Code_Aster
®
Version
8.1
Titrate:
SSNA116 - Triaxial compression test with the model of Hoek-Brown modified
Date:
15/02/06
Author (S):
C. CHAVANT, V. GERVAIS
Key:
V6.01.116-A
Page:
1/10
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A
Organization (S):
EDF-R & D/AMA, CS-SI
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
Document: V6.01.116
SSNA116 - Triaxial compression test with the model of Hoek-
Brown modified into axisymmetric
Summary
This test makes it possible to validate the elastoplastic law of behavior of Hoek-Brown modified in mechanics of
rocks. It is about a triaxial compression test for which calculations are carried out only on the solid part of the ground in
pure mechanics.
Two levels of containment are applied: 5 MPa and 12 MPa. Parameters
end
,
rup
and
LMBO
are taken
equal (what returns to a constant voluminal plastic deformation): one can in this case calculate one
analytical solution with the problem and thus to compare the results obtained with Code_Aster with this solution of
reference.
For reasons of symmetry, one is interested only in the eighth of a sample subjected to a triaxial compression test.
modeling is axisymmetric.
Code_Aster
®
Version
8.1
Titrate:
SSNA116 - Triaxial compression test with the model of Hoek-Brown modified
Date:
15/02/06
Author (S):
C. CHAVANT, V. GERVAIS
Key:
V6.01.116-A
Page:
2/10
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A
Contents
1
Problem of reference .......................................................................................................................... 3
1.1
Geometry ........................................................................................................................................ 3
1.2
Properties of the material .................................................................................................................... 3
1.3
Initial conditions, with the limits and loading ................................................................................ 4
2
Reference solution ............................................................................................................................. 4
2.1
Calculation of the reference solution ................................................................................................... 4
2.2
Results of reference ..................................................................................................................... 5
3
Modeling A ....................................................................................................................................... 6
3.1
Characteristics of modeling ................................................................................................. 6
3.2
Characteristics of the mesh ........................................................................................................... 6
3.3
Functionalities tested .................................................................................................................... 6
4
Results of modeling A .............................................................................................................. 7
4.1
Values tested ................................................................................................................................ 7
5
Modeling B ....................................................................................................................................... 8
5.1
Characteristics of modeling ................................................................................................. 8
5.2
Characteristics of the mesh ........................................................................................................... 8
5.3
Functionalities tested .................................................................................................................... 8
6
Results of modeling B .............................................................................................................. 9
6.1
Values tested ................................................................................................................................ 9
7
Summary of the results ......................................................................................................................... 10
Code_Aster
®
Version
8.1
Titrate:
SSNA116 - Triaxial compression test with the model of Hoek-Brown modified
Date:
15/02/06
Author (S):
C. CHAVANT, V. GERVAIS
Key:
V6.01.116-A
Page:
3/10
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A
1
Problem of reference
1.1 Geometry
One considers here a cube of dimension 1m
× 1m × 1m.
Co-ordinates of the points (in m):
WITH B C
D
X 0 1 0.5
1
y 0 0 0.5
1
Z 0 0 0.5
1
1.2
Properties of material
Parameters of the elastic law of behavior:
E
= 4500 MPa
= 0.3
Parameters of the law of Hoek-Brown modified:
rup
= 0.005
LMBO
= 0.017
end
C
S
)
(
2
= 225 MPa
2
rup
C
S
)
(
2
= 482.5675 MPa
2
end
C
m
)
(
= 13.5 MPa
rup
C
m
)
(
= 83.75 MPa
= 3 MPa
end
= 15°
rup
= 15°
LMBO
= 15°
= 3.3
Z
y
X
With
B
D
·
·
·
·
C
1 m
1 m
1 m
Code_Aster
®
Version
8.1
Titrate:
SSNA116 - Triaxial compression test with the model of Hoek-Brown modified
Date:
15/02/06
Author (S):
C. CHAVANT, V. GERVAIS
Key:
V6.01.116-A
Page:
4/10
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A
1.3 Conditions
initial,
with the limits and loading
The test breaks up into two phases:
1) Initially, one brings the sample in a homogeneous state
0
0
0
zz
yy
xx
=
=
. For
that, the corresponding confining pressure is imposed on the front faces (
1
=
Z
),
side straight line (
1
=
X
) and higher (
1
=
y
), while displacements are taken null on
the faces postpones (
0
0
=
=
Z
Z
U
), side left (
0
0
=
=
X
X
U
) and lower (
0
0
=
=
y
y
U
).
2) Once the homogeneous state obtained, displacements are maintained locked on the faces
rear, side left and lower and the confining pressure are always imposed on
front faces and side straight line. A displacement is imposed on the higher face (
)
(T
U
y
)
in order to obtain a deformation
yy
equalize with 25% starting from the beginning of the second phase,
by constant increments of deformation
4
5
.
