Mathematics and Mechanics of Solids Non-linear influence...

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Mathematics and Mechanics of Solids

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DOI: 10.1177/1081286516675662

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Non-linear influence of hydrostaticpressure on the yielding of asymmetricanisotropic sheet metals

Farzad MoayyedianDepartment of Mechanical Engineering, Eqbal Lahoori Institute of Higher Education (ELIHE),Mashhad, Iran

Mehran KadkhodayanDepartment of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran

Received 15 May 2016; accepted 17 September 2016

AbstractThe objective of the current research is the investigation into possible non-linear influence of hydrostatic pressure onyielding of asymmetric (exhibiting the so-called ‘‘strength-differential effect’’) anisotropic sheet metals. To reach this aim,two yield functions are developed, called here ‘‘non-linear pressure sensitive criteria I and II,’’ (NPC-1 and NPC-2). Inaddition, the non-associated flow rules are employed for these new criteria. The yield functions are defined as non-linearly dependent on hydrostatic pressure, while the plastic potential functions are introduced to be pressure insensi-tive. To calibrate these criteria, the yield functions need 10 directional experimental yield stresses and the plastic poten-tial functions need eight Lankford coefficients data points. Four well-known anisotropic sheet metals with differentstructures, namely AA 2008-T4, a Face Centered Cubic material (FCC), AA 2090-T3, a Face Centered Cubic material(FCC), AZ31, a hexagonal closed packed material (HCP) and high-purity a-titanium (HCP) are considered as case stud-ies. Finally, it is observed that NPC-1 and NPC-2 are more successful than previous criteria in anticipating directionalstrength and mechanical properties.

KeywordsAsymmetric anisotropic sheet metals, strength-differential effect, non-linear pressure sensitive criterion, non-associatedflow rule, directional yield stresses, directional Lankford coefficients

1. Introduction

According to the studies of Spitzig and Richmond [1] in the late 1970s on high-strength steels and aluminumalloys, the yielding behavior clearly depends upon hydrostatic pressure, but the materials exhibit nearlyplastically incompressible deformations. They stated that this dependency was linear. The main objective ofthe current research is an investigation into the possible non-linear effect of hydrostatic pressure on the yieldfunction of asymmetric anisotropic sheet metals. A literature review on yielding of anisotropic materialswith different structures and also employing non-associated flow rules (non-AFRs) is presented here.

Spitzig and Richmond [1] experimentally showed that the flow stress was linearly based on hydrostaticpressure in both iron-based materials and aluminum. Liu et al. [2] extended the Hill criterion to include

Corresponding author:

Mehran Kadkhodayan, Department of Mechanical Engineering, Ferdowsi University of Mashhad, Mashhad, Iran.

Email: kadkhoda@um.ac.ir

at University of Otago Library on November 22, 2016mms.sagepub.comDownloaded from

orthotropic plastic materials with different yield stresses in tension and compression. Barlat et al. [3] pro-posed a plane stress yield function named Yld2000-2d and validated it with experimental and polycrystaldata achieved on a binary Al-2.5% wt Mg alloy sheet. Stoughton and Yoon [4] suggested a non-AFRbased on a pressure sensitive yield criterion with isotropic hardening, which was consistent with Spitzigand Richmond’s results [1]. Hu and Wang [5] proposed a yield function to anticipate the strength-differential effects (SDEs) in tension and compression of materials. Hu [6] presented a yield function thatdefined the yield condition by considering the effect of both loading force and loading direction in aniso-tropic materials. Artez [7] modified a plane stress yield function based on the Hasford non-quadraticyield function called ‘‘Yld2003,’’ which was approximately as flexible as Balart yield function ‘‘Yld2000-2D’’ but had an easier mathematical formulation. Lee et al. [8] developed a yield function with a pressuresensitive term considering the high directional differences in initial yield stress and high asymmetry intension and compression as well. Stoughton and Yoon [9] proposed a model for proportional loadingfor any biaxial stress conditions. The model was shown to lead to a reduction in the error of predictionof the anisotropic stress–strain relationship in uniaxial and equi-biaxial tension. Hu and Wang [10]defined a plastic potential function to depict the characteristic of plastic flow of material to construct asuitable constitutive model. Huh et al. [11] computed the accuracy of anisotropic yield functions ofHill48, Yld89, Yld91, Yld96, Yld2000-2d, BBC2000 and Yld2000-18p based on the root mean squareerror (RMSE) of yield stresses and Lankford coefficients. Taherizadeh et al. [12] compared three aniso-tropic models for simulation of sheet metal forming processes in the form of Hill, an associated non-quadratic formulation (Yld2000) and a non-associated non-quadratic formulation in which the yieldand potential functions were based on Yld91 and Yld89, respectively. Lou et al. [13] presented anapproach to extend symmetric yield functions to consider the SDEs for incompressible sheet metals withassociated flow rule. Safaei et al. [14] presented a non-associated plane stress anisotropic constitutivemodel with mixed isotropic–kinematic hardening. The quadratic Hill 1948 and non-quadratic Yld-2000-2d yield criteria were considered in non-AFRs. Yoon et al. [15] proposed a yield function based on thefirst, second and third stress modified invariants of the stress tensor to depict SDEs of anisotropic mate-rials. Safaei et al. [16] presented a method to describe the evolution of anisotropy during plastic deforma-tion. A non-AFR based on the Yld2000-2d anisotropic yield model was used. They described two simplemethods for the relationship between equivalent plastic strain and compliance factor in a non-AFRmodel. Moayyedian and Kadkhodayan [17] combined von Mises and Tresca surfaces in place of yieldand plastic potential functions and vice versa. They showed that taking von Mises and Tresca surfacesas yield and plastic potential functions, respectively, predicted experimental results more accurately thanthe associated von Mises. Moreover, taking Tresca and von Mises surfaces as yield and plastic potentialfunctions, respectively, predicted experimental results more precise than associated Tresca. Oya et al.[18] offered a new expression for the plastic constitutive model for materials with initial anisotropy. Forthis purpose, a non-associated normality model, in which the plastic potential function was defined inde-pendently of the yield function, was adopted. An explicit expression for the equivalent plastic strain rate,which was plastic-work-conjugated with the defined equivalent stress corresponding to the proposedyield function, was also presented. Moayyedian and Kadkhodayan [19] introduced the ModifiedYld2000-2d II by inserting modified Yld2000-2d and Yld2000-2d in place of yield and plastic potentialfunctions, respectively, to depict the behavior of anisotropic pressure sensitive sheet metals more accu-rately. Moayyedian and Kadkhodayan [20] modified the Burzynski criterion used for pressure sensitiveisotropic materials to a criterion for anisotropic pressure sensitive sheet metals based on non-AFRs for abetter description of the asymmetric anisotropic sheet metal behavior. Ghaei and Taherizadeh [21] pro-posed a model to depict the anisotropic behavior of sheet metals in both yield stresses and plastic strainratios by using the non-AFR and quadratic yield and potential functions. In addition, to reproduce anaccurate prediction of cyclic plastic deformation phenomena, a two-surface mixed isotropic-non-linearkinematic hardening model was combined with the quadratic non-AFR anisotropic formulation.Moayyedian and Kadkhodyan [22] introduced an advanced criterion with non-associated flow rule(non-AFR) for depicting the behavior of anisotropic sheet metals to consider the SDEs for these materi-als. To verify the accuracy of the advanced criterion, three anisotropic sheet metals with different struc-tures were taken as case studies, namely AA 2008-T4 (a FCC material), AA 2090-T3 (a FCC material)and AZ31 (a HCP material).

