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Titre : Loi de comportement viscoplastique LETK Date : 25/09/2013 Page : 1/38Responsable : FERNANDES Roméo Clé : R7.01.24 Révision :
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Viscoplastic law of behavior LETK
Summary:
Law L&K describes a behavior élasto-visco-plastic of the rocks. Elastoplasticity is characterized by a positivework hardening in pre-peak and a lenitive behavior beyond resistance. Viscoplasticity translates the effect oftime on the behavior. It is described by a law in power of Perzyna.
The initiation of the phenomena elastoplastic or viscoplastic starts as of the crossing of the correspondingthreshold. The behavior related to each phase is described by the evolution of these various thresholds. Thisevolution is governed by functions of plastic or viscoplastic work hardening.
For the elastoplastic mechanism, a surface of load evolves through various thresholds:• A threshold of damage confused with the threshold of initial viscosity,• A macroscopic of peak, definite threshold starting from the laboratory tests,• An intermediate threshold, qualified of limit of cleavage, determined analytically,• A definite threshold characteristic as the envelope of the threshold of damage and limit of
cleavage, called also limiting of contractance/dilatancy (this limit is confused with the threshold ofmaximum viscoplasticity),
• A threshold of residual resistance.
For the viscous mechanism, a viscoplastic surface evolves through:• An initial threshold confused with the threshold of damage• A maximum threshold of viscosity considered confused with the limit of contractane/dilatancy
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Contents1 Notations............................................................................................................................... 3
1.1 General information.................................................................................................3
1.2 Convention of signs..................................................................................................4
1.3 Parameters of the model..........................................................................................5
2 Introduction........................................................................................................................... 7
3 Equations of model L&K.......................................................................................................7
3.1 Simplification of the model.......................................................................................7
3.2 Description of the mechanisms................................................................................9
3.3 Decomposition of the tensor of deformation............................................................9
3.4 Expressions of the criteria........................................................................................11
3.5 Functions of work hardening....................................................................................12
3.6 Laws of dilatancy......................................................................................................13
3.7 Derived from the criterion........................................................................................15
4 Integration in Code_Aster.....................................................................................................18
4.1 Internal variables......................................................................................................18
4.2 Diagram of integration clarifies ( SPECIFIC )..........................................................19
4.3 Implicit diagram of integration..................................................................................27
5 References............................................................................................................................ 31
6 Features and checking..........................................................................................................31
7 Description of the versions of the document........................................................................31
8 Appendices: Terms of the matrix jacobienne........................................................................33
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1 Notations
1.1 General information
indicate the tensor of the effective constraints in small disturbances, noted in the shape of thefollowing vector:
11
22
33
212
213
223
One notes:
I 1= tr first invariant of the constraints
s=−I 1
3I tensor of the constraints déviatoires
sII=s . s second invariant of the tensor of the constraintsdéviatoires
max major principal constraint
min minor principal constraint
=−Tr
3I diverter of the deformations
V=tr voluminal deformation
cos 3 =21/2 33/2 det s
s II3 being the angle of Lode
p= 23ij
p ijp cumulated plastic deviatoric deformations
vp= 23 ij
vp ijvp cumulated viscoplastic deviatoric deformations
p plastic parameter of work hardening
vp viscoplastic parameter of work hardening
G visc function controlling the evolution of the viscousdeformations and describing the direction of flow
G=G−Tr G
3I diverter of G
G=Tr G trace of GG II= G⋅G normalizes G
angle of dilatancy
f d elastoplastic surface of load
f vp viscoplastic surface of load
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1.2 Convention of signs
• In Code_Aster, the convention of signs is that of the mechanics of the continuous mediums:
In compression: 0 ; =∂ u∂ x
0
In traction : 0 ; =∂ u∂ x
0
• In the model LETK , the convention of sign is that of the soil mechanics:
In compression : 0
Contractance : v0
In traction : 0
Dilatancy : v0
Note:
To integrate this law in Code_Aster such as it is presented, it is necessary to change the sign of all thefields at the entrance of the routine corresponding to the law of behavior and its exit.
At the entrance of the routine:
L&K−
=−−
L&K− =−−
L&K=−
At the exit of the routine:
=−L&K
=−L&K
=− L&K
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1.3 Parameters of the model
Notation Description
Pa atmospheric pressure
c resistance in simple compression, intervening in the expression of the criteria
H 0ext parameter controlling resistance in extension, intervening in the expression of the
criteria
point1 min intersection enters the thresholds of peak and intermediary
xams parameter not no one intervening in the laws of work hardening pre-peak
parameter not no one intervening in the laws of work hardening post-peak
a0 value of a on the threshold of damage
m0 value of m on the threshold of damage
s0 value of S on the threshold of damage
a pic value of a on the threshold of peak
m pic value of m on the threshold of peak
pic level of work hardening necessary to p to reach the threshold of peak
ae value of a on the threshold of cleavage
me value of m on the threshold of cleavage
e level of work hardening necessary to p to reach the threshold of cleavage
mult value of m on the residual threshold
ult level of work hardening necessary to p to reach the residual threshold
mv−max value of m on the maximum viscoplastic threshold
v−max value of v for which the maximum viscoplastic criterion is reached
Av parameter characterizing the amplitude the speed of creep
nv exhibitor intervening in the formula controlling the kinetics of creep
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0,v parameter relating to dilatancy in pre-peak
0,v parameter relating to dilatancy in pre-peak
1 parameter relating to dilatancy in post peak
1 parameter relating to dilatancy in post-peak
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2 Introduction
This document presents rheological model L&K developed with the CIH by F. Laigle and A. Kleine. It isa model élasto-visco-plastic dedicated to the rocks. The specificity of elastoplasticity resides in themodeling of a non-linear behavior in phase pre-peak and of a behavior post peak softening. Viscositycharacterizes the effect of time on the behavior of the rock. The initiation of each one of thesephenomena starts as of the crossing of a threshold. The behavior related to each phase is describedby the evolution of these various thresholds governed by functions of plastic or viscoplastic workhardening.
3 Equations of model L&K
3.1 Simplification of the model
In order to describe as well as possible and in a concise way this version of the model, it is necessaryto give an outline on the original model. The difference between the two versions will be perceivedbetter.
3.1.1 Short outline on the thresholds of the original model
In the original version of model L&K such as it is developed under the software Splash with the CIH(cf. R 1) or the thesis of A. Kleine (cf. R 4 ), there exist three distinct mechanisms:• An elastoplastic mechanism pre-peak, governed by a positive work hardening, • A viscoplastic mechanism also governed by a positive work hardening, • An elastoplastic mechanism post peak governed by a negative work hardening describing the
fracturing.
The characteristic of this original model lies in the fact that the coupling of the two mechanisms prepeak thus starts the fracturing the mechanism post-peak. Indeed, the cracks of extension induce adegradation of the mechanical properties of materials with the increase in dilatancy.
For the elastoplastic mechanism, a surface of load evolves through various thresholds. For theviscous mechanism, a viscoplastic surface evolves of an initial threshold to a final threshold.
The various thresholds delimit fields associated with particular physical mechanisms:• A threshold of damage confused with the threshold of initial viscosity,• An intermediate threshold, qualified of limit of cleavage,• A definite threshold characteristic as the envelope of the threshold of damage and limit of
cleavage, called also limiting of contractance/dilatancy (this limit is confused with the threshold ofmaximum viscoplasticity),
• A macroscopic of peak, definite threshold starting from the laboratory tests,• A definite purely conceptual intrinsic threshold like extrapolation of the threshold of peak, (this
threshold is eliminated in the version simplified from the model),• A threshold of residual resistance.
