A Review About the Anisotropic Material Properties

8/13/2019 A Review About the Anisotropic Material Properties

http://slidepdf.com/reader/full/a-review-about-the-anisotropic-material-properties 1/14



2

Fig. 1 Compliance Matrix for Anisotropic Materials

Many materials, however, show more limited forms of anisotropy. For soils, the

anisotropic elastic properties depend on the depositional history. It is assumed that any

anisotropy would be of the horizontally layered type. This implies that the governing

parameters are independent of the horizontal directions and the vertical axis at any point

is an axis of radius symmetry. This special form of symmetry is known as cross-

anisotropy or transverse isotropy. The behavior of the cross-anisotropic materials could

be described by five independent elastic constants (Love 1927): Young’s modulus in any

horizontal direction Eh, Young’s modulus in a vertical direction E

v, Poisson’s ratio for

strain in the vertical direction due to a horizontal direct stress !hv

, Poisson’s ratio for

strain in any horizontal direction due to a horizontal direct stress at right angles !hh

, and

modulus of shear deformation in a vertical plane Ghv. The compliance matrix for the

cross-anisotropy materials is shown in Fig. 2.

The strain energy per unit volume can be written as the quadratic form

(1/2){"}T[C]{"}, and this quadratic form should be positive definite so that the strain

energy could be positive as required. The necessary and sufficient conditions for the

(1/2){"}T[C]{"} to be positive definite are that all the principle minors of [C] should be

positive, so putting (after Pickering 1970):

=

66

5655

464544

36353433

2625242322

161514131211

][

c

cc semmetryccc

cccc

ccccc

cccccc

C



3

+

−−

−−

−−

=

h

hh

hv

hv

hh

hv

h

hh

h

hv

vh

hv

h

hh

h

hv

h

E

G

G

E E E

E E E

E E E

C

)1(200000

01

0000

001000

0001

0001

0001

][

γ

γ γ

γ γ

γ γ

3

1

1

1

D

E E E

E E E

E E E

hh

hv

h

hh

h

hv

vh

hv

h

hh

h

hv

h

=

−−

−−

−−

γ γ

γ γ

γ γ

5

3

100

01

0

00

D

G

G

D

hv

hv

=

6

3

22

000

01

00

001

0

000

D

E

G

G

D

h

hh

hv

hv

=

+ γ

Fig. 2 Compliance Matrix for Cross-Anisotropic Materials

In the principle minors of the [C], the Ghv always appears by itself on the main

diagonal, and thus it requires it to be positive. The criteria for positive strain energy are:

(1) D1=1/Eh, therefore E

h must be positive;

(2) D2 Eh2 = E

h /Ev – !

hv2 must be positive;

(3) D3 Eh3 =[E

h /Ev(1– !

hh) – 2 !

hv2][1+ !

hh] must be positive;

1

1 D

E h=

4

3

10

0

D

G

D

hv

=21

1

D

E E

E E

vh

hv

h

hv

h=

−−

−

γ

γ



4

(4) Ghv must be positive;

(5) 2(1+ !hh

) must be positive, therefore !hh

must be greater than –1.

In condition (2), since !hv

2 and Eh2 are always positive, D2 can be positive only if

Eh/Ev is positive and greater than !hv

2. It has been clarified that Eh is always positive in

condition (1), so Ev also must be positive. Condition (5) requires 1+ !hh

be positive, so

the 1+ !hh

in (3) can be omitted, and the term [Eh/Ev(1– !hh

) – 2 !hv

2] in (3) remains to be

discussed. If !hh

is at its bounding value –1, this term equals to twice the value of

(Eh/Ev – !hv

2) in (2) and these two conditions coincide. If !hh

> –1, then [Eh/Ev(1– !hh

) –

2!hv

2] < 2(Eh/Ev – !hv

2). Therefore, within the permitted values of !hh

, condition (3) is

always more restrictive than condition (2), which can be therefore be omitted. The

remaining conditions are:

(a) Eh, E

v and G

hv must each be greater than 0;

(b) !hh

must be greater than –1;

(c) Eh/E

v(1– !

hh) – 2!

hv2 must be greater than 0.

