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Document: R7.01.03
Law of behavior to double Drücker criterion
Prager for cracking and compression
concrete

Summary

The model presented in this document is a nonlinear law of behavior for the concrete. It rests on
theory of plasticity, it is valid for the three-dimensional states of stress. Assumptions of
modeling retained are as follows:

· a field of reversibility of the constraints delimited by two criteria of the type Drücker Prager,
· a work hardening of each criterion,
· in compression, a positive work hardening to a peak, then a negative work hardening,
· in traction, a negative work hardening exclusively,
· a dependence of the shape of the curves post-peak in both cases (traction/compression) with the size
finite element (the shape of this curve is related on negative work hardening and the energy of cracking),
· normal plastic rules of flow (associated plasticity) and a formulation of work hardening
isotropic,
· the taking into account of the dependence of the thresholds of elasticity compared to the temperature,
· the taking into account of the dependence of the Young modulus compared to the temperature.
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Count

matters

1 Notations ............................................................................................................................................... 4
2 Introduction ............................................................................................................................................ 5
2.1 Principal characteristics of the model .......................................................................................... 5
2.2 Why two criteria of Drücker Prager ...................................................................................... 5
3 Field of reversibility and functions thresholds .......................................................................................... 6
3.1 Pace of the field and the thresholds of reversibility .............................................................................. 6
3.2 Mathematical expression of the field of reversibility .................................................................. 8
3.3 Criterion of rupture. choice of the coefficients has, B, C and D ........................................................................ 9
3.4 Analyze field of reversibility retained .................................................................................. 12
3.5 Work hardening .................................................................................................................................. 19
3.5.1 Functions of work hardening ...................................................................................................... 19
3.5.2 Curves of work hardening and modules post peak ........................................................................ 21
3.5.2.1 Model of cracking distributed ................................................................................. 21
3.5.2.2 Behavior of the concrete in traction and linear curve post-peak ............................. 24
3.5.2.3 Behavior of the concrete in traction and exponential curve post-peak .................... 24
3.5.2.4 Behavior of the concrete in compression and linear curve post-peak ..................... 25
3.5.2.5 Behavior of the concrete in compression and nonlinear curve post-peak .............. 26
4 plastic Flow .......................................................................................................................... 27
4.1 General form of the rule of normality ...................................................................................... 27
4.2 Expression of the plastic flow partly current ............................................................ 28
4.3 Expression of the plastic flow at the top of a cone ....................................................... 29
4.3.1 Demonstration by the general theory of standard materials ........................................ 29
4.3.2 Demonstration by plastic work ................................................................................. 32
4.4 Together equations of behavior (summarized) .................................................................. 33
5 numerical Integration of the law of behavior ............................................................................... 35
5.1 The total problem and the local problem: recalls ........................................................................ 35
5.2 Digital processing of the regular case. ......................................................................................... 37
5.3 Existence of a solution and condition of applicability ..................................................................... 40
5.4 Processing of the nonregular cases .................................................................................................. 41
5.4.1 Calculation of the constraints and plastic deformations .............................................................. 41
5.4.2 Acceptability ......................................................................................................................... 42
5.4.2.1 Acceptability a priori and a posteriori ......................................................................... 42
5.4.3 Existence of a regular solution and a singular solution. .......................................... 44
5.4.4 Inversion of the nodes of the cones of traction and compression ....................................... 45
5.4.5 Projection at the top of the two cones ................................................................................ 46
5.5 Determination of the tangent operator ............................................................................................ 47
5.5.1 Tangent operator of speed with only one active criterion ...................................................... 47
5.5.2 Tangent operator of speed with two active criteria ....................................................... 48
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5.5.3 Derivative successive of the criteria in traction and compression ...................................... 49
5.5.3.1 Successive drifts of the criteria compared to the constraint .................................. 49
5.5.3.2 Successive drifts of the criteria compared to the plastic multipliers ......... 49
5.6 Variables intern model ....................................................................................................... 50
5.7 Top-level flowchart of resolution ........................................................................................... 50
Appendix 1
snap-back with the initial values of the coefficients C and D .................................... 56
6 Bibliography ....................................................................................................................................... 60

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1 Notations

indicate the tensor of constraint, arranged in the form of vector according to convention:

11


22


33

12
13


23

One notes:

I

1 = Trace ()

= 1 tr

the hydrostatic constraint
H
()
3
1

S = - tr () I
the diverter of the constraints
3

= 1 tr
voluminal deformation
H
()
3

~ = - 1 tr () I
the diverter of the deformations
3

&
~
3


the rate of deformation is equivalent
eq =
trace (~.
2
& &~)
1

J = trace (2
S
the second invariant of the constraints
2
)
2

eq
3
= 3J =
trace (2
S
the equivalent constraint
2
)
2
2
trace (2
S)

=
J =

Oct.
3 2
3
I
trace

1
()


Oct. = H =
=
3
3

F


C
initial limit of rupture in simple compression
F
initial limit of rupture out of Bi compression
DC
F



C
elastic limit in compression
F


T
initial limit of rupture in traction
F

T
=

relationship between rupture limit in traction and compression
F C
F

DC
=

relationship between rupture limit in Bi-compression and simple compression
F C
p



plastic deformation in traction
T
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T
plastic multiplier in traction
p



plastic deformation in compression
C

plastic multiplier in compression
C

F (p

C
curve of work hardening in compression
c)

F (p

T
curve of work hardening in traction
T)
U



T
ultimate plastic deformation in traction
U



C
ultimate plastic deformation in compression
F

G
C
energy of rupture in compression (characteristic of material)
F

G
T
energy of rupture in traction (characteristic of material)

the maximum of the temperature during the history of loading

2 Introduction
2.1
Principal characteristics of the model

The model presented in this document is a nonlinear law of behavior for the concrete. It
be based on the theory of plasticity, it is valid for the three-dimensional states of stress.
assumptions of modeling selected partly take again the models developed per G. Heinfling
[bib2] and J.F. Georgin [bib1] and are as follows:

· there is a field of reversibility of the constraints delimited by two criteria of the Drücker type
Prager,
· each criterion is hammer-hardened, the field of rupture corresponds to the maximum of the field of
reversibility,
· in compression, work hardening is positive to a peak, then it becomes negative,
· in traction, work hardening is negative exclusively,
· the curves post-peak in both cases (traction/compression) vary with the size of
the finite element (the shape of this curve is related on negative work hardening and the energy of
cracking),
· the plastic flow is governed by a rule of normality (associated plasticity) the formulation
work hardenings is of isotropic type,
· the modulus of elasticity and the thresholds of reversibility vary with the temperature.

Note:

The terminology of criterion of traction and criterion of compression is debatable. Us
will use by practice, while being quite conscious that a state of tensile stresses
can lead to the activation of the criterion known as of compression.

2.2
Why two criteria of Drücker Prager

The authors of the theses referred to [bib1] and [bib2] use a criterion of Drücker Prager in
compression and a criterion of Rankine in traction. They justify these choices by considerations
physics by showing that the field of reversibility thus obtained is close to reality
experimental. On the other hand they limit their modelings in states of two-dimensional stresses.
We preferred to also replace the criterion of traction by a surface of the type Drücker Prager.
By this choice, one frees oneself from certain difficulties particularly in the formulations
three-dimensional. Surface 3D defining the working states of stresses with respect to
traction is not any more one pyramid (Rankine 3D) but a conical surface whose node is located on
the hydrostatic axis. The trace of the criterion “known as of traction” on the plan deviatoric is not any more one triangle,
but a circle. The formulation obtained is simpler. The difference between the two criteria is
tiny for states of stress close to states of plane stress. On the other hand, for the states of
constraint strongly confined, the two approaches (of Rankine and Drücker Prager) are different,
what is a limit of the model suggested.
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3
Field of reversibility and functions thresholds

3.1
Pace of the field and the thresholds of reversibility

The field of reversibility is the field of the space of the constraints inside of which ways of
constraint are reversible. In the space of the principal constraints (, they are two
1
2
3)
cones whose axis is the trisecting one of equation = =. [Figure 3.1-a] one gives some
1
2
3
chart.


