A New Yield Function on Isotropic Metals Included One to Six-Order Plastic Tensors

The yield function is very important in establishing the plastic constitutive relation and analyzing the plastic deformation. Hence this article derives the yield function on isotropic metals included one to six-order plastic tensor, expanded of the general yield function in its Taylor. As well, the plastic tensors are analyzed by traceless, totally symmetric and objectivity in this article. And the yield function for isotropic metals can be degenerated to the one for identical and different property on tension-compression yield. Finally, by means of the results of Lode test, it is proved that this yield function is very suitable in the metal materials, which had the identical and different property for tensioncompression yield. And there are 2 material parameters in this yield function, hence, the form is simple and had higher value in engineering.


INTRODUCTION
The yield function is vital to describe the plastic deformation of metal, and it is also the necessary supplement equations to deal with the metal plastic forming [1].
At present, there are hundreds of yield functions of metal, such as Hill, Hershey, Barlart, Hostford, Man C.-S and Huang function etla, to precisely describe the mechanical behavior of a material [2].However, these yield functions are suitable for anisotropic metal materials, the studies of isotropic metals are so far very few.Some investigators reported that both Tresca yield criterion and Von-mise ones are generally used to establish the plastic constitutive relation and analyze the plastic deformation [3][4][5].
Based on the results of simple tension test for mild steel, Tresca yield functionis showed in formula (1).And this function is simple and don't include the second principal stress.But under the condition of the unknown principal stress, the expression of Tresca yield criterion maybe extremely complicated.Because of the corner point in the plane, the Tresca yield criterion is difficult to be dealt with during the solving equations [6]. (1) Considering the second principal stress, Von-mise yield function is derived from the results of simple tension test and pure shear experiment.And the formula is Eq. ( 2).However, it is inadaptable for the identical and different property on tension-compression yield, and does not considerer the hydrostatic pressure for metal having no effect on yield [7]. ( Microscopically metal is an aggregate of numerous tiny crystallites, we consider metal as isotropic materials in engineering.Hence, Because of the hydrostatic pressure for metal having no effect on yield and isotropic hypothesis, this article derives the yield function on isotropic metals and corresponding six-order plastic tensor, expanded of the traditional yield function by Taylor and intercept the six order.As well, the new yield function of isotropic metals reduces to the one for the identical and different property on tension-compression yield.Finally, simple tension and compress and pure shear experiment can determine the parameters.

ESTABLISH OF THE YIELD CRITERION
When the metals were yielded, the yield function, f (σ) = 0 , must be expressed by stress.Then the smooth function , showed by Eq. (3), is expressed in its Taylor expansion at .
Because of , we obtain the yield functions by truncating the preceding expansion for each .Then the Eq. ( 3) can reduce to Eq. ( 4).(4) where, , , are respectively second-order, forthorder, sixth-order plastic tensor of isotropic metals, as follows: (5)

The Minor and Major Symmetries
Because of the reciprocal theorem of shear stress and formula 3, we can obtain the expression ( 6) from the invariance principle on the exchange order of derivation [8]. (6)

Traceless
The yield of metal is arouse from none recovery for the slip deformation during atoms, appeared on the shear stress.And the hydrostatic pressure has no effects on the shear stress [9].Hence, the yield function of metal sheet can satisfy with the formula (7), , When , based on the equality of the corresponding plastic tensor, the combination of ( 3) and (6) lead to (8)

Objectivity
The rotation of external force and object have no effect on the yield for metal, that is, When external force and object rotate, the yield function for isotropic metals must satisfy the objectivity [10].
Let R denote the rotation tensor, the Euler angle be , , , we have [11] Assumed that the external force and object rotate by R, the stress become (10) where the stress ( ) is before rotating, the stress ( ) is after rotating.
Hence, the yield function must be meet with Eq. (11).
Put ( 3) into (11), based on the equality of the corresponding polynomial, the second-order, forth-order, sixth-order plastic tensor respectively are where, , , , were the plastic tensor of referred grains before rotating; , , , were the plastic tensor of referred grains after rotating.

Isotropy
If the materials are isotropic, the yield must be satisfy that the material properties can not change after rotating by R, in , that is, the plastic tensor for isotropic materials is showed as follow [12]: Put ( 13) into (12), we can obtain (14).(14) The isotropic materials plastic tensors can obtain from the homogenization [13].
where, SO 3 is group of rotation tensor, and

The General Form of the Yield Function
When the plastic tensors meet the Eq. ( 5), ( 7), ( 11) and ( 13), they maybe become Eq.( 16). ( 16) where, Put (16) into (3), the yield function on isotropic metal materials is showed as follow.This yield function includes the stress of acquadratic term, cubic term, and have 2 material parameters measured by experiment.Hence, the form of the new yield function is simple and had higher value in engineering.

THE DISCUSSION OF THE DEGRADATION FORM FOR YIELD FUNCTION
Instead of the principle stress, the new yield function on isotropic metal materials is the Eq. ( 18).

The Yield Function on the Identical Property for Tension-Compression Yield
According to metal materials of the identical property for tension-compression yield, the yield strength of tension is , the one of compression is , the one of shear is .The state of the principle stress for uniaxial tensile, uniaxial compression and pure shear experiment are showed in Table 1.
According to kinds of the state of the principle stress, the Eq. ( 18) can become to (19) We can solve Eq. ( 19), and obtain M, P as follows.
(20) Therefore, the 2 material parameters from the new yield function can be identified by the tension-compression and pure shear experiment.

The Yield Function on the Different Property for Tension-Compression Yield
According to metal materials of the different property for tension-compression yield, the yield strengths of tensioncompression are respectively , , the one of shear is .The state of the principle stress for uniaxial tensile, uniaxial compression and pure shear experiment are showed in Table 2.

EXPERIMENTAL VERIFICATION
In 1926, W. Lode applied the axial force and internal pressure to the thin-wall cylinder made of mild steel, copper, nickel.And the 3 parameters, , , , are measured by the thin-wall cylinder yield experiment [14,15].And the formulas of , , are According to Eq. ( 23), when , the new yield function can be degenerated to Von-mise yield function.
The Fig. (1) showed that this yield function is very suitable for the metal materials, which had the identical and different property for tension-compression yield.And there are 2 material parameters in this yield function, hence, the form is simple and had higher value in engineering.
results of W.Lode experiments, the relation between and is analyzed by the yield criterion of Teresa, Von-mise and the new yield criterion on isotropic metals in this thesis.And the results are showed in Fig. (1).

Table 1 . The principal stress state of metals having the identical property for tension-compression yield on uniaxial tension, compression and shearing tests.
It is not agree with that Tresca and Von-mise yield function demand is the determined value in practical engineering.The new yield function included the degree term various one from four on the isotropic metals overcomes the shortcomings.Put Lode parameters into the yield function, is