Data-Driven Constitutive Model for the Inelastic Response of Metals: Application to 316H Steel


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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  1. 1.

    Lee MG, Kim SJ, Wagoner RH, Chung K, Kim HY (2009) Constitutive modeling for anisotropic/asymmetric hardening behavior of magnesium alloy sheets: Application to sheet springback. Int J Plast 25:70–104

    CAS  Article  Google Scholar 

  2. 2.

    Chaboche JL (2008) A review of some plasticity and viscoplasticity constitutive theories. Int J Plast 24:1642–1693

    CAS  Article  Google Scholar 

  3. 3.

    Chen B, Smith DJ, Flewitt PEJ, Spindler MW (2011) Constitutive equations that describe creep stress relaxation for 316H stainless steel at 550°C. Mater High Temp 28:155–164

    Article  CAS  Google Scholar 

  4. 4.

    Hyde TH, Becker AA, Sun W, Williams JA (2006) Finite-element creep damage analyses of P91 pipes. Int J Press Vessels Pip 83:853–863

    CAS  Article  Google Scholar 

  5. 5.

    Goyal S, Laha K, Das CR, Panneer Selvi S, Mathew MD (2013) Finite element analysis of uniaxial and multiaxial state of stress on creep rupture behaviour of 2.25Cr–1Mo steel. Mater Sci Eng, A 563:68–77

    CAS  Article  Google Scholar 

  6. 6.

    Hall FR, Hayhurst DR (1991) Continuum damage mechanics modelling of high temperature deformation and failure in a pipe weldment. Proc Math Phys Sci 433:383–403

    Google Scholar 

  7. 7.

    Frost HJ, Ashby MF (1977) Deformation-mechanism maps for pure iron, two austenitic stainless steels, and a low-alloy ferritic steel. In: Jaffee RI, Wilcox BA (eds) Fundamental aspects of structural alloy design. Battelle Institute Materials Science Colloquia. Springer, Boston, pp 27–65

    Google Scholar 

  8. 8.

    Chen B, Flewitt PEJ, Cocks ACF, Smith DJ (2015) A review of the changes of internal state related to high temperature creep of polycrystalline metals and alloys. Int Mater Rev 60:1–29

    CAS  Article  Google Scholar 

  9. 9.

    Wang Y-J, Ishii A, Ogata S (2011) Transition of creep mechanism in nanocrystalline metals. Phys Rev B 84:224102

    Article  CAS  Google Scholar 

  10. 10.

    Yang X-S, Wang Y-J, Zhai H-R, Wang G-Y, Su Y-J, Dai LH, Ogata S, Zhang T-Y (2016) Time-, stress-, and temperature-dependent deformation in nanostructured copper: creep tests and simulations. J Mech Phys Solids 94:191–206

    CAS  Article  Google Scholar 

  11. 11.

    Kloc L, Sklenička V (1997) Transition from power-law to viscous creep behaviour of p-91 type heat-resistant steel. Mater Sci Eng, A 234–236:962–965

    Article  Google Scholar 

  12. 12.

    Wen W, Kohnert A, Arul Kumar M, Capolungo L, Tomé CN (2020) Mechanism-based modeling of thermal and irradiation creep behavior: an application to ferritic/martensitic HT9 steel. Int J Plast 126:102633

    CAS  Article  Google Scholar 

  13. 13.

    Kloc L, Skienička V, Ventruba J (2001) Comparison of low stress creep properties of ferritic and austenitic creep resistant steels. Mater Sci Eng, A 319–321:774–778

    Article  Google Scholar 

  14. 14.

    Pahutová M (1980) Research report UFM CSAV (Brno)

  15. 15.

    Rabotnov YN (1965) Experimental data on creep of engineering alloys and phenomenological theories of creep. A review. J Appl Mech Technol Phys 6:137–154

    Article  Google Scholar 

  16. 16.

    Wilshire B, Scharning PJ (2008) Extrapolation of creep life data for 1Cr–0.5Mo steel. Int J Press Vessels Pip 85:739–743

    CAS  Article  Google Scholar 

  17. 17.

