international journal of plasticity, vol. 7, pp. 693-712, 1991

International Journal o f Plasticity, Vol. 7, pp. 693-712, 1991 0749-6419/91 $3.00 + .00 Printed in the U.S.A. Copyright © 1991 Pergamon Press plc

A S I X - C O M P O N E N T Y I E L D F U N C T I O N F O R A N I S O T R O P I C M A T E R I A L S

FR~Dt~RIC BARLAT, DANIEL J. LEGE, a n d JOHN C. B ~ r a

Alcoa Laboratories

Abstract- In classical flow theory of plasticity, it is assumed that the yield surface of a material is a plastic potential. That is, the strain rate direction is normal to the yield surface at the corresponding loading state. Consequently, when the yield surface is known, it is possible to predict its flow behavior and, associated with some failure criteria, to predict limit strains above which failure occurs. In this work a new six-component yield surface description for orthotropic materials is developed. This new yield function has the advantage of being relatively simple mathematically and yet is consistent with yield surfaces computed with polycrystal plasticity models. The proposed yield function is independent of hydrostatic pressure. So, except for such cases, strain rates can be calculated for any loading condition. Applications of this new criterion for aluminum alloy sheets are presented. The uniaxial plastic properties determined for 2008-T4 and 2024-T3 sheet samples are compared to those predicted with the proposed constitutive model. In addition, for 2008-T4, the predictions of the six-component yield function are compared to those made with the plane stress tricomponent yield criterion proposed by Barlat and Lian. Though rather good agreement between experiments and predicted results is obtained, some discrepancies are observed. Better agreement could result if the isotropic work-hardening assumption associated with the yield criterion were relaxed. Nevertheless, the proposed yield function leads to plastic properties similar to those computed with polycrystalline plasticity models and can be very useful for describing the behavior of anisotropic materials in numerical simulation of forming processes.

!. INTRODUCTION

This paper deals with the plastic behavior of metals subjected to deformation conditions such that temperature and strain rate effects can be neglected. In particular, this study can be applied to most aluminum alloys deformed at room temperature. When a tension test specimen is subjected to uniaxial loading, there is a given critical stress level under which the material behavior is elastic. Above this critical stress or yield stress, ir- reversible plastic deformation occurs. If the specimen is unloaded and reloaded, the same phenomenon can be observed, but the yield stress on reloading corresponds to the stress just prior to unloading. The questions which must be addressed in order to describe the elasto-plastic behavior of a material are (ZYczXowsra [1981]): What stress results in inelastic response (yield condition)? How are plastic deformation increments related to stress (flow rule)? How does the yield condition change with deformation (hardening)? The last two questions relate to the postyield conditions. Experimentally, yield stress measurement is not an easy task. Several different definitions of the yield stress have been proposed in the past (for instance, see HECI~R [1976]). This difficulty can be ex- plained by considering the two scales of microscopic and macroscopic yielding. Plastic flow in metallic materials occurs by dislocation glide on given crystallographic planes and directions. On a particular slip system, the movement of a dislocation can be reversible with or without energy dissipation. In addition, slip does not occur at the same time in all the grains of the polycrystalline aggregate because of their different orientations. Therefore, plastic deformation is nonhomogeneous at the small strain levels.

693

694 F. BARLAT et al.

In this work, the relatively large plastic strain range is of interest and the conventional 0.2% offset plastic strain is adopted as a definition of yielding. At this deformation level, all of the grains are assumed to be subject to plastic deformation.

When multiaxial loading is considered, the questions discussed for the uniaxial case are still valid. What are the yield and the postyield conditions? However, the yield condition does not reduce to a critical number, the yield stress. Instead, it is expressed by a function describing the yield surface. In the most general case, this function • is defined in the six-dimensional space of the stresses, and the material is

elastic if ~ ( o ~ ) < c, c = constant (la)

o r

plastic if ~ ( o ~ ) = c. (Ib)

• ( o ~ ) > c has no physical meaning. For multiaxial loading, the postyield conditions are not given by a simple stress-strain curve. One has to know the relationship between the stress and the strain-rate tensor (flow rule) and how the yield surface shape evolves during deformation. In the following section, the yield conditions and flow rules that characterize the behavior of isotropic and anisotropic materials are succinctly described.

I!. YIELD CONDITIONS AND FLOW RULES

The yield surface represents the limitation of the elastic range. Experimental evidence and theoretical considerations concerning the plastic behavior of materials have led to some restrictions on the mathematical representation of the yield surface. It is well known that yielding in metals does not depend on a superimposed hydrostatic pressure and that plastic deformation occurs with no volume change. These observations are not true for materials exhibiting high porosity, for materials subjected to structural transformations during plastic straining, for polymers, or for materials subjected to very high pressure. However, for heavily cold worked materials, the relative density change due to the multiplication of dislocations from the undeformed state is 1012 b 2 ~ 10 -3, where b is the Bfirgers vector (FRANCOIS [1976]). For commercial purity sheet metals, the relative density change due to void nucleation and growth at hard particles during plastic deformation is of the same order of magnitude, 10 -3 (ScHMITT & JALrNmR [1982]), and for plasticity flow rules, these corresponding volume changes can be neglected. For pressure independent material behavior, the yield surface is assumed to be a function of the stress deviator S defined by the tensor equation:

S = O - - Om I (2)

where Om is the mean pressure and I the identity tensor. Thus, for isotropic materials, the yield function can be defined as a function of two stress deviator invariants only.

Two different approaches can be used to derive the yield surface (ZYCZKOWSra [1981]). One approach assumes that the limitation of the elastic range is the result of a physical quantity which reaches a critical value. For instance, the isotropic von Mises and Tresca yield criteria are obtained if the elastic range is bounded by a certain value of the elastic distortion energy or by a critical level of the maximum shear stress, respectively. The von Mises yield function can be derived, as well, if yielding is assumed to occur when

Six-component yield function for anisotropic materials 695

the octahedral shear stress reaches a given value. The yield surface of a single crystal results f rom the assumption of a critical resolved shear stress on certain slip systems.

