Introduction

Strain hardening in single-phase metallic materials is controlled by the production, migration, and annihilation of defects [1]. Vacancies, interstitials or substitutional atoms, dislocations, and grain boundaries contribute to hardening by creating barriers that need to be overcome by dislocations. Point obstacles, such as isolated solute atoms, interstitials, and vacancies, correspond to weak obstacles; forest dislocations are medium strengthening obstacles, while precipitates are strong obstacles [2]. In engineering alloys, all obstacles coexist and contribute to macroscopic strength. However, strengthening in well-annealed single-phase metals is controlled by weak obstacles.

Molecular dynamics calculations have recently [3,4,5] been employed to estimate the dislocation glide energy barrier in simple metallic systems. For example, Esteban et al. [4] characterized the glide energy barrier from Guinier–Preston zones using molecular dynamics simulations. Similarly, Dong et al. [6] simulated the strengthening associated with the collaborative response of multiple obstacles in metals. However, the small scale and high deformation rate of atomistic models make it difficult to transfer quantitative data to coarser crystal plasticity models. Furthermore, the combinatorial nature of atomic configurations in alloys is still unresolved with molecular dynamics, whose results are often valid for a specific strengthening mechanism with a certain atomic order.

Other efforts [7, 8] have quantified the glide energy barrier by analyzing yield stress experimental data. For example, Frost and Ashby [2] characterized the dislocation glide barrier by studying the yield stress dependence on temperature. Building on this idea, Balasubramanian and Anand [9] proposed an elasto-viscoplastic model to parameterize thermal and athermal hardening using aluminum yield stress data at different temperatures. These approaches relied on constitutive models that do not distinguish strengthening from point defects, dislocation, and grain boundaries, which operate at different length scales. A further limitation of these efforts is their simultaneous estimation of multiple parameters, which sums up the uncertainty from various mechanisms across different scales. Indeed, not all mechanisms become active upon a change in loading conditions (e.g., monotonic, cyclic, etc.), so the results are not fully transferable across scales, models, and loading conditions.

This paper extends the work from Balasubramanian and Anand [9] by parameterizing multiple strengthening mechanisms independently, whose uncertainty is estimated from modeling and experimental data. These parameterizations inform Monte Carlo simulations that fit a crystal plasticity physics-based constitutive model to yield stress data at different temperatures. The approach results in a robust methodology to parameterize the dislocation glide barrier and can be used to inform crystal plasticity models. Moreover, the analysis of multiple FCC metals and alloys demonstrates a correlation between the glide barrier activation energy and the intrinsic stacking fault energy (SFE).

Mechanics of dislocation glide

Eyring [10] pioneer work recognized that inelastic deformation conforms to the principles of transition state theory and hypothesized that stochastic atomic perturbations control dislocations glide. Gibbs [11] further proposed that the probability of a dislocation to overcome an obstacle can be computed as,

$$ \Gamma = \nu \exp \left( {\frac{ - \Delta G\left( \tau \right)}{{k_{B} T}}} \right) $$
(1)

in which \(\nu \) is the attempt frequency, kB is the Boltzmann constant, T is the temperature, and ∆G is the Gibbs free activation energy required by a dislocation to glide, which depends upon the local shear stress (\(\tau \)). Furthermore, the probability of gliding can be related to the shear rate (\({\dot{\gamma }}^{\alpha }\)) along slip system \(\alpha \) following Orowan’s model [12],

$$ \dot{\gamma }^{\alpha } = \rho^{\alpha } bl_{v} \nu \exp \left( {\frac{ - \Delta G\left( \tau \right)}{{k_{B} T}}} \right) $$
(2)

in which \({\rho }^{\alpha }\) is the dislocation density on the primary slip system \(\alpha \), \(b\) is the Burgers vector, and \({l}_{v}\) is the average spacing between dislocation obstacles.

