Abstract
A significant barrier to broader implementation of magnesium alloys is their poor room temperature formability, a consequence of the anisotropic response of the Mg hexagonal closedpacked (hcp) crystal structure. Additions of rare earth (RE) elements, such as in the ZEK100 alloys, weaken the texture and improve formability. Room temperature forming limit analyses of REcontaining Mg alloys, particularly Mg ZEK100, have not been explored to any significant extent in the literature. In this paper, strainbased forming limit diagrams (FLDs) are derived for an Mg ZEK100O alloy (Zn1.34Zr0.23Nd0.182, wt.%) using an analytical method that combines the vertex theory of Storen and Rice (J Mech Phys Solids, 23:421441, 1979), the anisotropic yield criterion of Barlat and Lian (Int J Plast, 5:5166, 1989), and a hardening law. The method does not rely on assumptions about preexisting defects, is broadly applicable to sheet alloys exhibiting inplane anisotropy requiring a higherorder yield criterion, and requires only minimal experimental inputs. Results from the analytical method are compared with experimentally derived FLDs based upon the wellknown Nakajima test and tensile deformation, and with predictions from an existing analytical method for FLDs. Close agreement between the experimentally derived FLDs and the present theoretical method was obtained. Sheet materials where the theoretical method does not apply are also discussed.
Introduction
The compelling need for energy efficiency combined with mounting concerns for the environmental impact of greenhouse gas emissions is generating strong motivation for the increased use of lightweight engineering materials that will lead to weight reduction in transportation industries. The automotive industry in particular has been focused on development of magnesium (Mg) alloys to meet stringent vehicle mass reduction targets required by forthcoming CAFÉ requirements (Ref 1) without compromising occupant safety. As the lightest of all structural metals, Mg alloys hold great promise as replacement materials for heavier steel and aluminum. With high strengthtoweight ratios and low densities (~35% lighter than aluminum and ~78% lighter than steel), Mg alloys are potentially suitable for a large variety of component. However, broader implementation of Mg sheet alloys has been hampered by poor room temperature (RT) formability. The von Mises criterion (Ref 2) for strain compatibility requires the operation of at least five independent systems (slip and twinning), and these are not available at RT due to the very high stresses needed to operate nonbasal slip systems: fracture precedes plastic deformation resulting in low polycrystalline ductility. Magnesium sheet forming processes are typically conducted at warm temperatures (e.g., stamping, ~200300 °C) or hot temperatures (e.g., gas pressure forming, >300 °C). Examples are Ambrogio et al. (Ref 3) who investigated the hot incremental forming of Mg AZ31 alloy sheet, and Carter et al. (Ref 4) who formed Mg AZ31 at 450 °C by hot gas pressure forming. The addition of heat can add manufacturing cost and process intensity rendering RT forming the desirable alternative.
Substantial effort has therefore been focused on improving the RT ductility of Mg sheet alloys with both experimental and theoretical approaches. For example, Ha et al. (Ref 5) suggested a method for increasing the ductility of Mg alloy sheets by using a laser scanning treatment combined with a defocusing technique. Yasi et al. (Ref 6) developed a model of thermally activated crossslip in Mg based upon first principles density functional theory and showed that solutes such as K, Na, and Sc can lower the prismatic stacking fault energy providing a modest improvement in formability at lower temperatures. Another approach to improving RT ductility of Mg alloys is through the addition of rare earth (RE) elements, such as La, Ce, Nd, and Pr, often in the form of misch metal (Ref 7–10). Improved RT ductility results from softening of the recrystallization texture along with increased activity of otherwise inactive deformation mechanisms: \( \left\{ {10\bar{1}1} \right\} \) \( \left\langle {10\bar{1}2} \right\rangle \) compression twins, \( \left\{ {10\bar{1}1} \right\} \) \( \left\{ {10\bar{1}2} \right\} \) double twinning, and pyramidal \( \left\langle {c + a} \right\rangle \) dislocation slip (Ref 11–14). Zhang et al. (Ref 15) showed that addition of Er softens the Mg texture and increases RT ductility. AlSamman and Li (Ref 13) demonstrated that additions of Gd, Nd, Ce, La as misch metal elements weaken the Mg texture and anisotropy relative to conventional Mg sheet alloys (e.g., AZ31), especially after recrystallization annealing. Magnesium ZEK100 alloys, which have attracted particular attention, typically consist of Zn (for strength), RE (for reduced texture), and Zr (for grain refinement) (Ref 16). Potential applications of Mg ZEK100 alloys in the automotive industry are doors, decklids, liftgate inner panels, seat backs, and other reinforcements. Current drawbacks to the use of Mg ZEK100 are primarily related to material and manufacturing costs and the likelihood that a coating or paint would be needed to heal scratches if the Mg ZEK100 alloy were used as an autobody outer panel (for example).
