratory of Materials Science and Engineering, University of Cantabria (UNICAN), Avda. de los Castros,
s/n, 39005 Santander, Spain; sergio.arrieta@unican.es
*Correspondence: Igor.Simonovski@ec.europa.eu; Tel.: +31-(0)224-56-5072
Abstract:
A ratio of shoulder to gauge displacements (S2G) is calculated for three different fatigue
specimens in a pressurized water environment. This ratio needs to be known beforehand to determine
the applied shoulder displacements during the experiment that would result in the desired strain
amplitude in the gauge section. Significant impact of both the applied constitutive law and specimen
geometry on the S2G is observed. The calculation using the fully elastic constitutive law results in
the highest S2G values and compares very well with the analytical values. However, this approach
disregards the plastic deformation within the specimens that mostly develops in the gauge section.
Using the constitutive laws derived from actual fatigue curves captures the material behaviour under
cyclic loading better and results in lower S2G values compared to the ones obtained with the fully
elastic constitutive law. Calculating S2G values using elastic–plastic constitutive law based on the
monotonic uniaxial tensile test should be avoided as they are significantly lower compared to the
ones computed with elastic–plastic laws derived from hysteresis loops at half-life.
Keywords: environmental fatigue; 304 stainless steel; air; PWR primary water; 300 ◦C
1. Introduction
In the design of the current fleet of pressurized water reactors (PWR), environmentally
assisted fatigue (EAF) is a failure that was not originally taken into account by design
codes, e.g., ASME. In the meantime, experimental data have shown the significant negative
effect of the PWR environment on the fatigue life of common reactor steels and methods
for assessing EAF have been developed [
1
,
2
] and incorporated, e.g., into US Nuclear
Regulatory Commission Regulatory Guides (NUREG).
However, current guidance provided in NUREG CR-6909 [
1
] for assessing environmen-
tal fatigue predicts high-usage factors that are not reflected in actual plant experience [
3
].
This suggests that the most recent review of CR-6909 contains significant conservativisms.
Proper understanding of EAF and reduction of unnecessary conservativism are important
for the long-term operation (LTO) of current nuclear power plants (NPPs). Extension
of the life-time of current NPPs is an efficient means to provide low carbon energy and
contributes to the climate change fight. Accordingly, different proposals are currently being
discussed to further improve guidance for assessing EAF in NPPs [4–8].
Metals 2021,11, 376. https://doi.org/10.3390/met11030376 https://www.mdpi.com/journal/metals
Metals 2021,11, 376 2 of 13
To characterize environmental effects on fatigue, extensive test campaigns are per-
formed in which the fatigue life under reference conditions (usually air at room tempera-
ture) is compared to the fatigue life in the environment. The reference tests are normally
performed using solid specimens as described in the relevant ISO or ASTM standards. For
the tests in environment, two approaches are commonly used. One option is using hollow
specimens in which the water environment is flowing through the specimen. The other
option is to use the same solid specimens as used in the reference tests in air, and carry out
the test in an autoclave filled with the environment. The discussion how to compare results
from both types of specimens is still ongoing [9–12].
Both approaches have their inherent advantages and disadvantages. For hollow
specimens, the gauge section is easily accessible, so the strain in the gauge section can
be measured directly by means of an extensometer. However, having the pressurized
environment inside the specimen leads to additional stress components (like hoop stress),
which are not present in the reference tests. Depending on the experimental configuration,
temperature gradients through the specimen wall may exist that also induce strains and
stresses. The stress distribution in a hollow specimen is therefore quite different from the
membrane stress found in solid specimens during reference testing. These differences in
stress distribution could potentially have an impact on the sensitivities to the environment.
It is also difficult to machine the inner surface of hollow specimens, which limits their
usefulness for the study of parameters like surface roughness in EAF, and makes post-
mortem analysis more time-consuming.
Using solid specimens in an autoclave avoids these problems, since the stress state
is the same as for the reference tests in air, the specimen surface can easily be machined,
and by applying a sufficient soaking time before starting the test, a constant specimen
temperature can be achieved. On the other hand, the presence (and wear) of gaskets
at the point where the pull rods pass through the autoclave wall complicates the force
measurement. Furthermore, putting the specimen in an autoclave with a PWR environment
makes assuring proper strain measurement and/or control a challenge. Such test rigs can
use linear variable displacement transducers (LVDTs) that are connected to the specimen
shoulder [
13
]. Ideally, a LVDT would be placed at the gauge section. However, since
a crack can initiate at the points of contact between the extensometer and the specimen
in PWR water environments, tests are mostly done without such a LVDT. The strain in
the gauge section is then obtained from the displacement measured at the shoulder by
applying a shoulder to gauge (S2G) conversion ratio. One needs to obtain the S2G ratio
prior to the experiment, either by separate sets of tests in an air environment or by finite
element simulations.