2 Mathematics and Mechanics of Solids

In the present study the possible influence of non-linear pressure dependency on yielding of asym-metric anisotropic sheet metals is investigated. Two criteria are newly presented, called here NPC-1 andNPC-2. These criteria are based on non-AFRs with different yield and plastic potential functions. Theiryield functions non-linearly depend on modified hydrostatic pressure, while their plastic potential func-tions are pressure insensitive. The main differences between them are as follows: NPC-2 can handle ageneral three-dimensional (3D)-stress state; NPC-1 has only plane stress; NPC-2 has essentially asym-metric yield functions and plastic potential; NPC-1 is only asymmetric due to the influence of the hydro-static stress component. Four familiar anisotropic sheet metals, namely AA 2008-T4 (FCC), AA2090-T3 (FCC), AZ31 (HCP) and a-titanium (HCP) are employed as case studies. Finally, it is illu-strated that NPC-1 and NPC-2 in the presented form can successfully anticipate directional experimentalresults better than Yoon et al. [15] and Lou et al. [13] and this may be due to the use of the non-linearinfluence of modified pressure instead of linear pressure in their yield function formulations.

2. Non-linear pressure sensitive criterion I (NPC-1)

To describe the asymmetric behavior of an anisotropic material, a new non-AFR is proposed, called here‘‘NPC-1,’’ in which its yield function is non-linearly dependent on modified hydrostatic pressure whileits plastic potential function is pressure insensitive. In fact, NPC-1 is the extension of Lou et al. [13]. Thenumber of required experimental data points for calibrating its yield and plastic potential function are10 and eight, respectively. The axes of x, y and z represent the rolling direction (RD), transverse direc-tion (TD) and normal direction (ND) of cold rolled metals, respectively [13,15].

To consider the anisotropic effects in yield and plastic potential functions in a plane stress problem(i.e. szz = tyz = txz = 0), four linear transformation matrices (L0ij, L00ij for the yield function and M 0ij,M

00ij for

the plastic potential function), which are applied to the stress tensor, are employed. In this case, the firstand second modified deviatoric stress tensors (X 0ij,X 00ij for yield and Y 0ij, Y 00ij for plastic potential functions)are presented as equations (1) and (2)

9>=>;=

L011 L012 0

L021 L022 0

0 0 L066

X 00xx

X 00yy

X 00xy

9>=>;=

L0011 L0012 0

L0021 L0022 0

0 0 L0066

8>>>>>>>>><>>>>>>>>>:

9>=>;=

M 011 M 012 0

M 021 M 022 0

0 0 M 066

Y 00xx

Y 00yy

Y 00xy

9>=>;=

M 0011 M 0012 0

M 0021 M 0022 0

0 0 M 0066

8>>>>>>>>><>>>>>>>>>:

In which sij is the stress tensor and L0ij, L00ij and M 0ij,M00ij are expressed with 16 anisotropic unknown coeffi-

cients ai(i = 1� 8),bi(i = 1� 8) for yield and plastic potential functions, respectively, as shown in equa-tions (3) and (4)

Moayyedian and Kadkhodayan 3

8>>>><>>>>:

9>>>>=>>>>;

�1 0 0

0 �1 0

266664

377775

8>>>><>>>>:

9>>>>=>>>>;

�2 2 8 �2 0

1 �4 �4 4 0

4 �4 �4 1 0

�2 8 2 �2 0

0 0 0 0 9

266664

377775

8>>>><>>>>:

9>>>>=>>>>;

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

8>>>><>>>>:

9>>>>=>>>>;

�1 0 0

0 �1 0

266664

377775

M 0011

M 0012

M 0021

M 0022

M 0066

8>>>><>>>>:

9>>>>=>>>>;

�2 2 8 �2 0

1 �4 �4 4 0

4 �4 �4 1 0

�2 8 2 �2 0

0 0 0 0 9

266664

377775

8>>>><>>>>:

9>>>>=>>>>;

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

Hence, the first X 01,X 02� �

and the second X 001,X 002� �

modified principal deviatoric stresses for the yieldfunction and also the first Y 01, Y 02

� �and the second Y 001, Y 002

� �modified principal deviatoric stresses for the

plastic potential function are defined as in equations (5) and (6)

X 01,X02 = 1

2X 0xx + X 0yy6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX 0xx � X 0yy

� �2

+ 4X 0xy2

X 001,X 002 = 12

X 00xx + X 00yy6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiX 00xx � X 00yy

� �2

+ 4X 00xy2

r !8>>>><>>>>:

Y 01,Y02 = 1

2Y 0xx + Y 0yy6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiY 0xx � Y 0yy

� �2

+ 4Y 0xy2

Y 001,Y002 = 1

2Y 00xx + Y 00yy6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiY 00xx � Y 00yy

� �2

+ 4Y 00xy2

r !8>>>><>>>>:

From the above definitions and based on non-AFR, NPC-1 may be proposed as shown in equation (7),where F and G are the yield and the plastic potential functions, respectively, while s �epð Þ defines the iso-tropic hardening behavior of anisotropic materials in which �ep is the effective plastic strain

� �= hxsxx + hysyy

� �a+

X 01 � X 02�� a + 2X 002 + X 001

�� a + 2X 001 + X 002�� a

= s �epð Þ

� �=

Y 01 � Y 02�� b + 2Y 002 + Y 001

�� b + 2Y 001 + Y 002�� b

8>>>>><>>>>>:

Here hx, hy are unknown coefficients for modified hydrostatic pressure hxsxx + hysyy

� �in an anisotropic

sheet metal. The yield function is asymmetric in the sxx � syy plane is due to modified hydrostatic pres-sure, while the plastic potential function is a symmetric function. In NPC-1, a, b are material parametersthat can be adjusted for different anisotropic structures in the process of calibration of the criterion. It isassumed that a is an odd number (to consider the effect of modified hydrostatic pressure in tensile andcompressive tests) and b is an even number.