Using the other approach, a yield function can be chosen as a good approximation for a set of data generated either by experimental techniques or by physics-based modeling. For instance, the following phenomenological yield function proposed by HER- SHEY [1954] and HOSFORn [1972]:

,/i = IS l _ S 2 I m + IS= - S 3 l m + - s , I m = 2 ~ m (3)

is a particularly good approximation of yield surfaces calculated with polycrystal models for isotropic BCC and FCC metals when the exponent m is equal to 6 and 8, respectively (LOGAN & HOSFORD [1980]). In this equation, # is the effective stress. It is interesting to note that the function defined by eqn (3) reduces to the von Mises criterion for m = 2 and m = 4, and to the Tresca condition for m = 1 and m = 0o.

The plastic behavior of materials is not linear. So, in the classical flow theory of plasticity, the flow rule gives a relationship between the stress deviator and the strain-rate tensor. For instance, the Levy-Mises relations assume the identity between these two tensor quantities. If the von Mises yield condition # is associated with the Levy-Mises re- lation, then the flow rule can be written as follows:

~ij = A --.0# (4) o/j

The yield function is considered a plastic potential or, in other words, the strain rate vector is orthogonal to the yield surface. The associated flow rule given in eqn (4) can be generalized to represent the behavior of other isotropic and anisotropic materials. HECKER [1976], who reviewed numerous critical experiments to assess the yield surface shape, found that the normality rule was never violated. The yield surface shape and, consequently, the associated flow rule must be described as accurately as possible in order to model the plastic behavior of materials. BARLAT [1987] has shown that the theoretical sheet forming limit strain at necking for balanced biaxial stretching is roughly five times larger for a material whose yield surface is described by the von Mises criterion than for one which follows the Tresca criterion, when all other properties are identical. In addition, when using the phenomenological description (eqn (3)) with an exponent m = 8, the theoretical forming limit strains reasonably agree with experimental trends, whereas they are unrealistically too high or too low with the von Mises or the Tresca yield conditions, respectively.

DRtrCKER [1951, 1959] showed that, based on a stability postulate, the yield surface must be convex. If the yield function is smooth with no vertex, convexity assures the uniqueness of the strain rate for a given stress state. Mathematically, the yield surface

is convex if for a given stress and strain rate condition a o and ~o, any stress oi~ in- side or on the yield surface obeys the following relationship:

(ao - ai~)~ij -> 0. (5)

If the function # is twice differentiable, its Hessian matrix 3£ is defined by:

02# ~cu- aa, a~j" (6)

696 F. BARLAT el al.

is convex if the matrix ~ is positive semi-definite, that is, if its eigenvalues are positive or zero (Eca3I~STON [1958]; ROCr,.A~tJ~R [1972]). In addition, these authors showed that a linear transformation on the variables does not affect convexity. LIVPU_ANN [1970] showed that if a yield function is convex in the three-dimensional space of the principal stresses, then it is also convex in the more general six-dimensional stress space. There- fore, for isotropic yield functions, the convexity needs to be checked only in the three- dimensional principal stress space. It can readily be shown that the yield function given by eqn (3) is convex for m equal or larger than 1.

IIL YIELD FUNCTIONS FOR ISOTROPIC MATERIALS

Phenomenological yield functions do not result directly from microstructure-based calculations. However, they have some advantages over the yield surfaces calculated from the crystallographic texture of polycrystalline aggregates. Namely, phenomenological yield functions are easy to implement (in FEM codes for instance), and lead to fast computations. They can describe the complete anisotropy, whereas polycrystal models only account for crystallographic texture. Finally, they are easy to adapt to different materials. For instance, using eqn (3), the difference between a FCC and a BCC isotropic material can be modeled by a change of exponent m. However, using polycrystal models, the same difference requires profound modifications, such as a change in slip systems, for instance.

As mentioned previously, eqn (3) gives an excellent approximation of an isotropic polycrystal yield surface calculated using the TAYLOR [1938J/BisHoP and HILL [1951] model, or using a self-consistent model (H~Rsrmv [1954]). Therefore, the main purpose of this work is to generalize eqn (3) for anisotropic materials. However, eqn (3) does not possess shear stress terms, and this restricts its possible use for anisotropic gener- alizations to stress states where the anisotropy axes and principal stress axes are superimposed. Therefore, this section describes how shear stress terms can be introduced in the isotropic formulation. In a first approach, only plane stress states where three out of the six stress components are equal to 0 (azz = Ozx = Oyz = 0) are considered. The stress tensor is represented by a 2 x 2 matrix whose characteristic equation is a poly- nomial of the second order. Therefore, a simple calculation of the eigenvalues leads to the isotropic, tricomponent, plane-stress yield surface:

= IK, - KE] m + IK1 + K21 m + 12K21 m = 2a '~ (7a)

Kl -- Oxx + ayy (7b) 2

J/ / 2 g 2 - - - °xx -- Oryy 2 "~" "l- 0"xy. (7c)

This formulation was proposed by BARLAT and RICHMOND [1987] and it was extended to planar anisotropy by BARLAr et al. [1988], and BARLAT and LIAN [1989]. According to LIVPMANN'S work [1970], the convexity requirement is obviously fulfilled since the function describes the same yield surface as eqn (3) in another stress space. Figures 1 and 2 illustrate the excellent agreement between this yield function and the TAYLOR [1938]/BISHOP and HILL [1951] yield surface for isotropic FCC materials (from BARLAT & LtAN [1989]). In this type of plot, the solid lines represent the intersections of the yield


1.5

1.0

0.5

-0.5

-1.0 .0

~ / - - - p = 1

X A i

__i/'-,,, 7 - - . 'y'.ff /

I I

-0.5 0 0.5 1.0 1.5

Fig. 1. Tricomponent plane stress yield surface (oxx/#, Oyy/O, Oxy/O ) for an isotropic FCC sheet calculated with the Bishop and Hill model. The solid line represents the section of the yield surface by a plane parallel to (o~/O, oyy/o) for different values S = o~/O and projected onto the (o=/O, oyy/#) plane. Dashed lines represent the projection onto (o~/0, ayr/O) of curves defined by points having the same strain ratio o (~yy/~=).

surface with a p lane paral le l to (Oxx/#, oyy/#) for di f ferent values o f S = oxy/# and pro- j ec ted on to the (ox~/6, Oyy/0) p lane .

