Figure 1.
(a) Strong and (b) regularized discontinuities in an elasto-plastic solid.
Figure 1.
(a) Strong and (b) regularized discontinuities in an elasto-plastic solid.
Figure 2.
Kinematics of (a) strong and (b) regularized discontinuities.
Figure 2.
Kinematics of (a) strong and (b) regularized discontinuities.
Figure 3.
Yield criteria in Haigh–Westergaard (HW) stress space, lateral view from the hydrostatic axis: (a) Hill; (b) Parabolic Drucker–Prager; (c) Hoffman; (d) Tsai–Wu.
Figure 3.
Yield criteria in Haigh–Westergaard (HW) stress space, lateral view from the hydrostatic axis: (a) Hill; (b) Parabolic Drucker–Prager; (c) Hoffman; (d) Tsai–Wu.
Figure 4.
Definition of the localization angle .
Figure 4.
Definition of the localization angle .
Figure 5.
Cross sections of the yield criteria and strain localization angle under uniaxial tension and compression in plane stress and plane strain. (a) Plane stress with ; (b) Plane strain with ; (c) Plane stress with ; (d) Plane Strain with .
Figure 5.
Cross sections of the yield criteria and strain localization angle under uniaxial tension and compression in plane stress and plane strain. (a) Plane stress with ; (b) Plane strain with ; (c) Plane stress with ; (d) Plane Strain with .
Figure 6.
Geometry of a strip under vertical stretching.
Figure 6.
Geometry of a strip under vertical stretching.
Figure 7.
Strip under vertical plane stress tension (Parabolic Drucker–Prager): (a) identical to von Mises; (b) ; (c) .
Figure 7.
Strip under vertical plane stress tension (Parabolic Drucker–Prager): (a) identical to von Mises; (b) ; (c) .
Figure 8.
Strip under vertical plane stress compression (Parabolic Drucker–Prager): (a) identical to von Mises; (b) ; (c) .
Figure 8.
Strip under vertical plane stress compression (Parabolic Drucker–Prager): (a) identical to von Mises; (b) ; (c) .
Figure 9.
Strip under vertical plane strain tension (Parabolic Drucker–Prager): (a) , identical to von-Mises; (b) ; (c) .
Figure 9.
Strip under vertical plane strain tension (Parabolic Drucker–Prager): (a) , identical to von-Mises; (b) ; (c) .
Figure 10.
Strip under vertical plane strain compression (Parabolic Drucker–Prager): (a) identical to von-Mises; (b) ; (c) .
Figure 10.
Strip under vertical plane strain compression (Parabolic Drucker–Prager): (a) identical to von-Mises; (b) ; (c) .
Figure 11.
Strip under vertical plane stress tension (Cohesive–frictional models, ): (a) Parabolic Drucker–Prager; (b) Hoffman; (c) Tsai–Wu.
Figure 11.
Strip under vertical plane stress tension (Cohesive–frictional models, ): (a) Parabolic Drucker–Prager; (b) Hoffman; (c) Tsai–Wu.
Figure 12.
Strip under vertical plane stress compression (Cohesive–frictional models, ): (a) Parabolic Drucker–Prager; (b) Hoffman; (c) Tsai–Wu.
Figure 12.
Strip under vertical plane stress compression (Cohesive–frictional models, ): (a) Parabolic Drucker–Prager; (b) Hoffman; (c) Tsai–Wu.
Figure 13.
Strip under vertical plane strain tension (Cohesive–frictional models, ): (a) Parabolic Drucker–Prager; (b) Hoffman; (c) Tsai–Wu.
Figure 13.
Strip under vertical plane strain tension (Cohesive–frictional models, ): (a) Parabolic Drucker–Prager; (b) Hoffman; (c) Tsai–Wu.
Figure 14.
Strip under vertical plane strain tension (Cohesive–frictional models, ): (a) Parabolic Drucker–Prager; (b) Hoffman; (c) Tsai–Wu.
Figure 14.