In condition (c), the 2!hv

2 is positive, so Eh/E

v(1– !

hh) must be positive. Since

both Eh and E

v are positive, it requires that be !hh

< 1. By equating the terms of (c) to be

zero, the bounding surface for the elastic parameters could be obtained between !hh

= 1

and !hh

= –1 in Eh/E

v, !

hh, !hv space. The surface includes all the parameters except Ghv

,

which is independent of other parameters. It should be noticed that Eh and E

v are also

independent and only !hh and !

hv are bounded by the ratio of Eh/E

v. The shape of the

bounding surface is paraboloid, as shown in Fig. 3. Any vertical section of the surface



5

with constant !hh

is a parabola, and any horizontal section with constant Eh/E

v value is

also a parabola.

Fig. 3 Bounds for Elastic Constants (after Pickering 1970)

4. Anisotropic Plasticity

Many plastic models for anisotropic materials have been proposed (Hill 1948;

Bassani 1977; Gotoh 1977; Hill 1979; Hosford 1979; Hosford 1985; Barlat and Lian

1989; Barlat 1991). The major yield functions proposed to describe the anisotropic

behavior of orthotropic materials are listed in Table 1.



6

Constitutive relations for the plastic yielding and deformation of anisotropic

metals at a macroscopic level were proposed by Hill in 1948:

2f=F ( ! yy- ! zz )2+G ( ! zz - ! xx )2 +H ( ! xx- ! yy )2 +2L! yz 2 +2M ! zx

2+2N ! xy2="

where F, G, H, L, M and N were material constants.

Yield Criterion Type Shear Dimension

Tresca Isotropy ___ ___

von Mises Isotropy ___ ___

Hill (1948) Cross Anisotropy Yes 6

Hill (1979) Cross Anisotropy No 3

Hosford (1979) Cross Anisotropy, Plain Stress No 2

Barlat and Lian (1989) Cross Anisotropy, Plain Stress Yes 3

Barlat (1991) Anisotropy Yes 6

Table 1 Summary of Yield Functions

The theory described the yielding and plastic flow of an anisotropic metal on a

macroscopic scale. The type of anisotropy considered was that resulting from preferred

orientation. The yield criterion was postulated on general grounds which is similar in

form to the Huber Mises criterion for isotropic metals, but which contained six

parameters specifying the state of anisotropy. By using von Mises’ concept of a plastic

potential, associated relations were then found between the stress and strain-increment

tensors.

This theory was the simplest conceivable one for anisotropic materials. The yield

condition could be applied to an orthotropic material subject to any stress state, since it



7

concluded the shear stress terms. However, it gave a rather crude approximation of the

yield surface shape computed from polycrystal models and, therefore, did not provide

good predictions of materials behavior.

Hill’s 1979 yield function was proposed to account for the so-called “anomalous

behavior” of aluminum. It was assumed that:

(1) Every yield surface was convex;

(2) The material obeyed the postulate of normality (the plastic strain increment

vector is in the direction of the outward normal to the yield surface);

(3) The material obeyed the principle of equivalence of plastic work.

The expression of the function was written as:

mmmmmmmcbah g f σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ =−−+−−+−−+−+−+− 213132321211332 222

where f, g, h, a, b, and c were material constants.

In this yield criterion, the plasticity of material ploycrystals with preferred

orientations were considered from a phenomenological standpoint. This model could

lead to better results. However, this function did not include shear stress terms and was

confined to a three-dimensional stress space. This restricted the use of this function

considerably. In addition, it was shown that this formulation could lead to nonconvex

and sometimes nonbounded yield surface (Zhu et al. 1987; Lian et al. 1989). The

general form of the yield function could be reduced to four simple cases for cross-

anisotropy in plane stress loading conditions. Lian et al. (1989) thoroughly studied all the

four cases and their limitations; they found that all the cases of the yield function except

for one could provide reasonable predictions of the forming limits in the region of

positive strain ratios.



8

Hosford (1979) developed a yield criterion for plain stress loading case, which

was found to be an extension of Hill’s 1948 yield function:

M M M M Y r r )1(2121 +=−++ σ σ σ σ

where r was the normal anisotropy parameter (the ratio of yield stress in equal biaxial

tension to the average yield stress in uniaxial tension), Y was the yield stress of a bar in

uniaxial tension, and M was the parameter which characterized the shape of the yield

surface. The r value could be determined from standard tensile tests, but calculation of

the exponent M value required complicated testing procedures. Wagoner (1980)

proposed a method for obtaining the M value by comparing the uniaxial effective stress-

strain and plain-strain curves. From a numerical method for minimizing the standard

deviation between these two curves, the M value for a material could be determined.

This yield criterion could give good approximations, but it also didn’t include the

shear stress term in its expression. When the exponent M = 1 and r = 1, this criterion

reduced to Tresca yield function, and when M = 2 and r = 1, it reduced to von Mises yield

function.

