MicroROM: A Physics-Based Constitutive Model for Ni-Based Superalloys

Abstract

In this article, a physics-based constitutive model is developed for representing the stress–plastic strain response of Ni-based superalloys by considering the hardening mechanisms that contribute to the macroscopic yield stress and the subsequent strain-hardening response in order to derive the entire tensile stress–strain curve. It is demonstrated that the log stress vs log plastic strain curves of Ni-based superalloys such as 702 Li and ME3 exhibit a bilinear behavior with a lower strain-hardening exponent in the low plastic strain regime and a much higher strain-hardening exponent in the high plastic strain regime. These two hardening regimes can be modeled on the basis of self-hardening of individual slip planes and latent hardening of five operative independent slip systems due to cross-slip from {111} planes to {001} planes, leading to the formation of incomplete and complete Kear-Wilsdorf locks. The proposed physics-based constitutive model, dubbed as MicroROM, exhibits a form that is similar to the Ramberg–Osgood (RO) constitutive model, which is widely used in structural analyses of engineering designs and components. MicroROM is applied to predict the tensile stress–strain curves of 720 Li and ME3 with either a subsolvus or supersolvus microstructure for temperatures ranging from 24 °C to 815 °C. The agreement is good between model predictions and experiment data from the literature. The sensitivity of the predicted stress–strain response to individual microstructural parameters is highlighted and the relation to individual hardening mechanisms is elucidated.

Appendix A

This appendix highlights the six yield stress components that are comprised in the yield stress model of Kozar et al. Some of the terms are slightly different from those reported by Kozar et al.[1]

• 1. Hall–Petch hardening[41,42] of the γ matrix

  $$ \Delta \sigma_{1} = f_{m} \left[ {\sigma_{o} + k_{y} d_{m}^{ - 0.5} } \right], $$

  (A1)

  where f_m is the volume fraction and d_m is the average diameter of the γ grains, respectively; so is the friction stress and k_y is the Hall–Petch coefficient of the γ-matrix.

• 2. Solid solution strengthening of the γ matrix[43]

  $$ \Delta \sigma_{2} = f_{\text{m}} \Delta \sigma_{\text{ss}} \exp \left( { - {\beta_{1}} T} \right) $$

  (A2)

  with

  $$ \Delta \sigma_{\text{ss}} \, = \,\sum\limits_{i} {s_{i} \sqrt {C_{i} } }, $$

  (A3)

  where s_i and C_i are the strengthening coefficient and concentration of the ith solid solution element in the γ matrix.[43]. In Eq. (A2), β₁ is a constant.

• 3. Hall–Petch hardening[41,42] of the primary γ′ precipitates

  $$ \Delta \sigma_{3} \, = \;f_{1} \left[ {\sigma_{\text{ogp}} + k_{\text{yp}} d_{1}^{ - 0.5} } \right], $$

  (A4)

  where σ_ogp is the friction stress and k_yp is the Hall–Petch coefficient of the primary γ′.

• 4. Solid strengthening and anomalous hardening of γ′ precipitates[1]

  $$ \Delta \sigma_{4} = \left( {f_{1} + f_{2} } \right)\left[ {\Delta \sigma_{\text{gp}} + q_{1} T + q_{2} T^{2} + q_{3} T^{3} + q_{4} T^{4} } \right] $$

  (A5)

  with

  $$ \Delta \sigma_{\text{gp}} \, = \,\sum\limits_{i} {s_{\text{gi}} \sqrt {C_{\text{gi}} } }, $$

  (A6)

  where s_gi and C_gi are the strengthening coefficient and concentration of the ith solid solution element in the γ′. The q_i are empirical constants obtained by fitting to the yield strength of γ′, as a function of temperature. Tertiary γ′ precipitates are not included in this term because they are small and are easily sheared with the presence of cross-slip on (001) planes and the formation of K-W locks.

• 5. Shearing of tertiary γ′ by weak-pair coupling mechanism[44]

  $$ \Delta \sigma_{5} = M\left( {\frac{{\gamma_{\text{APB}} }}{2b}} \right)\left[ {\sqrt {\frac{{\gamma_{\text{APB}} d_{3} }}{{2T_{\text{L}} }}} \left( {\frac{{d_{3} }}{{L_{\text{s}} }}} \right) - \frac{\pi }{4}\left( {\frac{{d_{3} }}{L}} \right)^{2} } \right], $$

  (A7)

  where M is the Taylor factor, T_L is the line tension, L is the center and center particle spacing, and L_s is the mean planar spacing. L, L_s, and T_L are given by Eqs. [5a], [5b], and [11] in Kozar et al.[1]

• 6. Shearing of secondary γ′ by strong-pair coupling mechanism[44]

  $$ \Delta \sigma_{6} = M\left( {\frac{{2T_{L} }}{{\pi bL_{x} }}} \right)\left[ {\frac{{\pi d_{2} \gamma_{APB} }}{{2T_{L} }} - 1} \right]^{1/2}, $$

  (A8)

  where L_x is the strong-pair particle spacing given by Eq. [13] in Kozar et al.[1]