The current work demonstrates how the S2G ratio is obtained for a number of speci-
men geometries used in the INCEFA-PLUS project [
14
] and explores the sensitivity of the
conversion factor on the underlying constitutive material law. The authors are not aware
of literature dealing with the specific S2G issue.
2. Shoulder-to-Gauge (S2G) Ratio
Shoulder-to-gauge ratio, S2G, is defined as the ratio of the change of shoulder,
∆
L
shoulder
and gauge length,
∆
L
gauge
, as shown in Equation (1). Specimen dimensions
are given in Figure 1. For a fully elastic material, the change in the specimen length due
to the applied axial load Fis given by Equations (2)–(4), where D1, D3, L1, L2, L3 and R
stand for diameter, length and transition region radius dimensions (see Figure 1) and Efor
modulus of elasticity. The smaller cross section in the gauge section leads to higher stresses
than in the shoulder. In the presence of plasticity, the plastic strain is therefore higher in
the gauge than in the shoulder, and the fully elastic values provide the upper limit of the
S2G ratio.
S2G = ∆Lshoulder/∆Lgauge (1)
Metals 2021,11, 376 3 of 13
S2Gelastic =
∆Lshoulder
z }| {
∆L1elastic +∆L2elastic +∆L3elastic
∆L1elastic
| {z }
∆Lgauge
(2)
∆L1elastic =F·L1·4
π·D12·E(3)
∆L2elastic =F
π·E
2·L2
D1
R·(L22+D1·R)+2
D1·qD1
R
·arctan
L2
R·qD1
R
(4)
∆L3elastic =F·L3·4
π·D32·E(5)
Metals 2021, 11, 376 3 of 13
S2G =∆1+∆2+∆3
∆
∆1
∆
(2)
∆1 =∙1∙4
π∙1∙ (3)
∆2 =
π∙
2∙2
1
∙2+1∙+2
1∙
1
∙
2
∙
1
(4)
∆3 =∙3∙4
π∙3∙ (5)
Figure 1. Sketch of a fatigue specimen.
3. Finite Element Model
The finite element method is employed to compute the S2G ratios of different speci-
mens subjected to cyclic loading. Two-dimensional axisymmetric finite element models
(FEMs) are used with the ABAQUS solver (2017, Dassault Systèmes®, Paris, France).
Length-wise, only one half of the specimens is modelled due to the symmetry.
3.1. Specimen Geometries
Three different round solid specimen geometries, Figure 2, are analysed: (1) JRC, (2)
SCK-CEN and (3) CIEMAT. The JRC and CIEMAT specimens exceed the ASTM-E606 [15]
recommended (transition region radius)/(gauge diameter) ratio but conform with ISO
12106 [16], Table 1. SCK-CEN and CIEMAT specimens use a smaller gauge diameter than
the recommended minimum 6.35 (ASTM) and 5–10 (ISO) mm. This is due to the load re-
striction on the fatigue testing machines. The recommended gauge length to diameter ra-
tio is observed for all geometries. The specimen are manufactured using a turning process.
Figure 1. Sketch of a fatigue specimen.
3. Finite Element Model
The finite element method is employed to compute the S2G ratios of different speci-
mens subjected to cyclic loading. Two-dimensional axisymmetric finite element models
(FEMs) are used with the ABAQUS solver (2017, Dassault Systèmes
®
, Paris, France).
Length-wise, only one half of the specimens is modelled due to the symmetry.
3.1. Specimen Geometries
Three different round solid specimen geometries, Figure 2, are analysed: (1) JRC,
(2) SCK-CEN and (3) CIEMAT. The JRC and CIEMAT specimens exceed the ASTM-E606 [
15
]
recommended (transition region radius)/(gauge diameter) ratio but conform with ISO
12106 [16], Table 1. SCK-CEN and CIEMAT specimens use a smaller gauge diameter than
the recommended minimum 6.35 (ASTM) and 5–10 (ISO) mm. This is due to the load
restriction on the fatigue testing machines. The recommended gauge length to diameter
ratio is observed for all geometries. The specimen are manufactured using a turning
process. Two different surface finishes are tested: (a) fine laboratory finish (using a 9
µ
m
5230 microtec slurry coated paper first, 80 passes, followed by 5
µ
m 5230 microtec slurry
Metals 2021,11, 376 4 of 13
coated paper, 140 passes) and (b) worst case typical plant finish where grinding is applied
after the turning, obtaining a total height of the roughness profile Rtof 50 ±10 µm.
Metals 2021, 11, 376 5 of 13
(a)
(d)
(b)
(c)
Figure 2. JRC (a), SCK-CEN (b), CIEMAT (c) specimens and geometries of the 2D axisymmetric FEM models (d).
