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Fig. 5. FLC for different strain hardening exponents (n) derived from: (a) Swift diffusive necking theory, (b) Hill-Swift necking theory, and (c) schematic rep- resentation of Hill and Swift necking theory.

Figure 5 FLC for different strain hardening exponents (n) derived from: (a) Swift diffusive necking theory, (b) Hill-Swift necking theory, and (c) schematic rep- resentation of Hill and Swift necking theory.

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Fig. 1. Schematic illustration of forming limit diagram (FLD).
Fig. 1. Schematic illustration of forming limit diagram (FLD).
Fig. 2. (a) Schematic of forming limit diagram (FLD) in orthotropic axes, (b) schematic of FLD representing safe forming zone.
Fig. 2. (a) Schematic of forming limit diagram (FLD) in orthotropic axes, (b) schematic of FLD representing safe forming zone.
Fig. 3. (a) diffuse neck followed by localised neck formation during a uniaxial tensile test, (b) major (€,) and minor (€2) strain alteration with time. (Paul et al., 2018). Diffuse necking takes place in a bigger axial length, which is larger than the specimen thickness. The detection of a localised necking start point is painful and difficult. Generally, localised necking happens within the diffuse necking zone, and the axial length of localised necking is comparable to the sheet thickness. Within the localised necking zone, approximately plane strain deformation happens (i.e., transverse (width) strain maintains constant although axial strain and bulk strain rise). Commencement of localised necking can be directly detected from the local axial and transverse strain data, and it is explained with a scatch in Fig. 3(b).
Fig. 3. (a) diffuse neck followed by localised neck formation during a uniaxial tensile test, (b) major (€,) and minor (€2) strain alteration with time. (Paul et al., 2018). Diffuse necking takes place in a bigger axial length, which is larger than the specimen thickness. The detection of a localised necking start point is painful and difficult. Generally, localised necking happens within the diffuse necking zone, and the axial length of localised necking is comparable to the sheet thickness. Within the localised necking zone, approximately plane strain deformation happens (i.e., transverse (width) strain maintains constant although axial strain and bulk strain rise). Commencement of localised necking can be directly detected from the local axial and transverse strain data, and it is explained with a scatch in Fig. 3(b).
Fig. 4. Schematic diagram of the uniaxial tensile test (a) engineering and true stress-strain curve, (b) forming limit diagram (FLD). quantify the anisotropy in the sheet. The r-values are measured from the width and true longitudinal strains as per Eq. (3).
Fig. 4. Schematic diagram of the uniaxial tensile test (a) engineering and true stress-strain curve, (b) forming limit diagram (FLD). quantify the anisotropy in the sheet. The r-values are measured from the width and true longitudinal strains as per Eq. (3).
Variation of FLCo, €4, and € for various types of steel-grades. 3.4. Present model (Model 2) Table 1
Variation of FLCo, €4, and € for various types of steel-grades. 3.4. Present model (Model 2) Table 1
Fig. 7. Prediction of experimental FLC by models (model 1 & 2) for (a) EDD steel with a thickness of 1.25 mm, (b) EDD steel with a thickness of 2 mm, (c) DP980 steel with a thickness of 1.2 mm, and (d) DP780 steel with a thickness of 1.2 mm. Experimental FLCs are collected from the literature (Kumar, 2002; Chung et al., 2014; Panich, 2013).
Fig. 7. Prediction of experimental FLC by models (model 1 & 2) for (a) EDD steel with a thickness of 1.25 mm, (b) EDD steel with a thickness of 2 mm, (c) DP980 steel with a thickness of 1.2 mm, and (d) DP780 steel with a thickness of 1.2 mm. Experimental FLCs are collected from the literature (Kumar, 2002; Chung et al., 2014; Panich, 2013).
Fig. 8. Prediction of experimental FLC by models (model 1 & model 2) for (a) TWIP940 steel with a thickness of 1.47 mm, and (b) TRIP780 steel with a thickness of 1.2 mm. Experimental FLCs are collected from the literature (Chung et al., 2011; Panich, 2013).
Fig. 8. Prediction of experimental FLC by models (model 1 & model 2) for (a) TWIP940 steel with a thickness of 1.47 mm, and (b) TRIP780 steel with a thickness of 1.2 mm. Experimental FLCs are collected from the literature (Chung et al., 2011; Panich, 2013).
