Development of intergranular stresses in zirconium alloy cladding tubes: experimental and numerical studies
D. Gloaguen1,a, E. Girard1,b, R. Guillén1,c 1GeM, Institut de Recherche en Génie Civil et Mécanique, Université de Nantes, Ecole Centrale de Nantes, CNRS UMR 6183, 37 Boulevard de l’Université, BP 406, 44 602
Saint-Nazaire cedex, France. [email protected], bemmanuel.girard @univ-nantes.fr, [email protected]
Keywords: plastic anisotropy, self-consistent model, intergranular strains, residual stresses.
Abstract. Complementary methods have been used to analyse residual stresses in zirconium alloy
tubes which were manufactured by cold rolling : X-ray diffraction and scale transition model. A
modified elasto-plastic self-consistent model (EPSC) has been used to simulate the experiments and
exhibits agreement with experimental data. X-ray diffraction analysis in rolling direction shows
opposite stress values for 4110 and 2220 planes respectively. The measured strains were
generated by an anisotropic plastic deformation. Plastic incompatibility stress on X-ray
measurements should be taken into account so as to make a correct interpretation of the
experimental data.
Introduction
Pressurized heavy water reactors use zirconium alloys due to their low neutron absorption cross-
section, low irradiation creep and high corrosion resistance in reactor atmosphere. Due to their hcp
structure, Zr single crystals are usually highly anisotropic. Consequently, strong residual stresses
are generated in fabricated components of Zr alloy during mechanical treatments. These stresses
play an important role in subsequent deformation and modify the mechanical properties of the
reactor core components. This work concerns the manufacturing process of Zr alloy cladding tubes.
These tubes are manufactured by cold rolling. To reach its final size, the material undergoes three
cold-rolling passes in a particular mode called cold pilgering. To understand the influence of plastic
anisotropy on the mechanical properties, the last rolling pass has been studied. X-ray diffraction
techniques are used to characterize the mechanical state at the macro- and meso-scopic levels of the
specimens. The experimental data show the appearance of important second order stresses which
depend on crystallographic planes. In the first section, a novel approach is adopted to describe the
plasticity evolution and determine the active deformation systems with an EPSC approach. A
modified formulation of the crystal behaviour is proposed. This model is tested by simulating the
development of intergranular strains during uniaxial tension of a commercial purity aluminium.
Neutron diffraction measurements of the elastic strains are used as a reference [1]. In a second part,
experimental results have been compared with those given by the modified model for Zr alloy
tubes. The EPSC formulation provides an accurate description of the intergranular plastic residual
stresses induced by the anisotropic plastic behaviour of single crystals. An estimation of the first-
order stresses is also proposed.
Modelling and numerical validation
Modified self-consistent approach
The plastic flow can take place when the Schmid criterion is verified, i.e. slip occurs if the resolved
shear stress gτ on a system g is equal to the critical value g
cτ depending on the hardening state of
the slip system. A complementary condition which states that the increment of the resolved shear
Materials Science Forum Vols. 524-525 (2006) pp 853-858Online available since 2006/Sep/15 at www.scientific.net© (2006) Trans Tech Publications, Switzerlanddoi:10.4028/www.scientific.net/MSF.524-525.853
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stress must be equal to the incremental rate of the critical resolved shear stress (CRSS) has to be
verified simultaneously. In small strain formulation, one has:
gcτ
gτ == σ..
gR and g
c
g ττ &&& == σRg .. (1)
where gR is the Schmid tensor on a system g is the double scalar product. If
gγ& denotes the slip rate
on a system g, the Schmid criterion is thus given by:
0gg
c =γ⇒τ< &gτ (2.a)
0 and gg
c
gg
c
g =γ⇒τ<ττ=τ &&& (2.b)
0 and gg
c
gg
c
g >γ⇒τ=ττ=τ &&& (2.c)
The main problem is to determine which combination of slip systems will be really activated at each
step of the plastic deformation path. In this case, all possible combinations of potentially active
systems must to be scanned to find one that satisfies the two preceding conditions simultaneously.
