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7.3
Titrate:
Law of behavior CAM-CLAY
Date:
03/02/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key:
R7.01.14-B Page
: 1/34
Organization (S): EDF-R & D/AMA
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Document: R7.01.14
Law of behavior CAM-CLAY
Summary:
The Camwood-Clay model one of the elastoplastic models known and the most are the most used in mechanics of
grounds. It is especially adapted to argillaceous materials. There are several types of models Camwood-Clay, that
presented here is most current and is called modified Camwood-Clay. This model is characterized by surfaces of
load hammer-hardenable in the shape of ellipses in the diagram of the first two invariants of the constraints. With
the interior of these surfaces of reversibility, the material is elastic nonlinear. There exists moreover, in a point of
each ellipse, a critical state characterized by a null variation of volume. The whole of these points
constitute a line separating the zones from dilatancy and contractance from material as well as the zones
of negative and positive work hardening. Work hardening is governed by only one scalar variable and the rule of flow
normal is adopted.
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1 Notations
indicate the tensor of the effective constraints in small disturbances defined as being
difference between the total constraints and the pressure of water in the case of water-logged soils, noted under
the shape of the following vector:
11
22
33
212
2 23
2 31
One notes:
1
P = - tr ()
3
constraint of containment
S = + pi
diverter of the constraints
1
I = tr
second invariant of the constraints
2
(S.S)
2
Q = = 3I
equivalent constraint
eq
2
= 1
(U T
+ U)
total deflection
2
= + +
partition of the deformations (elastic, plastic, thermal)
E
p
HT
= - tr + -
v
() 3 (T T0) voluminal total deflection
p
= - tr
voluminal plastic deformation
V
(p)
= 1
~
+ I
diverter of the deformations
v
3
~
E
~ ~ p
= -
diverter of the elastic strain
p
p
1
~
p
= + I
deviatoric plastic deformation
v
3
E
2
=
tr ~ ~
.
equivalent elastic strain
eq
(E E)
3
p
2
=
tr ~ ~
.
equivalent plastic deformation
eq
(p p)
3
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E index of the vacuums of the material (report/ratio of the volume of the pores on the volume of the solid matter constituents)
E initial index of the vacuums
0
porosity (report/ratio of the volume of the pores on total volume)
coefficient of swelling (elastic slope in a hydrostatic test of compression)
M slope critical line of state
1
(+ E)
0
K =
0
P variable interns model, critical pressure equal to half of the pressure of consolidation
Cr
P
CON
coefficient of compressibility (plastic slope in a hydrostatic test of compression)
1
(+)
0
=
E
K
(-)
µ elastic coefficient of shearing (coefficient of Lamé)
F surfaces of load
plastic multiplier
D
I tensor unit of command 2 whose term running is
ij
D
I tensor unit of command 4 whose term running is
4
ijkl
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2 Introduction
The model describes here is the model known as of modified Camwood-Clay. The initial Camwood-Clay model was
developed by the school of soil mechanics of Cambridge in the Sixties. It predicted
too important deviatoric deformations under weak loading deviatoric, and was modified by
Burland and Roscoe in 1968 [bib1].
2.1
Phenomenology of the behavior of the grounds
The materials poroplastic such as certain clays are characterized by the behaviors
following:
· the strong porosity of these materials causes unrecoverable deformations under loading
hydrostatic corresponding to an important reduction of porosity. This mechanism
purely contractor is sometimes called “collapse”,
· under loading deviatoric, these materials show a contracting phase followed of one
phase where the material becomes deformed with constant plastic volume or dilates.
For the two types of loading, the energy blocked in material evolves/moves according to the number
of contact between the grains. For a hydrostatic loading, the number of contact increases, thus
that blocked energy, one thus has positive work hardening. For a loading deviatoric, the material
can become deformed without variation of volume to a number of intergranular contacts constant. Moreover,
one can observe in the tests of the localizations of deformations accompanied by strong
dilatancy. In these zones, the number of grains in decreasing contact, there is reduction in energy
blocked and thus softening.
These behaviors are highlighted primarily by triaxial compression tests of revolution. These
observations bring to postulate that there is a plastic threshold whose evolution is controlled by two
mechanisms: one purely contractor associated with the hydrostatic constraint, and a mechanism
deviatoric controlled by internal friction being held with constant volume and possibly
dilating with the approach of the localization.
All the interest of the model of Cam Clay lies in its faculty to describe these phenomena with one
minimum of ingredients and in particular only one surface of load and a work hardening associated with one
only scalar variable.
2.2
Behavior under hydrostatic compression
During a hydrostatic test of compression (E the initial index of the vacuums under loading equal to
0
atmospheric pressure P), the grounds present an index of the vacuums which decrease logarithmiquement
has
with the exerted hydrostatic pressure (cf [Figure 2.2-a]). Until a pressure 0
P
called
CON
pressure of consolidation, the behavior is reversible, the slope of the diagram (,
E Ln P) is
called elastic coefficient of swelling. 0
P
corresponds to the maximum pressure which underwent it
CON
material during its history. Beyond this preconsolidation, the diagram presents one
new slope (coefficient of compressibility) more marked and appearance of deformations
irreversible. 0
P
thus corresponds to an evolutionary elastoplastic threshold.
CON
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E
e0
Ln Pa
Ln 0
P
1
Ln P
CON
CON
Ln P
Appear 2.2-a: hydrostatic Essai of loading and unloading
Note:
The diagram above corresponds to a whole of measurements where the effective constraint is
stabilized. Indeed, in the process of consolidation of the grounds, it is the water contained in
the pores which takes again initially the hydrostatic pressure with very little deformation, front
to run out and let the skeleton become deformed. After consolidation of material and
stabilization of the pressure of water, the effective constraint (forced total minus pressure
water) is stabilized and deferred on the graph. Relations of behavior in
saturated porous environments are generally expressed with the effective constraints according to
the assumption of Terzaghi.
2.3
Behavior under loading deviatoric
The triaxial compression tests of revolution make it possible to control at the same time the deviatoric component Q and
spherical component P of the loading. According to the report/ratio of these two components, one observes
Q
Q
a plastic behavior purely dilating (
> M) or contracting (< M), the line Q = MP
P
P
Cr
representing the whole of the critical points on surfaces of load where the mechanical state evolves/moves
without plastic change of volume. The basic model of Cam Clay makes the assumption that the rates
~
p
F
p
F
plastic deformations are normal on the surface of load F (& = &
, & = &
). Of
v
P
Q
more, plastic work in an unspecified point of the surface of load is considered equal to work
plastic in a critical state. These considerations bring to the following equation for the plastic threshold:
Q
P
F (P, Q, P)
Ln
éq
2.3-1
Cr
=
+
(
) = 0
MP
Pcr
Note:
In Code_Aster, the adopted criterion is that of modified the Cam_Clay model [éq 3.2-1].
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3
Law of Cam Clay modified
The criterion of plasticity formulated above is not satisfactory for certain paths of loading.
In particular, for low values of Q/P, the model predicts deviatoric deformations too much
important. To cure it, a new expression of plastic work was adopted, which
conduit with the model known as of Cam Clay modified [bib1].
