3IVM: Three Internal Variable Model
Definition:Statistical strain hardening model based on three types of dislocation densities.
Explanation:Constitutive models based on dislocation densities as internal state variables were first introduced by Kocks (1976). The 3IVM (Roters 2000) combines it with the cell structure model introduced by Mughrabi (1983).

The model uses three types of dislocation densities associated with individual kinetics. Immobile dislocations occur both in the cell interiors and in the cell walls building the dislocation cell structure while mobile dislocations move through this cell structure. The Orowan equation is used as kinetic equation of state and the evolution of the dislocation densities is described by rate equations. These are formulated based on the elementary dislocation processes: multiplication, formation of locks and dipoles and annihilation by climb.

The model was originally developed for the description of aluminium alloys at elevated temperatures. It was later extended to better describe the temperature dependence of strain hardening and softening phenomena and render it also applicable at low temperatures (Prasad 2007). This extended formulation is called 3IVM+.
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(a) Schematic of the dislocation structure; ρw: dislocations in cell wall; ρi: dislocations in cell interior; ρm: mobile dislocations (b) Comparison of experimental and simulated stress (σ)- strain (ε) curve; AA2024 at 350°C, strain rate 10-3 s-1
SFB-Link:The TWIP model (Steinmetz 2013) developed within SFB 761 can be seen as an extension of the 3IVM here used for austenitic steels and including twinning as additional deformation and hardening mechanisms (→deformation mechanisms).
References:Kocks U.F.; J. Eng. Mater. Technol. 98, 1976, p 72
Mughrabi H.; Acta Metall. 31, 1983, p 1367
Prasad G.V.S.S.; PhD thesis, 2007, RWTH Aachen University
Roters F., Raabe D., Gottstein G.; Acta Mater. 48, 2000 p 4181
Steinmetz D.R., Jäpel T., Wietbrock B., Eisenlohr P., Gutierrez-Urrutia I., Saeed–Akbari A., Hickel T., Roters F. Raabe D.; Acta Mater. 61, 2013, p 494