Running time considerations become the main task of the model. Moreover, this method can give
several equivalent solutions for some hardening matrix [2]. Recently, Ben Zineb et al. [3] have
proposed a new formulation to resolve the problem of ambiguous selection of slip systems and
reduce running time computation. Their numerical results in the case of BCC single crystals present
a good agreement with the ‘classic’ crystal plasticity based on the CRSS. We propose to extend this
formulation in the polycrystalline model framework.
Base on the work of Ben Zineb et al., the relations (2a), (2b) and (2c) can be expressed with the
following equation:
gggM τ=γ && (3)
The slip rate is linked to the resolved shear stress through a function g
M . This formulation is based
on the crystalline rate-dependent flow rule. On the other hand, the temporal variable does not play
any role in this approach. The selection of active and non-active deformation systems is established
with g
M . This function depends on the ratio g
c
g
τ
τ and can describe the hardening behaviour during
the plastic regime. The hardening parameter g
M is given by:
( )( ) ( )( )
τ+
τ+
−
τ
τ+β= g
0
g
0g
c
g
0
g kth12
1kth1
2
11kth1
2
1M &
(4) Hyperbolic tangent function has been tested and used because it permits to reproduce the
mechanical and the hardening behaviour. β and k0 are material constants. With equation (4) and
after some algebric calculations, the constitutive relation which links the overall stress rate and
strain rate in the grain is then given by :
( ) σσRsεg
&l&& ..R1g −=
+= ∑ ......
g
gM (5)
l is the elastoplastic consistent tangent moduli tensor. This tensor depends on active systems,
elastic properties, stress rate and deformation history of the material. The other mechanical
variables are determined by the usual relations given by the EPSC model [4], taking into account
the equations (3) and (5).
Simulation results and discussion
The modified model has been used to predict the development of elastic lattice strains during
854 Residual Stresses VII, ECRS7
uniaxial loading. Clausen and Lorentzen [1] have measured lattice strains in commercial pure
aluminium loaded up to 3% strain with in-situ neutron diffraction measurements. Lattice strains
have been determined in the tension direction for the (111) and (220) reflections. The elastic single
crystal stiffness (GPa) for aluminium are: C11=107.3, C12=60.9, C44=28.3. The development of
elastic lattice strains has been simulated using the EPSC and the modified models. The input texture
was the same in all simulations: 2000 equally-weighted lattice orientations representing a random
texture (the initial experimental texture was very weak). A linear hardening matrix containing only
two terms H1 and H2 corresponding to weak and strong interactions between the slip systems has
been chosen. H1, H2, β, k0 and 0
cτ (initial value of CRSS) have been determined by fitting the
experimental macroscopic tensile curves plotted in Figure 1.a.
(a) (b) Figure 1: (a): simulated and experimental tensile curves for aluminium, (b): calculated and experimental elastic lattice strain for the (111) and (220) reflections for aluminium. The parameter values are listed in Table 1. The lattice strains predicted by applying the different
schemes are plotted in Figure 1.b. The experimental data are also shown in these plots. The
accuracy on the experimental strains is of the order of 10-4.
Table 1: Values of material parameters
H1(MPa) H2(MPa) 0
cτ (MPa) β k0
Aluminim 55 H1x1.1 10.2 3.4 104
5
The two approaches enable a very accurate representation of the measured macroscopic stress-strain
curves using the fitting parameters shown in Table 1. The modified model gives similar results to
the SC model. It should be noticed that the running time computation is considerably reduced with
this novel approach. The development of lattice strains is non-linear once the specimen reaches the
plastic regime. The SC model provides a good agreement with the experimental observations on a
mesoscopic scale. This model predicts the elastic lattice strain evolution for the (111) and the (220)
reflections and the numerical level of elastic strains is correctly described. The linear hardening law
is sufficient to reproduce the different experimental data. On the other hand, the modified model
reflects the elastic lattice strain evolution for the two crystallographic planes in the elastic and
plastic regimes. For the (220) reflection, the numerical predictions show a better agreement with
experimental results beyond a macroscopic strain of 1.5%. After this strain value, the model
underestimates the lattice strain evolution. At 3 % macroscopic strain for aluminium, the standard
deviation is 4%. Nevertheless, this discrepancy, owing to different experimental uncertainties, is
weak. For the (111) reflection, predictions with the modified model are especially accurate and
similar with the SC model and experimental results.
