Prediction of forming limit curves for nonlinear loading paths using quadratic and non-quadratic yield criteria and variable imperfection factor

Materials and Design 91 (2016) 248–255

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Prediction of forming limit curves for nonlinear loading paths using quadratic and non-quadratic yield criteria and variable imperfection factor Morteza Nurcheshmeh a,⁎, Daniel E. Green b a b

Engineering Department, Mechanical Engineering Program, Western Kentucky University, 1906 College Heights Blvd #21082, Bowling Green, KY 42101-1082, United States Department of Mechanical, Automotive and Materials Engineering, University of Windsor, 401 Sunset Avenue Windsor, Ontario N9B 3P4, Canada

a r t i c l e

i n f o

Article history: Received 28 January 2015 Received in revised form 24 November 2015 Accepted 25 November 2015 Available online 26 November 2015 Keywords: Forming limit curve Formability Yield criterion MK analysis Sheet metal Loading path

a b s t r a c t Industrial sheet metal forming processes often involve complex deformation modes and it is necessary to consider nonlinear loading path effects when predicting forming limit curves. Moreover, the yield criterion plays a critical role in the accuracy of predicted forming limits. In this work the MK analysis was modified to relate the initial imperfection factor to a physical property such as the surface roughness, and the orientation of the imperfection was also allowed to vary. This model was used to predict the strain-based and stress-based forming limit curves (FLC and SFLC) of sheet materials that are subject to either linear or non-linear strain paths. Two different yield criteria were employed in this study, Hill's 1948 quadratic yield criterion and Hosford's 1979 non-quadratic yield criterion. The theoretical model was validated by comparing predicted FLC and experimental FLC curves obtained from the literature. FLCs and SFLCs predicted with these two yield criteria were compared for both linear and nonlinear loading paths. Results showed that both the quadratic and non-quadratic yield criteria predict the FLC with acceptable accuracy however on the whole the non-quadratic yield criterion generally provides a slightly better correlation with experimental data, especially on the right side of the FLC. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction Significant progress in sheet metal formability evaluation occurred in Ref. [22] reported that localized necking in stretched sheets requires a critical combination of major and minor in-plane strains (along two perpendicular directions in the plane of the sheet). Subsequently, this concept was extended by Goodwin [11] to drawn sheet and the resulting curve is known as the Keeler–Goodwin curve, or more commonly, the forming limit curve (FLC). In other words, combinations of principal strains that lie above the FLC present some risk of necking, while those that lie below lead to a safe process. The FLC has become an essential tool to evaluate sheet formability, and it is typically obtained by stretching gridded sheet specimens of various widths over a hemispherical punch. However, the deformation behavior of metals is strongly dependent on the history of loading, in particular, on the specific strain path. The FLC, as a well established aid to either experimental or theoretical studies of the formability of sheet metal, should therefore be represented in terms of specific strain history.

⁎ Corresponding author. E-mail address: [email protected] (M. Nurcheshmeh).

http://dx.doi.org/10.1016/j.matdes.2015.11.098 0264-1275/© 2015 Elsevier Ltd. All rights reserved.

Although the FLC has been successfully used to evaluate sheet forming processes for many years, it has been shown that it is only valid for quasi-linear strain paths. Non-linear strain paths cause the FLC to translate in strain space, which can lead to erroneous interpretations of the forming severity for multi-stage processes in which the strain path is significantly non-linear and this has been investigated for all sheet materials including steel, copper and brass, as reported, for example, by Kleemola and Pelkkikangas [23]. It has already been shown by Stören and Rice [36] that in FLC prediction and generally when modeling the plastic behavior of metals, the yield function, which is usually assumed to take the same form as the plastic potential function in classical plasticity theory, plays an important role. It determines the direction of the plastic strain increment via the associated flow rule, and consequently the value of the effective plastic strain which in turn determines the work-hardening rate as defined by the work hardening function. Parmar and Mellor [33] investigated the discrepancy between theoretical and experimental results for aluminum alloys in metal forming calculations and concluded that it was due to the inadequacy of Hill's [12] yield criterion to represent materials with anisotropy coefficients (r0, r45, and r90) lower than unit. They recommended employing Hill's [14] non-quadratic yield criterion for the prediction of the FLC of aluminum and other alloys with lower r-values.