Fig. 10. Key mechanical factors can result in superior forming limit. The localised necking limit can be determined from a typical inter- national standards organization (ISO) method (2008). Fig. 11(a) shows the major strain versus distance along with the gauge length plot. The major local strain corresponds to just before the failure of the specimen. The succeeding steps can be followed using the ISO technique (2008) to calculate the localised necking limit. (i) Just before the failure, local strain data points i.e. local major strain is fitted with parabolic equation f(x) = ax? + bx +c. The vertex of the parabola denotes the crack initi- ation location. From the fitted parabolic curve, the constants a, b, and c can be calculated. (ii) On both sides of the crack position, the inner boundary (IB) is calculated from the second derivative of f(x) with respect to the position i.e. f’(Q0 = 2a. (iii) According to ISO standard
Fig. 10. Key mechanical factors can result in superior forming limit. The localised necking limit can be determined from a typical inter- national standards organization (ISO) method (2008). Fig. 11(a) shows the major strain versus distance along with the gauge length plot. The major local strain corresponds to just before the failure of the specimen. The succeeding steps can be followed using the ISO technique (2008) to calculate the localised necking limit. (i) Just before the failure, local strain data points i.e. local major strain is fitted with parabolic equation f(x) = ax? + bx +c. The vertex of the parabola denotes the crack initi- ation location. From the fitted parabolic curve, the constants a, b, and c can be calculated. (ii) On both sides of the crack position, the inner boundary (IB) is calculated from the second derivative of f(x) with respect to the position i.e. f’(Q0 = 2a. (iii) According to ISO standard
Fig. 9. Material factors that can affect the forming limit of sheet metals.
Fig. 9. Material factors that can affect the forming limit of sheet metals.
Fig. 12. Difference between true and engineering strain space plot of FLC. Experimental FLC in engineering strain space is collected from Kumar (2002). Finite element software is successfully used today in die design, sheet product geometry optimisation, and selecting the appropriate material
Fig. 12. Difference between true and engineering strain space plot of FLC. Experimental FLC in engineering strain space is collected from Kumar (2002). Finite element software is successfully used today in die design, sheet product geometry optimisation, and selecting the appropriate material
Fig. 11. Determination of limiting strain by (a) ISO method, (b) time-dependent method, (c) slope method, and (d) Flat valley method (Temporal evolution of Z (vertical displacement profile) (in top row), and the spatial derivative Of Z (in bottom row) in a section perpendicular to the failure region).
Fig. 11. Determination of limiting strain by (a) ISO method, (b) time-dependent method, (c) slope method, and (d) Flat valley method (Temporal evolution of Z (vertical displacement profile) (in top row), and the spatial derivative Of Z (in bottom row) in a section perpendicular to the failure region).
Fig. 13. Test setup for the Nakazima (left) and the Marciniak tests (right). All dimensions are in mm.
Fig. 13. Test setup for the Nakazima (left) and the Marciniak tests (right). All dimensions are in mm.
Fig. 14. FLCs for the DP980 steel in the Nakazima and Marciniak tests. Adopting the ISO-Method to measure the limiting strains. Experimental data are collected from Noder and Butcher (2019). stretching superimposed with bending according to the concave-side rule (CSR). As per CSR, the neck will form on the outer radius (i.e., convex surface) when the strain on the inner radius (i.e., concave surface) touches the plane-strain limiting strain (FLCo) for in-plane deformation. Kitting et al. (2009) were conducted angular stretch bend (ASB) tests on H340LAD micro-alloyed steel and reported that CSR is applicable for mild superimposed bending. Neuhauser et al. (2016) were conducted ASB tests on DP steels and noticed that the CSR is valid for severe superimposed bending. Noder and Butcher (2019) have suggested that the FLC determination from Marciniak tests should be preferred as strain paths remain almost linear, and the contact pressure rectifications can be avoided. Neuhauser et al. (2019) reported that FLC increases with a rise in t/R ratio (where t is the sheet thickness, and R is the punch radius). During bending, the top surface deformation is constrained by the lower surface deformation. Consequently, the forming limit strain increases i.e., instability of the top surface gets delayed with bending. Bending is superimposed with in-plane deformation in the Nakazima test case and pure in-plane deformation in Marciniak test. Consequently, a slight
Fig. 14. FLCs for the DP980 steel in the Nakazima and Marciniak tests. Adopting the ISO-Method to measure the limiting strains. Experimental data are collected from Noder and Butcher (2019). stretching superimposed with bending according to the concave-side rule (CSR). As per CSR, the neck will form on the outer radius (i.e., convex surface) when the strain on the inner radius (i.e., concave surface) touches the plane-strain limiting strain (FLCo) for in-plane deformation. Kitting et al. (2009) were conducted angular stretch bend (ASB) tests on H340LAD micro-alloyed steel and reported that CSR is applicable for mild superimposed bending. Neuhauser et al. (2016) were conducted ASB tests on DP steels and noticed that the CSR is valid for severe superimposed bending. Noder and Butcher (2019) have suggested that the FLC determination from Marciniak tests should be preferred as strain paths remain almost linear, and the contact pressure rectifications can be avoided. Neuhauser et al. (2019) reported that FLC increases with a rise in t/R ratio (where t is the sheet thickness, and R is the punch radius). During bending, the top surface deformation is constrained by the lower surface deformation. Consequently, the forming limit strain increases i.e., instability of the top surface gets delayed with bending. Bending is superimposed with in-plane deformation in the Nakazima test case and pure in-plane deformation in Marciniak test. Consequently, a slight
Fig. 15. Strain paths and limiting strains for the DP980 acquired from the (a) Nakazima and (b) Marciniak tests. Experimental data are collected from Noder and Butcher (2019).
