Code_Aster ®
Version
7.4
Titrate:
WTNV100 - Triaxial Essai not drained with model CJS (level 1)
Date:
05/07/05
Author (S):
C. CHAVANT, pH. Key AUBERT
:
V7.31.100-B Page:
1/14
Organization (S): EDF-R & D/AMA, CNEPE
Handbook of Validation
V7.31 booklet: Mechanical hydro Thermo in porous environment of voluminal structures
Document: V7.31.100
WTNV100 - Triaxial Essai not drained with the model
CJS (level 1)
Summary
This test makes it possible to validate level 1 of model CJS. It is about a triaxial compression test in not drained condition.
In the first two modelings, calculations are carried out only on the solid part of the ground,
the aspect not drained being modelled by a null voluminal deformation of the skeleton, they are modelings
3D which differs one from the other only by the grid.
In the third modeling, the hydraulic coupling is taken into account, the sample is completely saturated,
the skeleton and the fluid are supposed to be incompressible.
By reason of symmetry, one is interested only in the eighth of a sample subjected to a triaxial compression test.
The level of containment is of 100 kPa.
The results obtained with model CJS1 are compared with an analytical solution.
Handbook of Validation
V7.31 booklet: Mechanical hydro Thermo in porous environment of voluminal structures
HT-66/05/005/A
Code_Aster ®
Version
7.4
Titrate:
WTNV100 - Triaxial Essai not drained with model CJS (level 1)
Date:
05/07/05
Author (S):
C. CHAVANT, pH. Key AUBERT
:
V7.31.100-B Page:
2/14
1
Problem of reference
1.1 Geometry
Z
E
height:
H = 1 m
width:
L = 1 m
thickness: E = 1 m
C
H
With
B
y
X
L
Co-ordinates of the points (in meters):
With
B
C
X 0. 0.
0.5
y 0. 1.
0.5
Z 0. 0.
0.5
1.2
Material property
E = 22,4 103 kPa
= 0,3
Coefficient of biot b: 1
Water is supposed to be incompressible: UN_SUR_K = 0
Parameters CJS1: = - 0,03
= 0,82
Rm = 0,289
Pa = - 100 kPa
1.3
Initial conditions, boundary conditions, and loading
1.3.1 Pure mechanical modeling
Phase 1:
One brings the sample in a homogeneous state: 0
0
0
xx = yy = zz, by imposing the pressure of
containment corresponding on the front, side straight line and higher faces. Displacements are
blocked on the faces postpones (ux = 0), side left (uy = 0) and lower (uz = 0).
Phase 2:
One maintains displacements blocked on the faces postpones (ux = 0), side left (uy = 0) and
lower (uz = 0). One applies a displacement imposed to the higher face: U (T
Z
), in order to
to obtain a deformation zz = - 20% (counted starting from the beginning of phase 2). On the front faces
and side straight line, one imposes respectively displacements U (T
X
) and U (T
y
), in order to have one
null voluminal deformation for the sample, i.e. finally that one imposes
zz
xx = yy = -
. It is the manner of reproducing the behavior of the solid phase during a test
2
triaxial not drained.
Handbook of Validation
V7.31 booklet: Mechanical hydro Thermo in porous environment of voluminal structures
HT-66/05/005/A
Code_Aster ®
Version
7.4
Titrate:
WTNV100 - Triaxial Essai not drained with model CJS (level 1)
Date:
05/07/05
Author (S):
C. CHAVANT, pH. Key AUBERT
:
V7.31.100-B Page:
3/14
1.3.2 Modeling coupled with hydraulics
Phase 1:
One brings the sample in a homogeneous state of effective stresses: 0
0
0
xx = yy = zz, while imposing
corresponding total pressure on the front, side straight line and higher faces and while imposing
everywhere null water pressures. Displacements are blocked on the faces postpones (ux = 0),
side left (U y = 0) and lower (uz = 0).
Phase 2:
One maintains displacements blocked on the faces postpones (ux = 0), side left (uy = 0) and
lower (uz = 0).
On all the faces, hydraulic flows are null.