The mathematical nature of the Gibbs free activation energy is still a matter of debate, and different formulations have been proposed. One common approach quantifies ∆G by defining a dislocation–obstacle interaction potential, whose first derivate corresponds to the force (F) as a function of the distance traveled by the dislocation [11]. Several approaches including Seeger [13], Fleischer [14], and Mott and Nabarro [15] independently proposed different interaction potentials based on exponential, local tetragonal distortion, and sinusoidal formulations, respectively. These phenomenological interaction potentials can be generalized as,

$$ \Delta G = F_{0} \left( {1 - \left( {\frac{\tau }{{s_{o} }}} \right)^{p} } \right)^{q} $$
(3)

in which \({F}_{0}\) is the activation energy at 0 K, \({s}_{o}\) is the thermal stress, and p and q are profile parameters that range between 0 to 1 and 1 to 2, respectively.

A second common approach [16] quantifies the Gibbs free energy by subtracting the work carried out by the stress field (\(\tau \Delta V\)) to the Helmholtz free energy (\(\Delta F\)),

$$ \Delta G = \Delta F - \tau \Delta V $$
(4)

in which \(\Delta V\) corresponds to the thermodynamic activation volume. More recently, Langer [17] argued about the lack of physical understanding of the origin of both these mathematical formulations and proposed a thermodynamically consistent approach,

$$ \Delta G = F_{0}^{*} \exp \left( { - \frac{\tau }{{s_{0}^{*} }}} \right) $$
(5)

in which \({F}_{0}^{*}\) is the pinning energy at zero stress and \({s}_{0}^{*}\) is the Taylor stress according to Langer.

Equation (1) represents the thermally activated plastic deformation and assumes that the Gibbs energy does not depend on the temperature. Hence, parameters in Eqs. (3)–(5) should also be temperature independent. As noted by Kocks et al. [18], the temperature independence of parameters is a reasonable assumption for glide resistance profiles without a plateau, which is the case for most FCC metals and alloys up to moderate temperatures.

Equations (3)–(5) are up to a certain degree equivalent, with the exception that Eq. (3) requires four parameters rather than two. A larger number of parameters provides further flexibility for representing nonlinear responses but makes the estimation of the parameters more challenging. Hence, we first focus on determining the parameters for the hardest case (Eq. (3)), which has been extensively employed to model strain hardening in FCC single and polycrystals [9, 19, 20].

Following on, the combination of Eqs. (2) and (3) leads to [19],

$$ \dot{\gamma }^{\alpha } = \rho^{\alpha } bl_{v} \nu \exp \left[ { - \frac{{F_{0} }}{{k_{B} T}}\left\{ {1 - \left( {\frac{{\tau_{{{\text{eff}}}}^{\alpha } }}{{s_{o} \frac{\mu }{{\mu_{0} }}}}} \right)^{p} } \right\}^{q} } \right] $$
(6)

in which \({\tau }_{\mathrm{eff}}^{\alpha }\) is the effective shear stress, while \(\mu \) and \({\mu }_{0}\) correspond to the shear modulus at temperatures \(T\) and 0 K. Furthermore, \({\tau }_{\mathrm{eff}}^{\alpha }\) depends on the local resolved shear stress, \(({\tau }^{\alpha })\), the athermal stress \(({S}^{\alpha })\), and the long-range intragranular back stress \({(B}^{\alpha })\),

$$ \tau_{{{\text{eff}}}}^{\alpha } = \left\langle {\left| {\tau^{\alpha } - B^{\alpha } } \right| - S^{\alpha } } \right\rangle $$
(7)

We estimate the strength of dislocation pinning by point obstacles [21] (vacancies, impurities) by considering the deformation up to plastic yield. Indeed, other mechanisms (e.g., cross-slip or long-range back stresses) are less likely to be dominant at the onset of plastic yield in well-annealed materials. Even when many mechanisms are not active, the yield stress still carries a significant variability that should be taken into consideration. Thus, we reorganized Eq. (6) in terms of the yield stress as follows,

$$ \sigma_{y}^{\alpha } = \left[ {S^{\alpha } + \left( {s_{o} \frac{\mu }{{\mu_{0} }}} \right)\left[ {1 - \left\{ { - \frac{{k_{B} T}}{{F_{0} }}\ln \left( {\frac{{\dot{\gamma }_{y}^{\alpha } }}{{\rho^{\alpha } bl_{v} \nu }}} \right)} \right\}^{\frac{1}{q}} } \right]^{\frac{1}{p}} } \right] \times C_{f} $$
(8)

Here \({\dot{\gamma }}_{y}^{\alpha }\) is the strain rate at yield stress and Cf is a factor that projects the mean shear stress into normal stress [22]. This parameter has a value that typically ranges from the Taylor factor (3.06) as an upper bound and to the Sachs factor (2.238) as a lower bound. This range represents an independent and approximate estimate of the conversion factor and accounts for some crystallographic variability.