3.1
Assumptions of modeling
The model is written in small disturbances.
The coefficients of the model do not depend on the temperature.
3.2
Surface of load
The new assumptions lead to the following expression of the surface of load:
F (P, Q, P) = Q2 + m2 P2 - 2M2 PP
0
éq
3.2-1
Cr
Cr
In the plan (P, Q), the expression represents a family of ellipses, centered on P which is equal to
Cr
half of the pressure of consolidation (cf [Figure 3.2-a). P will be the parameter of work hardening of
Cr
model.
Q
Q=MP
Pcr1 Pcr2
Pcon1
Pcon2
P
Appear 3.2-a: Famille of hammer-hardenable surfaces of load
When F = 0 and P < P the material is dilating (p
&
) and P is decreasing (softening).
v < 0
Cr
Cr
When F = 0 and P > P the material is contacting (p
&
) and P is increasing (hardening).
v > 0
Cr
Cr
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3.3
Elastic law and law of work hardening
One makes the assumption of the decoupling of the partly hydrostatic and deviatoric elastic law and
the additional assumption that the modulus of rigidity is constant.
One thus considers an isotropic elastic law, with a linear deviatoric part and a part
voluminal non-linear:
Déviatoire part:
~e
S
=
éq 3.3-1
2µ
Voluminal part:
E
&
&
ln
éq 3.3-2
v = -
E
or E = e0 -
P if P < Pconsolidation
1+ e0
The law [éq 3.3-2] is in fact derived from a test oedometric where one measures the variation of the index of
vacuums according to the loading [Figure 2.2-a]. Let us recall that a homogeneous test oedometric
consist in increasing the axial effective constraint all while maintaining the constraint radial null on
a cylindrical test-tube.
Note:
The pressures P correspond to tests drained or not. Nevertheless, in one
modeling with Code_Aster constraints handled in the laws of behavior
are effective i.e. that one does not take into account the hydrostatic pressure of the fluid
who can circulate in the pores, this one being calculated in modelings THM.
The tests of voluminal loading (cf [Figure 2.2-a]) bring us to the following elastic law:
1 + E
P = P exp K (
p
-
)
with K =
éq
3.3-3
0
[0 v v]
(0)
0
In the same way, growth of the surface of load in phase of contractance (and decrease for
the experimental dilatancy) and results suggest writing:
p
p
+ E
0
P
with
éq
3.3-4
Cr = Pcr
[
exp K (v -)
K
v
=
0]
(1 0)
,
(-)
p
and E correspond to the voluminal deformation and the index of the initial vacuums, determined by
v0
0
extrapolation of the oedometric curve of the test to pressure 0
P (cf [Figure 2.2-a]).
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3.4
Plastic law of flow
The two plastic variables are the plastic deformation voluminal p
and the tensor deviatoric
v
plastic deformations p
~. The internal variable is also p
but associated the force
v
of P. work hardening the material standard is not generalized. The rule of flow is written:
Cr
p
F
p
F
& = &
, & = - &
,
éq
3.4-1
v
P
Cr
by breaking up the first term, one obtains:
F
F
~
F
p
p
p
& = &
& = &
& = - &
éq 3.4-2
v
v
P
S
P
Cr
knowing that:
1
P = - tr () and = - tr + -
éq
3.4-3
v
() 3 (T T0)
3
F is the plastic potential associated the phenomenon of work hardening. Let us note that the third part of
[éq 3.4-2] is only formal. Indeed, p is known
& by the first relation thus one knows
v
evolution of P.
Cr
3.5
Energy writing and plastic module of work hardening
One is thus within the not generalized “standard” material framework (one uses three then
potentials: the surface of load F, plastic potential F, and free energy. Even in this
configuration less favorable than the traditional framework of not generalized standard materials, one
is ensured to satisfy the second principle of thermodynamics [bib4]. Using the condition of
consistency (expressing that the point representative of the loading “follows” the surface of load) which
is written in the following way:
F
F
df =
dP +
+ F
dQ
dP
éq
3.5-1
Cr = 0,
P
Q
Pcr
the expression of the plastic multiplier is determined [bib4]:
1
F
1
F
=
D = -
dP éq
3.5-2
Cr
H
H
P
p
p
Cr
with [bib4]:
F
2
F
H =
, where H
of
modulate
is
E
écrouissag éq
3.5-3
p
2
p
p
P
P
Cr
Cr
v
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The identification of the first and third part of [éq 3.4-2] makes it possible to calculate F which is written:
F
F = -
dP = m2 P
- 2
éq
3.5-4
Cr
Cr (P
P
Cr
)
P
Concept of work hardening being associated that of blocked energy:
2
p
P
=
thus
dP =
D
éq
3.5-5
Cr
p
Cr
2
p
v
v
v
where is the density of free energy:
3
0
P
P
= µ (E
) 2
0
+
exp (
E
K)
Cr
+
exp (K (p
p
-))
éq
3.5-6
2
eq
0 v
v
v0
K
K
0
By using them [éq 3.4-2], [éq 3.5-4] and [éq 3.5-6], one can draw according to [éq 3.5-3] the expression from
modulate plastic work hardening:
F
2 F
H =
= 4KM 4PP P - P
éq
3.5-7
p
2
Cr (
Cr)
p
P
P
Cr
Cr
v
The module of work hardening is positive in phase of contractance (P > P and negative in phase of
Cr)
dilatancy (P < P. Pour P = P, the behavior is plastic perfect and proceeds with volume
Cr)
Cr
constant plastic.
3.6 Relations
incremental
The equation [éq 3.4-3] and the condition of consistency give the relations of flow:
1 1
1
p
Q
D
éq
3.6-1
v =
-
dP +
dQ
2
K P
P
Cr
Mr. PC
1
2
p
Q
Q
D
éq
3.6-2
eq =
dP +
dQ
K M2 PP
M 4 PP
Cr
Cr (P - Pcr)
~
3
p
p
S
D = D
éq 3.6-3
eq 2 Q
The rearrangement of [éq 3.6-1] and [éq 3.6-2] led to:
p
deq
Q
=
éq 3.6-4
p
D
M2
P - P
v
(
Cr)
i.e. with the equation [éq 3.6-3],
~ p
D
3
S
=
éq 3.6-5
p
D
M2
2
P - P
v
(
Cr)
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Particular case of the critical point:
For F = 0 and P = P: P&
, p
. One deduces some, by considering the elastic law: P & = K P
.
0
V
& =0
Cr = 0
Cr
V
&
The condition of consistency gives us Q & = 0.
3.7
Summary of the relations of behavior and the data of the model
3.7.1 Data and critical of the model
The data specific to the model are five:
The slope criticizes M,
the initial index of the vacuums E associated with an initial pressure equalizes in general with the pressure
0
atmospheric,
the elastic coefficient of swelling (which leads to K),
0
the plastic coefficient of compressibility: (which leads to K),
the initial critical pressure P equalizes with half of the pressure of preconsolidation,
Cr 0
which it is necessary to add the traditional coefficient of Lamé µ and the thermal dilation coefficient
. The coefficient of Lamé µ is in fact calculated starting from the two elastic coefficients E
, provided
in data.
