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# Decoding stress-strain diagrams to identify the evolution of crystal defects and their obstacle strengths using constitutive relations analyses

S Saimoto, M R Langille

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## To cite this version:

S Saimoto, M R Langille. Decoding stress-strain diagrams to identify the evolution of crystal defects and their obstacle strengths using constitutive relations analyses. IOP Conference Series: Materials Science and Engineering, 2019, 580, 10.1088/1757-899x/580/1/012043. [hal-04833490](https://hal.science/hal-04833490v1)

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Submitted on 12 Dec 2024

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# Decoding stress-strain diagrams to identify the evolution of crystal defects and their obstacle strengths using constitutive relations analyses

To cite this article: S Saimoto and M R Langille 2019 IOP Conf. Ser.: Mater. Sci. Eng. 580 012043

View the [article online](#) for updates and enhancements.

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# Decoding stress-strain diagrams to identify the evolution of crystal defects and their obstacle strengths using constitutive relations analyses

S Saimoto¹*, M R Langille²

1. Mechanical and Materials Engineering, Queen's University, Kingston, Canada

2. Univ. Grenoble Alpes, CNRS, Grenoble INP, SIMaP, F-38000 Grenoble, France

* E-mail: shigeo.saimoto@queensu.ca

**Abstract.** Recent advances in microstructural characterization from the mesoscopic and microscopic scales to the nanometric and atomic scale have revealed precise details regarding the formation of nanoparticles responsible for increased strength of alloys. The size and distribution of such features have been modelled ad hoc with limited success to replicate the work-hardening behaviour. The missing issue is that of including also the evolution of the crystal defects that arise as a result of plastic flow, such as the volume fraction of nano-voids and rotated lattice structures as observed by small angle X-ray scattering (SAXS). The constitutive relation analyses (CRA) approach assumes that all shape change is due to dislocation motion, and through use of the Taylor slip model, a functional relation has been formulated to replicate the measured curve in terms of shear stress and shear strain. The CRA parameters can be correlated to the evolving deformation products by quantitatively deriving a relation for the obstacle strength factor, $\alpha$, using the strain dependence of the mean slip distance, $\lambda$. Hence, the evolution of different crystal defects as straining proceeds can be assessed.

## 1. Introduction

The strength and ductility of an alloy can be defined by the yield stress, which corresponds to a dislocation-line force sufficient to cause inelastic motion of dislocations in the crystal matrix comprised of solutes and dispersed particles, and also by the subsequent strain to attain the ultimate tensile strength (UTS), which is determined by the nature of work-hardening. The main focus of study over many decades has been to characterize the size, shape and distribution of dispersed particles and their capacity to resist shearing. This depends on whether the particles are coherent or incoherent with the matrix crystal, which in-turn is dependent on their size and shape. Modern advances in electron imaging, together with the ability to characterize solute elements by use of atomic probe tomography have enabled

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Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.
the characterization of particles from the microscopic scale down to the atomistic scale, as demonstrated in many reviews [1, 2, 3]. The current challenge is to correlate the statistical response of the defined microstructure to the mechanical properties. The approaches used to perform this task generally rely on application of the additive rule, whereby the various strengthening mechanisms are summed and superposed with a work-hardening relation for strain evolution [4, 5]. The issue with this approach is that the defect species that are generated during plastic flow may change, such that the evolution of weaker defects results in a lower work-hardening coefficient and hence a reduction in the strain to attain the Considère relation, i.e., a lower ductility. A different approach is to reverse this process, whereby a functional constitutive relation is developed to replicate the measured stress-strain (σ - ε) curve, and from the parameters characterizing the work-hardening process to identify the evolving obstacles responsible for hindering dislocation motion over the applicable strain-range. The constitutive relation analyses (CRA) approach, developed over the last decade, has been applied to decode the σ - ε diagram, and the present work extends the analyses to determine the obstacle strength factor, α, of the evolving defect structures as function of strain. Thus the analyses elucidate the defect types and their generation process.

