(IUCr) Determination of the stress orientation distribution function using pulsed neutron sources

Volume 36
Part 1
Pages 14-22
February 2003

Received 1 June 2002
Accepted 23 September 2002

Determination of the stress orientation distribution function using pulsed neutron sources

Y. D. Wang,^a ^* X.-L. Wang,^a,^b A. D. Stoica,^a J. W. Richardson^c and R. Lin Peng^d,^e

^aSpallation Neutron Source (SNS), Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA,^bMetal and Ceramics Divisions, Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA,^cIntense Pulsed Neutron Source (IPNS), Argonne National Laboratory, Argonne, IL 60439, USA,^dStudsvik Neutron Research Laboratory (NFL), Uppsala University, S-61182 Nyköping, Sweden, and ^eDepartment of Mechanical Engineering, Linköping University, S-58183 Linköping, Sweden
Correspondence e-mail: wangy@ornl.gov

The stress orientation distribution function (SODF) was recently introduced as a mean-field representation to describe the grain-orientation dependence of intergranular stress. Pulsed neutron sources are ideally suited for the determination of the SODF since multiple reflections can be measured simultaneously with comparable precision. A method is developed for constructing the SODF from strain pole figures measured with a pulsed neutron source and demonstrated with cold-rolled interstitial-free steel. The experimental results are compared with those measured with a reactor-based constant-wavelength diffractometer. It is shown that access to a large number of reflections on a pulsed neutron source improves the precision of the experimental SODF and facilitates in situ studies of the evolution of the intergranular stress during deformation and annealing.

Keywords: stress orientation distribution function; neutron diffraction; texture.

1. Introduction

Intergranular stress is caused by stress or strain incompatibility between grains having different crystallographic orientations during mechanical or thermo-mechanical deformation. Unlike macroscopic stress, the intergranular stress is grain-orientation dependent.

The presence of intergranular stress leads to ambiguities in the determination of residual stress by conventional diffraction methods. In an X-ray experiment, the residual stress is typically determined from the slope of the d spacing versus sin² [psi] relationship assuming that the stress or strain is grain-orientation independent. When intergranular stress exists, however, the d versus sin² [psi] relationship may become strongly non-linear (Christenson & Rowland, 1953; Willemse et al., 1982; Hauk & Vaessen, 1985), rendering the slope meaningless. In a neutron diffraction experiment, the residual stress is determined by measuring the lattice strains from one or more reflections for several sample orientations. The stress values (and sometimes signs) obtained with various reflections differ when intergranular stress is present. Although the influence of texture on the diffraction elastic constants can also produce the non-linear relationship, a large discrepancy between the measured and calculated strains is observed in some materials, even if the crystallographic texture is incorporated into stress analysis (Wang et al., 1999a). This is why the high-multiplicity reflections, e.g. {211} in body-centred cubic (b.c.c.) and {311} in face-centred cubic (f.c.c.) materials, are mainly used for studying the residual macrostress, because of their relative insensitivity to the grain-orientation influence (Daymond et al., 2000).

The development of intergranular stress is also of fundamental interest. Over the past decade, intergranular stress, along with texture, has been used as fingerprints to understand how the grain-to-grain interactions occur in a polycrystalline material during and after deformation. In addition, the influence of intergranular stress on materials properties, such as fatigue behaviour or recrystallization texture, is a subject of active research. In spite of recent progress, however, quantitative simulation of the deformation behaviours in polycrystalline aggregates is far from being satisfactory, even for cubic materials (Kocks, 1998). Precise experimental data are critically needed to guide and further the development of modelling.

Recently, a method was developed for experimental evaluation of the grain-orientation-dependent residual stress from neutron or X-ray diffraction measurements. Analogous to texture representation by the orientation distribution function (ODF), the concept of the stress orientation distribution function (SODF) was introduced, which describes the intergranular or grain-orientation-dependent stress as a mean stress field in the Euler orientation space (Wang et al., 1999b; Behnken, 2000). To avoid confusion, the orientation distribution function, which describes the texture, is hereafter referred to as the crystal orientation distribution function (CODF). It has been shown that the SODF may be constructed from the experimental data using the spherical-harmonics approach (Wang et al., 2001), in which the SODF is expanded in terms of generalized spherical harmonics and the series coefficients are determined from the measured strain pole figures.