A critical step in sheet component manufacturing is the development of forming limit diagrams (FLDs) which consist of forming limit curves (FLCs). First introduced by Keeler (Ref 17) and Goodwin (Ref 18), FLCs are empirical relationships between critical (or limiting) inplane strains that describe the limit of sheet formability as now defined by ASTM E2218 (Ref 19). The FLCs divide combinations of major and minor strains that are “safe”, i.e., for which failure is not expected from a given strain combination in an FLD. The major strain is along the vertical (yaxis) and the minor strain is along the horizontal (xaxis) of the FLD (Ref 20). Localized necking is usually the failure mechanism of interest since necked regions are potential fracture sites in stamped automotive components (note that key distinctions between “diffuse” and “localized” necking are detailed in Hosford and Cadell (Ref 21)). The left side of the FLD corresponds to strain combinations appropriate for drawing and uniaxial tension, and the right side corresponds to strain combinations for biaxial stretching. The plane strain limit corresponds to the point where an FLC intersects the major strain axis. Data required to generate an FLD are produced in a set of ancillary experiments (i.e., in addition to those required for constitutive models of flow behavior) in which sheet specimens with different geometries are deformed with a punch. Strain fields on the surface of the deformed specimens are measured either with etched grids consisting of an array of circular elements or with digital image correlation, a whole field optical strain mapping technique (Ref 22). A useful review of strainbased FLDs may be found in Marciniak et al. (Ref 23). There is currently an important effort underway to develop stressbased FLDs (not considered in this paper). As discussed by Stoughton (Ref 24), the stressbased approach has some advantages over the ubiquitous strainbased approach that may well result in its being the preferred method in the future.
Since experimentation can be costly and time consuming, there has been some effort aimed at predicting strainbased FLDs using various theoretical methodologies. Theoretical prediction of FLDs for thin sheet materials of relevance to transportation industries has been based upon one of three existing theories. The first is due to Hill (Ref 25), and it assumes that a localized deformation band or neck forms along the zeroextension direction and an angle exists between the normal direction of the neck in the sheet plane and the major strain direction: this is the socalled “zeroextension hypothesis”. However, Hill’s zeroextension hypothesis is only applicable to the lefthand side (LHS) of the FLD, since no zeroextension direction exists in a stretched sheet with positive minor strain, and of course prediction of localized necking is not possible. The second theory is based on the groove (or initial thickness inhomogeneity) hypothesis by Marciniak and Kuczynski (Ref 26), also known as the “MK theory.” This postulates that localized necking develops from a geometric inhomogeneity or groove in the initial sheet thickness from which localized necking develops. The obvious physical significance of the MK theory and its simple mathematical form have contributed to its widespread application for more than four decades. For example, Cao et al. (Ref 27) used the MK theory to compute the FLDs of Al2008T4 and Al6111T4 alloys while McCarron et al. (Ref 28) investigated failure of NAPAC F50 and AKRephos sheet steels. Unfortunately, models based upon the MK theory are oversensitive to the assumed initial thickness inhomogeneity as discussed by Ghazanfari and Assempour (Ref 29) and Min et al. (Ref 30). In addition, there have been several published studies that question the extent to which an initial geometric inhomogeneity is the direct cause of localized necking. For example, Zhang and Wang (Ref 31) showed that localized geometric softening at a certain state of deformation is in fact the cause of localized necking in 2036T4 aluminum rather than an initial geometric inhomogeneity. The third theory is the vertex theory by Storen and Rice (Ref 32), also known as the “SR theory”. Here, it is assumed that localized necking corresponds to a corner or vertex that develops on the yield surface at the loading point, and a localized necking criterion is developed for the deformation on both sides of an FLD based on the \( J_{2} \) deformation theory of plasticity. The vertex theory was recently modified by Zhu et al. (Ref 33). Based on their modified theory, Chow and Jie (Ref 34) developed an analytical model coupled with material damage for the FLD of Al alloy AA6022 considering Hill’s 48 yield criterion, and Min et al. (Ref 30) derived an analytical model with the LoganHosford yield criterion and inplane isotropy (Ref 35) for the FLD of 22MnB5 steel at 800 °C. Jie et al. (Ref 36) deduced a model based upon the von Mises yield criterion and isotropy for a ratedependent material and validated their model with forming limit tests of a ratedependent AKDQ steel.
