viscoplastic law of behavior letk

Code_Aster Versiondefault

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

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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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.


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




79c03cf352a0

{ 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



∂ f d

∂ σ ij


−−∂ 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


visc σ− , ξ v− Δt ]

∂ f d

∂ σ ij


−−∂ 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



∂ f d

∂ σ ij


− −∂ 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




79c03cf352a0

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




79c03cf352a0

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−




79c03cf352a0

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




79c03cf352a0

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 :




79c03cf352a0

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




79c03cf352a0

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


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)


d R4 d Y 1 ij

=− 23⋅ t⋅∂ ⟨ f vp ⟩

+

ij

⋅G IIvp⟨ f vp ⟩

+⋅∂ G II

vp

ij if vpmax

vp −vp t -




79c03cf352a0

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:




79c03cf352a0

∂

∂ 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 .




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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


d Y 3

∂ G IIp

∂p =

G klp

G IIp mk nl−

13mnkl ∂Gmn

p

∂p


d Y 4 and

d R4d Y 4

∂ G IIvp

∂vp=Gkl

vp

G IIvp mk nl−

13mnkl ∂Gmn

vp

∂vp


viscoplastic law of behavior letk

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