Code_Aster
®
Version
7.3
Titrate:
SSNP116 - Coupling creep/cracking - uniaxial Traction
Date
:
30/09/04
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.03.116-B
Page:
1/18
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/04/005/A
Organization (S):
EDF-R & D/AMA
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
V6.03.116 document
SSNP116 - Coupling creep/cracking - Traction
uniaxial
Summary:
This case of validation is intended to check the model of coupling of the laws of creep of Granger with the laws of
plasticity/cracking. Coupling, initially restricted with some laws of the environment not
linear of Code_Aster, could be wide thereafter with more laws. Parameters of the models of
plasticity/cracking is selected in a particular way to modelize an elastoplastic behavior quasi
perfect, and to bring back itself to a problem presenting a relatively simple analytical solution.
The geometry consists of three linear elements (cubic and prisms in 3D, squares and triangles in 2D), and
three quadratic elements, connected to the precedents by linear relations. Modelings tested here are
modelings
3D
,
C_PLAN
, and
D_PLAN
.
The loading is a uniaxial traction in imposed displacement.
One tests the coupling of the model of creep of Granger with
BETON_DOUBLE_DP
(or
NADAI_B
in
C_PLAN
),
VMIS_ISOT_LINE
and
CHABOCHE
. One has the analytical solution in
3D
and
C_PLAN
, when there is not
variation in the temperature and drying. In the cases
3D
and
D_PLAN
, the solutions are also tested
obtained when there is variation in the temperature and the drying and activation of the corresponding withdrawals (with
opposite effect). They are then tests of not-regression.
Code_Aster
®
Version
7.3
Titrate:
SSNP116 - Coupling creep/cracking - uniaxial Traction
Date
:
30/09/04
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.03.116-B
Page:
2/18
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/04/005/A
1
Problem of reference
1.1 Geometry
1.2
Material properties
The parameters of the laws of behavior are as follows:
For the mechanical characteristics in linear elasticity (
ELAS
):
Young modulus:
E = 31.000 MPa
Poisson's ratio:
= 0.2
Thermal expansion factor:
= 10
- 5
Coefficient of withdrawal of desiccation:
= 10
- 5
For the nonlinear mechanical characteristics of the model
BETON_DOUBLE_DP
:
Resistance in uniaxial pressing:
f' C = 40 NR/mm ²
Resistance in uniaxial traction:
f' T = 4 NR/mm ²
Report/ratio of resistances in compression
biaxial/uniaxial pressing:
= 1.16
Energy of rupture in compression:
Gc =10 Nm/mm ²
Energy of rupture in traction:
WP =10000 Nm/mm ²
to simulate a work hardening
quasi no one
Report/ratio of the limit elastic to resistance
in uniaxial pressing:
33.33%
For the nonlinear mechanical characteristics of the model
NADAI_B
:
Resistance in uniaxial pressing:
f' C = 40 NR/mm ²
Resistance in uniaxial traction:
f' T = 4 NR/mm ²
Initial threshold of elasticity:
= 1
Plastic deformation with the peak in compression:
Epsp_p_c = 9.64705
Plastic deformation with rupture in compression: Epsp_r_c = 5
Deformation with rupture in traction:
Epsi_r_t =5000 to simulate a work hardening quasi
no one
For the mechanical characteristics of the model with linear work hardening
VMIS_ISOT_LINE
:
Yield stress:
Sy = 4 NR/mm ²
Slope of work hardening:
D_sigm_epsi =0.1 NR/mm ²
Relations
linear
X
y
Ux
A1
B1
C1
D1
F1
E1
Code_Aster
®
Version
7.3
Titrate:
SSNP116 - Coupling creep/cracking - uniaxial Traction
Date
:
30/09/04
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.03.116-B
Page:
3/18
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/04/005/A
For the mechanical characteristics of the model of
CHABOCHE
:
Initial radius of plasticity:
R_0 = 4.0 NR/mm ²
Radius of plasticity ad infinitum:
R_I = 4.1 NR/mm ²
Coefficient of evolution of the radius of plasticity:
Gc =0.1 NR/mm ²
The parameters characterizing kinematic work hardening are null to simulate a work hardening quasi
no one: A1 = A2 = C1 = C2 = W = 0 and K = 1.