Theoretical prediction of the FLD of sheet alloys requires a yield criterion and a hardening model. Various nonquadratic (or higherorder) yield criteria developed over the past 30 years or so are good candidates. For example, Hosford (Ref 35) developed a higherorder yield criterion, and suggested an order of 6 and 8 for bodycentered cubic (BCC) and facecentered cubic (FCC) metals, respectively. Barlat et al. (Ref 37–42) proposed highorder yield criteria known as Yld89, Yld91, Yld94, Yld97, Yld2000, and Yld2004, respectively. Recently, Cazacu et al. (Ref 43) developed an orthotropic yield criterion for hexagonal closepacked (HCP) metals. The yield criterion proposed by Hill (Ref 44), known as Hill’s 1948 criterion, is still widely employed since it has a relatively simple mathematical form. However, Naka et al. (Ref 45) and Min et al. (Ref 30) showed, respectively, that higherorder yield criterion must be adopted to obtain reasonable predictions for Mg alloys and hot sheet steels. Bohlen et al. (Ref 12) found that Mg sheet alloys exhibit orthotropic plasticity or inplane anisotropy. Nevertheless, most of the theoretical methods based on highorder anisotropic yield criterion do not consider inplane anisotropy due to mathematical complexity. For example, the Yld97 (Ref 40) and Yld2000 (Ref 41) include 6 and 8 fitting constants, respectively. A notable exception, however, is the theoretical FLD model of Zhu et al. (Ref 46) in which Hill’s 48 yield criterion (Ref 44) was employed to predict FLDs based upon Hill’s zeroextension hypothesis (Ref 25) and inplane anisotropy was also considered, They concluded that the Lankford coefficient \( r_{0} \) effects the FLC significantly on the RHS of the FLD, but has only minimal effect on the LHS of the FLD; however, both \( r_{45} \) and \( r_{90} \) have little effect on the FLC.
Most of the available information on forming limits for Mg sheet alloys has been generated at high temperatures, a consequence of their poor RT ductility. For example, AbuFarha et al. (Ref 47) performed forming limit tests for four Mg AZ31B sheets at 400 °C using elliptical dies. Siegert et al. (Ref 48) investigated the formability of an Mg AZ31B sheet material over the 200350 °C temperature range. Kim et al. (Ref 49) generated FLCs for Mg AZ31 using elevated temperature punch stretching tests. RT forming limit analyses of REcontaining Mg alloys have not been explored to any significant extent in the literature. In particular, there is no existing information on RT, strainbased FLDs for Mg ZEK100 sheet alloys.
In this paper, strainbased FLDs are derived for an Mg ZEK100O alloy (Zn1.34Zr0.23Nd0.182, wt.%) using an analytical method that combines the vertex theory of Storen and Rice (Ref 32), the anisotropic yield criterion of Barlat and Lian (Ref 37), and a hardening law. The model does not rely on assumptions about preexisting defects, is broadly applicable to sheet alloys exhibiting inplane anisotropy requiring a higherorder yield criterion, and requires only minimal experimental inputs. Uniaxial tensile tests of Mg ZEK100O sheets were conducted to obtain the Lankford coefficients (or \( r \) values) at 0°, 45°, and 90° with respect to the rolling direction (RD), and the wellknown Nakajima test was used to measure the FLDs for comparison with model predictions. Predictions of the analytical method are shown to be in closer accordance with experimental FLD data compared with the Zhu et al. (Ref 33) method for FLD prediction. In addition, the analytical method addresses the yield point elongation (YPE) exhibited by Mg ZEK100O tensile specimens with a 90° gage orientation. Sheet materials where the analytical method does not apply are also discussed.
Analytical Method for StrainBased Forming Limits
Vertex Theory
According to the vertex theory of Storen and Rice (Ref 32), the localized necking of a sheet material corresponds to the appearance of a vertex on the yield surface. The stress and strain across the necking band in a sheet are considered to be continuous, while the stress rate is discontinuous. It is assumed that the angle between the normal direction of the necking band and the direction of the major strain is θ (see Fig. 1): this corresponds to the zeroextension hypothesis proposed by Hill (Ref 44). Note that θ is 0 when considering the case of the righthand side (RHS) of the FLD since no zeroextension direction exists in a stretched sheet with positive minor strain. Hence, the discontinuity across the necking band can be expressed as
where \( n_{k} \left( {k = 1,2} \right) \) is the component of the unit normal vector \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {n} \) of the necking band, \( n_{1} = \cos \uptheta \), and \( n_{2} = \sin \uptheta \).
The gradient of the deformation rate inandout of the necking band is
where \( v_{i} \) is the velocity of a point on the material surface and \( g_{i} = f_{i}^{\prime } \left( {n_{k} x_{k} } \right) \) \( \left( {i,k = 1,2} \right) \). Using \( \dot{\upvarepsilon }_{ij} = \partial v_{i} /\partial x_{j} \), Eq 2 can be expressed in terms of principal axes
Here \( \upvarepsilon_{1} \) and \( \upvarepsilon_{2} \) are major and minor strains, respectively, and \( \dot{\upvarepsilon }_{1} \) and \( \dot{\upvarepsilon }_{2} \) are major and minor strain rates, respectively. By considering the moment equilibrium equation in the force equilibrium condition, the equilibrium equations in the modified vertex theory of Zhu et al. (Ref 33) have the following forms
As mentioned before, the necking band is normal to the major strain direction on the RHS of the FLD. Hence
Thus,
where \( \uprho_{\upvarepsilon } \) is the strain ratio
The simplification of the vertex theory makes possible the consideration of highorder yield criteria that are much more complex than the Hill’s quadratic yield criterion. Kim et al. (Ref 50), Zhang et al. (Ref 51), and Min et al. (Ref 30) have shown that incorporation of highorder yield criteria generally improves the FLD prediction relative to experiments for Al sheet alloys and for boron steel (for example) at elevated temperatures.