A n expression o f the yield surface descr ibed by eqn (3) in terms o f all six componen t s o f the stress tensor can be ob ta ined after solving the third-degree characterist ic equat ion:

/~3 - - 312A - 2/3 = 0 (8)

where - 3 / 2 and 2/3 are the second and the th i rd pr inc ipa l invar iants o f the stress devi- a to r , respect ively . Expressed with the BISHOP and HILL [1951] n o t a t i o n o f the stresses,

A = Oyy - Oz~ B = o~ - Oxx C = Oxx - Oyy (9a ,b ,c)

F = Oyz G = o= H = Oxy (9d ,e , f )

the stress dev ia to r invar ian t s t ake the fo l lowing fo rm:

F 2 + G 2 + H 2 ( A - C ) 2 + ( C - B ) 2 + ( B - A ) 2 12 = + (10a)

3 54


1.5

1.0

0.5

I ~yy/~ I

-0.5 -

- 1 . 0

-1.0 - 0 . 5 0 0.5 .0

Fig. 2. Tricomponent plane stress yield surface obtained with eqn (3) for m = 8.

~xx/~

1 . 5

( C - B ) ( A - C ) ( B - A ) ( C - B ) F 2 + ( A - C)G2+ ( B - A ) H 2 13 = + F G H -

54 6

(lOb)

Since eqn (8) is the characteristic equation of a symmetric matrix, the three solutions (principal stresses) are real. However, it is more convenient to express them in terms of complex numbers. Let us denote z the complex number of module p and argument 0:

z = 13 + i ( I 3 - 12) 1/2 = pe i° = p(cos 0 + i sin 0) (11)

where i is the imaginary unit number, p = 13/2, and 0 = arc c o s ( I 3 / I 3 / 2 ) . The principal deviatoric stresses are given by Cardan's solutions of eqn (8):

Sl = z l / 3 .~_ ~1/3 (12a)

S2 = o)zl/3 + (.~1/3 (12b)

S3 = ~_oZ 1/3 -~ 0oZ 1/3 (12c)


where co = e - 2 i x / 3 and the bar ( - ) stands for conjugate . The yield funct ion described by eqn (3) can be rewritten as follows:

= I ( 1 - co)z ~/3 + (1 - ~)Z1/3lm + I(co - - ~ ) z l / 3 -st- (¢~ - - c°)Zl/3lm

+ I(~ - 1)Z 1/3 + (co - 1)~ 1/31 m = 26 m (13)

or, alternatively:

~i15 = (312)m/2I 2COS~T]/2~'t-'ff~ m-F 2COS(20--371") m " J t ' ' 6 2COS( 20 6 571") m / = 2 6 " . (14)

It is interesting to examine the geometrical interpretat ion o f the quantities ment ioned above in the complex plane (Fig. 3). co, & and 1 are 120 ° apar t and their sum is zero. These properties also hold for the quantities 1 - co, ¢3 - 1 and co - t3.

Since the a rgument o f the produc t o f two complex numbers is equal to the sum o f their arguments , it is possible, using eqns (12), to visualize on Fig. 3 the complex numbers that generate the principal deviatoric stresses $1, $2 and $3. Then it is relatively

Imaginary axis i zl/3 (I - co!

zl/3(co 1 ) ~ ;";:"::';~"~ \ zl/3

Real axis

Z I/3 (f.l) -- ~)

Fig. 3. Stress deviator invariants in complex plane.


easy to find the position of the complex numbers whose real part ((R 0 gives the differences of the deviatoric stresses.

S1 - S2 = 2(RelZl/3( 1 -- w)} (15a)

$2 - $3 = 2 ( R e l z l / 3 ( o~ - ~)1 (15b)

$3 - $1 = 2(Relzl/3(& - 1)}. (15c)

Noting that sin 0 = (I~ - I 2 )1/2//132/2 is always positive or, equivalently that 0/3 is always between 0 ° and 60 °, the above mentioned complex numbers are located on the shaded area on Fig. 3. This implies that $1 - $2 as well as $2 - $3 are always positive, whereas $3 - SI is always negative.

Therefore, the absolute values in eqn (14) are not necessary and the yield function reduces to:

# = ( 3 1 2 ) m / 2 [ [ 2 C O S ( 2 0 _ ~ r ) ] m (2COS(20 . 6 ]m] + - - 3 r ) ] m + [ - - 2 c o s ( 2 0 + 5 7 r )

= 2 ~ m. (16)

Keeping in mind that -3•2 is the second invariant of the stress deviator and introduc- ing a constant k, the yield surface described by eqn (16) can be expressed by:

= 312 q'(0) = k 2. (17)

This is the generic form of the generalized isotropic yield criterion studied recently by BILLINGTON [1988]. Eqn (16) obviously reduces to the von Mises criterion when m = 2 or m = 4 and to the Tresca criterion when m = 1 or m = 0o.

Finally, it can be shown that the yield surface can be expressed with polar coordinates R(X) in the ,r-plane by

R = 2

I cos x - x/3 sin X m cos X + 4-3 sin X m 1 + I ~ c o s x l m

1/m

(18)

with X = 0/3 - 7r/6, when X is between -7r /6 and ~r/6. Figure 4 illustrates this representation for different values of m.