M. Nurcheshmeh, D.E. Green / Materials and Design 91 (2016) 248–255

Barlat [6] conducted a study on the effect of the shape of the yield surface on limit strains. In this investigation, Barlat listed critical characteristics of the yield surface and defined a new parameter, P, as the ratio of the yield stress in plane-strain to the yield stress in equibiaxial tension. Lian et al. and Xu et al. [24,41] employed Hill's [14,15] yield criteria and the MK analysis to predict the right side of the FLC where both major and minor in-plane strains are positive. They compared predicted forming limit results with corresponding experimental data for both aluminum and AK steel. Results showed that by using these yield criteria limit strains can be reasonably predicted. Also Asaro and Needleman [1] and Tvergaard and Needleman [37] introduced an alternate method to study the effects of plastic anisotropy on localized necking. They used an elastic-viscoplastic Taylor-type polycrystalline approach for initial texture representation and accounted for the texture evolution during on-going plastic deformation. This method was applied later by Wu et al. [38,39] to predict localized necking in rolled aluminum alloys. Hora et al. [16] made some improvements to Swift's diffuse necking criterion, and with the help of some experimental research, confirmed that the strain path, i.e. the ratio of the minor strain component ( ε2) to the major strain component (ε1), is the most important factor to determine the onset of necking in sheet metals. Kuroda and Tvergaard [21] used different orthotropic yield criteria in FLC prediction and they concluded that orthotropic axes disorientation may have an effect on predicted limit strains. Cao et al. [9] predicted limit strains using the MK analysis and the Karafillis–Boyce anisotropic yield criterion for the right side of the FLC and offered a new approach to specify yield criterion constants. Banabic and Dannenmann [3] used Hill's 93 yield criterion to study the influence of parameter a (defined as the ratio of the uniaxial yield stress to the biaxial yield stress) on limit strains using MK analysis and Swift's bifurcation instability theory. Using both methods they showed that the FLC translates upward, especially in equibiaxial tension, when parameter a is increased. Butuc et al. [7] examined the performance of two non-quadratic yield functions, Yld96 and BBC2000 in FLC prediction. The correlation of theoretical results and experimental data was shown to be satisfactory when using these yield criteria. Banabic et al. [2] compared the accuracy of a variety of FLC prediction methods using the orthotropic yield criterion BBC2003: in their work FLCs were predicted using Swift's diffuse necking criterion, Hill's bifurcation theory, the finite element method (FEM), the MK analysis and the method proposed by Hora et al. [16]. Banabic et al. [4] showed that the MK analysis and the FEM method gave a better correlation with experimental data than the other methods. In the current research both Hill's [12] quadratic yield function and Hosford's [17] non-quadratic yield function were employed in a modified MK analysis [26,27] to predict strain-based and stress-based forming limit curves following both linear and nonlinear loading paths. FLCs were calculated for AISI-1012 steel and AA-2008-T4 aluminum sheets and were compared with published experimental data [10,28].

249

Fig. 1. Thickness imperfection in the MK method.

a representation of the imperfection region in the MK method. These samples were then subjected to balanced biaxial tension, and the results of their investigation showed that there is no reduction of the limiting strains for very shallow grooves. In other words, when the ratio of the thickness in the groove to that of the sheet is greater than 0.992, the forming limit strains remain unchanged. This imperfection factor is considered equivalent to the microstructural defects that normally exist in as-rolled metal sheets. In the MK model, a sheet with a nominal thickness is assumed to have a band (in the shape of a groove) that is slightly thinner than the rest of the sheet; these two areas are denoted by (a) and (b), respectively (Fig. 1). In the current work, the initial imperfection factor of the groove, fo, was defined as the thickness ratio as follows: fo ¼

t bo t ao

ð1Þ

where ‘t’ denotes the sheet thickness, and subscript ‘o’ denotes the initial state. As deformation progresses the updated thickness imperfection can be determined from Eq. (1): df ¼ dε b3 −dε a3 f

ð2aÞ


 f ¼ f o exp ε b3 −ε a3

ð2bÞ

where ‘ε3‘denotes the true thickness strain. In this work, the imperfection factor was considered to change with the deformation of the sheet. In order to estimate the initial imperfection factor, it was thought reasonable to relate it to the surface roughness of the sheet. By assuming that the maximum thickness difference between regions (b) and (a) is equal to the surface roughness of the sheet, the initial imperfection factor can be written as follows: t ao −2Rzm t ao

2. Theory

fo ¼

2.1. MK analysis for prediction of FLC

where Rzm is the maximum surface roughness of the sheet. Research carried out by Stachowicz [35] shows that the surface roughness also changes with deformation and these changes depend upon the initial surface roughness, the grain size, and the strain according to the following empirical relation:

One of the most effective methods to predict the onset of localized deformation was introduced by Marciniak and Kuczynski [26,27] and is now commonly known as the MK method. This approach is based on the assumption that the inherent material heterogeneities in a thin sheet can be modeled by a very shallow groove. After a certain amount of deformation the strain in the groove increases more rapidly than elsewhere and a localized neck inevitably develops from this initial imperfection (Fig. 1). Due to its simplicity, the MK method has been used with different plasticity theories and hardening models to predict history-dependent forming limits [8,42]. McCarron et al. [29] presented the fundamentals of the MK analysis with a significant level of detail. These researchers machined grooves of different depths into samples made from two different grades of steel as

ð3Þ

0:5

Rzm ¼ Rzo þ Cdo εbe

ð4Þ

where ‘Rzo‘is the surface roughness before deformation, C is a material constant, εe is the effective strain, and do is the initial grain size. Combining Eqs. (2a), (2b), (3) and (4) yields:

fo ¼

h i 0:5 t ao −2 Rzo þ Cdo εbe t ao

ð5aÞ

250

f ¼

M. Nurcheshmeh, D.E. Green / Materials and Design 91 (2016) 248–255

h i 0:5 t ao −2 Rzo þ Cdo εbe t ao


 exp ε b3 −ε a3 :