## 2. Theory: Functional constitutive relation, mean slip distance and the derivation of $\alpha_D$

The basis of the Saimoto-Van Houtte constitutive relation derivation is calculation of the micro-slip areas required to comply with the imposed strain using Taylor slip analyses [6]. Each slip patch that is bounded by dislocation length can be calculated using a mean slip distance $\lambda$ together with its bounded area. Thus, the total energy to create the total dislocation lengths can be equated with the retained or stored dislocation lengths calculated using the Livingston relation [7]. For this the flow shear stress $\tau = \alpha \mu b \rho^{1/2}$, wherein $\alpha$ is the obstacle strength factor, $\mu$ is the shear modulus, $b$ is the Burgers vector, and $\rho = 1/\ell^2$, where $\ell$ is the forest dislocation spacing. Previous statistical analysis has shown that the intersection density $\rho$ (number per unit area) is related to the total line length, $\rho_t$, as $\rho_t = 2\rho$ (length per unit volume) [8]. Since the stored energy is typically less than 10% of the expended energy required for the specific shape change, an annihilation factor $A$ was introduced to balance the equation. The resulting model relation correlating the shear stress $\tau$ to shear strain $\gamma$ was in the power-law form earlier empirically determined by Hollomon [9]. However for this model at least two curves are required to replicate the measured data after conversion of tensile ($\sigma - \varepsilon$) data to that in terms of shear strain ($\tau - \gamma$). For this reason the intersection of the first curve, characterized by coefficient $\beta_1$, with that of $\beta_2$ is disjointed, with the transition strain interval at the intersection point of $\tau_3$ and $\gamma_3$. An iterative fitting process can be used to find the two fitting curves with least error such that the transition zone is minimized. The following analyses will show this disruption is due to modelling limitations. The fact that this model captures the essential aspects of plastic flow was demonstrated and correlated to the kinetics of thermally activated plastic flow [10]. The final result was that normalization of the ($\tau - \gamma$) curve using $\tau_3$ and $\gamma_3$ for identical specimens tested at the same strain rate at different temperatures resulted in the data collapsing onto a master curve independent of temperature.

The mean slip distance $\lambda$ (equation (1)), which was earlier defined for slip-based work-hardening models by Nabarro et al. [8] can be precisely determined for each curve by using a calibration factor $\phi$ [11] derived from $\theta_{cal} = \partial\tau/\partial\gamma$ at $\varepsilon_{0.2\%}$, given as $1/\phi = (\mu/\theta_{cal})/2\alpha$. Many trials have shown that use of the engineering standard proof strain of 0.2 % is convenient. However if a yield point phenomenon exists, the fitted data points are determined from near the ultimate tensile stress (UTS) back to the yield point elongation (YPE) position, and this truncated data set is used for the fitting. Following this procedure, the $\beta_1$ curve is back-extrapolated to 0.2 % to determine the calibration term. Hence the newly defined yield stress $\sigma_0^{final}$ correlates to the friction stress for plastic flow beyond the YPE position. Since the fitting is performed simultaneously for both curves, the defined value of $\sigma_0^{final}$ is common to both fit curves.

$$ \lambda = \frac{\phi b \mu^2}{2 \tau \partial \tau / \partial \gamma |_{\tau, \dot{\gamma}}} \qquad (1) $$
The derived constitutive relation given as equation (2) with fit parameters, $\sigma_0^{final}$, P/A, $\beta$ and $C_1$ from $\lambda = C_1\tau^\beta$. The A-factor accounts for the athermal dynamic annihilation of dislocations during plastic flow to result in reduced stored work. After converting the $(\sigma - \varepsilon)$ polynomial fits to $\tau = (\sigma - \sigma_0^{final})/M$ and $\gamma = M\varepsilon$, in which M is the Taylor factor, two loci, $\beta_1$ and $\beta_2$, are created in combination with the back-extrapolated value of $\sigma_0^{final}$. From the value of the term in brackets in equation (2), given in units of MPa$^{(2+\beta)}$, P/A is determined, wherein $C_1$ is derived from a plot of log $\lambda$ versus log $\tau$. Note that $\beta$ values are derived from the fitting process and not from the log $\lambda$ - log $\tau$ plot. In this relation, P is a ratio (y/x) derived by assuming the new segment lengths of expanding dislocations is given as $y\lambda$ and the encircling swept area as $x\lambda^2$

