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Document: R5.03.18
Law of damage of an elastic material
fragile
Summary:
This document describes the elastic model of behavior fragile ENDO_FRAGILE available in statics and in
dynamics. The damage is modelled in a scalar way; loadings in compression and in
traction are not distinguished. In addition to the local model, the nonlocal formulations with gradient
of damage and with regularized deformation are also supported to control the phenomena of
localization.
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1 Field
of application
Law ENDO_FRAGILE aims at modelling in the manner a simplest possible behavior
fragile rubber band. The material is elastic isotropic. Its rigidity can decrease in an irreversible way
when the deformation energy becomes important, without distinguishing traction from compression.
This loss of rigidity is measured by a variable interns scalar called damage which
evolve/move of 0 (healthy material) to 1 (completely damaged material, i.e. without rigidity). Moreover,
the constraint cannot exceed a threshold which also decrease him with the level of damage for
to reach 0 when the material is completely damaged. One will refer to [bib1] for one
description of this type of phenomenology.
The property of decrease of the threshold in constraint with the level of damage is called
softening and generally involves a loss of ellipticity of the equations of the problem. It results from it
a localization of the deformations and damage in tapes of which the thickness is
directly controlled by the size of the finite elements. To mitigate this deficiency of the model, two
nonlocal formulations are proposed, one founded on the introduction of the gradient of
the damage and activated by modeling * _GRAD_VARI [R5.04.01], the other resting on one
regularization of the deformations and activated by modeling * _GRAD_EPSI [R5.04.02]. In a case
as in the other, the width of the tapes of localization is henceforth controlled by a parameter
material, well informed in operator DEFI_MATERIAU under key word LONG_CARA of the key word factor
NON_LOCAL [U4.43.01]. However, obtaining a physical problem posed again well is not
obtained that at the price of an important overcost in time calculation. In addition, it should well be noticed that
only the relations of behavior are deteriorated and not the equilibrium equations. Consequently,
the constraints preserve their usual direction.
Lastly, that one activates or not these nonlocal formulations, softening character of the behavior
also involve the appearance of instabilities, physics or parasites, which result in
snap-backs on the total answer and returns the control of the essential loading in statics.
control of the type PRED_ELAS [R5.03.80] then seems the mode of control of the level of
the most suitable loading.
2
Local law of behavior
2.1
Relations of behavior
The state of material is characterized by the deformation and the damage D ranging between 0 and 1.
The forced relation deformation is elastic, rigidity is affected in a linear way by
the damage:
= (1 - D) E
éq
2.1-1
with E the tensor of Hooke. In addition, the evolution of the damage, always increasing, is
controlled by the following function threshold:
2
y
+
F (D) 1
,
= E - K (D)
where
(D)
1
K
= W
éq
2.1-2
2
1
+ - D
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The coefficients y
W and, both positive, is parameters of the model. The condition of
coherence then determines completely the rate of damage D &:
F (, D) 0
D & 0
D & F (, D) = 0 éq
2.1-3
The equations [éq 2.1-1] with [éq 2.1-3] are enough to entirely describe the law of behavior
ENDO_FRAGILE, indeed very simple. One can also notice that it fits in
formalism suggested by Marigo [bib2].
2.2
Identification of the parameters of the model
The parameters of this law of behavior are four. On the one hand, the module of
Young E and the Poisson's ratio who determine the tensor of Hooke by:
-
+
1
1
E =
-
(tr) Id
éq
2.2-1
E
E
In addition, y
W and which defines the lenitive behavior. They are determined by a test
of simple traction, cf [Figure 2.2-a]. To simplify the input of the data of the model, one informs not
not
y
W and but directly the tangent module T
E and the constraint with the peak y
under the key word
factor ECRO_LINE or ECRO_LINE_FO of operator DEFI_MATERIAU. As for E and, they are
given classically under the key word factor ELAS or ELAS_FO.