ð5bÞ

As shown in Fig. 1, the orientation of the imperfection band with respect to the minor stress direction, θ, was considered variable while plastic deformation takes place in the sheet metal, and the initial value for this orientation, θo, can be considered arbitrary at the beginning of deformation. In other words, the orientation of the imperfection band is updated at every plastic increment throughout the deformation. This effect was formulated by Barata da Rocha et al. [5] in an empirical equation for uniform deformation as a function of in-plane plastic strain increments in region (a) of the sheet as follows: tanðθ þ dθÞ ¼ tanðθÞ dεa1

1 þ dεa1 1 þ dεa2

ð6Þ

dεa2

where f is the yield function, F, G, H, L, M, and N are constants that describe the anisotropy of the material, and x, y and z are the orthogonal axes of anisotropy. Generally, ‘x’ is taken to be the rolling direction of the sheet material, ‘y’ is the transverse direction and ‘z’ is the normal direction to the sheet surface. Considering the state of plane-stress that exists in thin sheets, Hill's criterion reduces to:   2f σ ij ¼ ðG þ H Þσ 2x −2Hσ x σ y þ ðH þ F Þσ 2y þ 2Nτ2xy ¼ 1:

ð11Þ

In this work, Hill's [12] yield criterion was used to predict both the FLC and the SFLC of sheet materials, and considering the associated flow rule, the strain path can be expressed as follows: ρ¼

dε 2 ð F þ H Þα−H ¼ dε 1 1−Hα

ð12Þ

and are the major and minor principal strain increments where in the nominal area of the sheet, respectively. The basic compatibility equations for the MK analysis are:

where dε1 and dε2 are the principal strain increments and α is the ratio of principal stresses:

dε att ¼ dεbtt

α¼

ð7Þ

where dεatt and dεbtt denote the strain increment tangent to the groove in regions (a) and (b), respectively. The requirement of force equilibrium across the imperfection groove is written as: F ann ¼ F bnn

ð8aÞ

F ant ¼ F bnt

ð8bÞ

a b and Fnn denote the force per unit width in the direction norwhere Fnn a b and Fnt are mal to the groove in regions (a), and (b), respectively, and Fnt the shear forces per unit width in regions (a) and (b), respectively, i.e.:





F ann ¼ σ ann t a F bnn ¼ σ bnn t b

ð9aÞ

F ant ¼ σ ant t a F bnt ¼ σ bnt t b

ð9bÞ

Cold-rolled sheets are generally anisotropic and in the computation of limiting strains and stresses it was assumed that the principal axes of anisotropy are coincident with the principal stress directions in the sheet. 2.2. Anisotropic yield criteria According to the continuum theory of plasticity established by Hill [13], the plastic deformation of metals can be fully calculated when the following are defined: • a yield function to specify the transition from elastic to plastic deformation • a flow rule to describe the relation between plastic strain increments and stress components • a strain hardening rule to determine the evolution of the yield locus in stress space while plastic deformation takes place. 2.2.1. Hill's [12] yield criterion In 1948, Hill introduced the first yield criterion that accounts for the anisotropy of cold-rolled sheets, as follows:    2 2f σ ij ¼ F σ y −σ z þ Gðσ z −σ x Þ2  2 þ H σ x −σ y þ2Lτ2yz þ 2Mτ 2zx þ 2Nτ2xy ¼ 1

ð10Þ

σ 2 σ yy ¼ σ 1 σ xx

ð13Þ

where σ1 and σ2 are the principal stresses in the area of nominal thickness (a). The main advantage of Hill's [12] yield function is the simplicity with which the anisotropy coefficients can be determined from basic sheet mechanical properties. Moreover, only a limited number of material data are needed to fully define the yielding behavior, and only three independent material properties are sufficient to define the coefficients in Hill's [12] yield function in plane-stress applications. Therefore this yield function continues to be widely used in numerical simulations. Hill's [12] yield criterion does however have some drawbacks. Considering r0, r45, and r90 as the in-plane plastic anisotropy coefficients of the sheet at 0, 45, and 90° to the rolling direction, respectively, it is reported by many researchers that non-ferrous sheet materials, such as aluminum alloys, have an average anisotropic coefficient r ¼ ðr 0 þ 2r 45 þ r90 Þ=4 b 1.0 and, for such sheet materials, Hill's [12] yield function does not adequately represent the shape of the yield surface because it predicts that the yield locus of these metals will be located inside the one given by von Mises. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r Þ=2 ) which means Hill's [12] criterion predicts ( σ b ¼ σ u ð1 þ  that σ b /σ u b 1.0 but Woodthorpe and Pearce [34,40] showed that the yield locus of these materials should lie outside the von Mises surface (i.e. σ b /σ u N 1.0). This was termed “anomalous” behavior (of the yield function, not of the sheet material) but the yield locus can be predicted correctly using a non-quadratic yield criterion. Also, some sheet materials exhibit a planar anisotropy such that r0/r90 N 1 and σ0/σ90 b 1 at the same time. But once again Hill's [12] criterion is unable to correctly represent this type of behavior bepffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cause it predicts σ 0 /σ 90 = r 0 ð1 þ r90 Þ=ðr 90 ð1 þ r 0 Þ which means that σ0/σ90 N 1 when r0/r90 N 1. Therefore Hill's quadratic yield criterion is said to exhibit a second type of “anomalous” behavior. For these reasons, many anisotropic yield criteria have since been developed that are able to more accurately represent the anisotropic behavior of specific metal sheets. In particular, non-quadratic yield functions have been proposed for the analysis of anisotropic plasticity of sheet materials. 2.2.2. Hosford's yield criterion In 1979, Hosford introduced a non-quadratic yield criterion that is defined as follows:  a  a F σ y −σ z  þ Gjσ z −σ x ja þ H σ x −σ y  ¼ σ a