IV. YIELD FUNCTIONS FOR ANISOTROPIC MATERIALS

In this section, the isotropic formulation proposed previously is extended to the anisotropic case. This study is restricted to anisotropic materials exhibiting orthotropic symmetry, that is, to materials that possess three mutually orthogonal planes of symmetry at every point. Actually, this restriction is not a very significant constraint because most mechanically processed materials are orthotropic. Typical examples are roiled sheets or plates.

It is interesting to note that the definition of symmetry is related to a point in the material and not related to the whole part. Moreover, the three-dimensional material space


Von Mises (m-2 or 4)

Tresca (m=l or oo)

m - 8

Z

Fig. 4. Isotropic yield functions for m = 1 or ~, m = 2 or 4, and m = 8 in r-plane.

defined by x, y, and z should not be confused with the six-dimensional stress space which reduces to a three-d imensional space in the plane stress case. The or thot ropic symmetry of the material space implies some symmetry of the yield surface in stress space, but this symmet ry is no t or thot ropic symmetry.

Table 1 lists some of the ma jo r yield funct ions proposed to describe the anisotropic behavior of or thotropic materials. Some of the isotropic funct ions are also included for reference. B ~ A T [1987] has discussed the l imitat ions of each of these criteria. The ba-

sic problem can be summarized by considering the two yield condit ions proposed by HrtL [1948, 1979]. The first one can be applied to an orthotropic material subject to any stress

Table 1. Phenomenological yield functions

Yield Criterion Type Shear Dimension

Tresca lsotropy - - Von Mises lsotropy - - HttL [1948] Planar anisotropy yes 6 HE,stray [1954] lsotropy -- -- HOSFORD [1972] GOrOH [1977] Planar anisotropy yes 3 BAss~I [1977] Planar isotropy -- -- HILL [1979] Planar anisotropy no 3 Locxt~ & HOSFOgD [1980] Planar anisotropy no 2 BUDIANSKI [1984] Planar isotropy no 2

Shear indicates if shear terms appear in the formulation for anisotropic materials. Dimension indicates the number of stress components used in the formulation for anisotropic materials.

702 F. BARLAT e t al.

state. However, it gives a rather crude approximation of the yield surface shape computed from polycrystal models and, therefore, does not provide good predictions of material behavior. The second anisotropic yield function proposed by HILL can lead to better results. However, this function does not include shear stress terms and is confined to a three-dimensional stress space. As a result, orthotropic material axes and principal stress axes must be superimposed. This restricts the use of this function consider- ably. In addition, this formulation has been shown to lead to nonconvex and sometimes to nonbounded yield surfaces (Znu et al. [1987]; LIAN et al. [1989]).

In order to describe the plastic flow behavior of orthotropic polycrystals, the yield function must be expressed in six-dimensional stress space, be convex, and must be independent of hydrostatic pressure. In addition, according to previous findings, it must reduce to eqn (3) or another equivalent formulation in the isotropic case. BARLAT and LIA~ [1989] proposed a yield condition for the three-dimensional plane stress case, which is very often assumed in sheet forming problems:

eI, = a[Ki -- Kzl m + alKl + Kz[" + (2 - a)[2K2[ m = 2 0 m (19a)

Kl - O×x + hoyy (19b) 2

K 2 ~ / ( °xx -- htYyY) 2 2 2 + p Oxy (19c)

where a, h, p and m are material constants. This anisotropic yield function has been obtained by summing two convex functions, a I K1 - K21 m + a ] KI + K2 [ m and ( 2 - a)] 2K2 ] m, and using linear transformations of the stresses. Consequently, it is a convex function which generalizes the plane stress yield function described by eqn (7). LEGE et al. [1989] showed that this yield criterion was particularly accurate for the description of the constitutive behavior of a 2008-T4 aluminum alloy sheet sample. Hu~L [1990] has concur- rently proposed another description of the constitutive behavior of sheet metals. The implications of this description have not been considered in this paper. However, Hn.L's [1990] yield function applies to the three-dimensional plane stress case and, therefore, has the same limitations as the function defined by eqns (19).

In order to generalize the isotropic yield function (eqn (16)) for the six-dimensional case, it is necessary to consider the matrix representing the stress deviator, to express it in the Bishop and Hill notation:

Sz~ ] Sxx Sxy

Syz S = Sxy Syy =

Sz,, Syz Szz

C - B H G

3

A - C H F

3

B - A G F

3

(20)

and to replace A, B, C, F, G, H, by aA, bB, cC, fF , gG, h H in this matrix where a, b, c, f , g, and h are constants:


c C - bB h H

3

aA - cC h H

3

gG f F

gG

f F

bB - aA

(21)

This last matrix is still symmetrical and, therefore, has three real eigenvalues. In addition, its trace is zero and Cardan's solution used in the previous section for the isotropic case is still valid. As a consequence, due to this particular choice of the constant coefficients, z can always be defined. All the results obtained for the isotropic case ap- ply for the anisotropic case when aA, bB, cC, f F , gG, h H are substituted for A, B, C, F, G, H in 12 and 13 (eqns (10)):

= [ f F ] 2 -.[- [gG} 2 + [ h i l l 2

3 54

(aA - c C ) 2 + ( c C - bB) 2 + (bB - aA) 2 + (22a)

(cC - b B ) ( a A - c C ) ( b B - aA) I~ = + f g h F G H

54

( c C - b B ) [ f F I 2 + (aA - c C ) { g G I 2 + (bB - a A ) [ h H I 2

6

(22b)

o_ arccos( )

2 ~ m ,

[2cos(20 +]m / 2oos(2065 / (22d)

