The detailed interpretation which emerges in the rest of the paper relies upon an assessment of the relative contributions to the strengthening of the alloys. Increments to the yield stress will arise from three strengthening mechanisms: (1) Hall–Petch strengthening, (2) solid solution strengthening within the γ-matrix, and (3) precipitation hardening due to resistance to shearing of the secondary and tertiary γ′-precipitates. Orowan by-passing of the γ′-precipitates is neglected in these alloys. The Hall–Petch effect gives
$$\begin{aligned} \sigma _{\text{d}}\,=\,\frac{k_{\text{y}}}{\sqrt{d_{\upgamma }}}, \end{aligned}$$
(1)
where
\(k_{\text{y}}\) is the Hall–Petch coefficient and
\(d_{\upgamma }\) is the mean grain size. In this work,
\(k_{\text{y}} = 501\) MPa
µm
−1/2 was used, based on work performed on Nimonic 105 with a comparable γ′-phase volume fraction of
\(f_{\upgamma '}\) = 47 pct.[
18] The strengthening effect due to changes in the γ-matrix chemistry,
\(\sigma _\mathrm{ss}\), was determined using the method of Gypen and Deruyttere[
19] using the following equation:
$$\begin{aligned} \sigma _\mathrm{ss}=(1-f_{\upgamma '})\bigg (\sum _i k_i^{n}c_i \bigg )^{1/n}~~, \end{aligned}$$
(2)
where
\(c_i\) is the concentration of solute
i, and
\(k_i\) is the strengthening constant for solute
i. The value of
n, a constant, is taken to be 2/3.[
20] A further increase in the yield stress results from solute atoms acting as frictional obstacles for dislocation slip. The strengthening constants in Ni,
\(k_i\), have been estimated by Mishima
et al.[
21] The calculated chemical composition of the γ-matrix is summarized in Table
I for each ABD alloy. The final precipitation shearing stress was estimated using a model proposed by Galindo-Nava
et al.[
22] which allows one to calculate the γ′-precipitation hardening contributions using an estimate mean radius for the secondary and tertiary γ′-precipitate size,
\(r_{\text{s}}\) and
\(r_{\text{t}}\):
$$\begin{aligned} \sigma _{\text{p}}=M \frac{\upgamma _{{\text{APB}}}}{2b}\bigg (w_{{\text{s}}}\frac{l_{{\text{s}}}}{\Lambda _{{\text{s}}}+2r_{{\text{s}}}}+w_{{\text{t}}}\frac{l_{{\text{t}}}}{\Lambda _{{\text{t}}}+2r_{{\text{t}}}}\bigg )~~, \end{aligned}$$
(3)
where
M is the Taylor orientation factor;
\(\upgamma _{\text{APB}}\) is the anti-phase boundary energy of the γ′-phase;
b is the magnitude of the Burgers vector;
\(w_{\text{s}}\) and
\(w_{\text{t}}\) are the particle number fractions of secondary and tertiary γ′-precipitates;
\(l_{\text{s}}\) and
\(l_{\text{t}}\) are the average segment lengths of the leading dislocation acting in the cutting of a secondary and tertiary γ′-precipitate;
\(\Lambda _{{\text{s}}}\) and
\(\Lambda _{{\text{t}}}\) are the average effective lengths of the leading dislocations driving the secondary and tertiary γ′-precipitate cutting. Values for the Taylor orientation factor and the Burgers vector are taken to be 3 and 0.248 nm, respectively. The anti-phase boundary energy of the γ′-phase at 0K was obtained from theoretical predictions employing DFT and is as follows:
\(\upgamma _{\text{APB}}^{{\text{ABD-}}2}\) = 0.344,
\(\upgamma _{\text{APB}}^{{\text{ABD-}}4}\) = 0.349,
\(\upgamma _{\text{APB}}^{{\text{ABD-}}6}\) = 0.36 Jm
−2.[
14] These values were fitted to the following polynomial function to capture the effect of temperature at 700 °C as proposed by Galindo-Nava
et al.[
22]:
\(\upgamma _{\text{APB}}\,=\,\upgamma _{\text{APB}}^{0\,{\text{K}}}-(7\, \times\,10^{-6}~T-3 \, \times\, 10^{-8}~T^2)\). The
\(\upgamma '\)-precipitate shearing stress is controlled by strong-pair coupling since both
\(r_{\text{s}}\) and
\(r_{\text{t}}\) are greater than the particle radius when the maximum strength is defined by
\(r_{\text{m}} = \mu b^2 / 2 \upgamma _{\text{APB}}\), with
\(\mu = E/(2(1+v))\) and
\(v = 0.32\). The following analytical expressions were used to capture the effect of temperature
\(\mu = 87.3 - 9\, \times\, 10^{-4}~T - 1.9 \, \times\, 10^{-5}~T^2\).[
22] The length of the leading dislocation cutting the precipitates is expressed as
\(l=2(r^2 - (r-r_{\text{m}})^2)^{(1/2)}\) and the effective length of the leading dislocation driving
\(\upgamma '\)-precipitate cutting is given by
\(\Lambda = \mathrm {max} \{\lambda , L-l\}\).
\(\lambda \) is the Friedel sampling length which is defined by
\(\lambda = (2T_{{\text{ten}}}/r \, \times\, \upgamma _{\text{APB}})\), where
\(T_{\text{ten}}\) is the dislocation line tension and given by
\(0.5 \, \times\, \mu b^2\) and
L is the mean particle spacing
\(L=(2\pi /3f_{\upgamma '})^{1/2}\, \times\, r\).