3.2.2. Combined Isotropic and Kinematic Hardening
Fatigue hysteresis loops of AISI 304L at half-life, N25/2, performed at 300 °C in air
[19,20], are used for calibrating the isotropic and kinematic hardening components. Mod-
ulus of elasticity is taken as E = 163,400.5 MPa, while the Poisson ratio is ν = 0.30 [19,20].
One calibration is performed for Δε = 0.6% and another for Δε = 1.2%. The corre-
sponding results are labelled as “IsoKinHard, Δε = 0.6%” and “IsoKinHard, Δε = 1.2%”.
Δε stands for the total strain range. These strain ranges were selected during the INCEFA-
Figure 2. JRC (a), SCK-CEN (b), CIEMAT (c) specimens and geometries of the 2D axisymmetric FEM models (d).
Metals 2021,11, 376 5 of 13
Table 1. Comparison of specimens against the ASTM-E606 and ISO 12106 recommended values.
Specimen (Gauge Length 2*L1)/(Gauge
Diameter D1)
(Transition Region Radius
R)/(Gauge Diameter D1) Gauge Diameter D1 (mm)
ASTM-E606 3 ±1 4 ±2 6.35
ISO 12106 >2 >2 5–10
JRC 18/6 = 3 48/6 = 8 6
SCK-CEN 10/4.5 = 2.22 18/4.5 = 4 4.5
CIEMAT 10/4 = 2.5 50/4 = 12.5 4
3.2. Material
INCEFA-PLUS uses AISI 304L stainless steel as a common material [
17
]. Several types
of constitutive responses are evaluated, as defined in the coming sections.
3.2.1. Elastic and Elastic–Plastic
Monotonic tensile test of AISI 304L, performed at 300
◦
C in air [
18
] is used for obtaining
the modulus of elasticity, E= 185,667 MPa, Poisson ratio,
ν
= 0.29, and true strain versus
plastic strain data. The corresponding results are labelled as “Elastic” or “ElasticPlastic”.
Stress–strain data in tabular form are given in [18].
3.2.2. Combined Isotropic and Kinematic Hardening
Fatigue hysteresis loops of AISI 304L at half-life, N25/2, performed at 300
◦
C in
air [
19
,
20
], are used for calibrating the isotropic and kinematic hardening components.
Modulus of elasticity is taken as E= 163,400.5 MPa, while the Poisson ratio is
ν
= 0.30 [
19
,
20
].
One calibration is performed for
∆ε
= 0.6% and another for
∆ε
= 1.2%. The correspond-
ing results are labelled as “IsoKinHard,
∆ε
= 0.6%” and “IsoKinHard,
∆ε
= 1.2%”.
∆ε
stands
for the total strain range. These strain ranges were selected during the INCEFA-PLUS
project experimental campaign, see also section on loads and boundary conditions. The
isotropic hardening, i.e., expansion of the yield surface, is given by a table of equivalent
stress, defining the size of the elastic range and corresponding equivalent plastic strain
(Table 2). For the kinematic part, only the upper part of N25/2 loop is selected, up to the
unloading point: from point
ε0
p
to point (
σn
,
εn
) (see Figure 3). The plastic strains are then
calculated from an origin shifted to
ε0
p
, Equation (6), following the procedure defined in
ABAQUS documentation for calibrating to a stabilized loop.
εpl
i=εi−σi
E−ε0
p(6)
Table 2. Isotropic hardening properties, using the ABAQUS “*Cyclic Hardening” option.
∆ε= 0.6% ∆ε= 1.2%
Stress (MPa) Plastic Strain (/) Stress (MPa) Plastic Strain (/)
81.2 0.0 110.0 0.0
91.2 0.0592 120.0 0.0592
91.2 1.4798 120.0 1.4798
Metals 2021,11, 376 6 of 13
Metals 2021, 11, 376 6 of 13
PLUS project experimental campaign, see also section on loads and boundary conditions.
The isotropic hardening, i.e., expansion of the yield surface, is given by a table of equiva-
lent stress, defining the size of the elastic range and corresponding equivalent plastic
strain (Table 2). For the kinematic part, only the upper part of N25/2 loop is selected, up
to the unloading point: from point
to point (σn, εn) (see Figure 3). The plastic strains
are then calculated from an origin shifted to
, Equation (6), following the procedure
defined in ABAQUS documentation for calibrating to a stabilized loop.
=−
−
(6)
Table 2. Isotropic hardening properties, using the ABAQUS “*Cyclic Hardening” option.
Δε = 0.6% Δε = 1.2%
Stress (MPa) Plastic Strain (/) Stress (MPa) Plastic Strain (/)
81.2 0.0 110.0 0.0
91.2 0.0592 120.0 0.0592
91.2 1.4798 120.0 1.4798
Figure 3. Calibrating the kinematic hardening, (σi, εi) pairs to the N25/2 loop. Point (σ1, ε1) is the
end of the linear elastic region, following the un-loading half of the loop.
