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Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. Key LORENTZ
:
R5.03.16-C Page
: 1/14
Organization (S): EDF/AMA
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
Document: R5.03.16
Elastoplastic relation of behavior
with linear and isotropic kinematic work hardening
nonlinear. Plane modeling 3D and constraints
Summary:
This document describes an elastoplastic law of behavior to mixed, kinematic work hardening linear and
isotropic nonlinear. Equations to solve to integrate this relation of behavior numerically
are specified, as well as the coherent tangent matrix.
This behavior is usable for modelings of continuous mediums 3D, 2D (AXIS, C_PLAN, D_PLAN), and
for modelings DKT, COQUE_3D and TUYAU.
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Code_Aster ®
Version
5.7
Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. Key LORENTZ
:
R5.03.16-C Page
: 2/14
1 Introduction
When the way of loading is not monotonous any more, work hardenings isotropic and kinematic are not
more equivalent. In particular, one can expect to have simultaneously a kinematic share and one
isotropic share. If one seeks to precisely describe the effects of a cyclic loading, it is
desirable to adopt modelings sophisticated (but easy to use) such as the model of
Taheri, for example, cf [R5.03.05]. On the other hand, for less complex ways of loading,
one can wish to include only one linear kinematic work hardening, all nonthe linearities of
work hardening being carried by the isotropic term. That makes it possible to follow a curve precisely of
traction, while representing nevertheless phenomena such as the Bauschinger effect [bib1] (see
for example it [Figure 5-a]).
The characteristics of work hardening are then given by a traction diagram and a constant,
said of Prager, for the term of kinematic work hardening linear. They are introduced into
order DEFI_MATERIAU:
Linear isotropic work hardening
Nonlinear isotropic work hardening
DEFI_MATERIAU (
DEFI_MATERIAU
(
ECRO_LINE: (
TRACTION: (SIGM: curve of
SY: elastic limit
traction)
D_SIGM_EPSI
:
slope of the curve of PRAGER:
(C:
constant of Prager)
traction)
)
;
PRAGER: (C: constant of Prager)
);
These characteristics can also depend on the temperature, with the proviso of employing the words then
keys factors ECMI_LINE_FO and ECMI_TRAC_FO in the place of ECRO_LINE and TRACTION. The employment of
these laws of behavior is available in commands STAT_NON_LINE or
DYNA_NON_LINE:
Linear isotropic work hardening
Nonlinear isotropic work hardening
STAT_NON_LINE
STAT_NON_LINE
(
(
COMP_INCR
:
COMP_INCR
:
(
(
RELATION
:“VMIS_ECMI_LINE”
RELATION:“VMIS_ECMI_TRAC”
)
)
)
;
)
;
In the continuation of this document, one precisely describes the model of combined work hardening. One presents
then the detail of its numerical integration in link with the construction of the tangent matrix
coherent. Lastly, a tensile test uniaxial pressing illustrates the identification of the characteristics
material.
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Code_Aster ®
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Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. Key LORENTZ
:
R5.03.16-C Page
: 3/14
2
Description of the model
At any moment, the state of material is described by the deformation, the temperature T, the deformation
plastic p and the cumulated plastic deformation p. Les equations of state defines then in function
of these variables of state the constraint = H Id +
~ (broken up into parts hydrostatic and
deviatoric), the isotropic share of work hardening R and the kinematic X, so called share forced
of recall:
H 1
=
() =
(- HT) with HT = (ref.
tr
K tr
T-T
) Id éq 2-1
3
~
1
= - H Id = µ (~ - p
2
) ~
where = - tr () Id
éq
2-2
3
R = R (p)
éq 2-3
X
p
= C
éq 2-4
where K, µ, R
C
and are characteristics of material which can depend on the temperature.
More precisely, they are respectively the modules of compressibility and shearing, it
thermal dilation coefficient average (see [R4.08.01]), the isotropic function of work hardening and
constant of Prager. As for T ref., it is about the temperature of reference, for which
thermal deformation is null.
K, µ are connected to the Young modulus E and the Poisson's ratio by:
E
K = 3 + 2µ = 1 - 2
E
2µ = 1+
Note:
Concerning the kinematic share of work hardening [éq 2-4], one notes that it is linear in it
model. In addition, it is necessary to take guard with the fact that in certain references, one calls
constant of Prager 2C/3 and not C. In the same way, for the isotropic function of work hardening,
elastic limit is included there by R ()
0 = y, certain references treating it separately.