Even though Hill’s 1979 and Hosford’s 1979 yield criteria have been widely

applied to predict the plastic behavior of anisotropic materials, they have some

limitations in application. They only considered the principal stress components, in other

words, the loading were restricted to conditions in which the axis of principal stresses

coincide with axis of material symmetry.

Another non-quadratic yield criterion proposed by Barlat and Lian (1989)

included a shear stress term in the expression of the effective stress, and this criterion

made it possible to predict the plastic behavior for the complete range of strain ratios



9

without the trouble which may occur by using Hill’s 1979 or Hosford’s 1979 yield

function. This yield condition was for the three-dimensional plane stress case, which is

often assumed in sheet forming problems:

22

2

2

1

22121

2

2

22)2(

xy

yy xx

yy xx

mmmm

ph

K

h K

K a K K a K K a

σ σ σ

σ σ

σ

+

+=

+=

=−+++−=Φ

where K 1 and K 2 were modified stress tensors, a, h, p and m were material constants, and

σ was effective stress.

This anisotropic yield function had been obtained by summing two convex

function,mm

K K a K K a 2121 ++− andm

K a 2)2(2 − , and using linear transformations

of the stresses. The prediction of this function was in reasonable agreement with the

results from the plastic potentials calculated with the Taylor/Bishop and Hill’s theory of

polycrystalline plasticity. Lege et al. (1989) showed that this function was particularly

accurate for the description of the constitutive behavior of a 2008-T4 aluminum alloy

sheet sample. This criterion was used by Shi and Gerdeen (1991) and reasonable results

were obtained.

Barlat developed another ansisotropic yield criterion for six-dimensional

deformation in 1991. The yield function was defined as:

( )

6

))(())(())((

54

))()((

54

)()()(

3

)()(

222

2

3

222222

2

hH aAbB fF bBcC gGcC aA

fghFGH aAbBbBcC cC aA

I

aAbBbBcC cC aAhH gG fF I

−+−+−−

+−−−

=

−+−+−+

++=



10

m

mmm

m I

I

I

σ π θ π θ π θ

θ

26

52cos2

6

32cos2

6

2cos2)3(

arccos

2/

2

2/3

2

3

=

+−+

−+

+=Φ

=

where a, b, c, f, g, and h were constants.

The resulting anisotropic yield function was orthotropic and independent of

hydrostatic pressure. It reduced to the isotropic case when the constant coefficient a, b, c,

e, g and h were all equal to 1 (particularly to Tresca criterion for m = 1 and von Mises

criterion for m = 2 or 4), and reduced to Hill’s 1948 anisotropic criterion when m = 2.

This yield function was quite general and could be applied to orthotropic

materials for any stress state (except a hydrostatic pressure). Three of the six anisotropic

coefficients could be obtained from the three uniaxial yield stresses in the directions of

the symmetry axis. The exponent m could 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. This six-component yield function did

not reduce to the three-component one proposed by Barlat and Lian (1989) because the

linear transformations of the stresses, which must be made to get the anisotropic

mathematical expression from the isotropic one, were not the same for the two functions.

The yield criterion had the advantage of being relatively mathematically simple

and was consistent with yield surface computed with polycrystal plasticity models. The

function could be applied to any type of loading condition. This theory was used to

predict the uniaxial plastic properties for aluminum alloy sheets, and reasonable

agreement was found between theoretical and experimental results. However, better

results could be obtained if the isotropic work-hardening assumption associated with the



11

yield criterion were relaxed, which means the exponent m is allowed to vary with

different plastic strain rate.

5. Conclusions

Both the elastic and plastic behavior of the anisotropic materials were reviewed in

the article. If the anisotropic elasticity of the material is to be fully described, a total of

21 independent parameters are needed in the 6×6 compliance matrix [C]. However,

many materials show limited forms of anisotropy, such as cross-anisotropy. The

behavior of this special form of anisotropic materials could be described by five

independent elastic constants: Eh, Ev, !hv

, !hh

, and Ghv. The shape of the boundary

surface for the elastic constants is a paraboloid in the Eh/Ev, !hv

, !hh

space.

The cross-anisotropic model was assumed in most of the yield functions for the

anisotropic materials. The anisotropic yield criterion proposed by Hill in 1948 was the

simplest conceivable one, however, it gave a rather crude approximation of the yield

surface. Hill’s 1979 yield function could lead to better results, but it didn’t include shear

stress terms. Hosford (1979) developed an extension of Hill’s 1948 criterion under the

plain stress loading condition. It also only considered the principle stress components.