3.2.3. Multilinear Kinematic Hardening
In this section, N25/2 loops from a collection of cyclically tested 304 stainless steels at
300 °C in air are used [21]. Modulus of elasticity is taken as E = 174,000 MPa, while the
Poisson ratio as ν = 0.30. For the kinematic hardening, only the upper parts of N25/2 loops
are selected, following the same procedure as just described in the previous section. Once
the plastic strains have been calculated using Equation (6), a piecewise Ramberg–Osgood
curve, Equation (7), is fitted to the such-obtained stress–strain data. To capture the correct
hardening behaviour at a large range of strain amplitudes, different parameters above and
below a given strain threshold of amplitude are used: JACOBS used a 0.4% strain thresh-
old, while in the INCEFA project, a threshold of 0.33% was selected, Table 3. The corre-
sponding results are labelled as “JACOBS” and “INCEFA”.
=
/ (7)
Table 3. Ramberg–Osgood parameters for JACOBS and INCEFA constitutive laws.
Figure 3.
Calibrating the kinematic hardening, (
σi
,
εi
) pairs to the N25/2 loop. Point (
σ1
,
ε1
) is the
end of the linear elastic region, following the un-loading half of the loop.
3.2.3. Multilinear Kinematic Hardening
In this section, N25/2 loops from a collection of cyclically tested 304 stainless steels at
300
◦
C in air are used [
21
]. Modulus of elasticity is taken as E= 174,000 MPa, while the
Poisson ratio as
ν
= 0.30. For the kinematic hardening, only the upper parts of N25/2 loops
are selected, following the same procedure as just described in the previous section. Once
the plastic strains have been calculated using Equation (6), a piecewise Ramberg–Osgood
curve, Equation (7), is fitted to the such-obtained stress–strain data. To capture the correct
hardening behaviour at a large range of strain amplitudes, different parameters above and
below a given strain threshold of amplitude are used: JACOBS used a 0.4% strain threshold,
while in the INCEFA project, a threshold of 0.33% was selected, Table 3. The corresponding
results are labelled as “JACOBS” and “INCEFA”.
εplastic =σ
K1/n(7)
Table 3. Ramberg–Osgood parameters for JACOBS and INCEFA constitutive laws.
-ε≤0.4% ε> 0.4% - ε≤0.33% ε> 0.33%
-K(MPa) n(/) K(MPa) n(/) - K(MPa) n(/) K(MPa) n(/)
JACOBS 800 0.213 2800 0.425 INCEFA 770 0.234 2500 0.425
Full details of all the calibrations are given in [
22
]. Data sets used for calibration are
given in [19,20]. Comparison of the used constitutive models is given in Figure 4.
Metals 2021,11, 376 7 of 13
Metals 2021, 11, 376 7 of 13
- ε ≤ 0.4 % ε > 0.4 % - ε ≤ 0.33 % ε > 0.33 %
- K (MPa) n (/) K (MPa) n (/) - K (MPa) n (/) K (MPa) n (/)
JACOBS 800 0.213 2800 0.425 INCEFA 770 0.234 2500 0.425
Full details of all the calibrations are given in [22]. Data sets used for calibration are
given in [19,20]. Comparison of the used constitutive models is given in Figure 4.
Figure 4. AISI 304L material models, used for FEM models [22].
3.3. Loads and Boundary Conditions
During the INCEFA-PLUS experimental campaign, strain ranges of Δε = 0.6% and Δε
= 1.2% in the gauge section were selected to reach well into the low-cycle regime. To make
sure these strain ranges in the gauge section are obtained in the FEM model, a user-de-
fined amplitude (UAMP) displacement load in the vertical direction is applied to the top
edge of a specimen. The user subroutine monitors the displacement at the gauge length,
throughout the simulation. Once the displacement results in the target gauge strain range
(Δε = 0.6% or Δε = 1.2%), the displacement load is shortly kept constant and then reversed,
Figure 5. This results in a trapezoidal load waveform, keeping the gauge strain rate con-
stant during load-up and load-down for both gauge strain ranges. The displacement load
is applied on a reference point (RP) of which vertical and rotational displacements of the
RP are kinematically linked to the vertical and rotational displacements of the top edge of
the specimen. Vertical displacement of the RP node is therefore equal to the vertical dis-
placements of the nodes on the top edge of the specimen. The same is valid for the rota-
tional degree of freedom of the RP node and nodes on the top edge of the specimen. This
simplifies the extraction of the axial force during the post-processing phase. Vertical dis-
placements of the nodes on the bottom edge of the specimen are constrained. Horizontal
displacements of nodes on the specimen at the symmetry line do not need to be con-
strained as this is done automatically due to the application of 2D axisymmetric model.
Five load cycles are applied at a frequency of 0.5 Hz.
Figure 4. AISI 304L material models, used for FEM models [22].