The resulting anisotropic yield function is orthotropic and independent of hydrostatic pressure. It reduces to the isotropic case when the constant coefficients a, b, c, f , g, and h, are all equal to 1 (and particularly to Tresca for m = 1 or 2 and von Mises for m = 2 or 4). It accounts for the small radius of curvature of the yield surface near the uniaxial or the biaxial tension range that is observed with polycrystalline yield surface calculations. This yield function is quite general and can be applied to orthotropic materials for any stress state (except a hydrostatic pressure). Three of the six anisotropic coefficients can be obtained from the three uniaxial yield stresses in the directions of the symmetry axes using a Newton-Raphson numerical procedure. The other three coefficients are derived from the three shear yield stresses, each related to two of the symmetry axes. The exponent m can take any real value larger than 1 but practically, m should be larger than 6, depending on the severity of the texture, m can be determined such that it optimizes the predictions. Finally, it should be mentioned that the six-component anisotropic yield function (eqn (22)) does not reduce to the tr icomponent one (eqn (19)) because the linear transformations of the stresses, which must be made to get the an-


isotropic mathematical expression from the isotropic one, are not the same for the two cases.

Calculation of the strain rates is somewhat lengthy but straightforward when using the rule of composition of derivation:

o~ - I o~ os~ o_~_~ os____? o~ os3] o12 Oct~ . OS 1 012 + OS2 OIz + OS 3 012. Ooc,~

[ 0__~_~ OS_._2 0___~_~ 0S_...~2 0~I' 0S3] 013 -t- OSl 013 @ 0S2 013 + 0S3 Oh 00,~"

(23)

The case z = ~ (or 0 = 0 ° and 0 = 180°), which corresponds to shear in the isotropic case, is singular for the partial derivatives OSi/OIk. However, for this particular case, it is possible to calculate the strain rates as a limit of the general case when z - ~ --, 0.

V. APPLICATION TO ALUMINUM ALLOYS

In the present section, x, y, and z denote the rolling, transverse and normal directions in a rolled sheet. The six-component yield function is used to predict uniaxial properties for loading directions in the plane of the sheet for two different materials: 2008-T4 and 2024-T3. For each alloy, uniaxial yield stresses are used to calculate the material constants in the yield function. Then, R and F values defined by

~22 R = - - (24) ~33

p = --612 (25)

are predicted as a function of the tension loading direction. 1, 2, and 3 denote the lon- gitudinal, width, and thickness directions of a tensile specimen. Comparisons with experiments are made when data are available.

The chemical compositions (remelt analyses for the 2008 sample and typical data for the 2024 sample) and thicknesses of the samples are given in Table 2. For each of the sheet samples, specimens were taken at 0 °, 15 °, 30 °, 45 ° , 60 °, 75 °, and 90 ° to the rolling direction. Standard uniaxial tension tests were conducted to determine 0.2% offset yield strength, ultimate tensile strength, and total elongation. Single or duplicate (Alloy 2008) tests were also conducted in the seven sheet orientations at a constant crosshead speed of 0.1 in/rain to determine stress-strain behavior. Throughout these constant crosshead

Table 2. Thickness and composition a of aluminum alloy sheets

Alloy and Temper Thickness (mm) Si Fe Cu Mn Mg

2008-T4 1.24 0.60 0.13 0.93 0 .06 0.40

2024-T3 0.30 <0.50 <0 .50 4.40 0 .60 1.50

aComposition in weight%.


speed tests, extensometers measured strains in both the loading (~H) and width (d22) directions of the specimens. The plastic strain ratio (defined experimentally as R = C22//E33) was derived from the slope of the true width strain/true length strain curve evaluated from length strains of 0.01 to the uniform elongation assuming volume con- stancy in the specimen gage section.

A square grid (approximately 0.75 mm x 0.75 mm) was applied to each 2008-T4 tensile specimen using a photo-resist technique. After testing, grids in uniformly deformed regions of the specimens were analyzed with a View Engineering 1220 video measurement system to determine both the plastic strain ratio (R = c22//~33) and the shear strain ratio (/" = e12/~11 ). The grids were not analyzed prior to deformation because they were assumed to be of constant dimensions. This was found to be an erroneous assumption, however, and resulted in some uncertainty in the parameters calculated from the grid measurements. Nevertheless, the R values determined at a single point along the tensile specimen strain path with this suspect procedure showed the same directionality trends as those determined with extensometer measurements over the entire strain path.

X-ray techniques were used to determine the crystallographic texture of the 2008-T4 sheet. (l 11) and (200) poles figures were used to calculate the crystallographic orientation distribution function (CODF) using the expansion of the CODF in a series of generalized spherical harmonics (BuNGE [1965]; ROE [1965]). Corrections were made for the so-called "ghost peaks" of the CODF using the odd series coefficients resulting from the CODF positivity condition (BUNGE [1988]).

Figures 5a, 5b, and 5c present experimental and predicted results for the 2008-T4 sheet sample. Figures 5a and 5b show, respectively, the yield stress and R value variations with respect to the tension direction predicted with the six-component and the tricomponent yield functions. The uniaxial yield stresses measured at 0 °, 45 °, and 90 ° from the rolling direction where used to calculate the coefficients for the six-component yield function (Table 3), whereas the R values in these three directions were utilized to identify the parameters for the tricomponent yield function (Table 4). This explains why the re- spective experimental yield stresses and R values are almost superimposed on the theoretical curves. The R values predicted from the six-component model are slightly underpredicted for uniaxial tension in directions close to the transverse direction (90°), whereas the yield stress variation is underpredicted with the tricomponent yield function in the same range of tension directions. Perhaps it is not so obvious when looking at the figures, but from experience, the differences observed between experimental and theoretical results are significant for the yield stresses, whereas they are not for the R values. Figure 5c shows the F values predicted with the two models as well as experimentally measured values. Although LEGE et al. [1989] mentioned that the experimental data were not very accurate, the theoretical predictions result in a good trend. In fact, it can be assumed that the experimental point F ~ 0.037 in the rolling direction (0 °) is erroneous. This value should be zero because of symmetry in the rolled sheet.

