numerical simulation of damage in metal forming processes using abaqus

2002 ABAQUS Users’ Conference 1

Numerical Simulation of Damage in Metal Forming Processes Using ABAQUS

K. Saanouni, A. Cherouat, Ph. Lestriez , and H. Borouchaki

GSM/LASMIS Université de Technologie de Troyes, B.P. 2060

10010 Troyes Cedex email: [email protected]

Abstract: In this work, fully coupled thermo-elastoplastic-damage constitutive equations accounting for

both combined isotropic and kinematic hardening as well as the ductile damage at large plastic strain, are

implemented into the general purpose Finite Element code ABAQUS. First, the formulation of the fully

coupled anisotropic constitutive equations in the framework of Continuum Damage Mechanics is

presented. The particular case of the fully isotropic and isothermal flow is presented concerning both the

plastic flow and the damage. The associated numerical aspects concerning the local integration of the

coupled constitutive equations as well as the (global) equilibrium integration schemes are presented. The

local integration is performed thanks to the fully implicit iterative and asymptotic scheme of Newton type

together with a reduction in the number of integrated ordinary differential equations. This has been

implemented in both ABAQUS/Std and ABAQUS/Explicit using the user subroutines Umat and Vumat. With

ABAQUS/Std, which uses a fully implicit resolution strategy, a special care is given to the consistent

stiffness matrix calculation. For ABAQUS/Explicit, which uses a dynamic explicit resolution strategy, the

adaptive control of the time step is discussed within the same implicit local integration scheme as in

ABAQUS/Std. The numerical implementation of the damage is made in such a manner that calculations can

be executed with or without damage effect, i.e. coupled or uncoupled calculations. Some numerical results

are presented to show the capability of the proposed methodology to predict the damage initiation and

growth during various bulk and sheet metal forming processes.

1. Introduction

Ductile (or plastic) damage often occurs inside the deformed parts due to the large plastic flow (strains and rotations) experienced during metal forming processes. This leads to the initiation of highly localization zones and consequently to the onset of internal micro voids and micro cracks inside the deformed part. Accordingly, during the numerical simulation of metal forming processes, it is crucial to consider the damage occurrence and its effect on the other thermomechanical fields in order to improve the simulated results. This allows the construction of very useful numerical tools, which helps engineers to perform virtually any forming process before its physical realization, in order to avoid the damage occurrence and get an undamaged formed part as in forging or deep drawing. The same methodology can be used in order to enhance the damage occurrence (in intensity and direction) in order to simulate the metal cutting as in blanking or orthogonal cutting by chip formation processes.

2 2002 ABAQUS Users’ Conference

The ‘fully coupled’ terminology used here means that, at each load increment, the damage affects the other thermomechanical fields and modify deeply their distribution and evolution inside the deformed part (Lemaitre and Chaboche, 1990; Lemaitre, 1992; Saanouni et al, 1994). This kind of approach has been employed by many authors using damage models based either on Gurson's type theory (Gelin and Predeleanu, 1985; Onate and Kleiber,1988; Gelin, 1990; Aravas, 1986; Bontcheva and Iankov,1991; Boudeau and Gelin, 1994; Bennani and Oudin, 1995; Brunet et al,1996; Picart et al, 1998; Brunet et al, 2001), or on Continuum Damage Mechanics (CDM) ( Lee et al, 1985; Mathur and Dawson, 1987; Zhu et al, 1992; Zhu and Cescotto, 1995; Saanouni and Franqueville, 1999; Hammi, 2000; Saanouni et al, 2000; Saanouni et al, 2001a; Saanouni et al, 2001b; Saanouni et al, 2001c). These fully coupled approaches allow the prediction of, not only the large transformation of the processed part as large deformations, rotations, and evolving boundary conditions, but also indicate where and when the damaged zones can appear inside the formed part. This gives a powerful tool for the industrial metal forming optimization by choosing the process technological parameters in such a manner that the damage is avoided.

In some other processes, such as shaving, blanking or orthogonal cutting, ductile damage is designed into an operation by generating controlled damaged zones (or macroscopic cracks) through the intensively deformed zones. Again, the fully coupled approaches have been shown their ability to numerically simulate accurately these sheet metal operations (Abdali et al, 1995; Homsi et al, 1996; Bezzina and Saanouni,1996; Brokken et al, 1998; Fauron et al, 1998; Samuel, 1998; Saanouni and Franqueville,1999; Saanouni et al, 2000; Hambli, 2001; Saanouni et al 2001).