The athermal stress in Eq. (8) follows [19],

$$ S^{\alpha } = \alpha_{LE} \frac{\mu b}{{2d_{struc} }} + \mu b\sqrt {A_{ii} \rho^{\alpha } } $$
(9)

in which the first term conveys the stress required to bow-out dislocations and dislocation interaction stress. Here, \({\alpha }_{LE}\) corresponds to the line-energy scaling factor, while \({A}_{ii}\) is the interaction coefficient. In well-annealed metals, the contribution from latent hardening on the stress at the onset of plastic deformation (i.e., at 0.2% \({\epsilon }_{p}\)) is negligible [23]. Dislocation production on secondary slip systems promotes dislocation substructures and limits the dislocation free path of the dominant slip. Hence, the dislocation substructure length scale \({d}_{struc}\) in Eq. (9) follows a similitude relation [24] as,

$$ d_{{{\text{struc}}}} = K_{{{\text{struc}}}} \frac{\mu b}{\tau } $$
(10)

where \({K}_{\mathrm{struc}}\) is the similitude coefficient. At yielding, macroscopic annealed materials have sparse dislocations, and the contribution of the first term is negligible. (We assume crystals over 1 μm in size to neglect dislocation starvation hardening [25].) Instead, strengthening comes from self-hardening interactions.

The dislocation density at yield follows the initial density after annealing (\({{\rho }_{o}}^{\alpha }\)) and the density increase upon loading up to yield (\({\Delta {\rho }_{y}}^{\alpha }\)),

$$ \rho^{\alpha } = \rho_{o}^{\alpha } + \Delta \rho_{y}^{\alpha } $$
(11)

Hansen [26] demonstrated that the increase in dislocation density depends on grain size, which is due to differences in the dislocations’ mean free paths. Hence, \({\Delta {\rho }_{y}}^{\alpha }\) can be computed as,

$$ \Delta \rho_{y}^{\alpha } = \frac{{K_{m} \Delta \gamma^{\alpha } }}{{bd_{m} }} $$
(12)

where the dislocations’ production scaling parameter \({(K}_{m})\) has a value that ranges between 1 and 4 [26,27,28,29]. For well-annealed materials, the mean free path (\({d}_{m}\)) can be estimated as half the grain size [30], which corresponds to the average distance a dislocation can travel before encountering a grain boundary. Since the mean free path affects the calculation of the dislocation density and strengthening, Eq. (12) introduces a dependence on the grain size and accounts for Hall–Petch effects.

Finally, the jump frequency follows,

$$ \nu = \frac{{k_{B} T}}{h} $$
(13)

which comes from Eyring’s reaction rate theory [31] and corresponds to the atomic attempt frequency between 1010 and 1012 s−1. The obstacle spacing has a significant role in bypassing the energy barrier during thermal activation because a single event can create a cascade of unpinning events [32].

By combining Eqs. (9)–(13) with Eq. (8), we obtain

$$ \sigma_{y}^{\alpha } \left( T \right) = \left[ {\mu b\sqrt {A_{ii} \left( {\rho_{o}^{\alpha } + \frac{{K_{m} \Delta \gamma^{\alpha } }}{{bd_{m} }}} \right)} + \left( {s_{o} \frac{\mu }{{\mu_{0} }}} \right)\left[ {1 - \left\{ { - \frac{{k_{B} T}}{{F_{0} }}\ln \left( {\frac{{\dot{\gamma }_{y}^{\alpha } }}{{\left( {\rho_{o}^{\alpha } + \frac{{K_{m} \Delta \gamma^{\alpha } }}{{bd_{m} }}} \right) bl_{v} \frac{{k_{B} T}}{h}}} } \right)} \right\}^{\frac{1}{q}} } \right]^{\frac{1}{p}} } \right] \times \frac{{2C_{f} K_{{{\text{struc}}}} }}{{\left( {2K_{{{\text{struc}}}} - \alpha_{LE} } \right)}} $$
(14)

