Viscoplastic behavior with effect of memory []

Code_Aster Versiondefault

Titre : Comportement viscoplastique avec effet de mémoire [...] Date : 25/09/2013 Page : 1/23Responsable : GÉNIAUT Samuel Clé : R5.03.12 Révision :

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Viscoplastic behavior with effect of memory and restoration of Chaboche

Summary:

This document describes the integration of the model of behavior élasto-visco-plastic of Chaboche with anisotropic work hardening comprising a ratchet effect of work hardening and two work hardenings nonlinearkinematics, with taking into possible account of the restoration and viscosity. This model is usable by therelation VISCOCHAB keyword BEHAVIOR. The established model has an effect of work hardening on thetensorial variables of recall and takes into account all the variations of the coefficients with the temperature 1.This law of behavior is integrated by the resolution of a system of nonlinear equations. This model is availablein 3D, plane deformation, axisymetry. Modeling in plane constraint is taken into account.

1 except for the calculation of the jacobienneWarning : 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 Model élasto-visco-plastic of Chaboche with ratchet effect and restoration.........................................3

2 The model VISCOCHAB in Code_Aster...............................................................................................4

2.1 Description of the model................................................................................................................4

2.1.1 Surface of flow - potential viscoplastic.................................................................................5

2.1.2 Formulation of isotropic work hardening...............................................................................6

2.1.3 Formulation of kinematic work hardening.............................................................................7

2.2 Implicit integration.......................................................................................................................... 8

2.2.1 System of equations.............................................................................................................8

2.2.2 Outline general of resolution.................................................................................................9

2.2.3 Calculation of Jacobienne...................................................................................................11

2.3 Explicit integration....................................................................................................................... 12

2.4 Significance of the internal variables...........................................................................................14

3 Features and checking....................................................................................................................... 15

4 Identification of the parameters of the model.....................................................................................15

5 Bibliography........................................................................................................................................ 15

6 Description of the versions of the document......................................................................................16

7 Appendices......................................................................................................................................... 17

7.1 Expression of the terms of the Jacobienne matrix.......................................................................17

7.2 Calculation of tangent rigidity......................................................................................................22

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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1 Model élasto-visco-plastic of Chaboche with ratchet effectand restoration

The model elastoviscoplastic of J.L.Chaboche [bib1] most complete superimposes on isotropic workhardening two variables of kinematic work hardening, and makes it possible to take into account theeffects of cyclic hardening, creep, the restoration and the memory of largest work hardening. It is thuswell adapted to the cyclic loadings.

Under certain conditions of loading, the structures can be prone to a phenomenon of progressivedeformation, being able to harm the functional capacities of the device. In its original form, the modellargely over-estimates the progressive deformation; to improve its representativeness, it was modifiedby introducing terms of radial evanescence, in order to better model this phenomenon [bib2]. Theresulting model, named VISCOCHAB, is described in this document.

Several studies consisted in testing this law of behavior with respect to its capacity to model theprogressive deformations [bib3, bib4], while comparing with tests [bib5, bib6, bib7, bib8] and with othermodels [bib12] in particular that of Taheri [R5.03.05]. In particular, identifications of steels 316L and304L were made (resp. in [bib4] and [bib9]). Recently, the studies of thermal tiredness required the useof VISCOCHAB, in particular to take into account the effect of memory.

The model comprises 25 parameters (+ 2 elastic parameters) introduced into the orderDEFI_MATERIAU :

VISCOCHAB (VISCOCHAB_FO) = _F (# work hardening isotropic♦ K = k ,

◊ B = b ,

◊ A_R = α R , (defect: 1.)

# work hardening kinematic♦ C1 = C1 ,

♦ C2 = C2 ,

♦ G1_0 = γ 10 ,

♦ G2_0 = γ 20 ,

◊ A_I = a∞ , (defect: 1.)

# viscosity♦ K_0 = K0 ,

♦ NR = n ,

# exponential flow◊ A_K = α K , (defect: 0.)

◊ ALP = α , (defect: 0.)

# effect of memory◊ ETA = η , (defect: 0.5)

♦ DRIVEN = μ , (defect: 0.)

♦ Q_M = Qm ,

♦ Q_0 = Q0 ,

# progressive deformations (Burlet)




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◊ D1 = 1 , (defect: 1.)

◊ D2 = δ 2 , (defect: 1.)

# restoration◊ M_R = m r , (defect: 1.)

◊ G_R = γ r , (defect: 0.)

◊ M_1 = m1 , (defect: 1.)

◊ M_2 = m2 , (defect: 1.)

◊ G_X1 = γ X1 , (defect: 0.)

◊ G_X2 = γ X2 , (defect: 0.)