For the mechanical characteristics of the model of creep of
GRANGER
:
J1 coefficient:
J1 = 0.2 MPa
- 1
Coefficient
1:
1 = 4.320.000 S
Coefficient Q/R:
QsR_K = 0. K
The curve of desorption is worth 1 for all values of the hygroscopy, to simplify the solution
analytical.
1.3
Boundary conditions and loadings mechanical
For calculations in 3D:
·
Face in X = 0 of the first cube (its):
locked according to OX,
·
Nodes of the faces in y = 0:
locked according to OY,
·
Nodes of the faces in Z = 0:
locked according to OZ,
·
Linear relation (
LIAISON_DDL
) between the nodes end of the faces confused of
linear and quadratic elements adjacent (dependant nodes c1, c2, c3, c4 with the nodes d1,
d2, d3, d4),
·
Linear relation (
LIAISON_UNIF
) on the face sd to bind displacements according to X of
quadratic nodes of this vis-a-vis those of the nodes node,
·
Face in X = X
max
last cube (sf):
Traction exerted according to OX.
For calculations in 2D:
· Line in X = 0 of the first square (it):
locked according to OX,
· Nodes of the lines in y = 0:
locked according to OY,
·
Linear relation (
LIAISON_DDL
) between the nodes end of the lines confused of
linear and quadratic elements adjacent (dependant nodes c1, c2 with the nodes d1, d2),
·
Linear relation (
LIAISON_UNIF
) on the line ld to bind displacements according to X of
quadratic nodes of this line to those of the nodes node,
· Line in X = X
max
last square (lf):
Traction exerted according to OX.
The field of temperature is either constant (the first calculation), or crescent of 0°C with 20°C for all them
other calculations. If the temperature varies, it is supposed that the field of drying varies from 1
to 0. The characteristics material are constant. Moreover, one applies a coefficient of withdrawal of
desiccation not no one, in such way that the withdrawal of desiccation compensates for thermal dilation,
to check that these 2 phenomena are well taken into account.
Code_Aster
®
Version
7.3
Titrate:
SSNP116 - Coupling creep/cracking - uniaxial Traction
Date
:
30/09/04
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.03.116-B
Page:
4/18
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/04/005/A
2
Reference solution
2.1
Method of calculation used for the reference solution
To be able to calculate a simple analytical solution, the following choices were carried out, the objective being
to validate the coupling and not the laws of plasticity/cracking or creep:
·
a law of creep of Granger with only one model of Kelvin in series,
·
a law of plasticity/cracking modelizing a perfect elastoplastic law,
·
a loading of uniaxial traction.
The reference solution is calculated in an analytical way, knowing that in traction, only the criterion of
traction is activated. The equations of the model are brought back to scalar equations allowing
to calculate the analytical solution. The only difficulty comes from the determination of the beginning of plasticity
(moment and deformation of creep) which requires to solve by a numerical method an equation
nonlinear with an unknown factor.
If the temperature is not constant, creep is more complex to solve, the solution
analytical was not calculated. They are thus tests of not-regression. However, in the cases
3D
and
D_PLAN
, one can check that one obtains the same results with the 3 models.
The imposed deformation (displacement of an end of the structure) is a linear function of
time allowing to bring into play creep and plasticity.
2.2
Calculation of the reference solution
One notes
, component xx of the total deflection,
E
component xx of the deformation
rubber band,
fl
component xx of the deformation of creep of Granger, and
pl
component xx of
plastic deformation,
component xx of the stress, and
E
the Young modulus.
The model of creep selected comprises one model of Kelvin in series and the model of
plasticity/cracking is a nearly perfect law elastoplastic (quasi null slope of work hardening), which
allows to easily calculate the analytical solution of the coupling creep/plasticity, in the case of one
uniaxial simple traction. The nearly perfect elastoplastic law can be obtained starting from the laws of
Code_Aster
BETON_DOUBLE_DP
or
NADAI_B
,
VMIS_ISOT_LINE
or
CHABOCHE
, by choosing it
set of parameters which is appropriate (work hardening quasi no one).