    Brown SB, Kim KH, Anand L (1989) An internal variable constitutive model for hot working of metals. Int J Plast 5:95–130

    Article  Google Scholar 

  18. 18.

    Chaboche JL (1989) Constitutive equations for cyclic plasticity and cyclic viscoplasticity. Int J Plast 5:247–302

    Article  Google Scholar 

  19. 19.

    Watanabe O, Atluri SN (1986) Constitutive modeling of cyclic plasticity and creep, using an internal time concept. Int J Plast 2:107–134

    Article  Google Scholar 

  20. 20.

    Murakami S, Ohno N (1982) A constitutive equation of creep based on the concept of a creep-hardening surface. Int J Solids Struct 18:597–609

    Article  Google Scholar 

  21. 21.

    Moosbrugger JC, McDowell DL (1989) On a class of kinematic hardening rules for nonproportional cyclic plasticity. J Eng Mater Technol 111:87–98

    Article  Google Scholar 

  22. 22.

    Bammann DJ (1984) An internal variable model of viscoplasticity. Int J Eng Sci 22:1041–1053

    Article  Google Scholar 

  23. 23.

    Bammann DJ (1990) Modeling temperature and strain rate dependent large deformations of metals. Appl Mech Rev 43:S312–S319

    Article  Google Scholar 

  24. 24.

    Johnson GR, Cook WH (1983) A constitutive model and data for materials subjected to large strains, high strain rates, and high temperatures. In: Proceedings: seventh international symposium on ballistics, pp 541–547

  25. 25.

    Mecking H, Kocks UF (1981) Kinetics of flow and strain-hardening. Acta Metall 29:1865–1875

    CAS  Article  Google Scholar 

  26. 26.

    Norton FH (1929) The creep of steel at high temperatures. McGraw-Hill, New York

    Google Scholar 

  27. 27.

    Garofalo F (1963) An empirical relation defining stress dependence of minimum creep rate. Trans Metall Soc AIME 227:351

    Google Scholar 

  28. 28.

    Mukherjee A, Bird J, Dorn J (1969) Experimental correlations for high-temperature creep. ASM Trans Q 62:155

    CAS  Google Scholar 

  29. 29.

    Bari S, Hassan T (2002) An advancement in cyclic plasticity modeling for multiaxial ratcheting simulation. Int J Plast 18:873–894

    Article  Google Scholar 

  30. 30.

    Lu ZK, Weng GJ (1996) A simple unified theory for the cyclic deformation of metals at high temperature. Acta Mech 118:135–149

    Article  Google Scholar 

  31. 31.

    Nouailhas D (1989) Unified modelling of cyclic viscoplasticity: application to austenitic stainless steels. Int J Plast 5:501–520

    CAS  Article  Google Scholar 

  32. 32.

    Tanaka E (1994) A Nonproportionality parameter and a cyclic viscoplastic constitutive model taking into account amplitude dependences and memory effects of isotropic hardening. Eur J Mech Solids 13:155–173

    CAS  Google Scholar 

  33. 33.

    Keralavarma SM, Cagin T, Arsenlis A, Benzerga AA (2012) Power-law creep from discrete dislocation dynamics. Phys Rev Lett 109:265504

    Article  CAS  Google Scholar 

  34. 34.

    Patra A, McDowell DL (2012) Crystal plasticity-based constitutive modelling of irradiated bcc structures. Philos Mag 92:861–887

    CAS  Article  Google Scholar 

  35. 35.

    Beyerlein IJ, Tomé CN (2008) A dislocation-based constitutive law for pure Zr including temperature effects. Int J Plast 24:867–895

    CAS  Article  Google Scholar 

  36. 36.

    Krishna S, Zamiri A, De S (2010) Dislocation and defect density-based micromechanical modeling of the mechanical behavior of fcc metals under neutron irradiation. Philos Mag 90:4013–4025

    CAS  Article  Google Scholar 

  37. 37.