One applies a displacement forced to the higher face in order to obtain a deformation
zz = - 20% (counted starting from the beginning of phase 2). On the front faces and side straight line, one
impose boundary conditions in total constraint:
.n = 0 (= 100kPa)
Handbook of Validation
V7.31 booklet: Mechanical hydro Thermo in porous environment of voluminal structures
HT-66/05/005/A
Code_Aster ®
Version
7.4
Titrate:
WTNV100 - Triaxial Essai not drained with model CJS (level 1)
Date:
05/07/05
Author (S):
C. CHAVANT, pH. Key AUBERT
:
V7.31.100-B Page:
4/14
2
Reference solution
2.1
Reference solution for the water pressure into linear
0, 0 p0 indicating the constraints, water deformations and pressures obtained in the phase one a:
- 0 = tr (- 0) + 2µ (- 0) - (
B p - 0
p)
m
p - p
=
0 + btr
fl
(- 0)
M
1
In this writing, M indicates the module of biot and NR =
.
M
The boundary conditions of null flow and the conservation of the water mass give m = 0
Boundary conditions on the side walls and the fact that the state of stress is homogeneous
give:
xx - xx = 0
0
One has thus finally to solve the two equations:
(2
xx +
zz) + 2µ
xx = LP
(
p
B 2
xx +
zz) = -
= - Np
M
And one obtains:
2
zz
B + NR
xx = -
2 b2 + (+ µ) NR
µ
B zz
p = -
b2
+ (+ µ) NR
In our case,
zz
xx = -
;
p = -
µ zz
2
2.2
Development of analytical solution CJS
One has permanently:
zz
for the deformations: xx = yy = -
2
for the constraints: xx = yy
Elastic phase:
While writing the elastic law simply, it comes:
0
xx = xx - µ zz
0
zz = zz + 2 µ zz
Handbook of Validation
V7.31 booklet: Mechanical hydro Thermo in porous environment of voluminal structures
HT-66/05/005/A
Code_Aster ®
Version
7.4
Titrate:
WTNV100 - Triaxial Essai not drained with model CJS (level 1)
Date:
05/07/05
Author (S):
C. CHAVANT, pH. Key AUBERT
:
V7.31.100-B Page:
5/14
In addition, it is also known that during this I1 phase (= trace ()) remain constant bus v = 0. One in
deduced for the components from the diverter:
I
I 0
I
I 0
S
1
0
1
1
0
1
xx = xx -
= xx -
- µ zz = - µ zz and S = -
= -
+ 2 µ = 2 µ
3
3
zz
zz
zz
zz
zz
3
3
that is to say: software houses = - 6 µ zz and det (S) = 2 3 3
µ zz
1/6
Consequently: (
H = 1 -
S)
()
Thus when one reaches the criterion F D = 0, one a:
S (1) 1/6 R I 0 = - 6
1/6
0
-
+
1
µ (1 -) + R I
II
m
zz
m
= 0
1
I.e. the transition enters the states rubber band and perfectly plastic is done for one
axial deformation equalizes with:
0
trans
m
R I1
zz
=
µ (-) 1 6
6
1
/
The corresponding state of stresses is noted:
R I 0
R I 0
trans
0
m 1
trans
0
m 1
xx
= xx - µ
and zz
= zz + 2 µ
µ (-) 1 6
6
1
/
6 µ (1 -) 1/6
Plastic phase:
One notes s-d the diverter of the reverse of the tensor S
Generally, there are the following sizes:
1
1
- 3
- 3
S
= - (-) = S
-
- D
xx
zz
xx
yy S
=
S
=
3
xx
xx
zz -
xx
2 (zz - xx)
2
3
S
- 1
3
- D
zz =
(zz - xx) S =
S
=
3
zz
2 (
zz
-
zz - xx)
zz
xx
2
2
3
that is to say: software house = -
(zz - xx) and det (S) = (zz - xx)
3
33
1/6
Consequently: (
H S) = (1 -)
One deduces some:
1
2
Q
1/6
1/6
xx =
(1 -) and Qzz = - (1 -)
6
3
F D
1
D
1/6
F
2
1/6
