Figure 1.
Engineering stress-strain response of (a) DP1180, and (b) DDQ steel sheets determined in three orientations of RD, DD, and TD using a JIS No.5 standard tensile specimen at a strain rate of 0.001 s−1.
Figure 1.
Engineering stress-strain response of (a) DP1180, and (b) DDQ steel sheets determined in three orientations of RD, DD, and TD using a JIS No.5 standard tensile specimen at a strain rate of 0.001 s−1.
Figure 2.
Geometries of (a) plane strain notch specimen, and (b) NT20 notch specimen of Roth and Mohr (2016). All dimensions are in millimeters and GL stands for “gauge length” selected to report far-field elongation.
Figure 2.
Geometries of (a) plane strain notch specimen, and (b) NT20 notch specimen of Roth and Mohr (2016). All dimensions are in millimeters and GL stands for “gauge length” selected to report far-field elongation.
Figure 3.
(a) Universal tensile frame and DIC setup, and (b) plane strain notch, and (c) NT20 notch specimens after and before fracture.
Figure 3.
(a) Universal tensile frame and DIC setup, and (b) plane strain notch, and (c) NT20 notch specimens after and before fracture.
Figure 4.
Schematic of a yield locus displaying different stress states represented in the RD-TD plane. The following abbreviations were used: UT = uniaxial tension, UC = uniaxial compression, PST = plane strain tension, PSC = plane strain compression, EBT = equal biaxial tension, EBC = equal biaxial compression, SH = shear.
Figure 4.
Schematic of a yield locus displaying different stress states represented in the RD-TD plane. The following abbreviations were used: UT = uniaxial tension, UC = uniaxial compression, PST = plane strain tension, PSC = plane strain compression, EBT = equal biaxial tension, EBC = equal biaxial compression, SH = shear.
Figure 5.
Yield loci of DP1180 calibrated using conventional calibration and a calibration with the plane strain constraint enforced plotted in (a) principal stress, and (b) principal stress deviator planes. Symbols show the plane strain points with horizontal normal vectors (N2 = 0) in which only the calibration with the plane strain constraint enforced can satisfy s2 = 0.
Figure 5.
Yield loci of DP1180 calibrated using conventional calibration and a calibration with the plane strain constraint enforced plotted in (a) principal stress, and (b) principal stress deviator planes. Symbols show the plane strain points with horizontal normal vectors (N2 = 0) in which only the calibration with the plane strain constraint enforced can satisfy s2 = 0.
Figure 6.
Experimental evidence collected by Fast-Irvine et al. [
13] supporting the plane strain constraint for several body-centered cubic (BCC) and face-centered cubic (FCC) materials. For the plane strain deformation to occur, the normal vector should be either horizontal (
β2 = 0°) or vertical (
β2 = 90°) with the in-plane loading angles of
β1 = 26.6° and
β1 = 63.4° corresponding to loading along the RD and TD, respectively.
Figure 6.
Experimental evidence collected by Fast-Irvine et al. [
13] supporting the plane strain constraint for several body-centered cubic (BCC) and face-centered cubic (FCC) materials. For the plane strain deformation to occur, the normal vector should be either horizontal (
β2 = 0°) or vertical (
β2 = 90°) with the in-plane loading angles of
β1 = 26.6° and
β1 = 63.4° corresponding to loading along the RD and TD, respectively.
Figure 7.
Bezier curve is constructed between two reference points (σi and σj) using a hinge point σh.
Figure 7.
Bezier curve is constructed between two reference points (σi and σj) using a hinge point σh.
Figure 8.
Hardening response of the materials in (a) RD, (b) DD, and (c) TD. A good agreement was obtained with the modified Hockett-Sherby model that was strictly valid until the uniform elongation strain.
Figure 8.
Hardening response of the materials in (a) RD, (b) DD, and (c) TD. A good agreement was obtained with the modified Hockett-Sherby model that was strictly valid until the uniform elongation strain.
Figure 9.
Discretized models of (a) plane strain notch and (b) NT20 notch specimens considering symmetry with the symmetry axes depicted.
Figure 9.