which relates the yield stress with the temperature and strain rate. Hence, we can employ this equation to fit experimental data using minimum-square regression to estimate \({F}_{0}\) and \({s}_{o}\). For completeness, we apply the same approach but considering Eqs. (4) and (5) to obtain,

$$ \begin{aligned} \sigma_{y}^{\alpha } \left( T \right) = & \left[ {\mu b\sqrt {A_{ii} \left( {\rho_{o}^{\alpha } + \frac{{K_{m} \Delta \gamma^{\alpha } }}{{bd_{m} }}} \right)} + \left[ {\frac{\Delta F}{{l_{v} bd}} + \left\{ {\frac{{k_{B} T}}{{l_{v} bd}}\ln \left( {\frac{{\dot{\gamma }_{y}^{\alpha } }}{{\left( {\rho_{o}^{\alpha } + \frac{{K_{m} \Delta \gamma^{\alpha } }}{{bd_{m} }}} \right)bl_{v} \frac{{k_{B} T}}{h}}}} \right)} \right\}} \right]} \right] \\ & \times \frac{{2C_{f} K_{{{\text{struc}}}} }}{{\left( {2K_{{{\text{struc}}}} - \alpha_{{{\text{LE}}}} } \right)}} \\ \end{aligned} $$
(15)

in which we estimated the activation volume by \({(l}_{v}bd\)) where d represents the length of thermal activation, and

$$ \begin{aligned} \sigma_{y}^{\alpha } \left( T \right) = & \left[ {\mu b\sqrt {A_{ii} \left( {\rho_{o}^{\alpha } + \frac{{K_{m} \gamma^{\alpha } }}{{bd_{m} }}} \right)} - \left[ {s_{0}^{*} \times \ln \left\{ { - \frac{{k_{B} T}}{{F_{0}^{*} }}\ln \left( {\frac{{\dot{\gamma }_{y}^{\alpha } }}{{\left( {\rho_{o}^{\alpha } + \frac{{K_{m} \gamma^{\alpha } }}{{bd_{m} }}} \right)bl_{v} \frac{{k_{B} T}}{h}}}} \right)} \right\}} \right]} \right] \\ \times \frac{{2C_{f} K_{{{\text{struc}}}} }}{{\left( {2K_{{{\text{struc}}}} - \alpha_{{{\text{LE}}}} } \right)}} \\ \end{aligned} $$
(16)

Independent estimation of parameters and their uncertainty

A robust quantification of the glide energy barrier requires the estimation of the uncertainty of all the pre-assumed parameters in Eq. (14). We regard these parameters into two categories related to their uncertainties: the first parameters are atomistic fundamental quantities with low uncertainty such as \({k}_{B}\) and \(b\). The second category corresponds to high-uncertainty parameters that result from mesoscale stochastic processes such as dislocation jump frequency and dislocation–dislocation interactions. For these latter parameters, we identify the value ranges that have been reported in the literature.

The shear modulus (\(\mu \)) is a material parameter with relatively low uncertainty, which we accounted for by considering the Reuss and Voigt [33] models as lower and upper bounds, respectively. Overall, the uncertainty of elastic constants has a minor secondary effect as demonstrated in Appendix A using the elastic constants reported for various metals [34].