This article summarizes the work down in this field in the Laboratory of Mechanical Systems at the University of Technology of Troyes (France) since 1996 using the FE codes ABAQUS/Std and ABAQUS/Explicit. First the theoretical foundations of the finite elastoplastic constitutive equations coupled to the continuum damage are reviewed and effects of damage on both elastic and inelastic behaviour discussed. Even if multiple surfaces formulations have been worked out, we limit ourselves to the single surface formulation. Numerical formulation of the fully coupled problem is discussed on the light of its implementation into ABAQUS FE codes. Various selected examples dealing with the prediction of the ductile damage in sheet and bulk metal forming processes are presented.

2. Coupled constitutive equations for metal forming

2.1 Kinematical framework

The classical framework of the thermodynamics of irreversible processes with state variables is used to derive the finite thermo-elastoplastic constitutive equations fully coupled with the ductile damage. The transformation gradient F between the initial (undeformed and undamaged) configuration and the current (deformed and damaged) one is multiplicatively decomposed so that the following definitions hold :

TpeFF=BandFF=F ⋅⋅ (1)

W+D=FF=L1−

⋅� (2)

)L+L(2

1=D

T

and )L-L(2

1=W

T

(3)


where Fe and Fp stand for the elastic and plastic parts of the deformation gradient; B is the total Eulerian left Cauchy-Green deformation tensor; L is the spatial velocity gradient in the current configuration, D and

W are respectively the pure strain rate and the material spin tensors. The superimposed dot (� ) denotes the

usual time derivative.

To satisfy the objectivity requirement, the so-called (and widely used) Rotated Frame Formulation (RFF) is adopted. This leads to express the constitutive equations in a rotated configuration obtained from the current one by an orthogonal rotation tensor Q defined by :

Q

TW=QQ �

⋅ with (Q (t=0)= I ) (4)

where WQ defines the material rotation rate tensor to be properly chosen in order to define the desired objective rotational rate. Accordingly, for any second order tensor T, the objective rotational derivative with respect to the rotating frame is given by :

TWWTTtD

TD

QQ

Q

Q⋅−⋅+=

� (5)

from which the classical Jaumann and Green-Nagdhi rotational derivatives can be obtained. On the other

hand the rotated tensor Q

T is given by :

QTQTT

Q⋅⋅= (6)

Its time and rotational derivatives are related by :

QtD

TDQT

Q

QT

Q⋅⋅=

� (7)

Throughout this paper the constitutive equations are formulated in the rotated configuration and all the tensorial quantities are « rotated » by Eq. (6) without using any special notation, i.e. the subscript Q will be omitted from the tensorial notations.

Finally, by supposing that the elastic transformation gradient Fe is infinitesimal, the spatial total strain rate tensor (Eq. (3.a)) can be additively decomposed as :

pJ

eDD +ε≈ � (8)

where �εe

J is the Jaumann derivative of the elastic strain tensor defined by Eq. (5) in which WQ is taken

equal to the material spin defined by Eq. (3b) (the subscript J will be removed for simplicity). P

D is the

plastic strain rate tensor defined by the constitutive equation as will be presented in the next section (see Eq. (26)).


2.2 State variables versus effective state variables

In the present work the formulation based on a single (or unified) yield function and dissipation potential to derive all the dissipative phenomena supposed anisotropic is used (Saanouni et al, 1994; Hammi, 2000). Dealing with the simple first displacement gradient theory two couples of observable (or external) state variables are used :

� Total strain tensor associated with the Cauchy stress tensor (ε,σ);

� Absolute temperature associated with the specific entropy (T,s).

To describe the plastic behavior with damage in the framework of a single surface formulation, five couples of internal state variables are taken into account:

� Small elastic strain representing the elastic flow associated with the Cauchy stress tensor (εe,σ);

� Normalized heat flux vector associated to the gradient of the absolute temperature (q/T, g);

� Isotropic hardening variables (r, R) representing the size of the yield surface in strain space (r) and stress space (R);

� Purely deviatoric tensorial kinematic hardening variables (α, X) representing the displacement of

the center of the yield surface in strain space (α) and stress space (X),

� Second order tensorial damage variables (d, Y), where d is the damage tensor and Y is its generalized thermodynamic force (i.e. dual variable).