The validity of the SODF approach has been proved on simulated stress distributions (Wang et al., 2000). Moreover, this approach was successfully applied for analysing residual stresses at reactor neutron sources in a study revealing the effect of annealing on grain-orientation-dependent residual stress in two-phase cold-rolled stainless steel (Wang et al., 2001; Wang, Lin Peng et al., 2002). However, with constant-wavelength diffractometers on a reactor source, the measurement is time consuming and only a limited number of reflections may be measured because of the rapid deterioration of the instrument resolution at large diffraction angles, 2 [theta] . In addition, the low-index reflections that exhibit the largest anisotropy are usually measured at low angles with poor precision. These shortcomings cause uncertainty in the constructed SODF and prevent reliable evaluation of the SODF in regions of weak intensity caused by, for example, strong texture.

Pulsed neutron sources are better adapted for the determination of the SODF. On a pulsed neutron source, a white neutron beam is used and the detectors are held at a fixed angle. Neutron energy or wavelength is determined via time-of-flight, utilizing the pulsed nature of the source. A feature of time-to-flight diffraction is that the instrument resolution is nearly independent of the lattice spacing, so that a large number of reflections are measured simultaneously with each detector or detector bank. For instruments equipped with multiple detector banks, low-index reflections are measured at a variety of diffraction angles. This not only reduces the measurement time but also improves the precision of the SODF. In this paper, we describe a method to determine the SODF using time-of-flight diffractometers at pulsed neutron sources. The method is demonstrated with cold-rolled interstitial-free (IF) steel.

2. Method for constructing the SODF from time-of-flight data

In the time-of-flight technique, the neutron arrival time is proportional to the neutron wavelength,

$[\lambda = {ht/ {m\left( {L_0 + L_1 } \right)}} , \eqno(1)]$

where h is the Plank constant and m is the neutron mass. L₀ and L₁ are, respectively, the source-to-sample and the sample-to-detector distance. Considering Bragg's law,

$[\lambda = 2d\sin \theta, \eqno(2)]$

we obtain

$[d = {h \over {2m}} {1 \over { ( {L_0 + L_1 } )\sin \theta }}\, t = {\rm DIFC}\, t. \eqno(3)]$

DIFC is known as the diffraction constant and is usually determined with a calibration sample.

The strain measured with reflection (hkl) along sample direction y is given by

$[\tilde \varepsilon( {\bf h}, {\bf y}) = ( d_{}^{hkl} - d_0^{hkl} )/ d_0^{hkl}, \eqno(4)]$

where h is the normal direction of the (hkl) plane in the crystal coordinate frame and y is the measuring direction in the sample frame. d^hkl and d₀^hkl are the lattice spacing of the material under, respectively, stress and stress-free conditions. The tilde sign is used to denote the mean quantities, i.e. equation (4) estimates the average value over the distribution of strains introduced by the random fluctuation from grain to grain. We assume that this effective strain distribution is peak-shaped and the average estimation through a sample mean is not biased. Moreover, it will be considered that, in spite of the fact that the grain substructure is the main source of the peak broadening, its influence on the peak position is cancelled. This hypothesis is not generally true, but seems to be a good approximation for most of the plastically deformed materials showing symmetrical line broadening. The experimental peak shift from equation (4) is then the result of the elastic deformation of the grains under the influence of a long-range stress field. If this influence is considered to be grain-orientation dependent, then we have to rely on the existence of a mean strain tensor with its components, which are defined as functions in the Euler orientation space, {g}, relative to the sample coordinate system. These functions, $[\sigma _{ij} ( g )]$ , form the SODF (Wang et al., 1999b), or the stress function as was named by Behnken (2000).

The SODF may be constructed from the experimental strain pole figures using the spherical-harmonics approach. In this method, the SODF, $[\sigma _{ij} ( g )]$ , is expanded in terms of generalized spherical harmonics (Bunge, 1965; Roe, 1965),

$[\sigma_{ij}(g) = \textstyle\sum\limits_{l = 0}^{L^s_{\max}} \textstyle\sum\limits_{m = 1}^l \textstyle\sum\limits_{n = 1}^l\Gamma^{ij}_{lmn}Z_{lmn}(\cos\theta)\exp(-im\psi)\exp(-in\varphi),\eqno(5)]$

where ( [psi] [theta] [varphi] ) denotes Euler angles defined in Roe's notation. $[\Gamma _{lmn}^{ij} ]$ are the series coefficients of the SODF, Z_lmn ( x ) is the augmented Jacobi function, and L_max^s is the maximum order of series coefficients in the SODF. The l = 0 term is the grain-orientation independent part of $[\sigma _{ij} ( g )]$ , i.e. the macrostress. $[\tilde \varepsilon]$ (h, y) is related to the SODF by (Wang et al., 2001)