Table 3. Coefficients of the six-component yield function

Material m a b c f g h

2008-T4 11 1.222 1.013 0.985 1.0 1.0 1.0

2024-T3 8 1.378 1.044 0.955 1.0 1.0 1.210

706 F. BARLAT eta/.

1.1

1 . 0

tn == ¢0

a)

0.9

I I I I I I I I

0,6 i e0 > m

0,4

m = 11 (6 - comp. model) rn = 8 (3 - comp. model)

A Exp.

• - - ~ - ~

0.8 I I I I l I I I 0 10 20 30 40 50 60 70 80 90 Angle between tension axis and rolling direction

(a)

1,0 t I = J t = I t

rn = 11 (6-¢ornp. model) 0,8 "_ ~ Z ~ - - - - - - m = 8 (3-cornp. model)

- " ~ . A Exp,

0.2 I I i I I I [ I 0 10 20 30 40 50 60 70 80 00 Angle between tenslon sxls and rolling dlrectlon

(b)

0.10 I I I I t s o . J , , , ~ I I ~ m ~ ' ~ i i . t %.

0.08 / / x \

o.oe ,/ ' \

0.04 \ \

o.o2 -

0"00 ~ L~

-0.02 m = 11 (6-cornp. model) _ - - - - - rn = 8 (3-cornp. model)

-0.04 L~ Exp.

-0.06 i I I I I I I I 0 10 20 30 40 50 60 70 80 90 Angle between tenslon axls and rolllng dlrectlon

(c)

Fig. 5. (a) Normalized yield stress for 2008-T4 alloy sheet subjected to uniaxial tension in various directions (values normalized with respect to 0 ° yield stress, 157 MPa). Prediction from the six-component model ( - - ) ; and from the tricomponent model (--); experiments (4). (b) R value for 2008-T4 alloy sheet subjected to uniaxial tension in various directions. Prediction from the six-component model ( - - ) ; and from the tricomponent model (---); experiments (4). (c) F value for 2008-T4 alloy sheet subjected to uniaxial tension in various directions. Prediction from the six-component model ( - - ) ; and from the t r icomponent model (---); experiments (4).


Figure 6 shows the yield surfaces computed with the TAYLOR [1938]/BmHOP and Hr~L [1951] (TBH) model (Fig. 6a), the tricomponent (Fig. 6b), and the six-component (Fig. 6c) yield functions. For the 2008-T4 alloy sheet, the TBH yield surface results from the assumption that each grain in the polycrystal is subjected to the same plastic strain rate. There are only 56 possible stress states in each grain which can accommodate any imposed plastic deformation, and the maximum plastic work principle is used to determine the active one. For an imposed strain rate, the macroscopic stress state is obtained

a I - ._c

m o

e _N

[ o z

1.5

1.0

0.5

0.0

- 0 . 5 -

I I

s = o. / y S = 0.S- S = 0.4- S = 0.3" S =0 .2"

b

I I !

- 0 . 5 0.0 0.5 1.0

N o r m a l i z e d s t ress in RD

- 1 . 0 - 1.0 1.5

(a)

a I.,- a:

m m o

"O e N '3 [ o Z

1.5 ~ . 3

1.0

0.6

0.0

- 0 . 5

I I I

- 1 . 0 - 1 . 0 1.5

s:o..1 J y S = 0 . 5 S = 0 . 4 S = 0 . 3

S = 0 . 2 S : 0 /

I I I

- 0 . 5 0 .0 0 .5 1.0 N o r m a l i z e d s t ress in RD

(b)

Fig. 6. (a) Plane stress yield surfaces for 2008-T4 alloy sheet and computed with the Taylor/Bishop and Hill model. (b) Plane stress yield surfaces for 2008-T4 alloy sheet and computed with the tricomponent yield function. (Figure continued on the folio wing page.)


t,,- ===o +g 1 0 . = -

" 0 " 0

[=

0 0 e- Z "

C

1.5

1.0

0.5

0.0

-0 .5

-1 .0 -1 .I

I I ~3yy

s_os y s---ols S = 0.4 ~ .+.....---- -.~ S = 0 . 3 ~ ~ S = 0.2 - - - ~ S=O.O ~

I J I

(~xx

-0 .5 0.0 0.5 1.0 1.5 Normalized yield stress in the rolling direction

(c)

Fig. 6 continued. (c) Plane stress yield function for 2008-T4 alloy sheet and computed with the six-component yield function.

by averaging the stress in each grain using the crystallite orientation distribution function (CODF). These types of computations have shown the existence of rounded vertices in the uniaxial or the biaxial tension range for the different types of crystallographic textures found in aluminum alloys (BARLAX e, RlCm,tOND [1987]; BARLAX & FRIeKE [1988]). The TBH yield surface (Fig. 6a) resulting from this computation procedure for the 2008-T4 sheet exhibits a smaller yield stress in the transverse direction than in the rolling direction, in agreement with the experiments. The computed TBH yield surface shows a small radius of curvature, particularly in the biaxial tension range, as compared to the plane strain or the shear range. The phenomenological models (Figs. 6b and 6c) are able to precisely reproduce this yield surface shape. Therefore, these functions are compatible with polycrystal models, and they are much easier to use.