2
1
3
Appear 3.1-a


25
Oct.
C
20
15
10
Field of
reversibility
5
B
Oct.
0
-
-
-
0
10
P
P
T
C

Appear 3.1-b
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In a plan (,
the field of reversibility is determined by two lines as indicated on
Oct.
Oct.)
[Figure 3.1-b].
For a state of stress planes, the field of reversibility is the cut of the three-dimensional field
by a plan of equation = cste, as indicated on [Figure 3.1-c], the result in a plan
3
(, being represented on the figure [Figure 3.1-d].
1
2)


2
1
3
Appear 3.1-c


2
1

Appear 3.1-d

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3.2
Mathematical expression of the field of reversibility

It is defined by the inequation:

F (, A) 0










éq 3.2-1
in which A represents the thermodynamic forces associated with the variables intern (us
let us note the whole of the internal variables).
For the model concrete that we present here, the equation [éq 3.2-1] takes the particular form

has
F
With
F
With

éq
3.2-2
comp (
c)
Oct. +.
,
=
Oct. - C + C 0
B
C
F
With
F
With
éq
3.2-3
trac (
T)
Oct. +.
,
=
Oct. - t+ T 0
D
has
H
F
(A).
,
=
Oct.
F
With

éq
3.2-4
C
- C + C 0
comp
B
C
H
F (A)
.
,
=
Oct.
F
With

éq
3.2-5
T
- t+ T 0
trac
D

The equations [éq 3.2-2] and [éq 3.2-3] correspond respectively to the thresholds of “compression” and
of “traction”. The equations [éq 3.2-4] and [éq 3.2-5] limit the threshold of reversibility in the field
isotropic traction, they amount excluding the X-axis on [Figure 3.1-b] beyond
points P or P. It is clear that only one of these two last conditions is enough. For material not
T
C
hammer-hardened, the choice of the coefficients is such as COp < COp and the condition [éq 3.2-5] involves [éq 3.2-4].
T
C
We will see later that work hardening can reverse the command of the points P and P, returning the condition
T
C
[éq 3.2-4] more constraining than [éq 3.2-5].
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3.3
Criterion of rupture. choice of the coefficients has, B, C and D

When the state of stress reaches the edge of the field of reversibility, of the plastic deformations
develop and the thresholds move: they are hammer-hardened. The threshold of compression “increases” in
the first time, then decreases, whereas the threshold of traction can only decrease. The threshold of rupture
corresponds to the maximum field being able to be reached, it is represented on [Figure 3.3-a] in one
diagram of plane constraint:

Initial threshold of reversibility
in compression
2
1
Threshold of rupture
in compression

Appear 3.3-a

The work hardening of the thresholds results mathematically in the evolution of quantities A and A, them
C
T
thresholds of rupture corresponding to the maximum of the functions F = F - A and F = F - A. Dans them
C
C
C
T
T
T
models selected, these functions are such as: Max F = F and Max F = F;
C
C
T
T

The coefficients has, B, C, and D are thus defined from:

· ft': resistance in axial traction plain of the concrete,
· fc': resistance in axial compression plain of the concrete,
· fcc': resistance in axial compression Bi of the concrete,
F
F
One defines moreover coefficients:
T
=
and
DC
=

F
F
C
C
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To determine the coefficients has, B, C and D it is necessary to give itself 4 equations which express in fact that them
criteria are reached for states of stresses particular and judiciously selected.
A first possibility consists in writing that the two criteria are cut on the axes compression
simple (points C of [Figure 3.3-b]).


With
C
D

Appear 3.3-b

By recalling that:


2
In simple compression:
<
0;

Oct. =
;Oct. = -

3
3

2
Out of Bi compression <
0;

Oct. = 2
;Oct. = -

3
3


2
In simple traction >
0;

Oct. =
;Oct. =

3
3

The following relations then are obtained:

Number
State of stress
Criterion reached
relation obtained
of condition
1 Compression
simple
Compression
+ 3b = 2 has



3
2
Bi compression
Compression
2a + B = 2

3 Traction
simple Traction - C + 3D = 2
4 Compression
simple
Traction C + 3D = 2
Table 3.3-a
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F
F
Who gives, while posing:
T
=
and
DC
=

F
F
C
C
- 1
2
has =

2
B =

éq
3.3-1
2 - 1
3 2 - 1
1
C = 2
D = 2 2 1 éq
3.3-2
1+
3 1+
But this choice is problematic.

Indeed, after work hardening of the criterion of traction, and for a limit of traction become null it
field of admissibility takes the form indicated on [Figure 3.3-c], making nonacceptable of
Bi compressions states.


Appear 3.3-c

Moreover, with this choice of the coefficients, certain ways of simple traction compression presented
snap-back as indicated in appendix.

We then preferred to replace the condition number 4 of [Tableau 3.3-a] by a condition
expressing that, after the limit of traction fell down to zero, the field of reversibility is that
represented on [Figure 3.3-d].


Appear 3.3-d
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This resulted in replacing the relation C + 3D = 2 by C = 2
The choice of the coefficients has, B, C and D is finally:

- 1
2
has =

2
B =

éq
3.3-3
2 - 1
3 2 - 1
2 2
C = 2D =

éq
3.3-4
3


model selected

Appear 3.3-e

[Figure 3.3-e] shows the difference between the two models for a state of plane constraint.

3.4
Analyze field of reversibility retained

In this chapter, we give indications on the order of magnitude of working stresses to
feel criterion selected. We endeavor to give indications on tensile stresses,
in particular for three-dimensional states of stress.
[Figure 3.4-a] shows the initial fields (i.e. before work hardening) for the values
following of the parameters materials:

F


F = 40 Mpa
C
initial limit of rupture in simple compression:
C
F


F = 44 Mpa
DC
initial limit of rupture out of Bi compression
DC
F


DC
=

relationship between rupture limit in Bi-compression and simple compression
= 1.1
F C
F



=

C
elastic limit in compression;
33
,
0
F


F = 4 Mpa
T
initial limit of rupture in traction
T
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2
1
3
Appear 3.4-a

The figures [Figure 3.4-b], [Figure 3.4-c] and [Figure 3.4-d] the cuts of the field show
three-dimensional by plane = 0 and = -
Mpa

25
.
3
3


Plan = Mpa

0
3
2
1
3

Appear 3.4-b
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Plan = - Mpa

25
3
2
1
3

Appear 3.4-c


Plan = Mpa

0
3
Plan = - Mpa

25
3
2
1
3

Appear 3.4-d

[Figure 3.4-e] shows the fields of reversibility in a plan (, for states of
1
2)
constraint constant, fields parameterized by the value of. We represent the fields
3
3
for = - 25 Mpa, = 0 Mpa, = 4 Mpa, = 10 Mpa, = 15 Mpa. One sees there that for one
3
3
3
3
3
containment of 25 Mpa of compression, stresses tensile can reach 15 Mpa, and that,
in parallel, the field of reversibility for = 15 Mpa N `is not empty and corresponds to
3
compressive stresses and about - 25 Mpa. It is also seen, that, for a value
1
2
data of, the maximum value of traction
3
Obtained for and is reached with the intersection of the criteria of traction and compression.
1
2
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Field of rupture = - Mpa

25
3
Field of rupture = Mpa

0
3
Field of rupture = Mpa

4
3
Field of rupture =
Mpa

10
3
Field of rupture
= Mpa

15
3
Field of rupture = - Mpa

25
3
Appear 3.4-e

We thus study the place of intersection of the criteria of traction and compression. We note
(0 eq
, the point of intersection of the two criteria in the plan (
eq
, (not C of [Figure 3.4-f]).
H
)
H
0)


eq

C
eq
0
Field of
reversibility
H
0

P
P
H
T
C

Appear 3.4-f
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The place of intersection of the two criteria in the space of the constraints is given by:


2 eq



sin
1 =

+ + 0
0
H

3

6

2 eq



sin

2 =

- + + 0
0
H

3

6
= 0
3
3
H - 1 -

2

Where is a parameter.



Appear 3.4-g

[Figure 3.4-g] shows projections of this place in the plans (, and (.
2
3)
1
2)
One can easily calculate the maximum value of the constraint along this curve:

F
2
C


éq
3.4-1
max =
+
F T
3

3

This equation shows that, whatever the value chosen for the rupture limit in traction,
maximum constraint attack in traction is higher than the third of the rupture limit in compression.
[Figure 3.4-h] shows the three principal constraints according to the parameter.
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Appear 3.4-h

It is seen that one can reach a level of traction of 15 Mpa, but for a containment of
= - Mpa
25
and = - Mpa
25
.
2
3

To try to avoid this disadvantage, which is important, one can try to exploit the values of
resistance in compression and the parameter.
As example, we chose the following play of parameters:

F = 20 Mpa
C
F = 40 Mpa
DC
= 2
F = 4 Mpa
T

[Figure 3.4-i] shows the criteria with this choice of parameters. [Figure 3.4-j] the value shows of
principal constraints with the intersection of the two criteria for this new choice of parameters.
maximum traction obtained is weaker (8 Mpa), but it is reached for a level of containment
also low (- 7Mpa).
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Initial criterion of traction with = Mpa

0
3
Initial criterion of traction with = - Mpa

8
3
Limit of the initial elastic range
in compression with
= Mpa

0
3
Criterion of the initial peak of compression with
= Mpa

0
3
Criterion of the initial peak of compression with
= - Mpa

8
3
Appear 3.4-i



Appear 3.4-j
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3.5 Work hardening

As we already mentioned in the paragraph [§3.3], when the state of stress reaches the edge
field of reversibility, the plastic deformations and the variables intern develop, them
thresholds move: they are hammer-hardened. For our model, the variables intern are
two, they are noted p
for the internal variable “known as of compression” and p
for that “known as of
C
T
traction
”. These variables determine the evolution of the thresholds of compression and traction
respectively, the thermodynamic forces theirs are connected by the relations:

With = F - F
éq
3.5-1
C
C
C (p
c)
and
With = F - F








éq 3.5-2
T
T
T (p
T)
where F and F represent the values of resistances in compression and traction
T (p
T)
C (p
c)
respectively.

3.5.1 Functions
of work hardening

The function F is initially increasing then decreasing, the decreasing part being is linear
C (p
c)
[Figure 3.5.1-a], that is to say quadratic [Figure 3.5.1-b],


F
p

C (c)
f' C
F C
PC

U
E
C

Appear 3.5.1-a

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F
p

C (c)
f' C
F C
p

C
E
CPU

Appear 3.5.1-b

is a data of the model. Shape of the curve between E
and U
(negative work hardening) depends on
C
C
the element, and more precisely of its dimensions, according to a criterion similar to that chosen by
G. Heinfling, [Error! Source of the untraceable reference.] for the taking into account of the localization of
deformations.
In traction, shape of the curve giving the value of the elastic limit F according to
T (p
T)
cumulated plastic deformation p
do not comprise a part “pre-peak”, the part “post-peak” being is
T
linear [Figure 3.5.1-c], that is to say exponential [Figure 3.5.1-d].