Suppose that the current configuration at time t contains a given distribution of micro-defects as voids and/or micro-cracks. The concept of the effective stress (Chaboche, 1988; Lemaitre et Chaboche, 1990) together with the hypothesis of total energy equivalence (Saanouni et all 1994) can be used to define the effective state variables by:

eTe1:M

~and:M

~ε=εσ=σ

−

(10)

α=α=−

:N~

andX:NX~ T1

(11)

rd1r~

and

d1

RR~

−=

−

= (12)

where Mijkl(d)and Nijkl(d) are fourth order symmetric and positive definite operators describing the damage effect on Cauchy stress and kinematic stress respectively (Lemaitre and Chaboche, 1990; Voyiadjis and Kattan 1999; Hammi, 2000). They reduce to fourth order unit tensor if the damage vanishes (i.e d = 0) and they decrease as the damage tends to its critical (maximum) value at the final fracture of the RVE. The

notation d in Eq. (12) defines an appropriate Euclidean norm of the second order damage tensor.


In the next section, the above defined effective state variables will be used as arguments in the state and dissipation potentials in order to derive the complete set of fully coupled constitutive equations (Saanouni et al, 1994).

2.3 Fully coupled constitutive equations

The Helmholtz free energy ψ(εe ��,α�,r,�d,Τ�) is taken as a state potential, additively decomposed into two contributions; namely: thermo-elastic/damage, and plastic/damage terms:

)T;d,r,()T,d,( pd

e

ted αρψ+ερψ=ρψ (13)

whith

2

0

0

ve

0

eee

ted)TT(

T

C

2

1~:k)TT(~::~

2

1)T,d,( −ρ−ε−−εΛε=ερψ (14)

2

pd r~Q2

1~:C:~

2

1)T;d,r,( +αα=αρψ (15)

where ρ = ρt is the material density in the current deformed and damaged configuration and the variables

located after the (;) act as simple parameters. Λijkl is the symmetric fourth order elasticity operator, kij is the symmetric second order thermal operator, Cijkl is the symmetric fourth order kinematic hardening operator. They are all supposed symmetric, positive definite and temperature dependent parameters. Q is the scalar isotropic hardening modulus, T0 is the reference absolute temperature and Cv is the classical specific heat parameter.

From this potential, the state relations (or the stress-like variables) are obtained according to the first consequence of the Clausius-Duhem inequality (Saanouni et al, 1994):

k~)TT(:

~0

e

e−−εΛ=

ε∂

ψ∂ρ=σ (16)

)TT(T

C:k

~1

Ts

0

0

ve−+ε

ρ=

∂

ψ∂−= (17)

α=α∂

ψ∂ρ= :C

~X (18)

rQ~

rR =

∂

ψ∂ρ= (19)

)YYY(d

Yike

++−=∂

ψ∂ρ−= (20)


e

0

ee

e:

d

k~

)TT(:d

~

:2

1Y ε

∂

∂−−ε

∂

Λ∂ε= (20.a)

α∂

∂α= :

d

C~

:2

1Y

k (20.b)

2

ir

d

Q~

2

1Y

∂

∂= (20.c)

It is remarkable to note that in the state relations above the main physical properties of the material are affected by the ductile damage according to:

damaged elasticity moduli operator : T

M::M~

Λ=Λ (21)

damaged kinematic hardening moduli operator : T

N:C:NC~= (22)

damaged thermal properties : k:Mk~ T

= (23)

damaged isotropic hardening modulus : ( )Qd1Q~

−= (24)

It is also worth noting that at the virgin (undamaged) state of the RVE all these effective physical properties reduce to those of the virgin undamaged material. While at the final fracture of the RVE these properties decrease as indicated by the state relations above (Eqs. (21) to (24)).