For whatever purpose it may serve, here also expressions of the deformation with rupture R
in this test of
simple traction, as well as voluminal energy 0
K consumed to damage a point completely
hardware, this last expression being valid whatever the history of loading:
R
1
1 y
1 1
1
2
0
y
1 R y
y 1 +
= -
K =
-
= = W
éq 2.2-2
T
E
E
2
T
E
E
2
2
y
y
wy =
2nd
T
E < 0
E
AND
= -
E
R
Appear 2.2-a: Simulation of a simple tensile test
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2.3
Integration of the law of behavior
Temporal discretization of the equations [éq 2.1-1] with [éq 2.1-3] on a step of time [T - T] is
realized by a diagram of implicit Euler. For any function of time Q, one notes -
Q = Q (-
T) and
Q = Q (T). To integrate in time the law of behavior then means to determine the state of stress and
of damage solution of the following nonlinear system, where deformation and the state of material
at the beginning of the step of time (-
-
, D) are given:
= (1 - D) E
éq 2.3-1
F (, D) 0
D - -
D 0
(D - - D) F (, D) = 0
éq
2.3-2
A method of resolution was proposed by [bib3]. It starts by examining the solution without
evolution of the damage (also called elastic test) then, if necessary, proceeds to one
correction to check the condition of coherence. In this case, the existence and the unicity of
solution guarantee the correct operation of the method. Let us consider the elastic test:
D = -
D
solution if
F el () = F (, -
D) 0
éq
2.3-3
In the contrary case, the damage is obtained by solving F (, D) = 0:
y
= (
W
D
1 +)
-
where
W = 1
1
E
éq
2.3-4
W
2
As for the constraint, it is given by [éq 2.3-1] in all the cases.
It still remains to be made sure that the damage does not exceed value 1. In fact, when D = 1,
the rigidity of the material point considered is cancelled. Insofar as no technique of suppression
finite elements “broken” is not implemented (technical possibly delicate when them
finite elements have several points of Gauss), of the null pivots can appear in
stamp rigidity. This is why one introduces a numerical threshold beyond D which one is considered
C
elastic residual rigidity for the tangent matrix, the equations of remaining behavior
unchanged.
To preserve a reasonable conditioning of the matrix of rigidity, one chooses
5
D = 1 10
-
C
. One
indicator, arranged in the second internal variable, then specifies the behavior during the step
time running:
·
= 0 elastic behavior (deformation energy lower than the threshold)
·
=1 evolution of the damage
·
= 2 (saturated damage) (D = 1).
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2.4
Description of the internal variables
The variables intern are two:
·
VI (1) damage D
·
VI (2) indicating
3
Formulation with gradient of damage
3.1
Tally standard generalized
To mitigate the pathological localization of the deformations and the damage inherent in
softening character of material, a first alternative consists in extending the law of behavior
by taking of account the gradient of the damage on a material point scale [bib4]. This
approach is restricted with generalized standard materials. Model ENDO_FRAGILE enters
indeed within this framework; microscopic free energy µ and potential of dissipation
microscopic µ is written:
0el () = 1 E
0
2
µ (, D) = (1 - D) el () + bl (D)
+ I [0] 1 (D)
where
éq
3.1-1
0 D 1 -
bl (D)
(D)
= - K
1+ - D
= 0 + I
µ (D &)
K d&
éq
3.1-2
0 + (D &)
[
[
where I indicates the indicatrix of the convex unit K, null in K and being worth + elsewhere. One
K
the convexity of these potentials compared to the damage, strict convexity will notice for
free energy: that confirms the existence and the unicity of the solution for the integration of the law.
To preserve a simple model, one will be satisfied to introduce a quadratic term in gradient
of damage in the free energy. The potential of dissipation remains unchanged. Consequently, them
macroscopic potentials are written:
(
0
1
, D, D = 1 - D + D + I
D + C D D
·
)
(
· )
éq
3.1-3
el ()
bl ( · )
[0] 1 ( · )
2
(D &, &d = k0 D & + I
d&
·
)
·
éq
3.1-4
0 + ( · )
[
[
where the factor C depends on the length characteristic L of the material (distance characteristic
B
of interaction enters the microscopic cracks given under key word LONG_CARA of DEFI_MATERIAU) of
following manner:
2
2
C =
µ (
L
, 0) 4
8
B
0
2
=
K L
éq
3.1-5
2
D
13
13 (1+) 2
B
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The macroscopic potentials thus built then make it possible to define the total potentials in
the origin of the variational formulation of the law of behavior. In this relatively simple case,
one can interpret this variational formulation in term of laws of local state and evolution of which them
expressions are close to those for the local law:
= (1 - D) E
éq
3.1-6
F
E
éq
3.1-7
G (
1
, D) =
- K (D) + C D
2
F D
D &
d&
D =
éq
3.1-8
G (,
) 0
0
fG (,) 0
The essential difference with the local model lies in the definition of the function threshold which depends
henceforth Laplacian of the damage. This last confers on the condition coherence one
differential character. The interpretation of the variational formulation also provides the conditions
with the limits for the field of damage as well as the conditions of interface (or jump):
D N = 0
on
N
normal
of
edge
éq
3.1-9
[C D N] = 0
crosses
with
lies of
N
normal
of
interface
one
éq
3.1-10
3.2
Integration of the law of behavior
The integration of the laws of behavior to gradients of internal variables is described in the booklet
[R5.04.01]. It is based inter alia on a local stage which depends explicitly on the relation on
behavior treated. It is this one which one proposes to describe here.