    Needleman A, Asaro RJ, Lemonds J, Peirce D (1985) Finite element analysis of crystalline solids. Comput Methods Appl Mech Eng 52:689–708

    Article  Google Scholar 

  38. 38.

    Asaro RJ (1983) Crystal plasticity. J Appl Mech 50:921–934

    Article  Google Scholar 

  39. 39.

    Roters F, Eisenlohr P, Hantcherli L, Tjahjanto DD, Bieler TR, Raabe D (2010) Overview of constitutive laws, kinematics, homogenization and multiscale methods in crystal plasticity finite-element modeling: theory, experiments, applications. Acta Mater 58:1152–1211

    CAS  Article  Google Scholar 

  40. 40.

    Roters F, Raabe D, Gottstein G (2000) Work hardening in heterogeneous alloys—a microstructural approach based on three internal state variables. Acta Mater 48:4181–4189

    CAS  Article  Google Scholar 

  41. 41.

    Barton NR, Arsenlis A, Marian J (2013) A polycrystal plasticity model of strain localization in irradiated iron. J Mech Phys Solids 61:341–351

    CAS  Article  Google Scholar 

  42. 42.

    Castelluccio GM, McDowell DL (2017) Mesoscale cyclic crystal plasticity with dislocation substructures. Int J Plast 98:1–26

    Article  Google Scholar 

  43. 43.

    Arsenlis A, Wirth BD, Rhee M (2004) Dislocation density-based constitutive model for the mechanical behaviour of irradiated Cu. Philos Mag 84:3617–3635

    CAS  Article  Google Scholar 

  44. 44.

    Nes E (1997) Modelling of work hardening and stress saturation in FCC metals. Prog Mater Sci 41:129–193

    CAS  Article  Google Scholar 

  45. 45.

    Coble RL (1963) A model for boundary diffusion controlled creep in polycrystalline materials. J Appl Phys 34:1679–1682

    Article  Google Scholar 

  46. 46.

    Lebensohn RA, Hartley CS, Tomé CN, Castelnau O (2010) Modeling the mechanical response of polycrystals deforming by climb and glide. Philos Mag 90:567–583

    CAS  Article  Google Scholar 

  47. 47.

    Bishop JE, Emery JM, Field RV, Weinberger CR, Littlewood DJ (2015) Direct numerical simulations in solid mechanics for understanding the macroscale effects of microscale material variability. Comput Methods Appl Mech Eng 287:262–289

    Article  Google Scholar 

  48. 48.

    Miehe C, Schröder J, Schotte J (1999) Computational homogenization analysis in finite plasticity simulation of texture development in polycrystalline materials. Comput Methods Appl Mech Eng 171:387–418

    Article  Google Scholar 

  49. 49.

    Knezevic M, Savage DJ (2014) A high-performance computational framework for fast crystal plasticity simulations. Comput Mater Sci 83:101–106

    Article  Google Scholar 

  50. 50.

    Zecevic M, McCabe RJ, Knezevic M (2015) Spectral database solutions to elasto-viscoplasticity within finite elements: application to a cobalt-based FCC superalloy. Int J Plast 70:151–165

    CAS  Article  Google Scholar 

  51. 51.

    Smit RJM, Brekelmans WAM, Meijer HEH (1998) Prediction of the mechanical behavior of nonlinear heterogeneous systems by multi-level finite element modeling. Comput Methods Appl Mech Eng 155:181–192

    Article  Google Scholar 

  52. 52.

    Patra A, Tomé CN (2017) Finite element simulation of gap opening between cladding tube and spacer grid in a fuel rod assembly using crystallographic models of irradiation growth and creep. Nucl Eng Des 315:155–169

    CAS  Article  Google Scholar 

  53. 53.

    Forrester A, Sobester A, Keane A (2008) Engineering design via surrogate modelling: a practical guide. Wiley-Blackwell, Chichester

    Google Scholar 

  54. 54.

    Fast T, Kalidindi SR (2011) Formulation and calibration of higher-order elastic localization relationships using the MKS approach. Acta Mater 59:4595–4605

    CAS  Article  Google Scholar 

  55. 55.