The yield function of NPC-1 is the extension of Lou et al. [13] yield function to examine the non-linear effect of modified hydrostatic pressure on yielding of anisotropic sheet metal. In addition, aplastic potential function is introduced for NPC-1 in order to compute directional Lankford coeffi-cients properly. To hold the incompressibility condition (i.e. dep

xx + depyy + dep

zz = 0), it is proposed tobe pressure independent. To compute the directional Lankford coefficients for different anisotropicstructures (BCC [Body Centered Cubic material], FCC and HCP) more precisely, a material para-meter b is included in the plastic potential function equation, which can be different from materialparameter a in the yield function (equation (7)). To calibrate the plastic potential function of NPC-1,the first differentiations of G with respect to sij are needed. By introducing K 01,K 02,K 001 ,K 002 into equa-tion (9), the first differentiation is computed as equation (8)

∂sxx

=G1�b

2× ½ 2b1 + b2ð Þ× 2b1 + b2ð Þsxx � b1 + 2b2ð Þsyy

18K 022K 02� �b�1

2M 0011 + M 0021

� �� M 0011 �M 0021

� �×

M 0011 �M 0021

� �sxx + M 0012 �M 0022

� �syy

4K 002

� 3K 001 � K 002�� b�1

2M 0011 + M 0021

� �+ M 0011 �M 0021

� �×

M 0011 �M 0021

� �sxx + M 0012 �M 0022

� �syy

4K 002

� 3K 001 + K 002�� b�1�

∂syy

=G1�b

2× ½� b1 + 2b2ð Þ× 2b1 + b2ð Þsxx � b1 + 2b2ð Þsyy

18K 022K 02� �b�1

2M 0012 + M 0022

� �� M 0012 �M 0022

� �×

M 0011 �M 0021

� �sxx + M 0012 �M 0022

� �syy

4K 002

� 3K 001 � K 002�� b�1

2M 0012 + M 0022

� �+ M 0012 �M 0022

� �×

M 0011 �M 0021

� �sxx + M 0012 �M 0022

� �syy

4K 002

� 3K 001 + K 002�� b�1�

∂txy

=G1�b

K 022K 02s

� �b�1 �M 0066

� �2

K 0023K 001 � K 002�� b�1

+M 0066

� �2

K 0023K 001 + K 002�� b�1

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

K 01 =2b1 � b2ð Þsxx � b1 � 2b2ð Þsyy

K 02 =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2b1 + b2ð Þsxx � b1 + 2b2ð Þsyy

+ b7txy

� �2

K 001 =M 0011 + M 0021

� �sxx + M 0012 + M 0022

� �syy

K 002 =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiM 0011 �M 0021

� �sxx + M 0012 �M 0022

� �syy

+ M 0066txy

� �2

8>>>>>>>>>>>>>><>>>>>>>>>>>>>>:

3. Non-linear pressure sensitive criterion II (NPC-2)

To depict the asymmetric behavior of an anisotropic material, another new non-AFR is proposed, calledhere ‘‘NPC-2,’’ for which its yield function is non-linearly dependent on modified hydrostatic pressureand its plastic potential function is pressure independent. The yield function of NPC-2 is the develop-ment of the Lou et al. [15] function. The number of experimental data points that is required for

calibration of its yield and plastic potential functions are 10 and eight, respectively, as for NPC-1. Theaxes of x, y and z are similar to those for NPC-1.

To consider the anisotropy, four modified deviatoric stress tensors (s0ij, s00ij for the yield function and�s0ij,�s

00ij for the plastic potential function) are applied to the stress tensor (sij) in 3D stress space according

to equations (10) and (11), where c0i(i = 1, 6), c00i (i = 1, 6) and �c0i(i = 1, 6),�c00i (i = 1, 6) are unknown coeffi-cients that can be determined experimentally with different tests in different orientations from the RDto consider the anisotropic effects in yield and plastic potential functions, respectively

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

c02 + c033

� c033

� c023

� c033

c03 + c013

� c013

� c023

� c013

c01 + c023

0 0 0 c04 0 0

0 0 0 0 c05 0

0 0 0 0 0 c06

266666666666664

377777777777775

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

c002 + c0033

� c0033

� c0023

� c0033

c003 + c0013

� c0013

� c0023

� c0013

c001 + c0023

0 0 0 c004 0 0

0 0 0 0 c005 0

0 0 0 0 0 c006

266666666666664

377777777777775

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð10Þ

�s0xx

�s0yy

�s0zz

�s0yz

�s0xz

�s0xy

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

�c02 +�c033

��c033

��c023

��c033

�c03 +�c013

��c013

��c023

��c013

�c01 +�c023

0 0 0 �c04 0 0

0 0 0 0 �c05 0

0 0 0 0 0 �c06

266666666666664

377777777777775

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

�s00xx

�s00yy

�s00zz

�s00yz

�s00xz

�s00xy

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

�c002 +�c0033

��c0033

��c0023

��c0033

�c003 + c0013

��c0013

��c0023

��c0013

�c001 +�c0023

0 0 0 �c004 0 0

0 0 0 0 �c005 0

0 0 0 0 0 �c006

266666666666664

377777777777775

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð11Þ

Assuming the plane stress problem in the sxx � syy plane (i.e.szz = tyz = txz = 0) for anisotropic sheetmetals, s0ij, s00ij and �s0ij,�s

00ij can be expressed by equations (12) and (13)

s0xx =c02 + c03

3sxx �

s0yy = � c033

sxx +c03 + c01

s0zz = � c023

sxx �c013

s0xy = c06txy

8>>>>>>><>>>>>>>:

s00xx =c002 + c003

3sxx �

s00yy = � c0033

sxx +c003 + c001

s00zz = � c0023

sxx �c0013

s00xy = c006txy

8>>>>>>><>>>>>>>:

8>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>:

ð12Þ

�s0xx =�c02 +�c03

3sxx �

�c033

�s0yy = � �c033

sxx +�c03 +�c01

�s0zz = � �c023

sxx ��c013

�s0xy =�c06txy

8>>>>>>><>>>>>>>:

�s00xx =�c002 +�c003

3sxx �

�c0033

�s00yy = � �c0033

sxx +�c003 +�c001

�s00zz = � �c0023

sxx ��c0013

�s00xy =�c006txy

8>>>>>>><>>>>>>>:

8>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>:

ð13Þ

In the next step, modified stress invariants (�I1, J 02, J003 for yield and �J 02, �J 003 for plastic potential functions)

are defined for anisotropic sheet metals by equation (14), where �I1 is the first modified invariant of thestress tensor (modified hydrostatic pressure) while J 02 and J 003 are the second and the third modified invar-iants of the modified deviatoric stress tensor, respectively, in a plane stress problem for the yield func-tion, while the second and the third modified stress invariants of the proposed plastic potential functionare �J 02 and

�J 003 , respectively

�I1 = hxsxx + hysyy

J 02 = � s0xxs0yy � s0yys

0zz � s0xxs

0zz + s02xy

J 003 = s00xxs00yys00zz � s00zzs

�J02 = � �s0xx�s

0yy � �s0yy�s

0zz � �s0xx�s

0zz +�s02xy

�J003 =�s00xx�s

00yy�s00zz � �s00zz�s

8>>>>>><>>>>>>:

ð14Þ

In equation (14) hx and hy are unknown coefficients for determining modified hydrostatic pressure.Inserting the values of s0ij, s00ij and �s0ij,�s

00ij from equations (12) and (13) into equation (14), the invariants of

the modified deviatoric stress tensor are determined in terms of stress components for yield and plasticpotential functions as follows