moreover:
=
(1 -) +
= -
1 -
+
Rm and
(
)
Rm
xx
6
zz
3
S
R
Like one a: =
(
sign S
II
m
ij I
& J)
-
1
1 =
scII =
-
Rc
Handbook of Validation
V7.31 booklet: Mechanical hydro Thermo in porous environment of voluminal structures
HT-66/05/005/A
Code_Aster ®
Version
7.4
Titrate:
WTNV100 - Triaxial Essai not drained with model CJS (level 1)
Date:
05/07/05
Author (S):
C. CHAVANT, pH. Key AUBERT
:
V7.31.100-B Page:
6/14
then tensor N is written:
1
1
1
2
N
xx =
+
1 and N
-
+1
2
zz =
+ 6
3
2
+
3
3
It comes then for G D:
1/6
1
1/6
1
+ 3
D
m 1
xx =
(1 -)
()
R
G
+ Rm -
+
1
6
2
+ 3
6
1/6
2
1/6
1
+ 3
2
D
m
1
zz = -
(-)
()
R
G
+ Rm -
-
+1
3
2
+ 3
3
One also has according to the elastic law:
trans
trans
xx = xx
+
xx and zz = zz +
zz
where:
D
D
D
D
D
D
D
D
D
xx = 2 µ (xx - Gxx) + (v - tr (G) = - µ zz - 2 µ Gxx - (2 Gxx + Gzz)
D
D
D
D
D
D
D
D
D
zz = 2 µ (zz - Gzz) + (v - tr (G) = 2 µ zz - 2 µ Gzz - (2 Gxx + Gzz)
and with:
trans
trans
xx = xx - xx
and zz = zz - zz
maybe, according to what precedes, one has for software house:
2
S
trans
trans
trans
D
D
D
II = -
3 µ
2 µ
3 [(zz
- xx) +
(zz - zz) -
(G - G
zz
xx)]
2
= strans
trans
D
D
D
II
-
3 µ
2 µ
3 [
(zz - zz) -
(G - G
zz
xx)]
and for I1:
I = I trans - (+ µ)
D
(Gd +Gd
1
1
3
2
2 xx
zz)
One deduces from it that the function of load déviatoire is written:
1/6
2
F D = strans
trans
D
D
D
1/6
II
(1 -) -
3 µ
2 µ
1
3 [
(zz - zz) -
(G - G
zz
xx)] (-)
+R I trans
1
- R
D
D
D
m
m (3 + 2 µ)
(2 G +G
xx
zz)
By taking account of the fact that F D (trans
) =0, one finds then for the plastic multiplier:
3 µ 1 - 1/6
D
()
=
(trans
zz - zz
)
3
2 µ (D
D
D
D
Gzz - Gxx) -
Rm (3 + 2 µ) (2 Gxx + Gzz)
2
Handbook of Validation
V7.31 booklet: Mechanical hydro Thermo in porous environment of voluminal structures
HT-66/05/005/A
Code_Aster ®
Version
7.4
Titrate:
WTNV100 - Triaxial Essai not drained with model CJS (level 1)
Date:
05/07/05
Author (S):
C. CHAVANT, pH. Key AUBERT
:
V7.31.100-B Page:
7/14
what gives with the formulas of G D
D
xx and Gzz the preceding ones:
2 µ (1 -) 1/6 (2 +) 3
D =
3
(
trans
zz - zz
R
1/6
1/6
m - (1 -)
) (2µ (1) - (3+2µ) Rm) (
)
One concludes from it finally the analytical expression from the constraints:
While posing:
has = (1 -) 1/6;
B = (2
+) 3
One a:
trans
xx - xx
=
2
1
has + 3
1
R
- has
m
R
(m)
µ has B 2 µ
has +
1
3
m
R -
+ +
3
6
B
6
B
- µ
trans
+
-
(
zz zz
m
R -) (2 µa has - (3 + 2 µ) m
R)
(
)
trans
zz - zz
=
2
2
+ 3 R have
2
R
- has
m
(m)
µ has B 2 µ
-
+ R has
1 3
m -
-
+ +
3
3
B
3
B
2 µ
trans
-
-
(
zz zz
Rm - has) (2 µa - (3 + 2 µ) Rm)
(
)
2.3
Results of reference
Constraints xx, yy and zz at points A, B and C.
2.4
Uncertainty on the solution
Exact analytical solution for CJS1.
Handbook of Validation
V7.31 booklet: Mechanical hydro Thermo in porous environment of voluminal structures
HT-66/05/005/A
Code_Aster ®
Version
7.4
Titrate:
WTNV100 - Triaxial Essai not drained with model CJS (level 1)
Date:
05/07/05
Author (S):
C. CHAVANT, pH. Key AUBERT
:
V7.31.100-B Page:
8/14
3 Modeling
With
3.1
Characteristics of modeling
3D:
Z
B
y
X
Cutting: 1 in height, in width and thickness.