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max
Seuil viscoplastique initialSeuil d’endommagement
Seuil de résistance de pic
Seuil résiduel
Seuil intermédiaireou de clivage
min
Seuil viscoplastique maximalLimite de contractance/dilatance
Seuil intrinsèque
Figure 3.1.1-a. Thresholds of the original model presented in the plan min ,max
3.1.2 Characteristics of the simplified model
The simplified version proposed by the CIH (cf.R 2 ) rest only on two mechanisms: an elastoplasticmechanism and a viscoplastic mechanism. • The intrinsic threshold is eliminated in this version.• The threshold characteristic delimiting the fields of contractance and dilatancy in phase pre-peak,
is linearized to avoid any digital problem. It is supposed to be confused with the threshold ofmaximum viscosity.
max
Seuil viscoplastique initialSeuil d’endommagement
Seuil de résistance de pic
Seuil résiduel
Seuil intermédiaireou de clivage
min
Seuil viscoplastique maximalLimite de contractance / dilatance
Figure 3.1.2-a. Thresholds of the model simplified in the plan min ,max
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3.2 Description of the mechanisms
3.2.1 The viscoplastic mechanism
This mechanism is activated as soon as the point of load exceeds the initial viscoelastic threshold(compare to the initial elastic limit). Unrecoverable deformations are generated. These speeds ofunrecoverable deformations are proportional to the distance from the point of load compared to thethreshold of viscosity. Surface associated with the viscoplastic mechanism evolves of the initial elasticlimit to the maximum viscoplastic threshold according to the generated unrecoverable deformations.
3.2.2 The elastoplastic mechanism
3.2.2.1 pre peak
The plastic mechanism élasto is activated at the same time as the viscoplastic mechanism. As soonas the point of load exceeds the initial elastic limit, the surface of load starts with écrouire positively.
3.2.2.2 post peak
In the version simplified model, this mechanism is governed by:• a negative work hardening of the threshold of peak towards the intermediate threshold,• a negative work hardening of the intermediate threshold towards the residual threshold.
3.2.3 The voluminal behavior
The voluminal behavior, during the phase pre-peak, can be contracting or dilating.Below the initial elastic limit, the behavior is contracting.Below the limit contractance/dilatancy, the voluminal behavior is contracting plastic.Beyond this limit, the voluminal behavior is dilating.
N.B: In the simplified version of the model, the limit contractance/dilatancy is confused with themaximum viscoplastic threshold.
3.3 Decomposition of the tensor of deformation
The decomposition of the increment of total deflection is written:
=e
p
vp
where e ,
p and vp are the increments of the tensors elastic, irreversible instantaneous (plastic)
and irreversible differed (viscoplastic).
3.3.1 Hypo-elasticity
The selected elastic law is a law hypo-rubber band:
ije=
1E
ij−3E
p ij or ije=
12G
sij1
3Kpij
that one also notes: σ ij=De ij
e .
Moduli of rigidity G and of compressibility K depend on the state of stresses: K=K 0e [ I 1
−
3Pa]nelas
and G=G0e [ I 1
−
3Pa ]nelas
with I 1−=tr − .
I 1−=tr − being the trace of the constraints at the moment − .
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3.3.2 Plasticity
As in the simplified version of the model, only the mechanism deviatoric is taken into account, theinstantaneous unrecoverable deformation is written:
ijp=G ij
being the plastic multiplier and G is the function of flow.
That is to say f d the criterion of plasticity:
If f d≤0 then =0If f d
=0 then 0
The expression of G rest on several work quoted in the note R 2 and is form:
Gij=∂ f d
∂ij
− ∂ f d
∂ kl
nkl n ij , nij=
' sij
s II
− ij
' 23
, '=−
26sin 3−sin
Expressions of sin are detailed in the paragraph 3.6.1 and 3.6.2.
The calculation of fact the object of the paragraph 4.2.1.2.
The calculation of ∂ f∂ ij
is detailed in the paragraph 3.7.1
The evolution of elastoplasticity induces a plastic deformation: ε p connected through its deviatoric
component p with the parameter of work hardening p such as: p=∫ 23 p p dt
from where the relation p= 23G ijG ij= 2
3G II
3.3.3 Viscoplasticity
The calculation of the differed unrecoverable deformations vp be based on the theory of Perzyna.
εvp=⟨Φ f vp ⟩G ijvisc where Φ f vp and Gvisc the amplitude and the direction the speed of the
unrecoverable deformations characterize:
Φ f vp =Av f vp
Pa n v
and Gijvisc=
∂ f vp
∂ σ ij
− ∂ f vp
∂ σ kl
nkl n ij
f vp being the criterion of viscoplasticity, Av and nv are parameters of the model. Pa is theatmospheric pressure.
The evolution of viscoplasticity induces a viscous deformation: connected through its deviatoric
component εvp with the parameter of work hardening γ vp such as: γ vp=∫ 23εvpε vp dt .
Note:
That is to say Svp the surface defined within the space of constraints by: Svp={σ , f vp σ , ξ vp =0}The speed of creep for a state of stresses σ is proportional to the distance from σ with Svp . That is
to say Pσvp the projection of σ on Svp and d=∥σ−Pσ
vp∥ .
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One can also write d=∥∂ f vp
∂ σPσ
vp∥C . C being a constant which depends on the viscous
parameters. At first approximation, one writes: d=∥∂ f vp
∂ σσ ∥C . But this approximation poses a
problem insofar as the function f vp can not be defined for the value σ whereas it is for Pσvp . For
the moment, in Code_Aster this distance is not calculated. If the situation arises, a message of alarmwarns the user.
3.4 Expressions of the criteria
Expressions of the two criteria viscoplastic f vp and elastoplastic f d depend on the constraints and
functions on work hardening. In these expressions, one finds I 1 the first invariant of the constraints
and sII the second invariant of the tensor of the constraints déviatoires. In the two criteria, the samedefinitions are adopted for:
H θ =H 0
cH 0e
2H 0
c−H 0e
2 2h θ −h0ch0
e h0
c−h0e
h θ =1−γ cos3θ 16 , h0
c=H 0c=h 0° =1−γ
16 , h0
e=h 60° = 1γ 16 ,
H 0e is a parameter of the model. θ is the angle of Lode.
3.4.1 The viscoplastic criterion f vp
f vp σ =sII H θ −σ c H 0c [Avp ξ vp sII H θ Bvp ξ vp I 1D vp ξvp ]
avp ξ vp
with Avp ξ vp =−m vp ξ vp k
vp ξ vp6 σ c hc
0, Bvp ξvp =
mvp ξvp kvp ξ vp
3σc
, D vp ξ vp =svp ξ vp kvp ξ vp ,
kvp ξ vp = 23
1
2avp ξ vp
Functions of work hardening Avp ξ vp , Bvp ξvp and D vp ξ vp depend on the parameters of work
hardening avp ξ vp , mvp ξ vp and svp ξ vp whose expressions evolve with the variables of work
hardening ξ vp (see § 3.5). When ξ vp reached certain particular values, surface f vp reached thecorresponding thresholds.
Since the viscoplastic threshold is hammer-hardened only by viscosity, one always hasξ vp=Min [ γ vp , ξ v−max−ξ vp] . ξ vp−max corresponds to the maximum viscoplastic criterion and is a
parameter of the model
3.4.2 The elastoplastic criterion f d
f d σ =s II H θ −σ c H 0c [ Ad ξ p s II H θ Bd ξ p I 1Dd ξ p ]
a d ξ p
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with Ad ξ p =−md ξ p k
d ξ p 6σ c hc
0, Bd ξ p =
md ξ p kd ξ p
3σc
, Dd ξ p =sd ξ p kd ξ p ,
kd ξ p = 23
1
2ad ξ p
Functions of work hardening Ad ξ p , Bd ξ p and Dd ξ p depend on the parameters of work
hardening ad ξ p , md ξ p and sd ξ p whose expressions evolve with the variables of work
hardening ξ p (see § 3.5). When ξ p reached certain particular values, surface f d reached thecorresponding thresholds.
