Code_Aster ®
Version
8.1
Titrate:
SSNA116 - Triaxial Essai 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

Organization (S): EDF-R & D/AMA, CS-SI
Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
Document: V6.01.116
SSNA116 - Triaxial Essai 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. The 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.

Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A

Code_Aster ®
Version
8.1
Titrate:
SSNA116 - Triaxial Essai 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

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 grid ........................................................................................................... 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 grid ........................................................................................................... 8
5.3 Functionalities tested .................................................................................................................... 8
6 Results of modeling B .............................................................................................................. 9
6.1 Values tested ................................................................................................................................ 9
7 Summary of the results ......................................................................................................................... 10

Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A

Code_Aster ®
Version
8.1
Titrate:
SSNA116 - Triaxial Essai 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

1
Problem of reference

1.1 Geometry

One considers here a cube of dimension 1m × 1m × 1m.

y

1 m

·

D

· C

1 m

With
B
·
·

X

Z
1 m

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
2 end
(S) = 225 MPa2
C
2 rup
(S) = 482.5675 MPa2
C
end
(m) = 13.5 MPa
C
rup
(m) = 83.75 MPa
C
= 3 MPa
end
= 15°
rup
= 15°
LMBO
= 15°
= 3.3
Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A

Code_Aster ®
Version
8.1
Titrate:
SSNA116 - Triaxial Essai 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

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
= =. For
xx
yy
zz
that, the corresponding confining pressure is imposed on the front faces (Z = 1),
side straight line (X = 1) and higher (y = 1), while displacements are taken null on
the faces postpones (U
= 0), side left (U
= 0) and the lower (U
= 0).
Z z=0
X x=0
y y=0
2) Once the homogeneous state obtained, displacements are maintained blocked on the faces
back, 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 (U (T))
y
in order to obtain a deformation equalizes with ­ 25% starting from the beginning of the second phase,
yy
by constant increments of deformation
= - 5
.
2nd - 4.
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 constraints 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. end
rup
LMBO

=
=: it is then possible to calculate an analytical solution with the problem.
The criterion of plasticity and flow are written:



2


(
-) -

S

() -
m

(
)
- B
() 1
3
-
= 0
3
1
C
C
3


data base

3

p


- 1
& = &
(-)
1 =

1

&
+1

p
p
1


2 +1
& = =
+
=
3
&
&
(
)

2
2

&
(
2 +)
1
p


3
& =
3 & =



&
+1

An increasing situation of loading is considered for which the preceding equations can
to be written in a nonincremental way:

p
- 1
p
p

2 +1
p

=



,
= =


3

,
=

1
+1
3
2


(
2 +)
1
+1

The relations of elasticity give:

p
1

0
2
- = (-) -
(
0
-)
1
1
1
1
3
3
E
E

p
1

- =
(
0
-) - (
0
-)
3
3
3
3
1
1
E
E
Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A

Code_Aster ®
Version
8.1
Titrate:
SSNA116 - Triaxial Essai 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

i.e.: - 1 1

0
2
-
= (-) -
(
0
-)
1
+1
1
1
3
3
E
E

2 +1
1

-
=
(
0
-) - (
0
-)
3
2 (+)
1
3
3
1
1
E
E
with
0
and 0
values of and at the beginning of the loading. It thus remains to calculate in
3
1
1
3
1
function of by using the criterion of plasticity to obtain, and
.
1
3

1st case:
rup
While noting
2
S () = A + A
and m () = B + B
where
B are given in
C
1
2
C
1
2
1
With,
2
With, 1
B and 2
reference material of the law of behavior, is solution of the polynomial of degree 2:
2
- 1

- A - B
WITH - B
2
1
-




2
2
3
2
2
1
3 1

+




+ -
= 0,
+1
1

+1
2
1
2


E

E

with

in interval
[, 0 rup

].

2nd case: rup
LMBO


By taking again the notations of the reference material of the law of Hoek-Brown modified for A,
D, C and data base

, is solution of the polynomial of degree 2:
3
2 rup
rup
has


-


-


3
2
1
D

(S
3
c)
(m
3
c)
C

1


+ -
+ 1 -
+ +
+ 1
3
-
= 0

data base




+

data base
1


data base
E



1 E



E
E


3

3


3

with

in interval [rup LMBO

,
]

3rd case: LMBO
In this case, is constant:
1


= -
2 LMBO
(S) -
LMBO
(m) - LMBO
B
3
1

1
3
C
3
C

- data base

3
0
-
and
1
1
=
-.
1
E

2.2
Results of reference

Constraints (), () and () at point D.
xx
3
yy
1
zz
3
Displacements () and () at point D.
xx
3
yy
1
Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A

Code_Aster ®
Version
8.1
Titrate:
SSNA116 - Triaxial Essai 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

