Fig. 1. Stages of ductile fracture of high strength steels.

Fig. 2. Proposed multi-scale framework for the modeling of high strength steel showing the three nested domains centered at x. The square shaped particles are primary particles and the circular inclusions are secondary particles.

Fig. 3. Framework for the three-scale homogenization of high strength steel.

Fig. 4. Summary of the constitutive relation.

Fig. 5. Summary of the model and homogenization.

Fig. 6. Hierarchical framework for the determination of the macroscopic constitutive relation.

Fig. 7. One-dimensional tensile bar.

Fig. 8. Logarithmic tensile strain profile for different stages of deformation.

Fig. 9. Solutions for conventional continuum and multi-scale continuum.

Fig. 10. Profile of tensile strains for different particle sizes: (a) effect of the primary particles size; (b) effect of secondary particles size; and (c) effect of debonding peak stress at TiC–Fe interface (secondary particles).

Fig. 11. (a) Setting of the shear test experiment and finite element discretization used for (b) simulation with explicit microstructure and (c) three-scale micromorphic model.

Fig. 12. (a) Snapshots of the plastic deformation of the shear specimen with explicit microstructure; (b–d) snapshots of the plastic deformation, micro- and submicro-stresses of the shear specimen using the three-scale material model.

Fig. 13. Average shear stress–strain curve measured in the specimen.

Fig. 14. Geometry, boundary conditions and finite element discretization of the tensile specimen.

Fig. 15. Contours of effective plastic strain at the macro-, micro- and submicro-scales at different stages of deformation.

Fig. 16. Set up of the cell model for periodic boundary conditions.

Fig. 17. Set up of the cell model for boundary conditions with constant stress ratio.

Fig. 18. RVE for porous material subject to periodic boundary conditions and associated tensile stress–strain curves for various triaxialities.

Material properties, size and volume fraction of particles

Material properties of the matrix material

Material parameters used at the macroscale

Material parameters used at the microscale

Material parameters used at the submicro-scale