First of all, it is necessary to define the standard used to build the term of penalization of
R
R
Lagrangian increased, i.e. the matrix NR. As recommended, one adopts an estimate
diagonal of the matrix hessienne of energy compared to the variables of damage, which
conduit with:
NR ·
·
2
0
NR =
·
·
K
2
R
R
NR
(1+)
NR =
with
éq
3.2-1
NR
2
2
4
NR
=
0
B
L
K
NR
2
(1+)
13
In accordance with the general theory, it is now necessary to solve the following nonlinear system, in
which the state at the beginning of the step of time as well as the deformation, the multipliers of Lagrange µr,
R R
the coefficient of penalization R and the nodal damage evaluated at the point of current Gauss = B D
are given:
R
R
R
R R R
R R
- R (, D) + µ + R NR (- D) R (
-
D - D) éq
3.2-2
D
D
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R
where, once more, the notation compacts D = (D, D
·
) is used. Being given decoupling enters
the value and the gradient of the damage in the free energy and the potential of dissipation, it
system [éq 3.2-2] makes it possible to separately treat the gradient of the damage and its value:
-
(D
µ
R NR
D
) +
+
(
-
)
= 0 éq
3.2-3
D
-
(, D +µ + R NR - D
D - D
· )
·
·· ( ·
· )
µ (
-
·
· )
éq
3.2-4
D·
On the one hand, the equation [éq 3.2-3] makes it possible to determine the gradient of damage immediately:
µ + R NR
D
=
éq
3.2-5
C + R NR
As for the equation [éq 3.2-4], it is interpreted like an equation of coherence:
(FAr·) 0 D - D 0
·
·
(D - D
·
· ) (
F Ar· ) = 0
with
(Ar·) R 0
F
= A - K
·
éq
3.2-6
in which the thermodynamic force R
With· has as an expression:
Ar = -
, D + µ + R NR - D
·
( ·) ·
·· ( ·
· )
éq
3.2-7
D·
The solution of this equation of coherence is similar to that for the local model
[éq 2.3-1] - [éq 2.3-4]. Initially, an elastic test then a correction so necessary. Being given
strict convexity of the potentials, the solution is single. When the elastic test is not solution,
resolution of the equation (
F R
With· ) = 0 conduit with:
W + µ + R NR = R NR D +
1
·
·
·
·
·
·
·
K (D· )
where
W =
E
éq
3.2-8
2
After some handling, this equation is reduced to search roots of a polynomial of
degree 3 in D
D
· , which one knows that only one (that which one seeks) is higher than
-
· .
3.3
Estimate of the Laplacian of the damage
Like the local law, the law with gradients can lead to structural instabilities. Control by
elastic prediction PRED_ELAS can then appear essential. But this last requires
given of a threshold of elasticity point by point, whose existence is not necessarily acquired in
presence of nonlocal models. In fact, in this case, such a threshold exists thanks to
the interpretation [éq 3.1-6] - [éq 3.1-8] of the variational formulation of the behavior: it is the function
F.
G
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However, the evaluation of this threshold requires to know the Laplacian of the field of damage
who is not calculated by the algorithm of integration of the nonlocal laws. To know the nodal field
to damage is not enough besides because the functions to form, polynomials of degree 1,
do not allow to estimate a derivative directly second. Fortunately, the recourse to an artifice
allows despite everything to estimate the Laplacian with a precision sufficient for the algorithm of control.
Indeed, at the end of a step of time, the equation of coherence [éq 3.1-8] is checked, and this of as much
better than the precision requested by the user (key word RESI_DUAL_ABSO under the key word factor
LAGR_NON_LOCAL) is large. In each point of Gauss, one can thus be confronted with two cases of
appear. That is to say the threshold is not reached and, in this case, the damage in this point does not vary. One
suppose whereas it is the same for the Laplacian:
-
-
D = D
D = D
éq
3.3-1
In the contrary case, the damage evolves/moves and the equation F D = is checked, so that one
G (,
) 0
in deduced:
1
D =
K (D) 1
- E éq
3.3-2
C
2
In any rigor, this does not make it possible to determine the Laplacian when the damage is saturated.
However, that does not have importance since these points are not taken into account to control it
loading (one seeks points which dissipates).
