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SSNV156 - Column under voluminal loading
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E. LORENTZ
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Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/02/001/A
Organization (S):
EDF/AMA
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
Document: V6.04.156
SSNV156 - Column under voluminal loading.
Elastoplastic law has gradient
Summary:
In addition to the elastoplastic law with gradient which it tests only in its version with linear work hardening, this test has especially
for object to validate the algorithm of integration of the laws of behavior to gradient of internal variables. In
effect, the problem suggested, the setting in traction of a column under voluminal forces, led to a solution
nonhomogeneous which activates the various components of algorithm (Newton, linear search, BFGS) and
for which one can obtain an analytical expression. The results obtained are in agreement with the aforementioned, and
it with a high degree of accuracy.
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SSNV156 - Column under voluminal loading
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14/10/02
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E. LORENTZ
Key
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V6.04.156-A
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2/8
Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/02/001/A
1
Problem of reference
1.1 Geometry
The structure is a cylinder, with the most general direction, height
mm
2
=
L
and of which the form of
section does not influence the solution. So one will adopt a square section for modelings
in plane deformations and 3D (side
mm
1
,
0
=
has
) as well as a circular section for
axisymmetric modeling (radius
mm
1
,
0
=
R
).
1.2
Properties of material
The material follows an elastoplastic law of behavior to gradient whose work hardening is isotropic and
linear. Characteristics of material, respectively the Young modulus
E
, the coefficient of
Poisson
NAKED
, elastic limit
SY
, the slope of work hardening
D_SIGM_EPSI
and the length
characteristic
LONG_CARA
, are equal to:
MPa
000
100
=
E
3
,
0
=
MPa
100
=
y
MPa
000
10
=
T
E
mm
707
982
,
0
=
B
L
1.3
Boundary conditions and loading
Vertical displacements are locked on its higher face, horizontal displacements are
locked on the side faces and the lower face is free of any kinematic condition. By
elsewhere, the structure is subjected to a voluminal force vertical and directed to the bottom of intensity
increasing
T
F
T
F
0
)
(
=
where
3
0
NR/mm
1
=
F
and
T
is a parameter of loading (without unit).
Code_Aster
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SSNV156 - Column under voluminal loading
Date
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14/10/02
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E. LORENTZ
Key
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V6.04.156-A
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V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/02/001/A
2
Reference solution
2.1
Method of calculation of the reference solution
This problem admits an analytical solution. Sought sizes, namely the stresses
,
deformations
, plastic deformations
p
and cumulated plastic deformation
p
,
depend that level on loading
T
and of the dimension
Z
section considered, where
0
=
Z
indicate the lower face (free) cylinder and
L
Z
=
its higher face (locked).
Because of the blocking of the side faces, the sections do not become deformed horizontally, so that
the fields of displacements and deformations are written:
0
)
,
(
)
,
(
)
,
(
)
,
(
)
,
(
=
=
=
T
L
U
T
Z
T
Z
T
Z
U
T
Z
with
Z
Z
Z
E
E
E
U
éq 2.1-1
As for the tensor of stresses, it is diagonal. Its vertical component