ð14Þ

M. Nurcheshmeh, D.E. Green / Materials and Design 91 (2016) 248–255

251

Table 1 Material properties of AISI-1012, low carbon steel [28]. r0

r45

r90

F

G

H

K (MPa)

n

m

do (μ)

Rzo (μm)

C

t0 (mm)

1.4

1.05

1.35

0.432

0.417

0.583

238

0.30

0.01

25

6.5

0.104

2.5

where a is a positive integer greater than two. Hosford and coworkers related the value of this exponent to the crystallographic structure of the material [18–20,25] and proposed a = 8 for face-centered-cubic (FCC) materials and a = 6 for body-centered-cubic (BCC) materials as the most appropriate values to describe the shape of the yield surface [20]. When a = 2, Hosford's [17] yield criterion reduces to Hill's [12] yield criterion. In the current work, Hosford's [17] yield criterion was considered for plane-stress conditions and formulated in terms of the plastic anisotropy coefficients as follows:  a  a r 90 jσ x j þ r 0 σ y  þ r 0 r 90 σ x −σ y  ¼ r 90 ðr 0 þ 1Þσ a0 a

ð15aÞ

or equivalently:  a  a H σ x −σ y  þ F σ y  þ Gjσ x ja ¼ σ a0 :

ð15bÞ

In the MK analysis proposed in this work there are no shear stress components, therefore plastic strain increments predicted with Hosford's [17] yield function are very similar to those calculated with more complex non-quadratic yield criteria such as Yld2000. Moreover, with more recent yield criteria a large number of material parameters are required to define the yield locus, and it can be a real challenge for industrial users to carry out the experiments to obtain the required mechanical properties. Therefore the implementation of Hosford's [17] yield criterion in plasticity calculations is much more advantageous for industrial users. The ratio between the effective strain and the major principal strain (λ = εe/ε1) is: λ¼

1 ð1 þ αρÞ ξ

ð16Þ

where ξ is the ratio between the effective stress and the major principal stress (ξ = σe/σ1) and can be defined as follows in the case of normal anisotropy: ξ¼

 1  1 þ jα ja þ rj1−α ja 1þr

1=a ð17Þ

and the relation between the strain and stress path indicators (i.e. the relation between ρ = ε2/ε1 and α = σ2/σi) is: ρ¼

α a−1 −r ð1−α Þa−1

ð18aÞ

1 þ r ð1−α Þa−1

or also: ρ¼

Fα a−1 −H ð1−α Þa−1 H ð1−α Þa−1 þ G

ρ = ρ(α). There are seven solutions to this equation when a = 8 and five solutions when a = 6. However, only one of the solutions is real. Using the associated flow rule, the plastic strain increments can be written as: h i dε x ¼ dλ Hð1−α Þa−1 þ G σ a−1 x

ð19aÞ

h i dε y ¼ dλ Fα a−1 −H ð1−α Þa−1 σ a−1 : x

ð19bÞ

   dε z ¼ − dε x þ dε y ¼ −dλ G þ Fα a−1 σ a−1 x

ð19cÞ

Eqs. (15a) to (19c) were used to implement Hosford's non-quadratic yield criterion into the authors' numerical code that computes FLCs using the MK analysis. Material work hardening was defined using Swift's power law function: m σ e ¼ kε_ e ðεe þ ε0 Þn

ð20Þ

where σe and εe are the effective stress and strain values, respectively. Moreover, it was assumed that the yield surface expands isotropically in stress space as the material work hardens. The hardening law can also be expressed in differential form: dð ln σ e Þ ¼ mdð lnεe Þ þ ndð ln ðεe þ ε0 ÞÞ where ε0 is a uniform prestrain applied to the sheet prior to the current forming process; m is the strain-rate sensitivity coefficient; and n is the strain-hardening coefficient. Considering the MK equations, the associated flow rule, the yield criteria and work hardening relations, the governing equation for the strain between both regions can be developed. As biaxial stress increments are imposed in the nominal area, this causes the development of strain increments in both the nominal area (a) and the weaker band (b). Strain development in the thinner region (b) is greater than in the thicker region (a), and the strain rate difference between both regions becomes intensified as deformation progresses. The limit strains were calculated numerically [using a combination of Newton–Raphson and Runge–Kutta (4th order) methods] from the governing equation that expresses the development of necking under plane-stress conditions with the assumption that necking occurs once the effective strain rate in the groove exceeds 10 times that in the nominal area. In other words, the limit strains were obtained when εbe / εae N 10. The values of ε1 and ε2 in area (a) were determined for different linear strain paths in the range of ρ = ε2 / ε1 = −0.5 to + .0 and were used to plot the FLC. 3. Results