Discretized models of (a) plane strain notch and (b) NT20 notch specimens considering symmetry with the symmetry axes depicted.
Figure 10.
(a) Engineering stress-strain response of the DP1180 in the RD, (b) yield loci in the tensile region demonstrated that increasing the plane strain yield stress had a direct impact on the engineering stress-strain behavior of the material.
Figure 10.
(a) Engineering stress-strain response of the DP1180 in the RD, (b) yield loci in the tensile region demonstrated that increasing the plane strain yield stress had a direct impact on the engineering stress-strain behavior of the material.
Figure 11.
Engineering stress-strain response of plane strain notch tests extracted from models and experiments along with the yield loci for DP1180 in (a) RD, (b) DD, and (c) TD. A gauge length of 16 mm was used to calculate far-field strains in the models and experiments.
Figure 11.
Engineering stress-strain response of plane strain notch tests extracted from models and experiments along with the yield loci for DP1180 in (a) RD, (b) DD, and (c) TD. A gauge length of 16 mm was used to calculate far-field strains in the models and experiments.
Figure 12.
Engineering stress-strain response of plane strain notch tests extracted from models and experiments along with the yield loci for DDQ in (a) RD, (b) DD, and (c) TD. A gauge length of 16 mm was used to calculate far-field strains in the models and experiments.
Figure 12.
Engineering stress-strain response of plane strain notch tests extracted from models and experiments along with the yield loci for DDQ in (a) RD, (b) DD, and (c) TD. A gauge length of 16 mm was used to calculate far-field strains in the models and experiments.
Figure 13.
Measured DIC strain paths at the center of the plane strain notch tests compared to model predictions for DP1180 in (a) RD, (b) DD, and (c) TD.
Figure 13.
Measured DIC strain paths at the center of the plane strain notch tests compared to model predictions for DP1180 in (a) RD, (b) DD, and (c) TD.
Figure 14.
Measured DIC strain paths at the center of the plane strain notch tests compared to model predictions for DDQ in (a) RD, (b) DD, and (c) TD.
Figure 14.
Measured DIC strain paths at the center of the plane strain notch tests compared to model predictions for DDQ in (a) RD, (b) DD, and (c) TD.
Figure 15.
Engineering stress-strain response and local strain paths extracted from NT20 models and experiments for DP1180 in (a,b) RD, (c,d) DD, and (e,f) TD. A gauge length of 35 mm was used to calculate far-field strains in the models and experiments and the local strain paths were extracted from the center of the specimen.
Figure 15.
Engineering stress-strain response and local strain paths extracted from NT20 models and experiments for DP1180 in (a,b) RD, (c,d) DD, and (e,f) TD. A gauge length of 35 mm was used to calculate far-field strains in the models and experiments and the local strain paths were extracted from the center of the specimen.
Figure 16.
Engineering stress-strain response and local strain paths extracted from NT20 models and experiments for DDQ in (a,b) RD, (c,d) DD, and (e,f) TD. A gauge length of 35 mm was used to calculate far-field strains in the models and experiments and the local strain paths were extracted from the center of the specimen.
Figure 16.
Engineering stress-strain response and local strain paths extracted from NT20 models and experiments for DDQ in (a,b) RD, (c,d) DD, and (e,f) TD. A gauge length of 35 mm was used to calculate far-field strains in the models and experiments and the local strain paths were extracted from the center of the specimen.
Figure 17.
(a) The Yld2000 and Vegter yield loci and (b) normalized stresses and R-values predictions for DP1180. Symbols indicate the data used for calibration.
Figure 17.
(a) The Yld2000 and Vegter yield loci and (b) normalized stresses and R-values predictions for DP1180. Symbols indicate the data used for calibration.
Figure 18.
(a) The Yld2000 and Vegter yield loci and (b) normalized stresses and R-values predictions for DDQ. Symbols indicate the data used for calibration.
Figure 18.
(a) The Yld2000 and Vegter yield loci and (b) normalized stresses and R-values predictions for DDQ. Symbols indicate the data used for calibration.
Table 1.