The loading is a uniaxial traction in imposed displacement. A deformation is thus imposed
total proportional to passed time, of the form
T
xx
.
0
=
. Like there is no exerted effort
in the other directions, the stress field is uniaxial. One can thus bring back oneself to one
problem 1D for the resolution, which makes it possible to calculate in the second time of the deformations
in the transverse directions with the loading (yy and zz).
(
)
0
,
0
,
0
,
0
,
0
,
xx
=
and
(
)
0
,
0
,
0
,
,
,
zz
yy
xx
=
Code_Aster
®
Version
7.3
Titrate:
SSNP116 - Coupling creep/cracking - uniaxial Traction
Date
:
30/09/04
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.03.116-B
Page:
5/18
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/04/005/A
The equations of the model of creep and the model of plasticity merge with the equations
scalars following, by omitting index xx agent with the first component of the tensors:
=
0
.t
(imposed traction)
=
+
+
E
fl
pl
fl
fl
K
µ
+
= &
with
µ
=
S
S
J
and
K
J
S
= 1
= E
E
[
]
=
-
-
E
pl
fl
Resolution in linear elasticity
Before reaching the threshold of plasticity, the plastic deformation is null, which leads to:
=
0
.t
(imposed traction)
pl
= 0
=
+
E
fl
fl
fl
K
µ
+
= &
with
µ
=
S
S
J
and
K
J
S
= 1
= E
E
[
]
=
-
E
fl
One obtains the differential equation allowing to calculate the deformation of creep:
[
]
µ
=
-
=
+
E
K
fl
fl
fl
&
with
=
0
.t
The deformation of creep is thus expressed as the sum of a linear function of time and one
exponential function, of the type:
fl
T
T
T B has
E
()
.
.
=
+ +
-
who gives in the differential equation:
0
0
=
+
+
-
µ
. & ()
.
()
.
()
.
fl
fl
fl
T
K
T
E
T
E
T
That is to say:
(
)
[
]
(
)
[
]
(
)
[
]
0
0
=
+
+
+
+
-
+
+
-
-
K E B
has
K E has E
T
K E
E
T
µ
µ
.
.
.
.
from where:
has
E
K E
=
+
.
0
B
K E
E
K E
= - +
+
µ
.
0
µ
=
+
K E
At the initial moment, one starts from a null deformation of creep, which leads to:
µ
=
+
+
K E
E
K E
.
0
One obtains finally the expression of the deformation of creep according to time:
µ
µ
fl xx
fl
K E T
T
T
E
K E T
K E
E
()
()
.
=
=
+
-
+
-
- +
0
1
Code_Aster
®
Version
7.3
Titrate:
SSNP116 - Coupling creep/cracking - uniaxial Traction
Date
:
30/09/04
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.03.116-B
Page:
6/18
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/04/005/A
Component xx of the elastic strain is worth:
fl
E
-
=
That is to say:
(
)
-
+
+
+
=
=
+
-
T
E
K
E
xx
E
E
E
K
E
T
E
K
K
T
T
µ
µ
1
.
.
.
)
(
)
(
2
0
0
The components yy and zz of the elastic strain and creep are obtained by multiplication of
component xx by the Poisson's ratio.
Component xx of the stress is worth:
(
)
fl
E
E
E
-
=
=.
That is to say:
(
)
-
+
+
+
=
=
+
-
T
E
K
xx
E
E
K
E
T
E
K
E
K
T
T
µ
µ
1
.
.
.
.
)
(
)
(
2
2
0
0
Threshold of elasticity
The behavior remains elastic until one reaches limited of elasticity. In the case of one
uniaxial traction, the equivalent stress is equal to the nonnull component of the stress.
plasticity thus intervenes when
T
eq
xx
F
T
=
=
)
(
(resistance in traction), that is to say:
(
)
T
T
E
K
F
E
E
K
E
T
E
K
E
K
=
-
+
+
+
+
-
µ
µ
1
.
.
.
.
2
2
0
0
This equation, solved by a numerical method, makes it possible to obtain the moment of the beginning of
plasticization
T
plas
and deformation of creep at this moment:
-
+
-
+
=
=
+
-
plas
T
E
K
plas
plas
fl
plas
fl
E
E
K
T
E
K
E
T
µ
µ
1
.