3.3. Loads and Boundary Conditions
During the INCEFA-PLUS experimental campaign, strain ranges of
∆ε
= 0.6% and
∆ε= 1.2%
in the gauge section were selected to reach well into the low-cycle regime. To
make sure these strain ranges in the gauge section are obtained in the FEM model, a user-
defined amplitude (UAMP) displacement load in the vertical direction is applied to the top
edge of a specimen. The user subroutine monitors the displacement at the gauge length,
throughout the simulation. Once the displacement results in the target gauge strain range
(
∆ε
= 0.6% or
∆ε
= 1.2%), the displacement load is shortly kept constant and then reversed,
Figure 5
. This results in a trapezoidal load waveform, keeping the gauge strain rate
constant during load-up and load-down for both gauge strain ranges. The displacement
load is applied on a reference point (RP) of which vertical and rotational displacements
of the RP are kinematically linked to the vertical and rotational displacements of the top
edge of the specimen. Vertical displacement of the RP node is therefore equal to the vertical
displacements of the nodes on the top edge of the specimen. The same is valid for the
rotational degree of freedom of the RP node and nodes on the top edge of the specimen.
This simplifies the extraction of the axial force during the post-processing phase. Vertical
displacements of the nodes on the bottom edge of the specimen are constrained. Horizontal
displacements of nodes on the specimen at the symmetry line do not need to be constrained
as this is done automatically due to the application of 2D axisymmetric model. Five load
cycles are applied at a frequency of 0.5 Hz.
Metals 2021,11, 376 8 of 13
Metals 2021, 11, 376 8 of 13
Figure 5. Load and boundary conditions.
3.4. Mesh
Second order CAX8R elements are used with the JRC specimen model having 2860
elements, the SCK-CEN model 1425 elements and the CIEMAT model 3020 elements. Ta-
ble 4 provides the mesh quality check. Since the models are tens of mm in size and a target
element size of 0.2 mm is used, the resulting mesh is very fine. Higher mesh densities have
also been used to make sure the results with the above given meshes are mesh independ-
ent but were not used further due to higher computational cost. Validation of the FEM
model was performed by comparing the results of two independently developed FEM
models (JRC and CIEMAT) with the SCK-CEN experimental results [13]. The results
matched very well.
Table 4. Finite element mesh statistics.
Specimen Aspect Ratio
>10 (%) Average Worst
JRC 0 1.39 1.93
SCK-CEN 0 1.33 3.73
CIEMAT 0 1.57 2.03
4. Results
4.1. S2G Ratios
Displacements of nodes at the ends of gauge and shoulder sections of the model are
extracted from the simulation results as they represent changes of gauge and shoulder
lengths, enabling one to calculate the S2G values. S2G values, computed as average S2G
values at maximal/minimal load over the five load cycles are given in Table 5. One can see
a considerable dependence of S2G values on the chosen constitutive response. Elastic con-
stitutive response results in highest S2G values and is used for a comparison with the
linear elastic results Equation (1). FEM results match well with the linear elastic S2G val-
ues. Elastic–plastic properties from uniaxial (monotonic) tensile test were used initially
[13] and result in the lowest S2G values. These values can be considered representative of
the few initial cycles only. For N25/2 cycle, the isotropic kinematic hardening and JACOBS
Figure 5. Load and boundary conditions.
3.4. Mesh
Second order CAX8R elements are used with the JRC specimen model having 2860
elements, the SCK-CEN model 1425 elements and the CIEMAT model 3020 elements.
Table 4
provides the mesh quality check. Since the models are tens of mm in size and
a target element size of 0.2 mm is used, the resulting mesh is very fine. Higher mesh
densities have also been used to make sure the results with the above given meshes are
mesh independent but were not used further due to higher computational cost. Validation
of the FEM model was performed by comparing the results of two independently developed
FEM models (JRC and CIEMAT) with the SCK-CEN experimental results [
13
]. The results
matched very well.
Table 4. Finite element mesh statistics.
Specimen Aspect Ratio
>10 (%) Average Worst
JRC 0 1.39 1.93
SCK-CEN 0 1.33 3.73
CIEMAT 0 1.57 2.03
4. Results
4.1. S2G Ratios
Displacements of nodes at the ends of gauge and shoulder sections of the model are
extracted from the simulation results as they represent changes of gauge and shoulder
lengths, enabling one to calculate the S2G values. S2G values, computed as average S2G
values at maximal/minimal load over the five load cycles are given in Table 5. One can
see a considerable dependence of S2G values on the chosen constitutive response. Elastic
constitutive response results in highest S2G values and is used for a comparison with the
linear elastic results Equation (1). FEM results match well with the linear elastic S2G values.