The second consequence from the Clausius-Duhem inequality gives the so-called ‘residual’ inequality defining the total volumetric dissipation inside the RVE. This latter can be classically decomposed into a

mechanical (intrinsic) term and thermal term: 0thm≥℘+℘=℘ so that :

0T.and0d:YrR:XD:

thpm≥−=℘≥+−α−σ=℘

qg�� (25)

In this expression, the stress-like variables (i.e.�: σ, ��X�, R�, Y) are given by the state relations (Eqs. (16) to (20)), the fluxes or rate variables should be defined by using the generalized standard materials framework. This will be achieved by introducing both yield function and dissipation potential in the framework of the normal but non-associative formulation. Assuming the uncoupling hypothesis between mechanical and thermal dissipation one can derive the heat flux vector q from an appropriated dissipation potential (time dependent phenomena), and the flux variables concerning the plasticity, hardening and damage by:

nfF

Dp

λ=σ∂

∂λ=

σ∂

∂λ= �� (26)


αλ−=∂

∂λ−=α �� aD

X

F p

(27)

( )r~b1

d1R

Fr −

−

λ=

∂

∂λ−=

�� (28)

[ ]S

Y:JM::M

Y

Fd

Tβ

λ=∂

∂λ−= �� (29)

where the yield function f and the dissipation potential F are chosen as :

0R~

X~~

fy

s

<σ−−−σ= (30)

[ ] Y:J:YM::M2

1R~

Q

b

2

1X~

:C:X~

a2

1fF

T21β−

+++= (31)

in which σy is the initial radius of the yield surface, a and b are material constants governing the non-

linearity of the kinematic and isotropic hardening respectively. The parameter β affects the non-linearity of the damage evolution, while the symmetric fourth order operator Jijkl governs the anisotropy of the damage growth. The effective stress norm used in Eq. (30) is defined by the following quadratic form :

( ) ( ) ( ) ( )X:H~

:XX~~

:H:X~~

X~~

s

−σ−σ=−σ−σ=−σ (32)

where, for the sake of simplicity, we have supposed that the damage effect on both Cauchy stress σ and kinematic internal stress X is the same (i.e. Mijkl = Nijkl in Eq. 11). The fourth order symmetric and positive definite operator Hijkl define the anisotropy of the plastic flow for the undamaged RVE. For the damaged state this operator transforms to:

1T

M:H:MH~

−−

= (33)

which tends to zero as the damage approaches its critical value in a given direction. This leads to zero component of the stress inside the fully damaged RVE for the direction under concern.

The normal n to the yield surface f=0 used in Eq. (26) is defined at each time by :

1n:H:nnwithX~~

)X~~(:H:MFf

n1

s

T

==

−σ

−σ=

σ∂

∂=

σ∂

∂=

−

−

(34)

The plastic multiplier derives from the kuhn-Tucker condition; i.e. 0f0,0f =λ≥λ≤ �� and leads to :

THD:~

:nH

1

T

�� +Λ=λ (35)


where the scalars H (elastoplastic tangent modulus) and HT (thermal contribution) are given by :

( ) [ ]

01

d

d

)d1(

R~

2

1k:

d

M)TT(:C:

d

M:M2::

d

M:M2:n

X~~

)X(:d

H~

:)X(

2

1:

Y

FR~

bn:XaQn:C~~

:nH

T

0

TeT

s

>

∂

∂

−+

∂

∂−−α

∂

∂−ε

Λ

∂

∂

+

−σ

−σ∂

∂−σ

∂

∂++−++Λ=

(36)

TT

Q

Q

Rk~

k:T

M

T

k:M:n)TT(:M:

T

C:M

C:T

M:M2:n:M:

T:M:

T

M:M2:nH

y

T

T

0

T

TeTT

T

∂

σ∂−

∂

∂+

−

∂

∂+

∂

∂−−α

∂

∂

+

∂

∂+ε

∂

Λ∂+Λ

∂

∂=

(37)

The continuous elastoplastic tangent operator ep

C (usually fourth order non-symmetric tensor) and its

thermal contribution T

C (symmetric second order symmetric tensor) are obtained from the time derivative

of the Cauchy stress tensor (Eq. (16)). This gives:

TCD:CTep �� +=σ (38)

where the closed form of the tangent operators are given by :

( ) ( ) ( )

∂

∂−−ε

Λ

∂

∂

∂

∂⊗

Λ−

Λ⊗Λ−Λ= k:

d

M)TT(::

d

M:M2:

Y

F

H

n:~

H

n:~~

:n~C

T

0

eTep (39)

and

k~

k:T

M

T

k:M)TT(:M:

T:M:

T

M:M2

k:d

M)TT(::

d

M:M2:

Y

Fn:

~

H

HC

T

T

0

eTT

0

eTTT

−

∂

∂+

∂

∂−−ε

∂

Λ∂+Λ

∂

∂

+

∂

∂−−ε

Λ

∂

∂

∂

∂+Λ=

(40)


It is worth noting that the same type of tangent moduli can be obtained for the internal stresses X, Y and R. On the other hand, it is easy to check that, when the damage vanishes, the classical non associative thermo-elastoplastic constitutive equations are recovered (Saanouni et al, 1994).