)
(
0
Resolution in plasticity
The model of plasticity was selected in order to obtain a simple analytical resolution. It is about a law of
nearly perfect plasticity, obtained by taking a particular play of parameters for the model of
behavior leading to a quasi null slope of work hardening. Therefore, in plastic phase,
stress (component xx), equal to the equivalent stress is worth resistance in traction.
equations of the model are then:
=
0
.t
(imposed traction)
=
+
+
E
fl
pl
[
]
µ
µ
=
+
=
+
- -
&
&
fl
fl
fl
E
pl
K
K
with
µ
=
S
S
J
and
K
J
S
= 1
= E
E
[
]
=
-
-
=
E
F
pl
fl
T
with like initial conditions:
plas
T
T
=
plas
fl
plas
fl
T
=
)
(
what leads to the differential equation making it possible to calculate the deformation of creep:
fl
fl
T
K
F
µ
+
=
=
&
Code_Aster
®
Version
7.3
Titrate:
SSNP116 - Coupling creep/cracking - uniaxial Traction
Date
:
30/09/04
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.03.116-B
Page:
7/18
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/04/005/A
The deformation of creep is thus expressed in the form:
fl
T
T
has
E
()
.
= +
-
who gives in the differential equation:
[
]
[
]
0
=
+
-
=
-
+
-
-
µ
µ
.
& ()
.
()
.
.
.
.
fl
fl
T
T
T
T
K
T
F
K has
F
K
E
from where:
has
F
K
T
=
µ
= K
At the moment
T
plas
, the deformation of creep is worth
fl plas
, which leads to:
=
-
fl plas
T
T
F
K E
.
One obtains finally the expression of the deformation of creep according to time:
(
)
µ
fl xx
T
fl plas
T
K T T
T
F
K
F
K E
plas
()
=
+
-
-
-
with
=
0
.t
Component xx of the elastic strain is worth:
E
T
E
F
E
=
=
Component xx of the plastic deformation is worth:
pl
E
fl
E
fl
T
= - -
=
- -
0
.
That is to say:
(
)
µ
plas xx
T
T
fl plas
T
K T T
T
T
F
E
F
K
F
K E
plas
()
.
=
-
-
-
-
-
-
0
The components yy and zz of the elastic strain and creep are obtained by multiplication of
component xx by the Poisson's ratio.
Component xx of the stress is worth:
= F
T
Numerical application:
A deformation of 10 is imposed
3
in 100 seconds, which gives
0
= 10
5
The only difficulty consists in calculating the moment of plasticization
T
plas
, and deformation of creep
fl plas
who corresponds to him, by dichotomy for example.
One obtains finally the parameters:
T
plas
= 13.024296
fl plas
= 1.20969985.10
6
= 1.2903226.10
4
who allow to obtain the values of reference after plasticization of the concrete.
At 10 seconds, the behavior is a coupling creep/elasticity. At 100 seconds, the behavior
is a coupling creep/plasticity:
time 10 100
3.0778607 4.0
1.10
4
1.10
3
fl
7.1417140.10
7
1.7316168.10
5
E
9.9285829.10
5
1.2903226.10
4
pl
0.0
8.5365157.10
4
Code_Aster
®
Version
7.3
Titrate:
SSNP116 - Coupling creep/cracking - uniaxial Traction
Date
:
30/09/04
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.03.116-B
Page:
8/18
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/04/005/A
2.3
Uncertainty on the solution
It is negligible, about the precision machine.
2.4 References
bibliographical
The model was defined in the document of specification:
[1] CS
IF/311-1/420AL0/RAP/00.019
Version 1.1, “
Development of the coupling
creep/cracking in Code_Aster - Specifications “
Code_Aster
®
Version
7.3
Titrate:
SSNP116 - Coupling creep/cracking - uniaxial Traction
Date
:
30/09/04
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.03.116-B
Page:
9/18
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/04/005/A
3 Modeling
With
3.1
Characteristics of modeling
3D (1 HEXA8, 2 PENTA6, 1 HEXA20, 2 PENTA15)
They are a cube with 8 nodes and two prisms with 6 nodes bound by linear relations to a cube
with 20 nodes and of two prisms with 15 nodes. The unit is subjected to a uniaxial traction according to
direction X. Dimensions following y and Z are unit. Dimensions according to direction X are
chosen so that all the elements have the same characteristic length (the aforementioned is worth
cubic root of volume for the quadratic elements, and the cubic root of volume multiplied
by
2
for the linear elements).