Elastoplastic work hardening depends on the position of the point of load compared to the limitcontractance/dilatancy:
• if the point of load is below this limit, ξ p=γ p ,
• if the point of load is with the top of this limit, ξ p=γ p γvp .
3.5 Functions of work hardening
3.5.1 Functions of work hardening of the viscous criterion
The viscoplastic criterion is governed by the following functions of work hardening:
a ξ vp =a0a v−max−a0 ξ vp
ξ v−max
with av−max=1.
m ξ vp =m0mv−max−m0 ξ vp
ξ v−max
s ξ vp =s0 sv−max−s0 ξvp
ξ v−max
with sv−max=s0
3.5.2 Functions of work hardening of the elastoplastic criterion and their derivative
The expressions of the functions of work hardening which govern the elastoplastic criterion varyaccording to the value of the parameters ξ p :
Evolution enters the threshold of damage and the threshold of peak : If 0≤ξ pξ pic
a ξ p =a0ln 1 ξ p
xamsξ pic a pic−a0
ln 11 / xams ∂ a∂ ξ p
= a pic−a0
ln 11 / xams 1
ξ pxams ξ pic
m ξ p =m0ln1 ξ p
xams ξ pic m pic−m0
ln 11 / xams ∂m∂ ξ p
= m pic−m0
ln 11 / xams 1
ξ pxams ξ pic
s ξ p =s0ln 1 ξ p
xams ξ pic s pic−s0
ln 11/ xams
∂ s∂ ξ p
= s pic−s0
ln 11 / xams 1
ξ pxams ξ pic
with s pic=1 .
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Evolution enters the threshold of peak and the intermediate threshold or limit of cleavage: Ifξ pic≤ξ pξ e
a ξ p =a picae−a pic ξ p−ξ pic
ξ e−ξ pic
∂ a∂ ξ p
=ae−a pic
ξ e−ξ pic
s ξ p =1− ξ p−ξ pic
ξ e−ξ pic
∂ s∂ ξ p
=−1
ξ e−ξ pic
∂m∂ ξ p
=∂m∂ a
∂ a∂ ξ p
∂m∂ s
∂ s∂ ξ p
∂m∂ ξ p
=σ c
σ po int 1 [− a pic
a ξ p 2 mpic
σ po int1
σ c
s pica pic
a ξ p ln m pic
σ po int 1
σc
s pic ∂ a∂ ξ p
−∂ s∂ ξ p ]
Evolution enters the intermediate threshold and the residual threshold : If ξ e≤ξ pξ ult
a ξ p =a eln 11η
ξ p−ξ e
ξ ult−ξ e ault−ae
ln 11/η ∂ a∂ ξ p
= ault−ae
ln 11/η 1ξ pηξ ult−1η ξ e
s ξ p =0 ∂ s
∂p=0
m ξ p =σc
σ po int 2me
σ po int 2
σ c
ae
a ξ p
∂m∂ ξ p
=σ c
σ po int 2 [− ae
a ξ p 2 lnme
σ po int 2
σcme
σ po int 2
σ c
a e
a ξ p ] ∂a∂ ξ p
On the residual criterion : If ξ p≥ξ ult
a ξ p =ault=1. ∂ a∂ ξ p
=0
s ξ p =0 ∂ s∂ ξ p
=0
m ξ p =mult ∂m∂ ξ p
=0
3.6 Laws of dilatancy
The elastoplastic and viscoplastic mechanisms are non-aligned. Laws of evolution of ijp and of εij
vp
are controls respectively by a function G and a function G visc , such as:
Gij=∂ f d
∂ ij
− ∂ f d
∂ kl
nkl n ij and Gijvisc=
∂ f vp
∂ ij
− ∂ f vp
∂kl
nkln ij with
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n ij=
'sij
s II
− ij
' 2
3
and '=−26sin 3−sin
The calculation of the angle of dilatancy differ according to the viscous mechanisms orelastoplastic pre-peak and elastoplastic post-peak.
3.6.1 Elastoplastic pre-peak and viscoplastic angle of dilatancy of the mechanisms
sin =0,v max− lim
0,vmaxlim with 0,v and 0,v parameters of the model.
where
lim=min cmv−max
min
c
sv−max av−max
with sv−max=s0 and av−max=1 . c and mv−max
are parameters of the model.
There exist conditions on the parameters 0,v and 0,v who are:
• 0,v0,v or
• {0,v0,v
spic a pic
s0 a0
≤10,v
0, v−0,v
3.6.2 Angle of dilatancy of the elastoplastic mechanism post-peak
sin =1 −res
1res with 1 and 1 parameters of the model
where
=max
min and res=
max
min
=1mult
={c p tan p
si p≤e
0 si pe
with c p =c . s p
a p
21a p m p s p a p−1
and
p=2. arctg 1a p m p s ξ p a p −1
−π2
min and max are calculated starting from the invariants of the constraints:
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min=13 I 1− 3
2−
2H θ − H 0cH 0
e 2 H 0
c−H 0e 3
2sII
max=13 I 1 322H θ −H 0
cH 0e
2 H 0c−H 0
e 32
s II
3.7 Derived from the criterion
3.7.1 Calculation of ∂ f∂ ij
∂ I 1
∂ ij
=∂ tr ij ∂ ij
= ij
∂ sII H θ ∂ σ ij
=∂ sII H θ ∂ skl
∂ skl
∂σ ij
= ∂H θ ∂ skl
sIIH θ ∂ sII
∂ skl ∂ skl
∂ σ ij
∂ sII
∂ skl
=skl
s II
; sII= skl . skl
∂ skl
∂ σ ij
=
∂σ kl−13
tr σ δkl ∂ σ ij
=δik .δ jl−13
δ ij .δ kl
Note: skl .δik .δ jl−13
δij .δ kl=sij
∂H θ ∂ skl
= H0c−H 0
e
h0c−h0
e ∂ h θ ∂ skl
from where ∂ s II H
∂σij
= H 0c−H 0
e
h0c−h0
e ∂h ∂ skl
s IIH skl
s II .ik . jl−13 ij . kl
There is the relation: cos 3θ =54det s
sII3 (see Documentation R7.01.13-A: Law CJS in mechanics)
∂ h ∂ s kl
=16
1−γcos 3θ −
56 ∂ 1−cos 3θ ∂ s kl
=1
6h 5
∂
∂ skl sII
3 −54 det s
sII3
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∂h θ ∂ skl
=1
6h θ 5 {[∂ s II3
∂ skl
−γ 54∂det s ∂ skl ] s II
3
s II6− s II
3 −γ54det s 3skl s II
s II6 }
=1
6h θ 5 {3skl
sII2 −γ54∂det s
∂ skl1s II
3 −1−γ cos 3θ 3skl
s II2 }
=γ cos 3θ
6h θ 53skl
s II2−
γ 54
6h θ 5 s II3 ∂det s ∂ skl
One thus finds:
∂ s II H ∂ij
=
H 0c−H 0
e
h0c−h0
e cos 3θ
6h 53skl
sII2−54
6h 5 s II3 ∂ det s ∂ skl
s IIH skl
sII . ik . jl−13 ij . kl
Finally:
For the elastoplastic criterion :
∂ f d
∂σij
=
∂ s II H θ ∂σij
−a d ξ p σ c H 0c [Ad ξ p s II H θ Bd ξ p I 1Dd ξ p ]
adξ p −1
Ad ξ p∂ s II H θ ∂σ ij
Bd ξ p I d
and for the viscous criterion:
∂ f vp
∂σ ij
=
∂ s II H θ ∂σ ij
−avp ξ vp σ c H 0c [ Avp ξ vp s II H θ Bvp ξ vp I 1Dvp ξ vp ]
avp ξ pp −1
Avp ξ vp ∂ s II H θ ∂ σ ij
Bvp ξ vp I d
with ∂ s II H θ
∂ σ ij
=
H 0c−H 0
e
h0c−h0
e γ cos 3θ
6h θ 5
3skl
s II2−
γ 54
6h θ 5s II
3 ∂det s ∂ skl s IIH θ
skl
s II . δik .δ jl−13
δij .δ klWarning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2021 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)
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3.7.2 Calculation of ∂ f d
∂ ξ p
Expression of the threshold in constraints:
f d σ =s II H θ −σ c H 0c [ Ad ξ p s II H θ Bd ξ p I 1Dd ξ p ]
a d ξ p
with Ad ξ p =−md ξ p k
d ξ p6 σ c hc
0, Bd ξ p =
md ξ p kd ξ p
3σc
, Dd ξ p =sd ξ p k ξ p ,
kd ξ p = 23
1
2ad ξ p
∂ f d
∂ ξ p
=∂ f d
∂a d. a d ξ p
∂ f
∂md. md ξ p
∂ f
∂ sd. sd ξ p
∂ f d
∂ sd=−ad k d σ c H 0
c [ Ad s II H θ Bd I 1Dd ]ad−1
∂ f d
∂md=−ad σ c H 0
c [ Ad
mds II H θ Bd
mdI 1] [ Ad sII H θ Bd I 1Dd ]
a d−1
∂ f d
∂ ad=
σ c H 0c ad [ Ad s II H θ Bd I 1Dd ]
a d
.[ ln [ Ad s II H θ Bd I 1Dd ]−
sd
2ad ln 23
23
12a
[ Ad s II H θ Bd I 1Dd ] ]
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4 Integration in Code_Aster
The integration of the model LETK can be realized according to two distinct diagrams of integration.The first diagram of integration (“historical”) is described like SPECIFIC and corresponds to a diagramof explicit integration. The second diagram is built on the basis of diagram of implicit integration. It isaccessible under the keyword NEWTON_PERT .