Since it is difficult and time-consuming to quantify dislocation densities with experiments, their estimations carry a large uncertainty. Hence, we assume a range for the initial dislocation densities based on experimental reports for various metals. Mavlyutov et al. [35] studied the effect of annealing temperature on dislocation densities in ultrafine-grained aluminum and found values between 4 × 1012 and 1.5 × 1012 m−2 for annealing at room temperature and 423 K, respectively. Similarly, Williamson and Smallman [36] estimated dislocation densities between 1011 and 1012 m−2 for different annealed FCC metals. Here, we assume an initial dislocation density (\({{\rho }_{o}}^{\alpha }\)) along primary slip system between 1 × 108–1 × 1011 m−2, and \({\Delta {\rho }_{y}}^{\alpha }\) (contribution from yielding) is calculated using Eq. (12) subject to the grain size of material. Typically, \({\Delta {\rho }_{y}}^{\alpha }\) ranges between 5 × 1012 and 1 × 1010 m−2 for grain sizes between 1 and 250 μm [35]. In this analysis, the lower bound of dislocation density will remain 1 × 108 m−2; however, the upper bound will be modified for every material subject to its grain size.

The self-interaction coefficient (\({A}_{ii}\)) in Eq. (9) has been extensively estimated through experiments and dislocation dynamics simulations. The results obtained by multiple authors [37,38,39,40,41] reported a range between 0.1 and 0.3 for various materials and even for hydrogen pre-charged samples [42]. Furthermore, Fivel et al. [38] reported that interaction coefficients do not show significant change with the dislocation density and stress. As a result, we consider an average interaction coefficient between 0.1 and 0.3.

The line tension coefficient (\({\alpha }_{LE}\)) in Eq. (9) is related to the stress required to bow out and multiplicate dislocations. Szajewski et al. [43] investigated dislocation bow-out using molecular dynamics and quantified line tension coefficient in the range of 0.5 to 0.85. Tabata et al. [44] studied the effect of flow stress on dislocation behavior in aluminum [111] single crystal, assuming that the bow-out is pinned with forest dislocation in dislocation walls and used the line-energy coefficient as 1. Therefore, we assume that the line tension value should be in a range between 0.5 and 1.5. Furthermore, regarding the similitude coefficient (\({K}_{struc}\)) in Eq. (10), Sauzay and Kubin [24] showed that FCC metals follow the similitude relation under cyclic and monotonic loading. They demonstrated that the similitude coefficient under monotonic loading varies between 5 and 10, which corresponds to the range employed in this study.

Kocks et al. [18] bounded the profile parameters \(p\) and \(q\) in Eq. (3) between 0–1 and 1–2, respectively. Their calculations for ∆G considered various values and concluded that \(p=3/4\) and \(q=4/3\) represent an adequate but not unique solution. Instead, Fleischer [14] derived \(p=1/2\) and \(q=2\) for a dislocation interacting with local obstacles creating a tetragonal distortion, while Mott and Nabarro [15] proposed a sinusoidal interaction potential between dislocation and a particle in precipitate-hardened material and derived values of \(p=2/3\) and \(q=3/2\). More recently, Dong [45] used molecular dynamics to derive a polynomial expression for a dislocation-point obstacle interaction mechanism that resulted in \(p=2/3\) and \(q=3/2\). Hence, we initially assume \(p=2/3\) and \(q=3/2\), but we will later consider other values in the range proposed by Kocks [18].

Finally, Sobie et al. [32] studied the role of obstacle spacing on glide activation energy and proposed a spacing in the order of tens of nm. Thus, we assume an equivalent range for our analysis between 1 and 50 nm. Table 1 summarizes the ranges of all the parameters in Eq. (14) that were considered in the Monte Carlo analysis for evaluating \({F}_{0}\) and \({s}_{o}\). The references support that these parameters are unlikely to have values outside these ranges, but current epistemic uncertainty prevents us from making more precise estimations.

Table 1 Summary of different scaling-level parameters
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Quantification of glide activation from yield stress data

Monte Carlo simulations

To estimate the glide energy barrier parameters (\({F}_{0}\) and \({s}_{o}\)) and their uncertainty, we implement a Monte Carlo approach that fits Eqs. (14), (15) and (16) to yield stress as a function of temperature. The schematic of the process is shown in Fig. 1f or Eqs. (14). The analysis considers the ranges of parameters in Table 1 for the materials summarized in Table 2. The tabulated yield stress as a function of homologous temperature is shown in Fig. 2. These figures present a quasi-linear dependence of the yield stress with homologous temperature up to a value of 0.2, at which point a plateau becomes apparent.