$[\eqalignno{ \tilde \varepsilon ({\bf h},{\bf y}) = \hskip.2em& \textstyle\sum\limits_{i = 1}^3 \textstyle\sum\limits_{j = 1}^3 \textstyle\sum\limits_{l = 0}^{L_{\max }^{\,\prime} } \textstyle\sum\limits_{m = - l}^l \textstyle\sum\limits_{n = - l}^l \alpha _i \alpha _j qQ_{lmn}^{ij} P_l^m ( \cos \chi )\cr & \times\exp ( - im\eta ) P_l^n (\cos \Theta )\exp ( in\Phi ), &\hfill\llap{ (6)}}]$

with

$[\eqalignno{Q^{kl}_{lmn} =\hskip.2em& \textstyle\sum\limits_{l = |l_2-l_1|}^{|l_2+l_1|} \textstyle\sum\limits_{l_2 = |l_3-l_4|}^{|l_3+l_4|}(l_{1}l_{2}m_{1}m_{2}|lm)(l_{1}l_{2}n_{1}n_{2}|ln)\cr &\!\times (l_{3}l_{4}m_{3}m_{4}|l_{2}m_{2})(l_{3}l_{4}n_{3}n_{4}|l_{2}n_{2})\cr &\!\times W_{l_1m_1n_1} \Gamma^{ij}_{l_3m_3n_3} e^{klij}_{l_4m_4n_4} &\hfill\llap{(7)}}]$

and

$[\eqalignno{ q^{ - 1} =\hskip.2em& \textstyle\sum\limits_{l = 0}^{L_{\max } } \textstyle\sum\limits_{m = - l}^l \textstyle\sum\limits_{n = - l}^l W_{lmn} P_l^m ( \cos \chi )\exp ( - im\eta ) P_l^n (\cos \Theta )\cr &\!\times\exp ( in\Phi ) . &\hfill\llap{(8)}}]$

Here W_lmn are the series coefficients of the CODF; e_lmn^ijkl are the series coefficients of the elastic compliance of a single crystallite (Humbert et al., 1993); $[(\Theta ,\Phi)]$ are the polar and azimuth angles of h in the crystal coordinate system; [chi] and [eta] are the polar and azimuth angles of y in the sample coordinate system; $[\alpha _i]$ is the direction cosine of y relative to the ith sample axis; P_l^m (x) is the Legendre function; $[L_{\max}']$ is the maximum order of series coefficients determined by the combined production of three harmonics; denotes the Clebsch-Gordon coefficients. It is easy to see from (6), (7) and (8) that a linear relationship exists between $[\tilde \varepsilon ]$ (h, y) and $[\Gamma _{lmn}^{ij}]$ . This relationship allows a limited number of $[\Gamma _{lmn}^{ij}]$ to be determined from the measured lattice strains, which may be used to construct the SODF.

The representation of the measured lattice strain by equation (6) is model-free, but statistical and systematic errors in measurements drastically limit the number and accuracy of the coefficients $[\Gamma _{lmn}^{ij}]$ that can be determined from experimental data. The main concern is the smoothness of the solution. The variance of strain, or stress, or elastic energy has been proposed as an indicator of the SODF smoothness (Behnken, 2000). An equivalent way is to define the SODF index as

$[\Pi _{\rm index} = \textstyle\sum\limits_{i,j = 1}^3 \textstyle\sum\limits_{l = 1}^{L_{\max } }\textstyle\sum\limits_{m,n = - l}^l ( \Gamma _{lmn}^{ij} )^2 .\eqno(9)]$

Then, the condition of minimizing this index can be used to stabilize the numerical solution of equation (6) (Wang et al., 1999b). However, the smoothness criteria alone cannot produce a physically meaningful solution for the SODF. Primary information given by a deformation model should facilitate finding a stable numerical solution.

Here, the condition of stress or strain equilibrium between neighbouring grains is considered by implementing the elastic `inclusion' model proposed by Eshelby (1957) in the SODF analysis. A misfit function, [Lambda] , which measures the deviation of the real strain and stress state from the self-consistent state, has been introduced and defined as follows (Wang et al., 2001):

$[\Lambda = \textstyle\oint \{ \sigma _{ij} ( g ) - \tilde \sigma _{ij} - L_{ijkl}^* [ \tilde \varepsilon _{kl} - \varepsilon _{kl} ( g ) ] \}^2 f(g)\,{\rm d}g , \eqno(10)]$

where $[\tilde \sigma _{ij} ]$ and $[\tilde \varepsilon _{ij} ]$ are, respectively, the average (or macro) stress and strain. $[\varepsilon _{ij} ( g )]$ is the mean strain tensor for the grains with g orientation. L_ijkl^* is the Hill (1965) `constraint' tensor. The integration is carried out over all grain orientations, weighted by f(g), the CODF.