The evolution of the variables intern p and p is controlled by a normal law of flow associated
a criterion of plasticity F:
(
3 ~ ~
F, R, X) = (~ - X) - R
with
With
=
WITH A
eq
eq
éq
2-5
2
~
p
3
- X
& = & F
= &
éq 2-6
2 (~
- X) eq
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Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. Key LORENTZ
:
R5.03.16-C Page
: 4/14
&p = &
2
=
&p &p
éq 2-7
3
As for the plastic multiplier &
, it is obtained by the condition of following coherence:
if F<0 or &F < 0 & = 0
éq 2-8
if
F
= 0 and &F = 0 & 0
3
Integration of the relation of behavior
To numerically carry out the integration of the law of behavior, one carries out a discretization in
time and one adopts a diagram of implicit, famous Euler adapted for relations of behavior
elastoplastic. Henceforth, the following notations will be employed: Has, A E T A
represent
respectively values of a quantity A at the beginning and the end of the step of time considered thus that
its increment during the step. The problem is then the following: knowing the state at time T - thus
that increments of deformation and temperature T, to determine the state at time T like
constraints.
Initially, one takes into account the variations of the characteristics compared to
temperature by noticing that:
H
K
=
H + K tr (-
-
HT) éq
3-1
K
~
µ ~
=
- +
-
2µ (~
-
p) éq 3-2
µ
C
X
X
p
=
+
-
C
éq 3-3
C
Within sight of the equation [éq 3-1], one notes that the hydrostatic behavior is purely elastic.
Only the processing of the deviatoric component is delicate. To reduce the writings to come, one
introduced ~se the difference ~
- X in the absence of increment of plastic deformations, so that:
~
µ ~ - C -
~
- X =
-
X + 2µ -
p
-
(2µ +
-
C)
éq 3-4
µ
C
1444 2
4
3
4444
~se
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Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
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Author (S):
J.M. PROIX, E. Key LORENTZ
:
R5.03.16-C Page
: 5/14
The equations of flow [éq 2-6] and [éq 2-7] and the condition of coherence [éq 2-8] are written one
times discretized and by noticing that p =:
~
p
3
- X
= p
éq 3-5
2
(~ - X) eq
F 0 p 0 F p = 0
éq 3-6
The processing of the condition of coherence [éq 3-6] is traditional. One starts with a test
rubber band (p = 0) which is well the solution if the criterion of plasticity is not exceeded, i.e. if:
F = E - R (-
S
p
eq
) 0 éq 3-7
In the contrary case, the solution is plastic (p > 0) and the condition of coherence [éq 3-6] is reduced
with F = 0. To solve it, one starts by showing that one can bring back oneself to a scalar problem
by eliminating p. En effet, by taking account of [éq 3-4] and [éq 3-5], one notes that p is
colinéaire with ~se bus:
p
3
p
~e
p
=
2µ C
éq
3-8
2 (
S
~
-
+
- X) [
(
)]
eq
In addition, according to [éq 3-5], the standard of p is worth:
(
3
p) = p éq 3-9
eq
2
One thus deduces immediately the expression from it from p according to p:
~e
3
S
p = p
éq 3-10
2
seeq
It now only remains to replace p by its expression [éq 3-10] in the equation [éq 3-4]
3
(2µ + C) p
2
one obtains: ~
~
- X = S.E. 1
seeq
by deferring ~
- X in the equation F = 0, one is brought back to a scalar equation out of p to solve,
with knowknowing:
3
- (2µ + C) p
- R (p + p
eq
) = 0 éq
3-11
2
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Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
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Author (S):
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:
R5.03.16-C Page
: 6/14
Insofar as the function R is positive, which one will admit henceforth, there exists a solution
p with this equation, characterized by:
3
E
(
2
S
2µ +) + R (-
C
p
p + p
) =
where 0
eq
eq
< p
<
éq
3-12
2
3 2µ + C
Let us note that in the interval specified in [éq 3-12], the solution is single. For details as for
solution of this equation, one will refer to [R5.03.02].
The particular case of the plane constraints is studied with [§6].