The number of data is relatively low, which makes the model very simple. One of the limitations more
visible of the model is the assumption of the alignment of the critical points on a line of slope Mr.
This is besides the expression of the concept of internal friction. One can also interpret the size
M
3
M by connecting it to the angle of repose natural of Coulomb by the relation: sin =
. However one knows
6 + M
that for very cohesive materials, this angle varies when the average constraint decreases. One
note besides that for a chock of M on a triaxial compression test with a certain average constraint,
one simulates well with this model the triaxial ones realized with a average constraint step too different
but one cannot correctly consider the bearings plastic for a broad range of pressure of
containment (cf [bib2]). It is thus necessary to readjust M for several ranges of constraint
average.
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3.7.2 Summary of the relations of behavior of the model
Elasticity
~e
S = 2µ
éq 3.7.2-1
P = P exp
éq
3.7.2-2
0
(E
k0v)
Plasticity
The criterion: F (, P
Q
MR. P
MR. PP
with (Q =
eq)
Cr)
2
2
2
=
+
- 2 2 Cr = 0
F
1 F D 3 F S
= -
I +
éq
3.7.2-3
3 P
2 Q Q
1
4
4
4
4
2
4
4
4
4
3
thus:
~ p
& = 3 & S
éq 3.7.2-4
p
& = & m2
2
P - P éq
3.7.2-5
v
(
Cr)
Work hardening
P
= P
K -
éq
3.7.2-6
Cr (p
v)
exp
p
p
Cr 0
((v v 0)
Elastic behavior: If F < 0 or (F = 0 and F & 0) then:
P&
éq 3.7.2-7
Cr = 0
~ p
&
eq =,
0 p
&v = 0
éq
3.7.2-8
&s = µ&~
2
éq 3.7.2-9
P & = K
éq
3.7.2-10
0 & P
v
Elastoplastic behavior: If F = 0 and F & = 0 then:
p
P & 0
;
P & = K & P
éq
3.7.2-11
Cr
Cr
v
Cr
~ p
& = 3 & S
if
P P
éq
3.7.2-12
Cr
p
& = & m2
2
P - P
if P P
éq 3.7.2-13
v
(
Cr)
Cr
Note:
· From only unknown p
P &.
v
&, one can deduce the other unknown factors p
&~ and Cr
· If P = P: p
&
, Q & = P & =,
0 P & = K P.
0
v = 0
Cr
Cr
&v
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4
Numerical integration of the relations of behavior
4.1
Recall of the problem
For an increment of loading given and a whole of variables given (initial field of
displacement, constraint and variable intern), one solves the discretized total system (2.2.2.2-1 of [bib3])
who seeks to satisfy the equilibrium equations.
The resolution of this system gives us U
, therefore
. One thus seeks locally (in each
not Gauss) the increment of constraint and variable interns correspondent with
and which satisfies
law of behavior.
The following notations are employed: Has, A, A
for the quantity evaluated at the known moment T, with
the moment T + T
and its increment respectively. The equations are discretized in manner
implicit, i.e. expressed according to the unknown variables at the moment T + T
.
4.2
Calculation of the constraints and variables internal
The elastic prediction of the deviatoric constraint is written:
= -
S + µ ~
2 éq
4.2-1
however one can always write S at the moment + as being:
-
~e
S = S + 2µ
éq 4.2-2
These two equations enable us to deduce S according to E
S:
E
~
~e
S = S - 2µ
+ 2µ
éq
4.2-3
E
~ p
or S = S - 2µ
éq 4.2-4
While replacing
p
~
by its expression according to
p
, one obtains:
v
E
S
S =
éq 4.2-5
p
3
µ
1+ m2 (v
P - Pcr)
from where,
E
Q
Q =
éq
4.2-6
p
3
µ
1+ m2 (v
P - Pcr)
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One writes P according to the hydrostatic elastic prediction:
There is the equation:
P = P exp
éq 4.2-7
0
(k0 (
p
-
v
v)
By supposing that K is independent of the temperature, the incremental writing of this
0
equation is:
P = P exp [
-
-
E
K
éq 4.2-8
0
k0
v -
E
v
]
P = p exp [
E
K
éq 4.2-9
0 v]
P = -
P (exp [K E
éq
4.2-10
v -
0
]) 1
In the same way one can write the expression of E
P according to -
P:
E
P = p exp [K
éq 4.2-11
0 v]
from where the expression of P at the moment + is:
E
P = P exp [
p
- K
éq 4.2-12
0 v]
In the incremental writing of P, the coefficient K does not depend on the temperature, one thus finds
Cr
the following expression:
P = P exp
p
p
K -
éq
4.2-13
Cr
Cr 0
[(v v 0)]
P = p exp K
éq 4.2-14
Cr
Cr
[statement]
P
P
K
éq
4.2-15
Cr =
-
Cr [exp (p
v) -]
1
Summary:
F (E
S, E
P, -
P
in this case P
that is to say
-
E
S = S + S = S
Cr = 0
Cr) 0
E
P = P
F (E
S, E
P, -
P
in this case P
, ~
p
0 and p
v 0
Cr > 0
Cr) > 0
that is to say
E
~ p
S = S - 2µ
P = P exp - K
0
E
[
p
v]
P = p exp K
Cr
Cr
[statement]
Note:
The principal unknown factor is
p
.
v
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Law of behavior CAM-CLAY
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Key:
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4.3
Calculation of the unknown factor
p
v
By deferring in the criterion the expressions of P and Q according to E
P and of E
Q and while using
the equation [éq 4.2-6]:
p
2
µ
2
3
v
Q = - 1
+
M2 P P - 2P éq
4.3-1
E
2
(
Cr)
M (P - P)
Cr
p
2
2
µ
2
3
Q = - M 1+
exp
2
-
- 0
M
(
v
P
K
E
P exp - K
éq
4.3-2
0
- P exp K
E
[
p
v]
Cr
[statement]) E [
p
v]
(P exp - k0
- 2P- exp K
E
[
p
v]
Cr
[statement])
In under following paragraph one determines limits with this function which facilitate the resolution
equation [éq 4.3-2] with for example the method of the cords or by the method of Newton.
Some examples of paces of the preceding function are given in the following figures for
several data files.
Particular case: Q = 0 (hydrostatic test of compression)
The criterion is reached for P = 2P
Cr
From where: P exp - K
0
= 2P- exp K
E
(
p
v)
Cr
(statement)
E
1
P
Thus p
Ln
v =
-
K + K
2P
0
Cr
For: M = 9
,
0; µ = 4000; -
P
; K = 10; K = 30 and P
; then
- 3
p
v = 2.310
E = 1
Cr =
,
0 2
0
Appear 4.3-a: Allure of the function [éq 4.3-2] for Q = 0
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Example for: -
-
Q < MP (contractance)
That is to say following data: M = 9
,
0; µ = 4000; -
P
; K = 10; K = 30
Cr =
,
0 2
0
E
Q = 2; E
P = 6
,
0; ~
4
.