F
p

T (T)
ft
G ft
Pt
C

Appear 3.5.1-c
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F
p

T (T)
ft
U
p
T
T

Appear 3.5.1-d

3.5.2 Curves
of work hardening and modules post peak

3.5.2.1 Model of cracking distributed

The introduction of a behavior softening post-peak into the relations stress-strains
pose a major problem. Under statical stress, beyond a certain level of constraint,
corresponding to the starter of the lenitive behavior, equations governing the balance of
structure lose their elliptic nature. These equations of the mechanical problem form one then
system of partial derivative equations evil posed of which the number of solutions is multiple. It
problem results in an not-objectivity compared to the grid. It results from this a sensitivity
pathological of the numerical solution to the smoothness and the orientation of the grid.
In order to solve this problem, or at least, to limit the consequences of them on the reliability of the solution
predicted, it is necessary to use techniques known as of regularization. The object of these techniques is
to enrich the mechanical description of the medium, to be able to describe nonhomogeneous states of
deformation, and to preserve the mathematical nature of the problem. One operates this regularization in
introducing, in the law behavior, a characteristic length or internal length, connected to
width of the zone of localization. Several techniques are possible to improve description
mechanics of the lenitive medium. They constitute limitings device of localization. The implementation of
these techniques requires in general, of the delicate numerical developments. An approach
intermediary enters the use of the traditional models and the placement of these limitings device of
localization consists in making depend the slope post-peak on the relation stress-strain, of the size
element, so as to dissipate with the rupture a constant energy. This approach constitutes one
not towards a nonlocal description of the continuous medium.
Let us consider initially a real fissure of surface S whose measurement is A [Figure 3.5.2.1-a]. S is one
surface discontinuity of the field of displacement U. It is supposed that to create this discontinuity,
it is necessary to spend an energy W whose expression is: W = G (X) dS, G being a property of
S
F
F
material.
Let us consider now that one wants to represent the same phenomenon, while representing not
a discontinuity of displacement but a plastic deformation uniformly distributed in one
volume V.
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tr
D
ij
p
Dissipated energy will be: W = FD
ij dt, where one noted T the “time-to-failure”.
V

dt
R
0
S
V




Lver

Appear 3.5.2.1-a
By making the series of following assumptions:

· the fissure is plane,
·
G is constant along the fissure and thus W = A G
.
,
F
F
·
V is a basic cylinder S and a height L,
worm
tt
D p
·
ij
ij
G
dt is constant in V.
F =

dt
0

One leads finally to the relation:
tt
p

D
ij
ij
W = Vg = V

dt

= AG
.
éq
3.5.2.1-1
F
F

dt
0
Or:
tt
p

D
G
F
ij
ij
F
G =

dt

=
éq
3.5.2.1-2

dt
L
0
worm
It is seen easily that: G F = U F
, writing in which the quantities (F
G, F,
U)
0
() D
represent (F respectively
U
p
G, F, in traction and (F
U
p
G, F, in compression.
C
C
C
c)
T
T
T
T)
data of F
G thus determines, this in traction as in compression:
U
U
C

G
F
G =
F



=

C
(p D
C
c)
F
p
C

L
0
worm
U
T

G
F
G =
F



=

T
(p D
T
T)
F
p
T

L
0
worm
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The quantity F
G is thus related to the slope of the curve post peak in a diagram constraint-variable
of work hardening, which is related to the forced slope post peak in a diagram deformation.
Let us suppose for example that the forced relation deformation is linear in mode post peak.
Let us call E
the slope post peak in the diagram (,) and H < the 0 corresponding slope in
T < 0
EE
He
of diagram (F,) [Figure 3.5.2.1-b]. There is the relation H
T
=
E =
who show
E - E
T
E + H
T
that one must have:
- H < E, or else the diagram (,) presents a snap back.

ft

ft
EE
E
H
T
=
T
G F
E
E - E
T
T
T


Appear 3.5.2.1-b
The condition - H < E is known as condition of applicability, it will result in an inequality on F
G and thus
on L.
worm
Within the framework of a resolution by the finite element method, representative elementary volume
fissured medium can be compared to an element of the grid. The characteristic length (noted by
continuation LLC) introduced into the method of the energy of equivalent rupture corresponds to the Lver length.
During a calculation corresponding to an unspecified structure, determination this length
characteristic is delicate. It depends on the position of the plan of fissure, dimensions and the type
elements…
A simple estimate for the two-dimensional cases can be expressed in the form:
L = R A where A
C
E
E is the surface of the element considered, and R, a correct factor, being worth 1 for
quadratic elements, and 2 for the linear elements.
One can extend this formulation to the case 3D:
3
L = R V where V
C
E
E indicates the volume of the element.
Concerning the evolution of work hardening with the temperature, we regard as in [bib2] that
energies of rupture and resistances to rupture not depend on the current temperature T
material point considered at time T, but of the maximum temperature reached in this point since
the beginning of the loading until time T. When we need to show the dependence of
quantities compared to the temperature, we will note:

· the maximum of the temperature since the beginning of loading,
·
F
resistance in compression,
C ()
·
F (indicates resistance in traction,
T)
·
F, the curve of work hardening in compression,
C (
p
c)
·
F, the curve of work hardening in traction.
T (
p
T)
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3.5.2.2 Behavior of the concrete in traction and linear curve post-peak

In this modeling, the concrete is supposed to be elastic until its resistance in traction F.
T
curve F in traction is represented on [Figure 3.5.1-c] and is entirely defined by
T (p
T)
resistance in traction of material, the energy of cracking
F
G, and characteristic length LLC.
T
The mathematical expression of this curve is:
F
,
F
1

éq
3.5.2.2-1
T (
p
Pt) = ()


-
T
T

C ()

The equivalence of dissipated energy makes it possible to write:

C
U
F
G
L
F
,
L F
1
D

T (
T
p
) = C T (Pt)

= C T ()

T
p

-
0
0
C ()


T
from where


L F
F
G

éq
3.5.2.2-2
T (
U
)
.
C
T () .t ()
=
2
and
F

G
.
2
U
éq
3.5.2.2-3
T ()
T ()
=

L. F
C
T ()
The condition of applicability is written:
.
2nd (). F
WP ()
L

éq
3.5.2.2-4
C
2
ft ()

3.5.2.3 Behavior of the concrete in traction and exponential curve post-peak

In this modeling, the concrete is supposed to be elastic until its resistance in traction F.
T
curve F in traction is represented on [Figure 3.5.1-d] and is entirely defined by
T (p
T)
resistance in traction of material, the energy of cracking
F
G, and characteristic length LLC.
T
The mathematical expression of this curve is:

F
,
F
.exp
has

éq
3.5.2.3-1
T (
p
Pt) = ()



-
T
T

C ()
The equivalence of dissipated energy makes it possible to write:



F
G
L
F
L F
has
D

T (
p
) = C T (, Pt)

= C T ()

T
p


exp -
U
T
0
0

T ()



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From where:

U
.
.

G ft () L F
C
T () T ()
=
éq
3.5.2.3-2
has
and
C ()
F
WP ()
=

éq
3.5.2.3-3
has

LLC. ft ()


p
p
That is to say still: F
,
F
.exp
L. F

T (T) =
T (
)

- C T () T

F
WP ()

2
L. F
C
T

The maximum slope of the curve is then H

max ()
()
= -
F
WP ()
and the condition of applicability is written:

E (). F
WP ()
L
éq
3.5.2.3-4
C


2
ft ()

3.5.2.4 Behavior of the concrete in compression and linear curve post-peak

In this modeling, the behavior of the concrete is supposed to be elastic until the elastic limit,
data by a proportionality factor (noted) expressed as a percentage of resistance to the peak F
.
C ()
For the standard concretes is about 30%. The curve F in compression is represented
C (p
c)
on [Figure 3.5.1-a] and is entirely defined by resistance in traction of material, the energy of
cracking
F
G, and characteristic length LLC.
C
The mathematical expression of this curve is:



U
p 2
FC (, PC)
C

= F 'C () + (2 -
2)
C

+ - 1
if p
C E
éq 3.5.2.4 - 1





2
E ()
(
)
()
E ()






p
U
F
,
F '
(C - c)
C (c) = C ()
()
if E PC U

U

C
éq 3.5.2.4 - 2

(E () - C ())
()
()


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FC
Resistance in maximum compression is reached when:
2 2

E () = (
-) ()
E ()
The equivalence of dissipated energy makes it possible to write:
U
C
F
G = L
F, D
C ()
C C (
p
c)
p
C
0
from where
F
2
1
1
G
L. F

éq
3.5.2.4-3
C ()

= C C () + E () + CPU ()


6
2

and
F
U
G
.
2
2
1

éq
3.5.2.4-4
C ()
C ()
+
=
-
E ()

L. F
3
C
C ()


FC
The slope of the curve is then H ()
()
= -

CPU () - E ()
and the condition of applicability is written:

E (). F
Gc ()
L

éq
3.5.2.4-5
C
6


2
FC (
2
) 11 -
4 -
4

3.5.2.5 Behavior of the concrete in compression and nonlinear curve post-peak

In this modeling, the behavior of the concrete is supposed to be elastic until the elastic limit,
data by a proportionality factor (noted) expressed as a percentage of resistance to the peak F
.
C ()
For the standard concretes is about 30%. The curve F in compression is represented
C (p
c)
on [Figure 3.5.1-b] and is entirely defined by resistance in traction of material, the energy of
cracking
F
G, and characteristic length LLC.
C
The mathematical expression of this curve is:



p
p 2
FC (, PC)
C

= F C () + (2 -
2)
C

+ - 1
if p
C
E
éq 3.5.2.5 - 1



2
E ()
(
)

()