◊ QR_0 = Qr* , (defect: 0.)

)

These parameters are real constants. All these parameters can depend on the temperature (keywordsVISCOCHAB_FO) and the expected values are of type function. There is no value by default in thiscase. It will be noted that if C1 or C2 depends on the temperature, the calculation of the matrixjacobienne is not exact, which can make convergence local difficult if C1 or C2 strongly vary with T .

The use of this law of behavior is accessible in the orders STAT_NON_LINE or DYNA_NON_LINE bythe keyword VISCOCHAB of BEHAVIOR.

This complete model can be “degraded” by cancelling the effect of certain mechanisms (for examplethe effect of restoration). For example, by cancelling the effect of the restoration and the terms ofradial evanescence of kinematic work hardening (parameters materials by default), one finds themodel of Chaboche with effect of memory VISC_CIN2_MEMO [bib12].

For details concerning the choice of the parameters materials leading to the cancellation of certaineffects, one will be able to refer to the documentation of the order DEFI_MATERIAU.

In the continuation of this document, one describes the model precisely VISCOCHAB. One presentsthen the detail of his digital integration in link with the construction of the coherent tangent matrix.

2 The model VISCOCHAB in Code_Aster

2.1 Description of the model

At any moment, the state of material is described by the deformation , the temperature T ,

plastic deformation p , cumulated plastic deformation p and the tensor of recall X . The

equations of state then define according to these variables of state the constraint =H Id

(broken up into parts hydrostatic and deviatoric), the isotropic share of work hardening R and the

kinematic share X :




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H=

13

tr =K tr − th with

th=T−T ref

Id éq 2.1-1

=−H Id=2 − p

éq 2.1-2

R=R p éq 2.1-3

X=p , p éq 2.1-4

where K , , and coefficients of X p and R p are characteristics of material whichcan depend on the temperature. More precisely, they are respectively the modules of compressibilityand shearing, the thermal dilation coefficient, the functions of isotropic and kinematic work hardening.As for T ref , it is the temperature of reference, for which one regards the thermal deformation asbeing worthless.

2.1.1 Surface of flow - potential viscoplastic

The surface of flow associated with the model VISCOCHAB is represented within the space of principalconstraints by:•its center X , kinematic tensor of work hardening,

•its size RRk , k being its initial size and R the isotropic variable of work hardening giving the

evolution of this size, modulated by the coefficient R ,

•its form given by the criterion of Von Mises steady to -X . is the diverter of the constraints.

The evolution of the plastic deformation is controlled by a law of normal flow to a criterion of plasticityof Von Mises:

F , R ,X = −X eq−RR p −k avec Aeq= 32A: A éq 2.1-5

The viscoplastic potential of dissipation for the model of Chaboche is written:

p=K0 k R

n1 exp[ ⟨ F

K 0k R⟩n1

] éq 2.1-6

where ⟨F ⟩ is the positive part of F .

One from of deduced the law of evolution from the plastic deformation:

p=∂ p

∂= p

∂ F∂

=32p

−X

−X eqéq 2.1-7

while having posed:

p=⟨F

K 0k R⟩n

exp[ ⟨ FK 0k R

⟩n1

] éq 2.1-8

Classically, p is the speed of cumulated plastic deformation. Parameters K 0 , k and are

relating to the viscosity of the material (viscosity of Norton). The equation [éq 2.1-7] can be alsowritten in an equivalent way in the following form:

p= 23 p : p éq 2.1-9




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The unit normal on the surface of flow is noted:

n=∂F∂

∥∂ F∂

∥−1

= 23∂F∂

= 32

−X

−X eqéq 2.1-10

The laws of evolution of isotropic work hardening and kinematic work hardening are obtained startingfrom the thermodynamic potential and from the potential of dissipation.

2.1.2 Formulation of isotropic work hardening

Evolution of the isotropic variable of work hardening R is given by:

R=b Q−R pr∣Q r−R∣mrsgn Qr−R éq 2.1-11

This law of evolution of isotropic work hardening utilizes a first linear term according to the speed ofcumulated deformation viscoplastic p . This term is useful to describe the evolution of the loop(softening or hardening) in cyclic loading. This term provides an asymptotic value of R(corresponding at the stabilized state) equal to the variable Q . This variable Q represented cyclichardening (the effect of memory of the maximum deformations). It is not a constant but it depends onthe maximum amplitude of the deformation (effect of memory):