The yield function proposed by Barlat and Lian (1989) was for the three-

dimensional plane stress case. This function included a shear stress term in the

expression, which made it possible to predict the plastic behavior for the complete range

of strain ratios. Barlat developed another six-component yield surface description in

1990. It could be applied to any loading condition. Both these two criteria could give

quite good approximations.



12

References

F. Barlat, D. J. Lege and J. C. Brem (1991). “A Six-Component Yield Function for

Anisotropic Materials”. International Journal of Plasticity, Vol. 7, p. 693.

F. Barlat and J. Lian (1989). “Plastic Behavior and Stretchability of Sheet Metals Part I:

A Yield Function for Orthotropic Sheet Under Plane Stress Conditions”. International

Journal of Plasticity, Vol. 5, p. 51.

J. L. Bassani (1977). “Yield Characterization of Metals with Transversely Isotropic

Plastic Properties”. International Journal of Mechanical Sciences, Vol. 19, No. 11, p.

651.

J. Ghaboussi and H. Momen (1984). “Plasticity Model for Inherently Anisotropic

Behavior of Sands”. International Journal for Numerical and Analytical Methods in

Geomechanics, Vol. 8, p. 1.

M. Gotoh (1977). “Theory of Plastic Anisotropy Based on A Yield Function of Fourth

Order (Plane Stress State)”. International Journal of Mechanical Sciences, Vol. 19, No.

9, p. 505.

J. Graham and G. T. Houlsby (1983). “Anisotropic Elasticity of A Natural Clay”.

Geotechnique, Vol. 33, No. 2, p. 165.

R. Hill (1948). “A Theory of the Yielding and Plastic Flow of Anisotropic Metals”.

Proceedings of the Royal Society of London, Series A, Mathematical and Physical

Sciences, Vol. 193, Issue 1033, p. 281.



13

R. Hill (1979). “Theoretical Plasticity of Textured Aggregates”. Mathematical

Proceedings of Cambridge Philosophic Society, Vol. 55, p. 179.

W. F. Hosford (1979). “On Yield Loci of Anisotropic Cubic Metals”. 7 th

North

American Metalworking Research Conference Proceedings, SME Dearborn, Michigan, p.

191.

W. F. Hosford (1985). “Comments on Anisotropic Yield Criteria”. International

Journal of Mechanical Sciences, Vol. 27, No. 7/8, p. 423.

D. J. Lege, F. Barlat and J. C. Brem (1989). “Characterization and Modeling of the

Mechanical Behavior and Formability of A 2008-T4 Sheet Sample”. International

Journal of Mechanical Sciences, Vol. 31, p. 549.

J. Lian, D. Zhou and B. Baudelet (1989). “Application of Hill’s New Yield Theory to

Sheet Metal Forming # Part I. Hill’s 1979 Criterion and Its Application to Predicting

Sheet Forming Limit”. Journal of Mechanical Sciences, Vol. 31, No. 4, p. 237.

A. E. H. Love (1927). “A Treatise on the Mathematical Theory of Elasticity”. 4th

edn.

Cambridge University Press.

D. J. Pickering (1970). “Anisotropic Elastic Parameters for Soil”. Geotechnique, Vol. 20,

No. 3, p. 271.

M. F. Shi and J. C. Gerdeen (1991). “Effect of Strain Gradient and Curvature on

Forming Limit Diagrams for Anisotropic Sheets”. Journal of Materials Shaping

Technology, Vol. 9, No. 4, p 253.



14

Y. S. Suh, F. I. Saunders and R. H. Wagoner (1996). “Anisotropic Yield Function with

Plastic-Strain-Induced Anisotropy”. International Journal of Plasticity, Vol. 12, No. 3,

p. 417.

R. H. Wagoner (1980). “Measurement and Analysis of Plane Strain Work Hardening”.

Metallurgical and Materials Transactions A, Vol. 11A, p. 165.

D. Zhou and R. H. Wagoner (1994). “A Numerical Method for Introducing An Arbitrary

Yield Function into Rigid-Viscoplastic FEM Programs”. International Journal for

Numerical Methods in Engineering , Vol. 37, p. 3467.

Y. Zhu, B. Dodd, R. M. Caddell and W. F. Hosford (1987). “Convexity Restrictions on

Non-Quadratic Anisotropic Yield Crieteria”. International Journal of Mechanical

Sciences, Vol. 29, p. 733.

Documents

A Review About the Anisotropic Material Properties