Uncertainty in flow stress measurements using X-ray diffraction for sheet metals subjected to large plastic deformations

Abstract

X-ray diffraction techniques have been developed to measure flow stresses of polycrystalline sheet metal specimens subjected to large plastic deformation. The uncertainty in the measured stress based on this technique has not been quantified previously owing to the lack of an appropriate method. In this article, the propagation of four selected elements of experimental error is studied on the basis of the elasto-viscoplastic self-consistent modeling framework: (1) the counting statistics error; (2) the range of tilting angles in use; (3) the use of a finite number of tilting angles; and (4) the incomplete measurement of diffraction elastic constants. Uncertainties propagated to the diffraction stress are estimated by conducting virtual experiments based on the Monte Carlo method demonstrated for a rolled interstitial-free steel sheet. A systematic report on the quantitative uncertainty is provided. It is also demonstrated that the results of the Monte Carlo virtual experiments can be used to find an optimal number of tilting angles and diffraction elastic constant measurements to use without loss of quality.

1. Introduction

This article focuses on uncertainty estimation applied to an experimental X-ray diffraction technique that was developed to measure the multiaxial flow stress of sheet metals subjected to large plastic deformation (Foecke et al., 2007). Using this technique the multiaxial flow stress behaviors of engineering aluminium sheets have been successfully measured (Foecke et al., 2007; Iadicola & Gnäupel-Herold, 2012; Pham et al., 2015; Güaner et al., 2014). In the mentioned work, the sin² ψ method (Noyan & Cohen, 1986) was applied to analyze the multiaxial stress present on the surface layer of the aluminium alloys.

Recently, it was discussed that the sin² ψ method might not be adequate for assessing stress using {211} diffraction planes of an interstitial-free (IF) steel rolled sheet in the body-centered cubic crystal structure (Jeong et al., 2015). Instead, fully anisotropic diffraction elastic constants (DECs) were required to obtain the multiaxial flow stress behavior. The experimental measurement of DECs requires additional ex situ experiments using uniaxial specimens cut from the samples deformed to several increments along selected strain paths. Then, the DECs are determined by fitting the X-ray lattice strains to the macro stress values. In the work of Jeong (2015), the experimental DECs obtained at various plastic strain increments were used to estimate the biaxial flow stress for the IF steel during equal biaxial straining. The measured stress was in good agreement with an independent mechanical measurement by hydraulic bulge test. Recently, this method has been extended for plane-strain paths and compared with multiaxial predictions by various constitutive models (Jeong et al., 2016). However, the uncertainty in the measurement technique has not been clearly addressed owing to the absence of an appropriate method. For that reason, the reliability of the experimental data has not been fully assessed.

In the current article, a Monte Carlo method to estimate the uncertainties in the macroscopic flow stress tensor is presented. This method relies on an accurate constitutive model developed for the polycrystalline aggregate that is also suitable to describe the mechanical behavior under large plastic deformation. Moreover, the constitutive model should be able to capture the anisotropic elastic behavior of polycrystals and the evolving crystallographic texture during plastic deformation.

Various elasticity models have been developed in the literature (Dölle, 1979; Behnken & Hauk, 1986; Barral et al., 1987; Welzel, Fréour & Mittemeijer, 2005; Gnäupel-Herold et al., 2012). These models can describe the effects of crystallographic texture on the DEC through the use of the orientation distribution function (ODF), from which diffraction data can be extracted. However, the ODF is a state function with no direct relationship to the current state of plastic strain; thus, an elastic model based on a static texture cannot capture the evolving DECs as a function of plastic strain.

Therefore, a hybrid approach was implemented by Gnäupel-Herold et al. (2014). In that article, the calculation of DECs included the development of crystallographic texture. That texture development was accounted for by using predictions based on the viscoplastic self-consistent model (Lebensohn & Tomé, 1993; Tomé & Lebensohn, 2009) in addition to a self-consistent elasticity model (Gnäupel-Herold et al., 2012) to calculate the DEC. Gnäupel-Herold et al. (2014) compared the results of the calculations with experimental measurements of DECs and found that the calculations were able to capture much of the DEC variation but still had an undesirable level of disagreement.

A self-consistent elasto-plastic model can describe both the diffraction data on the basis of the elasticity constitutive law and the grain reorientation due to plastic deformation (Clausen et al., 1998). In a previous article by the current authors (Jeong et al., 2015), it was also shown that a similar self-consistent model called the elasto-viscoplastic self-consistent model (EVPSC) developed by Wang et al. (2010) is capable of capturing elastic anisotropy in DECs and their changes as a result of plastic deformation. In the current article, the EVPSC constitutive description is used to conduct a Monte Carlo virtual experiment to simulate the propagation of errors in diffraction experiments to the final flow stress of the IF steel sample as described by Jeong et al. (2015).

The structure of the current article is as follows. First, a brief review of the EVPSC modeling framework is given in §2. Then, detailed procedures for the diffraction stress analysis are given in §3. Four elements of error in diffraction experiments are addressed in detail in §4. In §5, the method using Monte Carlo virtual experiments is presented. A systematic report on the results obtained by the Monte Carlo virtual experiments is given in §6. The results from the Monte Carlo experiments are further discussed in §7 to find optimal conditions to improve the quality of diffraction experiments.

2. Elasto-viscoplastic polycrystalline aggregate

An extensive description of the EVPSC model is available elsewhere (Wang et al., 2010). In this section, selected elements of the EVPSC model are briefly discussed.