Loading of phase 1:
Confining pressure: 0
0
0
xx = yy = zz: 100 kPa.
Level 1 of model CJS
3.2
Characteristic of the grid
A number of nodes: 8
A number of meshs and types: 1 HEXA8 and 6 QUA4
3.3 Functionalities
tested
Commands
DEFI_MATERIAU CJS
STAT_NON_LINE COMP_INCR
RELATION
“CJS”
Handbook of Validation
V7.31 booklet: Mechanical hydro Thermo in porous environment of voluminal structures
HT-66/05/005/A
Code_Aster ®
Version
7.4
Titrate:
WTNV100 - Triaxial Essai not drained with model CJS (level 1)
Date:
05/07/05
Author (S):
C. CHAVANT, pH. Key AUBERT
:
V7.31.100-B Page:
9/14
4
Results of modeling A
4.1 Values
tested
Localization Number Deformation
Constraint
Reference
Aster %
difference
of command axial zz (%)
(kPa)
Not A and B
1
0.25
xx
78.461538 78.461538 <
105
2
0.50
xx
56.923077 56.923077 <
105
3
0.75
xx
53.606 53.606 <
105
4
1.0
xx
54.480 54.480 <
105
8
5.0
xx
68.467 68.467 <
105
23
20.0
xx
120.918 120.918
<
105
1
0.25
yy
78.461538 78.461538 <
105
2
0.50
yy
56.923077 56.923077 <
105
3
0.75
yy
53.606 53.606 <
105
4
1.0
yy
54.480 54.480 <
105
8
5.0
yy
68.467 68.467 <
105
23
20.0
yy
120.918 120.918
<
105
1
0.25
zz
143,07692 143,07692
<
105
2
0.50
zz
186.153846 186.153846
<
105
3
0.75
zz
196.818 196.818
<
105
4
1.0
zz
200.028 200.028
<
105
8
5.0
zz
251.383 251.383
<
105
23
20.0
zz
443.961 443.961
<
105
Handbook of Validation
V7.31 booklet: Mechanical hydro Thermo in porous environment of voluminal structures
HT-66/05/005/A
Code_Aster ®
Version
7.4
Titrate:
WTNV100 - Triaxial Essai not drained with model CJS (level 1)
Date:
05/07/05
Author (S):
C. CHAVANT, pH. Key AUBERT
:
V7.31.100-B Page:
10/14
5 Modeling
B
5.1
Characteristics of modeling
This modeling differs from the preceding one by the smoothness of the grid
3D:
Z
B
y
X
Cutting: 2 in height, in width and thickness.
Loading of phase 1:
Confining pressure: 0
0
0
xx = yy = zz: 100 kPa.
Level 1 of model CJS
5.2
Characteristic of the grid
A number of nodes: 27
A number of meshs and types: 8 HEXA8 and 24 QUA4
5.3 Functionalities
tested
Commands
DEFI_MATERIAU CJS
STAT_NON_LINE COMP_INCR
RELATION
“CJS”
Handbook of Validation
V7.31 booklet: Mechanical hydro Thermo in porous environment of voluminal structures
HT-66/05/005/A
Code_Aster ®
Version
7.4
Titrate:
WTNV100 - Triaxial Essai not drained with model CJS (level 1)
Date:
05/07/05
Author (S):
C. CHAVANT, pH. Key AUBERT
:
V7.31.100-B Page:
11/14
6
Results of modeling B
6.1 Values
tested
Localization Number Deformation
Constraint
Reference
Aster %
difference
of command axial zz (%)
(kPa)
Not A, B and C 5
0.2
xx
82.76923 82.76923 <
105
10
0.4
xx
65.53846 65.53846 <
105
20
0.8
xx
53.78079 53.78079 <
105
40
1.6
xx
56.578176 56.578176 <
105
60
5.6
xx
70.565109 70.565109 <
105
100
20.0
xx
120.918065 120.918065
<
105
5
0.2
yy
82.76923 82.76923 <
105
10
0.4
yy
65.53846 65.53846 <
105
20
0.8
yy
53.78079 53.78079 <
105
40
1.6
yy
56.578176 56.578176 <
105
60
5.6
yy
70.565109 70.565109 <
105
100
20.0
yy
120.918065 120.918065
<
105
5
0.2
zz
134.46154 134.46154
<
105
10
0.4
zz
168.92308 168.92308
<
105
20
0.8
zz
197.460849 197.460849
<
105
40
1.6
zz
207.731697 207.731697
<
105
60
5.6
zz
259.085935 259.085935
<
105
100
20.0
zz
443.961194 443.961194
<
105
Handbook of Validation
V7.31 booklet: Mechanical hydro Thermo in porous environment of voluminal structures
HT-66/05/005/A
Code_Aster ®
Version
7.4
Titrate:
WTNV100 - Triaxial Essai not drained with model CJS (level 1)
Date:
05/07/05
Author (S):
C. CHAVANT, pH. Key AUBERT
:
V7.31.100-B Page:
12/14
7 Modeling
C
7.1
Characteristics of modeling
3d_HM:
Z
B
y
X
Cutting: 1 in height, in width and thickness.