is fixed by the equation
of balance while its horizontal components, identical in the two directions, depend
law of behavior (effect of fastening):
(
)
)
(
)
,
(
)
,
(
)
,
(
)
,
(
T
F
Z
T
Z
T
Z
T
Z
S
T
Z
=
+
+
=
with
Z
Z
y
y
X
X
E
E
E
E
E
E
éq
2.1-2
One can notice that the evolution of the diverter of the stresses is radial:
(
)
S
eq
eq
-
=
+
+
-
=
and
Z
Z
y
y
X
X
E
E
E
E
E
E
3
2
3
1
D
éq 2.1-3
Having proposed a field of displacements kinematically acceptable and a stress field
statically acceptable, it any more but does not remain to show than they are bound by the law of behavior. Well
that it is about a nonlocal model, the law of flow preserves its usual form:
(
)
+
+
-
=
=
Z
Z
y
y
X
X
p
p
E
E
E
E
E
E
2
1
2
3
D
p
p
eq
&
&
éq
2.1-4
The same applies to the relation stress-strain, where
and
µ
are the coefficients of Lamé which
result from the Young modulus and the Poisson's ratio:
()
(
)
p
Id
-
µ
+
=
2
tr
éq
2.1-5
While deferring [éq 2.1-1], [éq 2.1-2] and [éq 2.1-4] in [éq 2.1-5], one deduces the expression from it from
stresses and of the deformations according to the only cumulated plastic deformation:
p
E
F
Z
F
Z
p
E
S
-
-
+
-
-
=
+
-
=
1
1
1
2
1
2
1
1
2
éq 2.1-6
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V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/02/001/A
The condition of coherence then makes it possible to determine
p
. Insofar as evolution of the diverter
stresses is radial and monotonous, one can directly determine the current state without having with
to integrate the rate of plastic deformation. Indeed, one can distinguish two areas in the structure:
one,
B
Z
0
, in which the plastic deformation is null and where the threshold of plasticity is not
reached, and the other,
L
Z
B
, where the plastic deformation is nonnull and the threshold reached. The border
B
between these two areas is a new unknown factor of the problem. As follows:
=
=
0
F
0
0
L
Z
B
p
B
Z
éq
2.1-7
According to [bib1], the threshold of plasticity has as an expression:
()
13
4
,
F
2
B
T
T
y
eq
L
H
C
E
E
E
E
H
p
C
p
H
p
=
-
=
+
-
-
=
and
where
éq
2.1-8
Moreover, always according to [bib1], the field of cumulated plastic deformation
p
is
1
C
and of derivative
null at the edge of the structure. That implies in particular:
()
()
()
0
0
0
=
=
=
L
p
B
p
B
p
éq
2.1-9
Taking into account [éq 2.1-7] and [éq 2.1-8],
p
check a linear differential equation of the second
command on the field
L
Z
B
:
(
)
y
F
Z
Z
p
H
E
Z
p
C
-
-
-
=
+
-
+
1
2
1
)
(
1
2
)
(
éq
2.1-10
The data of the 3 boundary conditions [éq 2.1-9] then makes it possible to determine completely
p
like
the position
B
free border.
To reduce the expressions, the following notations are introduced:
(
)
(
)
-
=
-
=
-
=
-
-
=
=
=
+
-
=
C
C
y
B
C
L
L
Z
Z
L
L
Z
Z
H
Z
Z
F
H
H
H
L
H
C
L
H
E
H
sinh
)
S (
cosh
)
C (
)
(
1
2
1
13
4
1
2
éq 2.1-11
Then, after some calculations, one obtains the following expression for
p
:
-
=
-
=
+
+
=
C
C
L
B
B
B
B
L
With
Z
B
Z
With
Z
Z
p
)
C (
)
(
)
S (
where
)
S (
)
C (
)
(
)
(
éq 2.1-12
As for the free border, it is given according to the level of loading by the equation
following (or conversely, the level of loading corresponding to a certain position of
free border):
(
)
)
C (
1
)
S (
)
S (
B
L
B
B
B
H
C
y
-
+
=
éq
2.1-13
Code_Aster
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Titrate:
SSNV156 - Column under voluminal loading
Date
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V6.04.156-A
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V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/02/001/A
2.2
Results of reference
The cumulated plastic deformation is examined
p
, total deflection
, the equivalent stress
eq
and the horizontal stress
S
at the point
L
Z
=
for various levels of loading which
correspond to various positions of the free border.