Orthotropic HighOrder Yield Criterion
The orthotropic, higherorder yield function proposed by Barlat and Lian (Ref 37), also known as Yld89, which contains only three parameters that need to be determined: a, c, and m, is employed in this study. It has a much simpler form than either Yld91 (Ref 38) or Yld94 (Ref 40) as well as the yield criterion developed for HCP metals by Cazacu et al. (Ref 43). The simple form of Yld89, which allows one to analytically deduce the necking criterion for a sheet alloy, is
where
Here, \( \upsigma _{ij} \left( {i,j = 1,2} \right) \) are the stress components and \( \upsigma _{\text{eq}} \) is the equivalent stress; \( m \) is recommended to be 6 and 8 for BCC and FCC metals by Barlat and Lian (Ref 37) and Dixit and Dixit (Ref 52), respectively. Note that \( K_{1} \) and \( K_{2} \) are the plane stress tensor invariants; a, c, h, and p are material constants that depend upon Lankford’s coefficients (or r values) r _{0}, r _{45}, r _{90}.
Let the stress axes be coincident with the principal coordinate system. Then \( \upsigma _{12} = 0 \), \( \upsigma _{11} = \upsigma _{1} \), \( \upsigma _{22} = \upsigma _{2} \); here, \( \upsigma _{1} \) and \( \upsigma _{2} \) are the major and minor stresses, respectively. For simplicity, let \( k = \sqrt {\frac{{r_{0} }}{{1 + r_{0} }} \times \frac{{r_{90} }}{{1 + r_{90} }}} \). Then the BarlatLian orthotropic yield criterion can be rewritten as
If \( m = 2 \) (where m is the order of the yield criterion), and \( r_{0} = r_{90} = r \) (without consideration of inplane anisotropy), then \( h = 1 \), \( k = r/\left( {1 + r} \right) \), and Eq 10 becomes
which is Hill’s quadratic yield criterion (\( \upsigma _{\text{eq}} \) is equivalent stress). Normality of plastic strain increments requires
where \( \uplambda \) is a scalar quantity. The major and minor strains, \( \upvarepsilon_{1} \) and \( \upvarepsilon_{2} \), respectively, can be obtained by integrating Eq 12 after substituting Eq 10 for \( f \)
where \( \upvarepsilon_{\text{eq}} \) is the equivalent strain. The strain ratio \( \uprho_{\upvarepsilon } \) can now be expressed as
Note that \( \upvarepsilon_{1} \) and \( \upvarepsilon_{2} \), which follow from the normality rule (Eq 12) and the BarlatLian orthotropic yield criterion can now be used to develop the analytical model for the FLD.
LocalizedNecking Criterion for Sheet Alloys
By differentiating Eq 13 with respect to time, the following two evolution equations can be derived
Let \( \dot{\upsigma }_{\text{eq}} = w\dot{\upvarepsilon }_{\text{eq}} \), where \( w = d\upsigma _{\text{eq}} /d\upvarepsilon_{\text{eq}} \) is the hardening modulus which can be derived from true stress vs. true strain curves. By eliminating \( \dot{\upsigma }_{2} \) in Eq 16 (see Appendix for additional details), \( \dot{\upsigma }_{1} \) can be expressed as
where \( B_{i} \left( {i = 1,2,3,4} \right) \) are
and
By taking the differential of both sides of Eq 17 we find
As shown by Min et al. (Ref 30), there is a zeroextension direction with respect to the deformation on the LHS of the FLD (for many sheet metal alloys) according to the zeroextension hypothesis proposed by Hill (Ref 25). The critical hardening modulus, \( w_{\text{cr}} \), which corresponds to localized necking and is used to solve for the forming limit strains, can be computed by substituting Eq 4 and the first of Eq 6 into Eq 23
If the necking band of the sheet specimen is normal to the major strain direction, e.g., on the RHS of the FLD, then \( w_{\text{cr}} \) can be computed by substituting Eq 4 and the second equation of Eq 6 into Eq 23. This gives
From the strain energy principle, (see Eq A4)
By employing Eq 14 and 21, \( \upvarepsilon_{\text{eq}} \) can be expressed as
If power law hardening is assumed
then the hardening modulus can be calculated as
An analytical model for the LHS of the FLD can be deduced by substituting Eq 2729 into Eq 24. This yields \( \upvarepsilon_{1}^{ * } \), which is the major limit strain corresponding to localized necking
For the RHS of the FLD, \( \upvarepsilon_{1}^{ * } \) can be deduced by substituting Eq 2729 into Eq 25.