Q=Q 0Qm−Q0 1−e−2 q éq 2.1-12

One defines within the space of deformations a surface of flow inside which Q is a constant (field ofnot-work hardening):

f p , , q = 23 p− eq−q≤0 éq 2.1-13

This criterion defines a field characterizing the maximum plastic deformations, of which qmeasurement the ray and the center, calculated according to a law of normality:

p :∂ f

∂ p0 éq 2.1-14

The unit normal on the surface of flow is noted:

n*=∂ f∂

∥∂ f∂

∥−1

= 32 p−

p− eqéq 2.1-15

Laws of evolution of the variables q and are given in the form:

q= H f ⟨n: n* ⟩ p éq 2.1-16

= 321− H f ⟨n :n*⟩ p n* éq 2.1-17

The parameter (which does not exist in the initial formulation 5) allows to partially take into

account the effect of memory. If it is equal to 1/2, the initial formulation is found. If it is worth 1, q isequal to the standard of the greatest plastic deformation reached. If it is much lower than 1/2, theeffect of memory is taken into account partly only.

Note:

Same expressions for Q , f , q and are used in the laws VMIS / VISC_CIN1 / 2_MEMO[bib11] who are versions simplified (and optimized) of VISCOCHAB cf [R5.03.04].




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The law of evolution of isotropic work hardening [éq 2.1-11] utilized a second term, allowing to takeinto account the effect of the restoration. The variable Q r is given by the equation:

Q r=Q−Q r* [1−Qm−Q

Qm 2

] éq 2.1-18

Let us note that in the initial model of Chaboche [bib1], the coefficient m r in the equation [éq 2.1-11]1 is worth.

2.1.3 Formulation of kinematic work hardening

Before giving the expression of the law of work hardening of the model VISCOCHAB, one points outthe various stages which allowed its development.

The simplest law is a linear work hardening of the form:

X=23C p

éq 2.1-19

With this law, one can add a non-linear term of recall providing an effect of evanescent memory of theway of loading (initial model of Chaboche):

X=23C p−X p éq 2.1-20

with

=0 [a∞ 1−a∞ e−bp ] éq 2.1-21

It was shown that such a law largely over-estimates the phenomenon of progressive deformation. Thisbrings to introduce a term with radial evanescence (term due to Burlet and Cailletaud):

X=23C p− X: n n p éq 2.1-22

In fact, this law underestimates the phenomenon of progressive deformation now. One can thencombine the two equations [éq 2.1.20] and [éq 2.1.22] with the parameter of weighting ∈ [0,1 ] , inorder to better consider the deformations progressive:

X=23C p

−[ X p 1− X : n n p ] éq 2.1-23

In the initial model of Chaboche, one finds also a term additional which makes it possible to introducethe effects of the restoration into kinematic work hardening, which gives to final the following law:

X=23C p

− [ X 1− X :n n ] p−X [ X eq ]m−1

X éq 2.1-24

with given by the equation [éq 2.1-21].

For an exact taking into account of the dependence of the parameters materials compared to thetemperature, it is necessary to add an additional term in T , which gives:

X=23C p

− [ X 1− X :n n ] p− X [ X eq ]m−1

X1C∂C∂T

X T éq 2.1-25

with given by the equation [éq 2.1-21].




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The model VISCOCHAB proposed comprises in fact 2 variables of kinematic work hardening X1 and

X2 whose laws of evolutions are given by the equation [éq 2.1-25].

Assessment:The law of evolution of work hardening is form:

{X=X1X 2

X i=23C i

p− i [i X i1−i X i :n n ] p

− Xi [ Xi eq ]mi−1

X i1Ci

∂C i

∂TX i ˙T i=1,2

éq 2.1-26

with

i=i0 [a∞1−a∞ e−bp ] i=1,2 éq 2.1-27

2.2 Implicit integration

To numerically carry out the integration of the law of behavior, one carries out a discretization in timeand one adopts a diagram of implicit, famous Euler adapted for elastoplastic relations of behavior. It isthe method used by default. It is also possible to carry out an explicit integration (see §2.3) bychoosing the keyword ALGO_INTE=' RUNGE_KUTTA'.

Henceforth, the following notations will be employed: A− , A and A represent respectively thevalues of a quantity at the beginning and the step of time considered thus that its increment during thestep. There is thus the relation: A=A−A− . The problem is then the following: knowing the state

at time t as well as the increments of deformation (resulting from the phase of prediction

(cf. STAT_NON_LINE [R5.03.01]) and of temperature T , to determine the state of the internal

variables at time t as well as the constraints .