4.1 Internal variables
V 1 : elastoplastic variable of work hardening ξ p
V 2 : plastic deviatoric deformation γ p
V 3 : viscoplastic variable of work hardening ξ vp
V 4 : viscoplastic deviatoric deformation γ vp
V 5 : 0 if contractance, 1 if dilatancy
V 6 : indicator of viscoplasticity
V 7 : indicator of plasticity
V 8 : Fields of behavior of the rock in plasticity
Five fields of behavior, numbered from 0 to 4 (cf appears), are identified to make it possible to have arelatively simple representation of the state of damage of the rock, since the intact rock to the rock in
a residual state. These fields are function of the cumulated plastic deformation déviatoire p and of
the state of stress. Each increment of number of field defines the passage in a field of higher damage.
• If the diverter is lower than 70% of the diverter of peak, then the material is in field 0;• If not:
• If p=0 then the material is in field 1;
1) If 0 p e then the material is in field 2;
• If e p
ult then the material is in field 3;
• If pult
then the material is in field 4.
V 9 : position of the state of stresses compared to the thresholds of viscosity.
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D o m a i n e E t a t d e l a r o c h e 0 I n t a c t e 1 E n d o m m a g e m e n t
p r é - p i c 2 E n d o m m a g e m e n t
p o s t - p i c 3 F i s s u r é e 4 F r a c t u r é e
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Four fields of behavior, numbered from 1 to 4 (cf appears), are identified to make it possible to have asimple representation of the viscous behavior of material, since the intact rock to the rock in a residualstate. These fields are function of the cumulated viscoplastic deformation déviatoire vp and of thestate of stress.
4.2 Diagram of integration clarifies ( SPECIFIC )
4.2.1 Update of the constraints
One expresses the constraints brought up to date at the moment + compared to those calculatedat the moment -:
σ=σ−D e Δε e ; s=s−2GΔ εe ; I 1=I 1−3KΔεv
e
σij=sijI1
3δij
; Δεij=Δ εijtr Δε
3δij=Δ εij
Δεv
3δ ij
; I 1=tr σ ; εv=tr Δε
Elastic prediction:
σ e=σ−De Δε ; se=s−2GΔ ε ; I 1e=I 1
−3KΔεv
K=K 0e [ I 1
−
3Pa ]n elas
and G=G0e [ I1
−
3Pa ]nelas
Note: The coefficient of compressibility K and it modulus of rigidity G are considered at the
moment -.
4.2.1.1 Hypoelasticity
Δσ ij=ΔsijΔI 1
3δij Δεij=Δ εij
Δεv
3δ ij Δσ ij=2GΔεijK− 2G
3 tr Δε δ ij
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{ 11
22
33
212
213
2 23
}=[4G3K K− 2G
3K− 2G
30 0 0
K−2G3
4G3K K−
2G3
0 0 0
K−2G3
K−2G3
4G3K 0 0 0
0 0 0 2G 0 00 0 0 0 2G 00 0 0 0 0 2G
]
De
.{Δε11
22
33
2 12
2 13
2 23
}
4.2.1.2 Plasticity and viscoplasticity
One expresses the stress field at the moment :
σ ij=σ ij−D ijkl
e Δεkle=σ ij
−Dijkl
e Δεkl−Δεklp−Δεkl
vp
who is written by replacing the increase in the plastic deformations and viscous by their expressions inthe form:
σ ij=σ ij−D ijkl
e Δεkl−ΔλGkl σ− , ξ p
−−⟨φ⟩Gklvisc σ− , ξ v
− Δt
The principal unknown factor is the plastic multiplier Δλ .
One seeks Δλ / f d σ , ξ p=0
f d σ ij−D ijkl
e Δεkl−ΔλGkl σ− , ξ p
−−⟨φ⟩G klvisc σ− , ξ v
− Δt , ξ p−Δξ p =0
with Δγ p=Δλ 23G ijGij=Δλ 2
3G II
One chooses to make an explicit resolution with a development of Euler:
f d σ ij−Dijkl
e Δεkl−Dijkle ⟨Φ ⟩Gkl
visc σ− , ξ v− Δt , ξ p
−−∂ f d
∂ σ ij
D ijkle G kl σ− , ξ p
− Δλ∂ f d
∂ ξ p
Δξ p=0
The two cases are distinguished: • Δξ p=Δγ pΔγvp (dilating case: the state of stresses exceeds the limit
contractance/dilatancy)
f d σ ij−Dijkl
e Δεkl−Dijkle ⟨Φ ⟩Gkl
visc σ− , ξ v− Δt , ξ p
−
−∂ f d
∂σ ij
Dijkle Gkl σ− , ξ p
− Δλ∂ f d
∂ ξ p Δγ pΔγvp =0
f d ij−D ijkl
e kl−Dijkle G kl
visc − ,v− t , p
−=
∂ f d
∂ij
D ijkle G kl
− , p−−∂ f d
∂p 23
G II − ,p
− −∂ f d
∂ p
vp
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Δλ=
f d σ ij− , ξ p
− ∂ f d
∂ ξ p
Δγvp∂ f d
∂ σ ij[Dijkl
e Δεkl−Dijkle ⟨Φ ⟩G kl
visc σ− , ξ vp− Δt ]
∂ f d
∂ σ ij
Dijkle G kl σ− , ξ p
− −∂ f d
∂ ξ p 23G II σ− , ξ p
−
• Δξ p=Δγ p (case contracting: the state of stresses is below the limit contractance/dilatancy)
Δλ=
f d σ ij− , ξ p
− ∂ f d
∂ σ ij[Dijkl
e Δεkl−Dijkle ⟨Φ ⟩G kl
visc σ− , ξ vp− Δt ]
∂ f d
∂ σ ij
Dijkle Gkl σ− , ξ p
−−∂ f d
∂ ξ p 23G II σ− , ξ p
−
with Φ=Av f vp σ e , ξ vp−
Pa nv
, Av and nv are parameters of the model.