L
B
(
)
3
NR/mm
F
p
(
)
MPa
eq
(
)
MPa
S
0, 75
104, 811 963
1, 165.975 E-4 1, 623.833 E-3 111, 456 702
98, 167 224
0, 50
146, 159 407
6, 125.415 E-4 2, 521.534 E-3 123, 286 355
169,032 459
0, 25
250, 078 993
1, 905.213 E-3 4, 804.152 E-3 149, 717 896
350, 440 090
0, 00
875, 079 453
9, 693.407 E-3 1, 854.027 E-2 307, 704.531 1.442, 454 356
2.3
Uncertainties on the solution
It is about an analytical solution.
2.4 References
bibliographical
[1]
Lorentz E., Andrieux S.: With variational formulation for nonlocal ramming models. Int. J. Plas.,
15, pp. 119-138 (1999)
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V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/02/001/A
3 Modeling
With
3.1
Characteristics of modeling
It is about a modeling 3D. The section of the cylinder is square. Linear work hardening is characterized
initially by a curve of work hardening (
VMIS_ISOT_TRAC
) then by its limit
of elasticity and its module (
VMIS_ISOT_LINE
), which makes it possible to test the two establishments.
3.2
Characteristics of the mesh
The mesh is carried out by GMSH. The meshs are of smaller size in the area where one examines
the results (higher face). On the whole, 1749 TETRA10 are counted.
3.3 Functionalities
tested
Controls
DEFI_MATERIAU
NON_LOCAL
LONG_CARA
AFFE_MODELE
AFFE
MODELING = “3d_GRADIENT”
STAT_NON_LINE
LAGR_NON_LOCAL
4
Results of modeling A
4.1 Values
tested
The various values are tested with the dimension
L
Z
=
, as presented to [§2.2]. It is about
cumulated plastic deformation (component `
V1'
CHAM_NO
`
VARI_NOEU_ELGA
'and component
`
VANL'
CHAM_NO
`
DEPL
'), of the vertical deformation (component `
EPZZ
'of
CHAM_NO
`
EPSI_NOEU_DEPL
'), of the stress of von Mises (component `
VMIS
'of
CHAM_NO
`
EQUI_NOEU_SIGM
') and finally of the horizontal stress (component `
SIXX
'of
CHAM_NO
`
SIEF_NOEU_DEPL
'). For recall, the analytical values are as follows.
(
)
3
NR/mm
F
p
(
)
MPa
eq
(
)
MPa
S
104, 811 963
1, 165.975 E-4
1, 623.833 E-3
111, 456 702
98, 167 224
146, 159 407
6, 125.415 E-4
2, 521.534 E-3
123, 286 355
169,032 459
250, 078 993
1, 905.213 E-3
4, 804.152 E-3
149, 717 896
350, 440 090
875, 079 453
9, 693.407 E-3
1, 854.027 E-2
307, 704 531
1.442, 454 356
The results obtained differ extremely little from the analytical solution (lower relative difference
with the precision owing to lack of 0, 1%).
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SSNV156 - Column under voluminal loading
Date
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14/10/02
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V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/02/001/A
5 Modeling
B
5.1
Characteristics of modeling
It is about a modeling 3D. The section of the cylinder is square. Linear work hardening is characterized
initially by a curve of work hardening (
VMIS_ISOT_TRAC
) then by its limit
of elasticity and its module (
VMIS_ISOT_LINE
), which makes it possible to test the two establishments.
5.2
Characteristics of the mesh
The mesh is carried out by GMSH. The meshs are of smaller size in the area where one examines
the results (higher face). On the whole, 169 TRIA6 are counted.
5.3 Functionalities
tested
Controls
DEFI_MATERIAU
NON_LOCAL
LONG_CARA
AFFE_MODELE
AFFE
MODELING = “D_PLAN_GRADIENT”
STAT_NON_LINE
LAGR_NON_LOCAL
6
Results of modeling B
6.1 Values
tested
The various values are tested with the dimension
L
Z
=
, as presented to [§2.2]. It is about
cumulated plastic deformation (component `
V1'
CHAM_NO
`
VARI_NOEU_ELGA
'and component
`
VANL
'of
CHAM_NO
`
DEPL
'), of the vertical deformation (component `
EPYY
'of
CHAM_NO
`
EPSI_NOEU_DEPL
'), of the stress of von Mises (component `
VMIS
'of
CHAM_NO
`
EQUI_NOEU_SIGM
') and finally of the horizontal stress (component `
SIXX
'of
CHAM_NO
`
SIEF_NOEU_DEPL
'). For recall, the analytical values are as follows.