The optimal solution of the SODF is determined by minimizing both the errors between the measured and re-calculated lattice strains and the misfit function, i.e.

$[\textstyle\sum\limits_h \textstyle\sum\limits_y [ \tilde \varepsilon ^E ({\bf h}, {\bf y}) -\tilde \varepsilon ^C ({\bf h}, {\bf y})]^2/ N\delta^2({\bf h}, {\bf y})+ \alpha \Lambda \Rightarrow {\rm Min} , \eqno(11)]$

where N is the total number of measurements, $[\tilde \varepsilon ^E ({\bf h}, {\bf y})]$ and $[\tilde \varepsilon ^C ({\bf h}, {\bf y})]$ are the measured and recalculated strains, respectively, and $[\delta ({\bf h}, {\bf y})]$ is the uncertainty of $[\tilde \varepsilon ^E ({\bf h}, {\bf y})]$ . [alpha] (>0) is a regularization parameter that can be regarded as a measure of the coupling between the two terms in equation (11). The selection of [alpha] is based on the smoothness criteria by considering the change of the index, $[\Pi _{\rm index} ]$ , defined by equation (9). This index is a decreasing function of [alpha] , reaching an asymptotic value for the solution corresponding to the self-consistent (SC) model. In the following, [alpha] was empirically chosen so that the SODF index became twice as much as the asymptotic value. This choice usually increases the first term of equation (11) by only 10%, but allows sufficient freedom for the numerical solution to deviate from the self-consistent model.

For time-of-flight diffractometers, an artificial strain is generated when the sample is displaced from the diffractometer centre, leading to an error in the measured strain values. Previous studies (Wang, Wang & Richardson, 2002) show that the observed error is a linear function of the sample displacement along, x, and normal, y, to the incident beam, i.e.

$[\varepsilon _i = xK_{i,x} + yK_{i,y} , \eqno(12)]$

where K_i,x and K_i,y are the trigonometric coefficients corresponding to the ith detector bank. Note that $[\varepsilon _i]$ becomes particularly pronounced for low-angle detector banks. To account for this error, equation (11) is therefore modified to become

$[\eqalignno{&\textstyle\sum\limits_h \textstyle\sum\limits_y [ \tilde \varepsilon ^E ({\bf h},{\bf y})+ xK_{i,x} + yK_{i,y} - \tilde \varepsilon ^C({\bf h},{\bf y})]^2/N\delta ^2({\bf h},{\bf y})+ \alpha \Lambda \cr &\quad\Rightarrow {\rm Min}.&\hfill\llap{(13)}}]$

In this approach, the neutron beam size was considered large enough to bathe the entire sample. In a general case, the effect of beam profile and centre location have to be evaluated more carefully. The beam absorption and extinction by the sample can also affect the strain measurements. However, preliminary evaluations of these effects produced a small influence on the strain values compared with the statistical error and, consequently, were neglected in the analysis of our experimental data.

3. Experimental details

3.1. Sample preparation

The material was commercial Ti-stabilized interstitial-free (IF) steel, with the following composition (wt%): Fe-0.003C-0.15Mn-0.01P-0.0073S-0.0022N-0.001Nb-0.011Si-0.034Al-0.084Ti. It was first hot-rolled to a thickness of 3.7 mm and then annealed for 5 h at 1023 K in nitrogen to relieve the residual stress caused by hot-rolling. The annealed sheets were further cold-rolled to a reduction of 70%. Disc-shaped coupons with a diameter of 10 mm were obtained by spark erosion cutting. These discs were then stacked, with the rolling direction (RD) and transverse direction (TD) carefully aligned, to form a cylindrical specimen of height 10 mm. The cylinder axis was parallel to the sample normal direction (ND).

A well-annealed iron powder sample was also prepared for calibrating the peak position and intensity for the strain and texture measurements. The iron powder was sealed in a cylindrical quartz container that had approximately the same dimensions as the cold-rolled sample.

