## Abstract

Predictions of the mechanical response of structural elements are conditioned by the accuracy of constitutive models used at the engineering length-scale. In this regard, a prospect of mechanistic crystal-plasticity-based constitutive models is that they could be used for extrapolation beyond regimes in which they are calibrated. However, their use for assessing the performance of a component is computationally onerous. To address this limitation, a new approach is proposed whereby a surrogate constitutive model (SM) of the inelastic response of 316H steel is derived from a mechanistic crystal plasticity-based polycrystal model tracking the evolution of dislocation densities on all slip systems. The latter is used to generate a database of the expected plastic response and dislocation content evolution associated with several instances of creep loading. From the database, a SM is developed. It relies on the use of orthogonal polynomial regression to describe the evolution of the dislocation content. The SM is then validated against predictions of the dead load creep response given by the polycrystal model across a range of temperatures and stresses. When the SM is used to predict the response of 316H during complex non monotonic loading, extrapolating to new loading conditions, it is found that predictions compare particularly well against those from the physics-based polycrystal model.

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## Acknowledgements

This work was sponsored by the U.S. Department of Energy, Office of Nuclear Energy, Nuclear Energy Advanced Modeling and Simulations (NEAMS). Los Alamos National Laboratory, an affirmative action/equal opportunity employer, is operated by Triad National Security, LLC, for the National Nuclear Security Administration of the U.S. Department of Energy under Contract No. 89233218CNA000001.

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## Appendix

### Appendix

*Dislocation shear rate* Given by Eq. 6 from “Mechanistic Constitutive Model” section is:

here \( {\text{sign}}\left( {\tau^{\text{s}} } \right) \) defines the direction of the shear rate to be the same as the direction of glide. \( b^{s} \) is the magnitude of the Burgers vector. \( v^{s} \) is the mean dislocation velocity, which is calculated from the dislocation mean free path, \( \lambda^{s} \), and the total time a dislocation spends traveling between obstacles. The total time is the sum of the waiting time at obstacles (\( t_{w}^{s} \)) and the travel time within the interspacing (\( t_{t}^{s} \)):

The mean dislocation interspacing associated with dislocation–dislocation interactions is given as in Franciosi and Zaoui [75, 76]:

where \( \bar{\alpha }^{{ss^{{\prime }} }} \) refers to the latent hardening matrix. Using the Kocks-type activation enthalpy law [77,78,79], the dislocation waiting time is written as:

here \( \Delta G_{0} \) is the thermal activation energy without any external stress. *k* is the Boltzmann constant. *T* is absolute temperature. \( p\left( {0 < p \le 1} \right) \) and \( q\left( {0 < q \le 2} \right) \) are the exponent parameters related to the shape of the obstacle resistance profile [78]. \( v^{s} = \chi_{e} C_{s} /\lambda^{s} \) is the attack frequency. Here, \( C_{s} \) is the shear wave velocity and \( \chi_{e} \) is an entropy factor (of the order of 1). \( \tau_{c}^{s} \) refers to the Critical Resolved Shear Stress (CRSS), which is expressed via use of a nonlinear superposition law [79,80,81]:

where \( \tau_{\text{cw}} = \mu b^{s} \sqrt {\sum\nolimits_{s} {\bar{\alpha }^{{ss^{\prime } }} } \rho_{\text{cw}}^{{s^{\prime } }} } \) denotes the cell wall-induced hardening and \( \tau_{h} \) denotes the strengthening due to solute pinning and precipitates. \( \tau_{\text{eff}}^{s} \) in Eq. 10 represents the effective driving stress acting on dislocations inside the cells and it is expressed as:

where \( \tau^{s} \) denotes the local resolved shear stress. \( \Delta \tau_{m}^{s} \) and \( \Delta \tau_{l}^{s} \) are associated with the local reduction in driving force acting on dislocations due to the presence of solutes and due to line tension, respectively. As per Wen et al. [12], \( \Delta \tau_{m}^{s} \) and \( \Delta \tau_{l}^{s} \) are written as:

\( t_{{a,{\text{local}}}}^{s} \) is the local aging time (pinning period). \( \alpha \) is associated with the energy variation along the core. \( \bar{w} \) denotes the core width and \( \Delta E^{\text{core}} \) is the binding energy of the solute to the dislocation.