$$ \tau = \left[ \left( \frac{P}{4A} \right) (2 + \beta) (\alpha \mu b)^2 \frac{1}{c_1 b} \right]^{\frac{1}{2+\beta}} \gamma^{\frac{1}{2+\beta}} \quad (2) $$

In order to derive a microstructure-based relation, equation (2) is modified by substituting $\lambda = C_1 \tau^\beta$:

$$ \tau^2 = \left[ \left( \frac{P}{4A} \right) (2 + \beta) (\alpha \mu b)^2 \frac{1}{\lambda b} \right] \gamma \quad (3) $$

This $\lambda$-based relation as for equation (2) encompasses modelling parameters P/A and $\alpha^2$. However the model only requires that the product (P/A) $\alpha^2$ be kept constant. In cases where $\alpha$ is a well-established value it saves computing time to assume a value, which can be checked after completion of the constitutive relations analyses (CRA) described above. However the operative strength factor denoted as $\alpha_D$ correlated to the model relation can be derived point by point, as follows. Differentiation of equation (3) results in equation (4), into which equation (1) is substituted. Simplifying results in equation (5).

$$ 2\tau \frac{\partial \tau}{\partial \gamma} = \left[ \left( \frac{P}{4A} \right) (2 + \beta) (\alpha \mu b)^2 \frac{1}{b} \right] \frac{\partial (\frac{\gamma}{\lambda})}{\partial \gamma} \quad (4) $$

$$ \alpha_D = \left( \frac{\mu}{2\theta_{cal}} \right) \left[ \left( \frac{P}{4A} \right) \alpha^2 (2 + \beta) \right] \left( 1 - \frac{\partial \ln \lambda}{\partial \ln \gamma} \right)_{\text{any } \gamma} \quad (5) $$

The limitation to the CRA modelling method becomes apparent as follows. The varying term in equation (5) is given as $(1 - \frac{\partial \ln \lambda}{\partial \ln \gamma})$, which can be derived by differentiation of the natural logarithm of equation (3) with respect to $\partial \ln \gamma$, whereby $(1 - \frac{\partial \ln \lambda}{\partial \ln \gamma}) = 2 (\frac{\partial \ln \tau}{\partial \ln \gamma}) = (\frac{2}{2+\beta})$. The factor $(\frac{2}{2+\beta})$ results from differentiating equation (2). Hence CRA modelling results in $\alpha_D$ being constant with strain, which is consistent with the modelling procedure where $\alpha$ was kept constant at 0.4 for aluminum (Al). The cause of this discrepancy is that the relation $\lambda = C_1 \tau^\beta$ does not exactly replicate log $\lambda$ versus log $\tau$ plot, which shows that the fitted curve $\beta$ only approximates the measured values and smooths out undulations. For mesoscale modelling this practice is acceptable but for detailed examination of the microstructure, $(\frac{\partial \ln \lambda}{\partial \ln \gamma})$ is required at any given $\gamma$ and is derived from $\ln \lambda$ versus $\ln \gamma$ plot, resulting in $\alpha_D$ versus $\gamma$. Moreover a new criterion for initiation of incipient diffuse necking is predicted, namely if $(1 - \frac{\partial \ln \lambda}{\partial \ln \gamma})$ becomes zero, the effective $\alpha$ becomes zero, implying that laminar slip initiates strain localization. The following data analyses show that this destabilizing process is incurred as the Considere condition is approached.