Elastic–plastic properties from uniaxial (monotonic) tensile test were used initially [
13
] and
result in the lowest S2G values. These values can be considered representative of the few
initial cycles only. For N25/2 cycle, the isotropic kinematic hardening and JACOBS and
Metals 2021,11, 376 9 of 13
INCEFA constitutive models are representative. JACOBS and INCEFA S2G values differ
only slightly and are considered by the authors to be the most representative since they are
based on material data from a collection of cyclically tested 304 stainless steels.
The highest S2G values are obtained for CIEMAT specimen. This can be explained
by: (a) its largest length and (b) smallest gauge to specimen length ratio (0.17). SCK-CEN
specimen are the shortest but still have lower gauge to specimen length ratio compared
to the JRC specimen (0.28 versus 0.35). Consequently, the elastic S2G values of SCK-CEN
specimen are slightly higher compared to the JRC ones. Once the plasticity effects are
accounted for, the differences in S2G values between the SCK-CEN and JRC specimens are
minimal.
Table 5. S2G values.
Constitutive Law JRC SCK-CEN CIEMAT
∆ε= 0.6% ∆ε= 1.2% ∆ε= 0.6% ∆ε= 1.2% ∆ε= 0.6% ∆ε= 1.2%
AnalyticElastic 2.20 2.20 - - 3.13 3.13
FEMElastic 2.22 2.22 2.37 2.37 3.16, 3.15 * 3.16, 3.15 *
FEMElasticPlastic 1.53 1.50 1.55 1.49 1.90, 1.88 * 1.81, 1.80 *
FEMIsoKin 1.71 1.60 1.71 1.61 2.20 2.00
FEMJacobs 1.84 1.82 1.87 1.82 2.40, 2.38 * 2.32, 2.31 *
FEMIncefa 1.83 1.84 1.85 1.84 2.35, 2.34 * 2.35, 2.34 *
Measured [13] - - 1.51 1.46 - -
* CIEMAT finite element model results. ANSYS solver used, monotonic load applied up to ε= 1.2%.
4.2. SCK-CEN Specimen, the Effect of the Extensometer Position
Within INCEFA-PLUS, SCK-CEN were the only ones measuring the gauge length in
the environment during the experiment. An extensometer at the gauge section is used to
control the displacements at the shoulder. However, since a crack can initiate at the points
of contact between the extensometer and the sample in the PWR water environment, the
extensometer is positioned slightly beyond the 10 mm gauge length, at the large R= 18 mm
curvature section, where the cross section is larger and stresses due to the axial load are
lower, decreasing the possibility of crack initiation at the contact points. The extensometer
length, L
ext
, which controls the displacement at the shoulder, is therefore higher, usually
between 11 and 11.5 mm. The effective/actual gauge length, L
g
, is therefore larger and
equal to L
ext
. Such extensometer positioning is used to prevent crack initialization at the
original (parallel) gauge length.
To study the impact of placing the extensometer above the gauge length, extensometer
points in the finite element model are placed 0.5, 0.75, 1.00 and 1.25 mm above the 10 mm
gauge length (Figure 6), while the geometry and strain range (
∆ε
= 0.6%, 1.2%) are kept
the same. Although the gauge length remains the same, the strain in the gauge section is
calculated using the extensometer length, mimicking the experimental setup. Since the
extensometer points are above the 10 mm gauge length, the S2G ratio decreases, as shown
in Table 6. Lower strain ranges result in higher S2G ratios (the material is closer to the
elastic state at which the S2G ratio is the highest), higher strain amplitude induces more
plasticity and reduces the S2G ratio. JACOBS constitutive model produces slightly higher
S2G ratios compared to the INCEFA one.
Metals 2021,11, 376 10 of 13
Metals 2021, 11, 376 10 of 13
Original +0.5 mm +0.75 mm +1.00 mm +1.25 mm
Figure 6. SCK-CEN specimen. Extensometer point positions (red dots).
Table 6. S2G values for SCK-CEN specimen: extensometer position effect.
Extensometer Position (mm) Effective Gauge Length (mm) S2G FEMJACOBS S2G FEMINCEFA
Δε = 0.6% Δε = 1.2% Δε = 0.6% Δε = 1.2%
Original 10 1.87 1.82 1.85 1.84
+0.5 11.0 = 10 + 2 × 0.50 1.71 1.66 1.68 1.68
+0.75 11.5 = 10 + 2 × 0.75 1.64 1.59 1.61 1.60
+1.00 12.0 = 10 + 2 × 1.00 1.57 1.52 1.55 1.54
+1.25 12.5 = 10 + 2 × 1.25 1.51 1.47 1.49 1.48
Measured [13] 10 + X 1.51 1.46 1.51 1.46
For studying the effect of surface roughness on fatigue life, some specimens have
been manufactured with a controlled rough surface finish by means of a grinding process.
This grinding reduced the specimen diameter so that ground specimens have gauge sec-
tions diameters below the nominal values of 4.5 mm, as shown in Table 7. All the specimen
have the same gauge length (10 mm). Additional FEM models were built to exactly match
the measured dimensions of these specimens, including the matching extensometer
lengths and, of course, the strain amplitudes (Table 7).