The stress fields and deformations are uniform.
In 3D, one validates the coupling of the law
BETON_DOUBLE_DP
with the law
GRANGER_FP
. One tests also it
coupling of the laws
VMIS_ISOT_LINE
and
CHABOCHE
with the law
GRANGER_FP
.
3.2
Characteristics of the mesh
A number of nodes: 46
A number of meshs and type: 1 HEXA8, 2 PENTA6, 1 HEXA20, 2 PENTA15
3.3
Functionalities tested
Controls Options
“MECHANICAL” AFFE_MODELE
“3D”
DEFI_MATERIAU “BETON_DOUBLE_DP”
DEFI_MATERIAU “VMIS_ISOT_LINE”
DEFI_MATERIAU “CHABOCHE”
AFFE_CHAR_MECA “SECH_CALCULEE”
STAT_NON_LINE “RELATION” “KIT_DDI”
CALC_ELEM “OPTION” “EPSP_ELNO”
CALC_ELEM “OPTION” “EPGR_ELNO”
X
y
Z
Relations
linear
Elements
linear
Elements
quadratic
Ux
A1
B1
C1
D1
E1
F1
Code_Aster
®
Version
7.3
Titrate:
SSNP116 - Coupling creep/cracking - uniaxial Traction
Date
:
30/09/04
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.03.116-B
Page:
10/18
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/04/005/A
4
Results of modeling A
4.1 Values
tested
One tests components xx of the stress field
SIEF_ELNO_ELGA
, of the field of deformations
of creep
EPGR_ELNO
, and of the field of plastic deformation
EPSP_ELNO
.
For the coupling with the law
BETON_DOUBLE_DP
, if the temperature and drying are
constant and the solution analytical known, these values were tested at the C1 point located at the interface
between the linear elements and the quadratic elements, and at the F1 point located at the end of
structure, where the displacement imposed (in X is applied
max
).
When the temperature and drying vary, the analytical solution was not calculated: one tests
thus same components as previously but only at the F1 point located at the end of
the structure. The solution obtained with
BETON_DOUBLE_DP
is tested as a not-regression, but
the values obtained are used then as reference for the other models (
VMIS_ISOT_LINE
and
CHABOCHE
).
The tests are carried out at moment 10, when plasticity did not start, only creep is
present, and at moment 100, after the beginning of the plasticization of the concrete.
4.2
Calculation with the law
BETON_DOUBLE_DP
at a constant temperature (Reference)
Coupling
GRANGER_FP/BETON_DOUBLE_DP
·
at the C1 point
Identification Reference
Aster
% difference
xx
for
xx
10
4
3.07786 3.07787
1.9.10
- 4
fl
xx
for
xx
10
4
7.14171 10
- 7
7.140035 10
- 7
- 0.023
xx
for
xx
10
3
4.0 3.999999
- 2.0.10
- 5
fl
xx
for
xx
10
3
1.73162 10
- 5
1.731596 10
- 5
- 0.001
p
xx
for
xx
10
3
8.53652 10
- 4
8.536546 10
- 4
3.1 10
- 4
·
at the F1 point
Identification Reference
Aster
% difference
xx
for
xx
10
4
3.07786 3.07787
1.9.10
- 4
fl
xx
for
xx
10
4
7.14171 10
- 7
7.140035 10
- 7
- 0.023
xx
for
xx
10
3
4.0 3.999998
- 6.0.10
- 5
fl
xx
for
xx
10
3
1.73162 10
- 5
1.731596 10
- 5
- 0.001
p
xx
for
xx
10
3
8.53652 10
- 4
8.536023 10
- 4
- 0.006
Code_Aster
®
Version
7.3
Titrate:
SSNP116 - Coupling creep/cracking - uniaxial Traction
Date
:
30/09/04
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.03.116-B
Page:
11/18
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/04/005/A
4.3 Calculation with the law
BETON_DOUBLE_DP
in nonisotherm (Not
regression)
·
at the F1 point
Identification
Aster
xx