(
)
3
NR/mm
F
p
(
)
MPa
eq
(
)
MPa
S
104, 811 963
1, 165.975 E-4
1, 623.833 E-3
111, 456 702
98, 167 224
146, 159 407
6, 125.415 E-4
2, 521.534 E-3
123, 286 355
169,032 459
250, 078 993
1, 905.213 E-3
4, 804.152 E-3
149, 717 896
350, 440 090
875, 079 453
9, 693.407 E-3
1, 854.027 E-2
307, 704 531
1.442, 454 356
The results obtained differ extremely little from the analytical solution (lower relative difference
with the precision owing to lack of 0, 1%).
Code_Aster
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Titrate:
SSNV156 - Column under voluminal loading
Date
:
14/10/02
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E. LORENTZ
Key
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Manual of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
HT-66/02/001/A
7 Modeling
C
7.1
Characteristics of modeling
It is about an axisymmetric modeling 2D. The section of the cylinder is necessarily circular.
Linear work hardening is characterized initially by a curve of work hardening
(
VMIS_ISOT_TRAC
) then by its elastic limit and its module (
VMIS_ISOT_LINE
), which allows
to test the two establishments.
7.2
Characteristics of the mesh
The mesh is carried out by GMSH. With the difference in two preceding modelings, it is about one
regulated mesh. The meshs are smaller sizes in the area where one examines the results (face
higher). On the whole, 50 QUAD8 are counted.
7.3 Functionalities
tested
Controls
DEFI_MATERIAU
NON_LOCAL
LONG_CARA
AFFE_MODELE
AFFE
MODELING = “AXIS_GRADIENT”
STAT_NON_LINE
LAGR_NON_LOCAL
8
Results of modeling C
8.1 Values
tested
The various values are tested with the dimension
L
Z
=
, as presented to [§2.2]. It is about
cumulated plastic deformation (component `
V1'
CHAM_NO
`
VARI_NOEU_ELGA
'and component
`
VANL
'of
CHAM_NO
`
DEPL
'), of the vertical deformation (component `
EPYY
'of
CHAM_NO
`
EPSI_NOEU_DEPL
'), of the stress of von Mises (component `
VMIS
'of
CHAM_NO
`
EQUI_NOEU_SIGM
') and finally of the horizontal stress (component `
SIXX
'of
CHAM_NO
`
SIEF_NOEU_DEPL
'). For recall, the analytical values are as follows.
(
)
3
NR/mm
F
p
(
)
MPa
eq
(
)
MPa
S
104, 811 963
1, 165.975 E-4
1, 623.833 E-3
111, 456 702
98, 167 224
146, 159 407
6, 125.415 E-4
2, 521.534 E-3
123, 286 355
169,032 459
250, 078 993
1, 905.213 E-3
4, 804.152 E-3
149, 717 896
350, 440 090
875, 079 453
9, 693.407 E-3
1, 854.027 E-2
307, 704 531
1.442, 454 356
The results obtained differ extremely little from the analytical solution (lower relative difference
with the precision owing to lack of 0, 1%).
9
Summary of the results
Because of the positive character of work hardening, the problem exhibe not of instabilities, which confirms
good convergence of calculations. The validation relates thus to the law of behavior itself and
on the algorithm of integration of the nonlocal laws, from which the various components are activated
(primal and dual iterations observed). The remarkable agreement enters the values of reference
and those calculated shows the capacities of the algorithm when a fine precision is necessary (criteria
of convergence severe) while the size of the dealt with problems, particularly in 3D, seems
to prove its robustness.