0
10
20
30
40
50
0 0,5 1 1,5 2 2,5 3Macroscopic strain (%)
Stress (MPa)
experimental
EPSC model
modified model
0
100
200
300
400
500
600
700
800
900
1000
0 0,5 1 1,5 2 2,5 3
Macroscopic strain (%)
Elastic strain (x 10-6)
experimental - (220) reflection
experimental - (111) reflection
EPSC model - (111) ref lection
EPSC model - (220) ref lection
modified model - (220) reflection
modified model - (111) reflection
Materials Science Forum Vols. 524-525 855
Experimental procedure
Evaluation of residual stresses
We present succinctly the principles of residual stresses evolution by X-ray diffraction and the key
role played by plastic anisotropy properties on the interpretation of the data. More details can be
found in references [5,6,7]. The average strain for diffracting grains can be written as :
dd V
IIpiI
ijijV)hkil,,()hkil,,(F)hkil,,( ψφε+σψφ=ψφε (6)
< >Vd is the average over diffracting grains for the hkil reflection. Fij(φ,ψ,hkil) are the X-ray elastic
constants and dV
IIpi )hkil,,( ψφε the mean value of strain incompatibilities between the diffracting
and the non diffraction volumes. σI is the macroscopic stress tensor.
The second term in the above equation can be approximated by :
dd V
IIth
V
IIpi )hkil,,(q)hkil,,( ψφε=ψφε (7)
Where dV
IIth )hkil,,( ψφε is calculated from the EPSC model and averaged for crystallites having
the (hkil) normal to the orientation characterized by the ψ and φ angles. q is a constant factor (a
fitting parameter) introduced in order to find the real magnitude of plastic strains. In this case, it is
possible to evaluate the magnitude of the first as well as the second order residual stresses using
eqs. (6,7) with a non-linear fitting procedure associated with the scale transition model.
the relation (6) shows clearly that the measured strain cannot be identified to the macroscopic strain
if the material presents anisotropic properties. The presence of intergranular strain after a
mechanical solicitation influences the measured strain. Generally, the interpretation of experimental
data is based on the unjustified assumption that dV
IIpi )hkil,,( ψφε = 0. In our study, we propose to
quantify the importance of these intergranular stresses and to show their influence in the
interpretation of the experimental results.
Stress measurements by X-ray diffraction technique
We have determinated the evolution of internal stresses due to plastic anisotropy in deformed
samples along the rolling direction in the finished tube after the last rolling step. These experiments
were carried out on a D500 SIEMENS goniometer with a Cr Kα radiation. An Ω goniometric
assembly with a scintillation detector were used. Two plane families were studied: 4110 at 2θ =
156.7° and 2220 at 2θ = 137.2°. The direction φ = 0° corresponds to the rolling direction
(parallel to the tube axis). The X-ray elastic constants (XREC) Fij(φ,ψ,hkil) are theoretically
calculated with an elastic self-consistent model. The influence of the texture on these constants is
taken into account by weighting single crystal elastic constants with the texture function. The
sample shows a typical texture of rolled zirconium. The basal poles are preferentially oriented at an
angle of 35° from the normal direction towards the transverse direction, while the prism poles
exhibit a weak maximum at the rolling direction.
Simulation data
The texture was introduced in the model by a set of 1000 grains characterized by Euler angles and
by weights which represent their volume fraction. The main active slip systems are assumed to be :
the 6 prismatic 02110110 systems, the 24 pyramidal 32111110 systems and 2110
856 Residual Stresses VII, ECRS7
twinning. The critical resolved shear stresses (CRSS) used here, τ = 118 MPa for prismatic mode, τ = 220 MPa for first-order pyramidal mode and τ = 260 MPa for tensile twinning, give a correct
description of the experimental data. We considered a linear hardening law [8], the coefficient Hgr is
equal to Hgg for any deformation mode r : ∑=
r
rγg
Hg
τ && . The hardening coefficients were found to
be 80 MPa for prismatic slip, 160 MPa for pyramidal slip and 240 MPa for twin mode. The strain
rate tensor adopts the form :
−
−=
34.000
066.03.0
03.01
E& (7)
where the tube axis parallel to x1, the hoop direction is parallel to x2 and the tube radius is along x3.
