mono Behaviors elastoviscoplastic and polycr [] · elasticity, the criterion and the model of flow....

Code_Aster Version 11

Titre : Comportements élastoviscoplastiques mono et polycr[...] Date : 23/10/2014 Page : 1/58Responsable : Jean-Michel PROIX Clé : R5.03.11 Révision : 12699

Behaviors elastoviscoplastic mono and polycrystalline

Abstract:

The goal of this document is to describe the integration of the mono and polycrystalline behaviors.

One treats here integration of these constitutive laws associated with sliding systems corresponding to the crystalline families usual or specified by the user. This integration can be done explicitly (method of Runge_Kutta with control of the accuracy and local recutting of time step) or in an implicit way (method of Newton with local recutting of time step).

These behaviors can be employed for the computation of microstructures (mesh of an aggregate, with geometrical representation of each single-crystal physical grain) or for the computation of the type POLYCRISTAL, “homogenized” medium having in any material point (or not integration or of computation) several simultaneous phases, in variable proportions.

One describes also the possible types of computation with these behaviors as well as the procedure of generation of meshes.

Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and is provided as a convenience.

Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)



Contents1 Introduction4 .........................................................................................................................

2 Formulation of the mono behaviors and polycristallins4 .......................................................

2.1 Behavior models of the monocristal4 .............................................................................

2.1.1 Examples of relations of écoulement5 ..................................................................

phenomenologic 2.1.1.1 Model MONO_VISC15 ..................................................

phenomenologic 2.1.1.2 Model MONO_VISC25 ..................................................

2.1.1.3 Model resulting from the Dynamics of Dislocations: MONO_DD_KR5 .....

2.1.1.4 Models MONO_DD_CFC/MONO_DD_CFC_IRRA /MONO_DD_FAT . . . 7

2.1.1.5 Models MONO_DD_CC/MONO_DD_CC_IRRA9 ....................................

the laws of evolution of the densities of dislocations ( and ) are: ...............................

11.2.1.2 Examples of relations of hardening cinématique11 .........................................

2.1.3 Examples of relations of hardening isotrope11 .....................................................

2.1.3.1 MONO_DD_CFC/MONO_DD_FAT ........................................................ 12.2.2

Sliding systems and total behavior of the monocristal12 .....................................................

2.2.1 Small strains, configuration initiale12 ...................................................................

2.2.2 Small strains, rotation of the network cristallin13 ..................................................

2.2.3 Large deformations .............................................................................................. 14.2.3

Behavior of the polycrystal homogénéisé15 ........................................................................

2.3.1 Behavior of type POLYCRISTAL15 ......................................................................

2.3.1.1 Relation of change of échelle15 ................................................................

3 Integration local and implemented numérique16 ..................................................................

3.1 System of equations at résoudre16 ................................................................................

3.1.1 Behavior of type MONOCRISTAL16 .....................................................................

3.1.2 Behavior of type POLYCRISTAL17 ......................................................................

3.2 implicit Resolution of behavior MONOCRISTAL ...........................................................

18.3.2.1 implicit Resolution – System of equations réduit19 .........................................

3.2.2 implicit Resolution – Integration of model MONO_DD_KR21 ...............................

3.2.3 implicit Resolution – models MONO_DD_CFC/MONO_DD_FAT .......................

23.3.2.4 implicit Resolution – models MONO_DD_CC /MONO_DD_CC_IRRA27 ........

3.2.5 Algorithm of implicit integration into large déformations29 ...................................

3.2.6 Convergence criteria used for the resolutions implicites31 ...................................

3.3 explicite31 Resolution ....................................................................................................

4 Variables internes32 .............................................................................................................

4.1 Cases of the monocristal33 ............................................................................................

4.2 Cases of the polycristal33 ..............................................................................................

5 numerical Establishment in Code_Aster34 ...........................................................................

6 Utilisation35 ..........................................................................................................................

6.1 Cases of the monocristal35 ............................................................................................

6.2 Cases of the polycristal37 .............................................................................................. Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and is provided as a convenience.




6.3 Exemple37 .....................................................................................................................

7 Bibliographie40 ....................................................................................................................

8 History of the versions of the document41 ...........................................................................

Appendix 1 Statement of the Jacobian of the equations élasto-visco-plastics intégrées42 . . .

Appendix 2 Evaluating of the coherent tangent operator .................................................... 45

Appendix 3 Evaluating of the coherent tangent operator – Case of the system réduit46 .......

Appendix 4 tangent Matrixes in large deformations: ............................................................. 47

Appendix 5 general Writing of the matrixes jacobiennes50 ...................................................

Appendix 6 Examples of matrixes of interaction52 ................................................................

Appendix 7 Procedure of creation of maillage54 ....................................................................





1 Introduction

the general purpose of the development of the “microphone-macro” features into Code_Aster is to be able to integrate in a modular way of the models into several scales (with a possibility of choice of the constitutive laws, rules of localization, types of microstructures). What can lead to different types of computation (polycrystalline computations, use of a model of the Berveiller-Zaoui type or a standard model “regulates in ”, computations of aggregates multi-crystalline lenses with mesh of a microstructure,…).

The approach presented here consists in allowing decoupling, by modularity, of the various elements which constitute a constitutive law. This flexibility is accessible directly to the user. Moreover, for the developer, it is possible to easily add a crystalline constitutive law by simply defining derivatives partial of the problem, in terms of local variables. This is sufficient if one is satisfied with an explicit integration; for an implicit integration, it is necessary to define in more the tangent operator.

More precisely, for the aspect behavior MONOCRISTAL, in each point of integration of a given finite element, the behavior is that of a monocrystal having a given directional sense, and a certain number of sliding systems. Each family of sliding systems has her own local constitutive law.

In the case of a polycrystalline model, one supposes that in a material point (not integration of a finite element), several metallurgical phases are present simultaneously, each phase being able to be made up of grains with directional senses given, each having grain a certain number of sliding systems (not inevitably the same ones for each phase). The representation of the material is founded on a self-coherent approach simplified 41 or possibly extended for nonradial or cyclic loadings 41. Each family of sliding systems has her own local constitutive law. One finds thus separately structure crystallographic, the model of crystalline viscoplasticity and the rules of transition from scales. This mode of separation is also extended to the model of viscoplasticity itself, with a separation between elasticity, the criterion and the model of flow.

2 Formulation of the behaviors mono and polycrystalline

2.1 Behavior models of the monocrystal

the behaviors relating to the system of sliding of a monocrystal are (in all the behaviors considered) of élasto-visco-plastic type. It can be built on phenomenologic bases or physical bases (dynamics of dislocations). For each direction of sliding, the behavior is mono dimensional. It can break up into three types of equations:

• Relation of flow: s represent the plastic sliding of the system s

s=g s ,s ,s , ps , with ps=∣s∣ and:

�elastoplastic case, a criterion of the type: F s ,s ,s , ps≤0 and F⋅ps=0�élasto-viscoplastic case, ps= f s , s , s , ps

• s in the case of represents kinematic hardening a phenomenologic model and the density of dislocations for a behavior with physical base.

Its evolution is the form: s=hs , s , s , ps

• for the phenomenologic models, isotropic hardening is defined by a function: R ps .

These relations become, after discretization in time:





•Relation of flow:

s=g s , s , s , ps , with, ps=∣s∣for an elastoplastic behavior a criterion of the type: F s ,s ,s , ps≤0 and F⋅ ps=0for a élasto-viscoplastic behavior, ps= f s , s , s , ps t

•Evolution of s : s=hs , s , s , ps

•Evolution of isotropic hardening: R ps

The quantities s ,s ,s , ps are evaluated at time running for an implicit discretization and at previous time for an explicit discretization.

To fix the ideas, here examples of relations of viscoplastic or elastoplastic flow, and hardening. The names of these relations correspond to their name in command DEFI_MATERIAU [U4.43.01].

2.1.1 Examples of phenomenologic flow relations

2.1.1.1 Model MONO_VISC1

s=g s ,s ,s , ps= ps

s−c s

∣s−c s∣

ps= t ⟨∣s−c s∣−Rs ps

k ⟩n

the parameters are: c , k , n .

2.1.1.2 Phenomenologic model MONO_VISC2

s=g s ,s ,s , ps= ps

s−c s−as

∣s−c s−as∣

ps= t ⟨∣s−c s−a s∣−Rs psd2cc s

2

k ⟩n

the parameters are: c , k , n , a , d .

2.1.1.3 Model resulting from the Dynamics of Dislocations: MONO_DD_KR

The model MONO_DD_KR (Kocks-Rauch) is applicable to steels of the type CC (Cubic crystalline structure Centered). Its principal local variable is the density of dislocations on each system of sliding. It in addition offers the advantage of being valid for a broad range of temperatures [bib7]. Flow is formulated in the following way:

So ∣s∣0

effs=⟨∣s∣−0−

s ⟩ ⟨ ⟩ positive part

s=

2∑uasu

u

∣s∣−0

with u=

u b2





if not effs=0 . In these statements, the shear modulus of the material indicates, b is the constant

of Burgers, u is the density of dislocations, asu is the matrix of interaction between the 24 sliding

systems (its particular form, dependant on 5 coefficients, is provided in appendix). The diagonal terms represent the auto--hardening of a system of sliding, the other terms represent latent hardening.

If ∣s∣0 and effs0

s=g s ,s=0exp −G effs

k B T ⋅s

∣s∣⋅ t

with Geffs=G01− ⟨eff

s ⟩R

p

q

if not

s=0

s=0

G effs is an activation of enthalpy corresponding to a free activation in an athermic field.

0,R , p , q are data materials, k B the Boltzmann constant and T the temperature.

The evolution of the density of dislocation is proportional to the square root of the sum of the densities of dislocations of all the others sliding systems:

s=∣s∣b 1

s−gc T s with 1

s=

1d∑

u≠s

u

K T

and gc T =g c0 exp [− E gc

k BT ] where E gc , k B , b , d , gc0 , K T are materials parameters,

from where

s=∣s∣ 1

bd∑

u≠s

u

b2K−

g cs

b3 and consequently:

s=∣s∣ bd

∑u≠ s

u

K−

gcs

b =∣g s ,s∣⋅hs

The parameters are: 0, k B ,T ,R ,G0, p ,q ,0, , b , d , K , gc

k B is the Boltzmann constant.

Note: The resolution of Newton can bring to negative values for u , it is necessary thus that the

terms under the root are protected. With this intention, we choose the following modification:

∑u≠ s

⟨u⟩ where ⟨ . ⟩ to it party represents positive: ⟨ x ⟩={x si x0

0 sinon





Also let us notice that the statement of s , which is function of G effs , led to a discontinuity in

the vicinity of effs=0 : for eff

s0 s=0 , and lim

effs0 *

s=0exp −G0

k BT ⋅s

∣s∣⋅ t

2.1.1.4 Models MONO_DD_CFC/MONO_DD_CFC_IRRA /MONO_DD_FAT

the formulation of models DD_CFC [8], [9] and DD_FAT [18] is built from computations of dynamics of dislocations. It applies to the materials with Cubic crystalline structure with Sides Centered ( CFC ) such as austenitic steels. A priori the model DD_CFC is not compatible with a change of way of loading, in particular the parameters are not adapted to a cyclic request (when a dislocation “reconsiders its steps” its kinetics is different because of a different interaction with the obstacles).

The equations below are written for each system of sliding s .

The direction of flow is given by:

s= ps

s

∣s∣= pssignes and intensity of flow by:

ps=0 ∣s∣ fs

forest n

−1 si ∣s∣≥ fsforest , sinon ps=0

with n large (representing a quasi elastoplastic model: for CFC, the dependence in time is indeed negligible).

Compared to the initial model [8], the threshold was introduced as well as the term – 0 to ensure continuity when the threshold is crossed.

In practice, the parameter 0 is small and the term ∣s∣ fs

forest n

becomes very quickly largely

higher than 1 , because of great values of the exhibitor n .

sforest is a function of hardening whose coefficients are provided by the key word MONO_DD_*

(*=CFC, FAT or DC) , and who is described with the § according to. In DD_CFC, the evolution of the density of dislocation is given by:

s=ps

b A∑

j∈ forest s asj

eff j

∑j=1,12

asjeff j

B ∑j∈coplas

asjeff j− y s

� j∈ forest s are the systems of distinct norms n j≠ns

� j∈copla s are the identical systems of norms n j=ns

the initial value of the density of dislocation is: st=0=s0

(field provided by the user).