2 10 -;
- 4
v = 3.10
eq =
Appear 4.3-b: Allure of the function [éq 4.3-2] for -
-
Q < MP
Example for: -
-
Q > MP (dilatancy)
That is to say following data: M = 9
,
0; µ = 4000; -
P
; K = 10; K = 30
Cr =
,
0 2
0
E
Q = 2; E
P =,
0 2; ~
- 4
;
- 4
v = 3.10
eq =
.
2 10
Appear 4.3-c: Pace of the function [éq 4.3-2] for -
-
Q > MP
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4.4
Determination of the terminals of the function
One poses
p
= X the unknown factor of the problem.
v
One thus has:
P (X) = EP exp (- K X éq
4.4-1
0)
P (X)
P -
=
exp (kx)
éq 4.4-2
Cr
Cr
X
(X) =
éq
4.4-3
2M2 (P (X) - Pcr (X)
Qe
Q (X) = 1+6µ
éq 4.4-4
(X)
F (X)
2
= Q (X)
2
2
+ MR. P (X) - 2 2
MR. P (X) P
éq
4.4-5
Cr (X) = 0
(X) 0 then two cases arise:
X 0 and P P P (0) P
and 0 X X; P (X
= P X
éq
4.4-6
sup)
Cr (sup)
Cr (0)
Cr
sup
X 0 and P P P (0) P
and X
X 0; P (X
= P X
éq
4.4-7
inf)
Cr (inf)
Cr (0)
Cr
inf
The first terminal is the 0 which is the terminal inf in the case of the contractance and limits it sup in
case of dilatancy.
Calculation of the second terminal X = X
= X:
B
sup
inf
P (X = P X P exp - K X
P
=
exp kx
b)
Cr (
)
E
B
(0 b) Cr
(b)
E
P = exp K + K X
-
(
0) B
Pcr
éq
4.4-8
E
1
P
X
Ln
B =
-
K + K
P
0
Cr
One will thus distinguish between the two fields:
Dilatancy: X [X 0
; and Contractance: X [0: X
B]
B
]
Values of the function at the boundaries:
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At the point
E
X = 0; P ()
0 = P; P ()
0 = p; 0 = 0; Q 0 = Q
éq
4.4-9
Cr
()
() E
Cr
F (0) = E
Q 2 +
2
E
MR. P (E
-
P - 2P
éq
4.4-10
Cr)
At the point
X = X P = P;
=
= and
= -
éq 4.4-11
B
Cr
(xb)
; Q (xb) 0
F (xb)
2
2
MR. P
4.5
Particular case of the critical point
Q
Q=MP
T
t+
Pcr
Pcon
P
Appear mechanical 4.5-a: State around the critical point
If at the moment -
T one reaches the critical state, then +
P
P
and
-
-
Q = MP. If F =,
0 F & = 0,
Cr =
-,
Cr
statement = 0
then the point (P, Q)
at the moment +
T moves on the initial ellipse (cf [Figure 4.5-a]). One deduces immediately from the law
rubber band and of the condition p
:
v = 0
-
P = K
éq 4.5-1
0 P
v
The criterion being checked at the moment +
T, one has while using [éq 4.5-1]:
+2
2
Q = Mr. P+ (2P- - P+)
2
= M (p + P
) (p - P
)
2
- (2)
= MR. P
1
(
2
2
- K
)
- (2)
= Q
1
(
2
2
- K
)
Cr
0
V
0
v
éq 4.5-2
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In addition the diverter of the constraints can be written:
~
E
p
E
F
S = S - 2µ = S - 2µ
= - 6µ S éq
4.5-3
S
One deduces some:
Qe
1+ 6 µ =
, éq
4.5-4
Q
and:
Q 1
(- K 2
2
0
V
E
S =
S
éq
4.5-5
E
Q
4.6 Summary
The discretization of the equations and the law of implicit behavior of manner leads to the resolution
equation [éq 4.3-2].
If -
-
P P, then one solves the equation [éq 4.3-2] whose unknown factor is
p
.
Cr
v
One deduces then:
E
-
-
S
p
p
P = P exp (K), P = P exp (K (
), then
0
-
S =
éq 4.6-1
Cr
Cr
v
v
v
p
3µ
1+ m2 (v
P - Pcr)
One deduces finally:
p
p
3
~
v
=
S
éq
4.6-2
2 M2 (P - P)
Cr
At the critical point:
p
-
,
0 P
P
éq
4.6-3
v =
Cr =
Cr
In this point, there is no evolution of work hardening, on the other hand the state of stress can continue with
to evolve/move either in contractance, or in dilatancy (the tangent with the criterion is horizontal). The new state
constraints moves on the surface of load of the preceding state.
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5 Operator
tangent
If the option is: RIGI_MECA_TANG, option used at the time of the prediction, the tangent operator calculated in
each point of Gauss is known as of speed:
elp
& = D
ij
ijkl kl
&
In this case,
elp
D is calculated starting from the not discretized equations.
ijkl
If the option is: FULL_MECA, option used when one reactualizes the tangent matrix with each iteration
by updating the internal constraints and variables:
D = A D
ij
ijkl
kl
In this case, A is calculated starting from the implicitly discretized equations.
ijkl
5.1
Nonlinear elastic tangent operator
The elastic relation of speed is written:
&
P
S
K Ptr
éq
5.1-1
ij = - & ij + &ij =
ij + µ &
&
~
2
0
2
& = (K P -
+
éq
5.1-2
0
µ tr
) &
µ
2
ij
ij
&ij
3
The tangent operator in elasticity of the law noted Cam_Clay
E
D is thus deduced from the matric writing
following:
4
2
2
K P
µ K P
µ K P
µ
0
0
0
&
11
0 +
0
-
0
-
11
&
3
3
3
2
4
2
&
22
K P
µ K P
µ K P
µ
0
0
0
&
0
-
0
+
0
-
22
3
3
3
&
33
33
&
=
2
2
4
K P
µ K P
µ K P
µ 0
0
0
éq 5.1-3
2
0
-
0
-
0
+
&
2
12
3
3
3
12
&
0
0
0
2µ
0
0
2&
2
23
&23
0
0
0
0
2µ
0
2
&
2
31
&31
0
0
0
0
0
2µ
1
4
4
4
4
4
4
4
4
4
4
2
4
4
4
4
4
4
4
4
4
4
3
E
D
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5.2
Plastic tangent operator of speed. Option RIGI_MECA_TANG
The total tangent operator is in this case K (the option
I 1
-
RIGI_MECA_TANG called with the first
iteration of a new increment of load) starting from the results known at the moment T [bib3].
I 1
-
If the tensor of the constraints with T is on the border of the field of elasticity, the condition is written:
I 1
-
F & = 0 which must be checked jointly with the condition F = 0. If the tensor of the constraints with T
I 1
-
is inside the field, F < 0, then the tangent operator is the operator of elasticity.