Table 7. S2G values for SCK-CEN specimen: extensometer position effect. Top part: Δε = 0.6%. Bottom part: Δε = 1.2%.
Specimen Surface Lext Dg Δεmeas S2G
(mm) (mm) (%) Meas@N25/2 FEMJacobs FEMIncefa
SC-2 Smooth 11.11 4.474 0.616 1.395 1.684 1.662
SC-5 Rough 11.40 4.230 0.626 1.448 1.621 1.600
SC-15 Smooth 11.16 4.474 0.610 1.642 1.679 1.656
SC-18 Rough 11.07 4.129 0.606 1.652 1.663 1.639
SC-31 Rough 12.00 4.155 0.626 1.454 1.538 1.519
SC-24 Smooth 11.15 4.481 0.618 1.506 1.679 1.657
SC-25 Smooth 11.18 4.475 0.622 1.591 1.673 1.651
SC-26 Smooth 11.82 4.476 0.640 1.430 1.583 1.564
Average - - - 0.620 1.51 1.64 1.62
SC-17 Rough 11.16 4.028 1.224 1.373 1.597 1.612
SC-19 Rough 11.16 4.071 1.244 1.441 1.601 1.617
SC-16 Smooth 11.23 4.475 1.228 1.498 1.624 1.642
Figure 6. SCK-CEN specimen. Extensometer point positions (red dots).
Table 6. S2G values for SCK-CEN specimen: extensometer position effect.
Extensometer Position (mm) Effective Gauge Length (mm) S2G FEMJACOBS S2G FEMINCEFA
∆ε= 0.6% ∆ε= 1.2% ∆ε= 0.6% ∆ε= 1.2%
Original 10 1.87 1.82 1.85 1.84
+0.5 11.0 = 10 + 2 ×0.50 1.71 1.66 1.68 1.68
+0.75 11.5 = 10 + 2 ×0.75 1.64 1.59 1.61 1.60
+1.00 12.0 = 10 + 2 ×1.00 1.57 1.52 1.55 1.54
+1.25 12.5 = 10 + 2 ×1.25 1.51 1.47 1.49 1.48
Measured [13] 10 + X1.51 1.46 1.51 1.46
For studying the effect of surface roughness on fatigue life, some specimens have been
manufactured with a controlled rough surface finish by means of a grinding process. This
grinding reduced the specimen diameter so that ground specimens have gauge sections
diameters below the nominal values of 4.5 mm, as shown in Table 7. All the specimen have
the same gauge length (10 mm). Additional FEM models were built to exactly match the
measured dimensions of these specimens, including the matching extensometer lengths
and, of course, the strain amplitudes (Table 7).
Table 7. S2G values for SCK-CEN specimen: extensometer position effect. Top part: ∆ε= 0.6%. Bottom part: ∆ε= 1.2%.
Specimen Surface Lext Dg∆εmeas S2G
(mm) (mm) (%) Meas@N25/2 FEMJacobs FEMIncefa
SC-2 Smooth 11.11 4.474 0.616 1.395 1.684 1.662
SC-5 Rough 11.40 4.230 0.626 1.448 1.621 1.600
SC-15 Smooth 11.16 4.474 0.610 1.642 1.679 1.656
SC-18 Rough 11.07 4.129 0.606 1.652 1.663 1.639
SC-31 Rough 12.00 4.155 0.626 1.454 1.538 1.519
SC-24 Smooth 11.15 4.481 0.618 1.506 1.679 1.657
SC-25 Smooth 11.18 4.475 0.622 1.591 1.673 1.651
SC-26 Smooth 11.82 4.476 0.640 1.430 1.583 1.564
Average - - - 0.620 1.51 1.64 1.62
SC-17 Rough 11.16 4.028 1.224 1.373 1.597 1.612
SC-19 Rough 11.16 4.071 1.244 1.441 1.601 1.617
SC-16 Smooth 11.23 4.475 1.228 1.498 1.624 1.642
SC-23 Smooth 10.78 4.463 1.228 1.542 1.690 1.708
Average - - - 1.231 1.46 1.63 1.64
Metals 2021,11, 376 11 of 13
4.3. JRC Specimen, L3 Length Effect
The JRC shoulder displacement holder device is constructed in such a way that it
measures the shoulder displacement at L3 equal to several mm, see Figure 7. The L3
section should be deformed only elastically (due to its larger diameter), therefore the L3
length should linearly increase the S2G ratio. The FEM results confirm this (Figure 7,
Tables 8and 9).