4
Calculation of tangent rigidity
In order to allow a resolution of the total problem (equilibrium equations) by a method of
Newton, it is necessary to determine the coherent tangent matrix of the incremental problem. For
that, one once more adopts the convention of writing of the symmetrical tensors of command 2 pennies
form vectors with 6 components. Thus, for a tensor a:
T
= [axx ayy azz has
2axy
2axz
2ayz]
éq
4-1
If one introduces moreover the hydrostatic vector 1 and stamps it deviatoric projection P:
1 =t [1 1 1 0 0]
0
éq 4-2
1
P = Id - 1 1
éq 4-3
3
Then the matrix of coherent tangent rigidity is written for an elastic behavior:
= K 1 1 + 2Μ P
éq 4-4
and for a plastic behavior:
µ
3
p
~
~
2
E
E
p
1
S
S
= K 1 1 + 2µ1-
P 9µ
éq 4-5
+
-
E
3
E
E
seq
R
S
S
eq
eq
eq
(p) + (2µ + C)
2
The initial tangent matrix, used by option RIGI_MECA_TANG is obtained by adopting it
behavior of the preceding step (elastic or plastic, meant by internal variable being worth 0
or 1) and by taking p = 0 in the equation [éq 4-5].
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Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
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Author (S):
J.M. PROIX, E. Key LORENTZ
:
R5.03.16-C Page
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Note:
RIGI_MECA_TANG is the operator linearized compared to time (cf [R5.03.01], [R5.03.05]) and
to the problem of speed corresponds what is called; in this case, the linearization by
report/ratio with U, out of U = 0, provides the same expression.
One now proposes to show the expression [éq 4-5]. By differentiating them [éq 2-1] and [éq 2-2] with
fixed temperature, one obtains immediately:
= [K 11+ µ P] - µ p
2
2
éq 4-6
If the mode of behavior is plastic, the incremental law of flow [éq 3-10] provides then:
~e
~e
3
S
3
S
p = p
+
p
éq 4-7
2
2
eq
eq
As for p, it is obtained by differentiating the implicit equation [éq 3-12]:
3 (
2µ + C) + (p) p
R
éq 4-8
2
= eq
Lastly, it any more but does not remain to provide the variations of ~se:
~e
~e
~e
~e
~
1
= 2µ ~ seeq = µ S
3
~ S
2µ µ S
S
E
3
~
S
éq
4-9
E
E
E
E
eq
seq =
-
seq
S
S
eq
eq
While replacing then [éq 4-7], [éq 4-8] and [éq 4-9] in [éq 4-6], one obtains well the expression [éq 4-5].
This expression is formally identical to that defined in R5.03.02: [éq 4-3] and is written:
µ
3
~e
~e
p
1
p
1
S
S
K1 1 2µ
1
Id
1 1
9µ2
=
+
-
-
3
+
-
E
3
E
E
seq
R+ (2µ + C) S
S
eq
eq
eq
2
with = 1 if led to a plasticization, and = 0 if not.
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Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
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:
R5.03.16-C Page
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While using [éq 3-12], one finds:
2
Dev.
Dev.
* R
R
*
9µ
R. p
1
= 1 1 + 2µ Id -
1
H (p)
R (p)
3
R+ (2µ + C) R (p) R (p)
2
2µ G (p)
G (p)
with * = K -
2
2
3 H (
* =
p)
µ
µ H (p)
for the option
FULL_ MECA
Dev. = ~
-
: X
for option RIGI_ MECA_ TANG Dev. =
~ - - -
X
:
3 (2µ + C) p
with H (p) = 1+ 2
(
R p)
3
p
and
G (p) = 1+ C
2
(
R p)
5
Identification of the characteristics of material
Let us consider a tensile test uniaxial pressing, [Figure 5-a]. One proposes to show
how it makes it possible to identify the constant of Prager and the isotropic function of work hardening. In such
test, the various tensors are with fixed directions, i.e.:
2 3
~
p
3 p
= D
X = X D
= D
with D =
-
1 3
5-1
2
- 1
3
As long as the loading is monotonous, therefore in phase of traction, one obtains them immediately
following relations:
3
3
p
p
X
C p
T
C p
R (p
=
=
=
+) 5-2
2
2
The constant of Prager is determined by the first plasticization in compression, since one a:
T
3
p
p
WITH = CA + R
2
(A)
T
C
WITH + A
C =
3
p
éq
5-3
C
p
p
3
With =
A.C. - R
2
(A)
With
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Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. Key LORENTZ
:
R5.03.16-C Page
: 9/14
T (p)
your
p
3C 2 A
p
= - E
C
C (p)
With
Appear 5-a: Essai of tensile uniaxial pressing
The curve of work hardening T = (p
F
) from the traction diagram T = F is deduced () provided by
the user under key words ECRO_LINE ((SY and D_SIGM_EPSI (linear work hardening)) or
TRACTION (unspecified work hardening). It finally makes it possible to obtain the isotropic function of work hardening
by [éq 5-2]:
3
R (p) = T (p) -
p
C
.