:

ð18bÞ

In this case, the inverse relation, α = α(ρ) cannot be given explicitly but must be numerically solved for each value of ρ using the equation

3.1. Material characterization The sheet materials considered throughout this investigation for model verification are a low carbon steel AISI-1012 [28] and a AA-

Table 2 Average mechanical properties of AA-2008-T4 [10]. r0

r45

r90

F

G

H

K (MPa)

n

m

do (μ)

Rzo (μm)

C

t0 (mm)

0.58

0.48

0.78

0.246

0.633

0.367

535

0.27

−0.003

8a

2.5a

0.70a

1.7

a

Data determined by calibration with the experimental as-received FLC.

252

M. Nurcheshmeh, D.E. Green / Materials and Design 91 (2016) 248–255

Fig. 2. Comparison of predicted and experimental FLCs of as-received AISI-1012 steel sheets.

Fig. 4. Comparison of predicted and experimental FLCs of AISI-1012 steel after 10% prestrain in uniaxial tension.

2008-T4 aluminum alloy described in the work of Graf and Hosford [10]. Tables 1 and 2 present the anisotropy coefficients, the corresponding yield stresses and other related material properties for AISI-1012 and AA-2008-T4 alloys, respectively. Based on Eqs. (1), (3), and (4), f 0 values were calculated to be 0.995 and 0.997 for the AISI-1012 steel and the AA-2008-T4 alloy, respectively. The fit of the experimental stress–strain curves using Swift's power law function is very good, as shown in another publication [31].

Following the implementation of both Hill's [12] quadratic yield criterion and Hosford's [17] non-quadratic yield criterion into the MK analysis code, the predicted FLCs were compared with corresponding experimental FLCs for both linear and bilinear loading paths. Fig. 2 shows good agreement between both theoretical predictions and experimental data for AISI-1012 sheet steel in its as-received state. In this figure, the FLC predicted with Hosford's yield function fits very well with the corresponding experimental curve in all regions of the diagram. Both yield criteria lead to the same prediction of the FLC in the region of plane-strain deformation which is a critical deformation mode in sheet metal forming. As anticipated, the FLC predicted with the non-quadratic yield function is in better agreement with the experimental data than that predicted with the quadratic yield function on the right side of the diagram (i.e. for positive minor strains) but the

two criteria predict the left side of the FLC (i.e. for negative minor strains) with a similar level of accuracy. The FLCs of AISI-1012 steel were also calculated for two nonlinear loading paths. In the first case sheet specimens were preloaded to 8% strain in equibiaxial tension and the FLC was determined following this prestrain by simulating a series of linear load paths in the range between ρ = − 0.5 and ρ = + 1.0 (i.e. between uniaxial tension and equibiaxial tension). In the second case the sheet material was subject to a 10% prestrain in uniaxial tension followed by a range of linear loading paths between uniaxial tension and equibiaxial tension. The FLCs predicted for these two types of bilinear strain paths with either Hill's or Hosford's yield criterion are shown in Figs. 3 and 4, respectively, along with the corresponding experimental data. In Fig. 3, the published experimental data was only available for the left side of the FLC, however, both plasticity models show good agreement with the experimental data after a prestrain in equibiaxial tension. Once again it can be observed that both yield criteria lead to the same prediction of limiting strains in the plane-strain region for both loading histories. But it can also be observed that Hosford's non-quadratic yield criterion gives a better prediction in the regions to the left and right of plane-strain for steel specimens prestrained in uniaxial tension, as shown in Fig. 4. Overall, it can be seen that the differences between the predictions using these two yield criteria are not significant for this particular grade of steel.

Fig. 3. Comparison of predicted and experimental FLCs of AISI-1012 steel after 8% prestrain in equibiaxial tension.

Fig. 5. Comparison of calibrated/predicted and experimental FLCs of as-received 2008-T4 aluminum sheets.

3.2. Forming limit curves

M. Nurcheshmeh, D.E. Green / Materials and Design 91 (2016) 248–255

Fig. 6. Comparison of predicted and experimental FLCs of 2008-T4 aluminum after 4% prestrain in equibiaxial tension.

253

Fig. 8. Comparison of predicted and experimental FLCs of 2008-T4 aluminum after 5% prestrain in uniaxial tension.