E ()






2

(PC - E)
FC (, c) = F C ()
()
1
if E PC CPU
éq 3.5.2.5 - 2


U
2


(C () - E ())
()
()


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FC
Resistance in maximum compression is reached when:
2 2

E () = (
-) ()
E ()
The equivalence of dissipated energy makes it possible to write:

U
C
F
G = L
F, D
C ()
C C (
p
c)
p
C
0
from where:
F
2
G
L. F

éq
3.5.2.5-3
C ()

= C C () CPU ()
+ E ()


3
3

and:
F
U
3 G

éq
3.5.2.5-4
C ()
C ()

=
- E ()
2

L F
2
.
C
C ()

.
2 FC
The maximum slope of the curve post-peak is then H

max ()
()
= - CPU () - E ()
and the condition of applicability is written:
F
3rd () .G ()
L
C
C
1


éq
3.5.2.5-5
2
2
FC (
2
) 4 - -

4 Flow
plastic

In this paragraph, we give the expression speeds of plastic deformation, while distinguishing
the case says general where the state of stress is located on a “regular” zone of the edge of the field of
reversibility and the case where it is at the top of one of the cones.

4.1
General form of the rule of normality

In space (, A), the inequalities [éq 3.2-2], [éq 3.2-3], [éq 3.2-4], [éq 3.2-5], define one
convex field which we will note C
the indicating function of this convex:
(
. We will note
, A)
C

(
if

0
, A) C
(, A)

éq
4.1-1
C
=
(, A)
if not

When the border of the field of reversibility is reached, of the irreversible plastic deformations
develop, according to the classical theory of plasticity.
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For a standard material, [bib4] the law of flow checks the principle of maximum plastic work,
what results in the equation: (p) éq 4.1-2
C
&, &
where
note under differential of the function. We point out [bib3] that under differential of one
C
C
convex function in an item X is the whole of vectors Z such as
:
F (*
X) F (X)
*
*
+ Z, X - X X

It is then seen easily that [éq 4.1-2] involves:


éq
4.1-3
C (*
*
, A) (, A) + (*
-) + (*
WITH - WITH)
*
*

With
and

&p
C
&

Taking into account the definition of the characteristic function, one to see easily that [éq
4.1-3]
is
equivalent with:

p
p
+ A
*
+
*
With
*
and

*
WITH C (
éq
4.1-4
, A)
& & &
&

In other words the plastic flow is such as the couple (, A) carries out the maximum of
plastic dissipation among the acceptable thermodynamic forces.

4.2
Expression of the plastic flow partly current

When the function F is differentiable at the point considered (,) the rule of normality is written
simply
p
F
=
& & éq
4.2-1

F
=
& & éq
4.2-2
With


& and F checking the conditions of Kuhn-Tucker:
0
&
F 0 éq
4.2-3

. F = 0
&

The variable of work hardening is related to the plastic multiplier by the law of work hardening. By using it
plastic work, one can write:
p
F


& = &.
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F

If F is a homogeneous function of command 1 compared to the tensorial variable, one has
= F, it
who leads to the equality: =
& & and thus finally with the equations:
F
p

éq
4.2-4
C = p
comp
C
& &

F
p
p


éq
4.2-5
T =
trac
T
& &


4.3
Expression of the plastic flow at the top of a cone

We give two presentations of the same result. The first presentation uses the theory of
standard materials generalized and under differentials, the second share of an equality posed a priori
on plastic work.

4.3.1 Demonstration by the general theory of standard materials

The field C (


consists of two cones. The function
is not differentiable is with
, A)
C
the intersection of these two cones, is at the top of each one of these cones. When the point (, A)
belongs to the intersection of the two cones, the preceding equations remain valid, with
precision that the deformations figure of compression and traction develop into same
time. This case known as “multi criterion” is remainder treated in [bib4]. We will be satisfied here to treat
the case where (, A) is at the top of a cone, and we will choose the most frequent case of the node of
cone of traction, knowing that the case of the node of the cone of compression is treated exactly
even way.
The criteria are rewritten by using the variables eq
and, more practical in the development
H
analytical.
2
C
F
(, A)
F
With

4.3.1-1
trac
=
eq
T
+ H - t+ T 0
3D
D
H
F (
C
, A
F
With

4.3.1-2
T) =
H - t+ T 0
trac
D
We thus consider a case where:
eq = 0


C









4.3.1-3

F
With
H -
t+ T = 0
D

On the basis of [éq 4.1-4], we will calculate plastic dissipation as being the maximum of
p *
*

With
& + & for all the couples *, * A C (
, A)

p
D = Max (p *
*

With)
4.3.1-4
,
C, A &
+ &
* With *
(
)
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By writing whereas this maximum is finished and reached when * = and A * = A, we will find
conditions on p
& and &. In fact, the finished character will be enough.
By using the partly isotropic decomposition of the tensors and déviatoire, and the particular form of
variables of work hardening, one finds easily:
p *
*
p *
*
p
p
*
+ A = ~
S + 3 + A
&
&
&

4.3.1-5
H H
T
T
& &
Then let us consider the whole of the vectors forced of null trace and of which the constraint
1
equivalent of Von Mises 1 is worth: = eq =
=
1
{,
,
1
(
trace)
} 0
=
eqs
I
1 +

H
s1
1
(, A)


2
C
C (

4.3.1-6
, A)

eq +
F
With
H -
t+ T 0
3D
D

C
F
With
H -
t+ T 0
D
In other words, the “direction” of the diverter of the constraints is unspecified for a couple
(, A) C (.
, A)
One can thus write:
p

*
eq
p *
*
p
p *
D =

Max ~
Max

4.3.1-7
S
& s1 +3H + A
H
T
T
& &
*


1 1

eq
*
**

,
,
, S

T
With
H
1
1
2
*
eq

+ C *
- F + * 0

3D
D H
T
With
T

C *
*

- F + 0
D H
T
With
T
~
It is clear that the maximum of
*
PS
& when * sest is reached “parallel” with p
&~ and that one has then:
1
1
p *
2 p
Max ~
S
.
1 =

1
S
1
&
~eq
3 &



[éq 4.3.1-7] can thus be written:
p
2
*
~ p eq
D =
eq +
Max
&
Max
(*
3
pH
H
)

+
&
*
3

*
eq
*
*
eq
*
*



,
,


,
,
T
With
H
T
With
H

2
*

eq
C *
*
2
*

F
0


+ - +


eq + C * - F +
* 0

3D
D H
T
With
T
3D
D H
T
With
T



C

*
*
C *
*

- F +

0

- F +

0
D H
T
With
T
D H
T
With
T




4.3.1-8
+
Max
(p * A
T
T)


&
*

eq *
*
,
, T
With
H

2
*
eq
C *
*



+ - F +

0
3D
D H
T
With
T


C *
*


- F +

0
D H
T
With
T



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Like ~
peq 0
&
, one has for the first term:
2
*
p
eq
2
~
p
3
~
D * 3D
3c *
Max

With
F
4.3.1-9
eq
= eq -
+
T

&
&
*
3
3

2 T
2
2 H
eq *
*
,
, T
With
H
2
*

eq + C * - F +
* 0
3D
D H
T
With
T

C *
*

- F +

0
D H
T
With
T
[éq 4.3.1-9] deferred in [éq 4.3.1-8] gives:

p
~ p
D = 2D F +
*
3
-
C ~
2
+
With * -
D ~
2
4.3.1-10
eq
T
&
Max ((p
p
H
eq
*
*
&
&
Max
*
*
&
&
H
)
(T (p
p
T
eq)
, your
,
H
T
With
H
C *
*
C
- F +

0
* - F +
* 0
D H
T
With
T
D H
T
With
T
Let us pose then:
p
~ p
m =
3
- 2c,
p
~ p
N = - 2D and Q =
D p

eq F
H
eq
&
&
T
eq
&
&
&
~
2


With these notations, [éq 4.3.1-10] becomes:

p
D = Q +
Max (*
*
m + Na
4.3.1-11
T
H
)
*
*
, your
H
C *
*
- F +

0
D H
T
With
T

It is about a problem of the type “simplex”. The field of *
*
, A is represented on
H
T
[Figure 4.3.1-a].


*
At
F T
*
*
Field, A
H
T
*
D
H
F T
C

Appear 4.3.1-a
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Like the field *
*
, A extends towards - at the same time for *
and *
With, so that p
D is finished, it is necessary
H
T
H
T
that m and N are positive. The maximum of
*
*
m + Na is reached for a couple *
*
, A located on
T
H
H
T
edge of the field of *
*
, A.
H
T
p
*
C
One has then: D = Q + nf

T + Max m - N

H


*


D
H
So that
p
D is finished, it is necessary that:
C
m = N
D
Taking again the definitions of m and N, this relation gives:
p
C p
=
H
T
&3










éq 4.3.1-12
D &
In addition, the constraints m 0 and N 0 give:
p
~ p

3
2c








éq 4.3.1-13
H
eq
&
&
and
p
~ p
2D








éq 4.3.1-14
T
eq
&
&
these two last inequalities being equivalent because of [éq 4.3.1-11].
The equations [éq 4.3.1-11] and [éq 4.3.1-12] define the plastic flow in the node of one of
cones of the field of reversibility.