Loading of phase 1:
Confining pressure: 0
0
0
xx = yy = zz: 100 kPa.
Level 1 of model CJS
Coefficient of biot: 1
UN_SUR_K of water: 0
7.2
Characteristic of the grid
A number of nodes: 20
A number of meshs and types: 1 HEXA20 and 6 QUA8
7.3 Functionalities
tested
Commands
DEFI_MATERIAU CJS
STAT_NON_LINE COMP_INCR RELATION “KIT_HM”
“RELATION_KIT”:
“CJS” “LIQU_SATU” “HYDR_UTIL”
Handbook of Validation
V7.31 booklet: Mechanical hydro Thermo in porous environment of voluminal structures
HT-66/05/005/A
Code_Aster ®
Version
7.4
Titrate:
WTNV100 - Triaxial Essai not drained with model CJS (level 1)
Date:
05/07/05
Author (S):
C. CHAVANT, pH. Key AUBERT
:
V7.31.100-B Page:
13/14
8
Results of modeling C
8.1 Values
tested
Localization Number Deformation
Constraint
Reference
Aster %
difference
of command axial zz (%)
(kPa)
Not A and B
1
0.25
xx
78.461538 7,8462E+01
<
105
2
0.50
xx
56.923077 5,6923E+01
<
105
3
0.75
xx
53.606 5,3606E+01 <
105
4
1.0
xx
54.480 5,4480E+01 <
105
8
5.0
xx
68.467 6,8467E+01 <
105
23
20.0
xx
120.918 1,2092E+02 <
105
1
0.25
yy
78.461538 7,8462E+01
<
105
2
0.50
yy
56.923077 5,6923E+01
<
105
3
0.75
yy
53.606 5,3606E+01 <
105
4
1.0
yy
54.480 5,4480E+01 <
105
8
5.0
yy
68.467 6,8467E+01 <
105
23
20.0
yy
120.918 1,2092E+02 <
105
1
0.25
zz
143,07692 1,4308E+02
<
105
2
0.50
zz
186.153846 1,8615E+02
<
105
3
0.75
zz
196.818 1,9682E+02 <
105
4
1.0
zz
200.028 2,0003E+02 <
105
8
5.0
zz
251.383 2,5138E+02 <
105
23
20.0
zz
443.961 4,4396E+02 <
105
1
0.25
pressure water 2,1538E+04
2.15385E+04
< 105
2
0.50
pressure water 4,3077E+04
4.30769E+04
< 105
For the water pressure, one with the reference as long as the behavior is elastic linear
Handbook of Validation
V7.31 booklet: Mechanical hydro Thermo in porous environment of voluminal structures
HT-66/05/005/A
Code_Aster ®
Version
7.4
Titrate:
WTNV100 - Triaxial Essai not drained with model CJS (level 1)
Date:
05/07/05
Author (S):
C. CHAVANT, pH. Key AUBERT
:
V7.31.100-B Page:
14/14
9
Summary of the results
The values of Code_Aster are in perfect agreement with the values of reference. Concerning
coupling with hydraulics, this test proves that by means of computer, coupling CJS/THM functions and
that the equations of hydraulics are at least able to give again the null variation of volume
when water is incompressible.
Handbook of Validation
V7.31 booklet: Mechanical hydro Thermo in porous environment of voluminal structures
HT-66/05/005/A
Outline document