The FLC of the AA-2008-T4 aluminum alloy sheet predicted with both Hosford's and Hill's yield criteria and the corresponding experimental data for the as-received condition are shown in Fig. 5. The parameters in Stachowicz's surface roughness equation (C, do, Rzo) were not available for the AA-2008-T4 aluminum sheet that was used in Graf and Hosford's [10]investigation. These parameters were therefore calibrated by fitting the theoretical FLC to the experimental asreceived FLC. The parameters were optimized one at a time by a series of FLC predictions in which one parameter was varied successively while the other two were held constant. For each case, the value of the variable parameter that gave the best overall fit of the predicted FLC with the experimental as-received FLC was selected as the calibrated value for this parameter. This procedure was repeated for all three parameters and the material constants thus obtained are C = 0.70, do = 8.00 μm and Rzo = 2.5 μm. Prediction of the FLC using Hosford's yield function was also performed with the same material constants. It can be observed in Fig. 5 that the predicted curves correlate very well with the experimental data, but the FLC predicted with the nonquadratic yield criterion appears to provide a better fit than the one predicted with the quadratic function. The FLC of AA-2008-T4 sheet was also predicted for bilinear loading paths in which a prestrain was applied in different modes of deformation. In the first case, the FLC was predicted for sheets prestrained to either 4% or 12% in equibiaxial tension. The curves calculated using both quadratic and non-quadratic yield functions are shown along with the corresponding experimental data in Figs. 6 and 7, respectively.

In Fig. 6, it appears that Hosford's yield criterion leads to a better prediction of the FLC for the samples with a 4% prestrain in equibiaxial tension. However, when a greater magnitude of prestrain is applied along the same strain path (ρ = 1), Hill's quadratic yield criterion seems to provide a slightly better correlation with experimental data (Fig. 7). Nevertheless, both yield functions give very similar predictions and both criteria lead to an acceptable level of accuracy for this aluminum alloy. Graf and Hosford also published experimental FLC data for this aluminum alloy for a prestrain of either 5% or 12% in uniaxial tension. In order to further validate the FLC predictions using these two different plasticity models, the FLC was predicted for both strain histories with the present MK model. The predicted FLCs and the corresponding experimental data are shown in Figs. 8 and 9. In Fig. 8, after a 5% prestrain in uniaxial tension, it can be seen that Hill's yield criterion gives a slightly better prediction of the right side of the FLC than Hosford's criterion; but this is one of the rare cases amongst those investigated where the quadratic function seems to give a better prediction than the non-quadratic function. It can also be pointed out that, in this case, Hosford's criterion still gives a better prediction in the region of plane-strain deformation and shows a more accurate trend on the left side of the FLC. For the AA-2008-T4 sheet samples deformed to a 12% prestrain in uniaxial tension (Fig. 9), the two plasticity models yield similar results, but the non-quadratic yield function clearly provides a more accurate prediction than the quadratic criterion for both sides of the FLC.

Fig. 7. Comparison of predicted and experimental FLCs of 2008-T4 aluminum after 12% prestrain in equibiaxial tension.

Fig. 9. Comparison of predicted and experimental FLCs of 2008-T4 aluminum after 12% prestrain in uniaxial tension.

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Fig. 10. Comparison of predicted SFLCs of AISI-1012 steel using Hill's quadratic and Hosford's non-quadratic yield functions.

It could be ideal to have consistent experimental data set for both materials in the verification of the theoretical work in strain space. Obtaining very consistent experimental FLC data on nonlinear loading paths was not possible for these materials. Knowing this we analyzed the experimental FLC data of these materials separately in verification of the theory. 3.3. Stress-based forming limit curves In order to observe the effect of the yield criterion on the stressbased forming limit curve (SFLC) prediction, SFLCs corresponding to the as-received AISI-1012 steel and AA-2008-T4 aluminum alloy were calculated using Hill's quadratic and Hosford's non-quadratic yield criteria and the results are shown in Figs. 10 and 11, respectively. For lower levels of prestrain, SFLCs for bilinear loading paths coincide with the as-received SFLC [30,42] therefore SFLC results corresponding to the bilinear loading paths are not shown in this paper. There is not also enough experimental data in literature on the SFLC. However further research is needed to determine the extent and limitations of SFLC path-independence. The current work will ensure that the MK analysis is able to calculate both stress-based and strain-based FLC simultaneously, so that verification of the proposed theory in strain space, verifies it in stress space as well. Figs. 10 and 11 show that the SFLCs predicted with different yield functions can be quite different. As it can be seen in Fig. 10, SFLCs predicted for AISI-1012 steel using the quadratic and non-quadratic yield criteria

are significantly different; however, the SFLCs for AA-2008-T4 aluminum predicted with these two yield criteria are very similar (Fig. 11). The differences between the SFLCs predicted with these two yield criteria may be surprising at first sight. Since the stresses and strains are calculated simultaneously in this MK model it is expected that the predicted SFLCs are correctly computed in as much as the FLC predictions were shown to correlate very well with the experimental FLC. The differences between the two SLFCs in Fig. 10 come from the fact that the shapes of the quadratic and non-quadratic yield loci are quite different; and since the normality rule requires that the plastic strain increment be perpendicular to the yield locus, it can be anticipated that for a given strain path, the corresponding stress path will depend on the orientation of the normal to the yield locus at this point. Therefore it is not surprising that different yield loci will lead to different orientations of the plastic strain vector and therefore to different limiting stresses. This work underscores the importance of using the most accurate yield locus when predicting the SFLC. In the absence of experimental SFLC data it is not possible to conclude which of these two yield loci leads to the better SFLC prediction, however, since Hosford's [17] criterion was shown to provide a better prediction of the FLC than Hill's [12] yield function, it is expected that the non-quadratic yield function will also lead to a more accurate prediction of SFLC than the quadratic yield function.