Figure 1
figure 1

Schematic of Monte Carlo approach to estimate \({F}_{0}\) and \({s}_{o}\)

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Table 2 Summary of materials and their properties used in Monte Carlo analysis. The SFE values are approximations that carry significant uncertainty (e.g., 25% [57])
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Figure 2
figure 2

Normalized yield stress vs temperature for a polycrystalline metals [46, 47, 51, 52] for aluminum, copper, and silver, yield data corresponds to 0.5% strain but is 0.2% for nickel and b polycrystalline alloys [53,54,55,56]. The yield stress was reported at 0.2% strain for all alloys. The normalization factor corresponds to the yield stress at room temperature. We consider each of the strain values as reported in experimental data in our analysis with Eq. (14)

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The plateau in the yield stress has been attributed [46,47,48,49] to dynamic strain aging (DSA) and is caused by the interference of impurities and solute atoms (e.g., carbon) with the mobility of dislocations [50]. Because the model does not have any special provision for modeling the interference of diffusing of point obstacles, we limit our analysis yielding at homologous temperatures below 0.2. Nevertheless, the analysis can still be used in models without the explicit provision of DSA, and Appendix B presents the estimates of \({F}_{0}\) and \({s}_{o}\) using yield data over full range of temperature. In this case, glide activation parameters are engineering approximations that could be dependent on the deformation rate.

For each Monte Carlo calculation, we employ a set of parameters randomly chosen within the ranges in Table 1, while \({F}_{0}\) and \({s}_{o}\) were bounded between 0–5 eV and 1–500 MPa, respectively. Each parameter was chosen assuming a flat distribution within the bounds of their intrinsic epistemic uncertainty. Furthermore, the fitting skill is taken into account and only those results with R-square above 0.8 are considered in the analysis.

Results

Figure 3 to Fig. 4 present the outcomes from the Monte Carlo analysis for different FCC metals and alloys. The results of \({F}_{0}\) and \({s}_{o}\) are further summarized in Table 3, which presents the mean and 95% confidence interval values computed from Monte Carlo distributions. A total of 10,000 calculations per material was sufficient to converge the results for \({F}_{0}\) and \({s}_{o}\) as demonstrated in Appendix C. Among all materials, \({F}_{0}\) results in values between 1 and 3 eV indicating, as expected, weak point obstacles [2]. Notably, \({s}_{o}\) resulted in a wide range between 15 and 350 MPa among all materials. Moreover, the average normalized variability for \({F}_{0}\) and \({s}_{o}\) is 65% and 96%, respectively, which demonstrates that the former has lower intrinsic uncertainty than the latter.

Figure 3
figure 3

a Glide activation energy and b thermal slip resistance of aluminum, nickel, copper, and silver

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Figure 4
figure 4

a Glide activation energy and b thermal slip resistance of SS 304, SS316, Cupro-Nickel, and Al 2024-T4

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Table 3 Mean values and 95% confidence interval of \({F}_{0}\) and \({s}_{o}\) computed for pure metals and alloys
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To further explore the significance of the results, we compare \({F}_{0}\) and \({s}_{o}\) with the intrinsic SFE, which often carries two-digit uncertainty. Although the parameters may correlate better with the energy that a leading partial dislocation must overcome (i.e., unstable SFE), the uncertainty of these magnitudes is even higher. The results in Figure 5 demonstrate that there is no apparent correlation between \({s}_{o}\) and the SFE. On the contrary, the activation energy presents an inverse proportionality with the SFE that can be parameterized as:

$$ F = Fi - a*SFE $$
(17)
Figure 5
figure 5

Correlation between glide activation and stacking fault energy of different FCC metals and alloys. The experimental data for SS 310 s alloy are given in Appendix D

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with \(Fi\) = 2.6 (\(\pm 0.3\)), \(a\) = 0.0067(\(\pm 0.0031\)). The relation between the SFE and \({F}_{0}\) was indeed alluded by Kocks for FCC metals [20] and can be used to obtain a first-order estimation of \({F}_{0}\) when limited experimental data are available. The value of Eq. (17) is further underpinned by the lower intrinsic uncertainty of \({F}_{0}\), which suggests that limited experimental data should be used to estimate \({s}_{o}\) before refining the estimations of \({F}_{0}\).