Like a ijeff=C

2.aij

, sf o re s t

=b C ∑j=1 , 1 2

a s j j and

s=p s

b A ∑j∈f o r e s t s

a s j j

∑j=1 , 1 2

a s j j

B C ∑j∈cop la s

a s j j−y s

to avoid any problem of conditioning of the jacobian matrix, one chooses adimensionner the system

by replacing the density of dislocation by : s=b2∗s and to solve in s . That gives:

s= ps hs avec hs= A∑

j∈ forest s a sj j

∑j=1,12

asj j

B C ∑j∈coplas

asj j−ybs

Note: physically, the density of dislocation must remain positive. Numerically, so that all the terms

are licit, it is necessary to test the positivity of and is redécouper time step if this value becomes negative (by “hesitation” of the algorithm during iterations) that is to say to introduce a positive part into the equations of evolution. This last solution is selected for explicit integration as for implicit integration.

All the differential equations governing the viscoplastic flow of model DD_CFC are thus written:

s= ps

s

∣s∣ (12 equations)

ps=0 ∣s∣ fs

forest n

−1 si ∣s∣≥ fsforest , sinon ps=0 (12 equations)

s= ps hs ⟨ ⟩ (12 equations) with: ⟨i ⟩={i si i00 sinon

C =0.20.8

ln ∑i=1,12⟨i ⟩

ln b ref

sforest=C ∑j=1,12

a sj ⟨ j ⟩

hs= A∑

j∈ forest s a sj ⟨ j ⟩

∑j=1,12

asj ⟨ j ⟩BC ∑

j∈coplas asj ⟨ j ⟩−

yb⟨s ⟩

with ωs=b2∗ρs

In case MONO_DD_CFC_IRRA, the evolution of the variables related to the irradiation is given by:

ρsloops=−ξ( ∑j∈copla (s)

∣γj∣)(ρsloops−ρsat) ϕs

voids=−ζ( ∑j∈copla (s)

∣γj∣) (ϕsvoids−ϕsat )

the statement dedevient sforest : τs

forest=μ√ ∑j=1,12

a sjeffωj+α

loopsϕloopsρsloops b2+αvoidsϕs

voidsρvoids b2

what, after multiplication by b2 : ωs

loops=b2∗ρsloops

:





ωsloops=−ξ( ∑j∈copla ( s)

∣γj∣) (ωsloops−ωsat ) and τs

forest=μ√ ∑j=1,12

a sjeffωj+α

loopsϕloopsωsloops+αvoidsϕs

voidsωvoids

one can notice on the way that the equations of evolution ofωsloop andϕs

voids are integrated analytically:

ωsloops(t+Δt )=ωsat

+(ωsloops(t )−ωsat )exp (−ξ ∑j∈copla ( s)

∣Δγj∣)

ϕsvoids(t+Δt )=ϕsat

+ (ϕsvoids(t )−ϕsat )exp (−ζ ∑j∈copla (s)

∣Δγj∣)

In case DD_FAT, the evolution of the density of dislocation is given by:

s=ps

b 1d∑u≠s

u

K−gc0s

While adimensionnant like above, i.e. while posing:

s=b2×s

one obtains:

s= ps hs with hs= bd∑u≠s

u

K−gc0

s

b From where by introducing the positive parts:

s= ps

s

∣s∣ (12 equations)

ps=0 ∣s∣ fs

forest n

−1 si ∣s∣≥ fsforest , sinon ps=0 (12 equations)

s= ps hs ⟨ ⟩ (12 equations) with: ⟨i ⟩={i si i00 sinon

C =1.

sf o r e s t

= ∑j=1 , 1 2

a s j ⟨ j ⟩

h s = bd ∑u≠s

⟨u ⟩

K−g c 0

⟨s ⟩b

with s=b2∗s

2.1.1.5 Models MONO_DD_CC/MONO_DD_CC_IRRA

the formulation of models DD_CC (with or without irradiation) is built from computations of dynamics of dislocations 41. It applies to the materials with Centered Cubic crystalline structure ( CC )

Two alternatives in the formulation of the behavior exist (case 1 and case 2).Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and is provided as a convenience.




The equations below are written for each system of sliding s .

Viscoplastic flow is obtained by harmonic mean of two modes (low temperature, and high temperature).

One chooses here to reveal the unknowns (representing all the densities of dislocation set irr

s )

and effs (who is an additional unknown a priori. It will be seen that the algorithm of resolution

proposed allows all to express according to ).

irrs indicate the density of dislocations induced by the irradiation (case of MONO_DD_CC_IRRA), and

is not used for behavior MONO_DD_CC in order to limit the number of local variables.

s being the cission solved on the system s :

� the plastic sliding is obtained by: 1

s=

1

nucs

1

probs (eq. CC_1)

� nucS=m

s b H⋅l s ,eff

s⋅exp −G eff

s

k B T sgn s (eq. CC_2)

� Geffs=G 01− ⟨eff

s ⟩0

0.5

(with : effs0 and 0GG0 (eq. CC_3)

� effs

s ,=∣s∣−F−LRs−LT

s ,eff

s case 1 (eq. CC_5)

� effs

s ,=∣s∣−c c=F LRs

2−LT

s ,eff

s

2 case 2

� LRs = bAT

s tots or (case 2) LR

s=b a AT

s

s (eq. CC_6)

� tots=∑

j

jirr

s(eq. CC_7) or (case 2) tot

s =∑

j≠s

jirr

s

� ATs=∑j

a ATsj

j

tots a irr

irrs

tots

(eq. CC_8)

LTs ,eff

s =max0 ;1−ATs b 1

s ,effs −

1

2ATs R s ,eff

s l c T (eq. CC_9)

or (case 2) LTs ,eff

s =max0 ;ATs b 1

s ,eff

s−

1

2ATsR s

,effsl c T

�1

s ,eff

sD

=min tots ;D2 R s

,effstot

s (eq. CC_10)





�R s ,eff

s=

b

20 1−Geffs

G 0

2(eq. CC_11)

� l s ,eff

s=max s

,effs−2AT

sR s

,effs ; l c (eq. CC_12)

� prob

s=0

∣s∣

FLRs 1−AT

s b 1

s ,eff

s

n

sgn s

(eq.CC_13)

or (case 2) probs =0 ∣

s∣c

n

sgns

the parameters are: ms , b , H ,G0,0,F , , , airr ,l c T , D , 0, n and stamps it interaction AT

sr

the laws of evolution of the densities of dislocations ( s and irrs ) are:

s=∣ s∣b 1

d lath

aself

s

K self

∑j≠ s

a sj

j

K f

− ys

s and ˙irrs=−irr

s ∣ s∣ (eq.CC_14)

1ys=

1y AT

s 2eff

s

b and asj

=a ATsj 1−eff

s

0 ,

with the paramètes d lath aself K self K f y ATs

Note: : physically, the density of dislocation must remain positive. Numerically, so that all the terms

are licit, it is necessary to test the positivity of and is redécouper time step if this value becomes negative (by “hesitation” of the algorithm during iterations) that is to say to introduce a positive part into the equations of evolution. This last solution is selected for explicit integration as for implicit integration.

2.1.2 Examples of relations of kinematic hardening

MONO_CINE1

s=h s ,s ,s , ps=s−d s ps

the parameter is: d .

MONO_CINE2

s=h s ,s ,s , ps=s−d s ps−∣c s∣M

m s

∣ s∣

parameters being then: d , M ,m ,c .

2.1.3 Examples of relations of isotropic hardening

isotropic hardening R s pr appearing in certain flow relations (MONO_VISC1, MONO_VISC2) can be form:





MONO_ISOT1

R s p=R0Q ∑r=1

ns

hsr 1−ebpr ; the parameters are: R0, Q ,b ,hsr .

MONO_ISOT2

R s p=R0Q1∑r

hsr 1−e−b1 p rQ2 1−e−b2 ps ; the parameters are: R0, Q1, b1, b2, hsr ,Q2 .

hsr a symmetric square matrix of interaction between sliding systems (see appendix 4).

If only one a coefficient H is provided, the matrix hsr takes a very simple form:

hrs=1 si r=s and hrs=H si r≠s , with by default H=0 (matrix identity).

Note: the combination of models MONO_VISC1, MONO_ISOT1, MONO_CIN1 corresponds to the known crystalline model under the name of Meric-Cailletaud 41 .

2.1.3.1 MONO_DD_CFC/MONO_DD_FAT

the model of hardening of models DD_CFC and DD_CC is: sforest= b ∑j=1,ns

asjeff j

and a ijeff=C 2.a ij avec C =0.20.8

ln b ∑i=1,12

iln b ref

where a ij represents the matrix of interaction enters sliding systems, whose particular form is provided in appendix 4. This one differs according to model MONO_DD_CFC or MONO_DD_CC.

Model DD_FAT is: sf o re s t

=b ∑j=1 , 1 2

a s j j , which corresponds to C =1.

2.2 Sliding systems and total behavior of the monocrystal

a monocrystal is composed of one or more sliding system families, (cubic, octahedral, basal, prismatic,…), each family including a certain number of systems (12 for the octahedral family for example).

A each family of system of sliding are associated a flow model, a kind of isotropic kinematic hardening and, and values of the parameters for these models. In other words, one does not envisage to vary the behavior models or the coefficients within the same family of sliding systems. On the other hand, from one family to another, the constitutive laws can change, as well as the values of the parameters.

A system of sliding is determined by a tensor of directional sense (or tensor of Schmid) built starting from the crystallographic definitions of:

•direction of sliding of the system s (defined by the unit vector m s )

•and of the norm to the slip surface (defined by the unit vector ns ).

2.2.1 Small strains, initial configuration

In small strains, the tensor of directional sense is defined by:





ijs=

12mi⊗n jn j⊗mi

From the point of view of the behavior at the material point, this tensor intervenes for the computation of the solved cission.

s= :s

and that the total viscoplastic strainrate E vp , definite from the knowledge velocities of sliding s for

all sliding systems:

E ijvp=∑

s∈g

sij

s

Moreover, the monocrystal can be directed compared to the total axes of definition of the coordinates. This directional sense is defined for each mesh or mesh group (typically for each grain) by the data of 3 nautical angles or 3 Eulerian angles. The components of the tensor of directional sense

s , defined in the reference related to the monocrystal, are then expressed in the total reference by means of these nautical angles. In small strains, they are fixed in the initial configuration.

Moreover, on the level of a Gauss point, one applies a relation of elasticity on the total tensors:

•total deflection macroscopic E

•forced macroscopic viscoplastic E vp

•strain macroscopic:

=E−E th−E vp

where the operator of elasticity represents (isotropic or orthotropic)

2.2.2 Small strains, rotation of the crystal lattice

a good approximation of the effect of the strain on the behavior of the monocrystal consists in taking into account the rotation of the crystal lattice. This rotation is interesting for two reasons:

•its effective taking into account in computations is coherent with strains not infinitely small (in current computations, it is not rare to reach 10%). It is a preliminary with a complete model of large deformations on the monocrystal.

•It is information interesting for postprocessing, because it can be confronted with experimental results.

One describes here (according to 41 41, 41) the method of taking into account of the rotation of crystal lattice in the general frame of the small strains. The principal physical assumptions are:

•knowing the tensor vorticity or rate of rotation, which is the skew-symmetric part of the tensor

L=F F-1 ), one applies an additive decomposition in a part known as plastic rotation p (left skew-

symmetric plastic rotation) and an “elastic” part e , which will make it possible to calculate the

rotation of the crystal lattice;•one admits for the moment that one can calculate the tensor rate of rotation, while carrying out the integration of the behavior with a tensor of the linearized strains. This limitation falls naturally with the model complete in large deformations, but it is supposed here that one can reason in a decoupled way. One thus has, in incremental form:

=12F F-1

−F F-1T and

e= −

p which is skew-symmetric tensors.

The computation of p is carried out at the end of the integration: it is stored and used with the

following iteration (a purely implicit computation would imply to solve the system of equations





described in paragraph 16 by adding all the equations on the directional sense of the network, which would weigh down much computation). Once the increment of rotation

e known, one extracts the “axial” vector [see for example

R5.03.40]: e such as

e∧U=

e⋅U and one calculates the swing angle and the vector

which corresponds to the rotational axis:

=∥ e∥ and n= e

.

The matrix of increment of rotation is then obtained by the formula of Eulerian-Rodrigues:

Q=Idsin n1−cos n⋅ n where n is the matrix skew-symmetric resulting

from n

What makes it possible to update the matrix of rotation of the network: Q+=Q⋅Q -

It makes it possible to bring up to date the tensor of directional sense by: ns=Q+ n0

s

ms=Q+ m0

s .

The behavior is then integrated with this definition of the crystal lattice.

It then remains to bring up to date the increment of plastic rotation:

p=

12∑s

sms⊗n s

−ns⊗ms

The additional local variables are: Q , p , e ,

In practice the rotation of network is taken into account via key word ROTA_RESEAU of the command DEFI_COMPOR [U4.43.06]. It is effective only for the implicit integration of behavior MONOCRISTAL.

2.2.3 Large deformations expected Properties: the selected formalism must be objective, and must preserve the plastic incompressibility ( det F p

=1 ). One uses multiplicative decomposition F=F eF p with the isoclinal slackened intermediate configuration of Mandel.