3.2. Strain measurements

Neutron diffraction experiments were carried out using the General Purpose Powder Diffractometer (GPPD) at the Intense Pulse Neutron Source of Argonne National Laboratory. The sample was fully immersed in the beam so that only the intergranular stress was measured. The detector arrangement for GPPD is shown in Fig. 1. As the precision for strain measurements becomes poor at low 2 [theta] , data collected with detector banks 13-14 were excluded for the determination of the SODF.

Figure 1
Experimental setup on the GPPD. The detector banks (shaded areas) are numbered and their nominal angles are indicated. The [varphi]

axis of the Kappa goniometer lies at 27° relative to the incident beam in the horizontal scattering plane. During the measurements, the sample was rotated about the [varphi]

axis from 0 to 90° at 10° intervals.

A Kappa goniometer was used for the measurement of strain pole figures. The Kappa goniometer was mounted with the [Omega] axis vertical and the [varphi] axis pointing at 27° relative to the incident beam in the horizontal scattering plane. The cylindrical specimen was mounted on a micro-goniometer attached to the Kappa goniometer head, with ND parallel to the axis. During the measurements, the sample was rotated about the axis from 0 to 90° at 10° intervals. As a result of the cylindrical geometry of the sample, a 90° rotation about the [varphi] axis covers the entire pole-figure space, as illustrated in Fig. 2. To minimize sample displacement during the measurements, translation adjustments were made to ensure that the cylindrical sample was concentric with the axis.

Figure 2
Pole-figure coverage with 12 detector banks on the GPPD. As a result of the sample symmetry, each detector bank maps out a 90° arc in the pole figure when the sample rotates about the [varphi]

axis from 0 to 90°.

For the purpose of comparison, additional measurements were made using REST, a high-resolution diffractometer at the reactor of the Studsvik Neutron Research Laboratory, Sweden. The nominal wavelength of the incident neutron beam was 1.70 Å. Elastic strain pole figures for (200) and (211) reflections were measured using an Eulerian cradle attached to REST. Details of the experimental setup have been described previously (Wang, Lin Peng et al., 2002). A 10° × 10° grid was used for the strain pole-figure measurements in the present experiment.

4. Data treatment

Analysis of the experimental data follows three steps: (i) fitting of individual peaks to determine the lattice spacings and hence the lattice strains; (ii) analysis of the texture; (iii) construction of the SODF.

4.1. Fitting of diffraction peaks

Individual diffraction peaks in the time-of-flight spectrum are fitted separately by convoluting a sample broadening function, V(t), with a pulse-shape function, H(t), of the neutron beam,

$[I(t) = \textstyle\int\limits_0^\infty V(t-t\,'\,) \, H(t\,' - t_{hkl} )\,{\rm d}t\,' , \eqno(14)]$

where t_hkl is the position of the peak (hkl). A back-to-back exponential decay profile was chosen for the pulse-shape function, i.e.

$[H(\Delta t) = N [ e^u {\rm erfc}(y) + e^v {\rm erfc}( z )] , \eqno(15)]$

where $[\Delta t]$ = t - t_hkl. The terms N, u, v, x, y are instrument dependent and determined by calibration with a standard silicon powder. Detailed descriptions on the pulse-shape function can be found in the GSAS manual (Larson & Von Dreele, 1986).

As mentioned before, the peak broadening induced by the sample is almost entirely the result of grain substructure, i.e. the line profile could be considered as a convolution of the grain size and micro-strain contribution. In principle, the orientation dependence of the strain can also influence the peak broadening obtained by superposing the independent contributions of grains with different orientations and, consequently, with different values of strain. This contribution can be evaluated backward from the SODF and is always at least one order of magnitude smaller than the observed broadening. Thus, a standard Voigt profile approximation was considered in the following.

No analytical form is available for equation (14) so the numerical method is used. Following Scardi et al. (2000), the sample-dependent term V(t) is expressed as a Fourier series,

$[V(t) = K\textstyle\int\limits_0^\infty T_V \cos ( 2\pi Lt )\,{\rm d}L, \eqno(16)]$

where T_V is the Fourier coefficient and K is the intensity factor. For a crystal lattice, L = n/t_hkl and equation (15) becomes a Fourier series,

$[V(t) = (K /t_{hkl} )\textstyle\sum\limits_{i = 1}^N T_V \cos ( 2\pi Lt ) . \eqno(17)]$

For a Voigt function (Scardi et al., 2000),

$[T_V = \exp ( - \pi \beta _{V,G}^2 L^2 - 2\beta _{V,C} L ), \eqno(18)]$

where $[\beta _{V,G}]$ and $[\beta _{V,C}]$ are the integral breadths of the Gaussian and Lorentzian components. The advantage of this approach is that the values of V(t) and their derivatives converge quickly as N increases, so that only a limited number of series coefficients need be included.