4.3.2 Demonstration by plastic work

The starting point is to consider that compared to the developments made in regular points,
they are primarily the relations [éq 4.2-1] and [éq 4.2-2], known as rules of normality, which cannot
to be written more. However the relation [éq 4.2-1] implies the equality p
F
, which can, it, being
T
T (p
T)
p
&
= &
maintained.
We will thus leave the equation:
p
F

éq
4.3.2-1
T
T (p
T)
p
&
= &
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We use the partly isotropic decomposition and déviatoire tensors and find:
p
F = ~
S +
3

&
&
éq
4.3.2-2
T
T (p
T)
p
H H
&
At the top of the cone of traction, there are the relations [éq 4.3.1-3], which, carried in [éq 4.3.2-2] give,
while also using [éq 3.5-2]:
p
D
F =
F
T
T (p
T)
T (p
T) H
&
3

éq
4.3.2-3
C
&
C
And one thus finds the relation [éq 4.3.1-12]:
p
=
H
T
&3
.
D &

4.4
Together equations of behavior (summarized)

One notes H the matrix of elasticity:

+ µ
2


0
0
0



+ µ
2

0
0
0


+ µ
2
0
0
0
H =

0
0
0
µ
2
0
0
0
0
0
0
µ
2
0



0
0
0
0
0
µ
2
With:
=
E
E
3 + 2µ
(
and µ =
, and K =

1 +) (1 -
2)
(21+)
3

The forced relations deformations are written finally:

= H (
p
p
- -
éq
4.4-1
C
T)

For a regular point of the cone of compression:
has
F
With
F
With

éq
4.4-2
comp (
) 2
,
=
eq
C
+ H - c+ C 0
3b
B
F
p F
0
éq
4.4-3
C
comp =
PC = p comp
C
&
;
& &

For a regular point of the cone of traction:
2
C
F
(, A)
F
With

éq
4.4-4
trac
=
eq
T
+ H - t+ T 0
3D
D
F
p
p
p
F
0

éq
4.4-5
T
trac =
T =
trac
T
&
;
& &

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For a point at the top of the cone of compression:
S = 0











éq 4.4-6
H
F
(
has
, A
F
With
éq
4.4-7
c) =
H - c+ C = 0
comp
B
p
p has

=
C H
C
&3











éq 4.4-8
B &
p
~ p

3
2a








éq 4.4-9
C
&
&
H
C eq
For a point at the top of the cone of traction:

S = 0











éq 4.4-10
H
F (
C
, A
F
With

éq
4.4-11
T) =
H - t+ T = 0
trac
D
p
C p

=
T H
T
&3











éq 4.4-12
D &
p
~ p

3
2c









éq 4.4-13
T
&
&
H
T eq

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5
Numerical integration of the law of behavior

5.1
The total problem and the local problem: recalls

For a given structure (geometry and material), and for a given loading, fields of
displacement, constraint and variables intern are by solving a whole of equations with
nonlinear derivative partial formed starting from and the law equilibrium equations of behavior.
The document [bib5] presents the algorithm of which we give a summary here:

U and known
0
0
Loop urgent T: loading L = L T
I
(I)
I
U known; calculation of the prediction
0
U

I 1
-
I
Iterations of balance of Newton N
N
U known; N
U
U
U
I =
N
I -
I
I 1
-
Loop elements el
Loop points of gauss G
N
calculation
el
el

=
U


G
G
I
(nor)
law of behavior:
N
N
N
el
calculation of: el
and el
starting from el
, el
and


G I
G I
G I 1
-
G I 1
-
G I
N
el

calculation of
G I (according to option)
N
el

G I
Accumulation in vectors and matrices assembled:
el
N
Accumulation of T
el
Q
T
G
.
in
N
Q
.
G I
I
N
el
el

Accumulation of calculation of T
G I
el
Q G
Q in N
K (according to option)
N
G
el
I

G I
Calculation of
N
U
by:
I
N
N
T
N
K. U
= Q
-
. + L
I
I
I
I
linear iteration of search to determine
Updating:
N
N
N
U
+1 = U
+ U

I
I
I
IF test convergence OK
end Newton: no time following I = i+1
If not
N = n+1
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N
The calculation of the constraints and variables intern el
N
el
with the iteration of Newton N and time T with
G I
G I
I
N
el
to leave the constraints and variables intern el
, el
at time T and value and

of
G I 1
-
G I 1
-
I 1
-
G I
the increase in deformation in the interval of time estimated with the iteration of Newton N consists with
to integrate the equations [éq 4.4-1], [éq 4.4-2] with [éq 4.4-5] or [éq 4.4-6] with [éq 4.4-9] or [éq 4.4-10] with
[éq 4.4-13] according to the cases with the initial conditions:
(T
éq
5.1-1
I
= el
1
-)
G I 1
-
(T
éq
5.1-2
I
= el
1
-)
G I 1
-
p
(T
= el
p

éq
5.1-3
I 1
-)
G I 1
-

With the condition of loading in imposed deformation:
(T =
éq
5.1-4
I)
N
el
G I
The result of this integration will provide:
N
el
= T
G
(I)
I
N
el
p
p
G = T
I
(
I)
N
el
= T
G
(I)
I

The object of this chapter is to present the numerical integration of these equations. It is about a system
nonlinear differential equations which we solve by a method of implicit Euler. To leave
of now, the quantity at the beginning of the step of time (known) will be noted with an index -, then
N
that unknown factors at the end of the step of time (all unknown factors except
el
=) will be noted without index.
G I
For an unspecified quantity one has notes
-
= has - A. has.
One always starts by calculating an elastic solution E
, by supposing that there is no evolution
plastic deformations and internal variables. So at least one of the criteria is violated by this
elastic solution, it is necessary to calculate plastic flows. The cases should then be distinguished
regular for which the solution is on the regular part of one of the cones or with their intersection
cases known as singular where the solution is at the top of one of the two cones. Logic allowing
to examine and choose these various cases, and the algorithm which results from this are relatively complex.
We thus present in first the processing of each case and explain their sequence
subsequently, in the chapter [§5.7].
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5.2
Digital processing of the regular case.

One presents in detail only the case where at the same time plastic deformations develop in
traction and in compression and where thus the solution belongs to the intersection of the two cones. Let us note
however that, even if E
violate at the same time the two criteria, for as much the final solution can very
to belong well finally only to one of the cones hammer-hardened. One is thus brought to seek balanced
which one postulates that they belong to one of the two cones or both. The case or it belongs to
only one of the two cones results easily from the more general case presented here. Equations that us
let us have to solve are finally:

µ
=
-
S +

µ ~
2

éq
5.2-1
µ-
E
K
=
- + 3K
éq
5.2-2
H
-
H
H
K
E
S = S - 2µ (~ p
~ p
+
éq
5.2-3
C
T)
E
= - 3K

+

éq
5.2-4
H
H
(p
p
H C
H T)
has
F
F

éq
5.2-5
comp (
PC)
2
,
=
eq + H - C (PC) = 0
3b
B
F
p

éq
5.2-6
C =

p comp
C

p
2
C
F
(,
F

éq
5.2-7
trac
)
T
=
eq + H - (tp)
T
= 0
3D
D
F
p
p

éq
5.2-8
T =


trac
T

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By taking the isotropic and deviatoric parts plastic deformations, the equations [éq 5.2-6]
and [éq 5.2-8] give:
p
~


p
S
C
=

éq
5.2-9
C
eq
B
2
p
p has


=









éq 5.2-10
H C
C
B
3
p
~


p
S
T
=








éq 5.2-11
T
eq
2D
p
PC


=









éq 5.2-12
H T
T
D
3
While deferring [éq 5.2-9] and [éq 5.2-11] in [éq 5.2-3], one finds:
p
p




E
S
C
T
S = S - µ
2
+


éq
5.2-13
eq

B
2
2D
who shows that S is parallel to E
S from where one deduces:
E
S = S éq 5.2-14
eq
eq
E


While deferring [éq 5.2-14] in [éq 5.2-9] and [éq 5.2-11] one finds:

p
E
~


p
S
C
=








éq 5.2-15
C
eq
E
B
2
p
E
~


p
S
T
=








éq 5.2-16
T
eq
E
2D
One defers then [éq 5.2-15] and [éq 5.2-16] in [éq 5.2-3] and [éq 5.2-4], and one expresses the criteria
[éq 5.2-5] and [éq 5.2-7] with these new results. That led to two equations having like
unknown factors
p

and
p

:
C
T

2
eq
has

E
E
p
2
2
µ Ka
p
Kac
+
F
éq
5.2-17
H -

C
+
-
T
+
- C -
p
C
+
PC =

2
2


(
) 0
3b
B
3b
B
3bd
data base
2
eq
E
C E
p
Kac

p
2
2
µ
Kc
+
F
éq
5.2-18
H -

C
+
-
T
+
- T -
p
T
+
Pt =


2
2
(
) 0
3D
D
3bd
data base
3D
D
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It is this system of two equations to two unknown factors which should finally be solved. If
the functions F and F are linear, i.e. if one is in linear mode post-peak in
C
T
compression as in traction, it acts of a linear system which will thus be solved in an iteration.
In the case, is mode pre-peak in compression (which is always nonlinear), that is to say of
modelings with nonlinear modes post peak, the system [éq 5.2-17] and [éq 5.2-18] is solved by one
method of Newton:
F is noted *