F
F & =
F
& +
P&cr = 0
éq
5.2-1
Pcr
P
like
Cr
p
P & =
, then:
Cr
p
v
&
v
F
F
F & =
P
& +
Cr
p
éq
5.2-2
p
v = 0
&
P
Cr v
In addition E
p
& = & - &
thus:
-
1 & =
& - F
Of
&
,
éq
5.2-3
i.e.:
E
E
F
& = D & - &D
éq
5.2-4
ij
ijkl kl
ijkl kl
The plastic module of work hardening is written according to the equation [éq 3.5-7] and by using the rule
of flow:
F
P
F
1
F
P
Cr
Cr
p
H =
= -
éq
5.2-5
p
p
p
v
P
&
P
& P
Cr
v
Cr
Cr
v
The equations [éq 5.2-1] and [éq 5.2-5] give:
F
&
& H
éq
5.2-6
ij -
p = 0
ij
F
Multiplication of the equation [éq 5.2-4] by
give:
ij
F
F
F
F
E
E
& =
D & - &
D
éq 5.2-7
ij
ijkl
ijkl
ij
ij
ij
kl
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The two preceding equations make it possible to find:
F
E
F
E
F
H & =
D & - &
D
éq
5.2-8
p
ijkl kl
ijkl
ij
ij
kl
from where the expression of the plastic multiplier:
F
E
D
ijkl kl
&
ij
& =
éq
5.2-9
F
F
E
D
+ H
ijkl
p
ij
kl
That is to say H the definite elastoplastic module like:
F
F
E
H =
D
+ H
éq
5.2-10
ijkl
p
ij
kl
The plastic multiplier is written:
F
Of
ijkl kl
&
ij
& =
éq
5.2-11
H
While replacing & by his expression in the equation [éq 5.2-4], one obtains:
1 F
F
E
E
E
& = D & -
D
& D
.
éq
5.2-12
ij
ijkl kl
mnop COp
ijkl
H
mn
kl
One thus deduces the elastoplastic operator from it
elp
E
p
D
= D - D:
F
F
E
E
E
& =
1
D -
D D
éq 5.2-13
ij
ijkl
ijop
mnkl
&kl
H
COp
mn
1
4
4
4
4
4
4
4
2
4
4
4
4
4
4
4
3
Delp
with,
1 F
F
p
E
E
D
=
D D
éq 5.2-14
ijkl
ijop
mnkl
H
COp
mn
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Calculation of H:
F
2
= - m2 (P - P + S
3,
éq
5.2-15
Cr) ij
ij
3
ij
who is written in vectorial notation:
2
2
- M (P - Pcr)
+ 3s11
3
- 2 2
M (P - Pcr) + 3s
22
3
- 2 2
M (P - P
éq
5.2-16
Cr)
+ 3s33
3
3 2s12
3 2s
23
3 2s31
from where the expression of:
-
2
2K MR. P P P
6µs
0
(- Cr) +
11
2
- 2K MR. P P P
6µs
0
(- Cr)
+
22
µ
E F
-
2
2K MR. P P P
6 S
0
(- Cr) +
Dijkl
:
33
éq
5.2-17
µ
kl
6
2s12
µ
6
2s23
6µ 2s31
and
F
E
F
4
2
2
D
= 4K MR. P (P - P) +12 Q
µ éq
5.2-18
ijkl
0
Cr
ij
kl
According to the equations [éq 3.5-7] and [éq 5.2-17], one can deduce the expression from H:
4
H = 4M P (P - P
-
+
+ µ
Cr) (K (P
Pcr) kPcr)
2
4 Q
éq 5.2-19
0
While posing:
To = 2
- K m2 P P - P
+
,
éq
5.2-20
0
6 S
µ
ij
(
Cr) ij
ij
one can write the following symmetrical plastic matrix:
2
With
WITH A
WITH A
6 2µA S
6 2µA S
6 2µA S
11
11
22
11
33
11 12
11 23
11 31
2
.
With
WITH A
6 2µA S
6 2µA S
6 2µA S
22
22
33
22 12
22 23
22 31
2
µ
µ
µ
D p = 1
.
.
With
6 2 A S
6 2 A S
6 2 A S
33
33 12
33 23
33 31 éq
5.2-21
H
2 2
2
2
.
.
.
36µ S
36µ S S
36µ S S
12
12 23
12 31
2 2
2
.
.
.
.
36µ S
36µ S S
23
23 31
2 2
.
.
.
.
.
36µ s31
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5.3
Tangent operator into implicit. Option FULL_MECA
To calculate the tangent operator into implicit, one chose by preoccupation with a simplicity to separate in first
place processing of the deviatoric part of the hydrostatic part for then combining them in order to
to deduce the tangent operator connecting the disturbance from the total constraint to the disturbance of
total deflection.
5.3.1 Processing of the deviatoric part
It is considered here that the variation of loading is purely deviatoric (P =)
0.
The increment of the deviatoric constraint is written in the form:
S
= µ ~
2
- ~
éq
5.3.1-1
ij
(
p
ij
ij)
Around the point of balance (- +
), one considers a variation S of the deviatoric part of
constraint:
S
= µ
~
2
-
~ éq
5.3.1-2
kl
(
p
kl
kl)
Calculation of ~ p
:
kl
It is known that:
~ p
=
3 S éq
5.3.1-3
kl
kl
By deriving this equation compared to the deviatoric constraint, one obtains:
~ p
= 3 S +3 S
éq
5.3.1-4
kl
kl
kl
Calculation of:
One a:
1 F
1 F
F
=
=
S
+
P
H
mn
H
S
mn
P
p
mn
p
mn
éq 5.3.1-5
= 1 [S
3
S
+ m2
2
-
mn
mn
(P Pcr) P]
H p
If one considers only the evolution of the deviatoric part of (P =)
0, then:
(H) = H + H =
3 S S + 3s S
- M2
2
PP éq
5.3.1-6
p
p
p
[mn mn
mn
mn]
Cr
However:
P
P = kP.
Cr
Cr
v
p
2
p
Like
= 2M (P - P),
has
one
=
2 M2 (P - P) - 2M2
P, éq
5.3.1-7
v
Cr
V
Cr
Cr
From where:
2
1
2 M (P - P) =
+ 2
2
M
P.