2.1. Self-consistent elasto-viscoplasticity laws

The local strain rate $\dot{\mathit{\epsilon }}$ for a crystallite is decomposed into the elastic and the viscoplastic terms (${\dot{\mathit{\epsilon }}}^{\text{e}}$ and ${\dot{\mathit{\epsilon }}}^{\text{vp}}$, respectively):

$${\dot{\epsilon }}_{ij}={\dot{\epsilon }}_{ij}^{\text{e}}+{\dot{\epsilon }}_{ij}^{\text{vp}}=-{\tilde{\mathbb{M}}}_{ijkl}^{\text{e}}\left({\dot{\sigma }}_{kl}-{\dot{\overline{\sigma }}}_{kl}\right)-{\tilde{\mathbb{M}}}_{ijkl}^{\text{vp}}\left({\sigma }_{kl}-{\overline{\sigma }}_{kl}\right),$$ (1)

where ${\tilde{\mathbb{M}}}^{\text{e}}$ and ${\tilde{\mathbb{M}}}^{\text{vp}}$ are the local interaction matrices for elastic and viscoplastic responses, respectively; $\mathit{\sigma }$ and $\overline{\mathit{\sigma }}$ are the local and the macroscopic stresses, respectively. Note that the upper dot (·) symbol is used to denote the rate terms. These interaction matrices are iteratively determined in order to simultaneously satisfy the force equilibrium and the strain compatibility. The above viscoplastic strain rate $\left({\dot{\mathit{\epsilon }}}^{\text{vp}}\right)$ follows the power law type constitutive response as suggested by Hutchinson (1976):

$${\dot{\epsilon }}_{ij}^{\text{vp}}={\dot{\gamma }}_0\sum _{s=1}^{24}{m}_{ij}^s{\left(\frac{m_{kl}^s{\sigma }_{kl}}{{\widehat{\tau }}_0^s}\right)}^n,$$ (2)

where m^s is the Schmid tensor for the slip system s. The exponent n and ${\dot{\mathit{\gamma }}}_0$ are the parameters associated with the strain rate sensitivity. The strain hardening is described as a function of accumulated plastic shear deformation on the individual slip systems. In the current model, the following strain hardening rule is used (Tomé et al., 1984):

$${\widehat{\tau }}_0^s={\tau }_0^s+\left({\tau }_1^s+{\theta }_1^s\text{Γ}\right)\left[1-\exp \left(-\text{Γ}\left|{\theta }_0^s/{\tau }_1^s\right|\right)\right],$$ (3)

where Γ is the cumulative plastic shear deformation on the slip systems pertaining to a crystallite. The strain hardening parameters ${\tau }_0,{\tau }_1,{\theta }_0\text{and}{\theta }_1$ are usually identified by fitting to a macroscopic experimental flow stress–strain curve. Throughout this study the hardening parameters in Table 1 were used. Since the IF steel has a body-centered cubic structure, the {110}[111] and {112}[111] slip system families were considered.

Table 1.

Material parameters [see equations (2) and (3)] used in the current study (Jeong et al., 2015).

n	${\dot{\gamma }}_0\left({\text{s}}^{-1}\right)$	${\tau }_0^s\text{(MPa)}$	${\tau }_1^s(\text{MPa})$	${\theta }_0^s(\text{MPa})$	${\theta }_1^s(\text{MPa})$
20	10−3	79.3	73.7	340	19.9

2.2. Polycrystalline aggregate consisting of discrete crystallites

The IF steel sheet is represented by a population consisting of 20 000 discrete crystallites. Each crystallite is treated as a spherical inclusion embedded in a homogeneous medium. Each of the discrete crystallites is characterized by crystallographic orientation and volume fraction, i.e. $f^c=V^c/{\sum }_c{V}^c$, where V^c is the volume for the crystallite c. Following the Bunge nomenclature, the orientation of each crystallite is represented by three sequential rotations. Each rotation sequence is quantified by one of the three Euler angles $\left({\phi }_1,\text{Φ},{\phi }_2\right)$ with respect to the orthogonal sample axes:

$$r^{(1)}=\left(\begin{array}{ccc}\cos {\phi }_1& \sin {\phi }_1& 0\\ -\sin {\phi }_1& \cos {\phi }_1& 0\\ 0& 0& 1\end{array}\right),$$ (4)

$$r^{(2)}=\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \text{Φ}& \sin \text{Φ}\\ 0& -\sin \text{Φ}& \cos \text{Φ}\end{array}\right),$$ (5)

$$r^{(3)}=\left(\begin{array}{ccc}\cos {\phi }_2& \sin {\phi }_2& 0\\ -\sin {\phi }_2& \cos {\phi }_2& 0\\ 0& 0& 1\end{array}\right).$$ (6)

Therefore the crystallographic orientation for crystallite c (i.e. r^c in the passive convention) is $r_{ij}^c=r_{ik}^{(3)}{r}_{kl}^{(2)}{r}_{lj}^{(1)}$.

As mentioned earlier, in the self-consistent crystal plasticity model the macroscopic strain (rate) and stress should be equivalent to the weighted averages of the local counterparts over the entire polycrystal aggregate. To that end, the macroscopic stress ${\overline{\sigma }}_{ij}$ should be equivalent to the volume-weighted average of the local stresses (Hutchinson, 1976):

$${\overline{\sigma }}_{ij}=\sum _c{f}^c{r}_{ik}^c{\sigma }_{kl}^c{r}_{jl}^c.$$ (7)

Throughout the current article, a pair of angle brackets is used to denote the volume-weighted average over the entire polycrystal aggregate, thus $\overline{\mathit{\sigma }}=\langle \mathit{\sigma }\rangle$. Later, a volume-weighted average over a subset of selected crystallites will be denoted with an appropriate superscript to the brackets.

3. Flow stress determination based on X-ray diffraction

3.1. Model description of properties measured by X-ray diffraction

According to Bragg’s law, the diffraction condition for a volume of lattice is determined by the orientation of the diffraction vector with respect to the orthotropic sample axes: see Fig. 1 for the illustration of material and X-ray coordinates, and the diffraction vector defined by the two independent rotation angles $(\phi ,\psi )$. During the diffraction experiment, usually a fixed set of diffraction vectors are used, whereas the crystallographic orientation may evolve as a result of plastic deformation.

Figure 1.