4.2.2 Tangent operator
The constraint with the state :
ij= ij−Dijkl
e kl
e= ij
−Dijkl
e kl− klp− kl
vp = ij−Dijkl
e kl−G kl − , p
−−⟨⟩Gkl
visc
− ,vp− t
∂ σ ij
∂ Δεkl
=D ijkle −Dijmn
e Gmn− ∂ Δλ∂ Δεkl
−Dijmne G imn
visc− ∂ ⟨φ⟩∂ Δεkl
Δt
The two cases are distinguished:
• Δξ p=Δγ p (case contracting: the state of stresses is below characteristic threshold)
•
Δλ=
f d σ ij− , ξ p
− ∂ f d
∂ σ ij[Dijkl
e Δεkl−Dijkle ⟨Φ ⟩G kl
visc σ− , ξ v− Δt ]
∂ f d
∂ σ ij
Dijkle Gkl σ− , ξ p
−−∂ f d
∂ ξ p 23G II σ− , ξ p
− and Φ=Av f vp σ e , ξ vp
− Pa
nv
∂ Δλ∂ Δεkl
=
∂ f d
∂ σ ij
.[Dijkle −Dijmn
e Gmnvisc σ− , ξ v
− ∂Φ∂Δεkl
Δt ] ∂ f d
∂ σ ij
Dijmne Gmn σ− , ξ p
−−∂ f d
∂ ξ p 23G II σ− , ξ p
−
∂Φ∂ Δεkl
=Av . nv
Patm f vp σ e , ξ vp−
Patm nv−1
.∂ f vp
∂σ ije .
∂ σ ije
∂Δεkl
=Av . nv
Patm f vp σe , ξvp−
Patm n
v−1
.∂ f vp
∂ σ ije . D ijkl
e
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∂ σ ij
∂ Δεkl
=Dijkle −D ijmn
e .Gmn−
∂ f d
∂σ ij
.[Dijkle −Dijmn
e Gmnvisc σ− , ξ p
− ∂Φ∂ Δεkl
Δt ]∂ f d
∂ σ ij
D ijkle Gkl σ− , ξ p
−−∂ f d
∂ ξ p 23G II σ− , ξ p
−−
Dijmne .Gmn
visc−Av .nv
Patm f vp σ e , ξvp−
Patm nv−1
.∂ f vp
∂ σ ije . D ijkl
e Δt
• p= pv (dilating case: the state of stresses exceeds the characteristic threshold)
Δλ=
f d σ ij− , ξ p
− ∂ f d
∂ ξ p
Δγvp∂ f d
∂ σ ij[Dijkl
e Δεkl−Dijkle ⟨Φ ⟩G kl
visc σ− , ξ vp− Δt ]
∂ f d
∂ σ ij
Dijkle Gkl σ− , ξ p
− −∂ f d
∂ ξ p 23G II σ− , ξ p
−
∂ Δλ∂ Δεkl
=
∂ f d
∂σ ij
.[Dijkle −Dijmn
e Gmnvisc σ− , ξ p
− ∂Φ∂ Δεkl
Δt ]∂ f d
∂ ξ p
.∂ Δγvp
∂ Δεklvp
.∂ Δεkl
vp
∂ Δσ ije
. Dijkle
∂ f d
∂σ ij
Dijmne Gmn σ− , ξ p
−−∂ f d
∂ ξ p 23G II σ− , ξ p
−
where:
∂ Δγvp
∂ Δεklvp=
12 2
3Δ ε ij
vp . Δ εijvp −
12 .
23
.2 . Δ εijvp .∂ Δ εij
vp
∂ Δεklvp=
23
.Δ εij
vp
Δγvp
. δik .δ jl−13
δij . δkl
∂ Δεklvp
∂ Δσ ije=
nv . Av
Patm. f vp σe , ξ v
− Patm
nv−1
.∂ f vp
∂ σ ije .G kl
visc− σ− , ξ vp− Δt
4.2.3 Algorithm of resolution in Code_Aster
• Change of sign of the constraints to the state − and of the increase in deformation:
σ L∧K− =−σ−
Δε L∧K=−Δε
• Calculation of the elastic constraint of prediction σ e : σ e=σ−De Δε• Checking of the sign of the viscous criterion with the viscous variable max: f vp σe , ξ vp−max
• If f vp σe , ξ vp−max 0 : dilating case and coupled work hardening 15 V . One regards as variable
of work hardening of the plastic criterion the office plurality between the plastic variable of work
hardening and the viscous variable: p= pvp
• If f vp σe , ξ vp−max 0 : case contracting and work hardening not coupled V 5=0 . The variable of
work hardening of the plastic criterion is the plastic variable: p= p
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For the viscous criterion: uvp=Avp ξ vp sII H θ Bvp ξ vp I 1D vp ξ vp
If uvp σ e , ξ vp− 0 :
• if −Dvp
Bvp−
Dd
Bd (Figure 4.2.3-a) then recutting of the step of time
−Dd
B d−Dv p
Bv p
s II
I1
Seuil élas
toplastiq
ue
Seuil viscoplastique
uvp<0
Figure 4.2.3-a. Schematic representation in the case uvp σ e , ξ vp− 0 , −
D vp
Bvp−
Dd
Bd
If not ( Figure 4.2.3-b ) two cases arise: • if
e is in the zone A then f vp σ e is not defined but one can make ageometrical projection (cf notices paragraph 3.3.3)
• if e is in the zone B then it is necessary to make a projection at the top.
One is satisfied then with the message: “ stop for coefficients non-cohesive materials of the law “.
−Dd
Bd −Dvp
Bvp
s II
I 1
Seuil
élastoplastique Seuil v
iscoplas
tique
zone A
zone B
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Figure 4.2.3-b. Schematic representation in the case uvp σ e , ξ vp− 0 , −
Dd
Bd−
Dvp
Bvp
For the elastoplastic criterion:
ud=Ad ξ p sII H θ Bd ξ p I 1Dd ξ p
If ud σ e , ξ p−0 then there is recutting of the step of time
For the viscous criterion:
If f vp σe , ξ vp− 0 then not of creep and Δεvp=0
•
If f vp σe , ξ vp− 0 then creep develops according to the following form:
Δεvp=A[ ⟨ f vp σe , ξ vp− ⟩
Patm ]nv
G visc σ− , ξ vp− Δt with G visc=
∂ f vp
∂ σ−∂ f vp
∂ σnn
: Hooks of Macauley
where sin =μ0,v max− lim
0, v σmax lim (see § 3.6.1 ) , β ' and n are deduced
one can deduce vp= 23 vp . vp0 where vp= vp−
tr vp 3
I d
reactualization of the variable of work hardening of the viscous criterion:
vp=vp−vp with vp=Min [vp ,v−max−vp
− ]
reactualization of the constraints: σ=σ e−De Δεvp
reactualization of the internal variables:
V 1=ξ p=ξ p−Δγvp
V 2=γ p=γ p−
V 3=ξ vp=ξ vp−Δξ vp
V 4=γ vp=γvp−Δγvp
For the elastoplastic criterion:
• If f d σ e−D e Δξ vp , ξ p− vp ≤0 then: = p=p=0
• If f d σ e−D e Δεvp , ξ p− vp 0 then: Δ ε p=ΔλG σ− , ξ p
− with
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sin Ψ =μ0, v σ max−σ lim
ξ 0,v σ maxσ lim (see § 3.6.1 ) if 0≤ξ p
−ξ pic
sin Ψ =μ1 α−α res
ξ1 α−α res (see § 3.6.2 ) if ξ p
−ξ pic
G=∂ f d
∂σ−∂ f d
∂ σnn β ' and n are deduced
One seeks Δλ0 such as f d σ , ξ p− =0
One deduces p0
If work hardening not coupled (contractance) f d σ e−De Δξ vp , ξ p
−Δγvp≤0 then:
p= p
if not coupled work hardening (dilatancy) f d σ e−De Δεvp , ξ p− vp 0 then:
p= pvp
One the table of the internal variables supplements:
V 1=ξ p=ξ p−Δξ p
V 2=γ p=γ p−Δγ p
Update of the constraints:
Δε irr=ΔεvpΔε p
Δεe=Δε−Δεirr
Δσ=De Δεe
σ=σ−Δσ
Summary of the algorithm•
e=
−De Δε
• if f vp σe , ξ v max 0 then contractance ( VARV=0 ) and the plastic variable is
p=
p
• if f vp σe , ξ v max 0 then dilatancy ( VARV=1 ) and the plastic variable is
Δξ p=Δγ pΔγvp
Checking of creep:
• calculation of f vp σe , ξ vp−
• if f vp σe , ξ vp− 0 (Pas de creep)
vp= vp=0
ξ vp=ξ vp−
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vp=vp−
• if f vp e ,vp− 0 (creep)
calculation of Δεvp and of Δγvp according to e and of ξ vp
−
vp=min vp ,v−max−vp−
ξ vp=ξ vp−Δξ vp
vp=vp− vp
• Adjustment of the elastic prediction: ne=
e−De Δε vp
Checking of plasticity :
• calculation of f d ne ,p
− vp • if f d n
e ,p− vp 0 (Elasticity)
p= p=0
p= p−
ξ p=ξ p−Δξ p with
p=0 if VARV=0
p= vp if VARV=1
update of the constraints:
=e−De Δεvp
• if f d ne ,p
− vp 0 (Plasticity)
calculation of , Δγ p and p
p= p if VARV=0
p= pvp if VARV=1
p= p− p
update of the constraints:
=−D e
− vp− p
table of the internal variables:
V1= p
V2= p
V3=vp
V4=vp
V5=VARV (0 if contractance or 1 if dilatancy)V6= indicator of viscosityV7= indicator of plasticity
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4.3 Implicit diagram of integration
The integration of the model LETK according to the implicit diagram of integration is realized underenvironment PLASTI . The integration of model LETK under implicit scheme is currently availableonly by calculation of a disturbed matrix jacobienne local ( ‘NEWTON_PERT’ ).
