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. agloaguen@lamm.univ-nantes.fr, bemmanuel.girard @univ-nantes.fr, cronald.guillen@univ-nantes.fr

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.

References

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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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