2.2.1 System of equations

Implicit discretization of the problem led to a system of 27 equations:

Relationstress-strain

−H − th− p∂ F

∂ =0 6 éq éq 2.2-1

Kinematicwork

hardening {i=1,2

Xi−

23Ci p

i

[i Xi1−i X i

:n n ] p

Xi [ Xi eq ]mi−1

X i t−

1C i

∂C i

∂TXi T=0

12 éq éq 2.2-2

Cumulatedplasticity

p− t ⟨F

K 0k R⟩n

exp[ ⟨ F

K 0 k R⟩n1

]=0 1 éq éq 2.2-3

Isotropic workhardening R−b Q

−R p− r∣Q r−R∣

mrsgn Q r−R t=0 1 éq éq 2.2-4

Effect ofmemory

q− H f ⟨n :n* ⟩ p=0 1 éqéq 2.2-5




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− 321− H f ⟨n :n* ⟩ p n*

=0 6 éq éq 2.2-6

In this system, i

is given according to p by the equation [éq. 2.1-27]; Q and Q r

are

obtained according to q by the equations [éq. 2.1-12] and [éq. 2.1-18].

The 27 unknown factors are: , X1 , X2 , p , R , q and .

Note:Contrary to VISC_CIN2_MEMO [R5.03.04], the equation of checking of the threshold of the maximum

deformations: f p , , q = 23 p− eq−q=0 (see [éq. 2.1-13]) is not taken into account.

2.2.2 Outline general of resolution

One calculates the constraint by making the assumption of a purely elastic increment ( p=0 ).

=H avec p=X i==0

R= p= q=0éq 2.3-1

The function threshold is calculated F= −X eq−RR

−k . If F≤0 then the increment is

purely elastic and integration is finished. If not of the viscoplastic corrections are carried out by solvingthe system (S1) according to:


−H − th− p∂ F

∂ =0 6 éq éq 2.3-2

Kinematicwork

hardening {i=1,2

Xi−

23Ci p

i

[i Xi1−i Xi

: n n ] p

Xi [ Xi eq ]mi−1

X i t−

1C i

∂C i

∂TX i T=0

12 éq éq 2.3-3

Cumulatedplasticity

p− t F

K 0k R

n

exp [ F

K0k R

n1

]=0 1 éq éq 2.3-4




Effect ofmemory

q=0 1 éq éq 2.3-6

=0 6 éq éq 2.3-7




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The 20 unknown factors are: , X1 , X2 , p , R . It is noticed that q and in

are not part of the unknown factors because their solution is commonplace.

It is also noticed that, at this stage, one can check that F0 , and like K 00 , k0 and

R0 , one formally replaced the hooks (left positive) of the equation [éq 2.2.3] by simple bracketsin the equation [éq 2.3-4].

One calculates then the dual function f “envelope surface of the maximum deformations”:

f =23ε p−ξ eq

−q éq 2.3-8

If f ≤0 then integration is finished. If not, one solves the system (S2) according to:


−H − th− p∂ F

∂ =0 6 éq éq 2.3-9

Kinematicwork

hardening {i=1,2

Xi−

23Ci p

i

[i Xi1−i Xi

: n n ] p

Xi [ Xi eq ]mi−1

X i t−

1C i

∂C i

∂TX i T=0

12 éq éq 2.3-10

Cumulatedplasticity

p− t F

K 0k R

n

exp [ F

K0k R

n1

]=0 1 éq éq 2.3-11




Effect ofmemory

q− ⟨n :n* ⟩ p=0 1 éq éq 2.3-13

− 321− ⟨n :n*⟩ p n*

=0 6 éq éq 2.3-14

The 27 unknown factors are: , X1 , X2 , p , R , q and .

It is noticed that since f 0 , then one could replace H f equations [éq 2.2-5] and [éq 2.2-6] by

1 in the equations [éq 2.3-13] and [éq 2.3-14].

The system (S1) (resp. (S2)) with 20 (resp is a non-linear implicit system. 27) equations and 20 (resp.27) unknown.One can formally write his systems: Y =0 , with:

• Y= , X1 ,X2 , p , R t

for (S1),

• and Y= , X1 ,X2 , p , R , q , t

for (S2).

These non-linear systems are solved by the iterative method of Newton (in environment PLASTIdescribes for example in [R5.03.10]):

Y k ∂∂ Y Yk

Y k1− Y k =0




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while reiterating in k until convergence and while starting with an initial solution ensuring thisconvergence.

2.2.3 Calculation of Jacobienne

According to whether one has to solve the system (S1) or (S2), one breaks up the system

Y =0 in subsystems:

Y =[g Y l Y j Y f Y r Y

] for (S1) and Y =[g Y l Y j Y f Y r Y h Y c Y

] for (S2).