Experimentally characterized R-values (R) and yield stresses in tension (σ) and shear (τ) normalized with respect to tensile stress in the RD. The subscript indicates the orientation (in degrees) of the major principal direction with respect to the rolling direction of the sheets. The values provided in brackets present the standard deviations. The variable wp is the plastic work per unit volume that was selected based on the tensile orientation with the lowest necking strain and was the plastic work at the start of diffuse necking for DP1180, and half of the plastic work at diffuse necking for DDQ to reduce potential distortional hardening effects.
Table 1.
Experimentally characterized R-values (R) and yield stresses in tension (σ) and shear (τ) normalized with respect to tensile stress in the RD. The subscript indicates the orientation (in degrees) of the major principal direction with respect to the rolling direction of the sheets. The values provided in brackets present the standard deviations. The variable wp is the plastic work per unit volume that was selected based on the tensile orientation with the lowest necking strain and was the plastic work at the start of diffuse necking for DP1180, and half of the plastic work at diffuse necking for DDQ to reduce potential distortional hardening effects.
Material | DP1180 | DDQ |
---|
wp [MJ/m3] | 61.11 | 34.76 |
σ0/σ0 | 1.000 (0.006) | 1.000 (0.006) |
σ15/σ0 | 0.995 (0.003) | - |
σ22.5/σ0 | - | 1.024 (0.007) |
σ30/σ0 | 0.996 (0.003) | - |
σ45/σ0 | 1.004 (0.007) | 1.042 (0.008) |
σ60/σ0 | 1.008 (0.008) | - |
σ67.5/σ0 | - | 1.019 (0.007) |
σ75/σ0 | 1.013 (0.003) | - |
σ90/σ0 | 1.025 (0.007) | 0.992 (0.008) |
τ0/σ0 | 0.600 (0.005) | 0.571 (0.010) |
τ22.5/σ0 | 0.600 (0.008) | 0.595 (0.021) |
τ45/σ0 | 0.612 (0.005) | 0.607 (0.009) |
R0 | 0.82 (0.01) | 2.09 (0.04) |
R15 | 0.84 (0.01) | - |
R22.5 | - | 1.82 (0.03) |
R30 | 0.90 (0.01) | - |
R45 | 0.95 (0.01) | 1.55 (0.03) |
R60 | 0.98 (0.01) | - |
R67.5 | - | 1.98 (0.04) |
R75 | 1.00 (0.00) | - |
R90 | 0.98 (0.01) | 2.46 (0.04) |
Rb | 0.94 (0.03) | 0.85 (0.03) |
Table 2.
DIC parameters used for strain measurements.
Table 2.
DIC parameters used for strain measurements.
Test | Material | Test Direction | Image Resolution (mm/pixel) | Step Size (pixel) | Filter Size | Subset Size (pixel) | VSG (mm) |
---|
Plane-Strain Notch | DP1180 | RD, DD, TD | 0.016 | 3 | 11 | 31 | ~1.0 |
DDQ | RD, DD, TD | 0.016 | 6 | 15 | 31 | ~2.0 |
NT20 Notch | DP1180 | RD, DD, TD | 0.019 | 3 | 9 | 27 | ~1.0 |
DDQ | RD, DD, TD | 0.019 | 6 | 15 | 27 | ~2.0 |
Table 3.
Coefficients of the hardening model determined from a uniaxial tensile test data. The inverse finite-element analysis was limited to local strains that were below the uniform elongation. The uniform elongation was enforced in the calibration process via the Considère criterion.
Table 3.
Coefficients of the hardening model determined from a uniaxial tensile test data. The inverse finite-element analysis was limited to local strains that were below the uniform elongation. The uniform elongation was enforced in the calibration process via the Considère criterion.
Materials | Modified Hockett-Sherby Model | Experiment Uniform Elongation Strain |
---|
A(MPa) | B (MPa) | C | D | E (MPa) |
---|
DP1180-RD | 704.51 | 1104.65 | 91.91 | 0.88 | 689.05 | 0.071 ± 0.002 |
DP1180-DD | 726.89 | 1127.60 | 75.11 | 0.87 | 598.47 | 0.066 ± 0.002 |
DP1180-TD | 723.51 | 1157.59 | 89.18 | 0.88 | 594.46 | 0.065 ± 0.004 |
DDQ-RD | 123.39 | 176.58 | 34.98 | 1.27 | 356.91 | 0.294 ± 0.001 |
DDQ-DD | 136.43 | 191.04 | 55.83 | 1.43 | 354.49 | 0.272 ± 0.001 |
DDQ-TD | 133.43 | 175.69 | 79.05 | 1.65 | 356.43 | 0.293 ± 0.004 |
Table 4.