4. Conclusions In this paper a numerical MK analysis code was developed to predict the FLC and SFLC of sheet metal using Hosford's [17] nonquadratic yield criterion. Forming limits were predicted for both linear and bilinear loading paths for AISI-1012 steel and AA-2008T4 aluminum sheets. The theoretical results that were obtained were compared with the corresponding experimental data and also with the FLCs predicted with Hill's [12] yield function at the same condition. Both anisotropic plasticity theories are able to predict the FLC of these two sheet materials very well, for both the as-received condition and also for samples prestrained in uniaxial or equibiaxial tension. Not only do the predicted FLCs follow the general shape of the experimental FLC, but their accuracy is also very good considering there is an estimated error of ± 2.5% strain on the vertical position of an experimental FLC [32]. However, on the whole, the prediction of FLC using Hosford's yield function is somewhat better than when Hill's quadratic yield criterion is employed, especially on the right side of the FLC. In most cases, the predictions made with these two yield criteria were very similar, with only minor variations on the left side of the FLC. There was only one case where the prediction using Hill's [12] criterion was more consistent with experimental data than Hosford's [17] criterion, and this was on the right side of the FLC of the 2008-T4 aluminum alloy after a 5% percent prestrain in uniaxial tension. Based on the observations made in this work, it may be concluded that Hosford's non-quadratic yield function generally leads to more accurate predictions of limiting strains than Hill's quadratic function, for as-received as well as prestrained material, whether it is prestrained in uniaxial tension or in equibiaxial tension, and for both steel and aluminum sheets. Different yield criteria lead to different SFLC and the discrepancy was more significant than for AISI-1012 steel for the 2008-T4 aluminum alloy.

Acknowledgments

Fig. 11. Comparison of predicted SFLCs of AA-2008-T4 aluminum using Hill's quadratic and Hosford's non-quadratic yield functions.

The authors would like to acknowledge the financial support of the Natural Sciences and Engineering Research Council (NSERC) of Canada APCPJ 418056-11 for this work.

M. Nurcheshmeh, D.E. Green / Materials and Design 91 (2016) 248–255

References [1] R.J. Asaro, A. Needleman, Texture development and strain hardening in rate dependent polycrystals, Acta Metall. 33 (1985) 923–953. [2] D. Banabic, H.J. Bunge, K. Pöhlandt, A.E. Tekkaya, Formability of Metallic Materials, Springer-Verlag, Berlin Heidelberg, 2000. [3] D. Banabic, E. Dannenmann, Prediction of the influence of yield locus on the limit strains in sheet metals, J. Mater. Process. Technol. 109 (2001) 9–12. [4] D. Banabic, H. Aretz, L. Paraianu, P. Jurco, Application of various FLD modeling approaches, Model. Simul. Mater. Sci. Eng. 13 (2005) 759–769. [5] A. Barata da Rocha, F. Barlat, J.M. Jalinier, Prediction of the forming limit diagrams of anisotropic sheets in linear and nonlinear loading, Mater. Sci. Eng. 68 (1985) 151–164. [6] F. Barlat, Crystallographic texture, anisotropic yield surfaces and forming limits of sheet metals, Mater. Sci. Eng. 91 (1987) 55–72. [7] M.C. Butuc, D. Banabic, A. Barata da Rocha, J.J. Gracio, J. Ferreira Duarte, P. Jurco, D.S. Comsa, The performance of Yld96 and BBC2000 yield functions in forming limit prediction, J. Mater. Process. Technol. 125–126 (2002) 281–286. [8] M.C. Butuc, J.J. Gracio, A. Barata da Rocha, An experimental and theoretical analysis on the application of stress-based forming limit criterion, Int. J. Mech. Sci. 48 (2006) 414–429. [9] J. Cao, H. Yao, A. Karafillis, M.C. Boyce, Prediction of localized thinning in sheet metal using a general anisotropic yield criterion, Int. J. Plast. 16 (2000) 1105–1129. [10] A. Graf, W. Hosford, Effect of changing strain paths on forming limit diagrams of Al 2008-T4, Metall. Trans. A 24A (1993) 2503–2512. [11] G.M. Goodwin, Application of strain analysis to sheet metal forming in the press shop, SAE technical paper 680093, 1968. [12] R. Hill, A theory of the yielding and plastic flow of anisotropic metals, Proc. R. Soc. Lond. A 193 (1948) 281–297. [13] R. Hill, The Mathematical Theory of Plasticity, University Press, Oxford, 1950. [14] R. Hill, Theoretical plasticity of textured aggregates, Math. Proc. Camb. Philos. Soc. 85 (1979) 179–191. [15] R. Hill, A user friendly theory of orthotropic plasticity in sheet metals, Int. J. Mech. Sci. 35 (1993) 19–25. [16] P. Hora, L. Tong, J. Reissner, A prediction method for ductile sheet metal failure in FEsimulation, Proc. Numisheet’96 Conf., Dearborn, Michigan 1996, pp. 252–256. [17] W.F. Hosford, On yield loci of anisotropic cubic metals, Proc.7th North American Metalworking Conf. (NMRC), SME, Dearborn, MI 1979, pp. 191–197. [18] W.F. Hosford, Comments on anisotropic yield criteria, Int. J. Mech. Sci. 27 (1985) 423–427. [19] W.F. Hosford, The Mechanics of Crystals and Textured Polycrystals, Oxford University Press, New York, 1993. [20] W.F. Hosford, On the crystallographic basis of yield criteria, Texture Micro. 26-27 (1996) 479–493. [21] M. Kuroda, V. Tvergaard, Forming limit diagrams for anisotropic metal sheets with different yield criteria, Int. J. Solids Struct. 37 (2000) 5037–5059. [22] S.P. Keeler, W.A. Backhofen, Plastic instability and fracture in sheet stretched over rigid punches, ASM Trans. Q. 56 (1964) 25–48.