Where:g represent the relation stress-strain

l represent the equations of kinematic work hardening X1j represent the equations of kinematic work hardening X2f represent the equation defining cumulated plasticity pr represent the equation defining isotropic work hardening R p h represent the equation defining the effect of memory qc represent the equations defining the effect of memory

The Jacobienne matrix of the system is the matrix J written per blocks:

J=[∂ g

∂ Δ σ ∂ g

∂ ΔX1 ∂ g

∂ ΔX2 ∂ g∂ Δp

∂ g∂ ΔR

∂ g∂ Δq

∂ g∂ Δ ξ

∂ l

∂ Δ σ ∂ l

∂ ΔX1 ∂ l

∂ ΔX2 ∂ l

∂ Δp ∂ l

∂ ΔR ∂ l

∂ Δq ∂ l

∂ Δ ξ

∂ j

∂ Δ σ ∂ j

∂ ΔX1 ∂ j

∂ ΔX2 ∂ j

∂ Δp ∂ j

∂ ΔR ∂ j

∂ Δq ∂ j

∂ Δ ξ

∂ f∂ Δ σ

∂ f∂ ΔX1

∂ f∂ ΔX2

∂ f∂ Δp

∂ f∂ ΔR

∂ f∂ Δq

∂ f∂ Δ ξ

∂r

∂ Δ σ ∂r

∂ ΔX1 ∂r

∂ ΔX2 ∂ r

∂ Δp ∂r

∂ ΔR ∂ r

∂ Δq ∂r

∂ Δ ξ

∂ h∂ Δ σ

∂ h

∂ ΔX1 ∂ h

∂ ΔX2 ∂ h∂ Δp

∂ h∂ ΔR

∂h∂ Δq

∂ h∂ Δ ξ

∂c

∂ Δ σ ∂c

∂ ΔX1 ∂c

∂ ΔX2 ∂ c

∂ Δp ∂ c

∂ ΔR ∂ c

∂ Δq ∂c

∂ Δ ξ

]




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Each term of this not-symmetrical matrix is clarified in Appendix [§7.1]. It will be noted that the terms∂ l

∂ X 1 ,

∂ l

∂ X 2,

∂ j

∂ X 1 and

∂ j

∂ X 2 do not take account of the dependence in T

(see [éq 2.3-10]).

2.3 Explicit integration

To carry out the explicit integration of the law of behavior, one uses the method of Runge-Kutta[R5.03.14]. One thus integrates directly by this method the system of 27 differential equationsaccording to:

Flow p=∂ p

∂= p

∂ F∂

=32p

−X

−X eq 6 éq éq 2.3-10

Kinematicwork

hardeningi=

p− i [ i i 1− i i :n n ] p− Xi [ X i eq ]mi−1

i i=1,2 12 éq éq 2.3-11

Cumulatedplasticity

p=⟨F

K 0k R⟩n

exp[ ⟨ FK 0 k R

⟩n1

] 1 éq éq 2.3-12

Isotropicwork

hardeningR=b Q−R pr∣Q r−R∣

mrsgn Qr−R 1 éq éq 2.3-13

Effect ofmemory

q= H f ⟨n:n*⟩ p 1 éq éq 2.3-14

= 321− H f ⟨n :n*⟩ p n* 6 éq éq 2.3-15

with:

=−H Id=2 − p

Aeq= 32A : A

F , R ,X = −X eq−R R p −k

X=X1X2

X1=23

C11 X 2=

23

C22

Q=Q 0Qm−Q0 1−e−2q




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Q r=Q−Q r* [1−Qm−Q

Qm 2

]

f p , , q = 23p− eq−q

n*= 32

p−

p− eq




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2.4 Significance of the internal variables

Internal variables of the model VISCOCHAB at the points of Gauss (VARI_ELGA) are:

in implicit 3D in 2D implicit Runge-Kutta

V1 = X 1 xx V1 = X 1 xx V1 = ε xxp

V2 = X 1 yy V2 = X 1 yy V2 = ε yyp

V3 = X 1zz V3 = X 1zz V3 = ε zzp

V4 = X 1 xy V4 = X 1 xy V4 = ε xyp

V5 = X 1 xz V5 = X 2 xx V5 = ε xzp

V6 = X 1 yz V6 = X 2 yy V6 = ε yzp

V7 = X 2 xx V7 = X 2 zz V7 = 1 xx

V8 = X 2 yy V8 = X 2 xy V8 = 1 yy

V9 = X 2 zz V9 = p V9 = 1 zz

V10 = X 2 xy V10 = R V10 = 1 xy

V11 = X 2 xz V11 = q V11 = 1 xz

V12 = X 2 yz V12 = ξ xx V12 = 1 yz

V13 = p V13 = ξ yy V13 = 2 xx

V14 = R V14 = ξ zz V14 = 2 yy

V15 = q V15 = ξ xy V15 = 2 zz

V16 = ξ xx V16 = ζ V16 = 2 xy

V17 = ξ yy V17 = 2 xz

V18 = ξ zz V18 = 2 yz

V19 = ξ xy V19 = ξ xx

V20 = ξ xz V20 = ξ yy

V21 = ξ yz V21 = ξ zz

V22 = ζ V22 = ξ xy

V23 = ξ xz

V24 = ξ yz

V25 = R

V26 = q




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V27 = p

V27 = 0

• the indicator ζ 1 is worth if the point of Gauss plasticized during the increment or 0 if not