Error (Equation (15)) in resolving measured engineering stress, major strain, and minor strain for the plane strain notch tests.
Table 4.
Error (Equation (15)) in resolving measured engineering stress, major strain, and minor strain for the plane strain notch tests.
Material-Direction | Error (Eng. Stress) [MPa] | Error (Major Strain) | Error (Minor Strain) |
---|
DP1180-RD | 4.01 ± 1.28 | 0.0006 ± 0.0003 | 0.0007 ± 0.0006 |
DP1180-DD | 4.17 ± 1.54 | 0.0015 ± 0.0009 | 0.0012 ± 0.0004 |
DP1180-TD | 4.95 ± 1.95 | 0.0008 ± 0.0001 | 0.0006 ± 0.0002 |
DDQ-RD | 2.49 ± 0.31 | 0.0026 ± 0.0003 | 0.0005 ± 0.0001 |
DDQ-DD | 2.02 ± 0.57 | 0.0008 ± 0.0003 | 0.0007 ± 0.0001 |
DDQ-TD | 2.38 ± 0.42 | 0.0016 ± 0.0005 | 0.0005 ± 0.0002 |
Table 5.
Normalized plane strain yield strengths in different orientations derived from the inverse analysis of notch tension tests.
Table 5.
Normalized plane strain yield strengths in different orientations derived from the inverse analysis of notch tension tests.
Material-Direction | Normalized Yield Strength | Normalized Yield Strength w.r.t RD |
---|
DP1180-RD | 1.10 | 1.10 |
DP1180-DD | 1.11 | 1.12 |
DP1180-TD | 1.12 | 1.14 |
DDQ-RD | 1.27 | 1.27 |
DDQ-DD | 1.17 | 1.22 |
DDQ-TD | 1.26 | 1.25 |
Table 6.
Error (Equation (15)) in resolving measured engineering stress, major strain, and minor strain for the NT20 notch tests.
Table 6.
Error (Equation (15)) in resolving measured engineering stress, major strain, and minor strain for the NT20 notch tests.
Material-Direction | Error (Eng. Stress) [MPa] | Error (Major Strain) | Error (Minor Strain) |
---|
DP1180-RD | 7.22 ± 3.35 | 0.0053 ± 0.0007 | 0.0019 ± 0.0005 |
DP1180-DD | 4.06 ± 2.10 | 0.0063 ± 0.0004 | 0.0053 ± 0.0004 |
DP1180-TD | 6.23 ± 2.05 | 0.0071 ± 0.0005 | 0.0073 ± 0.0004 |
DDQ-RD | 5.83 ± 1.92 | 0.0165 ± 0.0017 | 0.0158 ± 0.0016 |
DDQ-DD | 3.36 ± 1.50 | 0.0161 ± 0.0014 | 0.0127 ± 0.0007 |
DDQ-TD | 2.87 ± 1.94 | 0.0103 ± 0.0014 | 0.0097 ± 0.0013 |
Table 7.
Coefficients of the Yld2000 yield criterion for DP1180 and DDQ. A yield exponent of 6.0 was utilized as suggested for BCC materials.
Table 7.
Coefficients of the Yld2000 yield criterion for DP1180 and DDQ. A yield exponent of 6.0 was utilized as suggested for BCC materials.
Coefficient | DP1180 | DDQ |
---|
α1 | 0.9684 | 1.0125 |
α2 | 0.9864 | 1.0752 |
α3 | 1.0303 | 0.7966 |
α4 | 0.9855 | 0.8816 |
α5 | 1.0066 | 0.9032 |
α6 | 0.9582 | 0.8092 |
α7 | 0.9943 | 0.9883 |
α8 | 1.0158 | 1.0213 |
a | 6.0 | 6.0 |