255

[23] H.J. Kleemola, M.T. Pelkkikangas, Effect of pre-deformation and strain path on the forming limits of steel, copper and brass, Sheet Met. Ind. 63 (1977) 559–591. [24] J. Lian, D. Zhou, B. Baudelet, Application of Hill's new yield theory to sheet metal forming. Part I: Hill's 1979 criterion and its application to predicting sheet forming limit, Int. J. Mech. Sci. 31 (1989) 237–247. [25] R. Logan, W.F. Hosford, Upper-bound anisotropic yield locus calculations assuming (111) — pencil glide, Int. J. Mech. Sci. 22 (1980) 419–430. [26] Z. Marciniak, K. Kuczynski, Limit strains in processes of stretch-forming sheet metal, Int. J. Mech. Sci. 9 (1967) 609–620. [27] Z. Marciniak, K. Kuczynski, T. Pokora, Influence of the plastic properties of a material on the forming limit diagram for sheet metal in tension, Int. J. Mech. Sci. 15 (1973) 789–805. [28] B. Molaei, Strain path effects on sheet metal formability, Amirkabir University of Technology, Iran, 1999 (Ph.D. dissertation). [29] T.J. McCarron, K.E. Kain, G.T. Hahn, W.E. Flanagan, Effect of geometrical defects in forming sheet steel by biaxial stretching, Metall. Trans. 19A (1988) 2067–2074. [30] M. Nurcheshmeh, D.E. Green, Investigation on the strain-path dependency of stressbased forming limit curves, Int. J. Mater. Form. 4 (2011) 25–37. [31] M. Nurcheshmeh, D.E. Green, Implementation of mixed nonlinear kinematic-isotropic hardening into MK analysis to calculate forming limit curves, Int. J. Mech. Sci. 53 (2011) 145–153. [32] M. Nurcheshmeh, D.E. Green, Influence of out-of-plane compression stress on limit strains in sheet metals, Int. J. Mater. Form. 5 (2012) 213–226. [33] A. Parmar, P.B. Mellor, Prediction of limit strains in sheet metal using a more general yield criterion, Int. J. Mech. Sci. 20 (1978) 385–391. [34] R. Pearce, Some aspects of anisotropic plasticity in sheet metals, Int. J. Mech. Sci. 10 (1968) 995–1001. [35] F. Stachowicz, Effect of annealing temperature on plastic flow properties and forming limits diagrams of titanium and titanium alloy sheets, Trans. Jpn. Inst. Metals 29 (1988) 484–493. [36] S. Stören, J.R. Rice, Localized necking in thin sheets, J. Mech. Phys. Solids 23 (1975) 421–441. [37] V. Tvergaard, A. Needleman, Shear band development in polycrystals, Proc. Roy. Soc. London 443A (1993) 547–562. [38] P.D. Wu, K.W. Neale, E. Van der Giessen, On crystal plasticity FLD analysis, Proc. Roy. Soc. London 453A (1997) 1831–1848. [39] P.D. Wu, K.W. Neale, E. Van der Giessen, M. Jain, A. Makinde, S.R. MacEwen, Crystal plasticity forming limit diagram analysis of rolled aluminum sheets, Metall. Mater. Trans. 29A (1998) 527–535. [40] J. Woodthorpe, R. Pearce, The anomalous behaviour of aluminium sheet under balanced biaxial tension, Int. J. Mech. Sci. 12 (1970) 341–347. [41] S. Xu, K.J. Weinmann, Prediction of forming limit curves of sheet metals using Hill's 1993 user friendly yield criterion of anisotropic materials, Int. J. Mech. Sci. 40 (1998) 913–925. [42] K. Yoshida, T. Kuwabara, M. Kuroda, Path-dependence of the forming limit stresses in a sheet metal, Int. J. Plast. 23 (2007) 361–384.