    Becker R, Lloyd JT (2016) A reduced-order crystal model for HCP metals: application to Mg. Mech Mater 98:98–110

    Article  Google Scholar 

  56. 56.

    Kalidindi SR, Duvvuru HK, Knezevic M (2006) Spectral calibration of crystal plasticity models. Acta Mater 54:1795–1804

    CAS  Article  Google Scholar 

  57. 57.

    Knezevic M, Al-Harbi HF, Kalidindi SR (2009) Crystal plasticity simulations using discrete Fourier transforms. Acta Mater 57:1777–1784

    CAS  Article  Google Scholar 

  58. 58.

    Zecevic M, McCabe RJ, Knezevic M (2015) A new implementation of the spectral crystal plasticity framework in implicit finite elements. Mech Mater 84:114–126

    Article  Google Scholar 

  59. 59.

    Narula SC (1979) Orthogonal polynomial regression. Int Stat Rev Rev Int Stat 47:31–36

    Article  Google Scholar 

  60. 60.

    Wang H, Capolungo L, Clausen B, Tomé CN (2017) A crystal plasticity model based on transition state theory. Int J Plast 93:251–268

    Article  Google Scholar 

  61. 61.

    Wen W, Capolungo L, Patra A, Tomé CN (2017) A physics-based crystallographic modeling framework for describing the thermal creep behavior of Fe-Cr alloys. Metall Mater Trans A 48:2603–2617

    CAS  Article  Google Scholar 

  62. 62.

    Wang H, Clausen B, Capolungo L, Beyerlein IJ, Wang J, Tomé CN (2016) Stress and strain relaxation in magnesium AZ31 rolled plate: in situ neutron measurement and elastic viscoplastic polycrystal modeling. Int J Plast 79:275–292

    CAS  Article  Google Scholar 

  63. 63.

    Wen W, Capolungo L, Tomé CN (2018) Mechanism-based modeling of solute strengthening: application to thermal creep in Zr alloy. Int J Plast 106:88–106

    CAS  Article  Google Scholar 

  64. 64.

    Lebensohn RA, Tomé CN (1993) A self-consistent anisotropic approach for the simulation of plastic deformation and texture development of polycrystals: application to zirconium alloys. Acta Metall Mater 41:2611–2624

    CAS  Article  Google Scholar 

  65. 65.

    Lebensohn RA, Tomé CN, CastaÑeda PP (2007) Self-consistent modelling of the mechanical behaviour of viscoplastic polycrystals incorporating intragranular field fluctuations. Philos Mag 87:4287–4322

    CAS  Article  Google Scholar 

  66. 66.

    Hill R (1967) The essential structure of constitutive laws for metal composites and polycrystals. J Mech Phys Solids 15:79–95

    CAS  Article  Google Scholar 

  67. 67.

    Asaro RJ, Rice JR (1977) Strain localization in ductile single crystals. J Mech Phys Solids 25:309–338

    Article  Google Scholar 

  68. 68.

    Joseph VR (2016) Space-filling designs for computer experiments: a review. Qual Eng 28:28–35

    Article  Google Scholar 

  69. 69.

    Pronzato L, Müller WG (2012) Design of computer experiments: space filling and beyond. Stat Comput 22:681–701

    Article  Google Scholar 

  70. 70.

    Tang B (1993) Orthogonal array-based latin hypercubes. J Am Stat Assoc 88:1392–1397

    Article  Google Scholar 

  71. 71.

    Deutsch JL, Deutsch CV (2012) Latin hypercube sampling with multidimensional uniformity. J Stat Plan Inference 142:763–772

    Article  Google Scholar 

  72. 72.

    Moza S (2019) sahilm89/lhsmdu: first release for this code (Zenodo)

  73. 73.

    Khuri AI, Mukhopadhyay S (2010) Response surface methodology. Rev Comput Stat 2:128–149

    Google Scholar 

  74. 74.