Although both the tricomponent and the six-component anisotropic yield functions reduce to the same isotropic criterion, they are not identical. The tricomponent model was proposed after observing that the yield surfaces of textured polycrystals were roughly approximated by an expansion or a contraction of the isotropic yield surface in one or both directions ayy/# and Oxy/O. This is illustrated in Fig. 6b. However, there are many ways to introduce the anisotropic coefficients in the six-component function, and the particular solution proposed in this work was selected because it makes the function independent of the hydrostatic pressure. Since a different set of experimental data have been used to calculate the coefficients of both functions, it is not appropriate to compare both surfaces. It is clear, however, that the six-component anisotropic yield

Table 4. Coefficients of the tricomponent yield function

Material rn a h p

2008-T4 8 1.240 1.150 1.020


surface (Fig. 6c) does not result from the contraction of the isotropic one in the ors/# direction. Nevertheless, high curvature still occurs in the biaxial tension range of the yield surface predicted with the six-component model.

Figures 7a, 7b, and 7c show the yield stress, R value, and _r' value directionality of 2024-T3 sheet. In this case, the experimental results are compared to predictions made with the six-component model using a value of m that roughly optimizes the predictions and with HmL's [1948] anisotropic criterion. HmL'S [1948] criterion is compared with the six-component yield function because it is by far the most widely used for anisotropic calculations and because it can describe any type of loading condition, like the six-component model proposed in this work. Figure 7a shows the yield stress as a function of

1.1 I I I I I I I I I

I m = 8

I - - - - - - Hill's 1948 A Exp.

1.0

i '13 i A

~" 0.9

0.8 L I I I I I I I 0 10 20 30 40 50 60 70 80 90 Angle between tenslon axls and rolling dlmctlon

(a)

1 . 4 I I I I I I I I

1.2

_ A A 1.0 ~ , ~

o.a . . f a

n" 0.6 ] .~ , , , , , , , ,

0.4 m = 8 - - - - - - Hill's 1948

0 . 2 - A Exp.

0.0 I l I I I I I I 0 10 20 30 40 50 60 70 80 90 Angle between tenelon axle and rolllng dlrectlon

(h)

Fig. 7. (a) Normalized yield stress for 2024-T3 alloy sheet versus tensile direction (values normalized with respect to 0 ° yield stress, 327 MPa). Prediction with the six-component model for m = 8 (--); and with Hni's [1948] criterion (--); experiments (z~). (b) R value for 2024-T3 alloy sheet versus tensile direction. Predic- tion with the six-component model with m = 8 (--); and with Hn.L's [1948] criterion (---); experiments (a). (Figure continued on the following page.)


0.20

0.15

o.10

0.05 L.

0.00

-0.05 ~-

-0.1o 0

I I I I I I I I

..... 1 1 1 = 8 - - - - - - H i l l ' s 1 9 4 8

I I I I I I I I

10 20 30 40 50 60 70 80 90 Angle between rolling axis end tension direction

(c)

Fig. 7 continued. (c) F value for 2024-T3 al loy sheet versus tensile direction. Predict ion with the s ix-component mode l wi th m = 8 ( - - ) ; and wi th HILL'S [1948] cr i te r ion (---).

the tension direction. As mentioned previously, the yield stresses for different uniaxial stress states are used to calculate the constants in the yield function. Therefore, as ex- pected, the agreement between experiment and theory is quite good in both cases. The theoretical R values (Fig. 7b), which are fully predicted, are in reasonable agreement with the experiment. The curve calculated with H~L'S [1948] criterion tends to fit the experimental data better than the curve calculated with m = 8. The predicted/" value direc- tionalities (Fig. 7c) predicted by the two models are nearly identical. For this 2024-T3 sheet sample, the six-component yield function does not seem to result in better predictions than those made with HrLL'S [1948] yield criterion. However, the overall description of this material is more accurate with the proposed six-component yield function because the relatively high value of m leads to some rounded vertices near the uniaxial and biaxial range as polycrystal models would predict. For instance, this could result in better forming limit predictions.

This criterion can be very useful when implemented into FEM codes because it can account for anisotropy and, more specifically, for crystallographic textures in models of numerous forming processes. CaUNG and SaAa ]1990] have used a FEM code in con- junction with this yield function to model the hydraulic bulge test and the cup draw- ing test for the 2008-T4 alloy sheet. They have obtained results in excellent agreement with experiments.

It appears that the next step required to improve the predictive capabilities of the phenomenological model is the relaxation of the isotropic hardening assumption. Figure 8 illustrates this point for the extreme case of a material initially described by the Tresca criterion (thin full line). The R value is 1 in any tensile direction for this material. How- ever, if after some thermomechanical treatment the yield function is translated (thick full line) due to a Bauschinger effect, the R values for tension in the rolling and in the transverse directions are oo and 0, respectively. However, if the yield surface occupies a somewhat different position (dashed line), the R values are transposed: 0 in the rolling and ~ in the transverse direction! Of course, this is an extreme example because the yield surface has sharp vertices. However, this schematic model may explain why yield


t - O

0

" 0 4)

0 > p,

. - -

" 0

>-

1.5

1.0

0.5

0.0

-0.5

I

Oyy

J

I I

j (~xx

-1 .0 I i -1.0 -0.5 0.0 0.5 1.0 1.5

Yield stress in rolling direction

Fig. 8. Schematic anisotropic work-hardening.

stresses are quite well described by phenomenologica l or even polycrystal models , whereas R values are not .

VI. CONCLUSIONS

In this work , a new aniso t ropic yield funct ion for o r tho t rop ic mater ia ls has been proposed . This func t ion reduces to a very g o o d desc r ip t ion o f the yield sur face ca lcu la ted with polycrys ta l models for i so t ropic mater ia ls . Therefore , it is bel ieved to be a sui table ma thema t i ca l express ion for an i so t rop ic polycrys ta ls . This func t ion was used to predic t the uniaxia l plast ic proper t ies for a luminum al loy sheets, and reasonable agreement was found between theoret ical and exper imenta l results. The p roposed yield funct ion can be app l i ed to any type o f load ing cond i t i on and can, therefore , be a d v a n t a g e o u s l y subst i - tu ted for HILL's [1948] formula t ion , which is the s ix-component general anisot ropic yield func t ion cur ren t ly used in mos t F E M codes. As an ou t come , there are m a n y po ten t ia l app l i ca t ions o f the new yield func t ion , pa r t i cu la r ly in the s imula t ion o f any processes involv ing plas t ic d e f o r m a t i o n o f an i so t rop ic mater ia l s .