The temperature distribution inside the deformed part is defined by the classical heat equation derived from the Fourier thermal potential together with the energy balance. One gets:

∂

∂−

∂

∂+α

∂

∂+ε

∂

σ∂−χ−℘−ρ=− d:

T

Yr

T

R:

T

X:

TTTC)(div

em

v

��q (41)

where χ is the quantity of internal (volumetric) source of heat, while m

℘ is the intrinsic dissipation defined

by Eq. (25a). Appropriated thermal initial and limit conditions are required in order to define the heat exchange between the deformed part and the tools.

An important case for the bulk metal forming concerns the fully isotropic phenomena, i.e. the initial isotropy of the thermo-elastic behavior and the full isotropy of the plastic yield (Mises type), mixed hardening and damage yielding (represented by the scalars D and Y). In that case, the physical properties of the undamaged RVEs write:

� For the thermo-elastoplastic properties

1k,12

3H,1

3

2C

11K1G21112

dd

e

d

ee

α===

⊗+=⊗λ+µ=Λ

(42)

� For the damage properties

d1

3

2J = (43)

where 113

111

d

⊗−= is the deviatoric fourth order unit tensor and the isotropic elastic properties

(λe, Κe, Ge, µ) are expressed in terms of the Young’s modulus E and the Poisson’s ratio ν as:

3

23

)21(3

EK,

)1(2

EG,

3

G2K3

)21)(1(

Ee

ee

ee

e

µ+λ=

ν−=

ν+=µ=

−=

ν−ν+

ν=λ (44)

The damage effect tensors (see Eqs. (10), (11)) become:

( ) Ddand1D1NM =−== (45)

We close this section by recalling that the contact/friction between the deformed part and the tools plays an important role in metal forming. In this work, the Coulomb friction model together with the master slave surface methodology available in ABAQUS are used without any modification.


3. Numerical aspects

The above ‘fully’ coupled constitutive equations together with appropriated initial and limit conditions, define a highly nonlinear Initial and Boundary Value Problem (IBVP). This latter has to be solved numerically by combining the time and space discretizations in the framework of the FE code ABAQUS.

For the sake of shortness, instead of discussing the detailed aspects of the variational formulation, we limit ourselves to discussing briefly the implementation of the coupled constitutive equations (developed above) into ABAQUS/Std and ABAQUS/Explicit using the Umat and Vumat user’s subroutines. Just, lets recall that after the classical space and time discretizations neglecting the thermal effects (only mechanical phenomena will be considered in this work), the IBVP writes :

[ ]{ } { } { } { } { }extint

FFwith0UM −=ℜ=ℜ+�� (46)

where [ ] [ ]∑=

e

e

MM is the consistent mass matrix and { } { } { } { }( )∑∑ −=ℜ=ℜ

e

e

ext

e

int

e

eFF is the so-called

quasi-static equilibrium residual. These can be written on the reference element (Vr) under :

[ ] [ ] [ ] r

v

T

V

)e(dVJNNM

r

∫ρ= (47)

{ } [ ] { } [ ] { } r

cc

e

c

Te

c

r

v

eT

V

ee

intdJBdVJBF

r

c

r

Γσ+σ= ∫∫Γ

(48)

{ } [ ] { } [ ] { } r

s

eT

er

v

e

v

T

V

ee

extdJFNdVJfNF

rr

Γ+= ∫∫Γ

(49)

where the superscript (r) refers to the reference element expressed in the local (or natural) coordinates

system ),,( ςηξ , while Jv, JS and Jc are the determinants of the Jacobian matrices transforming Ve to Vr,

e

Γ to r

Γ and e

cΓ to r

cΓ respectively. The boundaries

uΓ ,

fΓ and

cΓ , are those where the displacement u,

the traction force F and the contact forces are prescribed. The field fv represented the body forces applied to the deformed part. [B] is the geometric or strain-displacement matrix in the current configuration, [N] is the

matrix of the nodal interpolation functions both associated to the solid element. The stress tensor σ is

obtained from the numerical integration of the coupled constitutive equations, while σc represents the contact stress tensor at nodes where the contact (with or without friction) takes place.

