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Microstructural Constitutive Model for Polycrystal Viscoplasticity in Cold and Warm Regimes Based on Continuum Dislocation Dynamics

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Microstructural Constitutive Model for Polycrystal Viscoplasticity in Cold and Warm Regimes Based on Continuum Dislocation Dynamics Microstructural constitutive model for polycrystal viscoplasticity in cold and warm regimes based on continuum dislocation dynamics S. Amir H. Motaman*, Ulrich Prahl Department of Ferrous Metallurgy, RWTH Aachen University, Intzestr. 1, D-52072 Aachen, Germany ARTICLE INFO ABSTRACT Manuscript history: Viscoplastic flow of polycrystalline metallic materials is the result of motion Revised and interaction of dislocations, line defects of the crystalline structure. In the microstructural (physics-based) constitutive model presented in this paper, the main underlying microstructural processes influencing viscoplastic deformation and mechanical properties of metals in cold and warm regimes Keywords: are statistically described by the introduced sets of postulates/axioms for continuum dislocation continuum dislocation dynamics (CDD). Three microstructural (internal) dynamics; state variables (MSVs) are used for statistical quantifications of different microstructural types/species of dislocations by the notion of dislocation density. Considering constitutive modeling; the mobility property of dislocations, they are categorized to mobile and dislocation density; (relatively) immobile dislocations. Mobile dislocations carry the plastic strain thermo-micro-mechanical (rate), while immobile dislocations contribute to plastic hardening. Moreover, modeling; with respect to their arrangement, dislocations are classified to cell and wall polycrystal viscoplasticity; dislocations. Cell dislocations are those that exist inside cells/subgrains, and nonlocal plasticity; wall dislocations are packed in (and consequently formed) the subgrain metal forming walls/boundaries. Therefore, the MSVs incorporated in this model are cell mobile, cell immobile and wall immobile dislocation densities. The evolution of these internal variables is calculated by means of adequate equations that characterize the dislocation processes dominating material behavior during cold and warm monotonic viscoplastic deformation. The constitutive equations are then numerically integrated; and the constitutive parameters are determined/fitted for a widely used ferritic-pearlitic steel (20MnCr5). Contents Nomenclature ...................................................................................................................................................................................................... 2 1. Introduction ............................................................................................................................................................................................... 4 Empirical constitutive models .................................................................................................................................................. 4 Microstructural constitutive models ..................................................................................................................................... 4 2. Background ................................................................................................................................................................................................. 6 3. Postulates and the constitutive model ........................................................................................................................................... 7 Fundamental postulates .............................................................................................................................................................. 7 Dynamic dislocation processes ............................................................................................................................................. 15 Static dislocation processes .................................................................................................................................................... 20 Temperature and strain rate dependencies .................................................................................................................... 21 4. Numerical integration and parameter identification ............................................................................................................ 25 Numerical integration ............................................................................................................................................................... 25 Identification of constitutive parameters ......................................................................................................................... 26 5. Results, discussion and validation ................................................................................................................................................. 27 Isothermal uniaxial tests .......................................................................................................................................................... 30 * Corresponding author. Tel: +49 241 80 90133; Fax: +49 241 80 92253 Email: [email protected] 2 Microstructural constitutive model for polycrystal viscoplasticity in cold and warm regimes based on CDD Evolution of dislocation densities at constant strain rate ......................................................................................... 33 GND density measurements ................................................................................................................................................... 34 Multistep uniaxial tests ............................................................................................................................................................. 35 Nonisothermal uniaxial tests ................................................................................................................................................. 36 6. Outlook ...................................................................................................................................................................................................... 38 7. Conclusion ................................................................................................................................................................................................ 38 Acknowledgement .......................................................................................................................................................................................... 39 References .......................................................................................................................................................................................................... 39 Nomenclature Symbol Unit Description 푏 m Burgers length (magnitude of Burgers vector) 푐 - Material coefficient associated with probability amplitude of a dislocation process 푑 m Critical distance for dislocation processes 푒 % Relative error, residual/objective/fitness function 푓 - Volume fraction 퐺 Pa Shear modulus 퐻 Pa Viscoplastic tangent modulus 푙 m Length of a dislocation segment 푚 - Strain rate sensitivity parameter 푛 - Number of active slip/glide systems 푀 - Taylor factor p - Probability amplitude (or frequency) of a dynamic dislocation process 푞 J.m-3 Volumetric heat generation 푟 - Temperature sensitivity coefficient 푅 m Radius 푠 - Temperature sensitivity exponent 푺 Nonlocal microstructural state (a set containing all MSVs) 푡 s Time 푇 K Temperature 푢 J.m-3 Volumetric stored energy 풗 m.s-1 Velocity vector 푤 J.m-3 Volumetric work 훼 - Dislocation interaction strength/coefficient 훽 - Dissipation factor, efficiency of plastic dissipation, or Taylor–Quinney coefficient 훾 - Shear strain in slip system, mean shear strain 휀 - Mean/nonlocal (normal) strain 휃 Pa Plastic/strain hardening 휑 Pa.s Viscous/strain-rate hardening 휅 - Material constant associated with dissipation factor Λ m Dislocations spacing m-2 Dislocation density Pa Mean/nonlocal (normal) stress 휏 Pa Resolved shear stress Index Description ac Accumulation an Annihilation S. Amir H. Motaman, Ulrich Prahl 3 d Dynamic gn Generation GN Geometrically necessary i Immobile loc Local m Mobile, melt (n) Time step index, previous time increment (n+1) Current time increment nc Nucleation c Cell p Plastic pn Pinning rm Remobilization s Static sat Saturated SS Statistically stored t Total (subscript), time (superscript) tr Trapping v Viscous w Wall x Cell, wall, or total (푥 = 푐, 푤, 푡) y Mobile, immobile, or total (푦 = 푚, 푖, 푡), yield/flow z Dislocation process (푧 = gn, an, ac, tr, nc, rm, 푠pn, 푠rm) 0 Reference, initial/undeformed state + Increase/gain - Decrease/Loss 푥 ̂ Normalized/dimensionless (푥̂ = ) 푥0 ̌ Function ̃ Statistical mean/average ̅ Equivalent Abbreviation Description CB Cell block CDD Continuum dislocation dynamics CMD Continuum microstructure dynamics DDD Discrete dislocation dynamics DDW Dense dislocation wall DSA Dynamic strain aging DTH Dynamic thermal hardening DTS Dynamic thermal softening EBSD Electron backscatter diffraction EVP Elasto-viscoplastic FE Finite element FEMU Finite element model updating GB Grain boundary GNB Geometrically necessary boundary GND Geometrically necessary dislocation IDB Incidental dislocation boundary MB Micro-band 4 Microstructural constitutive model for polycrystal viscoplasticity in cold and warm regimes based on CDD MD Molecular dynamics MSV Microstructural (internal) state variable NMS Nonlocal microstructural state RMV Representative material volume RVE Representative volume element SC Sub-cell SFE Stacking fault energy TMM Thermo-micro-mechanical,
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