for
xx
10
4
3.077193
fl
xx
for
xx
10
4
7.357178 10
- 7
xx
for
xx
10
3
3.999998
fl
xx
for
xx
10
3
2.140534 10
- 5
p
xx
for
xx
10
3
8.495132 10
- 4
4.4
Calculation with the law
VMIS_ISOT_LINE
in nonisotherm
·
at the F1 point
Identification
Aster
xx
for
xx
10
4
3.077193
fl
xx
for
xx
10
4
7.357178 10
- 7
xx
for
xx
10
3
4.000009
fl
xx
for
xx
10
3
2.140537 10
- 5
p
xx
for
xx
10
3
8.495621 10
- 4
4.5
Calculation with the law
CHABOCHE
in nonisotherm
·
at the F1 point
Identification
Aster
xx
for
xx
10
4
3.077193
fl
xx
for
xx
10
4
7.357178 10
- 7
xx
for
xx
10
3
4.
fl
xx
for
xx
10
3
2.140535 10
- 5
p
xx
for
xx
10
3
8.495624 10
- 4
Code_Aster
®
Version
7.3
Titrate:
SSNP116 - Coupling creep/cracking - uniaxial Traction
Date
:
30/09/04
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.03.116-B
Page:
12/18
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/04/005/A
5 Modeling
B
5.1
Characteristics of modeling
D_PLAN (1 QUAD4, 2 TRI3, 1 QUAD8, 2 TRI6)
They are a square with 4 nodes and two triangles with 3 nodes bound by linear relations to a square
with 8 nodes and of two triangles with 6 nodes. The unit is subjected to a uniaxial traction according to
direction X. Dimensions following there are unit. Dimensions according to direction X are
chosen so that all the elements have the same characteristic length (root of
surface for the quadratic elements, and root of the surface multiplied by
2
for the elements
linear).
The stress fields and deformations are uniform.
In 2D plane deformations (D_PLAN), one tests the coupling between law
BETON_DOUBLE_DP
with
the law
GRANGER_FP
. One tests also the coupling of the laws
VMIS_ISOT_LINE
with the law
CHABOCHE
and
GRANGER_FP
. The analytical solution was not calculated in D_PLAN.
5.2
Characteristics of the mesh
A number of nodes: 20
A number of meshs and type: 1 QUAD4, 2 TRI3, 1 QUAD8, 2 TRI6
5.3
Functionalities tested
Controls Options
“MECHANICAL” AFFE_MODELE “D_PLAN”
DEFI_MATERIAU “BETON_DOUBLE_DP”
DEFI_MATERIAU “VMIS_ISOT_LINE”
DEFI_MATERIAU “CHABOCHE”
AFFE_CHAR_MECA “SECH_CALCULEE”
STAT_NON_LINE “RELATION”
“KIT_DDI”
CALC_ELEM “OPTION” “EPSP_ELNO”
CALC_ELEM “OPTION” “EPGR_ELNO”
Relations
linear
Elements
linear
Elements
quadratic
X
y
Ux
A1
B1
C1
D1
F1
E1
Code_Aster
®
Version
7.3
Titrate:
SSNP116 - Coupling creep/cracking - uniaxial Traction
Date
:
30/09/04
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.03.116-B
Page:
13/18
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/04/005/A
6
Results of modeling B
6.1 Values
tested
One tests components xx of the stress field
SIEF_ELNO_ELGA
and of the field of
deformations of creep
EPGR_ELNO
, and of the field of plastic deformation
EPSP_ELNO
at the F1 point
located at the end of the structure, where the displacement imposed (in X is applied
max
).
The analytical solution was not calculated in plane deformation. One thus carries out only the same one
calculation with the 3 models of cracking at temperature and variable drying. The tests are of type not
regression.
The tests are carried out at moment 10, when plasticity did not start, only creep is
present, and at moment 100, after the beginning of the plasticization of the concrete.