2
For the effective calculation of R (p), according to the R5.03.02 document, one titrates party of linearity (ECMI_LINE)
or of the linearity per pieces of the traction diagram (ECMI_TRAC):
ECMI_LINE:
T
E.E
= F (p) =
T
y +
p
E - AND
E.E
3
R (p) =
T
y +
- C p = y + R. p
éq
5-4
E - AND 2
The equation [éq 3-12] becomes then:
3 (2µ +C) p +
E
.
éq
5-5
y + R (p +
p) = S
2
eq
ECMI_TRAC:
T
p
i+1 -
= F () =
I
I +
(p - IP), for IP p ip+1
I
p - 1 - I
p
éq 5-6
+1 -
3
+1 -
R (p)
I
I
I
=
I
I +
(p - IP) - CP = I -
I
p + R. p
I
p - 1 - I
p
2
I
p - 1 - I
p
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Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
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Author (S):
J.M. PROIX, E. Key LORENTZ
:
R5.03.16-C Page
: 10/14
Note:
For the use: the correspondence enters the model of behavior VMIS_CINE_LINE and it
behavior VMIS_ECMI_LINE is as follows:
For VMIS_CINE_LINE, it is necessary to introduce into DEFI_MATERIAU a linear work hardening of slope
And by:
D_SIGM_EPSI: And
With VMIS_ECMI_LINE, to reproduce same behavior with kinematic work hardening
linear, it is necessary to give in DEFI_MATERIAU.
· a linear work hardening of slope And: D_SIGM_EPSI: And
· The constant of Prager C: PRAGER: C
2 EE
C is determined by: C
T
=
3rd - AND
It should well be noticed that the identification of C and R (p
) have directions only in one field
deformations limited (small deformations). In particular, if T (p) presents an asymptote
tmax for p sufficiently large, then the kinematic contribution of work hardening does not have any more
significance. It is thus advised to restrict itself with the field where work hardening is strictly
positive.
6
Particular case of the plane constraints: calculation of p
It is necessary to add to the equations [éq 3-1] with [éq 3-4] the condition of plane constraints = 0, which
33
add an unknown factor (corresponding deformation):
H
K
=
H + K tr (-
-
HT) éq 6-1
K
~
µ ~
=
- +
-
2µ (~
-
p) éq 6-2
µ
C
X
X
p
=
+
-
C
éq 6-3
C
= 0
éq 6-4
33
Then, the equation [éq 3-4] becomes:
~
µ ~
C
-
-
~
- X =
C
p
y
E
p
y
-
- - X + 2
- (2 + C)
~
~
+ 2
= S - (2 + C)
~
µ
µ
µ
µ
µ
+ 2µ éq 6-5
C
where ~
C is entirely determined by the elastic behavior:
-
C
~ =
(~c + ~c), ~c = ~, ~c
33
11
22
11
11
22 = ~
-
22
1
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Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
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Author (S):
J.M. PROIX, E. Key LORENTZ
:
R5.03.16-C Page
: 11/14
0 0 0
and ~
y = 0 0 0 is unknown. It is also supposed that =
=
=
= 0.
13
23 13 23
0 0 Y
One always has:
~
p
3
- X
= p
éq 6-6
2
(~ - X) eq
F =
R p
0
p
0
p
0
éq
6-7
eq -
()
F
=
Elastic test:
· If
F = E - R (-
S
p
0
éq 6-8
eq
)
then
~
~
=, p = 0, Y = 0 éq 6-9
K
H =
-
tr
HT
- H + K
(C -
) éq
6-10
K
· If not, the solution is plastic: p > 0, Y 0. One can still bring back oneself to a problem
scalar in p.
By taking account of [éq 6-5] and [éq 6-6], one notes that ~
- X is colinéaire with ~
~
S.E. + 2µ
y bus:
3
(2µ +C) p
(~ - X) + 2
= (~ - X) H (p
) = [~
~
S.E.
