SIMULATION IN DER UMFORMTECHNIK ANISOTROPIC DAMAGE EVOLUTION OF TOOLS DURING FORGING PROCESS Kunio Hayakawa, Shizuoka University, Hamamatsu Abstract In forging analysis, the damage evaluation parameters such as Cockloft and Latham, Oyane models and so on are very useful for the evaluation of the limitation of the deformation of the workpiece during the process. For the tool failure, the proper damage parameter has not ever been proposed yet. The damage to the tool material generally has an anisotropic nature that is related to the direction of the principal stress, since the tool materials are very hard and brittle. In the present paper, an anisotropic damage evolution model proposed by the present author is modified and implemented to the simufact.forming by the user subroutines. Then the evolution of the anisotropic damage to the cold forging die during the process was calculated using the damage model. The calculated models are axisymmetric and 3-dimensional ones. The proposed anisotropic damage model can describe the damage to the tools during the process properly. 1. Introduction In cold forging, the tools are subjected to cyclic high loadings. As a result, we often experience premature failure of tool. For example, in cold forward extrusion, the die insert often cause fatigue cracks, wear and axial cracks due to high loadings [1-3]. Therefore, the exact estimation of the service life of forging tool is very important for the reduction of total cost of forging operations. Recently, cemented carbide material such as WC-Co has been often used as the tool material of cold forging for higher dimensional accuracy of the forgings [4,5]. Therefore, precise constitutive equation of the cemented carbide material is useful for the more precise calculation of the stress and service life by finite element method. In the present paper, the elastic-plastic constitutive equation of cemented carbide material is proposed with anisotropic damage behavior taken into account. The conventional framework of thermodynamic theory is employed for the formulation [6-8]. The anisotropic damage is considered to express the salient stress unilateral behavior of cemented carbide material. Uniaxial behavior and cold forward extrusion is calculated using the proposed equation for the validation. 2. Constitutive Equations of Tool Materials In the present paper, the behavior of the WC-Co cemented carbides material is modeled by the elastic-plastic constitutive equations coupled with anisotropic damage based on the framework of the irreversible thermodynamics theory for the constitutive equation [6-8]. Session: Schmieden
Transcript
Simulation in der umformtechnik
ANISOTROPIC DAMAGE EVOLUTION OF TOOLS DURING FORGING
PROCESS
Kunio Hayakawa, Shizuoka University, Hamamatsu
Abstract
In forging analysis, the damage evaluation parameters such as Cockloft and Latham, Oyane
models and so on are very useful for the evaluation of the limitation of the deformation of the
workpiece during the process. For the tool failure, the proper damage parameter has not ever been
proposed yet. The damage to the tool material generally has an anisotropic nature that is related to
the direction of the principal stress, since the tool materials are very hard and brittle.
In the present paper, an anisotropic damage evolution model proposed by the present author is
modified and implemented to the simufact.forming by the user subroutines. Then the evolution of
the anisotropic damage to the cold forging die during the process was calculated using the damage
model. The calculated models are axisymmetric and 3-dimensional ones. The proposed
anisotropic damage model can describe the damage to the tools during the process properly.
1. Introduction
In cold forging, the tools are subjected to cyclic high loadings. As a result, we often experience
premature failure of tool. For example, in cold forward extrusion, the die insert often cause fatigue
cracks, wear and axial cracks due to high loadings [1-3]. Therefore, the exact estimation of the
service life of forging tool is very important for the reduction of total cost of forging operations.
Recently, cemented carbide material such as WC-Co has been often used as the tool material of
cold forging for higher dimensional accuracy of the forgings [4,5]. Therefore, precise constitutive
equation of the cemented carbide material is useful for the more precise calculation of the stress
and service life by finite element method.
In the present paper, the elastic-plastic constitutive equation of cemented carbide material is
proposed with anisotropic damage behavior taken into account. The conventional framework of
thermodynamic theory is employed for the formulation [6-8]. The anisotropic damage is considered
to express the salient stress unilateral behavior of cemented carbide material. Uniaxial behavior
and cold forward extrusion is calculated using the proposed equation for the validation.