Coordinate systems for the sample (x^s, y^s, z^s) and X-rays (x^X, y^X, z^X) with the diffraction vector defined by two rotation angles $(\phi ,\psi )$. The X-ray diffraction vector is parallel to z^X.

Moreover, when determining whether or not a crystallite meets the diffraction condition, the reflection multiplicity factor pertaining to the associated diffraction plane should be considered. The diffraction strain associated with a certain diffraction vector (which is a function of {hkl}) can be calculated by collecting local elastic strains (ε^e,c) of only those crystallites that are oriented such that one of their diffraction vectors n^{hkl} is parallel to z^X as illustrated in Fig. 2(a). Owing to the discrete nature of the sampled population of crystallites, it is statistically more appropriate to select those crystallites within an appropriate angular range rather than those with a definite orientation. To that end, an angular window, ω, between n^{hkl} and z^X, illustrated in Fig. 2(b), is introduced. Throughout the current numerical investigation, an angular window ω of 5° is used. Note that the multiaxial experiments conducted previously (Foecke et al., 2007; Iadicola & Gnäupel-Herold, 2012; Pham et al., 2015; Jeong et al., 2015, 2016) also use some angular range to measure the diffraction peaks. The experimental value varies depending on the grain size, beam spot size and depth of penetration. The experimental angular range typically is between ±2 and ±4°. A slightly larger angular window is chosen for the model owing to the smaller number of grains in the model compared to experiment.

Figure 2.

Schematic illustrations of X-ray diffraction conditions. (a) A two-dimensional view of Fig. 1 at $\phi =0^{\circ }$. The X-ray beam path, the Bragg angle (θ^Bragg), and the normal direction and d spacing of the {hkl} plane (i.e. n^{hkl} and d^{hkl}, respectively) are shown. (b) The angular window ω is allowed when selecting the crystallites in the diffraction condition.

As described above, the diffraction strains are collected from only a subset of the total population. Furthermore, the particular grains in the subset may change as the individual crystallites follow different lattice rotations. A subgroup G was defined to specify the group of crystallites that meet the given diffraction condition for a specific plastic deformation. The subgroup G is expressed as

$$G^{\left\{hkl\right\},(\phi ,\psi )}=\left\{c\mid {\cos }^{-1}\left({\mathbf{z}}^{\text{X},(\phi ,\psi )}\cdot {\mathbf{n}}^{\{hkl\},c}\right)\le \omega \right\}.$$ (8)

Provided that a fixed set of diffraction vectors are used, only n^{hkl} evolves during the plastic deformation. The diffraction data pertaining to the selected group G are denoted using ${\langle \rangle }^G$. The total diffraction strain ${\langle {\epsilon }^{\text{t}}\rangle }^G$ of the group G is calculated using

$${\langle {\epsilon }^{\text{t}}\rangle }^G=\sum _c{f}^c{\epsilon }_{ij}^{\text{e},c}{z}_i^{\text{X}}{z}_j^{\text{X}},\text{with}c\in G.$$ (9)

Only the portion of the total diffraction strain that is proportional to the macroscopic stress (i.e. ${\langle {\epsilon }^{\text{d}}\rangle }^G$) should be used for stress analysis by subtracting the intergranular strain $\left({\langle \tilde{\epsilon }\rangle }^G\right)$ as below:

$${\langle {\epsilon }^{\text{d}}\rangle }^G={\langle {\epsilon }^{\text{t}}\rangle }^G-{\langle \tilde{\epsilon }\rangle }^G.$$ (10)

Note that the intergranular strain is also referred to as incompatibility strain (see e.g. Baczmański et al., 1994, 2008). Also in equation (9), the strains were calculated by projecting the elastic strain of each crystallite in G onto z^X instead of n^{hkl}. Consequently the contribution of the crystallites to the diffraction strain decreases as the misorientation of n^{hkl} from z^X increases. This condition may justify the use of the wide angular window introduced for the collection of diffraction strain. The intergranular strain ${\langle \tilde{\epsilon }\rangle }^G$ in equation (10) is calculated through

$${\langle \tilde{\epsilon }\rangle }^G={\langle {\epsilon }^{\text{t}}\rangle }^G-F_{ij}^G{\overline{\sigma }}_{ij},$$ (11)

where ${\overline{\sigma }}_{ij}$ is the macroscopic stress applied on the polycrystal aggregate. Note that the strain ${\langle {\epsilon }^{\text{t}}\rangle }^G$ in the above is also a function of the applied stress. The intergranular strain ${\langle \tilde{\epsilon }\rangle }^G$ was obtained at the fully unloaded state (i.e. ${\overline{\sigma }}_{ij}=0$). In equation (11), F^G denotes the generalized DECs for the associated subgroup G. In general, diffraction stress analyses assume that the diffraction strain ${\langle {\epsilon }^{\text{d}}\rangle }^G$ in equation (10) is linearly associated with the macroscopic stress through a set of DECs $\left(F_{ij}^G\right)$. Thus, $\langle \mathit{\sigma }\rangle$ can be associated with ${\langle {\epsilon }^{\text{d}}\rangle }^G$ through

$${\langle {\epsilon }^{\text{d}}\rangle }^G=F_{ij}^G\langle {\sigma }_{ij}\rangle .$$ (12)

Note that both the diffraction strain and DECs pertain to the same subgroup G with an associated dependence on {hkl} and $(\phi ,\psi )$, whereas the macroscopic stress $\langle \mathit{\sigma }\rangle$ does not. Each ij component of F^G can be measured by invoking the corresponding ij component of the macroscopic stress (Gnäupel-Herold et al., 2012; Jeong et al., 2015). Also F^G is dependent on the strain path and the amount of deformation applied to the polycrystalline aggregate as discussed by Jeong et al. (2015). Therefore the DECs described here are only truly constant for a given selection of the strain path and amount of applied deformation. Note that the EVPSC model in the current form does not produce reliable intergranular strains in comparison with experiments (Jeong et al., 2015), which may require an additional mathematical treatment as performed by Baczmański et al. (1994, 2008) for a similar model. Yet, since all necessary inputs for the stress analysis are consistently produced within the EVPSC model, the stress based on the model diffraction data should be equivalent to the volume-weighted average stress.