The model of (hyper-) selected elasticity uses the second tensor of Piola-Kirchhoff

S= J eF e−1 F e−T and the strain of Green-Lagrange associated with the elastic deformation

gradient:

EGLe=

12FeT Fe – I d S= : EGL

e .

On each system of sliding s , the solved cission is written with the tensor of Mandel M defined by

M=J eFeT F e-T in the following way: s=M :ms⊗n s (cf 41, 41).

Indeed, the power of the internal forces can be written: 1 :D=

1 :L=

1 : F F -1

and F F -1= F eF pF e F p F p-1F e -1=LeF eL pF e -1

thus 1 :D=

1[ : Le

: F eL pF e

-1 ]= 10

S : ˙EGLe

10

M :Lp





the plastically dissipated power is written: M :L p and also: ∑s

ss , therefore

M :L p=M :∑s

ss=∑s

sM :s , from where s=M :s

For each crystalline constitutive law, the equations of flow and evolution of the local variables are written classically on each system of sliding:

s=g s ,s ,s , ps or s=hs , s , s , ps

On the other hand the relation connecting the plastic strain to the slidings of the active systems is

written now: Fp=∑s sms⊗nsF

p

2.3 Behavior polycrystal homogenized

In the case of homogenized polycrystal, it is necessary to define each single-crystal phase by its directional sense, its proportion (voluminal fraction) and the associated behavior. A total directional sense of the polycrystal can be defined. It is necessary moreover define one rule of localization.The single-crystal behavior is built like previously starting from the behavior elasto - viscoplastic precedent and of the data of sliding system families.

2.3.1 Behavior of type POLYCRISTAL

Besides the single-crystal behavior describes previously, one adds a scale of modelization, which represents the assembly of the phases.

On the level of a Gauss point, one applies a relation of elasticity on the total tensors:

•total deflection macroscopic E

•forced macroscopic viscoplastic EVP

•strain macroscopic:

=E−E th−EVP

represent the operator of elasticity (isotropic or orthotropic)

•Moreover, knowing all the local variables relative to sliding systems of each phase, the parameters of behavior of each phase, the directional senses and voluminal fractions of each phase, and the type of method of localization,

•for each phase single-crystal (or “grain”), defined by a directional sense and a proportion f g , a

relation of localization of the stresses, general form (to be expressed in the local coordinate system of each phase)

g=L ,Evp , gvp ,g

and for each system of sliding of each phase, of the behavior models relative to each system of sliding, similar to the case of the monocrystal:

•Relation of flow:

s=g s ,s ,s , ps , with, ps=∣s∣ and ps= f s ,s ,s , ps

•Evolution of the kinematic hardening or the density of dislocations: s=h s ,s ,s , ps

•Evolution of the isotropic hardening defined by a function: R ps

•Viscoplastic strains of the phase: ijv pg=∑

s∈g

sij

s





There remain the equations of homogenization then: Evp=∑

g

f g gvp

2.3.1.1 Relation of scaling

Two relations of localization of the type g=L ,Evp ,gvp ,g are available in the current version:

•The relation of Berveiller-Zaoui [bib5] established on the notion of autocoherence. This relation is validated under certain conditions, namely: isotropy of the material, homogeneous elastic behavior and monotonic loading:

ijg= ij E ijvp−ijv p g

1=1

32∥E ij

vp∥J 2 ij

•The second relation, inspired of the preceding one, and developed more particularly for cyclic loadings [bib4] makes it possible to give a good description to schematize the interactions between the grains:

• ijg= ij Bij− ij

g Bij=∑g

f g ijg

• ijg=ij

v pg−D ijg−ijv pg ∥ ijv pg∥

where D and are parameters characteristic of the material and temperature.

3 Local integration and implementation numerical

3.1 System of equations to solve

3.1.1 Behavior type MONOCRISTAL

the local behavior of the monocrystal is defined, at one time given of the discretization in time and to the level of a point of integration of a finite element, by the data:

•stress tensor macroscopic at previous time t i−1 =− ,

•local variables at previous time, for each system of sliding: s ti−1 ,s t i−1 , ps ti−1 ,

•and of the tensor of increase in total deflection provided by iteration N of the total algorithm of

resolution E=E in

(with the notations of [R5.03.01]).

Integration consists in finding:

•the macroscopic stress tensor = t i

•and the local variables s=s ti s=s t i , ps= ps t i

checking the equations of behavior in each system of sliding (which are dimensional mono relations), and the relations of transition between the tensors macroscopic and all the directions of sliding. Notation: one writes the equations in the form discretized of way:

•clarify, if the noted quantities A+/ - are evaluated at time t i−1 : A+/ -=A−=A t i−1

•implicit, if they are evaluated at time t i : A+/ -=A=A t i

The equations to be integrated can be put in the following general form:Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and is provided as a convenience.




Being given, in a Gauss point, the tensors:

E : variation of strain at time t i

E t i−1 =E− : strain at time t i−1

t i−1 =−

: macroscopic stress at time t i−1

s ti−1 ,s t i−1 , ps ti−1 : local variables for each system of sliding with t i−1 ,

It is necessary to find:

= t i : macroscopic stress at time t i , in the reference corresponding to the total directional

sense

s=s ti

s=s t i

ps= ps t i

checking:

D−1=D−−1 −E−E th−E vp , where can depend on the temperature, and

can correspond to an orthotropic elasticity.

Evp=∑s

ss

for each system of sliding (of all the families of systems):ns flow relations: either in viscoplasticity s=g s

+/ - , s+/ - , s

+/ - , ps+/ -

with ps= f s+/ - , s

+/ - ,s+/ - , ps

+/ -

or in plasticity F s+/ - , s

+/ - ,s+/ - , ps

+/ - ≤0 F ps=0 ,

ps=∣s∣

ns equations of evolutions of kinematic hardening: s=h s+/ - , s

+/ - , s+/ - , ps

+/ -

ns equations of evolution of isotropic hardening: R s ps+/ -

This is solved either explicitly (Runge_Kutta), or implicit (Newton).

3.1.2 Behavior of type POLYCRISTAL

the discretized behavior models are:

Being given (in a Gauss point) total tensors:

•increase in total deflection E ,

•total deflection at previous time, E t i−1 =E−

•forced at previous time: t i−1 =− ,

•all the local variables s− , s

− , ps− relative to sliding systems of each phase,

•parameters of behavior of each phase, directional senses and voluminal fractions of each phase, and the type of method of localization.

It is necessary to find = t i , s= s t i , s=s t i , ps=ps t i checking:

•on the level of the Gauss point: =−−1 −E−E th− Evp , in the total

reference,





•for each phase (or “grain”), defined by a directional sense and a proportion f g , a relation of localization of the stresses, general form (to be expressed in the local coordinate system of each phase)

g=L ,E vp , gvp ,g

•and for each system of sliding of each phase:

• g vp=∑s

mss

• ns equations: s= ij : ij

s

• ns flow relations: s=g s , s , s , ps with ps=∣s∣

• ns evolutions of hardening: s=hs , s ,s , ps

• F s , s ,s , ps≤0, F⋅ ps=0 , (in plasticity independent of time)

There remain the equations of homogenization then: Evp=∑g

f g gvp

The viscoplastic behaviors relative to each system of sliding are identical to the case of the microstructure.

In the current version of Code_Aster, these behavior models are integrated only explicitly.

3.2 Implicit resolution of behavior MONOCRISTAL

It is thus necessary to solve a system of the following general form (cf.3.1.116):

RY ={s ,E vp , s

+ ,s+ , ps

+

e ,Evp , s+ , s

+ , ps+

ns {a ,E vp ,s

+ , s+ , ps

+

g ,Evp ,s+ , s

+ , ps+

p ,E vp , s+ ,s

+ , ps+}={

-1−-

- 1

-−E−E th

−E vp

Evp−∑s

s s

ns{s−hs

+ , s+ ,s

+ , ps+

s−g s+ , s

+ ,s+ , ps

+

Δps− f s+ ,s

+ , s+ , ps

+ }

=0

In more contracted way, one poses:

R y =0=[s y e y a y g y p y

] with y=

Evp

s

s

ps

solving this system of 663ns nonlinear equations (in 3D), one uses a method of Newton: one builds a vector series in the following way solution:

Y k1=Y k−dRdY k

−1 RY k





It is thus necessary to define the initial values Y 0 , and to calculate the jacobian matrix of the

system: dRdY k

(this one is detailed in appendix for the viscoplastic behaviors described previously). It

has the following form:

J=[∂ s∂ΔΣ

∂ s

∂ ΔEvp

∂ s∂ Δα s

∂ s∂ Δγs

∂ s∂ Δp s

∂e∂ΔΣ

∂ e

∂ ΔEvp

∂ e∂ Δα s

∂ e∂ Δγs

∂ e∂ Δp s

∂ a∂ΔΣ

∂a∂ ΔEvp

∂ a∂ Δα s

∂a∂ Δγs

∂ a∂ Δp s

∂ g∂ΔΣ

∂ g

∂ ΔEvp

∂ g∂ Δα s

∂ g∂ Δγs

∂ g∂ Δp s

∂ p∂ΔΣ

∂ p

∂ ΔEvp

∂ p∂ Δα s

∂ p∂ Δγs

∂ p∂ Δp s

]

3.2.1 Implicit resolution – System of equations reduced

In the form of the 6 components of the tensor of the stresses, Evp can be expressed according to

∑s

ss thus an equation can be eliminated from the total system to solve.

Like p=∣∣ , the equation can p about it be eliminated.

In the case of MONO_CINE1, expresses itself directly according to . The size of the system

to be solved is thus reduced to 6ns equations in , with ns the number of sliding systems.

In the case of MONO_CINE2, one calculates numerically according to , by a method of secant.

The equation referring to the components stresses is reduced to:

R1i i ,s ,s , s ,Rs pr =−1 i−−−1 i

−−th∑

s

s g s s , s , s , Rs pr =0

The equation referring to the hardening parameter is written in the form:

R2s i ,s ,s , s , Rs pr =s−g s s , s , s , Rs pr =0

with i=1à 6 , s=1 with ns and r=1 with ns

In the case of MONO_VISC1 :

g s= t ⟨∣s−c s∣−Rs pr

k⟩

ns−cα s

∣τ s−c s∣ with s= :s






g s= t ⟨∣τs−c s−as∣−Rs pr

d2c

c s2

k⟩

n

s−c s−a s

∣s−c s−a s∣

The jacobian matrix of the reduced system is written in the form:

J=[∂R1

i

∂ j

∂ R1i

∂s

∂R2s

∂i

∂R2s

∂Δγ r] of dimension 6ns×6ns

with ∂R1

i

∂ j

=−1 ij∑s

si∂ g s

∂ j

=−1 ij∑s

si∂ g s

∂ τ s

∂ τ s

∂ j

=−1 ij∑s

si s j

∂ g s

∂ τ s

∂R2

s

∂i

=∂−g s τ s , Δα s , Δγ s , Rs pr

∂i

=∂ −g s ∂ τ s

.∂ τ s

∂i

=−si .∂ g s

∂ τ s

∂R1i

∂s

=∑s

si .∂ g s

∂s

, ∂R2

s

∂rs

=sr−∂ g s

∂r

and ∂g s

∂r

=∂ g s

∂ s

.∂ s

∂s

sr∂ g s

∂ Rs

.∂ Rs

∂r


∂ g s

∂s

=n tkn

⟨∣s−c s∣−Rs pr ⟩n−1

.s−c s

∣τs−c s∣.∂∣s−c s∣

∂s

=H s s , s ,s ,Rs pr . sgn s . sgn s

that is to say: H s s , s ,s , Rs pr =n tk n

⟨∣s−c s∣−R s pr ⟩n−1

and sgn s=s−c s

∣s−c s∣


∂ g s

∂ τ s

=nk n⟨∣s−c s−as∣−R s p d

2c c s2⟩

n−1 s−c s−as

∣s−c s−as∣

that is to say:

H s τ s , s ,s , R s pr =n

k n⟨∣s−c s−as∣−Rs pr

d2c

c s 2⟩

n−1

and sgn s=s−c s

∣s−c s∣

In the case of MONO_VISC1:





∂ g s

∂ s

=−c . H s s ,s , R s

∂ g s

∂R s

=−1. H s s ,s , R s sgn s with sgn s=s−c s

∣s−c s∣

In the case of MONO_VISC2:

∂ g s

∂ s

=H s s ,s , R s −c sgn sd s s

∂ g s

∂Rs

=−1. H s s ,s , Rs sgn s with sgn s=s−c s

∣s−c s∣

In the case of MONO_CINE1:

∂ s

∂r

=sr

1−d s−sgn s

1d∣s∣2

In the case of MONO_CINE2:

∂ s

∂r

; This derivative requires the solution of a nonlinear equation by the method of Newton