The first term in equation 5 ($\mu/2 \theta_{cal}$) is derived from CRA as $\alpha/\phi = (\mu/\theta_{cal})/2$, where $\alpha$ is the prior established value. Whether this choice is consistent with the model can be checked after completion of CRA process as described above. This is a short-cut process since permitting $\alpha$ to become a free parameter during the fitting procedure entails much longer computing time but the resulting $\{ (P/A) \alpha^2 \}$ remains the same. On the other hand, $\lambda$ is determined point by point at any given $\gamma$, but $\partial\lambda/\lambda$ is independent of $\alpha$ due to cancellation. Thus, $\alpha_D$ is dependent on the change in $(\tau \partial\tau/\partial\gamma)$ factor from the $\tau - \gamma$ curve and the calibration slope, which may lead to ambiguity if a yield phenomenon exists. However the slope at back-extrapolated $\varepsilon_{0.2\%}$ is based on the slip model and not empirical, and hence should encompass the high strain deformation mode. Hence CRA determines the value of $\sigma_0^{final}$ which separates the yield phenomenon from the friction stress applicable beyond the YPE.
### 3. Experimental

The prime data acquisition method used for the CRA is standard ASTM tensile testing using digitalized recording of load and extension performed on any universal testing system. In the current case, testing to failure was performed at 5 x 10⁻⁴ s⁻¹ using a screw-driven Instron 5969 machine equipped with a 50 kN load cell. The strain measurements were recorded using a clip-on 50.8 mm 2630-115 extensometer as previously described [12]. All mechanical tests were performed at 22 °C. The specimen was prepared as previously reported from extrusion channels of AA6063 in the shape of 3-cell trapezoid with overall dimensions of 78 mm x 143 mm x 66 mm with gauge size of 50 mm x 12 mm x 2mm. The machined surfaces were finished using 600-grit SiC paper and heat treated at 185 °C according to industrial practice for 5 h (T6) to attain optimum strength after water quenching (WQ) from 540 °C. Table 1 lists the composition of the extrusions as specified by the supplier. A detailed description of the data acquisition for small angle scattering (SAXS), performed at the Cornell High Energy Synchrotron Source G1 beamline, and the transmission electron microscopy (TEM) studies at McMaster University, are reported elsewhere [12, 13].

<table><caption>**Table 1.** As-received AA6063 composition in weight %</caption><thead><tr><th>Al</th><th>Si</th><th>Mg</th><th>Cu</th><th>Fe</th><th>Mn</th><th>Ti</th></tr></thead><tbody><tr><td>Bal.</td><td>0.4</td><td>0.49</td><td>0.01</td><td>0.16</td><td>0.029</td><td>0.01</td></tr></tbody></table>

### 4. Result and discussion

#### 4.1 Decoding of optimum aged AA6063 extrusion specimens (5 h at 185°C)

The CRA procedure separates the yield phenomenon from the work-hardening beyond YPE such that $\sigma_0^{final}$ defines the friction stress primarily due to the dispersed aged products that underwent shearing during yielding and subsequent straining as shown in figure 1 [12].

<figure><img src="image_6.png" /><figcaption>**Figure 1.** Stress-strain diagram of extruded and optimum aged AA6063-T6 determined at 296 K and strain rate of 5 x 10⁻⁴ s⁻¹ from [12]. The original data (dashed curve) is optimum fitted using CRA to form the $\beta_1$ and $\beta_2$ loci. The back-extrapolated yield stress common to both modelled loci is $\sigma_0^{final}$.</figcaption></figure>

