Stress-strain relation simulation method, springback-amount prediction method, and springback analyzer
Displacement or a load is applied to an elastic-plastic material to deform the elastic-plastic material plastically to acquire experimental values of a stress-strain relation. With a kinematic-hardening incremental vector dα ij of a yield surface in an elastic-plastic constitutive model as a predetermined first equation, the elastic-plastic constitutive model being defined as a function of stress and back stress, a computer identifies material constants contained in the elastic-plastic constitutive model with the acquired experimental values. The computer identifies material constants contained in a predetermined second equation on the basis of the acquired experimental values and the predetermined first equation into which the identified material constants are substituted. The computer simulates the stress-strain relation of the elastic-plastic material with the predetermined first equation, the predetermined second equation, and the elastic-plastic constitutive model into which the identified material constants are substituted.
1. A springback-amount prediction method comprising:
determining a spring-back amount of a press-formed product including:
(i) an experimental-value acquisition step of plastically deforming an elastic-plastic material to acquire experimental values of a stress-strain relation;
(ii) a first material-constant identification step of, by a computer, with a kinematic-hardening incremental vector dα ij of a yield surface in an elastic-plastic constitutive model as Equation (1), the elastic-plastic constitutive model being defined as a function of stress and back stress, identifying material constants contained in the elastic-plastic constitutive model using the experimental values acquired at the experimental-value acquisition step;
(iii) a second material-constant identification step of, by the computer, based on the Equation (1) into which the material constants identified at the first material-constant identification step are substituted and based on the experimental values acquired at the experimental-value acquisition step, identifying material constants contained in Equation (2); and
(iv) a step of, by the computer, predicting the springback amount with the Equation (1) and the Equation (2) into which the material constants identified are substituted, and the elastic-plastic constitutive model,
modifying a die based upon the determined spring-back amount, and
pressing the die onto a metal sheet to produce the press-formed product,
wherein Equation (1) is defined as follows:
d
α
i
j
=
[
C
(
a
Y
)
(
σ
i
j
-
α
i
j
)
-
C
0
ρ
X
i
j
]
d
ɛ
eq
p
Equation (2) is defined as follows:
C
=
C
a
+
C
c
exp
(
-
X
eq
n
A
)
where:
a: maximum value of movement of yield surface
Y: yield stress
α ij : movement vector of yield surface
σ ij : stress vector
X ij : yield—surface kinematic—hardening amount after stress reversal
X eq : equivalent value of X ij
dε p eq : equivalent plastic—strain increment
C 0 , C C , A, n: material constants
ρ: variable such that ρ=1 or ½.
2. The springback-amount prediction method according to claim 1 , wherein variables X ij , ρ, A, and n in the Equations (1) and (2) are represented by Equation (3), wherein Equation (3) is defined as follows:
If
σ
eq
≥
σ
eq
max
,
σ
eq
=
σ
eq
max
X
i
j
=
α
i
j
,
ρ
=
1
,
A
=
A
1
,
n
=
n
1
If
σ
eq
<
σ
eq
max
X
i
j
=
α
i
j
tmp
-
α
i
j
,
ρ
=
1
2
,
A
=
A
2
,
n
=
n
2
}
where
σ eqmax : maximum value of equivalent stress when isotropic hardening is assumed
α tmp ij back stress at the time of stress reversal
A 1 , A 2 , n 1 , n 2 : material constants.
3. The springback-amount prediction method according to claim 2 , wherein:
as a method for applying plastic deformation to the elastic-plastic material at the experimental-value acquisition step, one method is used out of:
a method in which stress is applied to the elastic-plastic material in a tensile direction to deform the elastic-plastic material plastically and is then unloaded;
a method in which stress is applied to the elastic-plastic material in the tensile direction to deform the elastic-plastic material plastically and is then unloaded, and stress is applied in a compression direction to deform the elastic-plastic material plastically; and
a method in which stress is applied to the elastic-plastic material in the tensile direction to deform the elastic-plastic material plastically and is then unloaded, and stress is applied again to the elastic-plastic material in the tensile direction to deform the elastic-plastic material plastically.
4. The springback-amount prediction method according to claim 1 , wherein:
as a method for applying plastic deformation to the elastic-plastic material at the experimental-value acquisition step, one method is used out of:
a method in which stress is applied to the elastic-plastic material in a tensile direction to deform the elastic-plastic material plastically and is then unloaded;
a method in which stress is applied to the elastic-plastic material in the tensile direction to deform the elastic-plastic material plastically and is then unloaded, and stress is applied in a compression direction to deform the elastic-plastic material plastically; and
a method in which stress is applied to the elastic-plastic material in the tensile direction to deform the elastic-plastic material plastically and is then unloaded, and stress is applied again to the elastic-plastic material in the tensile direction to deform the elastic-plastic material plastically.
5. A manufacturing method comprising:
using a springback analyzer to determine a spring-back amount of a press-formed product, the springback analyzer including:
(i) a press-forming analysis unit that performs press-forming analysis to acquire shape, residual-stress distribution, and strain distribution of the press-formed product through analysis before die release; and
(ii) a springback analysis unit that performs springback analysis based on the shape, the residual-stress distribution, and the strain distribution of the press-formed product to acquire the springback amount of the press-formed product after die release,
wherein a kinematic-hardening incremental vector dα ij of a yield surface in an elastic-plastic constitutive model that the press-forming analysis unit and the springback analysis unit have is represented by Equations (1) and (2),
modifying a die based upon the determined spring-back amount, and pressing the die onto a metal sheet to produce the press-formed product,
wherein Equation (1) is defined as follows:
d
α
i
j
=
[
C
(
a
Y
)
(
σ
i
j
-
α
i
j
)
-
C
0
ρ
X
i
j
]
d
ɛ
eq
p
Equation (2) is defined as follows:
C
=
C
a
+
C
c
exp
(
-
X
eq
n
A
)
where:
a: maximum value of movement of yield surface
Y: yield stress
α ij : movement vector of yield surface
σ ij : stress vector
X ij : yield—surface kinematic—hardening amount after stress reversal
X eq : equivalent value of X ij
dε p eq : equivalent plastic—strain increment
C 0 , C C , A, n: material constants
ρ: variable such that ρ=1 or ½.
6. The manufacturing method according to claim 5 , wherein variables X ij , ρ, A, and n in the Equations (1) and (2) are represented by Equation (3), wherein Equation (3) is defined as follows:
If
σ
eq
≥
σ
eq
max
,
σ
eq
=
σ
eq
max
X
i
j
=
α
i
j
,
ρ
=
1
,
A
=
A
1
,
n
=
n
1
If
σ
eq
<
σ
eq
max
X
i
j
=
α
i
j
tmp
-
α
i
j
,
ρ
=
1
2
,
A
=
A
2
,
n
=
n
2
}
where
σ eqmax : maximum value of equivalent stress when isotropic hardening is assumed
α tmp ij back stress at the time of stress reversal
A 1 , A 2 , n 1 , n 2 : material constants.