Fig. 1. Nucleation occurs at the matrix–particle interface owing to tensile stress.²⁴

Fig. 2. Critical strain of void nucleation at particles.^27–31

Fig. 3. Schematic of void nucleation at a particle. Separation occurs at the poles of the particle.²⁷

Fig. 4. (a) 2D configuration of dislocation emission. (b) Stress state at the point of dislocation due to equal biaxial tension σ.²⁰

Fig. 5. Normalized critical emission stress vs normalized radius of void. The three curves represent three different dislocation widths: w = b, w = 1.5b, and w = 2b.²⁰

Fig. 6. 3D configuration of dislocation emission. Variables with subscript 0 are geometric parameters associated with the prismatic dislocation loop (PDL). z₀ = a cos θ_cr + w is the equilibrium position of the PDL, ρ₀ = a sin θ_cr is the radius of the PDL, and r0=z02+ρ02 and θ₀ = arctan(ρ₀/z₀) specify the position of the PDL.⁵³

Fig. 7. Effect of porosity f on dislocation emission. σ_cr/μ and a/b are the normalized critical emission stress and the normalized radius of the void, respectively.⁵³

Fig. 8. Effect of stress triaxiality on dislocation emission. η = 0 and η = 1 correspond to the cases of uniaxial tension and hydrostatic tension, respectively.⁵³

Fig. 9. Simulated free-surface velocity profiles. Two strategies of homogenization modeling are adopted: the p-model assumes that a uniform pressure is applied to all unit cells, while the d-model assumes that a uniform strain rate is prescribed on unit cells. For more details, see Czarnota et al.⁷⁶

Fig. 10. J-resistance curves for the growth of a ductile crack.^78,102 Results are presented for different tractions T_a = 1100 and 1500 MPa and initial void radii a₁ = 1.5 and 5 µm.⁷⁸

Fig. 11. Influence of strain rate on spall strength for aluminum samples with different purity.^19,103–110

Fig. 12. Material velocity profiles for different shock stress amplitudes.⁸²