The algorithm of resolution follows following logic. It uses an elastic prediction then iterations ofcorrection if the viscous and/or plastic thresholds are requested. The purpose of the diagram is toproduce the variation of the constraints and variables of work hardening under the effect of anincrement of deformation.
The local subdivision of the model is activable under this diagram of integration by the keywordITER_INTE_PAS keyword factor BEHAVIOR , cf [U4.51.11]).
4.3.1 Elastic phase of prediction
This phase is similar to that presented in section 4.2.1 . The going beyond the thresholds of plasticity and viscosity is tested compared to this state of stresses.The expression of the thresholds tested is clarified in the § 3.2 .
• If none the thresholds is requested, the prediction is regarded as valid compared to the models.An update of the internal variables is undertaken to display the state of activation of the variousthresholds.
• If a threshold among both to consider (plasticity and/or viscosity) is requested, the resolution of alocal system of nonlinear equations must be initiated. The defined mechanisms of dissipation aspotentially active must lead to the going beyond the associated thresholds (plasticity and/orviscosity)
4.3.2 Phase of correction: nonlinear equations to solve
This stage consists in solving the system of nonlinear local equations established on the basis as of viscousand/or plastic mechanisms. After convergence, the constraints and internal variables of the model are put up todate.
The unknown factors of the system of nonlinear equations are the constraints n1 , the plastic multiplier
n1p
, the plastic variable of work hardening n1p and the viscous variable of work hardening n1
vp .
The vector of the inconuu be thus comprises to the maximum for modelings 3D 9 unknown factors. The nonlinear equations to solve are the following ones:
• The incremental equation of state, E1 :
n1− n−C e n1 : −G p− f vp ⋅Gvp =0
• The condition of Kuhn-Tucker, E2 :
{Si f d0 alors =0
Si f d=0 alors 0
• Incremental evolution of the variable of work hardening plastics, E3 :
n1p −n
p−p=0 , with p evolving according to the conditions specified with the § 3.4.2 .
• Incremental evolution of the viscoplastic variable of work hardening, E4 :
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n1vp −n
vp−vp=0 , with vp evolving according to the conditions specified with the § 3.4.1 .
These equations constitute a square system R Y , where the unknown factors are
Y= , , p ,vp . With the iteration j loop of on-the-spot correction of Newton, one abstr. O C
the following matric equation:
d R Y j d Y j
⋅ Y j1=−R Y j
The matrix jacobienne d R Y j d Y j
, nonsymmetrical, builds itself in the following way:
d R Y j d Y j
=[∂E1∂n1
j
∂E1∂ j
∂E1∂p
j
∂ E1∂vp
j
E2
∂n1j
E2
∂ j
E2
∂pj
E2
∂vpj
E3
∂n1j
E3
∂ j
E3
∂pj
E3
∂vpj
E4∂n1
j
E4∂ j
E4∂p
j
E4∂vp
j
]
This matrix is evaluated today analytically or by disturbance (ALGO_INTE = ‘NEWTON’ or ALGO_INTE =‘NEWTON_PERT’).
With an aim of standardizing the scales between the various equations to be solved, one makes the choice toput at the level of deformations bearing the E1 equation on the incremental equation of state. One applies forthat the reverse of the modulus of nonlinear rigidity of elasticity. This choice makes it possible to ensure a moreuniform convergence on the whole of the system.
Convergence famous is acquired since ∣∣R Y j ∣∣ RESI_INTE_RELA. One also makes sure when the
mechanism of plasticity is active that the plastic multiplier is strictly positive. If it is not the case, localintegration is started again without taking account of the mechanism of plasticity. Only the mechanism ofviscosity can then be considered.
4.3.2.1 Expression of the terms of the matrix jacobienne
Derived terms associated with R1 are:
d R1 ijd Y 1 mn
=I ilkl−∂C ijkl
e
∂mn
: [ kl−⋅G klp−kl
vp ]C ijkle :
∂G klp
∂mn
C ijkle : Gkl
vp⊗∂ ⟨ f vp ⟩
+∂mn
t⟨ f vp⟩+⋅ t⋅C ijkl
e :∂G kl
vp
∂mn
d R1ijd Y 2
=C ijkle :G kl
p
d R1ijd Y 3
=⋅C ijkle :
∂Gklp
∂p
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d R1ijd Y 4
=C ijkle :[ ∂ ⟨ f vp ⟩
+
∂vp G kl
vp⟨ f vp ⟩
+⋅∂G kl
vp
∂vp ]⋅ t
Derived terms associated with R2 s.e distinguishes according to the expression taken to satisfy the condition
with Kuhn-Tucker:
If R2 = : S I R2 = f p :
d R2 d Y 1 ij
=0ij d R2
d Y 1 ij=∂ f p
∂ij
d R2d Y 2
=1 d R2
d Y 2 =0
d R2d Y 3
=0 d R2
d Y 3 =∂ f p
∂ p
d R2d Y 4
=0 d R2
d Y 4 =0
Derived terms associated with R3 are:
d R3 d Y 1 ij
=−⋅ 23⋅∂ G II
p
∂ij
(case contract ant: the state of stresses is below characteristic threshold)
d R3 d Y 1 ij
=−⋅ 23⋅∂ G II
p
∂ij
− 23⋅ t⋅[ G II
vp⋅∂ ⟨ f vp ⟩
+
∂ ij
⟨ f vp⟩+⋅∂ G II
vp
∂ij ] (dilating case: the state of
stresses exceeds the characteristic threshold)
d R3d Y 2
=−23⋅G II
p
d R3
d Y 3 =1−⋅2
3⋅∂ G II
p
∂ p
d R3d Y 4
=− 23⋅ t⋅∂ ⟨ f vp ⟩
+
vp ⋅G II
vp⟨ f vp ⟩
+⋅∂ G II
vp
vp (dilating case: the state of stresses exceeds the
characteristic threshold)
Derived terms associated with R4 are:
d R4 d Y 1 ij
=− 23⋅ t⋅∂ ⟨ f vp ⟩
+
ij
⋅G IIvp⟨ f vp ⟩
+⋅∂ G II
vp
ij if vpmax
vp −vp t -
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d R4d Y 2
=0
d R4d Y 3
=0
d R4d Y 4
=1− 23⋅ t⋅∂ ⟨ f vp ⟩
+
vp ⋅G II
vp⟨ f vp ⟩
+⋅∂ G II
vp
vp
The detailed expression of the whole of the put ends concerned depends on the derivative principal following:
d C ijkle
d mn
; d G ij
p
d kl
; d ⟨ f vp ⟩
+
d ij
; d G ij
vp
d kl
; d G II
p
d ij
; d G II
vp
d ij
;
d G ijp
d p ;
d G IIp
d p ;
d ⟨ f vp ⟩+
d vp ;
d G ijvp
d vp ;
d G IIvp
d vp .