3.2. Diffraction stress analysis

In this section, a generalized numerical procedure to estimate the full tensorial stress states based on the diffraction data is presented. The diffraction strain ${\langle {\epsilon }^{\text{d}}\rangle }^G$ is linearly correlated with the macroscopic stress in equation (12). Thus the macroscopic stress $\langle \mathit{\sigma }\rangle$ can be obtained using the diffraction strains ${\langle {\epsilon }^{\text{d}}\rangle }^G$ provided that relevant DECs (F^G) are known.

Throughout the current study, the stress obtained by using the diffraction data (i.e. diffraction strains ${\langle {\epsilon }^{\text{d}}\rangle }^G$ and DECs F^G) is called the diffraction stress (denoted as σ^d) to distinguish it from the weighted-average stress $\langle \mathit{\sigma }\rangle$. In order to obtain the diffraction stress, the objective function below should be minimized:

$${\chi }^2=\sum _{G_i}{w}^{G_i}\left|{\langle {\epsilon }_{\text{X}}^{\text{d}}\rangle }^{G_i}-{\mathbf{F}}^{G_i}:{\tilde{\mathit{\sigma }}}^{\text{d}}\right|\text{with}{G}_i=G_1,\cdots ,G_n,$$ (13)

where ${\langle {\epsilon }_{\text{X}}^{\text{d}}\rangle }^{G_i}$ denotes the measured diffraction strain for the subgroup $G_i;{\tilde{\mathit{\sigma }}}^{\text{d}}$ denotes the guessed stress, which leads to guessed diffraction strain when multiplied by F^G_i; and the subscript n is the total number of independent subgroups under consideration. The parameter w^G_i is a weighting factor, which is suggested as the inverse of the standard deviation in the measured strains (Welzel, Ligot et al., 2005). ${\tilde{\mathit{\sigma }}}^{\text{d}}$ should approach the diffraction stress (σ^d) as χ² is minimized.

The stress under plastic flow typically increases as the polycrystal undergoes a plastic deformation due to the strain hardening [see equation (3)]. Moreover, since the crystallographic texture may change under a large plastic deformation, one should account for changes in the relevant subgroups $G^{\left\{hkl\right\},(\phi ,\psi )}$ in the diffraction condition and DECs $\left(F_{ij}^G\right)$. Therefore, in order to obtain a flow stress behavior with respect to the macroscopic plastic strain ${\overline{\mathbf{E}}}^{\text{p}}$, equation (13) is extended such that

$${\chi }^2\left({\overline{\mathbf{E}}}^{\text{p}}\right)=\sum _{G_i}{w}^{G_i}\left|{\langle {\epsilon }_{\text{X}}^{\text{d}}\rangle }^{G_i}\left({\overline{\mathbf{E}}}^{\text{p}}\right)-{\mathbf{F}}^{G_i}\left({\overline{\mathbf{E}}}^{\text{p}}\right):{\sigma }^{\text{d}}\left({\overline{\mathbf{E}}}^{\text{p}}\right)\right|.$$ (14)

With ${\chi }^2$ in equation (14) being minimized through the history of ${\overline{\mathbf{E}}}^{\text{p}}$, the flow stress–strain relationship (i.e. σ^d versus ${\overline{\mathbf{E}}}^{\text{p}}$) is obtained. Note that the evolutionary behavior of F^Gi with respect to ${\overline{\mathbf{E}}}^{\text{p}}$ varies depending on the considered strain path. The stress analysis was conducted using the DiffStress package (Jeong, 2015).

4. Uncertainties in X-ray diffraction

Following the procedure discussed in §3.2, the resulting σ^d should be equal to the weighted-average stress $\langle \mathit{\sigma }\rangle$ at any plastic level provided that precise diffraction data are used. However, the experimental diffraction data usually contain various measurement uncertainties caused by a number of factors. Moreover, some uncertainties are difficult to assess experimentally, hence the motivation of the current numerical investigation. In what follows, four different types of experimental factor that affect the measurement accuracy are considered:

1. The counting statistics error (CSE) in the measured diffraction strain attributed to the finite amount of exposure time.

2. The use of a finite range of tilting angles (ψ).

3. The use of a finite number of diffraction vectors: diffraction patterns are obtained from only a portion of a single diffraction cone (pertaining to either the {211} or {310} diffraction planes) recorded on a line detector.

4. The finite number of DEC measurements taken with respect to the increment of plastic deformation.

Considering that only a finite exposure time is available, finding an optimal balance between the number (and the range) of diffraction vectors, the diffraction exposure time for a fixed level of measurement uncertainty and a selected diffraction peak is desired. As a general rule of thumb, longer exposure times will always decrease the stress measurement error by reducing the CSE in the d-spacing measurements. However, experimentally, the minimum possible diffraction vectors (both the number and the range) and counting times are preferred.

As opposed to elastically isotropic samples where DECs can be replaced by a reduced set of a few parameters based on the sin² ψ method (Dölle, 1979; Noyan & Cohen, 1986; Welzel & Mittemeijer, 2003), the current IF steel requires the use of experimental DECs measured at various plastic deformations (Jeong et al., 2015). Experimental measurement of DECs is a tedious task, and thus they are usually measured only at a few selected plastic levels.

There are two additional factors that can contribute to the uncertainty in diffraction strain: (a) beam shape and penetration depth change as the tilting (ψ) angle increases; and (b) intensity peak decrease (or increase) due to crystallographic texture development as a result of plastic deformation. These two factors will be discussed in §5.3, where the detailed procedure of the Monte Carlo experiment is given.