�I1 = hxsxx + hysyy

J 02 = a01s2xx + a02sxxsyy + a03s2

yy + a04t2xy

J 003 = a001s3xx + a002s2

xxsyy + a003sxxs2yy + a004s3

yy + a005sxxt2xy + a006syyt

�J02 = �a01s2

xx + �a02sxxsyy + �a03s2yy + �a04t2

�J003 = �a001s3

xx + �a002s2xxsyy + �a003sxxs

2yy + �a004s3

yy + �a005sxxt2xy + �a006syyt

(8>>>>><>>>>>:

ð15Þ

Here �I1 depends linearly on stress components, while J 02,�J 02 and J 003 ,

�J 003 depend in a quadratic and cubicway, respectively. The coefficients a0i(i = 1, 4), a00i i = 1, 6ð Þ and �a0i(i = 1, 4), �a00i i = 1, 6ð Þ are obtained in termsof coefficients c0i, c00i and �c0i,�c

00i for yield and plastic potential functions, as shown in equations (16)–(19)

a01 =c02

2 + c032 + c02c039

a02 =c01c02 � c01c03 � c02c03 � 2c03

a03 =c01

2 + c032 + c01c039

a04 = c026

8>>>>>>>>>><>>>>>>>>>>:

ð16Þ

a001 =c003c002

2 + c002c0032

a002 =c001c003

2 � c001c0022 � c003c002

2 � 2c002c0032

a003 =c002c003

2 � c002c0012 � c003c001

2 � 2c001c0032

a004 =c003c001

2 + c001c0032

a005 =c002c006

a006 =c001c006

8>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>:

ð17Þ

�a01 =�c02

2 +�c032 +�c02�c

�a02 =�c01�c02 � �c01�c

03 � �c02�c

03 � 2�c03

�a03 =�c01

2 +�c032 +�c01�c

9�a04 =�c06

8>>>>>>>><>>>>>>>>:

ð18Þ

�a001 =�c003�c0022 +�c002�c0023

�a002 =�c001�c0023 � �c001�c0022 � �c003�c0022 � 2�c002�c0023

�a003 =�c002�c0023 � �c002�c0021 � �c003�c0021 � 2�c001�c0023

�a004 =�c003�c0021 +�c001�c0023

�a005 =�c002�c0026

�a006 =�c001�c0026

8>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>:

ð19Þ

Now, NPC-2 is suggested for an asymmetric anisotropic sheet metal in the form of equation (20),where F,G are yield and plastic potential functions, respectively, while �ep is the effective plastic strainand s �epð Þ defines the isotropic hardening

� �= �Ia

1 + J 02a2 + J

� �1a

= s �epð Þ

� �= �J

b2 + �J

� �1b

8><>: ð20Þ

The yield function is asymmetric with respect to the sxx � syy plane due to the odd power dependencyof �I1 and J 003 to stress components, and the plastic potential function is asymmetric in this plane as welldue to the odd power dependency of �J 003 to stress components (equation (15)). In NPC-2, equation (20),a, b are material parameters that are obtained for different anisotropic structures in the process of cali-bration of the criterion. It is assumed that the material parameters a and b are odd numbers for the bestexperimental data point matching. In the yield function, different from Yoon et al. [15], the materialparameter a is included to more accurately compute the experimental yield stresses. This material para-meter accounts for different structures for anisotropic sheet metals (i.e. BCC, FCC and HCP).Moreover, the influence of non-linear dependency of the modified hydrostatic pressure to the yieldfunction is considered. Furthermore, in order to compute the directional Lankford coefficients moreaccurately, a plastic potential function is introduced for NPC-2. To calculate the directional Lankfordcoefficients more precisely for different anisotropic structures (BCC, FCC and HCP), a material para-meter b is included in the plastic potential function relation that can be different from material para-meter a in the yield function (equation (20)).

To calibrate the plastic potential function of NPC-2, its first derivatives with respect to the stress ten-

sor sij are required, given by equation (21), where∂�J02

∂sijand

∂�J003

∂sijare achieved from equations (22) and (23)

∂sxx

= G1�b 1

2�J02

b2�1ð Þ ∂�J

∂sxx

3�J 00

b3�1

3ð Þ ∂�J003

∂sxx

∂syy

= G1�b 1

2�J02

b2�1ð Þ ∂�J

∂syy

3�J 00

b3�1

3ð Þ ∂�J003

∂syy

∂txy

= G1�b 1

2�J02

b2�1ð Þ ∂�J

∂txy

3�J 00

b3�1

3ð Þ ∂�J003

∂txy

8>>>>>>>>><>>>>>>>>>:

ð21Þ

∂�J02

∂sxx

= 2�a01sxx + �a02syy

∂�J02

∂syy

= �a02sxx + 2�a03syy

∂�J02

∂txy

= 2�a04txy

8>>>>>>><>>>>>>>:

ð22Þ

∂�J003

∂sxx

= 3�a001s2xx + 2�a002sxxsyy + �a003s2

yy + �a005t2xy

∂�J003

∂syy

= �a002s2xx + 2�a003sxxsyy + 3�a004s2

yy + �a006t2xy

∂�J003

∂txy

= 2 �a005sxx + �a006syy

� �txy

8>>>>>>><>>>>>>>:

ð23Þ

4. Calibration of NPC-1 and NPC-2

To determine the unknown coefficients of NPC-1 and NPC-2, 10 experimental data points for the yieldcondition and eight points for plastic potential are needed. In the current study, tensile and compressiveyield stress experimental data points both in uniaxial and biaxial stress state for yield functions are rec-ommended, while for plastic potential functions these are tensile uniaxial and biaxial Lankford coeffi-cient experimental data points.

In tensile/compressive uniaxial/biaxial tests, the stress components are given by equations (24)–(27)where sT

u ,sCu are tensile/compressive uniaxial yield stresses in the u direction from the RD, while sT

b ,sCb

are tensile/compressive biaxial yield stresses

sxx = sTu cos 2u

syy = sTu sin

txy = sTu sinu cos u

8><>: ð24Þ

sxx = � sCu cos 2u

syy = � sCu sin 2u

txy = � sCu sin u cos u

8><>: ð25Þ

sxx = sTb

syy = sTb

txy = 0

8><>: ð26Þ

sxx = � sCb

syy = � sCb

txy = 0

8><>: ð27Þ

Because the non-AFR is employed, the increment of the plastic strain components is defined as in equa-tion (28), where dl is the plastic multiplier. Then Lankford coefficients can be introduced, expressed byequation (29)

depxx = dl

∂sxx

depyy = dl

∂syy

depxy = dl

∂txy

8>>>>>>><>>>>>>>:

ð28Þ

= �∂G∂sxx

sin 2u + ∂G∂syy

cos 2u� ∂G∂txy

sin u cos u

∂G∂sxx

+ ∂G∂syy

∂G∂syy

∂G∂sxx

8>>>>><>>>>>:

ð29Þ

For NPC-1, by inserting equations (24)–(27) into equation (7), tensile/compressive uniaxial yield stres-ses sT

u ,sCu and tensile/compressive biaxial yield stress sT

b ,sCb are obtained as equation (30) and, with

using equations (7)–(9), (24)–(27) and (29), Lankford coefficients are achieved as well. In equation (30),K 01p,K 02p,K 001p,K

002p are as in equation (31)

sTu = s �epð Þ

hx cos 2u + hy sin 2uð Þa +

2K 02p

�� a + 3K 001p � K 002p

�� a + 3K 001p + K 002p

�� a2

sCu = s �epð Þ

(�1)a hx cos 2u + hy sin 2uð Þa +

2K 02p

�� a + 3K 001p � K 002p

�� a + 3K 001p + K 002p

�� a2

sTb = s �epð Þ

hx + hyð Þa +

a1�a2

�� a + L0011 + 2L0021 + L0012 + 2L0022

�� a + 2L0011 + L0021 + 2L0012 + L0022

�� a2

sCb = s �epð Þ

(�1)a hx + hyð Þa +

a1�a2

�� a + L0011 + 2L0021 + L0012 + 2L0022

�� a + 2L0011 + L0021 + 2L0012 + L0022

�� a2

8>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>:

ð30Þ

K 01p =2a1 � a2ð Þ cos 2u� a1 � 2a2ð Þ sin 2u

K 02p =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2a1 + a2ð Þ cos 2u� a1 + 2a2ð Þ sin 2u

+ a7 sin u cos uð Þ2s

K 001p =L0011 + L0021

� �cos 2u + L0012 + L0022

� �sin 2u

K 002p =

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiL0011 � L0021

� �cos 2u + L0012 � L0022

� �sin 2u

+ L0066 sin u cos u� �2

8>>>>>>>>>>>><>>>>>>>>>>>>:

ð31Þ

For NPC-2, by substituting equations (24)–(27) into equation (20), tensile/compressive and uniaxial/biaxial yield stresses, and also by employing equations (21)–(23), (24)–(27) and (29), tensile uniaxial/biaxial Lankford coefficients are obtained as equations (32) and (33) where A, B, C, D, E, H and I arecomputed as equations (34) and (35)

s �epð ÞAa + B

a2 + C

a3½ �

s �epð Þ�1ð ÞaAa + B

a2 + �1ð ÞaC

a3½ �

s �epð Þ

hx + hy

� �a+ a01 + a02 + a03� �a

2 + a001 + a002 + a003 + a004� �a

s �epð Þ

�1ð Þa hx + hy

� �a+ a01 + a02 + a03� �a

2 + �1ð Þa a001 + a002 + a003 + a004� �a

8>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>:

ð32Þ

RTu = � D

8><>: ð33Þ

A = hxcos2u + hysin

B = a01cos4u + a02 + a04

� �cos2usin2u + a03sin

C = a001cos6u + a002 + a005

� �cos4usin2u + a003 + a006

� �cos2usin4u + a004sin

8<: ð34Þ

D = 12

�a01cos4u + �a02 + �a04

� �cos2usin2u + �a03sin

4u � b

2�1ð Þ

�a02cos4u + 2�a01 + 2�a03 � 2�a04

4u� �

�a001cos6u + �a002 + �a005

� �cos4usin2u +

�a003 + �a006� �

cos2usin4u + �a004sin6u

" # b3�1ð Þ

�a002cos6u + 3�a001 + 2�a003 � 2�a005 + �a006

� �cos4usin2u +

2�a002 + 3�a004 + �a005 � 2�a006� �

E = 12

�a01cos4u + �a02 + �a04

4u� � b

2�1ð Þ

2�a01 + �a02� �

cos2u + �a02 + 2�a03� �

sin2u �

�a001cos6u + �a002 + �a005

� �cos4usin2u +

�a003 + �a006� �

! b3�1ð Þ

3�a001 + �a002� �

cos4u + 2�a002 + 2�a003 + �a005 + �a006� �

cos2usin2u +

�a003 + 3�a004� �

H = 12

�a01 + �a02 + �a03� � b

2�1ð Þ �a02 + 2�a03� �

�a001 + �a002 + �a003 + �a004� � b

3�1ð Þ �a002 + 2�a003 + 3�a004� �

I = 12

�a01 + �a02 + �a03� � b

2�1ð Þ

2�a01 + �a02� �

�a001 + �a002 + �a003 + �a004� � b

3�1ð Þ

3�a001 + 2�a002 + �a003� �

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð35Þ

In NPC-1 and NPC-2, yield functions are calibrated with 10 experimental yield stress tests, such asuniaxial tensile sT

� �, uniaxial compressive sC

� �yield stresses in orientations of 08, 158, 458 and 908 from

the RD and also biaxial tensile sTb

� �and biaxial compressive sC

� �yield. Moreover, plastic potential

functions are calibrated with eight experimental results, namely as uniaxial tensile Lankford coefficients

� �in 08, 158, 308, 458, 608, 758 and 908 and the biaxial tensile Lankford coefficient RT

b =dep

� �.

Consequently, to calibrate the NPC-1 and NPC-2, 18 experimental tests should be available. In cur-rent research, four common anisotropic materials with different structures are studied. AA 2008-T4 (aFCC material), AA 2090-T3 (a FCC material), which are aluminum alloys, AZ31 at �ep = 3% (a HCPmaterial), which is a magnesium alloy, and finally, high-purity a-titanium (a HCP material) are investi-gated in four effective plastic strains �ep(%) = 0, 5, 10, 20ð Þ to show the ability of NPC-1 and NPC-2 forpredicting consequent yield stresses.

The directional mechanical properties of these materials are presented in Table 1. It is noted that iftensile/compressive biaxial yield stresses are not experimentally achieved, they can be determined from

uniaxial tensile stresses in 08, 458 and 908 directions (i.e. sTb =

+ 2sT45

+ sT90

0+ 2sC

45+ sC

4[15]). It is

mentioned that to the best knowledge of the authors, the directional Lankford coefficients have not beencomputed experimentally for AZ31 and high-purity a-titanium; thus, the plastic potential function ofNPC-1 and NPC-2 cannot be determined for these materials.