    Weisstein EW (2002) Legendre polynomial. MathWorld–Wolfram Web Resour

  75. 75.

    Franciosi P, Zaoui A (1982) Multislip in fcc crystals a theoretical approach compared with experimental data. Acta Metall 30:1627–1637

    Article  Google Scholar 

  76. 76.

    Franciosi P, Zaoui A (1982) Multislip tests on copper crystals: a junctions hardening effect. Acta Metall 30:2141–2151

    Article  Google Scholar 

  77. 77.

    Kocks UF, Argon AS, Ashby MF (1975) Thermodynamics and kinetics of slip. Pergamon Press, Oxford

    Google Scholar 

  78. 78.

    Austin RA, McDowell DL (2011) A dislocation-based constitutive model for viscoplastic deformation of fcc metals at very high strain rates. Int J Plast 27:1–24

    CAS  Article  Google Scholar 

  79. 79.

    Lloyd JT, Clayton JD, Austin RA, McDowell DL (2014) Plane wave simulation of elastic-viscoplastic single crystals. J Mech Phys Solids 69:14–32

    CAS  Article  Google Scholar 

  80. 80.

    Dong Y, Nogaret T, Curtin WA (2010) Scaling of dislocation strengthening by multiple obstacle types. Metall Mater Trans A 41:1954–1960

    Article  CAS  Google Scholar 

  81. 81.

    Lagerpusch U, Mohles V, Baither D, Anczykowski B, Nembach E (2000) Double strengthening of copper by dissolved gold-atoms and by incoherent SiO2-particles: how do the two strengthening contributions superimpose? Acta Mater 48:3647–3656

    CAS  Article  Google Scholar 

  82. 82.

    Kitayama K, Tomé CN, Rauch EF, Gracio JJ, Barlat F (2013) A crystallographic dislocation model for describing hardening of polycrystals during strain path changes. Application to low carbon steels. Int J Plast 46:54–69

    CAS  Article  Google Scholar 

  83. 83.

    Estrin Y (1998) Dislocation theory based constitutive modelling: foundations and applications. J Mater Process Technol 80–81:33–39

    Article  Google Scholar 

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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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Correspondence to Aaron E. Tallman.

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

$$ \dot{\gamma }^{\text{s}} = \rho_{\text{cell}}^{\text{s}} b^{\text{s}} v^{\text{s}} {\text{sign}}\left( {\tau^{\text{s}} } \right) $$

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 (\( t_{t}^{s} \)):

$$ v^{s} = \frac{{\lambda^{s} }}{{t_{w}^{s} + t_{t}^{s} }} $$

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

$$ \frac{1}{{ \lambda^{s} }} = \sqrt {\mathop \sum \limits_{s} \bar{\alpha }^{{ss^{{\prime }} }} \rho_{\text{cell}}^{{s^{{\prime }} }} } $$

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:

$$ t_{w}^{s} = \left\{ {\begin{array}{*{20}l} {\frac{1}{{v^{s} }}{ \exp }\left( {\frac{{\Delta G_{0} }}{kT}\left( {1 - \left( {\frac{{\left| {\tau_{\text{eff}}^{s} } \right|}}{{\tau_{c}^{s} }}} \right)^{p} } \right)^{q} } \right)} \hfill & {{\text{if }}\left| {\tau^{s} } \right| < \tau_{c}^{s} } \hfill \\ 0 \hfill & {{\text{if }}\left| {\tau^{s} } \right| \ge \tau_{c}^{s} } \hfill \\ \end{array} } \right. $$

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]:

$$ \tau_{c}^{s} = \tau_{0}^{s} + \left( {\tau_{\text{cw}}^{n} + \tau_{h}^{n} } \right)^{1/n} $$

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:

$$ \tau_{\text{eff}}^{s} = \tau^{s} - \Delta \tau_{m}^{s} - \Delta \tau_{l}^{s} $$