éq
5.3.1-8
Cr
Cr
kP
Cr
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In addition,
H = 4
4
km PP (P - P) and H = 4
4
km P (P - 2P) P. éq 5.3.1-9
p
Cr
Cr
p
Cr
Cr
By injecting this last equation in the equation [éq 5.3.1-6], one obtains:
H + 4KM 4P (P - 2P) + 2M2 P P =
3 S S + 3s S
éq
5.3.1-10
p
[
Cr
] Cr [mn mn mn mn]
While using the relation [éq 5.3.1-8], it comes then:
[3s S +3s S
mn
mn
mn
mn]
=
éq
5.3.1-11
(H +)
With
p
2
M (P P)
4
2
-
with A = [4k MR. P (P - 2P)
Cr
+ 2M P]
Cr
1
2
+ M
2kPcr
One then obtains immediately the variation of the deviatoric part of the plastic deformation:
~ p
=
9
9
6
S
S S + S S S +
S
S
S +
M2 P - P
P
S
kl
(mn mn kl mn mn kl)
mn
mn
kl
(
Cr)
kl
(H +
p
With)
H p
H p
éq 5.3.1-12
S is written then:
ij
18
~
µ
18µ
12µ
S = 2
µ -
S
S S + S S S -
S
S
S -
M2 P - P
P
S
ij
ij
([kl ij kl kl ij kl)]
kl
kl
ij
(
Cr)
ij
(H + A)
H
H
p
p
p
éq 5.3.1-13
who becomes by separating the terms in variation from constraints and the term in variation of deformation
total:
12µ
2
18µ
18µ
+
-
+
+
+
= 2µ~
ijkl
ijkl
M (P
Cr
P) P
(sklsij sklsij)
smn smn ijkl skl
ij
H
+
p
H p A
H p
éq 5.3.1-14
or in tensorial writing:
D
12µ
I 1
2
+
MR. P - P
éq 5.3.1-15
Cr P +
S S +
S + S S S =
4
(
)
18µ
18µ
:
(
)
µ~
2
H
H
(H
p
p
p +
)
With
that one can still write by symmetrizing the tensor (S + S
) S:
D
12µ
I 1
2
MR. P P
P
S S
+
- Cr +
+
S =
4
(
)
18µ
18µ
:
µ~
2
éq 5.3.1-16
H
H
(H
p
p
p +
)
With
1
with: = [
T
((S + S
) S) + (S (S + S
)) ]
2
Handbook of Référence
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Code_Aster ®
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Titrate:
Law of behavior CAM-CLAY
Date:
03/02/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key:
R7.01.14-B Page
: 25/34
Calculation of, while posing:
=
+
ij
T
sij
sij
T S
T S
T S
2T S
2T S
2T S
11 11
11 22
11 33
11 12
11 23
11 31
T S
T S
T S
2T S
2T S
2T S
22 11
22 22
22 33
22 12
22 23
22 31
éq 5.3.1-17
T
T S
T S
T S
2T S
2T S
2T S
S = 33 11
33 22
33 33
33 12
33 23
33 31
2T S
2T S
2T S
2T S
2T S
2T S
12 11
12 22
12 33
12 12
12 23
12 31
2T S
2T S
2T S
2T S
2T S
2T S
23 11
23 22
23 33
23 12
23 23
23 31
2T S
2T S
2T S
T S
2T S
2T S
31 11
31 22
31 33
31 12
31 23
31 31
1
= [
T
T
(S) + T
(S)]
éq 5.3.1-18
2
That is to say:
D
1
6
2
9
C =
9
I
MR. P P
P
S S
4
+
(- Cr)
+
: +
2µ
éq 5.3.1-19
H
H
(H
p
p
p +
)
With
one poses:
9
C =
(S
: S)
éq 5.3.1-20
H p
and
6
D =
M2 (P - P
Cr) P
éq 5.3.1-21
H p
The symmetrical matrix C of dimensions (6,6) is too large to be presented whole, one
break up into 4 parts C, C, C and C:
1
2
3
4
C
C
1
2
C =
C
C
3
4
with
1
9
9
9
+ C + D +
S T
(T S
T S)
(T S
T S)
11 11
11 22 + 22 11
11 33 + 33 11
2µ
(H p +)
With
2 (H p +)
With
2 (H p +)
With
9
1
9
9
1
C =
(22
T
11
S + 11
T s22)
+ C + D +
22
T s22
(22
T s33 + 33
T s22)
2 (H
µ
p +
)
With
2
(H p +)
With
2 (H p +)
With
9
9
1
9
(33
T 11
S + 11
T s33)
(22
T s33 + 33
T s22)
+ C + D +
33
T s33
2 (H
µ
p +
)
With
2 (H p +)
With
2
(H p +)
With
éq 5.3.1-22
Handbook of Référence
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Code_Aster ®
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Titrate:
Law of behavior CAM-CLAY
Date:
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Author (S):
J. EL GHARIB, G. DEBRUYNE
Key:
R7.01.14-B Page
: 26/34
9 2
9 2
9 2
(11
T 12
S + S T)
(
11 12
11
T s23 + S T)
(
11 23
11
T 13
S + S T)
11 13
(
2 H p +)
With
(
2 H p +)
With
(
2 H p +)
With
C2 =
9 2
9 2
9 2
(22
T 12
S + S T)
(
22 12
22
T s23 + S T)
(
22 23
22
T 13
S + S T)
22 13
(
2 H p +)
With
(
2 H p +)
With
(
2 H p +)
With
9 2
(33
T 12
S +
9 2
S T)
(
33 12
33
T s23 +
9 2
S T)
(
33 23
33
T 13
S + S T)
33 13
(
2 H p +)
With
(
2 H p +)
With
(
2 H p +)
With
éq 5.3.1-23
C =
3
C2
éq 5.3.1-24
1
18
9
9
+ C + D +
S T
(T S
T S)
(T S
T S)
12 12
12 23 + 23 12
12 23 + 23 12
2µ
(H p +)
With
(H p +)
With
(H p +)
With
9
1
18
9
C4 =
(23
T 12
S + 12
T s23)
+ C + D +
23
T s23
(23
T 13
S + 13
T s23)
(H
µ
p +
)
With
2
(H p +)
With
(H p +)
With
9
9
1
18
(13
T 12
S + 12
T 13
S)
(13
T s23 + 23
T 13
S)
+ C + D +
13
T 13
S
(H
µ
p +
)
With
(H p +)
With
2
(H p +)
With
éq 5.3.1-25
Calculation of the rate of variation of volume:
p
2
p
2
2
3B
= 2M (P - P), = 2M (P - P) - 2M P = B =
(S + S
). S
v
Cr
v
Cr
Cr
(H + A)
p
éq 5.3.1-26
2
M (P - P)
with: B = 2
2
M (P - P)
Cr
- 2 2
M
Cr
.
éq 5.3.1-27
1
2
+ M
2kPcr
or while using [éq 5.3.1-11]
p
3B
=
(S + S
). S éq
5.3.1-28
v
(H + A)
p
One thus has:
B
= (
-
(+)
ij
C
)
ijkl
S
S kl ij
skl
éq
5.3.1-29
(H +
p
With)
5.3.2 Processing of the hydrostatic part
It is considered now that the variation of loading is purely spherical (S = 0).