,

the criterion of compression regarded as function of the only variables
comp (
p
p
C
T)
p

and
p

, in the same way for traction:
C
T

*
2
has
µ
2
Ka2
µ
2
Kac
F

,

=
+ -

+
-

+
- F - +


comp (
p
p
C
T)
eq
E
E
p
p
H
C
2
2


T
C (p
p
C
c)
B
3
B
B
3
B
data base
3
data base
*
2
C
µ
2
Kac
µ
2
Kc2
F

,

=

+ -

+
-

+
- F - +


trac (
p
p
C
T)
eq
E
E
p
p
H
C
T
2
2
T (p
p
T
T)
D
3
D
data base
3
data base
D
3
D

The ième iteration of Newton for system [éq 5.2-17] - [éq 5.2-18] is:

i+1
I
p
p




,
- 1
F






comp

C
C
(p p
C
T) I

=
- J

p
p
I





T
T
F

,


trac (p
p
C
T)
The jacobien J is worth:
I
I
F


F


comp
comp


p
p







C
T
J

=

I
F

F

trac
trac


p
p







C
T
With:

F


µ
2
2

-
comp
Ka
F
+

C (p
p
C
c)
= -
+
-

p

2
2
p



B
3
B





C
C


fcomp
µ
2
Kac
= -
+



p
3
T
data base
data base

F
2
trac
µ
Kac
=
-
+



p
3
C
data base
data base
F


µ
2
Kc2
F
- +

trac
T (p
p
T
T)
= -
+
-

p


2
2
p



D
3
D



T
T

Initial Jacobien of the system results from the values of derived in
p
and
p
, which
T = 0
C = 0
amounts solving the nonlinear system on the basis of the null solution. Nonthe linearities are
introduced by the curves of softening. In the post-peak part, when they are linear,
convergence is done in an iteration. When they are nonlinear, convergence only requires
some iterations. To leave the null solution thus does not pose a problem of convergence. That
returns starting from the linearization of the criteria in the vicinity of the elastic prediction.
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5.3
Existence of a solution and condition of applicability

We point out that the solution of the problem [éq 5.2-17] and [éq 5.2-18] must check the conditions
[éq 4.2-3] and thus inter alia the positivity of the increases in the plastic multipliers.

p
éq
5.3-1
C 0

p
éq
5.3-2
T 0
Let us suppose that we are in a case of linear behavior post peak in traction as in
compression and let us call H and H respectively the slopes of the parts post peak. Increases
C
T
plastic multipliers are obtained by solving the linear system:

µ
2
Ka2
µ
2
Kac
p


+
+
2
has
H
F
2
2
C
+


C

eq
-


E + eH -
p


C (c)
B
3
B
data base
3
data base
B
3
B


éq
5.3-3
2

=
µ
2
Kac
µ
2
Kc



2
C
p
eq
-
E
E
p

+
+
+ H
F

2
2
T
data base
3
data base
D
3
D

T
+ H - T (T)

D
3
D


Since the criteria of traction and compression were activated in traction as in compression,
the second member of this system is positive. But nothing ensures in so far as the solution of
[éq 5.3-3] will be positive.
If one poses:
µ
2
Ka2
µ
2
Kac

+
+ H
2
2
C
+

HTC = B
3
B
data base
3
data base

éq
5.3-4
2


µ
2
Kac
µ
2
Kc


+
+
+ H

2
2
T
data base
3
data base
D
3
D


One a:
p



2
has
C

eq
E
+ E
F
H -
-
p


C (c)
- 1 B
3
B


= HTC

éq
5.3-5

2
C
p
eq
-


F
T

E + eH - T (Pt)
D
3
D


With:
H
9

T
HTC = 3
+ µ -
+ µ +
+
+
+
+
+
éq
5.3-6
2 (3
) H
K
T
6
(3K 2) H H
H K 9h K
16
9
C T
C
T
(H
H
K
C
T
)


4
4

3
9K
9K
µ
3
HT + (3K + µ)

-
+
- µ
3

-
HTC 1 =
1

4

2
2

2
éq
5.3-7
HTC 9K
9K
µ
3
K
(-)

1
3
H
9K 18
12

-
+
- µ
C +
-
+ µ

2
2

2


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It is seen that conditions of positivity [éq 5.3-1] and [éq 5.3-2] lead to relations relatively
complicated. If the solution of the problem [éq 5.2-17] and [éq 5.2-18] does not check the conditions of
positivity [éq 5.3-1] and [éq 5.3-2], that can correspond is with the fact that the coefficients H and H are
C
T
such as there is no positive solution (that would correspond to a snap-back in a diagram
(,)), that is to say with the fact that the solution activates finally only one of the two criteria.
Let us examine the simpler case of only one activated criterion. Let us suppose to fix the notations that only
activated criterion is the criterion of traction.
One must have:

2
p
µ
2
Kc



2
C
H
F
éq
5.3-8
T
+
+
=
eq
E
2
2
T
+ E
H
-
p




T (
-
T)
D
3
D

D
3
D
One sees reappearing the condition known as of applicability:
2
2
µ
Kc

+
+ H

éq
5.3-9
T
> 0

3 2
2


D
D

This condition is the generalization of the condition - H < E presented at the paragraph [§3.5.2.1] in
a particular case of axial stress plain.
The following strategy will thus be retained:
2
eq
has
-
2
eq
has
-
If
E + E
F
and
E + E
F

H -
C (PC) > 0
H -
C (PC) > 0
3b
B
3b
B
Activation a priori of the two criteria: resolution problem [éq 5.2-17] and [éq 5.2-18]
So not convergence or so not checking conditions of positivity [éq 5.3-1] and [éq 5.3-2]
Seek solution with only one activated criterion
So not convergence or not checking condition positivity
Stop on diagnostic of nonchecking of condition of applicability of the type [éq 5.3-9]

5.4
Processing of the nonregular cases

In this paragraph, we describe the discrete processing of the equations corresponding to projection
at the node of the cone of traction, [éq 4.4-10] with [éq 4.4-13], knowing that projection at the top of
cone of compression is done in the same way. The equations [éq 4.4-10] and [éq 4.4-12] define
the plastic flow in this case, whereas the equation [éq 4.4-13] is a condition of acceptability
projection at the top of the cone.

5.4.1 Calculation of the constraints and plastic deformations

Discrete forms of [éq 4.4-10] with [éq 4.4-13], are:
S = 0 éq
5.4.1-1
D
=
F - +


éq
5.4.1-2
H
T (p
p
T
T)
C
p
C
p


3
=


éq
5.4.1-3
T H
T
D
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The relation [éq 5.2-4] established in the regular case is always valid, one jointly uses it with
[éq 5.4.1-3] in [éq 5.4.1-1] and one obtains:

E
C
p
D
- K

=
F - +

éq
5.4.1-4
T
T
H
(p
p
T
T)
D
C
The relation [éq 5.4.1-4] is a nonlinear equation compared to the variable
p

that one solves by
T
an algorithm of Newton, which makes it possible to calculate
p


by [éq 5.4.1-3] and by [éq 5.4.1-2].
T H
H
Taking into account [5.4.1-1], the constraints are thus completely known. [éq 5.4.1-1] gives
still:
E
~
S = S - 2
µ
p

éq
5.4.1-5
T
= 0
This last equation makes it possible to calculate
p
~
and the plastic deformations are completely
T
known.

5.4.2 Acceptability

The discrete form of the relation [éq 4.4-13] is:
p
p
3


c~
2

éq
5.4.2-1
T H
T eq
[éq 5.4.1-5] gives:
eq
~ p
E
éq
5.4.2-2
T
=
eq
µ
2
While using [éq 5.2-4] and [éq 5.4.2-2], [éq 5.4.2-1] is written:
eq
E
cK
E

- éq
5.4.2-3
H
H
µ
2
5.4.2.1 Acceptability a priori and a posteriori
D
-
For the criterion of traction and the part post peak of the criterion of compression, =
F +


H
T (p
p
T
T)
C
is a decreasing function of the variable of work hardening
p

. One deduces from it that
T
E
-
E
- - and thus that:
H
H
H
H
eq cK
eq
E
E
-
E
cK
E

-


-
H
H
H
H
µ
2
µ
2
eq
The condition E
cK
E
-

- is known as condition of acceptability a priori because it can be calculated
µ
H
H
2
eq cK
as of the elastic prediction. The condition E
E

- is known as condition of acceptability has
H
H
µ
2
posteriori.
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Direction of projection
eq

(eq
E
E
,
H
)
zone of projection
at the node of the cone
Of traction
H
Pt

Figure 5-1

These conditions have a simple graphic interpretation. One can see easily that, in the case of one
regular solution, one a:
eq
E
eq
-
µ
= 2
E
-
cK
H
H
eq
That shows that the solution in constraint is obtained by projecting the point (E
E
,
parallel to one
H
)
direction (cK,
µ
2) in a diagram (
eq
, as indicated on [Figure 5.4.2-a]. Zones
H
)
of acceptability of projection at the top are cones of which the node and that of the cone of
reversibility and delimited on the one hand by the axis > COp and a half-line resulting from the same point and
H
T
of direction (cK,
µ
2).
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5.4.3 Existence of a regular solution and a singular solution.