The quantities mentioned above are presented in appendix of the document.
4.3.3 Phase of update
The update of the vector solution is carried out according to the following operation:
Y= Y j1= Y j
Y j1 This phase of update consists in deferring the evolution of the constraints, plastic deformations, viscoplasticdeformations and parameters of plastic and viscoplastic work hardening.
4.3.4 Tangent operator of speed
The tangent operator of speed was introduced right now within the framework of the explicit diagram ofintegration. This operator of rigidity is used at the time of the predictions on a total scale of Newton-Raphsonout of tangent matrix ( PREDICTION=' TANGENTE' ). Sources FORTRAN associated with the onstruction withthis operator are common to both diagrams of integration ( ALGO_INTE = (‘NEWTON’, ‘SPECIFIC’) ) . 4.3.5 Consistent tangent operator
On the basis as of analytical developments specified in the document [R5.03.12], it is possible to determine the
tangent operator M c=∂
∂ starting from the terms of the matrix jacobienne definite above, § 4.3.2 (
J=d Rd Y
).
Indeed, the system Y =0 is checked at the end of the increment and for a small variation of , by
considering this time like a variable, the system remains with balance and thus one checks d =0 . By differentiation, one obtains:
∂
∂ d ∂
∂d ∂
∂ d ∂
∂ p
d p∂
∂ vp
d vp =0
One rewrites the system by putting the ends in in the member of right-hand side:
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∂
∂ d ∂
∂d ∂
∂ d ∂
∂ p
d p=−∂
∂ vp
d vp
This system can then be written in the following form:
J⋅d Y =− ∂
∂ d with
∂
∂ = {−C e ,0 ,0 ,0}
Finally, one obtains: J⋅d Y = {C e : , 0,0 ,0}
One writes then the system per blocks while separating d other variables Z= , p , vp ,
which gives:
[J J Z
JZ
J ZZ]⋅Z =C
e d 0
The expression of the tangent operator becomes:
M c=∂
∂=
d d
=[ J − J
Z J ZZ −1
J Z ]−1
C e
Notice : The Jacobienne matrix not being symmetrical, the tangent operator M c is not it either.
5 ReferencesR 1 . Model “L&K” for Code_Aster. IH-HAVL-SIO-00015-A.
R 2 . Model “L&K” for Code_Aster. IH-HAVL-SIO-00015-B.
R3. Réunion on viscoplastic model L&K of the CIH simplified for Code_Aster. CR-AMA-2007-142.
R 4 . Digital modeling of the behavior of the underground works by a viscoplastic approach. Thesis presented to the INPL by A. Kleine, November 2007.
6 Features and checking
This document relates to the law of behavior LETK (keyword BEHAVIOR of STAT_NON_LINE) and theassociated material LETK (order DEFI_MATERIAU).
This behavior is checked by the two cases tests:• SSNV206 - Triaxial compression test with model LETK of the CIH - [V6.04.206] • WTNV135 - Triaxial compression test drained with model LETK of the CIH - [V7.31.135]
7 Description of the versions of the document
Version Aster
Author (S) Organization (S)
Description of the modifications
9.2 J.El-Gharib, C. Chavant, EDF-R&D/AMAF.Laigle, A.Kleine EDF-CIH
Initial text
11.2 A.Foucault Algorithm of integration by implicit scheme
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11.3 A.Foucault Analytical development of the matrixjacobienne DR/DY
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8 Appendices: Terms of the matrix jacobienne
Evaluation of the terms relative to d R1 ij
d Y 1 mn :
d R1 ijd Y 1 mn
=I ilkl−∂C ijkl
e
∂mn
: [ kl−⋅G klp−kl
vp ]C ijkle :
∂G klp
∂mn
C ijkle : Gkl
vp⊗∂ ⟨ f vp ⟩
+ ∂mn
t⟨ f vp⟩+⋅ t⋅C ijkl
e :∂G kl
vp
∂mn
∂C ijkle
∂mn
=nelas
I 1
⋅C ijkle ⊗mn
∂Gijp
∂ kl
=∂
kl∂ f p
∂ij− ∂ kl
∂ f p
∂ij: nmn⊗nij− ∂ f p
∂mn
:∂nmn
∂ kl⊗nij− ∂ f p
∂mn
: nmn⋅∂nij
∂ kl
with
∂ f d
∂ σ ij
=
∂ s II H θ ∂ σ ij
−ad ξ p σ c H 0c [ Ad ξ p sII H θ Bd ξ p I 1Dd ξ p ]
ad ξ p −1
Ad ξ p ∂ sII H θ ∂ σ ij
Bd ξ p I d that is to say
∂
kl∂ f p
∂ij= ∂
∂ kl∂ s II H
∂ ij−ad p c H 0
c [Ad p s II H Bd p I 1Dd p ]ad p−1
⋅Ad p ∂∂ kl
∂ s II H ∂ij
−ad p a d p −1c H 0c Ad p sII H Bd p I 1Dd p a
d p−2
Ad p ∂ s II H ∂ ij
Bd p ij⊗Ad p ∂ s II H ∂kl
Bd p kl
However ∂ sII H ∂ σ ij
= H 0c−H 0
e
h0c−h0
e ∂h ∂ skl
s IIH skl
sII . ik . jl−13 ij . kl from where
∂
∂ kl∂ s II H
∂ij=H 0
c−H 0
e
h0c−h0
e ∂
∂ kl∂h ∂ smn
s II
∂ smn
∂ ij
H 0c−H 0
e
h0c−h0
e ∂h ∂ smn
∂ smn
∂ij
∂ s II
∂ spq
∂ spq
∂kl
∂H ∂ kl
smn
s II
∂ smn
∂ij
H
sII
∂ smn
∂kl
∂ smn
∂ij
−H smn
s II2
∂ s II
∂ spq
∂ spq
∂ kl
∂ smn
∂ij
One has moreover ∂h θ ∂ skl
=γ cos 3θ
6h θ 53skl
s II2−
γ 54
6h θ 5 s II3 ∂det s ∂ skl
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∂
∂ kl∂h ∂ smn
= ∂
∂ s pq ∂h ∂ smn
∂ spq
∂kl
∂
∂ spq∂h ∂ smn
= 54
2h5 sII5
smn⊗∂det sij
∂ spq
−3 54 det sij
2 h5 s II7
smn⊗s pq cos3
2 h5 s II2
I mnpq−
cos 3
h5 s II4
smn⊗s pq−5 cos3
2h6 s II2
smn⊗∂h ∂ spq
5 54
6 sII3 h6
∂det sij