In what follows, the method to characterize the above-mentioned four factors is discussed. First, the uncertainty in diffraction strain is realized by superimposing a perturbation on the diffraction strain as below:

$${\langle {\tilde{\epsilon }}^{\text{d}}\rangle }^G={\langle {\epsilon }^{\text{d}}\rangle }^G+\text{Δ}\epsilon ,$$ (15)

in which Δε is drawn from a Gaussian distribution with a mean value of zero and a specified nonzero deviation. This deviation is a function of various parameters, including CSE, that will be discussed in §5.3. The perturbed diffraction strain [denoted as ${\langle {\tilde{\epsilon }}^{\text{d}}\rangle }^G$ in equation (15)] mimics the experimentally measured strain ${\langle {\epsilon }_{\text{X}}^{\text{d}}\rangle }^G$ in equations (13) and (14).

In previous experimental work of the current authors (Jeong et al., 2015, 2016), the maximum tilting angle ψ was confined to ±35° owing to experimental constraints. In the current investigation, the two cases of −35 ≤ ψ ≤ 35° and −60 ≤ ψ ≤ 60° are compared. The influence of the finite number of diffraction vectors is studied by changing the number of tilting angles (N^ψ) sampled within the given range of ψ, and the $\phi$ rotation is fixed to 0, 45 and 90°.

In the previous studies by the current authors, experimental DECs were measured from samples that are pre-deformed to a few plastic strains, and a piecewise linear interpolation function was used. Likewise, an incomplete set of EVPSC-calculated DECs was approximated following the same method (Jeong et al., 2015). Note that, along the balanced biaxial strain path, the subgroups G_i were continuously updated in accordance with texture evolution. The incomplete measurements in DECs were emulated by using DECs calculated at a finite number of plastic deformation levels (denoted as N^DEC) over the given deformation path.

The four different types of experimental factor are para-meterized as below:

1. s^CSE denotes the standard deviation of the CSE: the main source of uncertainty due to the finite amount of exposure time.

2. The maximum tilting angle is denoted as ψ^max.

3. N^ψ denotes the finite number of tilting angles in use.

4. N^DEC stands for the finite number of DEC calculations through the deformation.

Detailed simulation procedures to estimate the propagation of the above four parameters (i.e. s^CSE, ψ^max, N^ψ and N^DEC) to the final stress measured are discussed in the following section.

5. Uncertainty estimation on diffraction stress

5.1. Problem description: in situ diffraction experiments during in-plane deformation

As mentioned earlier, the uncertainty in diffraction stress is quantified using the EVPSC model with hardening parameters for a polycrystal sample of an IF steel. The statistical population of the polycrystalline aggregate was created to approximate the as-received crystallographic texture of the IF steel. The strain hardening parameters [see equation (3) and Table 1] from the previous work (Jeong et al., 2015) were used. The polycrystalline aggregate is used to simulate a balanced biaxial (BB) strain path.

The principal axes of the BB path are coincident with the orthotropic sample axes, so that ${\overline{\sigma }}_{12}$ is negligible. Also, the components concerning the sheet’s normal direction, i.e. ${\overline{\sigma }}_{13}$, ${\overline{\sigma }}_{23}$ and ${\overline{\sigma }}_{33}$, are negligible because of the specimen geometry. Therefore, only two in-plane stress components are unknown, i.e. ${\overline{\sigma }}_{11}$ and ${\overline{\sigma }}_{22}$.

The intrinsic elastic anisotropy of the crystallites is characterized by the single-crystal elastic constants shown in Table 2. While the polycrystalline sample deforms, the diffraction strains $\left({\langle {\epsilon }^{\text{d}}\rangle }^G\right)$ are collected according to equations (10) and (11). Throughout the current study, the {211} and {310} diffraction planes are considered for two reasons: (1) Reflections with lower multiplicities (such as {110}, {200} and {222}) are excluded owing to the evolving crystallographic texture expected in the plastic deformation levels of current interest. (2) According to the limited selection of wavelengths available to the current laboratory X-ray source, both {211} and {310} planes give an adequate range of tilting angles and sufficiently intense peaks.

Table 2.

Single-crystal elastic constants ${\mathbb{E}}_{ijkl}^{\text{e}}$ of body-centered cubic iron (in GPa) used for the polycrystalline sample (Rotter & Smith, 1966).

${\mathbb{E}}_{1111}^e$	${\mathbb{E}}_{1122}^e$	${\mathbb{E}}_{1212}^{\text{e}}$
231.4	134.7	116.4

5.2. Monte Carlo virtual experiments for uncertainty estimation

Owing to the random nature of the perturbation for the diffraction strain, a sufficiently large number of ensembles should be used in order to create a statistically representative estimation. In this study, 400 ensembles are created by repeating the perturbation following equation (15). In the ideal condition, the resulting stress obtained by equation (14) should be equivalent to the weighted-average stress over the entire polycrystal aggregate $\langle \mathit{\sigma }\rangle$. Therefore the difference between the diffraction stress and the weighted-average stress is a reasonable error estimator:

$${\mathit{\sigma }}^{\text{E}}=\langle \mathit{\sigma }\rangle -{\mathit{\sigma }}^{\text{d}}.$$ (16)

Note that the error estimator (σ^E) is a tensor. In the current study, only the balanced biaxial condition is investigated, where the two unknown stress components are not significantly different from each other. Therefore, the maximum norm of the error estimator is used, which is defined as

$$\parallel \mathbf{A}{\parallel }_{\infty }:=\max \left(\left|A_{ij}\right|\right),\text{where}i,j=1,2,3.$$ (17)

By using equation (17), the error estimator reduces to a scalar value:

$$\text{Err}[\text{\%}]=\mathrm{sgn}\frac{{\Vert {\sigma }^{\text{E}}\Vert }_{\infty }}{{\Vert {\sigma }^{\text{d}}\Vert }_{\infty }}\times 100.$$ (18)

where sgn denotes the sign of the maximum component chosen for the maximum norm. Then, the mean (μ^Err) and the standard deviation (S^Err) in the population of Err values sampled from the 400 ensembles are used to determine the uncertainties propagated to the diffraction stress σ^d. Note that the mean value μ^Err and the standard deviation S^Err relate to the bias and the precision in the diffraction stress values, respectively.