In the case of MONO_ISOT1:

∂ Rs

∂r

=

∂[Q∑k hsk 1−e−bp k ]

∂r

=∂

∂ pr [Q∑k hsk 1−e−bpk ]

∂ pr

∂r

=bQhsr e−bpr . sgn r

with sgn r=∂ pr

∂r

=∂∣r∣

∂r

=r

∣r∣

In the case of MONO_ISOT2:

∂ Rs

∂r

=

∂ [Q1∑k

hsk 1−e−b1 pk Q2 1−e

−b 2 p r ]∂r

=

∂

∂ pr[Q1∑

k

hsk 1−e−b1 pk ]

∂ pr

∂r

∂

∂ pr

[Q2 1−e−b

2p

r ]∂ pr

∂r

=

b1Q1 hsre−b1 prb2 Q2 e

−b2

pr sr sgn r

with sgn r=∂ pr

∂r

=∂∣r∣

∂r

=r

∣r∣





3.2.2 implicit Resolution – Integration of model MONO_DD_KR

the system to be solved is reduced to the equations relating to the components stresses

R1i i ,s ,s=−1 i−-

-1- i−iith∑

s

is gs s , s=0

and those relating to the hardening parameter:

R2s ∑i

, τ s , s= s−∣g s s ,s ∣.h s =0

with i = 1 to 6 , s = 1 with ns and r =1 with ns

This makes it possible to obtain a formulated system of equations in an entirely implicit way, where

R1i and R2

s depend only on i , s (because of being s calculated explicitly from i

g s s ,s =0exp −G effs

k BT . s

∣s∣. t

G effs =G0 1− ⟨eff

s ⟩

R

p

q

h s = bd∑

u≠s

u

K−

g cs

b

effs =⟨∣s∣−0−

s ⟩ ⟨ ⟩ : positive part

s=

2∑

u

asuu

∣s∣−0

with u=u b2

Note:: for each system of sliding s where the conditions effs 0 ∣s∣0 are not checked, one

obtains immediately: g s s ,s =0, s=0The jacobian matrix of the system is written in the form:

J=[∂R1

i

∂ j

∂R1i

∂ s

∂R2s

∂ i

∂R2s

∂r

] of dimension 6ns×6ns

•∂R1

i

∂ j

=−1 ij∑s

si∂ g s

∂ j

=−1 ij∑s

si∂ g s

∂s

∂s

∂ j

=−1 ij∑s

si s j

∂ g s

∂s

=> It is necessary to calculate ∂ g s

∂s

=sgn s .∂∣g s∣

∂s

•∂R2

s

∂i

=∂ −∣gs∣. hs ∂s

.∂s

∂ i

=−si . ∂∣gs∣

∂s

.hs∣g s∣.∂ hs

∂ τs

=> the term ∂ hs

∂s

being null, it remains to calculate ∂∣g s∣

∂s

.





•∂R1

i

∂ s

=∑s

si s

∣s∣.∂∣g s∣

∂ s

•∂R2

s

∂ r

=sr−∂∣gs∣

∂r

=sr− ∂∣g s∣

∂ r

hs∣gs∣∂ hs

∂r

=> It is also necessary to calculate ∂∣g s∣

∂r

and ∂∣g s∣

∂s

;

hs= s

∣ s∣ and

∂s

∂s

=−

s

∣s∣−0

. sgn s

Computation of ∂∣g s∣

∂r

:

∂∣g s∣

∂s

=− t . 0

k BTexp −G eff

s k B T .G eff

s s

=−∣g s s , s∣

k BT.∂G eff

s ∂s

where

∂G effs

∂s

=−G0 1− effs

R

p

q−1

.q . pR

. effs

R

p−1

.∂eff

s

∂ τ s

and

∂effs

∂s

=s

∣s∣−∂

s

∂s

=sgn s 2 ∑ asu

u .

sgn s

∣s∣−02=sgn s

s .sgn s∣s∣−0


∂r

:

∂∣g s∣

∂r

=− t . 0

k B Texp −G eff

s k B T .G eff

s r

=−∣g s s , s∣

k BT.∂G eff

s ∂r

where

∂G effs

∂ r

=−qpR

G0 1−eff

R

p

q−1

. eff

R

p−1

.∂eff

s

∂r

and

∂effs

∂r

=−∂

s

∂ r

=−2

∣s∣−0

∂ ∑u asuu

∂ r

=−2

∣s∣−0

asr


∂s

:





∂hs

∂ r

=1K

∂

∂r∑

u≠ s

u−gc

bsr=

12K

1−rs

∑u≠s

u−

gc

bsr

3.2.3 Implicit resolution – models MONO_DD_CFC/MONO_DD_FAT

Statement of the residue

Knowing the solution at time t i−1 , noted -= t i−1 , s-= s t i−1 ,s

-=s t i−1 , ps-=p s ti−1

, it is a question of finding, at time t i , the quantities = t i , s= s t i , γs=γ s t i , ps=ps t i

checking:

−1 i− --1- i=i−i

th−∑s sis , =0

The cission solved by system of sliding is: s= :s=- :s

The flow model after discretization implicit becomes:

Δγ s= γ0 Δt ∣s∣

fsforest

n

−1 . τ s

∣τ s∣= ps

τ s

∣τ s∣

ps ,=0 t ∣s∣

fsforest

n

−1 if ∣s∣≥ fsforest ; if not ps=0

the model of hardening is: sforest=s

forest ⟨-

⟩ =C ∑ j=1,12 ⟨ asj j ⟩

with C =0.20.8

ln ∑i=1,12⟨i ⟩

ln b ref

or, in case DD_FAT, C=1.

the evolution of the density of dislocation is given (always in implicit form) by:

s= ps hs = ps hs- with:

hs= A∑


∑j=1,12

asj ⟨ j ⟩B C ∑


ybs

or, in case DD_FAT, h s = bd ∑u≠s

⟨u ⟩

K−g c 0

⟨s ⟩b

By means of the vectorial notation of Kelvin [chapter 5 of 41) for the symmetric tensors ( , s ), and in taking into account the possible variations of the moduli of the elasticity with the temperature, one





leads to the system to solve, comprising 18 equations; i=1, 6 s=1,12 , 18 unknowns being 6 components of and the 12 values .

System to be solved in (6 components) and (12 components):

Ri1 ,=−1

i−--1

- i−iith∑

s si ps , .τs

∣τs∣=0

R s2 , =s− ps hs =0

with: s= :s=-:s

ps ,=0 t ∣s∣

fsforest

n

−1 if ∣s∣≥0sf ; if not ps=0

and if ps0 :

sforest=s

forest

-=C ∑ j=1,12

a sj ⟨ j ⟩

C =0.20.8

ln ∑i=1,12⟨i ⟩

ln b ref

hs= A∑


∑j=1,12

asj ⟨ j ⟩BC ∑


yb⟨s ⟩

In the case DD_FAT, this system becomes:

Ri1 ,=−1

i−--1

- i−iith∑

s si ps , .τs

∣τs∣=0

R s2 , =s− ps hs =0

with: s= :s=-:s

ps ,=0 t ∣s∣

fsforest

n

−1 if ∣s∣≥0sf ; if not ps=0

and if ps0 :

sf o r e s t

=sf o r e s t

-=∑ j=1 , 1 2

a s j ⟨ j ⟩

h s = bd ∑u≠s

⟨u ⟩

K−g c 0

⟨s ⟩b

The jacobian matrix of the system can be then calculated for integration by the method of Newton (R5.03.14).





Jacobian matrix This subparagraph relates to only models DD_CFC and DD_CC : for the model DD_FAT, the jacobian matrix was not programmed yet and one uses the matrix calculated by disturbance.

The jacobian matrix of the system is written in the form:

J=[ ∂ Ri

1

∂ j

i=1,6; j=1,6 ∂ Ri

1

∂s

i=1,6 ; s=1,12

∂R s2

∂ j

s=1,12 ; j=1,6 ∂ Rs

2

∂r

s=1,12 ;r=1,12] of dimension 18×18

•

∂Ri1

∂ j

=−1 ij∑s

s i

∂ ps

s

∣s∣∂ j

=−1 ij∑s

sis

∣s∣∂ ps

∂s

∂s

∂ j

=

−1 ij∑s

sis j

s

∣s∣∂ ps

∂s

with ∂ ps

∂s

=n ps0 t

s

indeed: ∂ ps

∂s

=n 0 t

fsforest ∣s∣

fsforest

n−1

⋅s

∣s∣=n

ps0 t ∣s∣

⋅s

∣s∣

because sforest does not depend on s and ps0 t=0 t ∣s∣

fsforest

n

•∂ Ri

1

∂s

=∑u=1,12

uiu

∣u∣

∂ pu

∂s

with ∂ pu

∂s

=−n 0 t∣u∣ ∣u∣

fuforest

n−1∂u

forest

∂s

fuforest

2

or

∂ pu

∂s

=−n 0 t ∣u∣ fu

forest

n∂u

forest

∂s

fuforest

∂ pu

∂s

= −n pu0 t

fuforest

∂uforest

∂s





it remains to express:∂u

forest

∂s

=∂C ∂s

⋅ ∑j=1,12

auj jC aus

2 ∑j=1,12

auj j

⟨s ⟩s

let us calculate ∂C ∂s

=0.8

2 ln b ref

1

∑k=1,12

k

⟨s ⟩s

what provides derivative: ∂ Ri

1

∂s

=−∑u=1,12

uiu

∣u∣n

pu0 t fu

forest

∂uforest

∂s

•

∂R s2

∂i

=−si∂ ps

∂s

hs

with ∂ ps

∂s

=n ps0 t

s

• ∂R s2

∂r

=sr− ps

∂ hs

∂r

−hs∂ ps

∂r

• One already calculated ∂ ps

∂r

. It remains to derive: ∂hs

∂r

:

∂

∂r A∑

j∈ forest s asj j

∑j=1,12

asj j

B C ∑j∈coplas

asj j−ybs=A

∂T 1

∂r

B∂T 2

∂r

−ybsr

⟨r ⟩r

∂

∂r

T 1= asr

∑j=1,12

a sj j

⋅I sforestr

⟨r ⟩r

−

asr ∑j∈ forest s

asj j

2 asrr ∑j=1,12 a sj j

2

⟨r ⟩r

∂

∂r

T 2=∂C∂r

∑j∈coplas

asj jC asr

2 a srr

I scoplar

⟨r ⟩r

I scopla r =1 if r∈copla s , =0 if not and I s

forest r =1 if r∈ forest s , 0 if not

3.2.4 implicit Resolution – models MONO_DD_CC/MONO_DD_CC_IRRA





the algorithm proposed (cf 41 ) consists in eliminating the dependence from R s with effs while

approaching G by:

1. G =minG 0 ;k T ln [ ms b H

totseq] .

with tots =F

s =∑j≠s

jirrs

(case 2) or tots =∑

j

jirrs

(case 1)

in this statement, eq can represent the total or plastic equivalent strain

2.R s=

b

201−G G0

2

3. 1

sD

=min tots; D2 Rs

tots

4. l s=max s

−2ATsR s

; l c

5. ATs =∑j

hsj j

tots a irr

irrs

tots

with hsj the commonly-used term of the matrix of interaction

6. LTs =max0 ; 1−AT

s b 1

s −

1

2ATs Rs l cT (case 1)

or LTs =max 0 ;AT

s b 1

s −

1

2ATs Rs lc T (case 2)

7. LRs =bAT

s tot

s (case 1) or LRs =b hsss (case 2)

8. effs =∣s∣−F−LR

s −LTs (case 1)

effs =∣s∣−c and c

s=F LR

s

2LT

s

2 (case 2)

9. nucS=m

s b H⋅l s ⋅exp −G0

k BT 1− <effs>

0 sgns knowing that eff

s0

10. prob

s=0

∣s∣

F LRs1−AT

sb 1

s

n

sgns

(case 1)





probs =0 ∣

s∣

cs

n

sgns (case 2)

11.1

s=

1

nucs

1

probs

12. s=∣ s∣b 1

d lath

a self

s

K self

∑j≠s

asj

j

K f

− ys

s with a sj=hsj1−<effs >

0 (case 1)

s=∣ s∣b 1

d lath

ass

s

K self

ATs 1−

eff

0

sFs

K f

− y s

s and asj=hsj1−<eff

s >0

2

(case

2)

and 1

ys=1

y ATs

2effs

b

13.in case DD_CC_IRRA irrs =−irr

s ∣s ∣ , is: irrst t =irr

st exp − ps

(11) a system of 12 differential equations in the bus s defines all the calculated quantities (from

1 to 10) are functions of s .