• p represent the cumulated equivalent plastic deformation (positive or worthless)

3 Features and checking

The law of behavior is defined by the keyword VISCOCHAB (keyword factor BEHAVIOR ordersSTAT_NON_LINE, DYNA_NON_LINE, SIMU_POINT_MAT,… ). It is associated with materialsVISCOCHAB and VISCOCHAB_FO (order DEFI_MATERIAU).

The law VISCOCHAB is checked in particular by the cases following tests:

COMP002I [V6.07.102] Elementary test of robustness and reliability viscoplastic behaviors.

COMP010I [V6.07.110] Elementary validation of the taking into account of the temperature in the viscoplastic behaviors.

HSNV125D [V7.22.125] Element of volume in traction/shearing and temperature variables (comparison with other codes)

SSND105B [V6.08.105] Law of behavior visco-élasto-plastic with effect of memory.Comparison with VISC_CIN2_MEMO

SSND111A [V6.08.111] Effect of memory in a cyclic test. Comparison withVISC_CIN2_MEMO

SSNV118 [V6.04.118] Tensile test shearing with the viscoplastic model of Chaboche.Comparison with the software SIDOLO

4 Identification of the parameters of the model

The model suggested being very close to that of Chaboche, one will be able to refer to [bib4] for theidentification of the parameters of the initial model of Chaboche.

For the identification of the additional parameters, the reference [bib4] presents the tests used tosupplement the identification of steel 316 SPH.

More recently, the identification of steel 304L was carried out without taking account of the phenomenaof restoration and progressive deformation [bib10].

5 Bibliography

1. J.LEMAITRE, J.L.CHABOCHE, Mechanics of solid materials. Dunod 1996

2. L. BONNAFOUX, “Modification of the viscoplastic model of Chaboche for the modeling of thephenomenon of progressive deformations and establishment in the code finiteelements Aster”, Report interns EDF R & D, 1993

3. P. GEYER, “Modeling of the phenomena of progressive deformation by the elastoplasticmodel of Chaboche modified by Burlet and Cailletaud”,Note HT-26/93/052/A (H-T20-1994-00254-FR in Eureka), 1994




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4. P. GEYER, “Study and elastoviscoplastic modification of the model of Chaboche to improvethe description of the ratchet 2d”. Note HT-26/95/032/A (H-T20-1995-05189-FR inEureka), 1995

5. P. GEYER, “Confrontation of the model of Chaboche modified by Burlet and Cailletaud withthe tests of pressure-traction of the ECA”, HT-26/94/006/A (H-T20-1994-02201-FR inEureka) Notes, 1994

6. P. GEYER, “Digital simulation of a test bi--tube with the elastoplastic models of Chaboche andTaheri”, HT-26/94/023/A (H-T20-1995-00954-FR in Eureka) Notes, 1995

7. P. GEYER, “Evaluation of the models of Chaboche and Burlet by the calculation of testCOTHAA nA8”, HT-C2/96/010/A (H-T00-1997-00320-FR in Eureka) Notes, 1997

8. P. GEYER, “Evaluation of the models of Chaboche and Burlet by the calculation of the castsolid sleeve tested on the loop CUMULUS in progressive deformation”, HT-C2/96/047/A (H-T00-1997-02043-FR in Eureka) Notes, 1997

9. P. GEYER, “Characterization of steel 304L used during the tests “progressive deformation” onCUMULUS and identification of the parameters of the model of Chaboche”, HT-26/93/040/A (H-T20-1993-03153-FR in Eureka) Notes, 1993

10. F. CURTIT, “Identification of a law of behavior of the Chaboche type with effect of memory ofwork hardening for steel 304L with 20°C and 300°C”, H-T26-2007-3264, 2008 Notes

11. J.M. PROIX, “Relations of behavior élasto-visco-plastic of Chaboche,Reference document of Code_Aster “, [R5.03.04], 2007

12. P.GEYER, J.M.PROIX, S.JAYET-GENDROT, S.TAHERI, “With 3D elastoplastic cyclicconsitutive law for the description of ratchetting of 316 stainless steel”, SMIRT 1995

6 Description of the versions of the document

Version Aster

Author (S) orcontributor (S),organization

Description of the modifications

9.4 S. GeniautEDF/R & D /AMA

Initial text, law VISCOCHAB

10.5 J.M.ProixEDF/R & D /AMA

Correction of the § significance of the internal variables andaddition of the § features and checking




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7 Appendices

7.1 Expression of the terms of the Jacobienne matrix

In calculations presented here, one will omit the exhibitor “+” to indicate the quantities at the currentmoment (fine of the step of time).