Effects of solute concentration and crystallographic orientation

Continuing with the analysis of thermal stress, we recall the work from Wille et al. [65], who studied the effect of solute concentration on activation energy parameters in Cu–Mn single crystal oriented for single slip. Their analysis used an empirical relation for the activation volume to quantify the sensitivity of activation energy and thermal stress to solute concentration. To validate our approach, we consider the shear stress data at different temperatures from Wille et al. [65] (as shown in Appendix D) to compute the activation energy parameters. Figure 6 presents the effect of solute concentration on glide activation and thermal stress of Cu–Mn alloy with a 95% confidence interval. Our results agree with the trends from Wille et al. [65] and demonstrate that an increase in solute concentration increases primarily the thermal stress rather than the activation energy. We highlight that our analysis does not require the empirical relation proposed by Wille et al. [65], but fully relies on physics-based mechanisms parameterized independently.

Figure 6
figure 6

Activation energy and thermal stress of Cu–Mn alloy with different solute concentrations. Error bars correspond to a 95% confidence interval

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Further validation proceeds from an analysis of a single crystal oriented for single slip, which does not promote cross-slip at low plastic strain amplitude. Hence, to ascertain that the estimated activation energies relate indeed to the gliding process rather than cross-slip, we consider aluminum, nickel, and copper single crystals [66,67,68]. Figure 7 compares the activation energy for these single crystals (SC) and polycrystals (PC); the overlapping of activation energies between single- and polycrystals supports our methodology. These results also highlight the variability conveyed by polycrystals. A second consideration is that the thermal stresses in polycrystalline analysis seem consistently higher than that in single crystals. One source for such effect is their difference in the level of impurities as shown before. (These materials are effectively not pure when considering the thermal stress.)

Figure 7
figure 7

Activation energy and thermal stress of a aluminum, b nickel, and c copper single (SC) and polycrystals (PC). Experimental data of polycrystals [47] are shown in Fig. 2, and data for single crystals [66,67,68] are given in Appendix D. The dimensions correspond to the grain size for polycrystals and crystal size for single crystals

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Another aspect that requires consideration is the effect of grain size on yield stress [69], which may affect the estimation of the glide parameters. Figure 8 presents the effect of different grain sizes on activation energy parameters. The overlapping of results in Fig. 8 shows that glide activation energy and thermal stress are not significantly influenced by the grain size. This further validates the incorporation of the grain size effect in Eq. (14).

Figure 8
figure 8

Effect of grain size on glide activation energy and thermal stress in a copper b silver. The upper bound of the dislocation density range is different for each case subject to the grain size

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Considerations for different activation energy formulations

We now estimate the activation energy parameters with a similar analysis but considering different p and q values for Eq. (3). Figure 9 demonstrates that the choice of p and q can influence the values for \({F}_{0}\) and \({s}_{o}\). Moreover, the average quality of the least-square fitting represented by the R coefficient in Table 4 saturates for p > 2/3 and q < 3/2. Hence, we support Kocks [20] recommendation that p and q should be fixed between 2/3–1 and 1–3/2, respectively, and only \({F}_{0}\) and \({s}_{o}\) should be adjusted to match experimental data. This strategy is likely to work due to the linear nature of the yield stress data below 0.2 homologous temperature, which suggests that the use of four parameters overdetermines the problem.

Figure 9
figure 9

Effect of different p and q values on a glide activation energy and b thermal slip resistance of different FCC metals

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Table 4 Effect of p and q on average R-square computed from fitting results of Fig. 9
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Next, we perform a similar analysis considering the parameterization of Gibbs energy in Eqs. (4) and (5) The results in Fig. 10a show that the Helmholtz free energy is almost independent of the SFE, while the thermal activation length scale correlates with the SFE. Similar to the results for Eq. (3) in Fig. 5, Fig. 10b demonstrates that the pinning energy correlates with the SFE, but not the thermal stresses.