If projection at the top of the cone is acceptable a posteriori, it may be which exists too
a regular solution as one can see it on [Figure 5.4.2-b].

zone of projection
at the node a posteriori
eq

zone of projection
at the node a priori
(eq
E
E
,
H
)
H
2
1
-
P
P
P
T
T
T
Direction of projection for
regular case

Appear 5.4.2-b
The node of the cone of traction before work hardening is noted -
P, that of the cone hammer-hardened with one
T
1
increase in variable of work hardening
p


1 is noted
P, that of the cone hammer-hardened with one
T
T
2
1
increase in variable of work hardening
p
p


>

is noted 2
P. It is seen that there is a solution
T
T
T
1
2
regular with
p


and a solution with projection at the top of the cone for
p


. Since
T
T
the regular solution corresponds to a less work hardening, in the process of evolution, it will be
met before the solution with projection at the top: it is thus the regular solution which it is necessary
to retain.
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For this reason the sequence of regular search of solution and with projection at the top is
the following:
eq
I projection at the acceptable top a priori: E
cK
E
-

-
µ
H
H
2
Calculation of the solution with projection at the top:
p

by [éq 5.4-4]
T
So not
Seek regular solution
So not convergence or not checking condition positivity
Calculation of the solution with projection at the top:
p

by [éq 5.4-4]
T
eq cK
Checking of acceptability a posteriori: E
E

-
H
H
µ
2
eq cK
If not acceptable: E
E

> -
H
H
µ
2
Stop on diagnostic of nonchecking of condition of applicability

5.4.4 Inversion of the nodes of the cones of traction and compression

A priori, the node of the cone of compression corresponds to a hydrostatic pressure of traction
much larger than that the node of the cone of traction. But, as one can see it on
[Figure 5.4.4-a], one can find a history of loading which never activates the criterion of traction,
who activates and strongly hammer-hardens the criterion of compression until it to return strictly included in
field of reversibility of the criterion of traction.

Way of constraint
eq

Criterion of compression
H
Pt
Criterion of compression hammer-hardened

Appear 5.4.4-a
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When the two criteria were thus reversed, but the criterion should not intervene a priori any more of
compression. It may be whereas the solution is a projection at the top of the cone of compression,
who treats himself exactly like projection with the node of the cone of traction.

5.4.5 Projection at the top of the two cones

If the two cones were inverted and if the elastic prediction violates both
criteria, it may be which are finally acceptable at the same time the solution of projection at the top of
cone of compression and at the top of the cone of traction. In these situations, no criterion allows
to select a solution rather than the other and one will thus seek a simultaneous projection with
node of the two cones, which will have to thus share the same node, as indicated on
[Figure 5.4.5-a].

eq
Direction of projection
Criterion of
Criterion of
the criterion of compression
compression
traction
Elastic prediction
initial
initial
(E
H, eq E) to project
Solution with projection
on the criterion of traction
Criteria of
compression and
traction hammer-hardened
H
Solution with projection
at the node of the two cones

Appear 5.4.5-a
The solution with projection at the two tops is obtained by solving the system:
E
has
p
C
p
B
- K
- K

= F - +


éq
5.4.5-1
C
T
C
H
(p
p
C
c)
B
D
has
E
has
p
C
p
D
- K
- K

=
F - +


éq
5.4.5-2
C
T
T
H
(p
p
T
T)
B
D
C
The state of stress is given by:
S = 0
D
-
B
=
F +

= F - +


éq
5.4.5-3
H
T (p
p
T
T)
C (p
p
C
c)
C
has
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5.5
Determination of the tangent operator

During iterations of the algorithm of Newton-Raphson, it is necessary to calculate the matrix of
tangent stiffness. The construction of this one plays an important part in stability, the speed and
precision of the method of resolution. To preserve these properties, the matrix of tangent stiffness
must be built starting from an operator binding the increment of constraint to the increment of deformation
in a precise way at the end of the process of return on surfaces of load. The matrix of Hooke, thus
that the thermal deformations intervene like constants at the time of the determination of
the coherent tangent operator, built at the end of the iteration in the increment concerned.
The calculation of the operator of coherent tangent behavior takes into account the deformations
plastics. For reasons of simplicity, we chose to calculate the operator of behavior
tangent of speed.

5.5.1 Operator
tangent
in
speed with only one active criterion

In the case of only one active criterion, for example, the criterion in compression, the calculation of the operator of
tangent behavior speed is as follows:
One thus uses the equations of speed, in elastoplastic load:

F
p
comp
- H - C
= 0


&

éq
5.5.1-1
& &



T
F
F
comp

+ comp p

éq
5.5.1-2
p
C = 0

&


&
C
The tangent operator of speed is defined by:

D
& = &
éq
5.5.1-3
While identifying [éq 5.5.1-3] with [éq 5.5.1-1] and [éq 5.5.1-2], one finds classically:
1

T
F
F
comp

D = H - H
comp H éq
5.5.1-4





with:
T
F

F

F

comp
comp
comp
=
H
-

éq
5.5.1-5
p

C
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5.5.2 Operator
tangent
in
speed with two active criteria

If the two criteria are activated, the criterion in compression and the criterion in traction, it
calculation of the operator of tangent behavior speed is as follows:
One leaves:

F
F
p
comp
p

- H - C
-
trac
T
= 0


&

éq
5.5.2-1
& &

&


T
F
F
comp

+ comp p

éq
5.5.2-2
p
C = 0

&


&
C

T
F
F
trac

+ trac p

éq
5.5.2-3
p
T = 0

&


&
T
One leads to:
F

F
F
F
F
comp

T
T

comp

F
trac


T
T
D = H - H


trac
comp
trac

éq
5.5.2-4
DC
+ ct
+
Tc
+





tt













with:

T
F
F
F
trac
trac


H
- trac

p

T


DC =
T
F
F
F
F
F
F
F
F
F
F
comp
comp


comp

T
trac
trac

T
trac
comp



trac

T



H
-


H
-

-
H


trac
H comp

p




p
C




T










éq 5.5.2-5


T
F

comp
ftrac
-
H





ct =
T
F
F
F
F
F
F
F
F
F
F
comp
comp


comp

T
trac
trac

T
trac
comp



trac

T






H
-
H
-
-
trac
comp

p





p


H




H




C



T






éq 5.5.2-6


T
F
F

- trac H comp






Tc =
T
F
F
F
F
F
F
F
F
F
F
comp
comp


comp

T
trac
trac

T
trac
comp



trac

T



H
-


H
-

-
H


trac
H comp

p




p
C




T










éq 5.5.2-7
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T
F
F
F
comp
comp


comp

H
-

p

C



tt =
T
F
F
F
F
F
F
F
F
F
F
comp
comp


comp

T
trac
trac

T
trac
comp



trac

T



H
-


H
-

-
H


trac
H comp

p




p
C




T










éq 5.5.2-8

The expression seems expensive to express in term of products of matrix and calculation. But, when
the operations are made in the order which is appropriate, it is enough to calculate terms little. Moreover, it is
the same terms which intervene on several occasions. It is necessary to calculate the derivative of the criteria by
report/ratio with the constraint, and the plastic multipliers, then sums and products
with the actual values, to finish by the constitution of the matrices and theirs let us be.
Lastly, the resulting matrix with the advantage of being symmetrical, which is appropriate for the standard resolution
with Code_Aster.

5.5.3 Derivative successive of the criteria in traction and compression
5.5.3.1 Successive drifts of the criteria compared to the constraint

The derivative of the isotropic and deviatoric components of the constraints compared to the tensor of
constraints are expressed in the following way:
1

1
1
By defining the vector

0 =
0
0


0
the derivative of the criteria compared to the tensor of constraints are expressed in the following way:
F
comp
S
has
=
+

0

B eq
2
B
3
F

S
C
trac =
+

0

2D eq

D
3

5.5.3.2 Successive drifts of the criteria compared to the plastic multipliers

Derived from the criterion of compression in the case of a linear curve post-peak:
p
F

4

1

comp
C
p
= -. F.
-
if

p
C
2
C
E





3


C
E
E

F


1

comp = F.
if
p
C



-
C
(C
E)
p
U
E
C
C

Derived from the criterion of compression in the case of a nonlinear curve post-peak:
p
F

4

1

comp
C
p
= -. F.
-
if

p
C
2
C
E





3


C
E
E

p
F



comp
-
=.
2 F

.
if
p
C
2
C
(C
E
U
-
C
E)
p
U
E
C
C

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Derived from the criterion of traction in the case of a linear curve post-peak:
F

F
trac
T
p
U
=
if
p
U
T
T


T
T
Derived from the criterion of traction in the case of an exponential curve post-peak:
a. p
-
T
F
has
trac

U

T
= F
E

p
T U
T
T

5.6
Variables intern model

We assemble here the internal variables stored in each point of Gauss in the implementation of
model

Internal number of variable
Feel physical
1
p
: plastic deformation cumulated in compression
C
2
p
: plastic deformation cumulated in traction
T
3
: maximum temperature attack at the point of gauss
4 Indicator
of
plasticity