Metals 2021, 11, 376 11 of 13
SC-23 Smooth 10.78 4.463 1.228 1.542 1.690 1.708
Average - - - 1.231 1.46 1.63 1.64
4.3. JRC Specimen, L3 Length Effect
The JRC shoulder displacement holder device is constructed in such a way that it
measures the shoulder displacement at L3 equal to several mm, see Figure 7. The L3 sec-
tion should be deformed only elastically (due to its larger diameter), therefore the L3
length should linearly increase the S2G ratio. The FEM results confirm this (Figure 7, Ta-
bles 8 and 9).
Figure 7. JRC specimen, the effect of L3 length on the S2G ratio. Different responses at given Δε are due to different K and
n material parameters (Table 3).
Table 8. JRC specimen (L3 length effect on S2G values, FEMJACOBS).
Case L3 (mm)
0 1 2 4 8 12
Δε = 0.6% 1.842 1.854 1.865 1.887 1.932 1.977
Δε = 1.2% 1.818 1.825 1.833 1.848 1.879 1.909
Table 9. JRC specimen (L3 length effect on S2G values, FEMINCEFA).
Case L3 (mm)
0 1 2 4 8 12
Δε = 0.6% 1.826 1.836 1.845 1.864 1.903 1.941
Δε = 1.2% 1.842 1.850 1.857 1.871 1.900 1.928
5. Conclusions
It is challenging to directly measure the gauge strain of a solid specimen in a PWR
environment. The shoulder to gauge (S2G) ratio is often used for converting displace-
ments at the shoulder to those at the gauge. In this work, finite element method is used to
calculate this ratio for different specimen geometries. By knowing the S2G, the shoulder
and corresponding cross head displacement for a required strain in the gauge section can
be determined and set up for the experiment without the need for using the extensometer.
Figure 7.
JRC specimen, the effect of L3 length on the S2G ratio. Different responses at given
∆ε
are due to different Kand n
material parameters (Table 3).
Table 8. JRC specimen (L3 length effect on S2G values, FEMJACOBS ).
Case L3 (mm)
0 1 2 4 8 12
∆ε= 0.6% 1.842 1.854 1.865 1.887 1.932 1.977
∆ε= 1.2% 1.818 1.825 1.833 1.848 1.879 1.909
Table 9. JRC specimen (L3 length effect on S2G values, FEMINCEFA).
Case L3 (mm)
0 1 2 4 8 12
∆ε= 0.6% 1.826 1.836 1.845 1.864 1.903 1.941
∆ε= 1.2% 1.842 1.850 1.857 1.871 1.900 1.928
5. Conclusions
It is challenging to directly measure the gauge strain of a solid specimen in a PWR
environment. The shoulder to gauge (S2G) ratio is often used for converting displacements
at the shoulder to those at the gauge. In this work, finite element method is used to
calculate this ratio for different specimen geometries. By knowing the S2G, the shoulder
and corresponding cross head displacement for a required strain in the gauge section can
be determined and set up for the experiment without the need for using the extensometer.
Several constitutive laws are considered. The elastic calculation results in the highest
S2G values and compares very well with the analytical values. However, this approach
Metals 2021,11, 376 12 of 13
disregards the plastic deformation within the specimen, which mostly develops in the
gauge section. The results with the constitutive law based on the JACOBS and INCEFA
N25/2 fatigue curves are more representative and result in S2G values of 1.82–1.87 for
both the JRC and the SCK-CEN specimen, for both
∆ε
= 0.6% and
∆ε
= 1.2%. Since the
CIEMAT specimen is significantly longer, its S2G values are between 2.32 and 2.40 at the
same strain ranges. The FEM models are able to quantitatively capture the uncertainties
of the S2G related to the uncertainties of the extensometer contact point at the shoulder
(JRC specimen) or transition region (SCK-CEN specimen). The calculated S2G values for
the SCK-CEN specimen are still somewhat higher compared to the available experimental
results, in spite of accounting for the effective gauge length. In the previous work [
13
],
S2G values were calculated with a nominal gauge length, which is lower than the effective
one, and this resulted in lower S2G values. Calculating S2G ratio using elastic–plastic
constitutive law based on the monotonic uniaxial tensile test should be avoided as they are
significantly lower compared to the ones computed with elastic–plastic laws derived from
hysteresis loops at half-life, N25/2.
Author Contributions:
Conceptualization and original draft preparation: I.S. Formal analysis:
I.S., A.M., K.M., P.G., N.P., M.B., J.L.W., G.B.M., S.A.G. Experimental measurements: M.V., R.N.
Manuscript review: M.B., M.V. All authors have read and agreed to the published version of the
manuscript.
Funding:
The INCEFA-PLUS project has received funding from the Euratom Research and Training
Programme 2014-2018 under Grant Agreement No. 662320.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement:
Requests to access the data presented in this study can be submitted
to the data owner(s). The data are stored in a database and traceable through DOIs [
19
,
20
] but are
not publicly available due to the data policy of the project.
Conflicts of Interest: The authors declare no conflict of interest.
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