1
+ 2µ y] éq 6-11
R (p)
Thus:
(~
E
4
-
X
H p
=
éq
6-12
33 +
Y
33
33)
() ~s
µ
3
We will only express the equation [éq 6-12] according to p. According to [éq 6-4]:
K
= 0 ~
~
=
+
=
+ K Y
-
, with H
H
C
=
tr
HT
-
+ K
-
éq
6-13
E
(
)
E
+.
33
33
33
K
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Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. Key LORENTZ
:
R5.03.16-C Page
: 12/14
While using [éq 6-5], [éq 6-6] and the incompressibility of the plastic deformations, one can show that:
~
C
S.E. +
= -
X -
éq 6-14
33
E
C
33
Then:
~
~
C
= S.E.
éq 6-15
33 - K.
Y +
-
33
X
-
C
33
As according to [éq 6-3]:
C
C
3
~
-
-
- X
X =
X
.
.
-
+ C
p =
X
-
+ C
p 33
33
éq
6-16
33
C
33
33
C
33
2
R (p)
C
3
~
3
p
X .G
-
33
=
G p
= 1+ C
-
+
, with ()
éq 6-17
33
(p)
X 33
C p
C
2
R (p)
2
R (p)
From [éq 6-12], [éq 6-15], [éq 6-17], one obtains an equation flexible p and Y:
4
H (p)
H
~
(p)
Y. µ + K
= S.E.
- 1 éq
6-18
3
G (p)
33
G
(p)
The equation [éq 6-11] makes it possible to obtain the scalar equation out of p to be solved, namely:
(~ -) H (p
) = R (-
X
p + p
) H (p
) = [~
~
S.E.
y
2
µ
éq
6-19
eq
+
] eq
Equation in which Y is related to p by the equation [éq 6-18].
As in the case of isotropic work hardening, one obtains a scalar equation out of p, always not
linear, which is solved by a method of secant.
Once p known, the calculation of ~
and X is carried out starting from the expression of Y, therefore of
entirely known, by a step similar to that of the equation [éq 3-10].
~e
~ y
~
p
3
S + 2
µ
3
- X
= p
p
éq
6-20
2
(
=
~e
S + 2
µ ~ y)
2
H (p) ~
~ y
µ
eq
(be +2) eq
~
µ ~
=
- +
-
2µ (~
-
p) éq 6-21
µ
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A
Code_Aster ®
Version
5.7
Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. Key LORENTZ
:
R5.03.16-C Page
: 13/14
One obtains by eliminating p from [éq 6-6], [éq 6-3] and [éq 6-2]:
~
µ ~
G p
-
~
() 3
p
C
=
-
2µ
2µ
X
- +
+
éq
6-22
µ
H (p) 2
R (p) H (p) -
C
3
p
µ
3
p
~
~
C
X = C
2
µ
1
C
éq 6-23
2
R (p) H (p
)
-
-
-
+
+ -
µ
2
R (p) H (p
)
X
-
C
7
Significance of the internal variables
The internal variables of the model at the points of Gauss (VARI_ELGA) are for all them
modelings:
· VARI_1 = p: cumulated plastic deformation (positive or null)
· VARI_2 =: being worth 1 if the point of Gauss plasticized during the increment or 0 if not.
X: tensor of recall:
For modeling 3D:
· VARI_3 = X
11
· VARI_4 = X
22
· VARI_5 = X
33
· VARI_6 = X
12
· VARI_7 = X
13
· VARI_8 = X
23
For modelings D_PLAN, C_PLAN, AXIS
· VARI_3 = X
11
· VARI_4 = X
22
· VARI_5 = X
33
· VARI_6 = X
12
For modelings of hulls (DKT, COQUE_3D), in local reference mark and each point of integration
of each layer:
· VARI_3 = X
11
· VARI_4 = X
22
· VARI_5 = X
33
· VARI_6 = X
12
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A
Code_Aster ®
Version
5.7
Titrate:
Linear and isotropic work hardening mixed kinematic nonlinear
Date:
30/12/02
Author (S):
J.M. PROIX, E. Key LORENTZ
:
R5.03.16-C Page
: 14/14
8 Bibliography
[1]
J. LEMAITRE, J.L. CHABOCHE: “Mechanical of solid materials”. Dunod 1992
Handbook of Référence
R5.03 booklet: Nonlinear mechanics
HT-66/02/004/A
Outline document