When considering an isotropic case and the von Mises yield criterion, namely, \( r_{0} = r_{90} = 1 \) and, \( m = 2 \) the \( B_{i} \left( {i = 1  4} \right) \) in Eq 30 and 31 become \( B_{1} = \frac{3}{{4g\left( {\uprho_{\upsigma } } \right)}} \), \( B_{2} = \frac{3}{4} \), \( B_{3} = 1 \), \( B_{4} = 1/2 \). Equations 30 and 31 then reduce to, respectively,
which are the same as those reported by Zhu et al. (Ref 33) who validated their model for various sheet alloys using the von Mises yield criterion and inplane isotropy at RT. However, the models deduced in the present paper (Eq 30 and 31) consider both highorder yield criterion and planar anisotropy, which is more generalized than the models by Zhu et al. (Ref 33) and can be applied to more materials. We also note that the other studies have assumed various hardening models in the course of developing theoretical methods for FLD prediction. A notable recent example is the use of the Voce hardening model by Li et al. (Ref 22) for FLD prediction of 5182 aluminum subject to preform annealing.
For a ratedependent material, the following law can be employed
where \( M \) is the strain rate sensitivity. Corresponding expressions for \( \upvarepsilon_{1}^{ * } \) can be deduced according to the method introduced by Min et al. (Ref 53).
Up to this point, a generalized localized necking criterion based upon Eq 30 and 31 for metal sheet alloys has been developed based on the Storen and Rice (Ref 32) vertex theory and the BarlatLian orthotropic yield criterion. When predicting the FLCs of sheet metal alloys that follow a power law hardening equation (Eq 28), the \( n \) and \( r \) values in the 0° and 90° orientations and the \( m \) value are needed as inputs. The former can be obtained from uniaxial tensile tests, and the latter can be obtained from references or experiments. In addition, the fracture mode of the metal sheet alloy, especially in the negative minorstrain region, should be noted. If fracture follows the Hill’s zeroextension hypothesis (Ref 25), Eq 30 and 31 are used to predict the FLC on the LHS and RHS, respectively. If fracture follows the zeroangle necking hypothesis proposed by Min et al. (Ref 53), Eq 31 is used to predict both sides of the FLC.
Experimental Validation
The material used to validate the theoretical developments in the preceding sections is Mg ZEK100 rolled sheet in the F temper (asfabricated), with a 1.5 mm thickness, and a hardness of ~55 HRF. The chemical content of the alloy, which is listed in Table 1, consists of Zn (for strength), Nd (for texture control), Zr (for grain size refinement), and trace amounts of Ce and La. The microstructure of the starting material, which is shown in Fig. 2, exhibits numerous twins from sheet manufacturing.
Uniaxial Tensile Tests
Tensile specimens were water jet cut at 0°, 45°, and 90° to the RD. The dimensions are shown in Fig. 3. The specimens were annealed at 500 °C for 15 min to remove the “cold work”. The annealing process was carried out in an electric furnace which was filled with argon gas to protect the tensile specimens from oxidization. After annealing, the specimens were air cooled. The annealed material is hereinafter denoted as ZEK100O and its microstructure is shown in Fig. 4.
Three tensile specimens in both 0° and 90° orientations were quasistatitically elongated at a strain rate of 0.005/s and at RT until fracture, and single specimens in 0°, 45°, and 90° orientations were elongated to 10% strain to determine the Lankford coefficients \( r_{0} \), \( r_{45} \), and \( r_{90} \), which were used to calculate the averaged \( r \) value (\( \bar{r} \)). To facilitate determination of the \( r \) value in each orientation, five locations were marked with lines drawn on each tensile specimen, and the crosssectional dimensions were measured at each of these locations before and after elongation to an engineering strain of 10%. The \( r \) value in each orientation was calculated as the average of the five values via,
where \( w_{\text{i}} \) and \( t_{\text{i}} \) are the initial width and thickness, respectively, at each location on the specimens before elongation; \( w_{\text{f}} \) and \( t_{\text{f}} \) are the final width and thickness, respectively, at each location on the specimens after elongation. All tensile tests were performed on an Instron 5582 universal tensile testing machine at RT. An extensometer with a 25 mm gage length was used to measure the strain and the crosshead speed was 10 mm/min. All Mg ZEK100O specimens exhibit a very brief period of diffuse necking followed by an even shorter period of localized necking and fracture.
Forming Limit Tests
The specimens for the forming limit tests were cut at 0° and 90° (45° specimens were precluded since the axes of stresses are not coincident with the principal coordinate system) with respect to the RD and are hereinafter referred as the RD specimens and the TD specimens, respectively. The dimensions of the specimens are shown in Fig. 5. The lower left inset figure shows a sampling of the circle grids used to measure strain. The circular elements in each grid pattern had a diameter of 2.5 mm (\( d_{0} \)) and a centertocenter spacing of 3 mm. The overall patterns were laser etched onto one surface of each specimen: these are denoted by the gray regions in each specimen geometry in Fig. 5. For the 180 mm wide specimen, the grid was arranged in a pattern that followed a cruciform profile in which there were 15 circular elements and 33 circular elements in the short edge direction and in the long edge direction, respectively, as illustrated in Fig. 5. For all remaining specimens, there were 33 circular elements in the length direction. However, there were 17, 15, 11, and 5 circular elements along the width direction for specimens with widths of 140, 120, 80, and 30 mm, respectively.