Fig. 15. Strain paths and limiting strains for the DP980 acquired from the (a) Nakazima and (b) Marciniak tests. Experimental data are collected from Noder and Butcher (2019).
Fig. 16. Effect of pre-straining on FLC. Experimental data are collected from Basak and Panda (2019).
Fig. 16. Effect of pre-straining on FLC. Experimental data are collected from Basak and Panda (2019).
Fig. 17. Schematic representation of the procedure to calculate path independent limiting criteria. Paul (2016)
Fig. 17. Schematic representation of the procedure to calculate path independent limiting criteria. Paul (2016)
Fig. 19. Effect of strain rate on FLC. Experimental data are collected from Chu et al. (2014). Equivalent strain (€eq), stress ratio (a), and triaxiality (n) can be determined from a suitable yield criterion (Fig. 17). While, equivalent stress (eq), major (o,) and minor (02) strains and hydrostatic/mean stress (6m) can be calculated by the use of yield criterion and strain- hardening law of the material at the same time (Fig. 17). The selection of yield criteria will affect the result i.e. error for all path independent
Fig. 19. Effect of strain rate on FLC. Experimental data are collected from Chu et al. (2014). Equivalent strain (€eq), stress ratio (a), and triaxiality (n) can be determined from a suitable yield criterion (Fig. 17). While, equivalent stress (eq), major (o,) and minor (02) strains and hydrostatic/mean stress (6m) can be calculated by the use of yield criterion and strain- hardening law of the material at the same time (Fig. 17). The selection of yield criteria will affect the result i.e. error for all path independent
Fig. 18. For EDD steel at different pre-straining conditions, using Hill yield theory the analytical mapping of (a) FLSD, (b) XFLSD, (c) SFLD, and (d) PEPSD
Fig. 18. For EDD steel at different pre-straining conditions, using Hill yield theory the analytical mapping of (a) FLSD, (b) XFLSD, (c) SFLD, and (d) PEPSD
Fig. 20. Effect of temperature on FLC. Experimental data are collected from Chu et al. (2014). One assumption, “the cumulative equivalent strain is not changing with various pre-straining paths” is made in all strain path independent strain-based limiting criteria (TFD, XFLD, ESRFLD, and PEPSD). Simi- larly, the assumption “the cumulative equivalent stress remains unal- tered in various pre-straining paths” is made for all the strain path independent stress-based limiting criteria (TFSD, FLSD, SRFLSD, XFLSD, and PESSD). The cumulative equivalent stress is calculated from the cumulative equivalent strain with an appropriate introduction of yield criteria. Therefore, all the strain path independent limiting criteria are assumed that the cumulative equivalent strain is not altering with different pre-straining paths. Paul (2016) concluded in his study that all the stated limiting criteria are roughly straining path independent when
Fig. 20. Effect of temperature on FLC. Experimental data are collected from Chu et al. (2014). One assumption, “the cumulative equivalent strain is not changing with various pre-straining paths” is made in all strain path independent strain-based limiting criteria (TFD, XFLD, ESRFLD, and PEPSD). Simi- larly, the assumption “the cumulative equivalent stress remains unal- tered in various pre-straining paths” is made for all the strain path independent stress-based limiting criteria (TFSD, FLSD, SRFLSD, XFLSD, and PESSD). The cumulative equivalent stress is calculated from the cumulative equivalent strain with an appropriate introduction of yield criteria. Therefore, all the strain path independent limiting criteria are assumed that the cumulative equivalent strain is not altering with different pre-straining paths. Paul (2016) concluded in his study that all the stated limiting criteria are roughly straining path independent when
Fig. 21. Correlation among FLCo and uniaxial tensile properties: (a) ultimate tensile stress (oyrs), (b) total elongation (e;), (c) coefficient of normal anisotropy, (r), (d strain hardening exponent (n), and (e) thickness (t) (Paul, 2015).
Fig. 21. Correlation among FLCo and uniaxial tensile properties: (a) ultimate tensile stress (oyrs), (b) total elongation (e;), (c) coefficient of normal anisotropy, (r), (d strain hardening exponent (n), and (e) thickness (t) (Paul, 2015).
Related topics:
Analysis of Sheet Metal Forming ProcessesHydroforming of sheet metal
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