Although NPC-1 and NPC-2 are presented here as two-dimensional criteria, NPC-2 can bealso employed for 3D conditions. Using the material constants identified above, NPC-2 can be success-fully applied to describe the plastic behavior of metals under the plane stress condition. However, itcannot be used to model anisotropic-asymmetric plastic deformation under 3D loading as the through-thickness parameters of c04, c05, c004, c

005 have not been calibrated yet. Normally, these materials’ constants are

computed based on uniaxial tensile and compressive yield stresses in the x� z and y� z planes along thedirection at angle 458 from the RD. The uniaxial tensile and compressive yield stresses in the x� z planealong direction 458 are represented by sT

xz45 and sCxz45, which are calculated by the yield function of NPC-2

(equation (36)). Similarly, the uniaxial tensile and compressive yield stresses in the y� z plane in direction458denoted by sT

yz45 and sCyz45 are assessed by the yield function of NPC-2 (equation (37))

sTxz45 =

s �eð Þ

hx + hy + hz

� �a

3+ c02

3+ 9c02

� �a2

1+ c00

3ð Þ c001c00

3�6c002

5ð Þ216

� a3

sCxz45 =

s �eð Þ

�1ð Þa hx + hy + hz

� �a

3+ c02

3+ 9c02

� �a2

+ �1ð Þa c001+ c00

3ð Þ c001c00

3�6c002

5ð Þ216

� a3

8>>>>>>>>>><>>>>>>>>>>:

ð36Þ

sTyz45 =

s �eð Þ

hx + hy + hz

� �a

22 + c0

3+ c02

3+ 9c02

� �a2

2+ c00

3ð Þ c002c00

3�6c002

4ð Þ216

� a3

sCyz45 =

s �eð Þ

�1ð Þa hx + hy + hz

� �a

22 + c0

3+ c02

3+ 9c02

� �a2

+ �1ð Þa c002+ c00

3ð Þ c002c00

3�6c002

4ð Þ216

� �

8>>>>>>>>><>>>>>>>>>:

ð37Þ

Then the material constants c05, c005 are identified by sTxz45 and sC

xz45 from equation (36), while c04, c004 areevaluated by sT

yz45 and sCyz45 by equation (37). However, these tests are difficult for sheet metals due to

the fact that sheet metals are normally rather thin to manufacture specimens for these tests. Thus,these four parameters related with through-thickness properties can be assumed to be a value suchthat the material properties in the thickness direction are identical with those of in-plane ones, that is,c04 = c05 = c004 = c005 = c006. In addition, it has to be mentioned that setting c04, c05, c004, c005 as unity does notmean that the through-thickness behavior is isotropic or identical with in-plane plastic behavior,which is different from other yield functions, such as Barlat et al. [3]. This is because the anisotropicmaterial constants here also affect the asymmetric behavior of metals. It is worth stating that asimilar procedure can be employed to determine the values of �c04,�c

05,�c004,�c005 in the plastic potential func-

tion of NPC-2 for non-plane problems. Moreover, the anisotropic pressure sensitivity parameters ofhx, hy, hz are computed by uniaxial tensile tests in RD, TD and ND under different hydrostaticpressure [15].

Finally, with having all these 18 experimental results for an anisotropic sheet metal, Table 1, 10unknown coefficients in yield functions and eight unknown ones in plastic potential functions can bedetermined by minimizing the proposed error functions E1,E2ð Þ, given by equations (38) and (39), foryield and plastic potential functions, respectively, with the Downhill Simplex Method

E1 =sT

� �exp :

� �pred:

� �exp :

� �pred:

� �exp :

� �pred:

� �exp :

� �pred:

� �exp :

� �pred:

� �exp :

� �pred:

� �exp :

� �pred:

� �exp :

� �pred:

� �exp :

� �pred:

� �exp :

� �pred:

ð38Þ

E2 =RT

� �pred:

� �exp:

� �pred:

� �exp:

� �pred:

� �exp:

� �pred:

� �exp:

� �pred:

� �exp:

� �pred:

� �exp:

� �pred:

� �exp:

� �pred:

� �exp:

ð39Þ

The unknown coefficients are ai i = 1� 8ð Þ, hx, hy and bi i = 1� 8ð Þ for NPC-1 and c0i(i = 1, 2,3, 6), c00i (i = 1, 2, 3, 6), hx, hy and �c0i(i = 1, 2, 3, 6),�c00i (i = 1, 2, 3, 6) for NPC-2.

Directional stresses are computed from the yield function and calibrated with tensile/compressiveexperimental yield stress results, while directional Lankford coefficients are evaluated from the plasticpotential function and are calibrated with tensile/compressive experimental Lankford coefficient data indifferent orientations from the RD. The main advantageous of asymmetric yield or plastic potentialfunctions is the ability of estimating the tensile/compressive yield stresses or Lankford coefficients in dif-ferent orientations from the RD. However, the compressive experimental data of Lankford coefficients

Table 1. Experimental data points for AA 2008-T4, AA 2090-T3, AZ 31 and high-purity a-titanium from Lou et al. [13] and Yoonet al. [15].

(a) Tensile yield stresses of AA 2008-T4 (FCC) and AA 2090-T3 (FCC).

Material sT0 sT

15 sT30 sT

45 sT60 sT

75 sT90 sT

AA 2008-T4 211.67 211.33 208.50 200.03 197.30 194.30 191.56 185.00AA 2090-T3 279.62 269.72 255.00 226.77 227.50 247.20 254.45 289.40

(b) Compressive yield stresses for AA 2008-T4 (FCC) and AA 2090-T3 (FCC)

Material sC0 sC

15 sC30 sC

45 sC60 sC

75 sC90 sC

AA 2008-T4 213.79 219.15 227.55 229.82 222.75 220.65 214.64 222.02AA 2090-T3 248.02 260.75 255.00 237.75 245.75 263.75 266.48 247.50

(c) Lankford coefficients for AA 2008-T4 (FCC) and AA 2090-T3 (FCC)

Material RT0 RT

15 RT30 RT

45 RT60 RT

75 RT90 RT

AA 2008-T4 0.870 0.814 0.634 0.500 0.508 0.506 0.530 1.000AA 2090-T3 0.210 0.330 0.690 1.580 1.050 0.550 0.690 0.670

(d) Tensile and compressive yield stresses for AZ31 (HCP)

�ep(%) sT0 sT

45 sT90 sT

b sC0 sC

45 sC90 sC

3% 170.82 177.13 191.83 179.23 96.58 94.45 103.38 97.47

(e) Tensile and compressive yield stresses for high-purity a-titanium (HCP)

�ep(%) sT0 sT

90 sTb sC

0 sC90 sC

0% 149.837 179.567 236.459 148.208 179.567 228.5985% 229.642 266.355 329.569 228.013 278.816 302.52110% 254.865 280.186 359.138 271.104 311.146 317.97620% 294.118 300.31 417.495 387.25 362.229 337.623

has not been reported so far. Consequently, only directional tensile Lankford coefficients are anticipatedin the current work and the small values of errors in Table 2 show the validity of the error function inequation (39). It is worth stating that, with having the experimental compressive Lankford coefficientsin hand, the error function in equation (40) can be employed rather than that in equation (39) with theasymmetric plastic potential function of NPC-2. The plastic potential function of NPC-1 is symmetricand cannot compute compressive Lankford coefficients

E2 =RT

� �pred:

� �exp:

� �pred:

� �exp:

� �pred:

� �exp:

� �pred:

� �exp:

� �pred:

� �exp:

� �pred:

� �exp:

� �pred:

� �exp:

� �pred:

� �exp:

ð40Þ

Table 2. The obtained computation errors for AA 2008-T4 compared with experimental results (in percentages)