Equation (14) is fitted to the diffraction profiles to yield four parameters characterizing the peak position, diffraction intensity, integral breadths of Gaussian and Lorentz functions, i.e. t_hkl, K, $[\beta _{V,G}]$ and $[\beta _{V,C}]$ , for each diffraction peak. A maximum N of 40 is used for the evaluation of V(t) and the associated derivatives.

4.2. Analysis of the texture

A modified maximum entropy method algorithm (Wang et al., 1997) was used to construct the CODF from the intensity factors obtained from the least-squares fitting and calibrated using the iron powder sample. During the processing of the texture data, the differences in angular acceptance by different detector banks were also considered. A high-precision CODF is required to construct the SODF and the use of the maximum entropy method has turned out be to quite important for the heavily textured sample in the present study.

4.3. Construction of the SODF

The SODF was constructed using equation (13), modified to take into account possible sample displacement. Although a large number of reflections were measured, only seven of them, i.e. (110), (200), (211), (220), (310), (222) and (321), were used to construct the SODF. The single elastic constants of pure iron, with C₁₁= 236.9, C₁₂= 140.6 and C₄₄= 116.0 GPa (Simmons & Wang, 1971), were used for the calculation. The value of the maximum order of the series coefficients, L_max in equation (5), depends on the quality of the measured strain pole figures, which include the resolution, accuracy and the number of the independent data sets. It has been shown that the series expansions in the stress or strain space, with L_max = 6, are already adequate to describe some plastic deformation behaviours (Van Bael et al., 1996). Of course, a large number of high-quality strain pole figures, collected simultaneously by the time-of-flight technique, provides the possibility to expand the series coefficients to higher order. The maximum order of series coefficients, L_max^s , was set to 10 in the present study.

5. Results

5.1. Strain pole figures

The strain pole figures for the (110), (200), (211), (220) and (310) reflections are shown in Fig. 3. The left panel shows the raw data, whereas the right panel shows the recalculated strain pole figures from the constructed SODF. The strain pole figures for (222) and (321) are almost isotropic and therefore not displayed here (however, they were used for the construction of the SODF). Indications of intergranular stress are readily seen as the strain-pole-figure patterns are completely different for different reflections.

Figure 3
Strain pole figures determined for cold-rolled IF steel with various reflections: (110), (200), (211), (220), (310). The left panel shows the `raw' strain pole figures obtained with GPPD, whereas the right panel shows the corresponding recalculated ones. Data in the middle panel were measured with REST (see text). All strain values are in units of 10^-6.

The middle panel of Fig. 3 shows the (200) and (211) strain pole figures measured with a four-circle diffractometer on a constant-wavelength neutron diffractometer at the Neutron Research Laboratory in Studsvik, Sweden. These two reflections were chosen because their diffraction angles with [lambda] = 1.70 Å were 76 and 93°, respectively. This is the region where the instrument is optimized for strain measurements, so the strain pole figures thus obtained should be of high precision. As can be seen, the recalculated strain pole figures obtained with GPPD are in good agreement with those measured on a reactor.

To assess further the SODF approach, the lattice strains obtained in the present work are compared with a previous investigation by Pintchovius et al. (1987) on low-carbon steel. In their study, the low-carbon steel sheet was obtained by cold-rolling to 60% reduction, and the measurements were made using a triple-axis spectrometer on a reactor source. Fig. 4 gives lattice strain distributions for the (200) and (110) reflections in the RD-ND and ND-TD planes as a function of angle relative to the ND. The data reported in this study and by Pintchovius et al. (1987) show a similar trend and agree within the standard uncertainties. The larger deviations seen at some angular positions can be ascribed to the differences in the chemical composition of the samples and in the ways they were fabricated.

Figure 4
Comparison of lattice strain distribution determined with different techniques. The solid symbols are recalculated strains obtained with the GPPD for 70% cold-rolled IF steel. The open symbols are data from Pintchovius et al. (1987

[Pintchovius, L., Hauk, V. & Krug, W. K. (1987). Mater. Sci. Eng. 92, 1-12.]

) for 60% cold-rolled low-carbon steel, measured with a triple-axis spectrometer on a reactor neutron source. (a) (200) reflection in the ND-RD plane as a function of tilt angle relative to ND; (b) (110) reflection in the ND-RD plane as a function of tilt angle relative to ND; (c) (200) and (110) reflections in the ND-TD plane as a function of tilt angle relative to ND.