The width and depth of each circular element was ~50 μm (see Fig. 5) and ~20 μm, respectively. These dimensions did not lead to damage of the base metal, namely, specimens did not fracture early along the circular elements. The grids were free of distortion even after the 500 °C heat treatment. The forming limit tests were conducted with the Nakajima test apparatus in the 200 ton capacity press shown in Fig. 6.
The punch speed was 2 mm/s. Square Teflon® sheets, with dimensions of 100×100 mm^{2} and 0.1 mm thickness, were used as the lubricating medium between the punch and the test specimens. Figure 7(a) shows the bulged specimens that correspond to the specimens shown in Fig. 5. For the narrow specimens (i.e., with a width smaller than 100 mm), fracture always occurred at the drawbeads or die radius even though the binder force was decreased in proportion to the specimen width (see Fig. 7a). This may have been due to the poor (combined) drawing and bending response of the Mg ZEK100O material at RT. To address this issue, the narrow specimens (see Fig. 7b) were elongated on a tensile machine to fracture to obtain the limit strain data with negative minor strains. During deformation, the circular elements became ellipses with a major axis length of \( d_{1} \) and minor axis length of \( d_{2} \) (see Fig. 7c). Major and minor strains were then calculated via
The circular elements close to the fracture were regarded as critical elements, and the strains were measured along the RD and TD (see Fig. 7c).
Results and Discussion
Mg ZEK100O Mechanical Properties
True stress versus true strain curves of the Mg ZEK100O sheet material cut along the three orientations are shown in Fig. 8. The material shows an obvious YPE in the TD, an observation that has previously been reported for Mg ZEK100O by Min et al. (Ref 54). Barnett et al. (Ref 55) investigated Lüders phenomenon during compression of extruded Mg AZ31 (along the extrusion direction) with a fine (515 μm) grain size and attributed it to deformation twinning rather than dislocation glide. It is believed that it is also responsible for the YPE shown in Fig. 8. The work hardening exponent \( n \) values for Mg ZEK100O resulted from fits of Eq 28 to the flow data for the 0° and 90° specimens in the 0.020.16 and 0.020.23 strain ranges, respectively. The \( n \) values, the R ^{2} values, and the \( r \) values, are summarized in Table 2, where each value is averaged using data from the three specimen sets.
From Table 2, it can be seen that the Mg ZEK100O sheets show different \( n \) values and \( r \) values in the 0° and 90° orientations, which suggests inplane anisotropy. The average \( r \) value (\( \bar{r} \)) from
is 0.79, which will be used as an input when calculating forming limit strains without consideration of inplane anisotropy.
Forming Limits
The necking band of the Mg ZEK100O sheet material is always normal to the major strain direction. This is seen from all of the fractured RD and TD specimens in Fig. 7. Hence, Hill’s zeroextension hypothesis is not applicable. Rather, it follows the hypothesis of zeroangle necking in Min et al. (Ref 53) where it is shown that the Hill’s zeroextension hypothesis will overestimate the forming limit strains on the LHS of the FLD. Therefore, Eq 31 is employed to predict the forming limit strains on both sides of the FLD of the Mg ZEK100O sheet.
Forming Limit Diagram in the RD
For the RD specimens (see Fig. 5), the major strain (\( \upvarepsilon_{1} \)) and minor strain (\( \upvarepsilon_{2} \)) directions are along the RD and the transverse direction (TD), respectively. Note that the order (\( m \) value) of the BarlatLian yield criterion for Mg ZEK100O alloy is not known, and it cannot be obtained from the literature. Therefore, comparisons between the experimentally determined forming limits and predictions from the Zhu et al. (Ref 33) model and our analytical model are presented in Fig. 9. A range of yield criterion orders (\( m \)) is examined. This enabled determination of which model resulted in the closest agreement with the experimental results. Note that \( \upvarepsilon_{1} \) and \( \upvarepsilon_{2} \) are denoted as \( \upvarepsilon_{\text{RD}} \) and \( \upvarepsilon_{\text{TD}} \), respectively. The \( n \) value and \( r_{0} \) and \( r_{90} \) values used in Eq 31 for the FLC calculation are taken from Table 2. The comparisons in Fig. 9 suggest that the Zhu et al. (Ref 33) model, which considers the von Mises yield criterion and isotropy, gives predictions that are significantly in error compared with experimental data; however, an order of 12 in the yield criterion of Eq 8 leads to the smallest error between the analytical model in this paper and the experimental data (the error of the predicted limit strain data relative to the experimental data is within 4%). When \( m = 6 \) and \( m = 8 \), which are, respectively, recommended for BCC and FCC materials, the theoretical predictions overestimate the forming limits in both the LHS and RHS and the errors relative to experimental data reach ~32% and ~16% at the uniaxial tensile and biaxial states, respectively. For \( m = 10 \), the prediction is very close to the experimental forming limit strain data on the RHS, while on the LHS, it overestimates the forming limits by ~9% at the uniaxial tensile state. For \( m > 12 \), the predicted limit strains are underestimated especially on the RHS of the FLD, and the error relative to experimental data at the balanced biaxial tensile state reaches ~6%. For Al or steel sheets, the FLC on the RHS typically increases with increasing minor strain suggesting good stretchability. Alternatively, the FLC on the RHS in Fig. 9 for Mg ZEK100O decreases with increasing minor strain which suggests poor stretchability relative to Al and steel.