The increment of P is written in the form:
-
P = P exp (K
éq
5.3.2-1
0 E
v)
-
- P
Handbook of Référence
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HT-66/05/002/A
Code_Aster ®
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7.3
Titrate:
Law of behavior CAM-CLAY
Date:
03/02/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key:
R7.01.14-B Page
: 27/34
The derivation of this equation gives:
P = K P
0
(
p
-
v
v)
éq 5.3.2-2
Calculation of
p
:
v
It is known that:
p
=
2
2
-
v
M (P
Cr
P)
éq 5.3.2-3
By differentiating this equation, one obtains:
p
=
2
2
-
+ -
v
M (
(P Cr
P)
(P
Cr
P)
éq
5.3.2-4
One knows the expression of:
2M2 (P -
+ 3
Cr
P) P
S S
B
=
=
éq
5.3.2-5
H p
H p
while posing
B = 2M2 (P - P + 3
éq
5.3.2-6
Cr) P
S S
While differentiating, it comes:
2M2
=
([
4kM 4b
P -
+ -
-
-
+
-
Cr
P) P (P
Cr
P) P]
[Cr
PP (2P
Cr
P)
Cr
P P (P 2 Cr
P)]
H
2
p
H p
éq 5.3.2-7
One seeks the expression of P according to
:
Cr
One has
p
P = kP
éq
5.3.2-8
Cr
Cr
v
One can write:
Cr
P = 2M2 (P -
+
2
2
-
Cr
P)
M (P
Cr
P)
éq 5.3.2-9
Cr
kP
1+ 2M2 kP
P
Cr
= 2M2
2
-
+ 2
Cr
(P Pcr)
MR. P
éq 5.3.2-10
kP
Cr
2M2 (P - P
2 2
Cr) kP
M kP
P
Cr
Cr
=
+
P
Cr
éq 5.3.2-11
1+ 2kP
M2
1+ 2kP
M2
Cr
Cr
Handbook of Référence
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Titrate:
Law of behavior CAM-CLAY
Date:
03/02/05
Author (S):
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Key:
R7.01.14-B Page
: 28/34
One poses
2M2 Cr
kP (P - Cr
P)
C = [
,
éq 5.3.2-12
1 + 2M2 Cr
kP]
2M2 Cr
kP
has = [
éq 5.3.2-13
1+ 2M2 Cr
kP]
One has then:
P
éq 5.3.2-14
Cr =
p + C has
By replacing the expression of P in
[éq 5.3.2-7], one finds:
Cr
= [2
2
1
2M (P - P P + 2M P - C - aP P
.
Cr)
(
)] HP éq 5.3.2-15
4kM 4b
-
-
+ +
-
2
[PP 2P P
C
P P P 2P has
Cr (
Cr)
(
) (
Cr)]
H p
By gathering the terms in
and those out of P, one finds:
F
= P
éq 5.3.2-16
E
with,
1
F =
[2M2 (P - P + 2M2 P - 2AM 2P
Cr)
]
H p
éq
5.3.2-17
4kM 4b
-
-
+
-
2
([2P P P aP P 2P
Cr) Cr
(
Cr)]
H p
2cm2 P
bckM
4
4
E = 1+
+
P
-
éq
5.3.2-18
2
(P 2Pcr)
H
H
p
p
The expression of
p
thus becomes:
v
p
2
F
F
= 2M - has - C +
-
v
(P Pcr) P éq 5.3.2-19
E
E
from where the expression of according to P:
v
K P
P
0
=
v éq 5.3.2-20
G
2
G = 1+ 2M K P
0
- - F
has
+ F
C
(P - Cr
P)
éq 5.3.2-21
E
E
Handbook of Référence
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HT-66/05/002/A
Code_Aster ®
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7.3
Titrate:
Law of behavior CAM-CLAY
Date:
03/02/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key:
R7.01.14-B Page
: 29/34
Calculus of the variation of deviatoric deformation:
~
~ p
F
= = =
ij
3
S 3 Psij
éq 5.3.2-22
E
One thus has finally:
= F P
ij
éq 5.3.2-23
ij
with
3 F
G
D
F =
S -
1
éq
5.3.2-24
E
K
3 P
0
5.3.3 Operator
tangent
The tangent operator connects the variation of total constraint to the variation of total deflection. Being
given that the increment of the total deflection under loading deviatoric is written:
B
= C
(
-
(S + S
) D1
)
,
éq
5.3.3-1
ij
ijkl
kl
ij
klmn
mn
(H + A)
p
with:
2/3 - 1/3 - 1/3 0 0 0
- 1/3 2/3 - 1/3 0 0 0
1
- 1/3 - 1/3 2/3 0 0 0
D =
éq
5.3.3-2
0
0
0
1 0 0
0
0
0
0 1 0
0
0
0
0 0 1
projection in space deviatoric,
and that under spherical loading one a:
2
=
ij
ij
F Dkl
kl
éq
5.3.3-3
with:
- 1/
3
- 1/
3
2
- 1/
3
D =
éq
5.3.3-4
0
0
0
hydrostatic projection, one has then:
=
ij
ijkl
With kl éq
5.3.3-5
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A
Code_Aster ®
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Titrate:
Law of behavior CAM-CLAY
Date:
03/02/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key:
R7.01.14-B Page
: 30/34
with:
1
-
B
1
2
ijkl
With
= (Cijmn -
(S + S)
)
mn ij Dmnkl +
ij
F Dkl éq
5.3.3-6
(H p +)
With
the discretized tangent operator.
5.3.4 Tangent operator at the critical point
If the point of load is at the critical point (P = P, the general expression of the tangent operator is not
Cr)
more valid. This appears in particular by divide by 0 (see the equations of [§ 5.3.1]). One
detail in what follows the coherent tangent operator to the critical point while passing as for the case
General by the partly deviatoric and partly hydrostatic decomposition.
5.3.4.1 Processing of the deviatoric part
Let us recall that to the critical point, the expressions of the plastic multiplier and its derivation
are written in the following way:
Qe
E
E
=
-
Q
Q Q
1/6µ
and =
-
éq
5.3.4.1-1
Q
2
6µQ 6µQ
with,
E
E
3
3
E
S S
S S
Q =
and Q =
éq
5.3.4.1-2
E
2 Q
2 Q
from where the expression of:
1 3 sese
Qess
=
-
éq
5.3.4.1-1
E
3
6Μ 2 Q Q
Q
Let us point out in the same way the expression of S:
S = µ ~
2
-
3 S -
3 S
ij
(ij
ij
ij)
While replacing and by their expressions, one can write:
E
E
E
E
3 S
~
S
3 Q
Q
kl
kl
S = 2µ -
S +
S S S -
- 1 S
éq
5.3.4.1-2
ij
ij
E
ij
3
kl
kl ij
ij
2 Q Q
2 Q
Q
E
E
E
Q
3 Q
3 S.S
kl
ij
S
+
- -
S.S = 2µ
-
~ éq 5.3.4.1-5
kl
ijkl
ijkl
ijkl
3
kl
ij
ijkl
E
kl
Q
2 Q
2 Q Q
or in tensorial writing:
E
E
E
Q
D
3 Q
D
S
S
S
I -
S S
=
I -
éq
5.3.4.1-6
4
µ
3
2
3
4
~
Q
2 Q
2
E
Q Q
1
4
4
4
2
4
4
4
3
1
4
4 2 4
4 3
G
H
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A
Code_Aster ®
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7.3
Titrate:
Law of behavior CAM-CLAY
Date:
03/02/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key:
R7.01.14-B Page
: 31/34
As S does not depend on, one can confuse ~ with.
v
By using the tensor of projection in the space of the deviatoric constraints 1
D [éq 5.3.3-2], one
can write:
1
1
-
D.G.
=
H
.
éq
5.3.4.1-7
2µ
5.3.4.2 Processing of the hydrostatic part
In tensorial writing, one with the following relation:
D
I P = K P
.