The limit of this model at high temperature is: R s→∞ because G →G0

s → tot

s and l s →l c , nucSm

s b H⋅l c sgn s

LTsLR

s→AT

sb 1

s therefore prob

s=0 ∣s∣

FATs b tot

s n

sgns

3.2.5 implicit Algorithm of integration in large deformations

From U n and U , one can calculate Fn , F and Fn1=F F n . moreover, one knows the

results of the increment of time n : stresses of Cauchy n local variables.

It is a question of determining n1 local variables and the sn1 (or sn1 ) solutions of the

system of 6ns equations:



Sn1= .12F n1

e T F n1e – I d

and for each system of sliding:

sn1= t g s ,s ,s , ps or s= t h s ,s ,s , ps



with sS =M n1 :m s⊗ns=F en1T F en1S n1 :ms⊗ns=[ 2-1SI d S ] :ms⊗ns

indeed according to 41and 41, one considers here Piola-Kirchhoff stresses measured from the intermediate configuration, therefore only the elastic part of the gradient of transformation intervenes.

Fn1e=Fn1Fn1

p

-1=F Fn

eF p

-1

Several choices are possible for the computation of F p-1 :

� F p

-1=exp −∑s S ,sms⊗ns 41 who has the advantage null of preserving the trace

plastic strains, but which can present a overcost;

� F p

-1= I d∑sS ,sms⊗ns

-1 det I d∑ sS ,sms⊗ns

13 41

� F p

-1= I d−∑sS ,sms⊗ns det I d−∑sS ,sms⊗ns

13

The gradientssont F e stored as local variables (by removing the tensor identity)

In postprocessing one calculates the stresses of Cauchy: n1=1

det F n1e Fn1

eS n1F n1

e T

or stresses of Kirchhoff: n1=det F n1n1=Fn1eS n1F n1

e T

because det F n1=det F n1e det F n1

p =det F n1e

the purpose of the problem to be solved in each point of integration is thus, knowing the variables

internes.de sn sn psn Fne to determine local variables n1 and the ( sn1 , sn1 )

Fn1e solutions of the system with equations 6ns : and

sS =[ 2-1 S I d S ] :ms⊗ns

Fn1e=F Fn

e F p -1

the nonlinear resolution of this system of equations can be done classically by the algorithm of Newton [R5.03.14].

The vector of the unknowns is composed of: (6 Sn1 components) and ( s components ns ). The computation

of the jacobian matrix is described in appendix. Initialization

,

the algorithm is initialized in the following way: One

poses what Fn1e-trial

=Fn1F np

-1 amounts supposing that One Fn1p=F pF n

p= I d F n

p



R1=-1 Sn1−

12F n1

e T F n1e – I d =0

for each system of sliding: or

R2=sn1− t g s , s , s , ps=0 with R2=s− t⋅hs , s ,s , ps=0



calculates with R1 then Fn1e=Fn1

e-trial all the equations If R2 no threshold is reached, then.

Fn1e=Fn1

e-trial If not the system should be solved. R1 R2 One

can choose to keep like local variable the tensor. Fn1e to compute: more easily In practice

Fn1e=F F n

e F p -1

one stores in F e− I d the local variables with V 63∗ns1 One V 63∗ns9

also preserves in F p−I d the local variables at V 63∗ns10 In V 63∗ns19

postprocessing, the tensors and m s can ns be updated by means of rotation resulting from polar

decomposition: F e=R eU e For m s=R ems0 ns=R ens0

the computation of polar decomposition, an exact algorithm is proposed in. 41According to

, 41it can also be transported in the present configuration by: and

m s=Fems0 Convergence criteria n s=Fe−Tns0

3.2.6 used for the implicit resolutions

the stopping criteria of the iterations relate to the relative nullity of the residue: . ∣∣R Y ∣∣h∣∣Rref∣∣h

c where

c the tolerance given by RESI_INTE_RELA represents . The problem consists with well choosing

as well as Rref the norm By ∣∣.∣∣h homogeneity with the total algorithm of Newton used in STAT_NON_LINE [R 5.03.02], and to avoid useless iterations when the initial residue is very small, one chooses: with

maxi=1,6

∣Ri Y ∣

maxi=1,6

∣E itr∣c E tr=-1

-E−E th

maxi=7,n

∣RiY ∣

maxi=7, n

∣Y i-Y i∣

c

the two preceding criteria must be checked to obtain convergence. If

convergence is not reached after the maximum number of iterations, the stationarity of the solution is also tested:

∣Y k1−Y k∣

The method used time step allows a local recutting of, either systematic, or in the event of nonconvergence (key word ITER_INTE_PAS ): this allows an easier integration, but by then losing the benefit of a coherent tangent operator at the conclusion of the integration of the behavior. It

is often preferable, in the event of nonlocal convergence at the end of ITER_INTE_MAXI iterations , to carry out a total recutting of time step (by means of DEFI _LISTE_INST), in order to preserve a coherent tangent matrix. Resolution





3.3 clarifies Another

method of resolution, very simple to implement to solve the differential equations of the single-crystal behavior is the explicit resolution. So that it is effective numerically, it is essential to associate an automatic control of step to him. As in [R5.03.14], one uses the method of Runge and Kutta. The

computation local variables at time ht + is function only values of their derivatives: dYdt=F Y ,t

with

Y={

s

s

ps

Evp

={hs

- , s- , s

- , ps-

g s- ,s

- ,s- , ps

-

∣ s∣= f s- ,s

- ,s- , ps

-

∑s

ms s

One s= :s=- E−E th−Evp :s

integrates according to the following diagram: if

Y th=Y (2) the criterion of accuracy is satisfied with

Y2=Y

h2[F Y , t F Y

1 , th ] Y 1

=Yh F Y , t

the difference between (diagram Y 2 with order 2) and (diagram Y 1 of order 1, Eulerian) provides

an estimate of the error of integration and allows to control the size of time step which h is initialized

with for t i the first attempt. The strategy of the control of the step is defined from of the difference

between the two integration methods: and ∥Y 2 −Y 1∥ of the accuracy required by the user (key

word : RESI_INTE_RELA ). The criterion selected is the following, where one notes:

)...,,,( 21 NyyyY =

Y t = supj=1,N { ∣y i

2− y i

1 ∣max [ ,∣y j

t ∣] }

The parameter is fixed at 0,001. The accuracy of desired integration must η be coherent with the level of accuracy necessary for the total stage. If

the criterion is not checked, time step Re-is cut out according to a discovery method (number of under-PAS defined by the user via key word ITER _INTER_PAS). When time step becomes too weak (H < 1.10-20), computation is stopped with an error message. Explicit

algorithm of integration in large deformations At every moment

and each point of integration, one can calculate, Fn and F . Fn1=F F n One knows the

stresses of Cauchy and the n local variables at time It n

acts to determine local variables S and the, s at s F p time. n1 One can with this intention use a method of Runge-Kutta [cf R5.03.14] on the system of equations differentials: for





F p=∑s sms⊗nsF

p

s=g s ,s ,s , ps each system of sliding for

s=hs , s , s , ps each system of sliding with

: and s=M :ms⊗ns=FeT F e S :ms⊗ns=[ 2-1 SI d S ] :m s⊗ns

S = .12F eT F e – I d Local variables F e=F F p

-1

4 Case

4.1 of the monocrystal

the local variables in Code_Aster are named V 1 ,… V 2 . V p

The six first are the 6 components of the viscoplastic strain: : E ijvp with

E vp=∑t

E vp Evp=∑

s

s Δγs

V 1=E xxvp V 2=E yy

vp V 3=E zzvp V 4= 2 E xy

vp V 5= 2E xzvp V 6= 2E yz

vp

V 7 , V 8 are V 9 the values of for 1 1 p1 the system of sliding 1=sV 10 , V 11 correspond V 12 to the system, 2=s and so on, where: represent

� s the kinematical variable of the system in the case of s the phenomenologic models, and the density of

dislocations in a model resulting from the DD; represent � s the plastic sliding of the system represents s

� p1 the cumulated plastic sliding of the system Taken s

into account of the irradiation: in �case DD _CC_IRRA, it is necessary to add local variables n irra=12 : with

V 63ns1 contain V 63ns12 for each system of sliding the density of dislocations related to

the irradiation in sirr

�case DD _CFC_IRRA, it is necessary to add local variables n irra=24 : with

V 63n s1 contain V 63n s12 for each system of sliding with sloops∗b2

V 63n s13 contain V 63n s24 for each system of sliding One svoids

stores then cissions them for each system of sliding: , 1 … If ns

one takes into account the rotation of the crystal lattice, local variables should be added nrota=16 : with

� V 63n s1 are V 63n s9 the 9 components of the matrix of rotation, Q with

� V 63n s10 are V 63n s12 the 3 components of, p with

� V 63n s13 are V 63n s15 the 3 components of, e represents

� V 63n s16

the antepenultimate local variable is the stress of cleavage: maxs

Σ . n : n





Before last local variable contains the total cumulated plastic strain, defined by: with

V p−1=∑Eeqvp E eq

vp= 2

3Evp :Evp

the last local variable, Vp (, p=63nsnrota3 being ns the nombre total of sliding systems) is an indicator of plasticity (threshold exceeded in at least a system of sliding to time step running). If it is null, there no was increase in local variables at current time. If not, it contains the nombre of iterations of local Newton (for an implicit resolution) which were necessary to obtain convergence. Cases

4.2 of the polycrystal

the local variables in Code_Aster are named V 1 ,… V 2 . V p

The number of local variables is, p=76m ∑g=1, m

3ns g 6m1 being m the number of

phases and being nsg the number of sliding systems of the phase) g . If one takes into account the irradiation, the nombre total of local variables is:

p=76m ∑g=1, m

3ns g 12m6m1

�The first six local variables are the components of the macroscopic viscoplastic strain: E vp

V 1=E xxvp V 2=E yy

vp V 3=E zzvp V 4= 2 Exy

vp V 5= 2E xzvp ; V 6= 2E yz

vp

� the seventh is the cumulated equivalent viscoplastic strain macroscopic: P with

V 7=∑E eqvp ; Eeq

vp= 23Evp:Evp then

� , for each phase, one finds the 6 components of the viscoplastic strains or the tensor of the phase: ;

{xxvpg , yy

vpg ,zz

vpg , 2xy

vp g , 2xz

vpg , 2 yz

vpg }g=1, m

then

� , for each phase, and each system of sliding of the phase, one finds the values of; s s p s if

� the irradiation is taken into account, for each phase, one finds the 12 densities of dislocation related to the irradiation; irr

s then

� , for each phase, one finds the 6 components of the stresses of the phase: ;

{ xxg , yy g , zz g , 2 xy g , 2 xz g , 2 yz g }g=1,m

� the last local variable is an indicator of plasticity (threshold exceeded in at least a system of sliding to time step running). Numerical

5 establishment in Code_Aster Generally

, the single-crystal behaviors are integrated into the methods of Runge-Kutta for explicit integration, and into the environment “plasti” for implicit integration [R5.03.14]. The tensors of directional sense of sliding systems are as for them all defined in routine LCMMSG providing the tensor in total reference for the nth system of the provided family whose name is provided by the appealing routine.

To add a new behavior of monocrystal, or simply a new model of flow or hardening, it is advisable to define its parameters in DEFI _MATERIAU. According to the case (flow, isotropic hardening or kinematics), it is necessary to add the reading of these parameters in routines LCMAFL , LCMAEI , LCMAEC . For integration, it is enough to write the definition of the increases in local variables in





routines LCMMFE (flow ), LCMMFC (kinematic hardening) and LCMMFI (isotropic hardening), so that explicit integration functions.

Implicit integration also uses routines LCMMFE , LCMMFC and LCMMFI . She asks besides defining derivatives of the equations compared to the various variables. The derivatives are to be written in routines LCMMJF (derivatives of L “equation D” flow), LCMMJI (derivatives of the relation of isotropic hardening) and LCMMJC (derivatives of the relation of kinematic hardening). For

more details on architecture of the resolution of the crystalline behaviors, to see [D5.04.02]. Use

6 These

models are accessible in Code_Aster in 3D , plane strains (D_PLAN), plane stresses (C_PLAN, by the method of Borst [R5.03.03]) and axisymetry (AXIS). Case

6.1 of the monocrystal In the case ofmicrostructures with a grid, the various grains of a monocrystal being represented by mesh groups, it is necessary and to assign the parameters of the materials behaviors of the monocrystals like their directional senses to the various grains.