As modelled by Foreman and Makin [14], shearing of needle-shaped nano-precipitates which align with the cube axis of the matrix, follows a percolation process whereby subsequent hardening is due to the increase in forest dislocations due to multiple slip such that more or less homogeneous microstructure results at YPE. However if other larger unshearable micro-meter (µm) sized particles exist, at a specific shear stress due to stress concentration in the adjacent region, new dislocation arrays arise to maintain strain compatibility by forming rotated structures, as modelled by Brown [15]. This model predicts a parabolic behaviour that can be observed using CRA, and from the fit parameters in table 2 the size of the particles can be assessed [12]. The formation of a rotated lattice structure changes after some strain due to changes in local Schmid factor. Thus a counter rotation can occur resulting in a region of duplex slip which generates various deformation debris, such as dislocation loops and vacancies, resulting in
the change in work-hardening coefficient that is represented by the third deformation stage in figure 1. The formation of rotated structures has been shown to occur using SAXS and the nucleation of nano-voids by vacancies has also been observed. The measured volume fraction of nano-voids $C_{void}$ is comparable to that of the calculated volume fraction of vacancies $C_V$ using the CRA parameters [13]. To extend this decoding process, one way would be to identify the obstacle strength factor in each of the three deformation stages as presented in the next section.

<table><caption>**Table 2:** Input and fit parameters from CRA of AA6063 aged at 185 °C for 5 h</caption><thead><tr><th>Treatment</th><th>α*</th><th>μ (MPa)</th><th>P/A</th><th>σ<sub>o</sub><sup>final</sup> (MPa)</th><th>σ<sub>0.2%</sub> (MPa)</th><th>β<sub>1</sub> (μm)</th><th>β<sub>2</sub> (nm)</th><th>τ<sub>3</sub> (MPa)</th><th>γ<sub>3</sub></th></tr></thead><tbody><tr><td>AQ</td><td>0.4</td><td>32012</td><td>0.069</td><td>30.4</td><td>41.7</td><td>-0.27</td><td>0.95</td><td>1.10</td><td>21.3</td><td>24.8</td><td>0.175</td></tr><tr><td>T6 As-Rec</td><td>0.4</td><td>32012</td><td>0.084</td><td>187.4</td><td>205.1</td><td>0.00</td><td>0.97</td><td>0.91</td><td>110.1</td><td>16.0</td><td>0.126</td></tr><tr><td>T6 Control</td><td>0.4</td><td>32012</td><td>0.072</td><td>195.6</td><td>211.1</td><td>0.00</td><td>1.23</td><td>0.49</td><td>443.2</td><td>12.3</td><td>0.109</td></tr></tbody></table>

## 4.2 Identification of microstructure components using the derived alpha factor, α<sub>D</sub>

In order to determine α<sub>D</sub> according to equation (5), a plot of ln λ versus ln γ is shown in figure 2, from which α<sub>D</sub> versus γ in figure 3 can be derived. The α<sub>D</sub>-profile most similar to that of pure Al at 78 K [16] is that for WQ specimen in figure 3, which shows that α<sub>D</sub> decreases from near 0.4 to below 0.2 near γ<sub>3</sub>, where an abrupt change occurs due to the switch in the fit-locus as described earlier. Thus this transition strain interval should be ignored. Beyond γ<sub>3</sub>, the profile dips to near zero and increases again to near 0.4. It must be kept in mind that for a two obstacle model τ = τ<sub>For</sub> + τ<sub>Deb</sub>, wherein the debris contribution is only about 1/9 of the forest one. However since α<sub>D</sub> is primarily a measure of the most rapidly increasing defect density, the low value for vacancies and nano-defects are reasonable. It can be shown that this strain corresponds to the formation of a cell structure, at which point λ becomes constant as indicated in figure 2. This observation indicates that the λ parameter as function of stress or strain is more sensitive to microstructure evolution than the model prediction of a constant α<sub>D</sub>. As the cell structure takes form, dislocation density in the walls start to increase and α<sub>D</sub> attains a value of 0.4 again prior to the start of diffuse necking, and instability due to slip band formation starts to occur.