Figure 10 shows another comparison of the analytical model predictions based upon inplane isotropy and the BarlatLian anisotropic yield criteria with experimental data. The dashed line corresponds to thickness anisotropy only, i.e., it employs \( r_{0} \) and \( r_{90} \) both equal to 0.79 (\( \bar{r} \)), calculated from Eq 37. The solid line shows the predicted FLC considering inplane anisotropy, namely, the applied yield criterion is orthotropic, and it uses the measured \( r_{0} \) and \( r_{90} \) in Table 2. The order \( m \) used in both predicted FLCs is 12. Clearly, the FLC prediction based on the orthotropic yield criterion is in better agreement with experimental data. The prediction based on the planarisotropic yield criterion gives an overestimated FLC, especially on the LHS of the FLD, and the error reaches ~6% at the uniaxial tensile state.
Forming Limit Diagram in the TD
For the TD specimens, the \( \upvarepsilon_{1} \) and \( \upvarepsilon_{2} \) directions are consistent with the TD and RD, respectively; hence, \( \upvarepsilon_{1} \) and \( \upvarepsilon_{2} \) in the conventional FLD are denoted as \( \upvarepsilon_{\text{TD}} \) and \( \upvarepsilon_{\text{RD}} \), respectively. As Fig. 8 shows, material in the TD exhibits YPE and with an \( n \) value as high as 0.43. The predicted FLC in the TD using this \( n \) value is indicated by the dotted line in Fig. 11, in which a comparison of forming limit data in both the RD and the TD is also shown, and the point where the FLC in the RD intersects with the FLC in TD indicates the forming limit strains in the balanced biaxial tensile state. The predicted FLC in the TD is clearly much higher than the experimental results. The strain range used to fit the \( n \) value is 0.02 to 0.23; this includes the sigmoidal hardening region, the strain range from 0.01 to ~0.1, on the true stress versus true strain curve in Fig. 8. As Wu et al. (Ref 56) stated, a twinning dominated deformation mechanism results in the sigmoidal hardening on the true stress versus true strain curve for Mg AZ31 alloy sheet. However, a dislocation slip mechanism gives rise to power law hardening as shown in the strain range from ~0.1 to 0.23 for the TD tensile specimen and in the strain range from 0.02 to 0.18 for the RD tensile specimen (see Fig. 8). To exclude the influence of the YPE and sigmoidal hardening associated with twinning dominated deformation, a 0.150.23 strain range was used to refit the \( n \) value in the TD specimen. This gives an \( n \) value of 0.183, and the R ^{2} value is 0.999, which is much higher than the R ^{2} value (0.988) in Table 2 for the fit over a 0.020.23 strain range. This suggests that power law hardening more accurately describes the hardening behavior in the strain range 0.150.23 relative to the strain range 0.020.23. The predicted FLC in the TD using this new \( n \) value is shown in Fig. 11 as indicated by the dashed line. It can be observed that the dashed FLC is much closer to the experimental data in the TD.
For materials exhibiting sigmoidal hardening due to twinning dominated deformation, care should be taken when choosing the strain range over which to compute the \( n \) value and then using the \( n \) value to evaluate formability of Mg alloys sheets. If the strain range used to fit the \( n \) value includes the sigmoidal hardening region, a high \( n \) value will be obtained which does not reflect the real formability of the material. In addition, it should be noted that the deduced analytical model (Eq 30 and 31) for forming limits based on the vertex theory and orthotropic highorder yield criterion is applicable to many sheet metals appropriate for automotive component forming and hence it is not limited to Mg ZEK100O alloy sheets. Examples are various aluminum alloys (Ref 22, 57, 58) and steels (Ref 59,60). However, there are some notable exceptions. If the material undergoes diffuse necking followed by ductile rupture with no obvious localized necking, then the present theory should not be used. A poignant example is 900 grade fully martensitic steel which undergoes a ductile rupture process with no obvious localized neck. This is shown in Fig. 12. As discussed in Savic et al. (Ref 61), 1300 grade fully martensitic steel (for example) does display a localized neck and the present theoretical approach to FLD prediction can indeed be applied.
Conclusions

(1)
A new analytical method for strainbased FLD prediction, which combines the Storen and Rice (Ref 32) vertex theory, the BarlatLian highorder yield criterion (Ref 37) including inplane anisotropy, and a hardening law, was developed and applied to an Mg ZEK100O alloy (Zn1.34Zr0.23Nd0.182, wt.%). The method does not rely on assumptions about preexisting defects, requires only minimal experimental inputs (\( r \) values, \( n \) value, \( m \) value, fracture type). The predicted FLDs are in close agreement with experimentally measured FLDs (based upon the wellknown Nakajima test) at RT. The method is by no means limited to Mg alloys since it can be readily applied to predict FLDs in other ferrous and nonferrous sheet materials.