(a) AA 2008-T4

Criterion ETs ET

b ECs EC

Yoon et al. [15] 0.4460 0.0694 0.8255 0.1170 -Lou et al. [13] 0.2704 0.0778 1.5915 6.0219 3.9852NPC-1 1.0180 0.9342 1.1877 0.7523 0.3901NPC-2 0.2414 0.9695 0.2607 0.7801 0.4507

(b) AA 2090-T3

Criterion ETs ET

b ECs EC

Yoon et al. [15] 1.0599 0.0009 1.2441 0.0005 -Lou et al. [13] 0.7350 0.0015 2.4651 8.2141 12.8890NPC-1 0.8926 2.9112 1.0212 2.8711 1.4512NPC-2 0.6068 3.4022 1.2415 3.2407 3.3953

(c) AZ31 in �ep = 3%

Criterion ETs ET

b ECs EC

Yoon et al. [15] 0.0007 0.0014 0.0005 0.0051NPC-2 0.0043 0.0073 0.0149 0.0007

(d) High-purity a-titanium, NPC-1

�ep %ð Þ ETs ET

b ECs EC

0% 5.7876e-05 0.0079 5.3874e-04 0.00345% 1.1139 0.0068 1.1708 7.1354e-0410% 2.9225 0.0646 3.2317 0.127020% 7.1660 0.2061 9.3393 0.5595

(e) High-purity a-titanium, NPC-2

�ep %ð Þ ETs ET

b ECs EC

0% 0.0010 0.0033 0.0007 0.00265% 0.0010 0.0028 0.0022 0.002910% 0.0005 0.0012 0.0012 0.000620% 0.0006 0.0034 0.0018 0.0018

Table 3. Determined coefficients in NPC-1 and NPC-2

(a) Coefficients of NPC-1 for AA 2008-T4, AA 2090-T3

AA 2008-4 AA 2090-T3a1 2.1728 0.3641a2 20.6395 1.3098a3 1.1202 1.0224a4 1.1540 1.1029a5 20.0035 1.1151a6 21.3325 0.9338a7 0.7349 1.2734a8 21.4472 1.0860hx 0.1329 20.9157hy 0.5740 20.0482a 3 7

AA 2008-T4 AA 2090-T3

b1 1.4478 0.9648b2 0.0940 0.3833b3 0.3676 0.2615b4 0.9111 0.7903b5 0.8613 0.9439b6 1.5443 1.5769b7 0.8939 0.7375b8 1.3155 20.0001b 6 6

(b) Coefficients of NPC-2 for AA 2008-T4, AA 2090-T3 and AZ31 �ep = 3%ð Þ

AA 2008-T4 AA 2090-T3 AZ31

c01 2.3407 1.8924 2.3407c02 2.7187 1.7239 2.7187c03 2.3128 1.7386 2.3128c06 2.5491 2.1006 2.5491c001 20.2131 4.6551 20.2131c002 20.2709 0.4968 20.2709c003 213.2943 0.5377 213.2943c006 29.9399 21.7863 29.9399hx 20.6423 20.5633 20.6423hy 20.6699 20.3379 20.6699a 3 5 3

AA 2008-T4 AA 2090-T3

�c01 1.2338 0.5277�c02 1.4122 1.0691�c03 0.3152 20.3533�c06 0.4778 1.0034�c001 1.4176 0.6647�c002 2.2194 0.5051�c003 20.1905 3.7997�c006 20.0011 0.7759b 3 3

(continued)

In equation (40), RTu and RC

u are tensile/compressive directional Lankford coefficients and RTb and RC

are tensile/compressive biaxial Lankford coefficients, respectively. In Table 3, the unknown coefficientsof NPC-1 and NPC-2 are determined for AA 2008-T4, AA 2090-T3, AZ 31 and high-purity a-titanium(it is pre-deformation using a certain test and evaluation of the material in this deformed state [23]). Tocheck the accuracy of predicting the subsequent directional yield points, high-purity a-titanium is stud-ied in four over-strains, that is, �ep = 0%, 5%, 10%, 20%. The plastic potential of AA 2008-T4 and AA2090-T3 can be determined due to the reported Lankford coefficients in Table 1.

To consider the non-linear effect of hydrostatic pressure on yielding of asymmetric anisotropic sheetmetals, the material parameters a and b in both NPC-1 and NPC-2 are applied. These parameters arecomputed to be best fitted with experimental results. In this particular case, we have for NPC-1 for AA2008-T4 a = 3 and b = 6 and for NPC-2 for AA 2090-T3 a = 5 and b = 3.

To verify the accuracy of NPC-1 and NPC-2 in comparison to the experimental results, the RMSEsof tensile ET

� �, tensile equi-biaxial ETb

� �, compressive EC

� �yield stresses and compressive equi-biaxial

� �tensile Lankford coefficients ET

� �are calculated according to equations (41)–(45)

sT0ð Þexp:� sT

0ð Þpred:sT

0ð Þexp:

15ð Þexp:� sT15ð Þpred:

sT15ð Þexp:

sT30ð Þexp:� sT

30ð Þpred:sT

30ð Þexp:

sT45ð Þexp:

sT60ð Þexp:� sT

60ð Þpred:sT

60ð Þexp:

sT75ð Þexp:

sT90ð Þexp:� sT

90ð Þpred:sT

90ð Þexp:

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA

× 100 ð41Þ

(c) Coefficients of NPC-1 and NPC-2 for high-puritya-titanium �ep = 0%, 5%, 10%, 20%ð Þ

�ep 0% 5% 10% 20%

a1 1.1408 1.2768 1.4412 1.4987a2 0.6937 20.4037 20.3984 1.2571a3 1.7167 1.9261 1.8833 1.7129a4 0.1891 0.2335 0.2995 1.4934a5 0.1519 0.0366 0.0129 0.8955a6 1.3884 1.4884 1.5063 1.7644a7 27.7960 210.0353 9.0188 1.6465a8 2.7614 10.4271 14.7848 0.1013hx 20.4894 20.4501 20.3365 20.1675hy 0.0962 20.0850 20.2577 0.1245a 5 5 5 5

�ep 0% 5% 10% 20%

c01 0.7516 1.0345 1.1321 1.4987c02 1.4457 1.4788 1.4870 1.2571c03 1.9975 1.9735 1.8959 1.7129c06 6.7734 1.6581 1.5832 1.4934c001 0.9243 2.5235 1.0346 0.8955c002 0.2987 0.2095 1.0224 1.7644c003 0.5709 20.2431 0.8649 1.6465c006 21.5636 1.3891 1.1837 0.1013hx 20.0442 20.0217 0.0454 20.1675hy 20.0582 0.1799 0.0767 0.1245a 3 3 3 3

Table 3. Continued

sC0ð Þexp:� sC

0ð Þpred:sC

0ð Þexp:

15ð Þexp:� sC15ð Þpred:

sC15ð Þexp:

sC30ð Þexp:� sC

30ð Þpred:sC

30ð Þexp:

sC45ð Þexp:

sC60ð Þexp:� sC

60ð Þpred:sC

60ð Þexp:

sC75ð Þexp:

sC90ð Þexp:� sC

90ð Þpred:sC

90ð Þexp:

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA

× 100 ð42Þ

RT0ð Þexp:� RT

0ð Þpred:RT

0ð Þexp:

15ð Þexp:� RT15ð Þpred:

RT15ð Þexp:

RT30ð Þexp:� RT

30ð Þpred:RT

30ð Þexp:

RT45ð Þexp:

RT60ð Þexp:� RT

60ð Þpred:RT

60ð Þexp:

RT75ð Þexp:

RT90ð Þexp:� RT

90ð Þpred:RT

90ð Þexp:

0BBBBBBBBBBBB@

1CCCCCCCCCCCCA

× 100 ð43Þ

ETbs =

� �exp:� sT

� �pred:

�� sT

� �exp:

× 100 ð44Þ

ECbs =

� �exp:� sC

� �pred:

�� sC

� �exp:

× 100 ð45Þ

5. Results and discussion

In this section, coefficients and material parameters from Table 3 are employed for NPC-1 and NPC-2to show the yield functions in sxx � syy plane, tensile yield stresses, compressive yield stresses andLankford coefficients in different angles from the RD in Figures 1–4, respectively. To compare theresults to Yoon et al. [15] and Lou et al. [13], the relative errors are computed in Table 2 as well.

In Figure 1, yield functions are shown in the sxx � syy plane for AA 2008-T4, AA 2090-T3, AZ31 andhigh-purity a-titanium. NPC-1 and NPC-2 are compared with Yoon et al. [15], Lou et al. [13] and theexperimental results for AA 2008-T4 and AA 2090-T3 in Figures 1(a) and (b) and it is observed that theexperimental data points are well predicted. NPC-2 is compared to Yoon et al. [15] and the experimentalresults for AZ31 in Figure 1(c). It has to be mentioned that NPC-1 could not anticipate the yield func-tion for this material. To show the ability of NPC-1 and NPC-2 in predicting the consequent yield stres-ses, high-purity a-titanium is investigated in four effective plastic strains of �ep = 0%, 5%, 10%, 20%. It isobserved that both criteria, and especially NPC-2, can properly predict experimental data points(Figures 1(d) and (e)).

In Figures 2 and 3, the directional tensile and compressive yield stresses of AA 2008-T4, AA 2090-T3and AZ31 in different angles from the RD are shown. The difference among NPC-1 and NPC-2, Yoonet al. [15] and Lou et al. [13] can be studied more clearly in these figures. In Figure 4, the Lankford coef-ficients for AA 2008-T4 and AA 2090-T3 are determined with NPC-1 and NPC-2 and compared withLou et al.’s [13] results.

To show the ability of the non-linear influence of hydrostatic pressure in predicting the directionalmechanical properties of anisotropic sheet metal with NPC-1 and NPC-2, RMSEs in equations(41)–(45) for AA 2008-T4 (FCC), AA 2090-T3 (FCC), AZ31 (HCP) and high-purity a-titanium areemployed (Table 2). In Table 2(a), the relative errors of AA 2008-T4 for NPC-1 are less than 1.2{\rm

%} and for NPC-2 are less than 1.0%. Indeed, it can be said that these criteria are well adopted for AA2008-T4 (a FCC material) due to the small number of relative errors. It is observed that NPC-2 in pre-dicting tensile and compressive yield stresses with relative errors of ET

s = 0:2414% and ECs = 0:2607% is

the better criterion and the tensile/compressive yield stresses are predicted with nearly the same accu-racy. In predicating Lankford coefficients, NPC-1 with a relative error of ET

R = 0:3901% is the mostappropriate one. Therefore, via considering non-linear pressure sensitivity, equations (1)–(9) could becapable of estimating the directional mechanical properties with better accuracy for AA 2008-T4. Table2(b) shows the relative errors of AA 2090-T3 (a FCC material), which are slightly larger than those ofAA 2008-T4 (a FCC material), due to further differences between yield stresses in tension and compres-sion tests (Table 1). However, the computed relative errors are small enough to be adopted for estima-tion of the directional mechanical behaviors of AA 2090-T3 for NPC-1 and NPC-2. NPC-2 with relativeerror of ET

s = 0:6068% is the best criterion for predicting tensile yield stresses. However, for computingcompressive yield stresses and Lankford coefficients, NPC-1 with relative error of EC

s = 1:0212% andET

R = 1:4512%, respectively, is the most successful criterion.The computed relative errors for predicting tensile and compressive yield stresses of AZ31, a HCP

material, at �ep = 3%, in NPC-2 are very small (Table 2(c)). In this case, the unknown coefficients ofNPC-1 cannot be determined.

(a) (b)

(c) (d)

Figure 1. Comparison of yield functions in the sxx � syy plane for (a) AA 2008-T4, (b) AA 2090-T3, (c) AZ31, (d) high-purity a-titanium (NPC-1) and (e) high-purity a-titanium (NPC-2).

The error values for predicting tensile/compressive equi-biaxial yield stresses are small enough in threematerials for NPC-1 and NPC-2 (Tables 2(a)–(c)).

Finally, to validate NPC-1 and NPC-2 for predicting subsequent directional yield stresses, high-puritya-titanium (HCP) at �ep = 0%, 5%, 10%, 20% is studied.

The relative errors are shown in Tables 2(d) and (e). As is observed, both criteria, especially NPC-2,are successful in predicting subsequent directional yield stresses of high-purity a-titanium.

(a) (b)

Figure 2. Comparison of the tensile yield stress directionality for (a) AA 2008-T4, (b) AA 2090-T3 and (c) AZ31.

(a) (b)

Figure 3. Comparison of the compressive yield stress directionality for (a) AA 2008-T4, (b) AA 2090-T3 and (c) AZ31.

6. Conclusion

Two novel criteria researches were developed in the current study in order to investigate the possiblenon-linear influence of modified hydrostatic pressure on yielding of asymmetric anisotropic sheet metals.To reach this objective, Lou et al. [13] and Yoon et al.’s [15] yield functions (AFR) were extended toNPC-1 (non-AFR) and NPC-2 (non-AFR), respectively. The yield functions are non-linearly dependentto modified hydrostatic pressure, while the proposed plastic potential functions are pressure independentto hold the incompressibility flow rule. Two material parameters, a, b, were added to the yield and plasticpotential functions, which can be determined for different structures of anisotropic materials to predictdirectional yield stresses and Lankford coefficients more accurately. To validate the accuracy of the pro-posed criteria, four well-known anisotropic materials, namely AA 2008-T4, AA 2090-T3, AZ 31 andhigh-purity a-titanium, were studied. Finally, it was shown that the proposed criteria can predict direc-tional experimental data points better than of Yoon et al. [15] and Lou et al. [13].

Acknowledgement

The authors of this research would like to express their sincere appreciation to Dr Pawe1 Szeptynski of the Faculty of Civil

Engineering, Cracow University of Technology, Krakow, Poland, for his support and valuable advice.

Funding

This research received no specific grant from any funding agency in the public, commercial or not-for-profit sectors.

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