2
-
-
=
E
yy
.
2
Reference solution
2.1
Calculation of the reference solution
One places here in the case of a triaxial compression test for which the stresses of containment are
applied in directions X and Z and for which the direction of imposed deformation is the direction
y. One supposes moreover than the parameter
is independent of the parameter of work hardening
,
i.e.
LMBO
rup
end
=
=
: it is then possible to calculate an analytical solution with the problem.
The criterion of plasticity and flow are written:
&
&
&
&
&
&
&
&
&
&
1
3
3
)
1
(
2
1
2
)
2
1
(
1
1
)
1
(
0
1
)
(
)
(
)
(
)
(
2
3
1
3
3
3
2
1
3
+
=
=
+
+
=
+
=
=
+
-
=
-
=
=
-
-
-
-
-
-
p
p
p
p
D
B
C
C
B
m
S
An increasing situation of loading is considered for which the preceding equations can
to be written in a nonincremental way:
1
3
,
)
1
(
2
1
2
,
1
1
2
3
1
+
=
+
+
=
=
+
-
=
p
p
p
p
The relations of elasticity give:
)
(
)
(
1
)
(
2
)
(
1
0
1
1
0
3
3
3
3
0
3
3
0
1
1
1
1
-
-
-
-
=
-
-
-
-
=
-
E
E
E
E
p
p
Code_Aster
®
Version
8.1
Titrate:
SSNA116 - Triaxial compression test with the model of Hoek-Brown modified
Date:
15/02/06
Author (S):
C. CHAVANT, V. GERVAIS
Key:
V6.01.116-A
Page:
5/10
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A
i.e.:
)
(
)
(
1
)
1
(
2
1
2
)
(
2
)
(
1
1
1
0
1
1
0
3
3
3
0
3
3
0
1
1
1
-
-
-
-
=
+
+
-
-
-
-
=
+
-
-
E
E
E
E
with
0
3
and
0
1
values of
1
and
3
at the beginning of the loading. It thus remains to calculate
1
in
function of
by using the criterion of plasticity to obtain
3
1
and
,
.
1
Er
case:
rup
While noting
2
1
2
)
(
With
With
S
C
+
=
and
2
1
)
(
B
B
m
C
+
=
where
1
With
,
2
With
,
1
B
and
2
B
are given in
reference material of the law of behavior,
is solution of the polynomial of degree 2:
0
1
1
2
1
1
2
1
3
1
2
1
2
2
3
2
1
2
2
=
-
-
+
-
+
+
-
-
+
-
E
B
With
E
B
With
,
[
]
.
,
0
rup
interval
in
with
2
ème
case:
LMBO
rup
By taking again the notations of the reference material of the law of Hoek-Brown modified for A,
D, C and
D
B
-
3
,
is solution of the polynomial of degree 2:
[
]
LMBO
rup
D
B
rup
C
rup
C
D
B
D
B
E
C
E
m
S
E
D
E
has
,
0
1
)
(
)
(
1
1
1
1
3
3
3
2
1
3
3
2
3
3
interval
in
with
=
-
+
-
+
+
-
+
+
-
-
+
-
-
-
-
3
ème
case:
LMBO
In this case,
1
is constant:
-
-
-
-
=
- D
B
LMBO
LMBO
C
LMBO
C
B
m
S
3
3
3
2
3
1
1
)
(
)
(
and
1
0
1
1
-
-
=
E
.
2.2
Results of reference
Stresses
)
(
3
xx
,
)
(
1
yy
and
)
(
3
zz
at point D.
Displacements
)
(
3
xx
and
)
(
1
yy
at point D.