The values of the parameters of the behavior models are provided using command DEFI _MATERIAU. This is defined starting from key keys MONO_VISC1 , MONO_VISC2 , MONO_DD_KR , MONO_DD_CFC for flow, MONO_ISOT1 , MONO_ISOT2 , MONO_DD_CFC for isotropic hardening and MONO_CINE1 , MONO_CINE2 for kinematic hardening [U4.43.01]. For example [V6.04.172]: MATER

1=DEFI_MATERIAU (/ELAS = _F (E= …, NU=…,/ELAS_ORTH=_F (E_ L=192500, E_ T=128900, NU _LT=0.23, G_ LT=74520,), #

RELATIONS OF ECOULEMENT/ MONO_VISC1=_F (N=10, K=40, C=6333),/ MONO_VISC2=_F (N=10, K=40, C=6333, D=37, A=121),/ MONO_DD_KR=_F (...), #

HARDENING ISOTROPIC/ MONO _ISOT1=_F (R_0=75.5, Q=9.77, B=19.34, H=2.54),/ MONO _ISOT2=_F (R_0=75.5, Q1=9.77, B1=19.34, H=2.54, Q2=-33.27, B2=5.345,), #

KINEMATIC HARDENING/

MONO_CINE1=_F (D=36.68),/MONO _CINE2=_F (D=36.68, GM=, PM=,),); One

can thus dissociate, on the level of the data, the flow of isotropic hardening and kinematic hardening. It

is now necessary to define it (or them) standard of studied monocrystal. For that, one defines the external behavior of way to STAT_NON_LINE , via operator DEFI _COMPOR, for example: MONO





1=DEFI_COMPOR (MONOCRISTAL = (_F (MATER=MATER1, ECOULEMENT =MONO_VISC1, MONO _ISOT=MONO_ISOT1, MONO _CINE=MONO_CINE1, FAMI _SYST_GLIS= (“CUBIQUE1”,), _F

(MATER=MATER1, ECOULEMENT =MONO_VISC1, MONO _ISOT=MONO_ISOT2, MONO _CINE=MONO_CINE2, FAMI _SYST_GLIS=' CUBIQUE2',),), _F

(MATER=MATER2, ECOULEMENT =MONO_VISC1, MONO _ISOT=MONO_ISOT2, MONO _CINE=MONO_CINE2, FAMI _SYST_GLIS=' PRISMATIQUE',), _F

(MATER=MATER2, ECOULEMENT =MONO_VISC1, MONO _ISOT=MONO_ISOT2, MONO _CINE=MONO_CINE2, FAMI

_SYST_GLIS=' BCC24',),), )

the produced data structure contains names of sliding systems, associated with names of parameters of material, for each behavior of monocrystal. “

CUBIC ”FAMI _SYST_GLISMATE_SYSTTYPE_LOIECOULEMENTMONO_ISOTMONO_CINE1 MATER1 VISCMONO

_VISC1MONO_ISOT1MONO_CINE1 “BASAL” MATER1 VISCMONO _VISC1MONO_ISOT1MONO_CINE1 “PRISMATIQUE” MATER1 VISCMONO _VISC1MONO_ISOT2MONO_CINE2 “BCC24” MATER1 VISCMONO _VISC1MONO_ISOT2MONO_CINE2

.........

Operator DEFI _COMPOR calculates the nombre total of local variables associated with the monocrystal. Lastly,

to carry out a computation of microstructure, it is necessary to give, grain by grain, or mesh group (representing sets of grains) a directional sense, using key word MASSIF of AFFE_CARA_ELEM . For example: ORIELEM

= AFFE_CARA_ELEM (MODELS = MO_MECA, MASSIF = (_F (GROUP_MA=' GRAIN1', ANGL_REP= (348.0, 24.0, 172.0),), _F (GROUP_MA=' GRAIN2', ANGL_REP= (327.0,126.0,335.0),), _F (GROUP_MA=' GRAIN3', ANGL_REP= (235.0,7.0,184.0),), _F (GROUP_MA=' GRAIN4', ANGL _REP= (72.0,338.0,73.0),…) Notices

1: The directional senses of sliding systems can be indicated in nautical angles under the key word of ANGL_REP or in Eulerian angles under key word ANGL_EULER . Notice

2: Posur

•the same monocrystal, the values of the parameters can be different from one family from sliding systems to another. This is why one can define factor key word a material different by occurrence from the MONOCRISTAL . But in this case, how to provide to transmit to STAT_NON_LINE information stipulating that in a point of gauss (all those of the mesh





group concerned), one has several materials present? This is possible thanks to an evolution of AFFE_MATERIAU [U 4.43.03] and data structure material [D4.06.18]): MAT

=AFFE_MATERIAU (MAILLAGE=MAIL, AFFE = _F (GROUP_MA=' GRAIN1', MATER = (MATER1, MATER2),),);

The other data of computation are identical to a usual structural analysis. Lastly,

in STAT_NON_LINE , the behavior resulting from DEFI _COMPOR is provided, under key word COMP_INCR via key word COMPOR , compulsory with the key word RELATION = ' MONOCRISTAL'. COMP_INCR= _F (RELATION = ' MONOCRISTAL', COMPOR = COMP1 Precise details

that for explicit integration, (ALGORITHME_INTE=' RUNGE_KUTTA'), it is useless to ask the reactualization of the tangent matrix since this one is not calculated. To begin from the iterations of Newton of the total algorithm, it can be useful to specify PREDICTION = ' EXTRAPOLE' [U 4.51.03]. One

will be able to find examples of use in the tests: SSNV 171, SSNV172, SSNV194. Case

6.2 of the polycrystal In the case ofmultiphase polycrystals, each phase corresponds to a monocrystal. One will thus use the parameters preset previously in DEFI _MATERIAU for the monocrystal. Here, it is a question of laying, for each phase, down the direction, the voluminal fraction, the monocrystal used, and the type of model of localization. This is carried out under factor key word the POLYCRISTAL of DEFI _COMPOR. MONO

1=DEFI_COMPOR (MONOCRISTAL=_F (MATER=MATPOLY, ECOULEMENT = ' MONO_VISC2', MONO _ISOT=' MONO_ISOT2', MONO _CINE=' MONO_CINE1', ELAS = ' ELAS', FAMI _SYST_GLIS=' OCTAEDRIQUE',),); POLY

1=DEFI_COMPOR (POLYCRISTAL= (_F (MONOCRISTAL=MONO1, FRAC _VOL=0.025, ANGL _REP= (- 149.676, 15.61819, 154.676,),), _F (MONOCRISTAL=MONO1, FRAC _VOL=0.025, ANGL _REP= (- 150.646, 33.864, 55.646,),), _F (MONOCRISTAL=MONO1, FRAC _VOL=0.025, ANGL _REP= (- 137.138, 41.5917, 142.138,),), ...... _F (MONOCRISTAL=MONO1, FRAC _VOL=0.025, ANGL _REP= (- 481.729, 35.46958, 188.729,),),), LOCALIZATION = ' BETA', DL =321.5, DA =0.216,);

Key word POLYCRISTAL makes it possible to define each phase by the data of a directional sense, (provided by ANGL_REP or ANGL_EULER ), of a voluminal fraction, a monocrystal (i.e. a model of behavior and sliding systems).





Key word LOCALIZATION makes it possible to choose the method of localization for all the phases of the polycrystal. Key word MASSIF makes it possible to choose a total directional sense of the polycrystal under key word ANGL_REP if it is in nautical angles or under key word ANGL_EULER if it is in Eulerian angles. Lastly,

in STAT_NON_LINE , the behavior resulting from DEFI _COMPOR is provided, under key word COMP_INCR via key word COMPOR , compulsory with the key word RELATION = ' POLYCRISTAL'. COMP_INCR= _F (RELATION = ' POLYCRISTAL', COMPOR

= COMP1) Example

6.3 As

example of implemented, one presents here briefly a computation of aggregate, of cubic form (elementary volume) including 100 single-crystal grains, defined each one by a mesh group. The nombre total of elements is 86751. With meshes of order 1 (TETRA4) it comprises 15940 nodes. With meshes of order 2 (TETRA10), it comprises 121534 of them. The loading consists of a homogeneous strain, applied via a normal displacement imposed to a face of the cube (direction). z One reaches a strain of 4% in 1s and 50 increments. ACIER

=DEFI_MATERIAU (ELAS=_F (E =145200.0, NU=0.3,), MONO_VISC1=_F (N=10., K=40., C=10.,), MONO_ISOT2=_F (R_0=75.5, B1

=19.34, B2=5.345, Q1=9.77, Q2=33.27, H=0.5), MONO

_CINE1=_F (D=36.68,),); COEF

=DEFI_FONCTION (NOM_PARA = ' INST', VALE = (0.0, 0.0, 1.0, 1.0,),); MAT=AFFE_MATERIAU (MAILLAGE=MAIL, AFFE=_F (TOUT = " YES”, MATER= (ACIER),),); COMPORT

=DEFI_COMPOR (MONOCRISTAL= (_F (MATER =ACIER, ECOULEMENT= " MONO_VISC1”, MONO_ISOT = " MONO_ISOT2”, MONO_CINE = " MONO_CINE1”, ELAS= " ELAS”, FAMI_SYST_GLIS=' OCTAEDRIQUE',),),

); ORIELEM

= AFFE_CARA_ELEM (MODELS = MO_MECA, MASSIF = (_F(GROUP_MA=' GRAIN1', ANGL_REP= (348.0, 24.0, 172.0),), _F(GROUP_MA=' GRAIN2', ANGL_REP= (327.0,126.0,335.0),), _F(GROUP_MA=' GRAIN3', ANGL_REP= (235.0,7.0,184.0),), ................._F(GROUP_MA=' GRAIN99', ANGL_REP= (201.0,198.0,247.0),), _F(GROUP_MA=' GRAIN100', ANGL_REP= (84.0,349.0,233.0),),))FO

_UZ = DEFI_FONCTION (NOM_PARA = “INST”, VALE = (0.0,0.0,1.0,0.04,),) CHME

4=AFFE_CHAR_MECA_F (MODELE=MO_MECA, DDL_IMPO=_F (GROUP_NO=' HAUT', DZ=FO_UZ,),) LINST

= DEFI_LIST_REEL (DEBUT= 0. , INTERVALLE = (_F (JUSQU_A = 1. , NOMBRE= 50),)) SIG

=STAT_NON_LINE (MODELS =MO_MECA, CARA_ELEM=ORIELEM, CHAM_MATER =MAT, EXCITWarning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and is provided as a convenience.




= (_F (CHARGE=CHME1), _F(CHARGE=CHME2), _F(CHARGE=CHME3), _F(CHARGE=CHME4),), COMP

_INCR=_F (RELATION = ' MONOCRISTAL', COMPOR =COMPORT), INCREMENT=_F (LIST_INST=LINST),))

The following figures represent isovaleurs of the strains the stresses according to. z To it not homogeneity of the values is noted, and one can even distinguish the contour of the grains.

To be able to exploit this kind of results, one can for example calculate average fields by grains. On the following figure, one represented the equivalent stresses according to plastic strains equivalent for all the grains. Bibliography





7 MERIC

•L., CAILLETAUD G.: “individual extremely structural hook modeling calculations” in Newspaper of Engineering Material and Technology, January 1991, flight 113, pp171-182. LECLERCQ

•S., DIARD, O., PROIX J.M.: “Impact study of the establishment of a library of constitutive laws and rules of transition from scales” Notes EDF R & D HT 3/26/053 /A. CAILLETAUD

•G.: “A micromechanical approach to inelastic behavior of metals”, Int. J. of Plasticity, 8, pp. 55-73, 1992. PILVIN

•P.: “The contribution of micromechanical approaches to the modelling of inelastic behavior of polycrystals”, Int. Conf . one Biaxial/Multiaxial tires, France, ESIS/SF2M, pp. 31 - 46, 1994. BERVEILLER

•MR., ZAOUI A.: “Year extension of the coil-consistent design to plasticity flowing polycrystal” J. Mech. Phys . Solids, 6, pp. 325-344, 1979. G.

•MONNET “A crystalline plasticity law for austenitic stainless steels”, EDF Notes R & D H-B 60 - 2008 - 04690 N.

•Toff “implementation of has new constitutive law based one dislocation dynamics for FCC materials” Note EDF-R&D: Note EDF R & D HT24 - 2010 - 01128. C.

•Petry “Procedure of generation of meshes of polycrystalline aggregates” Notes EDF R & D H T24-2008-03481 A.

•ZEGHADI: “Effect of morphology 3D and the size of grain on the structural mechanics behavior of polycrystalline aggregates”. Thesis of the École des Mines of Paris, 2005. N.

•Toff: “Hot two-phase metal Strain. Theoretical modelizations and experimental confrontations”. Thesis of the Polytechnic school, 2007. Mr.