5.7
Top-level flowchart of resolution

The flow chart includes/understands the various stages of the resolution, with the processing of projections
at the nodes of the cones of compression and traction in the following way:
at the beginning of algorithm,
one carries out a projection at the top of the cone of traction:
· when the elastic prediction checks the condition of projection a priori in traction,
· when the elastic prediction checks the condition of projection a priori in compression and
that the nodes of the cones of traction and compression were inverted on the axis
hydrostatic,
one carries out a simultaneous projection with the nodes of the cones of traction and compression:
· when the elastic prediction checks the condition of projection a priori in compression and
that the nodes of the cones of traction and compression were inverted on the axis
hydrostatic, and that projection at the top of the cone of traction did not give a solution
validate,
one carries out a projection at the top of the cone of compression:
· when the elastic prediction checks the condition of projection a priori in compression and
that the nodes of the cones of traction and compression were inverted on the axis
hydrostatic, and that projection at the top of the cone of traction did not give a solution
validate, and that simultaneous projection with the nodes of the two cones did not give
valid solution,
in medium of algorithm,
one carries out one, two or three standards resolutions with projection on the criterion of compression or
on the criterion of traction or the two criteria at the same time,
and at the end of the algorithm,
when that the standards resolutions with activation of a criterion (traction or compression) or
two criteria at the same time did not give a solution,
one carries out a projection at the top of the cone of traction:
· when the elastic prediction checks the condition of projection a posteriori in traction,
· when the elastic prediction checks the condition of projection a posteriori in compression
(and that the nodes of the cones of traction and compression were inverted on the axis
hydrostatic,
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one carries out a simultaneous projection with the nodes of the cones of traction and compression:
· when the elastic prediction checks the condition of projection a posteriori in compression
and that the nodes of the cones of traction and compression were inverted on the axis
hydrostatic, and that projection at the top of the cone of traction did not give a solution
validate,
one carries out a projection at the top of the cone of compression:
· when the elastic prediction checks the condition of projection a posteriori in compression
and that the nodes of the cones of traction and compression were inverted on the axis
hydrostatic, and that projection at the top of the cone of traction did not give a solution
validate, and that simultaneous projection with the nodes of the two cones did not give
valid solution.

With the exit of each resolution having converged, one carries out the checks of conformity of the solution
following:

· validity of the solution compared to the second criterion, when the resolution was made with one
only criterion. In all the cases, it is enough to check that the two computed criterions with
final constraint, are negative or null,
· conformity of the solution: one calculates in the course of resolution the final equivalent constraint. It
arrive sometimes that the solution is beyond the node of the cone which is hammer-hardened, which leads to
an equivalent constraint “negative”. Numerically, that results in a final criterion
strictly positive. To check the conformity of the solution, it is enough to check that both
computed criterions with the final constraint, are negative or null,
· validity of projections at the tops of the cones. It should be checked that after resolution, when one
knows the work hardening of the criterion, the slope of the straight line connecting the elastic prediction to
projection is lower than the slope of the direction of projection. (condition of projection has
posteriori at the node of the cones). In the contrary case, that means that there is a solution
with standard resolution.

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Note:

In the case of projections at the tops of the cones, one starts systematically with
projection at the top of the cone of traction. If this solution is valid, this one is preserved. In
the contrary case, if the criterion of traction is activated, one carries out a resolution with projection with
nodes of the two cones. If the new solution is valid, that one is preserved. If not, one
carry out a resolution with projection at the top of the cone of compression alone.
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Note:

If the conditions of projection at the tops of the cones are activated, one of the three
solutions must be valid, but for particularly important elastic jumps, it may be
that the resolution does not succeed. The solution is then to carry out a recutting of the step of
time.

Note:

Projection at the tops of the two cones simultaneously supposes that the criterion of traction is
activated. It may be very well that in the event of permutation of the nodes, only the criterion of compression
finds itself activated. It is necessary to then make a projection at the top of the cone of compression alone.
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Appendix 1 snap-back with the initial values of the coefficients C
and D

We show in this appendix the problem of snap-back met in the simulation of a tensile test
follow-up compression simple, if the choice of the coefficients C and D of the criterion of traction corresponds to
a situation where the criterion of traction cuts the axes in a diagram of constraint plane, i.e. it
choice of the coefficients [éq 3.3-1] and [éq 3.3-2] leading to a field of reversibility represented on
[Figure 3.3-b]:
The assumptions are as follows:

· one takes into account only the criterion of traction,
· the marrow of work hardening is of the type: F = F + H,
T (p
T)
p
T
T
· one notes simply:
p
= so that the curve of work hardening is written:
F
,
T = F
T +

H
T
· the null Poisson's ratio,
· work hardening is negative,
· the condition of applicability is met: - E < H < 0.

One considers an axial plain test controlled in deformation according to X, as indicated on Figure 5 -


xx
xx
1
xx
P0
0

P1
xx
Time T
2
xx
0
xx

1

xx
xx
2
xx
P2

Appear 5-a

Under the other the imposed directions y and Z, conditions are conditions of null constraints:

.
yy = zz = 0
One starts by imposing a deformation of traction 1
such that there is a plasticization in traction, but
xx
without the limit of traction falling down to 0. It is the point P in the diagram stress-strain. One notes
1
0
the deformation for which the rupture limit in traction appears for the first time. Beyond the point
xx
P, one imposes a decrease of the deformation, which involves an elastic unloading of material,
1
up to value 2

deformation for which one has a plasticization again, but this time under
xx < 0
compressive stress. The object of this appendix is primarily to study the behavior of the model
not retained (that corresponding to the formulas [éq 3.3-1] and [éq 3.3-2]) beyond the point P.
2
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A1.1 Calcul of the constraints and the deformations during the loading

Taking into account the assumptions pointed out higher, the invariants of constraint are worth:

I










éq A1.1-1
1 = xx
xx
J =









éq A1.1-2
2
3
eq
=








éq A1.1-3
xx
The plastic flow is calculated by:
p
C +
=
2
&








éq A1.1-4
xx
&
D
3
A1.1.1 Chemins 0P0 and P0P1

Taking into account the relations [éq A1.1-1] with [éq A1.1-4], and the fact that along this way them
constraints are positive, one finds easily
xx



C + 2 '

H xx +
F
1
3D
T








éq A1.1.1-1
xx = E
1

(c+ 2) 2
E

H +
2



9d

A1.1.2 Chemin P1P2

By definition, P1P2 is an elastic path of discharge, the P2 point being such as the criterion is there with
new reached
3D
=







éq A1.1.2-1
2
xx
1
C - 2 xx
3D 1
xx
=

+
éq
A1.1.2-2
2
xx
1
C - 2
xx
E
A1.1.3 Au delà du not P2

One is interested now in the slope of the curve at the P2 point in the reference mark (. More
xx
xx)
precisely one is interested in the slope of this curve for a dissipative solution.
By writing that the material remains plastic beyond the P2 point, i.e. that the state of stress
remain on criterion-which is hammer-hardened, one finds:
=
D
3







éq A1.1.3-1
xx

&
h&
C - 2
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In addition, by calculating the increase in flow plastic, and by deferring it in the calculation of
the increase in constraint, one finds:
xx


&
&
xx =
+
(C 2)
&






éq A1.1.3-2
E
3D
One can then eliminate & between [éq A1.1.3-1] and [éq A1.1.3-2] and one obtains:


He
xx
& =

éq
A1.1.3-3
&
(
= AND
xx
C - 2) 2
2
H + E
2
9d
This formula gives the slope of the response in the plan (,

xx
xx)
The numerator is always negative since H is negative.

The sign of xx
& = E depends on the sign of the denominator, thus two cases are posed:
2
T
xx 2
&
(C 2) 2

If H < - E
then xx
& = E is positive one has a configuration of snap-back.
2
9d
2
T
xx 2
&

(C 2) 2

If H > - E
then xx
& = E is negative and there is no snap-back.
2
9d
2
T
xx 2
&

A new condition appears to avoid the snap back, condition which we already compare with that
evoked, but which related to in fact possible the snap back at the P1 point.
He
Slope at the P1 point in the reference mark (:
E =
,
xx
xx)
T1
H + E
He
Slope at the P2 point in the reference mark (:
E =
.
xx
xx)
2
T
(C 2) 2
H + E
2
9d
(C 2) 2
2
'
F
However
2
= 3 = 3 T
.
2

'
9


D
FC
One can for example express E according to E by eliminating H:
2
T
1
T

E E
1
E
.
T =
T
2
2
3rd + AND -
1 (
2
1 3
)
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For example,
3 2
E
E =
for E
.
T = -
2
T

1
1
(- 3 2
)
As example, for E=32000 Mpa, F '= Mpa
3
and F '=
Mpa
3
,
38
E is found
. Thus
T
601
-
T
C
1
E is very weak compared to E. as illustrated on [Figure A1.1.3-a].
1
T

xx
xx

Appear A1.1.3-a

Thus, a condition implying that there is no snap back at the P2 point would be too restrictive and
would lead to practically choose a material not fragile in traction.
For this reason we preferred to modify the expression of the coefficients C and D like
indicated at the paragraph [§3.3].

This said, and even if the adopted solution, consisting in modifying the coefficients C and D seems
reasonable, the example treated in this appendix shows that a very simple problem can finally be
a problem of structure: there is in this example of the equilibrium conditions, they are the conditions:

.
yy = zz = 0
Handbook of Référence
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6 Bibliography

[1]
J.F. GEORGIN “Contribution with the modeling of the concrete under stress of dynamics
rapid. The taking into account of the effect speed by viscoplasticity “- Thèse (15/01/98).
[2]
G. HEINFLING “Contribution with the numerical modeling of the behavior of the concrete and of
concrete structures reinforced under mechanical thermo stresses at high temperature “-
Thesis (14/01/98)
[3]
R.T. ROCKAFELLAR “Convex analysis” Princeton University Press 1972
[4]
B. HALPHEN and NGUYEN QUOC SON, “Sur the generalized standard materials” Journal
of Mécanique Vol 14 n° 1, 1975
[5]
NR. TARDIEU, I. VAUTIER, E. LORENTZ “Algorithme nonlinear quasi static” document
of reference Aster [R5.03.01].

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