5.2. Orientation and stress analysis

Having established the reliability of the experimental data and the analysis method, we began the evaluation of the intergranular stress. Table 1 gives the residual stress values for major texture components as well as the macrostress, $[\tilde \sigma _{ij}]$ , in the sample. Since the sample was fully bathed in the neutron beam and in an equilibrium stress state, the calculated macrostress tensor should be zero. The residual values of about 10 MPa reveal the precision of the method. This result further validates the SODF and the associated spherical-harmonics approach. The non-vanishing stress values exceeding 10 MPa for the major texture components are, therefore, entirely intergranular stress in nature.

Table 1
Stress tensor for major texture components in cold-rolled IF steel

Texture component	Euler angle {, , }	₁₁ (MPa)	₂₂ (MPa)	₃₃ (MPa)	₂₃ (MPa)	₁₃ (MPa)
{100}<110>	(90°, 0°, 45°)	-113.8	20.5	-0.3
{311}<110>	(90°, 25°, 45°)	-63.5	28.2	6.9	±9.5
{211}<110>	(90°, 30°, 45°)	-46.0	19.6	8.8	±27.4
{111}<110>	(90°, 54.7°, 45°)	22.0	6.0	-28.8	±35.3
{111}<112>	(0°, 54.7°, 45°)	4.1	29.2	42.1	±38	±3.8
Macrostress	-	-9.0	3.7	7.7

Fig. 5(a) shows the [varphi] = 45° cross section of the experimentally determined CODF. The ideal principal orientations for a b.c.c. crystal in the = 45° cross section are indicated in Fig. 5(b). A comparison of Figs. 5(a) and 5(b) shows that the major texture components in cold-rolled IF steel are [alpha] -fibre (<110>||RD) and [gamma] -fibre (<111>||ND). These are typical texture components found in cold-rolled IF steel (Ray et al., 1994). The changes of the principal stress components along [alpha] - and [gamma] -fibres are shown in Fig. 6. A large variation is seen for the [sigma] ₁₁ component along the [alpha] -fibre. On the other hand, the stress variation along the [gamma] -fibre is limited.

Figure 5
(a) ODF cross sections of cold-rolled IF steel with [varphi]

= 45°. (b) Ideal principal b.c.c. orientations in the [varphi]

= 45° cross section.

Figure 6
Residual stress in cold-rolled IF steel for texture components along (a) [alpha]

-fibre and (b) [gamma]

-fibre.

The distributions of residual stress along the RD for grain orientations from {001}<110> to {111}<110> can be explained by considering the anisotropy of hardening in b.c.c. metals during deformation. For steel, the hardening of those grains near {001}<110> is lower than that near {111}<110> (Hutchinson, 1999). The former grains may be regarded as the `soft phase' and the latter as the `hard phase'. As the `hard phase' takes on a larger share of the stress during rolling deformation than the `soft phase', it would have a larger elastic strain to release after deformation. As a result, the `hard phase' ({111}<110> grains) will be under a tensile residual stress, whereas the `soft phase' ({001}<110> grains) will be under a compressive residual stress. The detailed modelling results will be given elsewhere.

6. Discussion

Strain pole figures provide fingerprint information for understanding the deformation in polycrystalline materials. In general, the strain pole figures obtained for different reflections are not independent. Their interdependence serves as constraints that minimize the error in the constructed SODF and hence the recalculated strain pole figures. As an example, consider the pole figures for the (110) and (220) reflections (Fig. 2). The raw strain pole figures are not quite the same; the differences between them indicate systematic or statistical errors in the measurement of individual pole figures. As might be expected, the recalculated (110) or (220) pole figures (Fig. 2, right panel) are identical and fall between the raw data. Thus, in general, the use of multiple reflections improves the precision of the experimentally determined SODF. A corollary is that the recalculated strain pole figures are of higher precision than the raw data since errors incurred in the measurement of individual strain pole figures are minimized in the analysis process. Therefore, whenever possible, the recalculated strain pole figures, rather than the raw data, should be used to represent the measurement results.