Strip under vertical plane strain tension (Cohesive–frictional models, ): (a) Parabolic Drucker–Prager; (b) Hoffman; (c) Tsai–Wu.
Figure 15.
Geometry Prandtl’s punch test. The bottom edge is fixed in both directions, while the left and right edges are constrained along the horizontal direction.
Figure 15.
Geometry Prandtl’s punch test. The bottom edge is fixed in both directions, while the left and right edges are constrained along the horizontal direction.
Figure 16.
Prandtl’s punch test: (a) von Mises; (b) Parabolic Drucker–Prager, ; (c) Hoffman, ; (d) Tsai–Wu.
Figure 16.
Prandtl’s punch test: (a) von Mises; (b) Parabolic Drucker–Prager, ; (c) Hoffman, ; (d) Tsai–Wu.
Figure 17.
Directions of principal stresses below the rigid footing of Prandtl’s punch test (Parabolic Drucker–Prager, ): (a) Elastic stage; (b) Final plastic stage.
Figure 17.
Directions of principal stresses below the rigid footing of Prandtl’s punch test (Parabolic Drucker–Prager, ): (a) Elastic stage; (b) Final plastic stage.
Figure 18.
Stresses of the slide lines of Prandtl’s punch test (Plane Strain, ): (a) von Mises; (b) Parabolic Drucker–Prager; (c) Hoffman; (d) Tsai–Wu.
Figure 18.
Stresses of the slide lines of Prandtl’s punch test (Plane Strain, ): (a) von Mises; (b) Parabolic Drucker–Prager; (c) Hoffman; (d) Tsai–Wu.
Figure 19.
Results comparison where the slide lines cross (Plane Strain, : (a) Stress Invariant ; (b) Lode angle .
Figure 19.
Results comparison where the slide lines cross (Plane Strain, : (a) Stress Invariant ; (b) Lode angle .
Table 1.
Coefficients for matrix and vector ().
Table 1.
Coefficients for matrix and vector ().
| | | | | | | | | |
---|
DP | 1/3 | 1/3 | 1/3 | F | G | H | 1/3 | 1/3 | 1/3 |
Hoffman | 1/3 | 1/3 | 2/3 | F | G | H | 1/3 | 0 | 0 |
Tsai–Wu | 1/3 | 1/3 | 2/3 | | | 0.5 | 1/3 | 0 | 0 |
Table 2.
Plane Stress: stress state and slip-line angles for uniaxial tension and compression.
Table 2.
Plane Stress: stress state and slip-line angles for uniaxial tension and compression.
| Tension | Compression |
---|
| | | | | |
---|
DP | 1.0000 | 0.0000 | | 1.0000 | 0.0000 | |
Hoffman | 1.0000 | 0.0000 | ±22.2077° | 1.0000 | 0.0000 | |
Tsai–Wu | 1.0000 | 0.0000 | | 1.0000 | 0.0000 | |
Table 3.
Plane Strain: stress state and slip-line angles for uniaxial tension and compression.
Table 3.
Plane Strain: stress state and slip-line angles for uniaxial tension and compression.
| Tension | Compression |
---|
| | | | | |
---|
DP | 1.0275 | 0.2638 | | 2.0275 | 1.2638 | |
Hoffman | 1.3416 | 0.8944 | | 1.3416 | 0.8944 | |
Tsai–Wu | 1.1547 | 0.5774 | | 1.1547 | 0.5774 | |
Table 4.
Analytical and numerical Lode and strain localization angles for isotropic models under plane stress tension.
Table 4.
Analytical and numerical Lode and strain localization angles for isotropic models under plane stress tension.
| | | | |
---|
VM 1.0 | 0.0000° | 0.6459° | 35.2644° | 35.4699° |
DP 1.25 | 0.0000° | 0.3803° | 30.0000° | 30.4342° |
DP 1.5 | 0.0000° | 0.7800° | 24.0948° | 24.2277° |
Table 5.
Analytical and numerical Lode and strain localization angles for isotropic models under plane stress compression.
Table 5.