The nonlinear algebraic system Eq. (46) expresses the dynamic equilibrium of the workpiece and tools at each time step. It can be solved either by iterative Static Implicit methods (SI) using ABAQUS/Std if the acceleration is neglected, or by a Dynamic Explicit method (DE) using ABAQUS/Explicit. In both cases the local integration of the constitutive equations developed above should be performed. This will be briefly discussed, in this work, only in the case of the fully isotropic model defined by the Eqs. (42) to (45) for the sake of simplicity.


3.1 Local integration scheme

In what follows, we shall restrict our attention to the most widely used implicit incremental integration scheme named Return Mapping Algorithm RMA (see the recent monographs by Crisfield, 1991 and 1997; Simo and Hughes, 1998 and references given there for more details). In this scheme, the incremental time

integration of the time-independent constitutive equations over the time step [ ]nn1nnttt,t ∆+=

+ is

regarded as a strain-driven process in which the total strain is the basic variable. From the FEA point of view, this integration procedure is performed at each time step and for each Gauss point inside each element.

Keeping in mind the isotropy of the elasticity, the elastic predictor stage is defined by the so-called ‘trial’ state in which the trial stress, using the classical compact tensorial notation, is defined by:

( ) [ ]d

ne1nen

p

n1nn

trial

1nG21:K)D1(:)D1( ε−ε−=ε−εΛ−=σ

+++

�

(50)

in which nn1n

ε∆+ε=ε+

is completely known as all other variables containing the subscript (n) relative to

the beginning of the time increment. The deviatoric strain tensor p

n

d

1n

d

n ε−ε=ε+

�

is also entirely known

over the time step.

When replaced in the single yield surface (see Eq. 39) one get the ‘trial’ yield function:

y

n

ns

n

trial

1n

trial

D1

RX

f σ−

−

−−σ

=

+

(51)

If ftrial < 0, then the trial solution holds (elastic step) and a new time step is outworked with the updated

variables: p

n

p

1n

p

n

p

1n

p

n

p

1n

p

n

p

1n

trial

1n1nDDandRR,XX,, ===ε=εσ=σ

++++++.

If ftrial is equal or greater than zero, the trial solution above cannot be a solution to the incremental problem. This means that over the actual time step the problem is incrementally plastic characterized by

0)D,R,X,(f1n1n1n1n

=σ++++

and ∆λn+1>0. The objective is then to determine the unknowns

1n1n1n1n1nandD,R,X,

+++++λ∆σ at the end of the time step by using the RMA.

This procedure can be directly applied to the overall constitutive equations discussed in section 2.3. However, an important simplification of the problem can be made according to the idea originated by Simo and Taylor, 1985 and widely used since that (Hartmann and Haupt, 1993; Doghri, 1993 and 1995; Chaboche and Cailletaud, 1996, Hammi, 2000; Hammi and Saanouni, 2000 among many others). This simplification aims to exploit some special properties of the used constitutive equations to reduce the number of equations to be solved, from 15 to only 2 scalar equations in this fully isotropic case (see Hammi,2000 for the general case of fully anisotropic two surfaces formulation). After some algebraic

calculation one obtain the following system of two scalar equations )D,∆λ(g1n1n1 ++

and )D,∆λ(g1n1n2 ++

:


( )

( )( )

( )

=−

λ∆

−−=

=

λ−−+

−

−

+−−−

−−=

φ

+

++

+++

++

+

+

+

+++

0D1S

YDD)D,∆λ(g

0a∆exp1a

C∆λG3

D1

1

σ)D1bR1(1b

Q

D1

1Z)D,∆λ(g

1n

1n

s

1n

n1n1n1n2

1n1ne

1n

p1nn

1n

n1n1n1n1

(52)

in which the following notations have been used :

( )

( )( )

λ−−+

−

+

+−−−

−=

++

+

+

+

+

1n1ne

1n

p1nn

1n

s1n

a∆exp1a

C∆λG3

D1

1

σ)D1bR1(1b

Q

D1

1Z

(53)

( )

( )( )