∂ smn
⊗∂h ∂ spq
54
2h5 s II5
∂det sij
∂ smn
⊗spq−54
6 h5 sII3
∂2 det sij
∂ smn∂ s pq
with
∂2 det sij
∂ smn∂ spq
=[0 s33 s22 0 0 −2 s23
s33 0 s11 0 −2 s13 0
s22 s11 0 −2s12 0 0
0 0 −2 s12 −s33 s13 s23
0 −2 s13 0 s13 −s11 s12
−2 s23 0 0 s23 s12 −s22
]
It is pointed out that n ij=
' s ij
sII
−ij
' 23
and '=−
26sin 3−sin
∂nij
∂ kl
=[ ∂
'
∂ kl
sij
s II
'
sII
∂ sij
∂ kl
−' sij
s II2
∂ s II
∂ kl] '23 − '' sij
sII
− ij⊗ ∂'
∂ kl
'23 '2
3
∂'
∂ kl
=∂'
∂ smn
∂ smn
∂ kl
∂'
∂ I1
∂ I 1
∂ kl
∂'
∂ smn
=−6 6
3−sin 2
∂sin∂ smn
and ∂ '
∂ I 1
=−66
3−sin 2
∂ sin∂ I 1
Distinction of the expressions between the viscoplastic behavior pre-peak or and post-peak
Expression of the derivative in pre-peak or viscoplastic
∂ sin∂ smn
=∂sin∂max
∂max
∂ smn
∂sin∂ lim
∂ lim
∂ smn
= 0, v 10,v lim
0,vmaxlim 2
∂max
∂ smn
−10,v max
0,vmaxlim 2
∂ lim
∂ smn
and
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∂ sin∂ I 1
=∂sin∂max
∂max
∂ I 1
∂sin∂ lim
∂ lim
∂ I 1
=0, v
3 10,v lim
0,vmaxlim 2−
1mv,max 10,v max
0,vmax lim 2
∂max
∂ smn
=1
6 [ sII
H 0c−H 0
e
∂H ∂ smn
32
2 H −H 0cH 0
e 2 H 0
c−H 0e smn
s II ] and ∂max
∂ I1
=13
∂lim
∂ smn
=1mv,max
6 [ s II
H 0c−H 0
e
∂H ∂ smn
− 32−2 H −H 0cH 0
e 2 H0
c−H 0e smn
sII ] and ∂lim
∂ I 1
=1mv,max
3
Expression of the derivative in post-peak for the plastic law of dilatancy
∂ sin∂ smn
=∂ sin∂ ∂∂min
∂min
∂ smn
∂
∂max
∂max
∂ smn
and
∂ sin∂ I 1
=∂ sin∂ ∂∂min
∂min
∂ I 1
∂
∂max
∂max
∂ I 1
∂ sin∂ smn
=1
11 res
1res 2 − max
min 2
∂min
∂ smn
1
min
∂max
∂ smn
and
∂ sin∂ I 1
=1 res 11 min−max
3 1res 2min
2
∂min
∂ smn
=1
6 [ s II
H 0c−H0
e
∂H ∂ smn
− 32−2 H −H 0cH 0
e 2 H 0
c−H 0e smn
s II ] and ∂min
∂ I 1
=13
The evaluation of the term ∂Gij
vp
∂ kl
is identical to that of ∂Gij
p
∂ kl
with the distinctions close already specified
above.
One approaches now the evaluation of the term ∂ ⟨ f vp ⟩
+ ∂mn
with Φ f vp =Av f vp
Pa nv
.
One obtains then:
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∂ ⟨ f vp⟩+
∂mn
=∂ f vp
∂mn
⋅Av nv
Patm f vp
Patmnv−1
Evaluation of the terms relative to d R1ij
d Y 3 =⋅C ijkl
e :∂G kl
p
∂p :
∂Gijp
∂p=∂
p ∂ f p
∂ij− ∂p ∂ f p
∂ij: nmn⊗nij− ∂ f p
∂mn
:∂nmn
∂p ⊗nij− ∂ f p
∂mn
: nmn⋅∂nij
∂ p
with
∂
p ∂ f p
∂ij=∂ad
∂pc H 0
c Ad s II H Bd I 1Dd ad−1⋅Ad ∂ s II H
∂ ij
Bd ij −a dc H 0
c {∂ad
pln Ad sII H Bd I1Dd ...
...ad−1
Ad s II H Bd I 1Dd ∂ Ad
∂p
s II H ∂Bd
∂p
I 1∂Dd
∂p }⋅
Ad s II H Bd I 1Dd ad−1 [Ad ∂ s II H
∂ij
Bd ij]
−a dc H 0
c [Ad sII H Bd I1Dd ]a
d−1⋅∂ Ad
∂ p
∂ s II H ij
∂Bd
∂ p
and
∂nkl
p=
∂nkl
∂ '∂ '∂ p
=skl
s II
' 23 −2 ' 2 skl
s II
2 ' kl ' 23
3/ 2
∂ '
∂p
= s kl
s II
' 23−2 ' 2 s kl
s II
2 ' kl ' 23
3/ 2
−66sin
3−sin 2
∂ sin
∂p
• if the law of followed dilatancy corresponds to the pre-peak field, ∂ sin∂ p
=0
• if the law of followed dilatancy corresponds to the post-peak field, the following operations arenecessary:
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∂ sin
∂p =
∂ sin∂
∂
∂p
= 1
11 res
1 res2
∂
∂p
= 1
11 res
1 res2
min−max
min 2
∂
∂p
∂
∂p=
∂ ∂ c
∂ c
∂p
∂
∂ tan ∂ tan ∂
p
=1
tan ∂ c∂ p
−c
tan 2
∂ tan ∂p
with
∂ c
∂ p=c s
d a
d
[∂ad
∂ pln sd a d
sd
∂ sd
∂ p ]2 1ad md sd
ad−1
...
...
c sd
ad
∂a d
∂pmd sd
ad−1ad ∂md
∂psd
ad−1a d md ∂ad
∂pln sd
ad−1
sd
∂ sd
∂p sd a
d−1
4 1ad md sd a
d−1
3/ 2
∂ tan
∂p = 1tan2
∂∂
p
=
1tan2 [ ∂ad
∂pmd sd
ad−1a d ∂md
∂ p sd
ad−1ad md sd
ad−1∂a d
∂pln sd
a d−1sd
∂ sd
∂p ]2ad md sd
ad−1 1ad md sd ad−1
Calculation of the terms relative to d R1ij
d Y 4 =C ijkl
e :[ ∂ ⟨ f vp ⟩+
∂vp G kl
vp⟨ f vp ⟩
+⋅∂G kl
vp
∂vp ]⋅ t :
The evaluation of the term ∂G kl
vp
∂vp is identical in its form to preceding calculation for
∂G klp
∂ p .
∂ ⟨ f vp ⟩
+
∂vp =
Av nv
Patm f vp
Patm
nv−1∂ f vp
∂vp
The evaluation of the term ∂ f vp
∂vp identical in its form at the end is previously calculated
∂ f p
∂ p .
Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2021 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)
Code_Aster Versiondefault
Titre : Loi de comportement viscoplastique LETK Date : 25/09/2013 Page : 38/38Responsable : FERNANDES Roméo Clé : R7.01.24 Révision :
79c03cf352a0
Evaluation of the terms relative to d R3
d Y 1 ij and
d R4 d Y 1 ij
∂ G IIp
∂ij
=∂ G II
p
∂ G klp
∂ G klp
∂Gmnp
∂Gmnp
∂ij
=G kl
p
G IIp mk nl−
13mnkl ∂Gmn
p
∂ij
and
∂ G IIvp
∂ ij
=∂ G II
vp
∂ Gklvp
∂ Gklvp
∂Gmnvp
∂Gmnvp
∂ij
=G kl
vp
G IIvp mknl−
13mnkl ∂Gmn
vp
∂ij
Evaluation of the terms relative to d R3
d Y 3
∂ G IIp
∂p =
G klp
G IIp mk nl−
13mnkl ∂Gmn
p
∂p
Evaluation of the terms relative to d R3
d Y 4 and
d R4d Y 4
∂ G IIvp
∂vp=Gkl
vp
G IIvp mk nl−
13mnkl ∂Gmn
vp
∂vp
Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2021 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)
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