5.3. Application of Monte Carlo results to reflection X-ray method

In this section, an application specific to the reflection X-ray equipment used by Jeong et al. (2015, 2016) is discussed. The experimental time required for in situ X-ray diffraction for flow stress measurement can be estimated by the number of diffraction vectors ($N^{\phi }$ and $N^{\psi }$) and the diffraction exposure time at each diffraction vector $(\phi ,\psi )$. Therefore, the total amount of time can be expressed as

$$\text{Total experimental time}=N^{\psi }{N}^{\phi }{t}^{\exp },$$ (19)

where t^exp denotes the amount of time required for a single exposure at each diffraction vector $(\phi ,\psi )$. In the current investigation $N^{\phi }=3$ is used, so that the total experimental time reduces to a function of N^ψ and t^exp. The uncertainty in diffraction strain due to counting statistics error is assumed to follow the Poisson distribution such that

$$s^{\text{CSE}}=A/{\left(t^{\exp }\right)}^{1/2},$$ (20)

where A is a proportionality factor. Assuming that an exposure time of $\overline{t}$ is required when s^CSE = 50 μm m⁻¹, the factor A is expressed as

$$A=50{\overline{t}}^{1/2}\left[\mu {\text{m m}}^{-1}\right].$$ (21)

Substituting equation (21) into equation (20) gives

$$t^{\exp }=\overline{t}{\left(\frac{s^{\text{CSE}}}{50\left[\mathrm{\mu }{\text{m m}}^{-1}\right]}\right)}^2.$$ (22)

Meanwhile, the standard deviation (s^total) of the Gaussian distribution associated with the perturbation Δε in equation (15) is extended to account for additional factors as below:

$$s^{\text{total}}=\text{Θ}{\left[{\left(s^{\text{CSE}}\right)}^2/{\eta }^G\right]}^{1/2},$$ (23)

where $\text{Θ}$ and ${\eta }^G$ account for the geometric distortion of the projected X-ray beam on the specimen surface and the changes in diffraction intensity due to crystallographic texture, respectively. The parameter η^G is approximated as the ratio between the intensity of the uniform distribution (denoted as f_u) and that of the subjected polycrystalline aggregate at a certain plastic deformation $\left({\langle f^c\rangle }^G\right)$ as below:

$${\eta }^G={\langle f^c\rangle }^G/f_u.$$ (24)

As a result, s^total becomes a function of {hkl} and the diffraction vector $(\phi ,\psi )$. The geometric factor $\text{Θ}$ is a function of the tilting angle $\psi$ and the Bragg angle (θ^Bragg):

$$\text{Θ}=\frac{1}{1-\tan \left|\psi \right|\cot {\theta }^{\text{Bragg}}}.$$ (25)

Note that the total uncertainty in equation (23) reduces to s^CSE when the subjected material has a uniform texture, i.e. ${\eta }^G=1$ and $\psi =0^{\circ }$ (thus no tilting). The value of s^total is used to describe the Gaussian distribution that is selected for Δε [see equation (15)].

6. Results

Fig. 3 shows the DECs and volume fractions $\left({\langle f^c\rangle }^G\right)$ of the {211} and {310} planes that are calculated at $\phi =0^{\circ }$ after the equivalent strain of ${\overline{\text{E}}}^{\text{eq}}=0.5$. The gray vertical lines correspond to ψ = ±35°, which are the maximum tilting angles allowed in the experimental system considered by Jeong et al. (2015, 2016). Fig. 4 schematically illustrates the influence of counting statistics error. Note that under an ideal case without any flaw in the experimental data all three types of elastic strain shown in the figure (i.e. ${\langle {\tilde{\epsilon }}^{\text{d}}\rangle }^G,\mathbf{F}:\langle {\sigma }^{\text{c}}\rangle$ and $\mathbf{F}:\langle {\sigma }^{\text{d}}\rangle$) should be mutually equivalent. That is confirmed in Fig. 4(a) and its resulting flow stress shown in Fig. 4(c), where the diffraction stress and weighted-average stress $\langle {\sigma }^c\rangle$ are equivalent. In contrast, when the diffraction strains $\left({\langle {\tilde{\epsilon }}^{\text{d}}\rangle }^G\right)$ are perturbed as shown in Fig. 4(b), the diffraction stress may deviate from the weighted-average stress as shown in Fig. 4(d), which provides the basis of the uncertainty estimator in equation (16). The shaded area in Fig. 4(b) represents the perturbation in the diffraction strain: the vertical spread therein corresponds to ${\langle {\epsilon }^{\text{d}}\rangle }^G\pm 2s^{\text{total}}$ Note that the spread caused by s^total varies with respect to the ψ angle, which results from equation (23) where the magnitude of the perturbation is dependent on the ψ angle and the crystallographic texture.

Figure 3.

Diffraction elastic constants (F₁₁ component) and volume fractions of the {211} and {310} planes at $\phi =0^{\circ }$ after a strain of ${\overline{E}}^{\text{eq}}=0.5$. Vertical broken lines in gray correspond to ψ = ±35°, which are the maximum tilting angles considered by Jeong et al. (2015, 2016).

Figure 4.

Illustration of stress analysis with and without perturbation: (a) without perturbation of diffraction strain; (b) with perturbation using one standard deviation of 25 μ strain [see equation (15)]. The gray area in (b) denotes the range of ${\langle {\epsilon }^{\text{d}}\rangle }^G\pm 2s^{\text{total}}$ [see equations (15) and (23)]. (c) and (d) show the resulting diffraction stresses associated with (a) and (b), respectively, in comparison with the weighted-average stress.