2. Constitutive Equations of Tool Materials
In the present paper, the behavior of the WC-Co cemented carbides material is modeled by the
elastic-plastic constitutive equations coupled with anisotropic damage based on the framework of
the irreversible thermodynamics theory for the constitutive equation [6-8].
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In the present study, the strain of the tool material can be decomposed into the elastic and the
plastic parts as follows, as the deformation of the forging tools can be assumed infinitesimal.
=e+
p
(1)
2.1. Material Damage and Damage Variable
The debonding of the interface between the carbides and the matrix, the fracture of the carbides
and/or matrix as well as the plastic deformation of the matrix will cause the microscopic material
damage of cemented carbides.
The initiation and growth of the material damage depends the direction of the applied stress.
Moreover, the effects of the damage are diminished under compression because of the closure
effect of microcracks. As a result, the strength and toughness under tension are known to be lower
than those under compression.
In the present paper, we employ the second rank symmetric damage tensor D for the description
of the mechanical effect of the three dimensionally distributed microcracks in the material as [7, 8]
D = DI
pI
pI( )
I=1
3
(2)
where D
I and
p
I (I = 1, 2 and 3) are the principal value and principal direction of the damage
variable, respectively.
2.2. Description of Unilateral Characteristic of WC-Co Cemented Carbides
Some researchers have proposed descriptions of the unilateral property of material damage. In the
present paper, we introduce the modified Cauchy stress tensor as [7, 8]
=I I I( )
I=1
3
(3)
where is the Macauley bracket, and I
and I
(I = l, 2 and 3) are the principal values and
principal direction of stress tensor , respectively.
The modified stress tensor can be written in the global Cartesian coordinate system xi (i = l, 2 and
3) as
ij = Bijkl kl (4)
Bijkl = h K( )K=1
3
QiKQjKQKkQKl (5)
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where h(
K) is the Heaviside unit function for the principal stress
K, and Q is the direction
cosine between the global Cartesian coordinate system xi and the principal stress coordinate
system.
2.3. Elastic-Damage Constitutive Equation and Thermodynamic Conjugate Forces
According to the conventional procedure of the irreversible thermodynamic formulation [6-8], the
elastic constitutive equation of the damaged material can be obtained as follows:
e =ged( )
=1+
0
E0
+1 2
0
E0
M + 21trD( ) M :
M
+2
D + D( ) : (6)
where E0
and 0 are Young’s modulus and Poisson’s ratio at the initial undamaged state.
Furthermore, M and are hydrostatic stress tensor and deviatoric stress tensor, respectively.
Moreover, 1
and 2
are material constants. The first and second terms of the right side of eq.
(10) correspond to the common linear isotropic elastic constitutive equation. The third and fourth
terms express the effects of the anisotropic damage on the elastic behavior of the material.
The thermodynamic conjugate forces, Y, R, XN and B of the internal state variables D, r,
N and
on the other hand, can be derived as
Y =
1M M( ) + 2 ( ) , (7)
R = R 1 exp b
rr( ){ } , (8)
X N =2
3C N N (N = 1, 2, 3), (9)
B = K
b. (10)
2.4. Plastic-Damage Constitutive Equation
For the cemented carbide material considered in the present paper, the effect of the damage on
plastic deformation is considered by the use of the effective stress that describes the enhanced
stress effect by the existence of the damage.
The plastic potential Fp and yield surface
f p is given as follows by use of the effective stress
as
Fp = 3J2
X( ) y R +C N
N mN + 2( )N=1
3 N
C NXeq
N
mN +2
, (11)
f p = J2 X( ) y R = 0 , (12)
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J2
X( ) =1
2X( ) : X( ) , (13)
= M D( ) : , (14)
M D( ){ }ijkl
=1
2ik jl + il jk( ) + c p Dik jl + ikDjl + Dil jk + ilDjk( ) . (15)
where y is the yield stress under the non-damaged state, and N , mN and c p are material
constants.