•Fivel, S.Forest “crystalline Plasticity and transition from scale: case of the monocrystal”. Techniques of engineer M4 016 “

•Computational methods for plasticity” Wiley 2008, EA of Souza Neto, D. Peric, DRJ. Owen “

•A computational procedure for failure-independent hook plasticity “L.ANAND & M.KOTHARI, J.Mech.Phys.Solids, flight 44, n°4, pp.525-558, 1996 C. Stolz. Continuums

•in transformations fi deny: hyperelasticity, fracture, elastoplasticity. Editions of the Polytechnic school, 2002. Homogenization

•in mechanics of the materials”, volume 2. M.Bornert, P.Gilormini, T.Bertheau J. SCHWARTZ:

•“Nonlocal Approach in crystalline plasticity: application under investigation of the structural mechanics behavior of steel AISI 316LN in fatigue oligocyclic”. Thesis of the Central School of Paris, June 2011. G.Monnet: “

•DELIVERABLE D1-2.9. Hook plasticity constitutive law for irradiated RPV steel” Note EDF R & D HT 27 - 2011 - 02738, December 2011. History of





8 the versions of the document Version Aster

Author (S) or

contributor (S), organization Descriptionof

modifications J.M.PROIX, O.DIARD, T.

7.4 KANIT EDF/R & D initial Version J.EL- GHARIB, J.M.PROIX, EDF/R & D

8.4 Complements J.EL-GHARIB, J.M.PROIX, EDF /R & D Addition of

9.2 behavior MONO_DD_KR . J.EL-GHARIB , J.M.PROIX, EDF/R & D Optimization of behavior

9.4 MONO_DD_KR. J.M.PROIX notices on the matrix of interaction N.RUPIN, J.M.PROIX,

10.2 EDF/R & D Addition of behavior DD_CFC N.RUPIN, J.M.PROIX,

10.3 EDF/R & D Addition of the rotation of crystal lattice N.RUPIN

10.5 , F.LATOURTE, J.M.PROIX, EDF /R & D Addition

of large deformations N.RUPIN, J.M.PROIX, EDF

11.1 /R & D Files 16373 (convergence criterion ) and 17422: correction local variables

11.1 of the POLYCRISTAL . P. OF BONNIERES, J.M. PROIX Card-indexes 14586: addition of behavior DD_FAT J.M. PROIX Card-indexes 18398

11.1 : resorption of ALGO_C_PLAN J.M. PROIX J Card-indexes 18692 addition of DD_CC.

11.2 Mr. PROIX Card-indexes 19021 addition of the local variables for

11.2 DD_CC_IRRA Statement of the Jacobian of

11.3 the equations élasto-visco-plastics integrated the system to solve





Annexe 1 is form: That is to say thus to evaluate the terms of the hypermatrice

jacobienne at time With regard to

RY =R ,Evp ,s+ , s

+ , ps+= {

s ,Evp , s+ ,s

+ , ps+

e ,E vp ,s+ , s

+ , ps+

ns {a , Evp ,s

+ , s+ , ps

+

g ,Evp , s+ ,s

+ , ps+

p ,Evp , s+ ,s

+ , ps+ }

={

-1−-

- 1

-−E−E th

−E vp

Evp−∑s

s s

ns{s−hs

+ , s+ ,s

+ , ps+

s−g s+ , s

+ ,s+ , ps

+

Δps− f s+ ,s

+ , s+ , ps

+ }

=0 avec s+=

+ :s

the first line of the matrix, independently of the equations J of hardening t t

J=[∂ s∂ΔΣ

∂ s

∂ ΔE vp

∂ s∂Δαs

∂ s∂ Δγ s

∂ s∂ Δps

∂e∂ΔΣ

∂e

∂ ΔE vp

∂ e∂Δαs

∂e∂ Δγ s

∂ e∂ Δps

∂ a∂ΔΣ

∂ a∂ ΔE vp

∂a∂Δαs

∂ a∂ Δγ s

∂a∂ Δps

∂ g∂ΔΣ

∂ g

∂ ΔE vp

∂ g∂Δαs

∂ g∂ Δγ s

∂ g∂ Δps

∂ p∂ΔΣ

∂ p

∂ ΔE vp

∂ p∂Δαs

∂ p∂ Δγ s

∂ p∂ Δps

]

and flow, one a: the second-row forward can be written also independently of flow and hardenings

∂ s∂ΔΣ

=D−1 ∂ s∂ΔE vp

= Id∂ s∂ Δα s

=∂ s∂ Δγ s

=∂ s∂ Δp s

=0

: The first column of the lines corresponding to the equations (A), (G) and (p) is written: with





∂ e∂ΔΣ

=0∂e

∂ΔE vp=Id

∂e∂ Δα s

=0∂ e∂Δγ s

=− s∂ e∂ Δps

=0

the second column is identically null (because of equation (E): the relations

∂ a∂ΔΣ

=∂ a∂ Δτ s

Δτ s

ΔΣ

∂ g∂ΔΣ

=∂ g∂ Δτ s

Δτ s

ΔΣ

∂ p∂ΔΣ

=∂ p∂Δτ s

Δτ s

ΔΣ

of

Δτ s

ΔΣ=s

T

flow and hardening can be expressed only according to and not. The last block of equations, depends as for

him on the selected behaviors s : Example Let us choose ΔE vp

the viscoplastic flow relation MONO_VISC1 with isotropic

∂a∂ Δαs

∂ a∂ Δγ s

∂a∂ Δps

∂ g∂ Δαs

∂ g∂ Δγ s

∂ g∂ Δps

∂ p∂ Δαs

∂ p∂ Δγ s

∂ p∂ Δps

hardening

MONO_ISOT1: , and a kinematic hardening defined by MONO_CINE1

g Δγ s−Δps

τ s−cαs

∣τ s−cαs∣=0

p Δps−Δt .⟨∣τ s−cαs∣−R s ps

k⟩

n

=0

then: and, concerning the interaction Rs ps =R0Q ∑r=1

N

hsr 1−e−bpr srsrsrh δδ +−= )1(

enters sliding systems, it there only one non-zero terma Δα s−Δγ sdαs Δps=0

: Evaluating





∂ a∂ Δτ s

=0

∂ g∂ Δτ s

=0

∂ p∂ Δτ s

=−nΔt

K n⟨∣τ s−cα s∣−Rs p s⟩

n−1 τ s−cα

∣τ s−cαs∣

∂ a∂ Δα s

= 1dΔp s

∂ g∂ Δα s

= 0

∂ p∂ Δα s

=nc ΔtK n

⟨∣τs−cαs∣−Rs ps ⟩n−1 τ s−cα s

∣τ s−cα s∣

∂ a∂ Δγ s

= −1

∂ g∂ Δγ s

= 1

∂ p∂ Δγ s

= 0

∂ a∂ Δp s

= dαs

∂ g∂ Δp s

=τs−cαs

∣τ s−cαs∣

∂ p∂ Δp s

= 1nΔt

K n⟨∣τ s−cα s∣−Rs ps ⟩

n−1dRs ps dΔp s

dR s ps dΔp s

=Qbh sse−bp

s

of the coherent tangent operator It acts to find the operator tangent coherent

∂ p∂ Δpr

=1nΔt

K n⟨∣τ s−cα s∣−Rs ps ⟩

n−1dRs ps dΔpr

dR s ps dΔp r

=Qbh sr e−bpr





Annexe 2 , i.e. calculated from

the solution of at the end of the increment. For a small variation of, by regarding this time variable ( )( )0=YR

and not as parameter, one obtains: This system R can be written: By writing E the jacobian matrix in the form: With:

∂R∂Σ

δΣ∂R∂ΔE

δΔE∂R

∂ΔE vpδΔE vp

∂R∂ Δαs

δΔα s∂ R∂ Δγ s

δΔγ s∂R∂Δps

δΔps=0

The submatrices have as dimensions

∂R∂Y

δ Y =X ,avec Y=[Σ

ΔE vp

Δα s

Δγ s

Δps

] et X=[δΔE

0000]

: While operating by successive eliminations and

J . δY=[Y 0 [1 ] [0 ]

[0 ] [1 ] Y 1

Y 2 [0 ] Y 3][

ΣΔE vp

ΔZ ]

substitutions

Y 0=−1

ΔZ={Δαs

Δγs

Δps}×ns

, the third block of the system of equations

dim Y 0=−1=[ 6,6 ]

dimY 1=[6,3∗ns ]dimY 2=[3∗ns ,6 ]dimY 3=[3∗ns ,3∗ns]

gives: the required tangent operator can thus be written directly: Evaluating of the coherent

ΔZ=−Y 3−1

Y 2 Σ

ΔE vp=−Y 1 ΔZ=Y 1 Y 3 −1 Y 2 Σ

Y 0Y 1 Y 3−1Y 2 Σ=ΔE

tangent operator – Case of the reduced system It acts to find

∂ Σ∂E tΔt

= ∂Σ∂ ΔE tΔt

=Y 0Y 1Y 3−1 Y 2

−1





Annexe 3 the operator tangent coherent, i.e. calculated from

the solution of at the end of the increment. For a small variation of, by regarding this time variable R Y =0

and not as parameter, one obtains: This system R can be written: with and By E writing the jacobian matrix in the form: With

∂ R∂

∂R∂ E

E∂R∂s

s=0

: The submatrices have as

∂R∂Y

Y =X dimensions Y=[ s] X=[E0 ]

: While operating by successive eliminations and

J .Y=[Y 0 Y 1

Y 2 Y 3][ Z ]

substitutions

Y 0=−1

Z={s}×ns

, the third block of the system of equations

dim Y 0=−1=[ 6,6 ]

dimY 1=[6, ns ]dimY 2=[ns ,6 ]dimY 3=[ns , ns ]

gives: the required tangent operator can thus be written directly: Tangent matrixes in large deformations

Z=−Y 3−1

Y 2

Y 0−Y 1 Y 3−1

Y 2=E

: Local system solved by Newton: with and Computation

∂∂ E t t

= ∂∂ E t t

=Y 0−Y 1Y 3−1Y 2

−1





Annexe 4 of the tangent matrix: By expressing

the variation of and compared to and

R1S ,=-1 .S−12F eTF e

−I d =0

R2S ,=s−hssS ,s=0

, one sS =[ 2-1 S I d S ] :ms⊗ns obtains Fn1e=F Fn

e F p -1

the system: while replaçant

resulting from the second group of equations R1 R2 in the first S , F (static condensation) one

∂ R1

∂ S S

∂R1

∂

∂ R1

∂FF=0

∂R2

∂S S

∂ R2

∂=0 car

∂R2

∂F=0

obtains: thus : this makes it possible to calculate this statement intervening in the tangent operator only

∂ R1

∂S−∂ R1

∂∂ R2

∂-1∂R2

∂S S=−∂ R1

∂FF according to

SF

=− ∂ R1

∂ S−∂ R1

∂∂ R2

∂

-1∂ R2

∂ S -1∂ R1

∂F

the jacobian matrix of the local system of equations. The complete tangent operator can be written, in present configuration (cf R5.03.21): with

the tensor of the stresses of Kirchhoff, deduced from: one from of deduced: it remains

F to calculate

and It remains to calculate the terms of the jacobian matrix S =F e S F eT

, specific F

=∂F e

FS F eT

F e ∂ SF

F eTF e S

∂ F eT

F

to the large deformations ∂F e

F=∂F F n

eF p

-1

F

∂F ije

F kl

=ik Fet i−1lbF

pbj

-1 ∂F ij

eT

F kl

=∂ F ji

e

F kl

= jk F et i−1la F

pai

-1

∂ R1ij

F kl

=−12 [ ∂F

eT

FF eF eT ∂F

e

F ]ijkl

=−12 [ ∂F mi

e

F kl

F mjeFmi

e ∂F mje

F kl]





. with what is written out of components: . The terms, are specific to each single-crystal

� ∂R1S ,

∂S=

-1−

12 ∂F

eT

∂SF eF eT ∂F

e

∂S

∂F e

∂S=F F n

e ∂Fp

-1

∂ S=F Fn

e∑s [ ∂F

p

-1

∂ s

∂s

∂s

∂s

∂ S ]

∂s

∂S=[ 2-1 s S 2 -1 S ss ] où s=ms⊗ns

behavior, and are calculated ∂s

∂ Sab

=[ 2abmn-1 sS mn2 akmn

-1 S mn skbsab ]

∂R1ij∂S kl

=ijkl-1 −

12 ∂F mi

e

∂S klF mj

e F mie ∂F mj

e

∂ Skl ∂F ij

e

∂S kl=F imF mn

en∑s [∂F p

nj-1

∂ s

∂ s

∂s

∂s

∂Skl ]

�∂R1S ,

∂s

=−12 ∂F

eT

∂s

F eF eT ∂F e

∂s

∂F e

∂s

=F F ne ∂F

p

-1

∂s

=F F ne∑

r [ ∂Fp

-1

∂r

∂r

∂s]

�∂R2 S ,

∂S=−

∂ hs

∂S=−∑

r

∂ hs

∂r

∂r

∂S=−

∂ hs

∂s

∂s

∂S in a way identical

∂ s

∂s

∂r

∂s

∂ hs

∂s

to the

small strains. It remains to calculate. According to the statements chosen for the computation of: conduit with calculating