In Fig. 5, the results from Monte Carlo experiments that are conducted over the space of (s^CSE; N^DEC; N^ψ) are visualized for the {211} planes. Each cell in the figure corresponds to a particular set of (N^DEC; N^ψ). In each cell of the figure in the top row (i.e. N^ψ = 3), the levels of strain at which DECs are acquired are denoted by crosses. Three different levels of s^CSE = 10, 25 and 50 μm m⁻¹ are represented by the three different grayscale shades (from dark to light, respectively). In the figures, the error in diffraction stress [equation (16)] is plotted as a function of plastic strain $\left({\overline{E}}^{\text{eq}}\right)$. The upper and lower boundaries of each shaded region denote μ^Err + S^Err and μ^Err − S^Err, respectively. Note that the precision and bias in the measurement correspond to the spread of the region and the systematic deviation from 0, respectively. For the results presented in Fig. 5 the tilting angle ψ was confined to ±60° and the weighting factor w^G_i was fixed as 1. The template shown in Fig. 5 was also used for the other conditions (Figs. 6–10) considered in the Monte Carlo simulations (see Table 3).

Figure 5.

Uncertainty in diffraction stress based on the {211} planes estimated by the Monte Carlo experiment. The three different shades in the grayscale (from dark to light) correspond to counting statistics errors of s^CSE = 10, 25 and 50 μm m⁻¹, respectively. The maximum tilting angle ψ is bounded by ±60° and w^Gi = 1 in equation (14).

Figure 6.

The Monte Carlo results obtained for the {211} planes with ${\psi }^{\max }=\pm {60}^{\circ },w^{G_i}=1/s^{\text{total}}$.

Figure 10.

The Monte Carlo results obtained for the {310} planes with ${\psi }^{\max }=\pm {35}^{\circ },w^{G_i}=f^{G_i}$.

Table 3.

Conditions considered in the Monte Carlo experiments and the corresponding figures presented.

		{211 planes	{310} planes
${\psi }^{\max }=\pm {60}^{\circ }$	$w^{G_i}=1$	Fig. 5	–
	$w^{G_i}=1/s^{\text{total}}$	Fig. 6	–
	$w^{G_i}=f^{G_i}$	Fig. 7	–
${\psi }^{\max }=\pm {35}^{\circ }$	$w^{G_i}=1$	Fig. 8	–
	$w^{G_i}=f^{G_i}$	Fig. 9	Fig. 10

To better quantify the uncertainty as a function of only N^DEC and N^ψ, the respective arithmetic averages for the bias $(\overline{B})$ and the precision $(\overline{P})$ over all plastic levels are estimated. These averaged precision and bias values are defined as below:

$$\overline{P}=1/k\sum _i^k{S}_i^{\text{Err}},$$ (26)

$$\overline{B}=1/k\sum _i^k\left|{\mu }_i^{\text{Err}}\right|,$$ (27)

where k is the number of levels of plastic strain investigated. $\overline{P}$ and $\overline{B}$ are illustrated in Figs. 11 and 12, respectively, for both {211} and {310} planes.

Figure 11.

Precision maps of (N^ψ, N^DEC).

Figure 12.

Bias maps of (N^ψ, N^DEC).

The precision $\overline{P}$ can be further investigated to find an optimal $N^{\psi }$ value considering the associated experimental time as estimated in §5.3. For example, Fig. 13 illustrates the total experimental time [as defined in equation (19)], which is mapped in the space of (N^ψ, s^CSE) with N^DEC = 3 and ψ^max = ±35° for the {211} planes. In the figure, the experimental time is quantified as multiples of $\overline{t}$, that is the unit exposure time required for s^CSE = 50 μm m⁻¹ as discussed earlier in §5.3.

Figure 13.

Time map of (N^ψ, s^CSE, N^DEC = 3) for the {211} plane. The values in grid coordinates of (N^ψ, N^CSE) denote the associated arithmetic average of precision $\overline{P}$.

7. Discussion

It is important that the EVPSC model can address experimental constraints such as the counting statistics error, the finite number and the range of diffraction strain measurements, and incomplete measurement of the evolving DECs, as well as a range of material characteristics including crystallographic texture, anisotropy in DECs and its evolution with respect to plastic deformation. Since some parameters are coupled it is difficult to obtain an analytical solution to calculate the uncertainty in the current X-ray stress analysis. Moreover, the bias in the measurement uncertainty is difficult to obtain without knowing the stress present in the deforming sample. Therefore, the Monte Carlo method was adapted to consider various factors that may affect the measurement accuracy. The EVPSC model captures the anisotropy of DECs and texture as shown in Fig. 3, where two distinctive behaviors pertaining to the {211} and {310} planes are illustrated. The F₁₁ component of the {211} planes shows the typical undulation at around ψ = 29–32°, while that of {310} is nearly linear against sin² ψ.

The results from the Monte Carlo experiments are displayed in Figs. 5–10, and provide an overview of the uncertainty as a function of the experimental parameters, i.e. ψ^max, N^ψ, N^DEC and s^CSE, as well as plastic deformation $\left({\overline{E}}^{\text{eq}}\right)$. Figs. 5, 6 and 7 demonstrate that the uncertainty significantly reduces when using the weighting factor. Both w^G_i = 1/s^total and w^G_i = f^G_i (in Figs. 6 and 7, respectively) led to a significant reduction in precision values (i.e. s^Err) in comparison with the case of w^G_i = 1 (Fig. 5). The observation that w^G_i = 1/s^total and w^G_i = f^G_i conditions lead to virtually equivalent uncertainty is useful since the experimental measurement of f^Gi is much easier than that of 1 = s^total.

Figure 7.

The Monte Carlo results obtained for the {211} planes with ${\psi }^{\max }=\pm {60}^{\circ },w^{G_i}=f^{G_i}$.

Comparison between Figs. 7 and 9 shows that the uncertainty is also sensitive to the range of ψ angles in use. When N^ψ ≤ 5, the case with ψ^max = ±35° (i.e. Fig. 9) leads to more pronounced bias (i.e. μ^Err) in comparison with ψ^max = ±60° (Fig. 7). This observation confirms that the experimental condition of N^ψ = 13 used in the experimental studies of the current authors (Jeong et al., 2015, 2016) provides a level of uncertainty less than ±4% even though the tilting angle was constrained to ψ^max = ±35° therein.