By using the non-associate flow rule and the normality law, the plastic-damage constitutive
equation, evolution equations of the isotropic and kinematic hardening variables r and N can be
obtained as
p = p
Fp= p
3
2J2
M D( ) : X( ) , (16)
r = p
Fp
R( )= p
, (17)
N = p
Fp
X N( )= p
3 X( )2J
2
3 N
2C N
N
C NXeq
N
mN
X N , (18)
where the unknown multiplier p can be determined from the consistency condition as
f p =J
2
ij
ij +J
2
Dij
Dij +J
2
Xijl
XijN
N=1
3
+f
RR = 0 . (19)
2.5. Damage Evolution Equation
We consider two different mechanisms of damage evolution. One is the damage that develops by
movement of dislocation. This damage is in relation to the fatigue fracture, plastic deformation.
Another is the damage by the initiation and development of microcracks or debonding of the
interface between WC carbide and Co matrix. We assume that this damage develops in case that
the present load only exceeds the maximum load that the material have been ever experienced.
For this purpose, a damage surface fd inside which damage does not develop, which is like as
yield surface in plasticity, is introduced.
The damage potential F
d and
fd are assumed as
Fd = Yeq = Y :Y (20)
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fd = Yeq B( ) + B
0{ } = 0 (21)
Then, the evolution of damage can be prescribed as
D =d
fd
Y+
Fd
Y (22)
where d and are the unknown multiplier determined by the consistency condition on the
damage surface f
d, and magnitude of the damage development related to the fatigue, respectively.
In order to describe the fatigue damage behavior properly, we assume
as
=nd
Kd ep( )Yeq Y
0
Kd ep( )
nd 1
Yeq, (23)
Kd ep( ) =2Kd0
exp bdep( ) + exp bde
p( ), (24)
ep =2
3
p : p
1/2
dt . (25)
where n
d , Kd0 and bd are material constants. The value of Y0 is the threshold of the evolution of
damage by fatigue. Therefore, the Y0 can be calculated by eqs.(11) and (22) with the fatigue limit
f .
An equivalent damage variable Deq is introduced in the present study as
Deq= D :D . (26)
The material is assumed to attain to final fracture when the equivalent damage variable Deq
reaches the threshold value Dcr .
3. Uniaxial loading of cemented carbides material
3.1. Cemented Carbides Material
Uniaxial loading behaviors of WC-Co cemented carbide material are calculated by use of the
proposed constitutive equation. In the present case, x3 axis is the direction of loading, and x1 and x2
axes are on the cross section of the test specimen.
The cemented carbide material G7 (WC: 75%, Co: 25%, San Alloy Industry Company, Japan) was
selected in order to compare the experimental results by Brøndsted et. al [4] and Skov-Hansen et.
al [5]. This material was GT55 type hot isostatiscally pressed. From the literatures, Initial Young’s
modulus E0 and Poisson’s ratio n0 were selected as
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E0 = 470GPa, 0 = 0.3 . (27)
The material constants in eqs. (7) - (17) were determined so that the proposed constitutive
equation can describe the uniaxial tension and compression of the cemented carbides G7 as
follows:
1 = 2.61 10 5, 2 = 3.03 10 5,
y = 2.00 102, R = 0, Kb = 0.80, c p = 0.3
C1 = 7.20 105, C2 = 2.20 106, C3 = 1.00 105,
1 = 8.00 102, 2 = 2.00 103, 3 = 50.0,
m1 = 10.0, m2 = 10.0, m3 = 10.0
nd = 6.00, Kd0 = 3.0, f = 6.50 102
(28)
3.2 Results and Discussion
Figure 1 shows the experimental and calculated results of stress-strain curves under uniaxial
tension. The development of damage components is also shown. The tensile strength is about
1700MPa. We can observe the good agreement between the experimental and calculated results.
We can also observe the damage component D33 is larger than D11 = D22.
Figure 2 shows the experimental and calculated results of stress-strain and damage components-
strain curves under uniaxial compression. The compressive strength of the material is
approximately 3200MPa. We can observe that the damage components D11 = D22 is dominant in
the compression case. It can be predicted from the result that microcracks parallel to the loading
direction are predominantly developed and cause the final fracture under the uniaxial compression,
which will occur to most brittle materials.
Figure 1: Stress – Strain relation of WC-Co Figure 2: Stress – Strain relation of WC-Co
material under uniaxial tension. material under uniaxial compression.