The Jacobienne matrix of the system is the matrix J written per blocks:

J=[∂ g

∂ ∂ g

∂ X1∂ g

∂ X2 ∂ g

∂ p ∂ g

∂ R ∂ g

∂ q ∂ g

∂

∂ l∂

∂ l∂ X1

∂ l∂ X2

∂ l∂ p

∂ l∂ R

∂ l∂ q

∂ l∂

∂ j

∂ ∂ j

∂ X1∂ j

∂ X2 ∂ j

∂ p ∂ j

∂ R ∂ j

∂ q ∂ j

∂

∂ f∂

∂ f∂ X1

∂ f∂ X2

∂ f∂ p

∂ f∂ R

∂ f∂ q

∂ f∂

∂r

∂ ∂ r

∂ X1∂ r

∂ X2 ∂ r

∂ p ∂ r

∂ R ∂ r

∂ q ∂ r

∂

∂ h∂

∂ h∂ X1

∂h∂ X2

∂ h∂ p

∂ h∂ R

∂ h∂ q

∂h∂

∂ c

∂ ∂ c

∂ X1∂ c

∂ X2 ∂ c

∂ p ∂ c

∂ R ∂ c

∂ q ∂ c

∂

]

terms related to the elastic relation stress-strain:

g Y =−H − th− p∂F

∂

∂ g∂

= I dH ∂2 F∂2 p

∂ g

∂ Xi =H ∂

2F∂X i∂ p

∂ g

∂ p =H ∂F∂

∂ g

∂ R =0,

∂ g

∂ q =0,

∂ g

∂ =0




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terms related to kinematic work hardening:

k i Y =X i−23Ci

pi

[i X i1−i Xi

: n n ] p Xi [ Xi eq ]m i−1

X i t i=1,2 with:

n= 23∂F∂

= 32

−X −X eq

and p= p

∂F∂

∂ k i∂

=−23C i p

∂2 F∂ 2

i231− i p [X i :

∂ F∂

∂2F∂ 2

∂2 F∂ 2

:Xi⊗∂ F∂ ]∂ k i

∂ Xi =Id−

23C i p

∂2F∂X i∂

i p {i Id23 1− i [Xi :∂ F∂ ∂

2 F∂X i∂

∂ F∂

⊗∂ F∂

∂2F∂X i∂

:X i⊗∂ F∂ ]} Xi t [ X i eq]

mi−1Id Xi t mi−1

∂ X i eq∂Xi

[ X i eq ]mi− 2

Xi

∂ k i∂ X j j≠i

=−23C i p

∂2F∂ X j∂

i231−i p [Xi :∂ F∂ ∂

2 F∂X j∂

∂2 F∂X j∂

:X i⊗∂F∂ ]∂ k i

∂ p =−

23C i∂ F∂

i' p pi p [iXi

231−i X i:

∂F∂

∂ F∂ ]

∂ k i

∂ R =0,

∂ k i∂ q

=0,∂ k i

∂ =0




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terms related to cumulated plasticity:

f Y = p− t F

K0k R

n

exp [ F

K0 k R

n1

]

∂ f

∂ =− t F

K 0k R n−1

exp[ FK 0k R

n1

] 1K0k R [n n1 F

K 0k R n1

]∂F∂∂ f

∂ X i =− t F

K 0k R n−1

exp[ FK 0k R

n1

] 1K0k R [n n1 FK 0k R

n1

] ∂ F∂Xi∂ f

∂ R = t F

K 0k R n−1

exp[ FK 0k R

n1

] 1K 0kR

×[n n1 FK 0k R

n1

]RkF

K 0k R ∂ f

∂ p =1, ∂ g

∂ q=0, ∂g

∂ =0

terms related to isotropic work hardening:

r Y = R−b Q−R p−r∣Q r

−R∣

mr sgn Qr−R t

∂ r

∂ =0

∂ r

∂ Xi =0

∂ r∂ p

=−b Q−R

∂ r

∂ R =1b prmr∣Q r−R∣

mr−1 t

∂ r

∂ q =−b p Q' q −rmr∣Qr−R∣

mr−1Q r

' q t

avec {Q ' q =2 Qm−Q0 e

−2muq

Q r' q =Q ' q [1−2Qr

*Qm−Q

Qm2 ]

∂ r

∂ =0




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dependent terms in keeping with field relating to the effect of memory:

h Y = q− ⟨n :n* ⟩ p with n*= 32

p−

p− eqHere, one makes an approximation, by considering that ⟨n :n* ⟩= n :n*

.