�derivative of ∂F p

-1

∂r

the exponential one : This can be calculated thanks to F p-1 a development

1. F p

-1=exp −∑ss in series, described in. with or with

∂F p

-1

∂r

=−

∂ exp −∑s

s s

∂−∑ssr

: or out of components: with conduit with or out of components: General writing 41

2. Fp

-1= A

det A13

-1

of the jacobian matrixes A=I d∑sm s⊗ns

∂F p

-1

∂r

=−F p

-1⊗F p

-T ∂F

p

∂r





∂F p

-1ij

∂r

=−F p-1ik Fp-T jl

∂F pkl

∂r

∂F p

∂r

=det A-13 mr⊗nr−

13A-T :mr⊗nrA

general Forms

∂F pij

∂r

=det A- 13 mr⊗nrij−

13Akl

-Tmr⊗nrkl Aij

1. (ΔF p)

-1=(B)(det B)

−13 B=I d−∑sms⊗ns of the local

∂F p

-1

∂r

=−det B13 mr⊗nr

13B-T :mr⊗nr B

system solved by

∂F p

-1ij

∂r

=−det B13 mr⊗n rij

13Bkl

-Tm r⊗nrkl Bij





Annexe 5 Newton: Small strains: where the unknown

is: with Large deformations where the unknown is:

with and And, according to

R1 ,=-1.−th∑s

s s=0

R2 ,=s−k ss ,=0 the behavior considered Y=[ ]

, s= :s

and corresponds to the sign

R1S ,=-1. S−12F eTF e

−I d =0

R2S ,= s−k ssS ,=0 of flow represents Y=[ S]

sS =[ 2-1 S I d S ] :ms⊗ns Fn1e=F Fn

e F ps

-1

either the plastic increment of sliding

� s= pss , s s , s=

s

∣s∣ou

s− f

∣s− f ∣ for models MONO_VISC*, or the variation

� s of density of dislocations for models MONO s _DD_ * the derivatives of these equations for the

computation of the jacobian matrix s can be written

in a general way: HP GDEF In HP: and in large deformations and where and are calculated into 47 In HP: and

in large deformations

J 11=∂ R1 ,i

∂ j

J 12=∂ R1 ,i

∂s

J 11=∂ R1S ,i

∂S j J 12=

∂ R1S ,i∂s

J 21=∂ R2 ,s

∂i J 22=

∂ R2 ,s∂r

J 21=∂ R2S ,s

∂ S i J 22=

∂ R2S ,s∂r

� J 11

�∂R1

i

∂ j

=−1 ij∑s

si∂s

∂ τ s

∂ τ s

∂ j

=−1 ij∑s

si s j

∂s

∂ τ s

�and, where is calculated ∂R1S ,

∂S=

-1−

12 ∂F

eT

∂SF eF eT ∂F

e

∂S in

∂F e

∂S=F F n

e∑s [ ∂F

p

-1

∂s

∂s

∂s

∂s

∂ S ] Appendix ∂s

∂S ∂F p

-1

∂ s

4 In HP: and in 48





� J 12

�∂Ri

1

∂ s

=∑r

ri∂r

∂s

� . is calculated in ∂R1S ,

∂s

=−12 ∂F

eT

∂s

F eFeT ∂Fe

∂ s Appendix

∂F e

∂ s

=F F ne∑

r [ ∂Fp

-1

∂r

∂ r

∂s] 4.

∂F p

-1

∂r

identical term 48

�

� J 21

� large deformations ∂R s

2

∂i

=−si∂ k s

∂s

� : or ∂R2 S ,

∂ S=−

∂k s

∂S=−∑

r

∂k s

∂r

∂r

∂S=−

∂ k s

∂s

∂s

∂S (

∂s

∂S according to the order of 48

� J 22

�of the jacobian matrix): L be the derived preceding ones ∂ R2

s

∂r

=sr−∂ k s

∂r

in common have narrower

terms with each behavior, which

∂ R2r

∂s

=rs−∂ kr

∂s

�are detailed for each one in this document: , Examples of matrixes of interaction general Case: It is a question of defining a symmetric square

� ∂ s

∂s

=∂ ps

∂s

s

� ∂r

∂s

=∂ pr

∂s

r

� ∂ k s

∂s

�∂ k r

∂s





Annexe 6 matrix of interaction enters sliding systems

. The simplest form (usable for all the single-crystal behaviors) consists in using

only one coefficient making it possible to distinguish the diagonal terms from the others: the matrix is then defined by H : if and if, with by default. For the families with 12 sliding systems hsr only,

hrs=1 and the r=s behaviors hrs=H r≠s MONO_VISC* H=0 ,

the matrix of interaction can be built from 6 coefficients. h1 H2 H2 h4 h5 H 5 h5 h6 h3 h5 h3 h6 H2 h1 H2 h5 h3 h6 h4 h5 h5 h5 h6 h3 H2 H2 h1 h5 h6 h3 h5 H

3:6 H 4:5 H 5:4 H 5:5 H 1 H2

H2 H 6:5

H 3:6 H 3:5 H 5:3 H 6 H2

H 1 H2

H 3:5

H 6:5 H 5:4 H 5:6 H 3 H2

H2 H 1:5 H

4:5 H 3:6 H 5:5 H 4:5 H 6:3 H 5:1 H2H2 H 6:5 H 3:6 H 5:3 H 5:5 H 4

H2H

1 H2

H 3:5 H 6:3 H 5:6 H 3:6 H 5 H2

H2

H 1:5 H 4:5 H 5:5 H 4:6 H 5:3 H 6:3H 5:1 H2 H2 H 3:6 H 5:3 H 5:6 H 5:5H 4

H2H 1

H2H 6:3 H 5:5 H 4:5 H 3:6

H 5 H2

H2 H 1 MONO_DD_KR

the shape

of

the matrix

of interaction

used

with

the model

MONO_DD_KR

and adapted

to the 24

sliding systems associated to the two centered cubic families (BCC24 ) is the following one: MONO_DD_CFC and MONO_DD_FAT For constitutive law MONO_DD_CFC, the matrix of interaction





hrs=[h2 h2 h2 h2 h1 h1 h1 h1 h1 h1 h1 h1 h2 h2 h2 h1 h1 h1 h1 h1 h1 h1 h1 h1

h2 h2 h2 h1 h1 h1 h2 h1 h1 h1 h1 h1 h2 h2 h2 h1 h1 h1 h1 h1 h1 h1 h1 h1























]

adapted to the 12 sliding systems

are obtained by the dynamics of discrete dislocations (DDD). These values are functions of the type of dislocations which interact, the following table recapitulates this information. The parameter setting of the CFC systems used is that of Schmid and Boas. For model MONO_DD_FAT, one currently uses the same matrix of interaction as that of model MONO_DD_CFC . Normal Direction Designation A2 A3 A6 B2 B4 B5 C1 C3 C5 D1 D4 D6 the correspondence between

the numbers 1 11 111 111 111

of sliding systems

011 1 0 1 11 0 01 1 10 1 1 10 0 1 1 1 0 1 1 10 0 1 1 10 1 11 0

of the octahedral

family and

the parameter setting of Schmid and Boas is the following one: Normal Direction Number (SB) 1 2 3 4 5 6 7 8 9 10 11 12 Designation A2 A3 A6 B2 B4 B5 C1 C3

C5 D1 D 1 11 111 111 111

4 D6 Numéro 011 101 110 011 10 1 110 011 101 1 10 0 1 1 1 0 1 11 0

Aster 7 9 1 2 3 4 5 6 7 8 9 10 11 12

of interaction is made of block , each

block being 7 9 8 2 1 3 12 11 10 5 4 6

here this matrix in the order of sliding systems 4×4 defined in Aster 3×3 : Sliding systems 1, 2,3 4, 5,6 7, 8,9 10, 11,12 1, 2,3 4, 5,6 7, 8,9





10, the 11,12 matrixes is in the following way

defined

: MONO_DD_CC A0 A1 A2 A3

For A0 A3 A2

constitutive law A0 A1

MONO_DD_CC A0

, 3×3 the matrix of interaction adapted to the 12 sliding systems

A0={a* a* a*

a* a* a*

a* a* a*} A1={acolinéaire aglissile aglissile

a glissile a Hirth aLomer

a glissile aLomer a Hirth}

A2={a Hirth a glissile a Lomer

aglissile acolinéaire aglissile

aLomer a glissile a Hirth} A3={

aHirth aLomer a glissile

aLomer aHirth a glissile

aglissile a glissile acolinéaire}

of type CUBIQUE1 is obtained by the dynamics of discrete dislocations (). These values are functions of the type of dislocations which interact, the following table recapitulates DDD this information. The parameter setting of the systems used is: Direction [] [] Normal () () () () () () Number 1 2 3 4 5 6 Direction CC [] [] Normal

() () () ( 111 ) 111 (

) () Number 11 0 10 1 7 8 011 9 10 011 11 101 12 110

the matrix 1 2 3 4 5 6

is made 111 of 111

block, each 110 101 block 011 being 110 101 011 a matrix

7 8 9 10 11 12

here this matrix in the order of sliding systems 4×4 defined in Aster 3×3 (family CUBIQUE1): Sliding systems 1, 2,3 4, 5,6 7, 8,9 10, 11,12 1, 2,3 4, 5,6 7, 8,9

10, the 11,12 matrixes is in the following way

defined

: Procedure A0 A4 A3 A2

of creation A4 A0 A5 A5T

of mesh A3 A5T

A0 A1

the creation A2 A5 A1 A0





of meshes 3×3 rests on a random distribution

A0={h0 h1 h1

h1 h0 h1

h1 h1 h0} A1={

h4 h3 h2

h3 h5 h3

h2 h3 h4} A2={

h4 h2 h3

h2 h4 h3

h3 h3 h5}

A3={h5 h3 h3

h3 h4 h2

h3 h2 h4} A4={

h3 h5 h3

h4 h3 h2

h4 h3 h2} A4={

h3 h3 h5

h2 h4 h3

h4 h2 h3}

Annexe 7 of germs (centers of the cells

of Voronoï). One builds the grains in the form of cells of Voronoï (using the library qhull, confer http://www.qhull.org/) But when L” one starts from a random distribution of germs, one notes that one obtains , without reprocessing, of the cells of Voronoï comprising of the very small edges sometimes, which leads to a mesh either of poor quality, or impossible to carry out being given the dispersion of the local densities. Appear 8-a. geometry of an aggregate generated with cells of Voronoï For the elimination

8-a the small edges one uses the following property: the presence of

a small edge is related to the fact that a germ is close to the sphere circumscribed with the tetrahedron having for tops 4 other germs, without being exactly on this sphere. In this case, it is enough to project this germ on the sphere to eliminate the small edge. The algorithm thus builds a random distribution of germs in an iterative way, by taking care that the position of each new germ compared to the existing circumscribed spheres is correct. (with possible projection). Then it is necessary to restrict the geometry obtained with the cube representing the V.E.R. For that one uses

the library PyXL (http://www.tc.cornell.edu/~myers/PyXL) which proposes tools (written in python and into C) to truncate cells of Voronoï on a given geometry. After this operation, it is necessary to check again that one did not create too small edges. In this case, one locally adds germs on the cells comprising the small edges, then one recomputes with these new germs the cells of Voronoï and their envelopes convex. At the end of the generation amongst required germs, one calculates the number of small edges of which the length is lower than a certain threshold like the length of the smallest edge, and one eliminates the tops from the cells quasi-confused (following the projection of a germ on a circumscribed sphere). It is then enough to generate the data necessary to a software of mesh. The geometries

of the aggregates are written with format SALOME. The geometry with format SALOME (file written in python) offers useful functionalities like the computation of volumes of the cells. The mesh is carried out with the modulus MESH. In short the creation of a mesh thus consists of the call to a single procedure python

which connects the following actions: Creation of the geometry: The procedure connects calls to moduli written in FORTRAN

�and to gain in time computation (projections on the spheres) libraries Qhull and PyXL as well as procedures python to create the geometry with the number of desired germs. This creates a file of



http://www.tc.cornell.edu/~myers/PyXL

http://www.qhull.org/



geometry to format SALOME, as well as directives of mesh. Reading of the file python and creation of the mesh in SALOME, with computation of the volume of each grain

�and the position of the centers of gravity of the grains. The mesh is written with med format for computation with Code_Aster. A complete description of this procedure is provided in [10]. It functions in 2D and 3D

� : Some examples of mesh are provided below: Appear 8-b. mesh of an aggregate with�988 grains comprising 311510 Tetra4 tetrahedrons Appears





8-c. mesh 8-bof an aggregate 2D comprising 70 cells Appears 8-d. mesh 3D of 135 cells

8-c

8-d



mono Behaviors elastoviscoplastic and polycr [] · elasticity, the criterion and the model of flow....

Documents

Transcript of mono Behaviors elastoviscoplastic and polycr [] · elasticity, the criterion and the model of flow....