For the optimum aged specimen for which β<sub>1</sub> = 0, α should be 0.4 as the model demands. However during initial yielding, CRA, as indicated in figure 1, separates the yield phenomenon from the higher strain plastic flow. Consideration of the back-extrapolated value of α<sub>D</sub> to γ = 0.006, the proof strain, suggests that the nano-precipitates are stronger and requires larger α as predicted [15]. The higher efficiency of the quenching procedure in the laboratory compared to that in industry is represented by α<sub>yield</sub> for the T6 control being larger than of the T6 As-Rec sample. At γ<sub>3</sub> the plastic response changes to that of deformation around micro-scale particles. At this point the aforementioned duplex slip occurs with generation of nano-defects. Due to space restriction, extensive discussion is not possible but beyond γ<sub>3</sub>, the derived α should apply to the secondary much weaker defects. Since its density is responsible for only about 1/10 of the flow stress, the main load carrying structure is the forest density, which is approaching saturation and does not detectably affect the operative strength factor due to the high formation rate of nano-defects. On the other hand, the model encompasses the total flow stress and hence the predicted α<sub>D</sub> is constant.

## 5. Conclusion

The present analyses of optimum aged AA6063, analogous to that of pure Al at 78 K, has demonstrated that nano- and micro-particles embedded in the matrix gives rise to more complex α<sub>D</sub> profiles. Defect identification is difficult since the portion due to forest dislocation density needs to be accounted for as approximately 90 % of the flow stress. Applying the derived value of α<sub>D</sub> to the remaining 10 % of flow stress should allow prediction of the undulating densities of nano-defects as function of strain, which implies their dynamic generation and annihilation.

<footnote>* α for forest dislocations was used for initial modelling.</footnote>
<figure><img src="image_7.png" /><figcaption>**Figure 2.** Natural logarithm of $\lambda$ versus $\ln \gamma$ for three heat treatments of AA6063.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 3.** Derived obstacle strength, $\alpha_D$, versus $\gamma$ using slopes in figure 2 and equation (5) showing the deviation from the model value of 0.4. Note that the $\alpha_D$ magnitude at $\gamma = 0.006$ (proof strain) for the T6 control specimen is about 0.75 compared to 0.63 for the T6 As-Rec specimen.</figcaption></figure>

## Acknowledgements

The authors thank the Natural Sciences and Engineering Research Council of Canada for support and to Queen's University for the use of facilities. We are indebted to Professor B.J. Diak for consultation.

## References

[1] Edwards G A, Stiller K and Dunlop G L and Cooper M J 1998 *Acta Mater.* **46** 3893-04.

[2] De Geuser F, Dorin T, Lefebre W, Gault B and Deschamps A 2014 *Mater. Sci. Forum* **794-796** 926-32.

[3] Buchanan K, Colas K, Ribis J, Lopez A and Garnier J 2017 *Acta Mater.* **132** 209-27.

[4] Fisk M, Ion J C and Lindgren L-E 2014 *Comp. Mater. Sci.* **82** 531-9.

[5] Myhr O R, Grong O and Schafer C 2015 *Metall. Mater. Trans.* **A 46A** 6018-39.

[6] Saimoto S and Van Houtte P 2011 *Acta Mater.* **59** 602-12.

[7] Livingston J D 1962 *Acta Metall.* **10** 229-239.

[8] Nabarro F R N, Basinski Z S and Holt D B 1964 *Adv. Phys.* **13** 193-323.

[9] Hollomon J H 1945 *Trans. AIME* **162** 268-290.

[10] Saimoto S 2019 *Acta Mater.* **174** 43-52.

[11] Saimoto S 2006 *Philos. Mag.* **86** 4213-4233.

[12] Saimoto S, Singh M A, Langille M R, Levesque J, Inal K, Niewczas M and Woll A R 2018 *Mater. Sci. Eng.* **A 709** 9-16.

[13] Singh M A, Saimoto S, Langille M R, Levesque J, Inal K and Woll A R 2017 *Phil. Mag.* **97** 2496-513.

[14] Brown L M, 1997 *Phil Trans. R. Soc. Lond.* **A 355** 1979-90.

[15] Foreman A J E and Makin M J 1967 *Can. J. Phys.* **45** 511-7; 1966 *Phil. Mag.* **14** 911-24.

[16] Saimoto S, Langille M R and Niewczas M 2018 *Mater. Sci. Forum* **941** 2270-7.