3.4
Description of the internal variables
In the case of the formulation with gradient of damage, the variables intern are now with
numbers of six:
·
VI (1) damage D
·
VI (2) gradient of the damage D, component X (R into axisymmetric)
·
VI (3) gradient of the damage D, component y (Z into axisymmetric)
·
VI (4) gradient of the damage D, component Z (0 in 2D)
·
VI (5) estimate of the Laplacian of the damage D
(as long as D <1)
·
VI (6) indicating
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4
Formulation with regularized deformation
4.1
Formulation continues in time
The approach with regularized deformation [R5.04.02] also makes it possible it to control the phenomena of
localization and for this reason seems an alternative to the formulation with gradient
of damage. But at the difference in the latter, this formulation has the advantage of
to resort to the standard algorithms for the nonlinear problems. Indeed, the only difference by
report/ratio with the local law of behavior lies in the data of two deformations instead of one,
local deformation which intervenes in the forced relation deformation and the deformation
regularized which controls the evolution of the damage. This one results from the local deformation
by resolution of the system of partial derivative equations according to:
- 2 = 0
structure
in
B
L
éq
4.1-1
N = 0
on
normal
of
edge
N
where the characteristic length L is again indicated under the key word
B
LONG_CARA of
DEFI_MATERIAU. Finally, the relation of behavior is written in the following way, where
function threshold F was already defined in [éq 2.1-2]:
= (1 - D) E
éq 4.1-2
F (, D) 0
D & 0
D & F (, D) = 0 éq
4.1-3
4.2
Integration of the law of behavior
One of the advanced advantages for the nonlocal formulation with regularized deformation is the little of
modifications which it involves in the construction of the law of behavior. Indeed, the integration of
internal variables is completely controlled by the regularized deformation. They thus are found
expressions of the local law:
el
-
-
if F () = F (, D) 0
D = D
1
with
W =
E
y
éq 4.2-1
W
if F el () = F (
-
, D) > 0
D = (1+)
1
2
W
The constraint is then obtained directly by the relation [éq 4.1-2]. Moreover, one preserves well-sure
the introduction of a damage criticizes [éq 2.3-5] to preserve a residual rigidity.
4.3 Variables
interns
They are the same internal variables as for the local law:
·
VI (1) damage D
·
VI (2) indicating
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5
Control by elastic prediction
The control of the type PRED_ELAS controls the intensity of the loading to satisfy some
equation related to the value of the function threshold el
F during the elastic test [bib5]. Consequently, only
the points where the damage is not saturated will be taken into account. The algorithm which takes in
charge this mode of control, cf [R5.03.80], requires the resolution of each one of these points of Gauss of
the following scalar equation in which
is a data and the unknown factor:
~el
F () =
éq 5-1
~
The function el
F provides the value of the function threshold during an elastic test when the field of
displacement breaks up in the following way according to the scalar parameter:
U = U + U
éq 5-2
0
1
where U and U are given. Thanks to the linearity in small deformations of the operators deformation
0
1
(calculation of the deformations starting from displacements) and regularized deformation, one also obtains
following decompositions:
= 0 + 1
and
= 0 + 1
éq
5-3
Consequently, whatever the adopted, local, nonlocal modeling with gradient
~
of damage or not local with regularized deformation, the function el
F can be put under
following form:
~el () = 1
F
(E +
0
1
E) E (E +
0
1
E) - S
2
local
law
E =
E =
S = K
0
0
1
1
(- D)
éq
5-4
gradient of
endommagem ent
E =
E =
S = K
0
0
1
1
(- D) - C - D
déformatio regularized
N
E
E =
E =
S = K
0
0
1
1
(- D)
The equation [éq 5-1] is reduced thus to search roots of the polynomial of degree 2 following:
1
P = E E E -
0
0
0
2
P ()
2
= P + 2 P +
1
0
1
2
P
where
P = E E
1
0
1
E
éq
5-5
2
1
P = E E E
0
1
1
2
One provides to the algorithm control the linear approximation of P in the vicinity of the roots (real)
when they exist or if not the value of for which P reaches its minimum.
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6 Bibliography
[1]
LEMAITRE J., CHABOCHE J.L. : Mechanics of solid materials. Dunod: Paris, 1988.
[2]
MARIGO J.J.
: Formulation of a law of damage of an elastic material.
Report of Académie of Sciences, Paris 1981; series II, 292 (19): 1309-1312.
[3]
SIMO J.C., TAYLOR R.L.: Consist tangent operators for misses-independent elastoplasticity.
Methods computer in Applied Mechanics and Engineering 1985; 48: 101-118.
[4]
LORENTZ E., ANDRIEUX S.: With variational formulation for nonlocal ramming models.
International Journal off Plasticity 1999; 15: 119-138.
[5]
LORENTZ E., BADEL P.: With load control method for ramming finite element simulations.
International Journal for Numerical Methods in Engineering, submitted 2002.
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