6.2 Calculation with the law
BETON_DOUBLE_DP
in nonisotherm (Not
regression)
At the C1 point
Identification
Aster
xx
for
xx
10
4
3.205409
fl
xx
for
xx
10
4
7.357178 10
- 7
xx
for
xx
10
3
4.382555
fl
xx
for
xx
10
3
2.307822 10
- 5
p
xx
for
xx
10
3
8.217046 10
- 4
At the F1 point
Identification
Aster
xx
for
xx
10
4
3.205409
fl
xx
for
xx
10
4
7.357178 10
- 7
xx
for
xx
10
3
4.382555
fl
xx
for
xx
10
3
2.307822 10
- 5
p
xx
for
xx
10
3
8.217039 10
- 4
Code_Aster
®
Version
7.3
Titrate:
SSNP116 - Coupling creep/cracking - uniaxial Traction
Date
:
30/09/04
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.03.116-B
Page:
14/18
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/04/005/A
6.3 Calculation with the law
VMIS_ISOT_LINE
in nonisotherm (Not
regression)
At the F1 point
Identification
Aster
xx
for
xx
10
4
3.205409
fl
xx
for
xx
10
4
7.357178 10
- 7
xx
for
xx
10
3
4.614209
fl
xx
for
xx
10
3
2.195353 10
- 5
p
xx
for
xx
10
3
8.429335 10
- 4
6.4
Calculation with the law
CHABOCHE
in nonisotherm (Not regression)
At the F1 point
Identification
Aster
xx
for
xx
10
4
3.205409
fl
xx
for
xx
10
4
7.357178 10
- 7
xx
for
xx
10
3
4.614199
fl
xx
for
xx
10
3
2.195350 10
- 5
p
xx
for
xx
10
3
8.429339 10
- 4
Code_Aster
®
Version
7.3
Titrate:
SSNP116 - Coupling creep/cracking - uniaxial Traction
Date
:
30/09/04
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.03.116-B
Page:
15/18
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/04/005/A
7 Modeling
C
7.1
Characteristics of modeling
C_PLAN (1 QUAD4, 2 TRI3, 1 QUAD8, 2 TRI6)
The stress fields and deformations are uniform.
In 2D C_PLAN, one validates the coupling of the law
NADAI_B
with the law
GRANGER_FP (
because
BETON_DOUBLE_DP
is not available in plane stresses)
.
One tests also the coupling of the laws
VMIS_ISOT_LINE
and
CHABOCHE
with the law
GRANGER_FP
.
7.2
Characteristics of the mesh
A number of nodes: 20
A number of meshs and type: 1 QUAD4, 2 TRI3, 1 QUAD8, 2 TRI6
7.3
Functionalities tested
Controls Options
“MECHANICAL” AFFE_MODELE “D_PLAN”
DEFI_MATERIAU “BETON_DOUBLE_DP”
DEFI_MATERIAU “VMIS_ISOT_LINE”
DEFI_MATERIAU “CHABOCHE”
AFFE_CHAR_MECA “SECH_CALCULEE”
STAT_NON_LINE “RELATION”
“KIT_DDI”
CALC_ELEM “OPTION” “EPSP_ELNO”
CALC_ELEM “OPTION” “EPGR_ELNO”
Relations
linear
Elements
linear
Elements
quadratic
X
y
Ux
A1
B1
C1
D1
F1
E1
Code_Aster
®
Version
7.3
Titrate:
SSNP116 - Coupling creep/cracking - uniaxial Traction
Date
:
30/09/04
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.03.116-B
Page:
16/18
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/04/005/A
8
Results of modeling C
8.1 Values
tested
Were tested components xx of the stress field
SIEF_ELNO_ELGA
and of the field of
deformations of creep
EPGR_ELNO
. For the coupling with the law
NADAI_B
, these values were tested
at the C1 point located at the interface enters the linear elements and the quadratic elements, and at the F1 point
located at the end of the structure, where the displacement imposed (in X is applied
max
).
For the coupling with the other laws, (
VMIS_ISOT_LINE
and
CHABOCHE
), the fields were tested
only at the F1 point located at the end of the structure.