Figure 9.

The Monte Carlo results obtained for the {211} planes with ${\psi }^{\max }=\pm {35}^{\circ },w^{G_i}=f^{G_i}$.

Comparison between Figs. 9 and 10 shows that the measurement uncertainty is also significantly sensitive to the choice of diffraction plane {hkl}. The dependency on {hkl} is summarized as below:

1. For N^ψ ≥ 5 and N^DEC = 2, the bias μ^Err is commonly high for both {211} and {310} planes but the trend significantly differs: underestimation (Err > 0) for {211} planes but over-estimation (Err < 0) for {310} planes.

2. When N^ψ = 3 (the first rows of Figs. 9 and 10), the s^Err value of the {310} planes is much higher than that of the {211} planes. This is explained by the fact that the volume fractions pertaining to the {310} planes at those selected ψ angles (i.e. −35, 0 and +35°) are lower than those of the {211} planes, which is accounted for in equation (24) (see Fig. 3).

3. Nevertheless, where N^ψ ≥ 5, the difference in terms of precision between the two planes generally diminishes.

In Figs. 9 and 10, fixed values of N^ψ and N^DEC were used regardless of the plastic deformation. The results shown in the figures indicate that changes in the experimental parameters specific to the level of plastic deformation could lead to improvements in the measurement quality. For example, in the case of the {211} planes (Fig. 9), a combination of N^ψ = 9 and N^DEC = 3 led to a bias level lower than ±3% at ${\overline{E}}^{\text{eq}}\ge 0.4$. However, for ${\overline{E}}^{\text{eq}}<0.4$, the results recommend increasing N^DEC in order to obtain the same (or lower) level of uncertainty.

The results shown in Fig. 11(a) indicate that the average precision $\overline{P}$ is governed mainly by s^CSE and N^ψ, with an exception only for the case of N^ψ= 3 and N^DEC = 2 for the {211} planes. This exceptional behavior at N^ψ = 3 and N^DEC = 2 is probably due to the numerical instability over the minimization in equation (14) when insufficient data are available (only three data points at each $\phi$ rotation) with a relatively large scatter (s^CSE = 50 μm m⁻¹). Likewise, Fig. 11(b) shows that the {310} planes follow the same trends: precision $\overline{P}$ depends mainly on s^CSE and N^ψ; and N^ψ = 3 leads to exceptionally large $\overline{P}$ values. For both planes, it is commonly found that the decrease in s^CSE improves the measurement precision regardless of N^DEC and N^ψ. Such improvement in precision induced by decreasing s^CSE becomes more dramatic as N^ψ decreases. On the other hand, when s^CSE is as small as 10 μm m⁻¹, the precision stays below 3% for the {211} planes regardless of the (N^ψ, N^DEC) combinations under current consideration (see Fig. 11a).

Fig. 12 demonstrates that both N^ψ and N^DEC have a significant influence on the bias $\overline{B}$, whereas the influence of s^CSE is not noticeable. Nevertheless, when N^DEC ≥ 3 and N^ψ ≥ 5, the bias significantly reduces to the order of 3% (or lower) for both {211} and {310} planes.

The collection of results from Monte Carlo experiments shown in Fig. 13 can be used to choose a certain N^ψ value considering the experimental time. Depending on the intrinsic level of s^CSE pertaining to a choice of {hkl}, one can evaluate which N^ψ value is most efficient from the results shown in Fig. 13. For example, the coordinates denoted as circles in Fig. 13 correspond to the case of $\overline{P}\le 3\text{\%}$ and total experimental time less than $40\overline{t}$. Considering that an N^ψ value as low as 3 might lead to significant bias that exceeds 10% (see Fig. 12a), 5 ≤ N^ψ ≤ 17 seems a reasonable choice.

8. Conclusions

A stress measurement technique based on diffraction analysis developed for thin metal sheets under large plastic deformation was evaluated. Particularly, the uncertainty of the measurement technique has been quantified by perturbing the elastic diffraction strain and diffraction conditions (such as the number and the range of diffraction vectors, and the level of counting statistics error) in a Monte Carlo virtual experiment for an IF steel. Additionally, the peak intensity changes due to crystallographic texture and distortion of the X-ray beam shape projected on the specimen surface were considered. The method was applied to an IF steel under balanced biaxial deformation.

The current systematic parametric study successfully demonstrated that there are two types of uncertainty pertaining to the measurement technique: (1) bias and (2) precision. The bias was mainly due to either the insufficient number of DEC measurements (N^DEC) or the number of tilting angles to probe (N^ψ). On the other hand, the precision was attributed mainly to the counting statistics error and N^ψ. The correct weighting factor seems necessary in the objective function used for stress analysis to greatly reduce the uncertainty. The three observations below were commonly found for both {211} and {310} planes with the tilting angle confined to −35 < ψ < 35°.

1. To reduce the bias below 3%, the experimental conditions should be such that N^DEC ≥ 3 and N^ψ ≥ 5.

2. The influence of the level of counting statistics error is commonly manifested such that the increase in CSE leads to a less precise measurement.

3. The more tilting angles that are probed, the more precise the measurement is. However, such improvement becomes less pronounced when N^ψ ≥ 11 under the conditions investigated in this study.

The results presented in this study led to a better understanding of uncertainties in the diffraction stress measurement technique. The current framework for uncertainty estimation can be applied to other material systems under arbitrary loading paths. Moreover, the results of Monte Carlo experiments were used to assess an optimal diffraction condition to reduce the uncertainty in the diffraction stress considering that only a finite amount of total experimental time is allowed.

Figure 8.

The Monte Carlo results obtained for the {211} planes with ${\psi }^{\max }=\pm {35}^{\circ },w^{G_i}=1$.