The strain rate tensor is taking account the plastic shearing due to the process [9]. The total strain
reaches 175% after the rolling pass.
Results
Applying eq. (6) and (7) and fitting the results from the model to the experimental data, the
longitudinal stress I
11σ and the q factor have been found (Table 2). In order to visualize the results
of the calculation (and to give an example), the dV
)hkil,,( ψφε values evaluated with eq. (6) have
been compared in Fig. 2 with those measured by X-ray diffraction technique for the 2220 plane.
The agreement between the experiment and the model is qualitatively correct. I
11σ value reaches 50
MPa.
Table 2: results of evaluation of σI and q parameter with two different assumptions, i.e.
dV
IIpi)hkil,,( ψφε =
0 or ≠ 0
q I
11σ (MPa)
dV
IIpi)hkil,,( ψφε ≠ 0 0.302 ± 0.061 50 ± 10
4110 plane 2220 plane
dV
IIpi)hkil,,( ψφε = 0
-
189 ± 16 -175 ± 24
In order to express the influence of the second order plastic strain on the correct interpretation of
experimental data an additional test has been performed. The procedure presented above was
applied for the X-ray measurements but with the assumption that dV
IIpi )hkil,,( ψφε = 0. The results
are presented in table 2. For σ11, the two planes exhibit an opposite behaviour : traction for 4110
plane and compression for 2220 plane. σ value reaches 189 MPa for 4110 plane and –175 MPa
for 2220 plane. X-ray measurements show the effective existence of plastic anisotropy. As can be
seen from eq. (6), the measured stress depends on a function of the analysed plane family. Strain
incompatibilities are present at the mesoscopic level in the material and consequently, the stresses
obtained by XRD depend on the plane. The diffracting crystals are not the same for each case,
which allows us to deduce that different second order stresses exist, linked to a strong anisotropic
plastic deformation for these two plane families. The term dV
IIpi )hkil,,( ψφε plays a key role in a
Materials Science Forum Vols. 524-525 857
correct interpretation of different measurements of residual stresses. The method employed in this
study enables the magnitude of the macro- and meso-scopique residual stresses to be evaluated
using information from an EPSC approach.
-0,0005
0
0,0005
0,001
0,0015
0 0,1 0,2 0,3 0,4 0,5 0,6
sin2 ψψψψ
residual strain
Figure 2: Comparison of the calculated
dV)hkil,,( ψφε (◊) with experimental data (∆) for the 2220 plane
with φ = 0°.
Conclusions
A modified algorithm has been proposed for computing the mechanical response of a single crystal.
The new formulation of the crystal plasticity has been validated at the meso- and macro-scopic
levels with published experimental results and a good agreement between theory and experiment
was found. Numerical results, obtained at the different scales, show the relevance of this approach.
This approach has been developed in order to simulate the mechanical response of Zr alloy tube
after cold pilgering. Stresses observed by X-ray diffraction are significantly different from one
plane family to another. These results can be explained by the presence of mesoscopic stresses of
plastic in the material. Self-consistent model had been used to quantify the stress differences
between these plane families. The choice of prismatic slip as principal deformation mode can
explain the opposite pseudo-macrostresses values for the two studied planes. These predicted results
can only be obtained for a single set of slip modes and hardening parameters. This analysis shows
that X-ray diffraction stress analysis could constitute an effective validation or identification of a
model. Nevertheless, the influence of second order strain must be taken into account to get a correct
interpretation of X-ray diffraction results for hexagonal material.
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858 Residual Stresses VII, ECRS7
Residual Stresses VII, ECRS7 10.4028/www.scientific.net/MSF.524-525 Development of Intergranular Stresses in Zirconium Alloy Cladding Tubes: Experimental and
Numerical Studies 10.4028/www.scientific.net/MSF.524-525.853
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