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4. AXISYMMETRIC FINITE ELEMENT ANALYSIS ON COLD FORGING DIE DURING COLD
FORWARD EXTRUTION PROCESS
4.1 Finite Element Model and Material Properties of Workpiece and Shrink Ring
Mechanical and damage behavior of the cemented carbide used as a die-insert of cold forward
extrusion die set is evaluated by FEM in the present paper.
Figure 3 shows the schematic example of typical fatigue, fracture and wear of cold forward
extrusion die (Engel, 1994). From the figure, the different anisotropic damage components can be
expected to describe the different fracture behaviors; forced rupture (axial crack) and fatigue
fracture (radial crack), as the directions of the crack propagation are different.
Figure 3: Typical failure of cold forward extrusion die-insert [1].
In the present chapter, therefore, the proposed constitutive equation will be applied to the
calculation of the damage state of cold forward extrusion die made of WC-Co cemented carbide.
A commercial FE code MSC. Marc 2005 was used for the calculation. The constitutive and
damage evolution equations of the die material are implemented by the user subroutines provided
in the code, HYPELA2, ELEVAR and so on (MSC Software, 2005).
Figure 4 shows the geometries and discretization of the die set and workpiece used in the present
calculation. In the finite element analysis, axisymmetric model was used. The x1, x2 and x3 axes of
Cartesian coordinate system correspond to the axial, radial, and circumferential directions of the
die-insert, respectively. This figure shows the case of the die-angle = 120°. In the present
calculations, both the die-angle of = 90° and 120° were performed in order to examine the effect
of the die-angle on the damage behavior of the die-insert. Punch and knockout-pin were modeled
as rigid boundaries
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For materials of die insert and shrink ring, WC-Co G7 and SKD61 (in JIS) were selected. Only
elastic property is used in the SKD61, Young’s modulus and Poisson’s ratio are set to 206GPa and
0.3.
Figure 4: Geometry and discretization of analyzed model of cold forward extrusion (in case of
die-angle = 120°).
As the workpiece material, SS400 (in JIS) was used. The elastic and plastic properties of the
material are given as follows;
E = 206GPa, = 0.29 , (29)
= 285.0+ 461.0 ep 1.38 10 4( )0.286
(30)
where ep is the equivalent plastic strain of the workpiece material calculated by Equation (25).
The usage of the shrink ring is effective avoiding the onset of forced rupture (axial cracking). The
interference of shrink fit between the die-insert and shrink ring was set to 0.1mm.
The axisymmetric triangular and quadrilateral elements were employed for the extrusion die set
and the workpiece, respectively. The numbers of elements of the die-insert, shrink ring and
workpiece are 3093, 114 and 2739, respectively. The numbers of nodes of the die insert, shrink
ring and workpiece are 1606, 72 and 4442, respectively.
4.2 Results of Development of Damage Components and Discussion
Figure 5 shows the distribution of damage components D11, D22, D12 and D33 in case of the die-
angle = 120° when the length of the extruded region attains 5.3mm; the corresponding punch
stroke is S = 2.1mm. Although the D11, D22 and D12 show almost same values, their distributions are
slightly different. This means that one principal damage value is in the plane of x1-x2, and the
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principal direction is around 45° from x1 axis. The D33 is another principal damage value, and it can
describe the possibility of the forced rupture (axial cracking) in Figure 6. In this case, the D33 is
dominant damage component. Therefore, the axial cracking may occur in spite of the shrink fitting.
Table 1 shows the principal damage value D1, D2 and D3 (= D33), and the angle of D1 from the x1-
axis at each and S. In case of die-angle = 90°, the damage component D33 is very small value of
6.013×10-5 at even the punch stroke of S = 2.95mm. This means the possibility of the forced
rupture (axial cracking) can hardly be encountered. It can be also observed that the principal
damage D1, which stands for the fatigue damage in the region of die radius, is 0.368 and 0.324 in
each a. Therefore, the residual life to the onset of the fatigue damage will be similar in each die-
insert. Furthermore, the angle of principal axis 1 from x1 axis, which stands for the direction of the
propagation of fatigue crack, can be found almost same in each case.
(a) (b)
(c) (d)
Figure 5: Distribution of damage component during extrusion in case of punch stroke S =