*Dislocation climb rate* Given by Eq. 7 of “Mechanistic Constitutive Model” section is:

here \( \rho_{{{\text{cell}},{\text{edge}}}}^{s} \) denotes the edge dislocation density. In the present work \( \rho_{{{\text{cell}},{\text{edge}}}}^{s} = 0.1 \rho_{\text{cell}}^{s} \) is assumed. \( v_{\text{climb}}^{s} \) in Eq. 26 represents the climb velocity, which depends on the net flux of point defects *I*^{s}. The climb velocity \( v_{\text{climb}}^{s} \) depends on the imbalance between vacancies and interstitials being trapped by the dislocation, which can be written using a classic expression of rate theory, as:

where \( D_{v} = D_{v}^{0} \exp \left( { - E_{m}^{v} /kT} \right) \) denotes the vacancy diffusivity and \( z_{v}^{s} \) is the rate-theory parameter representing the dislocation capture efficiency for vacancies. \( \varOmega \approx b^{3} \) represents the atomic volume and \( c_{v}^{\text{th}} \) is the thermal equilibrium vacancy concentration.

*Dislocation density evolution* A recently developed dislocation density law for Fe–Cr–Mo alloy is employed here [61]. As mentioned before, the dislocation content can be divided into two populations: dislocations in the cell (subgrain) and in the cell walls (subgrain boundary). The evolution of the dislocation density in the cell is expressed as:

where \( \dot{\rho }_{{{\text{cell}},{\text{g}}}}^{s, + } , \dot{\rho }_{{{\text{cell}},{\text{a}}}}^{s, - } , \,{\text{and}}\, \dot{\rho }_{{{\text{cell}},{\text{trap}}}}^{s, - } \) denote the dislocation generation, dynamic recovery and trapping at the subgrain boundaries. The dislocation generation rate is associated with the area swept by the moving dislocations. The term \( \dot{\rho }_{{{\text{cell}},{\text{g}}}}^{s, + } \) can be expressed as [82]:

where \( \frac{{\lambda^{s} }}{{k_{1} }} \) is the effective mean free path. The dynamic recovery involves several mechanisms, such as cross-slip and climb, that allow the dislocation to move to another slip plane and annihilate with dislocations with opposite Burger vector. Estrin [83] proposed a general expression of the dynamic recovery rate:

where \( \dot{\varepsilon }_{0} \) is a reference strain rate. Estrin suggested that the parameter \( n_{0} \) should be associated with the dominant mechanism and it can vary between 3 and 5 [83]. The dislocation trapping rate at the subgrain boundaries is related to the subgrain size \( \lambda_{\text{sg}} \):

The trapped dislocations will essentially become part of the wall structure. Meanwhile, the dislocations in the cell wall will also annihilate. Thus, the rate of \( \rho_{\text{cw}}^{s} \) can be written as:

Dislocation annihilation in the subgrain boundaries is complex and for the sake of simplicity, the annihilation rate is written as:

The parameters \( k_{1} \), \( k_{2} \), \( k_{3} \) and \( k_{4} \) are material constants calibrated using the experimental data.

*Relative activity of deformation mechanisms* Relative contribution of glide, climb and Coble creep modes to the predicted creep responses is shown in Fig. 10 as a function of imposed stress for three different temperatures. At the early stages of creep, dislocation glide dominates the deformation for all the stress and temperature cases. Within a few hours, contribution of glide decreases and, depending on the stress and temperature, climb and Coble creep modes starts to activate. The relative activity of climb increases with the temperature and decreases with increasing imposed stress. Similarly, the contribution of Coble creep also decreases with imposed stress.

*Constitutive model parameters* The calibrated parameter values of the constitutive model are shown in Table 4.

*LHSMDU Example* The two samples shown below (Fig. 11) portray the effective difference in the sampling method employed here and a simple Monte Carlo random sampling.

*Uncertainty from anisotropy* The error induced with the assumption of an isotropic polycrystal is quantified using a batch of 20 CP-VPSC simulations, run with the same von Mises stress, temperature, microstructure, and initial dislocation densities (\( \sigma_{\text{vm}} = 300 {\text{MPa}}, T = 973.0 {\text{K}}, \rho_{\text{cell}} = 7.0 \times 10^{12} {\text{m}}^{ - 2} \,{\text{and}}\, \rho_{\text{wall}} = 7.0 \times 10^{11} {\text{m}}^{ - 2} \)), only varying the relative contribution of individual stress tensor components. The results of this set of VPSC simulations are shown in Fig. 12. The effective strain of each simulation is shown as a function of time. It can be seen that the simulations present variability in the strain rate resulting from the same level of stress. The variability would decrease if an increasing number of orientations were considered for representing the aggregate (50 orientations were used in the calculation). The low relative scatter in the simulations is used to support the J2 assumption.

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Tallman, A.E., Kumar, M.A., Castillo, A. *et al.* Data-Driven Constitutive Model for the Inelastic Response of Metals: Application to 316H Steel.
*Integr Mater Manuf Innov* (2020). https://doi.org/10.1007/s40192-020-00181-5

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### Keywords

- Crystal plasticity
- Reduced order modeling
- Creep
- Surrogate modeling