As for the case 3D, one carries out the first calculation with temperature and constant drying, which allows
to validate
NADAI_B
in plane stresses. All the models are then tested with temperature and
drying variables and activation of the corresponding withdrawals. It is checked that the same ones well are found
results with
VMIS_ISOT_LINE
and
CHABOCHE
that with
NADAI_B
.
The tests are carried out at moment 10, when plasticity did not start, only creep is
present, and at moment 100, after the beginning of the plasticization of the concrete.
8.2
Calculation with the law
NADAI_B
at a constant temperature (Reference)
·
at the C1 point
Identification Reference
Aster
% difference
xx
for
xx
10
4
3.07786 3.07787
1.9.10
- 4
fl
xx
for
xx
10
4
7.14171 10
- 7
7.140035 10
- 7
- 0.023
xx
for
xx
10
3
4.0 3.999999
- 2.0.10
- 5
fl
xx
for
xx
10
3
1.73162 10
- 5
1.731597 10
- 5
- 0.001
·
at the F1 point
Identification Reference
Aster
% difference
xx
for
xx
10
4
3.07786 3.07787
1.9.10
- 4
fl
xx
for
xx
10
4
7.14171 10
- 7
7.140035 10
- 7
- 0.023
xx
for
xx
10
3
4.0 3.999999
- 2.3.10
- 5
fl
xx
for
xx
10
3
1.73162 10
- 5
1.731597 10
- 5
- 0.001
Code_Aster
®
Version
7.3
Titrate:
SSNP116 - Coupling creep/cracking - uniaxial Traction
Date
:
30/09/04
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.03.116-B
Page:
17/18
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/04/005/A
8.3
Calculation with the law
NADAI_B
in nonisotherm (Not regression)
At the F1 point
Identification
Aster
xx
for
xx
10
4
3.077192
fl
xx
for
xx
10
4
7.357178 10
- 7
xx
for
xx
10
3
3.999942
fl
xx
for
xx
10
3
2.140718 10
- 5
8.4
Calculation with the law
VMIS_ISOT_LINE
in nonisotherm
At the F1 point
Identification
Aster
xx
for
xx
10
4
3.077193
fl
xx
for
xx
10
4
7.357178 10
- 7
xx
for
xx
10
3
4.000009
fl
xx
for
xx
10
3
2.140745 10
- 5
8.5
Calculation with the law
CHABOCHE
in nonisotherm
At the F1 point
Identification
Aster
xx
for
xx
10
4
3.077192
fl
xx
for
xx
10
4
7.357178 10
- 7
xx
for
xx
10
3
4.000001
fl
xx
for
xx
10
3
2.140743 10
- 5
Code_Aster
®
Version
7.3
Titrate:
SSNP116 - Coupling creep/cracking - uniaxial Traction
Date
:
30/09/04
Author (S):
S. MICHEL-PONNELLE
Key
:
V6.03.116-B
Page:
18/18
Manual of Validation
V6.03 booklet: Nonlinear statics of the plane systems
HT-66/04/005/A
9
Summary of the results
If one knows the analytical solution (not-variation of the temperature and drying), it
case test offers very satisfactory results with a lower deviation than 0.02% for all the cases of
calculation. The iteration count for the plastic phase is generally about ten; for
the model of
CHABOCHE
, convergence is better with an elastic matrix. This is explained by
the choice of the law of nearly perfect plasticity, obtained with models
VMIS_ISOT_LINE
and
CHABOCHE
, and with particular plays of parameters. In fact, these same models used without
coupling creep/cracking, under the same conditions of loading and with the same ones
parameters, present the same difficulties of convergence.
It is checked that under the effect of the increase in the temperature the deformations of creep are
increased (+ approximately 3% in the elastic phase, +23% in the plastic phase).
Lastly, in
3D
and in
C_PLAN
, it is checked that the 3 models which were degenerated give many
quasi-similar results. On the other hand, in
D_PLAN
, the model
BETON_DOUBLE_DP
is not
equivalent with the two other models because of writing of the criterion which depends on the trace of the tensor
deformations and is thus not equivalent to the perfectly plastic model.