Analytical and numerical Lode and strain localization angles for isotropic models under plane stress compression.
| | | | |
---|
VM 1.0 | 60.0000° | 59.3541° | 35.2644° | 35.4699° |
DP 2.0 | 60.0000° | 59.6989° | 45.0000° | 45.0000° |
DP 3.0 | 60.0000° | 59.6006° | 48.1897° | 48.8141° |
Table 6.
Analytical and numerical Lode and strain localization angles for isotropic models under plane strain tension.
Table 6.
Analytical and numerical Lode and strain localization angles for isotropic models under plane strain tension.
| | | | |
---|
VM 1.0 | 30.0000° | 30.1669° | 45.0000° | 45.0000° |
DP 1.25 | 22.3378° | 23.6244° | 38.2626° | 39.0939° |
DP 1.5 | 14.3077° | 15.9519° | 30.4411° | 31.4875° |
Table 7.
Analytical and numerical Lode and strain localization angles for isotropic models under plane strain compression.
Table 7.
Analytical and numerical Lode and strain localization angles for isotropic models under plane strain compression.
| | | | |
---|
VM 1.0 | 30.0000° | 30.1669° | 45.0000° | 45.0000° |
DP 2.0 | 19.1066° | 19.6359° | 54.7356° | 54.2934° |
DP 3.0 | 17.1330° | 17.6788° | 56.6531° | 57.5289° |
Table 8.
Analytical and numerical Lode and strain localization angles for frictional–cohesive models under plane stress tension,
Table 8.
Analytical and numerical Lode and strain localization angles for frictional–cohesive models under plane stress tension,
| | | | |
---|
Drucker–Prager | 0.0000° | 0.7800° | 24.0948° | 24.2277° |
Hoffman | 0.0000° | 0.5258° | 22.2077° | 22.1355° |
Tsai–Wu | 0.0000° | 0.3457° | 26.1746° | 26.5651° |
Table 9.
Analytical and numerical Lode and strain localization angles for frictional–cohesive models under plane stress compression,
Table 9.
Analytical and numerical Lode and strain localization angles for frictional–cohesive models under plane stress compression,
| | | | |
---|
Drucker–Prager | 60.0000° | 59.6006° | 48.1897° | 48.8141° |
Hoffman | 60.0000° | 58.6061° | 35.2644° | 35.4699° |
Tsai–Wu | 60.0000° | 59.8503° | 38.2620° | 37.7757° |
Table 10.
Analytical and numerical Lode and strain localization angles for frictional–cohesive models under plane strain tension,
Table 10.
Analytical and numerical Lode and strain localization angles for frictional–cohesive models under plane strain tension,
| | | | |
---|
Drucker–Prager | 14.3077° | 15.9519° | 30.4411° | 31.4875° |
Hoffman | 40.8934° | 42.5043 | 41.3843° | 41.5891° |
Tsai–Wu | 30.0000 ° | 30.7240° | 38.3075° | 38.2204° |
Table 11.
Analytical and numerical Lode and strain localization angles for frictional–cohesive models under plane strain compression,
Table 11.
Analytical and numerical Lode and strain localization angles for frictional–cohesive models under plane strain compression,
| | | | |
---|
Drucker–Prager | 17.1330° | 17.6788° | 56.6531° | 57.5288° |
Hoffman | 8.9483° | 7.8626° | 51.6975° | 50.7106° |
Tsai–Wu | 30.0000° | 29.4419° | 44.4488° | 44.6397° |
Table 12.
Stresses and localization angle in Prandtl’s punch test.
Table 12.
Stresses and localization angle in Prandtl’s punch test.
| | | | | | |
---|
VM | 18286 | 29832 | 24061 | 29.9902° | 45.0000° | 45.0000° |
Drucker–Prager | 104260 | 220560 | 171920 | 24.6035° | 50.1944° | 49.9512° |
Hoffman | 57783 | 97541 | 92394 | 6.8372° | 48.9909° | 48.4646° |
Tsai–Wu | 72007 | 69365 | 56219 | 8.9246° | 41.1859° | 40.6354° |