(2)
The analytical method for strainbased FLD prediction detailed in this report is summarized as follows:

(a)
Failure of the sheet material of interest must occur via localized necking followed by fracture. This can occur after a period of diffuse necking (such as occurs in various sheet steels), or, it may be the only obvious necking process prior to fracture (such as 1300 grade martensitic steel or press hardened boron steel). If the material undergoes diffuse necking followed by ductile rupture with no localized necking (as occurs, for example, in 900 grade martensitic steel), then the present theory should not be used (see Hosford and Caddell (Ref 21) for the distinction between diffuse and localized necking).

(b)
Obtain \( n \) values (work hardening exponent) and \( r \) values (Lankford coefficients) in the 0° and 90° orientations via uniaxial tensile tests or from tabulated values.

(c)
Obtain the \( m \) value (i.e. the order in BarlatLian yield criterion (Ref 37)) from references or experiments. Suggested values (from the following reliable literature sources in the reference list below: Barlat and Lian (Ref 37) and Hosford (Ref 35)) are 6 and 8 for steel (BCC) and aluminum (FCC) alloys, respectively, and 12 for magnesium (HCP) alloys.

(d)
Check the fracture type of the tensile specimens. If the fracture follows Hill’s zeroextension hypothesis (Ref 25), where the normal direction of the necking band in the sheet plane makes an angle with the major strain direction (or tensile direction), then Eq 30 and 31 are used to predict the FLC on both the LHS and the righthand side (RHS), respectively, of the FLD. If the fracture follows the zeroangle hypothesis (Ref 53), where the normal direction of the necking band in the sheet plane is always parallel with the major strain direction, then Eq 31 is used to predict the FLC on both the LHS and RHS.

(a)

(3)
The analytical method for FLD prediction was validated against the experimentally derived FLDs for Mg ZEK100O sheet in the RD only. This material exhibits a very brief period of diffuse necking followed by rapid localized necking and fracture in RT, quasistatic tensile tests. The \( m \) value in the BarlatLian yield criterion (Ref 37) which gives the closest agreement with experimental data for Mg ZEK100O is 12.

(4)
The Hill’s zeroextension hypothesis (Ref 25) on the negative minorstrain region does not accurately account for the necking mode of the Mg ZEK100O alloy sheets. Rather, the Mg ZEK100O alloy sheet necking mode is accurately described by the zeroangle necking hypothesis (Ref 53). Hence, the theoretical approach to FLD prediction detailed in this report is applicable to both sides of the Mg ZEK100O FLD.

(5)
The analytical method in the present paper shows much better agreement with experimental FLD data compared with the theoretical model of Zhu et al. (Ref 33). Consideration of orthotropic plasticity enables the analytical model to be in close accord with experimental data. However, when orthotropic plasticity is not considered, the model overestimates the forming limits for Mg ZEK100O by ~6% at the uniaxial tensile state on the LHS of FLD.

(6)
The YPE exhibited by Mg ZEK100O specimens with a 90° orientation in uniaxial tension, and the almost sigmoidal hardening beyond the YPE in the TD, and high work hardening exponent or n value of ~0.43 (for the TD only), results in unrealistically high formability prediction, and leads to overestimation of forming limits. A new n value resulting from a fit of the power law hardening model in the 0.150.23 strain range, which excludes the influence of the YPE, enables the predicted results to agree well with the experimental forming limit data for Mg ZEK100O.

(7)
The present analytical method for strainbased FLD prediction would require a modification to the material constitutive model if either forming at temperatures higher than RT were required or power law hardening is insufficient to describe the flow response of the sheet material.
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The authors are grateful to Drs. A.K. Sachdev and M.W. Verbrugge for their thorough review of this manuscript.
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Appendix: Derivation of Eq 17
Appendix: Derivation of Eq 17
Rearranging Eq 12 yields
Multiply the first equation in Eq A1 by \( \left[ {\left( {1  k} \right)\upsigma _{2}^{m  2} + kh^{2} \left( {\upsigma _{1}  h\upsigma _{2} } \right)^{m  2} } \right] \), and the second equation in Eq A1 by \( kh\left( {\upsigma _{1}  h\upsigma _{2} } \right)^{m  2} \). Adding the resulting expressions, followed by elimination of \( \dot{\upsigma }_{2} \) gives
Equation A2 is rearranged to
According to the strain energy principle
Substitution of Eq A4 into Eq A3 gives
Substitution of Eq 1821 into Eq A5 gives
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Min, J., Hector, L.G., Lin, J. et al. Analytical Method for Forming Limit Diagram Prediction with Application to a Magnesium ZEK100O Alloy. J. of Materi Eng and Perform 22, 3324–3336 (2013). https://doi.org/10.1007/s1166501305823
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DOI: https://doi.org/10.1007/s1166501305823
Keywords
 formability
 forming limit diagram (FLD)
 inplane anisotropy
 yield point elongation
 ZEK100 magnesium alloy