∂h

∂ =− p

1

p− eq[∂2F

∂2: p− p

∂2 F

∂ 2:∂ F

∂

−32

p

[ p− eq ]2 ∂F∂ : p− ∂

2 F

∂2: p− ]

∂h

∂ Xi =− p 1

p− eq[∂2 F∂X i∂

: p− p ∂2 F∂X i∂

:∂F∂

−32

p

[ p− eq]2 ∂F∂ : p− ∂

2 F∂ X i∂

: p− ]∂h

∂ p =−

1

p− eq[∂F

∂: p− p ∂ F∂ :

∂F

∂ −32

p

[ p− eq]2 ∂ F∂ : p−

2]

∂h

∂ R =0

∂h

∂ q =1

∂h

∂ =

p− eq p[∂ F∂ 3

2n :n*

p−

p− eq ]




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terms related to the center of the field relating to the effect of memory:

c Y =− 321− ⟨n :n*⟩ p n*

Here, one makes an approximation, by considering that ⟨n :n* ⟩= n :n*

.

∂c∂

=−32

1− p

p− eq[∂

2F∂ 2

: p−⊗ p− p− eq

p n :n*∂2F∂ 2

−3 p

[ p− eq ]2n :n*∂

2F

∂2: p−⊗p− p

p− p− eq

⊗∂ F∂ :∂2 F

∂ 2 ]∂ c

∂ Xi =−

32

1− p

p−eq[ ∂2 F∂Xi∂

: p− ⊗p− p− eq

p n :n*∂2 F∂X i∂

−3 p

[ p− eq ]2n :n*∂

2F∂X i∂

: p− ⊗p− pp− p− eq

⊗∂F∂ :∂2F∂X i∂ ]

∂ c∂ p

=−32

1− 1

p− eq[ n :n* p− p

p− eq∂F∂ :∂F

∂ p− p n :n* ∂F

∂−3 p

p−eqn :n*

2 p− ]

∂ c∂ R

=0

∂ c∂ q

=0

∂ c

∂ =Id

321− p

p− eq [∂F∂

⊗p−p−eq

n:n*Id−3 n :n*p−p−eq

⊗p− p− eq ]

=Id32

1− p

p− eq[ n⊗n*n:n* I d−3 n :n* n*⊗n* ]




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7.2 Calculation of tangent rigidity

The iterative diagram of total Newton (to ensure balance) requires to know the tangent operator of the

system assembled at the end of each increment. The tangent operator is noted Mc=∂

∂ . It is

possible to determine this operator starting from the terms of the jacobienne J local system, alreadypreviously calculated (see §7.1).

Indeed, the system Y =0 is checked at the end of the increment and for a small variation of

, by considering this time like variable, the system remains with balance and thus one checks

d =0 .By differentiation, one obtains:

For the system (S1):

∂

∂ d ∂

∂ d ∂

∂ X i d Xi

∂

∂ p d p ∂

∂ R d R =0

For the system (S2):

∂

∂ d ∂

∂ d ∂

∂ X i d Xi

∂

∂ p d p ∂

∂ R d R ∂

∂ q d q ∂

∂ d =0

In the continuation, the presentation is limited to the system (S1), but the approach is identical for thesystem (S2).

One rewrites the system by putting the ends in in the member of right-hand side:

∂

∂ d ∂

∂ Xi d X i

∂

∂ p d p ∂

∂ R d R =− ∂

∂ d

With the notations defined previously, this system is written in matric form:

J . d Y =− ∂

∂ d .

However ∂

∂ =

∂ g

∂ ∂ k i∂ ∂ f

∂ ∂ r

∂

=−H000

where H represent the elastic module.




222ffbb8f526

From where J .d Y =H d 000

.

The idea then consists in writing this system per blocks, while separating d other variables

Z=d Xi , d p , d R t

, which gives:

[J J Z

J Z JZZ]d Z =Hd

0

By calculating the complement of Schur of JZZ , it is found that:

[J −J Z JZZ −1JZ ] d =H d ,

from where the expression of the tangent operator:

Mc=∂

∂ =d d

=[J −J Z JZZ−1JZ ]

−1H

Note:

Since J is not symmetrical, the tangent operator Mc is not it either. It however is

symmetrized in Code_Aster.


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Viscoplastic behavior with effect of memory []