The spherical-harmonics analysis has been implemented in GSAS (Larson & Von Dreele, 1986) for the determination of the CODF from experimental data. It is possible to implement the spherical-harmonics analysis of the SODF in a Rietveld program, whereby the coefficients $[\Gamma _{lmn}^{ij} ]$ are determined from the refinement of all data. Popa & Balzar (2001) have recently explored this possibility. They defined a texture-weighted strain orientation distribution function (WSODF) and performed the spherical-harmonics analysis of this function for all crystal symmetries. The SODF can be calculated from the WSODF and, in principle, the result is not invariant to the crystal symmetry. We caution, however, that the SODF is a tensor with six components, and thus the number of coefficients required to construct the SODF reliably from the WSODF is much larger and there is no guarantee that the solution will be physically credible. For this reason, it is important to incorporate constraints like the self-consistent model [equation (10)] in the SODF analysis. Here we considered each grain as an ellipsoidal inclusion with a specified aspect ratio for the RD:TD:ND directions. In this case, the SODF should be invariant to the crystal symmetry operations, as we assumed in equation (5). The numerical results show a low sensibility to the variation of the aspect ratio and any attempt to consider a shape distribution will involve a dramatic increase of calculation complexity without a decisive improvement in the reliability of the final results.

The analysis method presented in this paper is quite general. The only requirement is knowledge of the single-crystal elastic constants of the material. Once the SODF is constructed, the strain and stress values can be reliably evaluated for any grain orientations. In Fig. 4(b), for example, no (200) strain data are available between the tilt angles of 50 and 90° for the low-carbon steel sample because these are the orientations of extremely weak intensities due to the strong rolling texture. Similarly, direct measurements for (200) were also imprecise for the IF steel sample at these orientations. However, the construction of the SODF from the measurements of seven reflections at these and other orientations has allowed the strain values for the (200) reflection to be determined. In fact, the IF steel data show a dramatic rise and fall in this region of interest. Thus, important information would be missing without invoking the SODF method.

Time-of-flight diffractometers at a pulsed neutron source provide an efficient way to construct the SODF. In the present experiment, the measurements for each sample took a day to complete. About 20 reflections were measured as a function of sample orientation, although only seven of them were used for the construction of the SODF and these data alone yielded a SODF of adequate precision. Similar measurements on a reactor source took three to seven days. With the more than two orders of increase in data rate that will be available with the new diffractometers at the next-generation spallation neutron sources, measurements of the SODF may be completed in 5-10 min. This will greatly facilitate in situ studies of the evolution of the intergranular stress during deformation and annealing.

The large number of peaks recorded in the same measurement cycle at a time-of-flight diffractometer delivers a huge amount of information, some of which is redundant and, strictly speaking, not necessary to recover the SODF. However, the knowledge of the mean strain value for a large number of directions in the reciprocal space can significantly increase the reliability of the constructed SODF compared with the results obtained at reactor neutron sources with comparable instrument resolution. A practical application of the technique is reliable measurements of the macrostress fields in engineering materials.

On the scientific implication of this technique, a major application is that the real boundary and hardening conditions during plastic deformation simulations can be obtained by determining the SODF. In the past, the Taylor model, the self-consistent model and the finite element (FE) model were most commonly used in three-dimensional crystal plastic deformation simulations at large strains. In spite of the success of these models in capturing the main characteristics of deformation texture components, discrepancies still exist between the measured textures and the simulation results. Notably among these differences are the presence or absence of certain ideal texture components, different ratios of ideal texture components as a function of strain, and a much slower texture evolution than predicted by the models (Hughes et al., 2000). Actually, most of these models either `explicitly' or `implicitly' assume certain stress/strain boundary conditions and neglect the detailed evolution of microstructures during plastic deformation. From the SODF, rich information can be obtained on the influence of microstructures on the micro-hardening parameters and the real boundary condition of plastic deformation in a polycrystalline material. Such experimental results should be accounted for in future crystal plastic deformation simulations.

7. Concluding remarks

The SODF provides a unique mean-field representation of the intergranular stress in grain-orientation space. Time-of-flight diffractometers at a pulsed neutron source are ideally suited to the determination of the SODF. The measurement speed is greatly increased with wide angular detector coverage. The theoretical basis and the practical implementations for the construction of the SODF from time-of-flight neutron diffraction data have been discussed. The method has been demonstrated with a cold-rolled IF steel sample. A comparison with measurements at reactor sources has confirmed the validity of the method.

Acknowledgements

Oak Ridge National Laboratory is managed by UT-Battelle, LLC, for the US Department of Energy under contract DE-AC05-00R22725. The authors are grateful to Professors L. Zuo and X. Zhao of Northeastern University (China) for help with the preparation of IF steel samples. This research was supported in part by an appointment (YDW) to the Oak Ridge National Laboratory Postdoctoral Research Associates Program administered by the Oak Ridge Institute for Science and Education and Oak Ridge National Laboratory.

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