Acknowledgement -The authors would like to thank the following people from Alcoa Laboratories: Drs. R. C. Becker and K. Chung for valuable comments on the manuscript, Dr. S. Panchanadeeswaran for computations related to crystallographic texture analysis, and Dr. O. Richmond for his support to this work.

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don, A193, 281. 1951 Bmnop, J.W.F. and HILL, R., "A Theory of the Plastic Distortion of a PolycrystaUine Aggregate under

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1951 DRUCKER, D.C., "A More Fundamental Approach to Plastic Stress-Strain Relations," in Proc. First U.S. Nat. Congr. Applied Mechanics, ASME, New York, pp. 487-491.


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1959 DRUCKER, D.C., "A Definition of a Stable Inelastic Material," J. Appl. Mech. Trans. ASME, 26, 101. 1965 BUNGE, H.J., "Zur DarsteUung Allgemeiner Texturen," Z. Metallkunde, 56, 872. 1965 ROE, R.J., "Description of Crystallite Orientation in Polycrystalline Materials Ill: General Solution

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des M&aux et Alliages, Editions du CNRS, Paris, pp. 49-57. 1976 HECKER, S.S., "Experimental Studies of Yield Phenomena in Biaxially Loaded Metals," in STRICK-

LIN, J.A. and SACZALSKI, K.J. (eds.), Constitutive Equations in Viscoplasticity: Computational and Engineering Aspects, ASME, New York, pp. 1-33.

1977 BASSANI, J.L., "Yield Characterization of Metals with Transversely Isotropic Plastic Properties," Int. J. Mech. Sci., 19, 651.

1977 GOTOH, M., "A Theory of Plastic Anisotropy Based on a Yield Function of Fourth Order (Plane Stress State)," Int. J. Mech. Sci., 19, 505.

1979 HILL, R., "Theoretical Plasticity of Textured Aggregates," Math. Proc. Cambridge Philos. Soc., 85, 179.

1980 LOGAN, R.W. and HOSVORD, W.F., "Upper-Bound Anisotropic Yield Locus Calculations Assuming (111)-Pencil Glide," Int. J. Mech. Sci., 22, 419.

1981 ZYCZKOWSKI, M., Combined Loadings in the Theory of Plasticity, Polish Scientific Publishers, War- saw, Poland.

1982 SCHMITT, J.H. and JALINIER, J.M., "Damage in Sheet Metal Forming- I . Physical Behavior," Acta Met., 30, 1789.

1984 BUDIANSKI, B., "Anisotropic Plasticity of Plane-lsotropic Sheets," in DvOlOa¢, G.J. and SHrELD, R.T. (eds.), Mechanics of Material Behavior, Elsevier, Amsterdam, pp. 15-29.

1987 BARLAT, F., "Crystallographic Textures, Anisotropic Yield Surfaces and Forming Limits of Sheet Met- als," Mat. Sci. Eng., 91, 55.

1987 BARLAT, F. and RICHMOND, O., "Prediction of Tricomponent Plane Stress Yield Surfaces and Asso- ciated Flow and Failure Behavior of Strongly Textured FCC Polycrystalline Sheets," Mat. Sci. Eng., 95, 15.

1987 ZHO, Y., DODD, B., CADDEtL, R.M. and HosroRo, W.F., "Convexity Restrictions on Non-Quadratic Anisotropic Yield Criteria," Int. J. Mech. Sci., 29, 733.

1988 BARLAT, F. and FmCKE, W.G., JR., "Prediction of Yield Surfaces, Forming Limits and Necking Di- rections for Textured FCC Sheets," in KALLEND, J.S. and GOTTSTEIN, G. (eds.), Proc. 8th Int. Conf. Texture of Materials, Santa Fe, NM, 1987, The Metallurgical Society, pp. 1043-1050.

1988 BARL~, F., LtA~, J. and BAUDELET, B., "A Yield Function for Orthotropic Sheets under Plane Stress Conditions," in Proc. 8th Int. Conf. Strength of Metals and Alloys, Tampere, Finland, 1988 August, Vol. 1, Pergamon Press, New York, pp. 283-288.

1988 BmLINGTOr~, E.W., "Generalized Isotropic Yield Criterion for Incompressible Materials," Acta Mechan- ica, 72, 1.

1988 BUNGE, H.J., "Calculation and Representation of Complete ODF," in KALLEND, J.S. and GOTTSaTEIN, G. (eds.), Proc. 8th Int. Conf. Texture of Materials, The Metallurgical Society, pp. 69-78.

1989 BARLAT, F. and LIA2q, J., "Plastic Behavior and Stretchability of Sheet Metals. Part !: A Yield Func- tion for Orthotropic Sheets Under Plane Stress Conditions," Int. J. Plasticity, 5, 51.

1989 LEGE, D.J., BARLAT, F. and BREM, J.C., "Characterization and Modeling of the Mechanical Behav- ior and Formability of a 2008-T4 Sheet Sample," Int. J. Mech. Sci., 31, 549.

1989 LIAN, J., ZHOO, D. and BAUDELET, B., "Application of Hill's New Theory to Sheet Metal Forming - Part I. Hill's 1979 Criterion and its Application to Predicting Sheet Forming Limit," Int. J. Mech. Sci., 31,237.

1990 CHUNG, K. and SHAH, K.N., "Finite Element Simulation of Sheet Metal Forming for Planar Aniso- tropic Metals," accepted for publication in Int. J. Plasticity.

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Alcoa Laboratories Alcoa Center, PA 15069, USA.

(Received 26 March 1990; in f inal revised form 28 July 1990)