3 Modeling
With

3.1
Characteristics of modeling

Axisymmetric modeling 2D

y

D

Z
X

Cutting: 1m in height, 1m in width
Loading of phase 1: 0
0
0
= = = - MPa

5
(confining pressure)
xx
yy
zz
Boundary conditions: U
= U
= U
= 0
X x=0
y y=0
Z z=0

3.2
Characteristics of the grid

A number of nodes: 4
A number of meshs and types: 1 QUAD4 and 4 SEG2

3.3 Functionalities
tested

Commands



DEFI_MATERIAU HOEK_BROWN


STAT_NON_LINE COMP_INCR
RELATION
“HOEK_BROWN”

Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A

Code_Aster ®
Version
8.1
Titrate:
SSNA116 - Triaxial Essai 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

4
Results of modeling A

4.1 Values
tested

Localization Number Forced
Code_Aster
Solution of
Relative variation
of command
(MPa)
reference
Not D
12

- 5 - 5
0
xx
70

- 5 - 5
0
xx
12

- 5 - 5
0
zz
70

- 5 - 5
0
zz
12

- 18.50 - 18.50 0
yy
16

- 22.5676 - 22.5675778 0
yy
32

- 30.8798 - 30.8797526 0
yy
41

- 34.9342 - 34.9342281 0
yy
42

- 32.9137 - 32.9136722 0
yy
46

- 26.8215 - 26.8215156 0
yy
52

- 22.7560 - 22.7560224 0
yy
70

- 20.7512 - 20.721512 0
yy

Localization Number Deformation
Code_Aster
Solution of
Relative variation
of command
reference
Not D
12

0.9 E-3
0.9 E-3
0
xx
16
1.24644 E-3
1.24644 E-3
0
xx
32
3.48682 E-3
3.48682 E-3
0
xx
41
4.81373 E-3
4.81373 E-3
0
xx
42
5.22653 E-3
5.22653 E-3
0
xx
46
6.66403 E-3
6.66403 E-3
0
xx
52
8.27551 E-3
8.27551 E-3
0
xx
70
12.0186 E-3
12.01865 E-3
0
xx
12
- 0.003 - 0.003 0
yy
70
- 0.0175 - 0.0175 0
yy

Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A

Code_Aster ®
Version
8.1
Titrate:
SSNA116 - Triaxial Essai 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

5 Modeling
B

5.1
Characteristics of modeling

Axisymmetric modeling 2D

y

D

Z
X

Cutting: 1m in height, 1m in width
Loading of phase 1: 0
0
0
= = = - MPa

12
(confining pressure)
xx
yy
zz
Boundary conditions: U
= U
= U
= 0
X x=0
y y=0
Z z=0

5.2
Characteristics of the grid

A number of nodes: 4
A number of meshs and types: 1 QUAD4 and 4 SEG2

5.3 Functionalities
tested

Commands



DEFI_MATERIAU HOEK_BROWN


STAT_NON_LINE COMP_INCR
RELATION
“HOEK_BROWN”

Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A

Code_Aster ®
Version
8.1
Titrate:
SSNA116 - Triaxial Essai 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

6
Results of modeling B

6.1 Values
tested

Localization Number Forced
Code_Aster
Solution of
Relative variation
of command
(MPa)
reference
Not D
16

- 12 - 12 0
xx
80

- 12 - 12 0
xx
16

- 12 - 12 0
zz
80

- 12 - 12 0
zz
16

- 30 - 30 0
yy
20

- 33.4287 - 33.4287301 0
yy
36

- 43.5095 - 43.5095082 0
yy
49

- 50.4230 - 50.4230084 0
yy
52

- 48.4776 - 48.4775526 0
yy
56

- 46.4936 - 46.4935733 0
yy
60

- 45.0479 - 45.0479008 0
yy
70

- 43.1175 - 43.1174944 0
yy
80

- 42.8023 - 42.8023313 0
yy

Localization Number Deformation
Code_Aster
Solution of
Relative variation
of command
reference
Not D
16

1.2 E-3
1.2 E-3
0
xx
20
1.61504 E-3
1.61504 E-3
0
xx
36
3.66549 E-3
3.66549 E-3
0
xx
49
5.46863 E-3
5.46863 E-3
0
xx
52
6.265 E-3
6.265 E-3
0
xx
56
7.26131 E-3
7.26131 E-3
0
xx
60
8.19982 E-3
8.19982 E-3
0
xx
70
10.3653 E-3
10.36527 E-3
0
xx
80
12.3573 E-3
12.35726E-3
0
xx
16
- 0.004 - 0.004
0
yy
80
- 0.02 - 0.02 0
yy

Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A

Code_Aster ®
Version
8.1
Titrate:
SSNA116 - Triaxial Essai 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

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.
Handbook of Validation
V6.01 booklet: Nonlinear statics into axisymmetric
HT-62/06/005/A