Code_Aster
®
Version
8.1
Titrate:
SSNA116 - Triaxial compression test with the model of Hoek-Brown modified
Date:
15/02/06
Author (S):
C. CHAVANT, V. GERVAIS
Key:
V6.01.116-A
Page:
6/10
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A
3 Modeling
With
3.1
Characteristics of modeling
Axisymmetric modeling 2D
Cutting: 1m in height, 1m in width
Loading of phase 1:
MPa
5
-
0
0
0
=
=
=
zz
yy
xx
(confining pressure)
Boundary conditions:
0
0
0
0
=
=
=
=
=
=
Z
Z
y
y
X
X
U
U
U
3.2
Characteristics of the mesh
A number of nodes: 4
A number of meshs and types: 1 QUAD4 and 4 SEG2
3.3 Functionalities
tested
Controls
DEFI_MATERIAU HOEK_BROWN
STAT_NON_LINE COMP_INCR
RELATION
“HOEK_BROWN”
Z
y
X
D
Code_Aster
®
Version
8.1
Titrate:
SSNA116 - Triaxial compression test with the model of Hoek-Brown modified
Date:
15/02/06
Author (S):
C. CHAVANT, V. GERVAIS
Key:
V6.01.116-A
Page:
7/10
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A
4
Results of modeling A
4.1 Values
tested
Localization Number
of command
Stress
(MPa)
Code_Aster
Solution of
reference
Relative variation
Not D
12
xx
- 5 - 5
0
70
xx
- 5 - 5
0
12
zz
- 5 - 5
0
70
zz
- 5 - 5
0
12
yy
- 18.50 - 18.50 0
16
yy
- 22.5676 - 22.5675778 0
32
yy
- 30.8798 - 30.8797526 0
41
yy
- 34.9342 - 34.9342281 0
42
yy
- 32.9137 - 32.9136722 0
46
yy
- 26.8215 - 26.8215156 0
52
yy
- 22.7560 - 22.7560224 0
70
yy
- 20.7512 - 20.721512 0
Localization Number
of command
Deformation
Code_Aster
Solution of
reference
Relative variation
Not D
12
xx
0.9 E-3
0.9 E-3
0
16
xx
1.24644 E-3
1.24644 E-3
0
32
xx
3.48682 E-3
3.48682 E-3
0
41
xx
4.81373 E-3
4.81373 E-3
0
42
xx
5.22653 E-3
5.22653 E-3
0
46
xx
6.66403 E-3
6.66403 E-3
0
52
xx
8.27551 E-3
8.27551 E-3
0
70
xx
12.0186 E-3
12.01865 E-3
0
12
yy
- 0.003 - 0.003 0
70
yy
- 0.0175 - 0.0175 0
Code_Aster
®
Version
8.1
Titrate:
SSNA116 - Triaxial compression test with the model of Hoek-Brown modified
Date:
15/02/06
Author (S):
C. CHAVANT, V. GERVAIS
Key:
V6.01.116-A
Page:
8/10
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A
5 Modeling
B
5.1
Characteristics of modeling
Axisymmetric modeling 2D
Cutting: 1m in height, 1m in width
Loading of phase 1:
MPa
12
-
0
0
0
=
=
=
zz
yy
xx
(confining pressure)
Boundary conditions:
0
0
0
0
=
=
=
=
=
=
Z
Z
y
y
X
X
U
U
U
5.2
Characteristics of the mesh
A number of nodes: 4
A number of meshs and types: 1 QUAD4 and 4 SEG2
5.3 Functionalities
tested
Controls
DEFI_MATERIAU HOEK_BROWN
STAT_NON_LINE COMP_INCR
RELATION
“HOEK_BROWN”
Z
y
X
D
Code_Aster
®
Version
8.1
Titrate:
SSNA116 - Triaxial compression test with the model of Hoek-Brown modified
Date:
15/02/06
Author (S):
C. CHAVANT, V. GERVAIS
Key:
V6.01.116-A
Page:
9/10
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A
6
Results of modeling B
6.1 Values
tested
Localization Number
of command
Stress
(MPa)
Code_Aster
Solution of
reference
Relative variation
Not D
16
xx
- 12 - 12 0
80
xx
- 12 - 12 0
16
zz
- 12 - 12 0
80
zz
- 12 - 12 0
16
yy
- 30 - 30 0
20
yy
- 33.4287 - 33.4287301 0
36
yy
- 43.5095 - 43.5095082 0
49
yy
- 50.4230 - 50.4230084 0
52
yy
- 48.4776 - 48.4775526 0
56
yy
- 46.4936 - 46.4935733 0
60
yy
- 45.0479 - 45.0479008 0
70
yy
- 43.1175 - 43.1174944 0
80
yy
- 42.8023 - 42.8023313 0
Localization Number
of command
Deformation
Code_Aster
Solution of
reference
Relative variation
Not D
16
xx
1.2 E-3
1.2 E-3
0
20
xx
1.61504 E-3
1.61504 E-3
0
36
xx
3.66549 E-3
3.66549 E-3
0
49
xx
5.46863 E-3
5.46863 E-3
0
52
xx
6.265 E-3
6.265 E-3
0
56
xx
7.26131 E-3
7.26131 E-3
0
60
xx
8.19982 E-3
8.19982 E-3
0
70
xx
10.3653 E-3
10.36527 E-3
0
80
xx
12.3573 E-3
12.35726E-3
0
16
yy
- 0.004 - 0.004
0
80
yy
- 0.02 - 0.02 0
Code_Aster
®
Version
8.1
Titrate:
SSNA116 - Triaxial compression test with the model of Hoek-Brown modified
Date:
15/02/06
Author (S):
C. CHAVANT, V. GERVAIS
Key:
V6.01.116-A
Page:
10/10
Manual of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A
7
Summary of the results
The results obtained make it possible to validate the model of Hoek-Brown modified integrated in Code_Aster
in the particular case of a constant voluminal plastic deformation.