Figure 10
figure 10

Estimation of glide parameters for a Eq. (15) and b Eq. (16)

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Discussion

This study characterized the dislocation glide energy barrier in FCC metals and alloys by combining physics-based crystal plasticity models and parameter uncertainty. We considered three formulations for the Gibbs free energy as a function of the effective stress, which was computed using parameters that were estimated independently. The analysis used least-square fitting of experimental data to determine only two parameters at a time rather than multiple coefficients [9, 65]. As a result, we mitigated the spurious cancelation of error among parameters and we were able to estimate the uncertainty of the activation energy parameterization.

The comparison among Gibbs free energy formulations showed that Kocks approach (Eqs. (14)) provides the best results when fitting the yield stress dependence on temperature, even when parameters p and q were fixed. Equations (15) and (16) represent fixed linear and exponential dependence of yield stress on temperature, respectively, which limits their applicability in fitting all materials in Figure 5 and Table 5 further compares the fitting quality from using Eqs. (14), (15) and (16) by presenting the average R-square coefficient from all materials and simulations. The results demonstrate a better fitting with Eq. (14) as compared to other formulations. Hence, our analysis supports the quantification of Gibbs free energy using Kocks formulation (Eqs. (14) with profile parameters p and q fixed in the ranges between 2/3–1 and 1–3/2, respectively.

Table 5 Comparison of average R-square between different formulations computed from fitting results of Fig. 5 and Fig. 10
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Multiscale crystal plasticity models are usually calibrated without sufficient data to quantify all parameters independently. In practical terms, a spurious increase \({F}_{0}\) can be counteracted by reducing \({s}_{o}\) to match the same experimental data. This makes it difficult to estimate both parameters independently and affects the model prediction (e.g., when used for temperatures and deformation rates outside the calibration database). However, the relation found between the activation energy and the intrinsic SFE in Eq. (17) can be employed as a first-order estimation of \({F}_{0}\). This estimation is independent of \({s}_{o},\) which can be subsequently quantified by fitting the response to the available experimental data. This methodology provides an independent and robust estimation of the glide activation parameters and mitigates spurious errors.

Our analysis has focused on yield stress at low temperatures due to the lack of a special provision of DSA in the crystal plasticity model. \({F}_{0}\) and \({s}_{o}\) can still be calculated using the entire temperature range as shown in Appendix B, but we regard this calculation as an engineering approximation that obscures the fact that \({s}_{o}\) should change due to DSA. Indeed, our work has shown that \({s}_{o}\) depends strongly on the solute concentration. Hence, small temporary changes in solute concentrations around dislocations due to DSA should affect the effective \({s}_{o}\). This analysis further suggests that DSA can be modeled by introducing a dependence for \({s}_{o}\) on the effective solute content around dislocations, which would depend on the deformation rate, temperature, and diffusivity of species.

Finally, our approach relied on experimental data readily available in the literature to predict the glide activation energy in monolithic FCC metals with weak point obstacles. Future efforts can further explore the extension of the analysis to metals with medium- and high-strength obstacles. These approaches should add additional strengthening mechanisms to the athermal stress as well as reconsider the dependences of the parameters involved.

Conclusions

A physics-based predictive approach is presented to estimate the dislocation glide energy barrier in metals while considering parameter uncertainties independently. We explored the roles of weak point obstacles, dislocation strengthening, and grain size to predict the dependence of yield stress on the temperature in annealed metallic materials.

We employed Monte Carlo simulations to fit the formulations to experimental data and quantify glide activation parameters and their intrinsic epistemic uncertainty. Our calculations identified the effect of impurities on solid solution strengthening and found a correlation between the glide activation energy and the SFE.

Our analysis suggests that Kocks parameterization of the Gibbs free energy can estimate nonlinear yield stress responses, even if parameters p and q are fixed between 2/3–1 and 1–3/2, respectively. Although the activation energy and thermal stress depend on p and q, the fitting quality does not in this range. Furthermore, we proposed the use of the relation between the glide activation energy and the SFE to estimate multiscale model parameters when limited experimental data are available for calibration.