éq 5.3.4.2-1
0
v
according to the equation [éq 5.3.2-2] with
p
at the critical point.
v = 0
As P then does not depend on ~ one can confuse with.
v
I dP = K
éq
5.3.4.2-2
0
P
By using the tensor of projection in the space of the hydrostatic constraints
2
D [éq 5.3.3-4], one
can write:
I D
=
D
2
éq
5.3.4.2-3
K P
0
5.3.4.3 Tangent operator
By combining the contributions of the two parts deviatoric and hydrostatic, one finds the writing of
the tangent operator who connects the variation of the total constraint to the variation of the total deflection to
not criticizes:
1
- 1
D G H
I D
.
2
=
+
D
.
2
µ
K P
0
or
= A
éq
5.3.4.3-1
ij
ijkl
kl
with
1
1
1
-
D.G. -
H
I D
2
Aijkl =
+
D
2
µ
éq
5.3.4.3-2
K P
0
Handbook of Référence
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HT-66/05/002/A
Code_Aster ®
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7.3
Titrate:
Law of behavior CAM-CLAY
Date:
03/02/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key:
R7.01.14-B Page
: 32/34
6
Examples of implementation of the model
6.1
Initialization of calculation
In model CAM_CLAY, the non-linear elastic law reveals a hydrostatic constraint
for a null voluminal deformation [éq 3.3-4]. One thus needs at the beginning for calculations to initialize
hydrostatic constraint with a strictly positive value. With this intention, one can proceed of two
different ways:
To carry out a linear elastic design by affecting boundary conditions such as the field of
forced in the structure is a uniform hydrostatic compression equal to pressure Pa
data in DEFI_MATERIAU. Pressure Pa corresponds to the initial index of the vacuums and is
generally equalizes with the atmospheric pressure (the latter is given positively to be
coherent with conventions of the civil engineering). One extracts from this calculation the stress field with
points of Gauss. This stress field is regarded as the initial state of the constraint
hydrostatic necessary to law CAM_CLAY in calculation STAT_NON_LINE using the model
CAM_CLAY.
To use operator CREA_CHAMP to create with operation “AFFE” a stress field
hydrostatic to the points of Gauss of value Pa, the constraint in this case is given of sign
negative (Aster convention for compressions) and the initial state in the STAT_NON_LINE constitutes
according to.
Numerical results for triaxial compression tests.
The following figures show triaxial ways of loading with evolutions of
axial deformation according to the diverter Q. They result from numerical calculations carried out
with model CAM-CLAY established in Code_Aster. These test were carried out by using one
modeling of the type KIT_HM in not drained condition (this condition allows us easily
to charge in a purely deviatoric way, the hydrostatic part of the loading being taken again by
pressure of water). The shapes of the curves obtained numerically with Code_Aster are very with
fact comparable with the diagrammatic curves presented in the paper of Charlez [bib2].
In the first test, the material is normally consolidated, i.e. hydrostatic pressure
of departure is equal to the pressure of consolidation (in this case
5
10
.
6
Pa). Work hardening (positive)
start at the beginning of the deviatoric phase, without preliminary elastic phase. Hardening
continue to a bearing of perfect plasticity when the critical point is reached (Q=MP).
As for the three other tests, the deviatoric phase starts for a value of the constraint
effective average lower than the pressure of consolidation, the material is of this surconsolidé fact.
If P is higher than P equal to
5
10
.
3
Pa, the specific point of the loading cross
Cr
surface of load before the critical line. There will be thus three specific phases: a phase
rubber band, a contracting plastic phase then a perfect plastic phase.
If P = P, the behavior is plastic perfect right after the elastic phase.
Cr
In the case where P is lower than P, the point representative of the loading cuts the critical line
Cr
before the surface of load which it reaches during a purely elastic way. In this
configuration, the behavior is lenitive and dilating and blocked energy decreases. The point
representative of the loading joined then the critical state where the material will enter in perfect plasticity.
The Cam_Clay behavior cannot produce a behavior continuement contractor/dilating.
not representative of the loading is obliged to pass by the critical state where the whole of the parameters
of work hardening (plastic voluminal deformation, critical pressure, blocked energy) become
stationary [bib2].
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A
Code_Aster ®
Version
7.3
Titrate:
Law of behavior CAM-CLAY
Date:
03/02/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key:
R7.01.14-B Page
: 33/34
6,00E+05
6,00E+05
5,00E+05
5,00E+05
Q=MP
4,00E+05
4,00E+05
Critical State
)
has)
3,00E+05
3,
P 00E+05
Q (Pa
Q (
2,00E+05
2,00E+05
hardening
1,00E+05
1,00E+05
0,00E+00
0,00E+00
0
100000 200000 300000 400000 500000 600000 700000
0, E+00 5, E-02
1, E-01
2, E-01
2, E-01
3, E-01
3, E-01 4, E-01
P (Pa)
eps1
6,00E+05
6,00E+05
5,00E+05
5,00E+05
4,00E+05
4,00E+05
radoucissement
critical state
3,00E+05
3,00E+05
Q (Pa)
Q (Pa)
2,00E+05
2,00E+05
1,00E+05
1,00E+05
rubber band
0,00E+00
0,00E+00
0
100000 200000 300000 400000 500000 600000 700000
0, E+00
5, E-02
1, E-01
2, E-01
2, E-01
3, E-01
P (Pa)
eps1
6,00E+05
6,00E+05
5,00E+05
5,00E+05
Q=MP
4,00E+05
4,00E+05
)
critical state
3,00E+05
Pa 3,00E+05
Q (Pa)
Q (
2,00E+05
2,00E+05
hardening
1,00E+05
1,00E+05
rubber band
0,00E+00
0,00E+00
0
100000 200000 300000 400000 500000 600000 700000
0, E+00
5, E-02
1, E-01
2, E-01
2, E-01
3, E-01
P (Pa)
eps1
6,00E+05
6,00E+05
5,00E+05
5,00E+05
4,00E+05
4,00E+05
Q=MP
)
critical state
3,00E+05
3,00E+05
Q (Pa)
Q (Pa
2,00E+05
2,00E+05
1,00E+05
1,00E+05
rubber band
0,00E+00
0,00E+00
0
100000 200000 300000 400000 500000 600000 700000
0, E+00 5, E-02 1, E-01 2, E-01 2, E-01 3, E-01 3, E-01 4, E-01
eps1
P (Pa)
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A
Code_Aster ®
Version
7.3
Titrate:
Law of behavior CAM-CLAY
Date:
03/02/05
Author (S):
J. EL GHARIB, G. DEBRUYNE
Key:
R7.01.14-B Page
: 34/34
7 Bibliography
[1]
I.B BURLAND, K.H. ROSCOE: One the generalized stress strain behavior off wet clay,
Engineering plasticity Cambridge Heyman-Leckie, 1968.
[2]
PH. A. CHARLEZ (Total Report/ratio): Example of model poroplastic: the model of
Camwood-Clay.
[3]
NR. TARDIEU, I. VAUTIER, E. LORENTZ
: Quasi-static nonlinear algorithm.
Reference material Aster [R5.03.01].
[4]
J. LEMAITRE, J.L. CHABOCHE: mechanics of solid materials, Dunod 1985
Handbook of Référence
R7.01 booklet: Modelings for Génie Civil and the géomatériaux ones
HT-66/05/002/A
Outline document