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:

$$ \Delta \tau_{m}^{s} \left( {t_{{a,{\text{local}}}}^{s} } \right) = \frac{{\alpha \Delta E^{\text{core}} \left( {t_{{a,{\text{local}}}}^{s} } \right)}}{{\bar{w}b^{s} }}\quad {\text{and}}\quad \Delta \tau_{l}^{s} = \mu b^{s} \sqrt {\mathop \sum \limits_{s} \bar{\alpha }^{{ss^{{\prime }} }} \rho_{\text{cell}}^{{s^{{\prime }} }} } $$

\( t_{{a,{\text{local}}}}^{s} \) 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:

$$ \dot{\beta }^{s} = \rho_{{{\text{cell}},{\text{edge}}}}^{s} b^{s} v_{\text{climb}}^{s} $$

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 Is. 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:

$$ \bar{v}_{c}^{s} = \frac{\varOmega }{b}\left( {z_{v}^{s} D_{v} c_{v}^{\text{th}} \left[ {\exp \left( {\frac{{\varOmega \bar{\tau }_{\text{climb}}^{s} }}{kT}} \right) - 1} \right]} \right) $$

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:

$$ \dot{\rho }_{\text{cell}}^{s} = \dot{\rho }_{{{\text{cell}},{\text{g}}}}^{s, + } - \dot{\rho }_{{{\text{cell}},{\text{a}}}}^{s, - } - \dot{\rho }_{{{\text{cell}},{\text{trap}}}}^{s, - } $$

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]:

$$ \dot{\rho }_{{{\text{cell}},{\text{g}}}}^{s, + } = \frac{{k_{1} }}{{b\lambda^{s} }}\left| {\bar{\dot{\gamma }}^{s} } \right| $$

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:

$$ \dot{\rho }_{{{\text{cell}},{\text{a}}}}^{s, - } = k_{2} \left( {\frac{{\dot{\varepsilon }_{0} }}{{\dot{\varepsilon }}}} \right)^{{\frac{1}{{n_{0} }}}} \rho_{\text{cell}}^{s} \left| {\bar{\dot{\gamma }}^{s} } \right| $$

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 \( \lambda_{\text{sg}} \):

$$ \dot{\rho }_{{{\text{cell}},{\text{trap}}}}^{s, - } = \frac{{k_{3} }}{{\lambda_{\text{sg}} }}\left| {\bar{\dot{\gamma }}^{s} } \right| $$

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:

$$ \dot{\rho }_{\text{cw}}^{s} = \dot{\rho }_{{{\text{cell}},{\text{trap}}}}^{s, - } - \dot{\rho }_{{{\text{cw}},{\text{a}}}}^{s, - } $$

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

$$ \dot{\rho }_{{{\text{cw}},{\text{a}}}}^{s, - } = k_{4} \rho_{cw}^{s} \left| {\bar{\dot{\gamma }}^{s} } \right| $$

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.

Fig. 10

Relative contribution of individual deformation mechanisms (glide, climb and Coble creep) as a function of imposed stress for a 650 °C, b 700 °C and c 750 °C. Contribution of climb increases with temperature. The relative activity of both the climb and Coble creep modes is inversely proportional to imposed stress

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

Table 4 The calibrated constitutive model parameter values for 316H steel

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.

Fig. 11

A comparison of a Monte Carlo (random) sampling with a LHSMDU sampling in 2 dimensions. The average minimum distance between points for the two method examples are 0.25 and 0.35, respectively

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.

Fig. 12

The CP simulations made to quantify the uncertainty associated with the use of a Prandtl–Reuss flow rule in the SM formulation in predicted plastic strain. Each solid line reflects a different stress direction with respect to a fixed microstructure orientation. Initial parameters are, for all runs: \( \sigma_{\text{vm}} = 300 {\text{MPa}}, T = 973.0 {\text{K,}} \rho_{\text{cell}} = 7.0 \times 10^{12} {\text{m}}^{ - 2} {\text{and}} \rho_{\text{wall}} = 7.0 \times 10^{11} {\text{m}}^{ - 2} \)

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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).

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  • Crystal plasticity
  • Reduced order modeling
  • Creep
  • Surrogate modeling