( )( )

2

1n

1n

1nn

2

1n1n

1nn

d

n1n1n

1n

2

1nd

n

d

ne1n

b∆exp1D-1b

1

)b∆exp(r2

Qna∆exp1

a

1

)a∆exp(3

C:n∆

D12

)3(∆:G2Y

λ−−+

+λ+

λ−−

+λα+

ελ−

−

λ+εε=

+

+

+++

+++

+

+

+

��

(54)

Note that in obtaining Eqs. (52) to (54), use has been made of a closed form asymptotic integration applied to both kinematic and isotropic hardening evolution equations (see details in Hammi, 2000 or Saanouni et al 2001).

The system given by Eq. (52) should be solved by using the iterative Newton-Raphson procedure in order

to determine the unknowns ∆λn+1 and Dn+1 (see Hammi, 2000 for detailed calculations), from which the

variables 1n1n1n

R,X,+++

σ at the end of the time step are easily calculated.

Recall that this local integration procedure is required with both the SI global resolution scheme and the DE resolution strategy. However, with the SI scheme the material Jacobian matrix is also required. This will be discussed in the next section.

3.2 Consistent Stiffness matrix

To complete the algorithmic procedure for the SI resolution strategy, there only remains to compute the closed-form expression for the consistent tangent matrix. This can be done by differentiating the expression of the stress at the time tn+1 :


( ){ }( )1n

1n1n1ne

d

ne1n

1n

1n

1n d

n∆λD1G2σ)1:(KD1d

d

σdC

+

++++

+

+

+ε

−−+ε−

=

ε

=

�

(55)

in which n should be replaced by Z depending on ∆λn+1 and Dn+1. According to the fact that ∆λn+1 and Dn+1

are dependent on the total deformation, then the differential in Eq. (55) requires the differentiation of ∆λn+1

and Dn+1 with respect to the total strain. These latter can be obtained from the differentiation of

)D,∆λ(g1n1n1 ++

and )D,∆λ(g1n1n2 ++

defined in the system Eq.(52) with respect to εn+1. Despite the

apparent complexity of the fully coupled constitutive equations and after some fastidious algebraic manipulations (see Hammi, 2000) one arrive to:

( )( )1n

1n

1n1ne

d

n1ne

1n

1n

1n

1n

1n

1n1ne1n1n

εd

dDn∆2Gσ11:εK

εd

nd∆λ

εd

dλn)D1(G2Λ

~C

+

+

+++

+

+

+

+

+

++++

⊗λ−+

−

+⊗−−=

�

(67)

where n has to be replaced by Z which is function of ∆λn+1 and Dn+1 themselves solution of the linearized system defined by Eq. (52). While the partial derivatives entering the Eq. (56) are obtained by taking the differential of both g1 and g2 defined in Eq. (52) (See details in Hammi, 2000).

4. Application to some metal forming processes

In this section numerical simulations of some 2D and 3D sheet metal forming processes are presented. The main goal is to illustrate the capability of the proposed methodology to predict when and where ductile damage zones may take place inside the deformed part during the process. Moreover, one can show the possibility to act on the process parameters (tools velocity, friction, material ductility, ….) in order to avoid damage as in forging, deep drawing, hydroforming, …; or to enhance damage in order to simulate some sheet metal cutting operations.

Note that, when implementing the damage effect in ABAQUS/Std (Umat) and ABAQUS/Explicit (Vumat), a flag parameter (icoupl) has been introduced allowing to perform a fully coupled analysis when icoupl=1 and an uncoupled analysis when icoupl=0. This allows the comparison between fully coupled and completely uncoupled calculations, using exactly the same analysis computational tools, in order to show the advantages of the fully coupled procedure against the uncoupled one.

The material constants used in what follows are E=84.0 GPa, ν=0.38, σy=120.0 MPa, Q=600.0 MPa, b=3.0, C=a=0 (no kinematic hardening), b=s=1., S=100.0 MPa defining a material with a large ductility (about

130 %). The Coulomb friction parameter have been taken as η=0.3 which correspond to the metal-metal friction. On the other hand, only linear isoparametric elements with adaptive mesh size are used (Borouchaki et al, 2002).


4.1 2D processes

For the simulation of 2D metal working processes a special procedure has been developed in order to execute ABAQUS step by step as shown in Figure 1. At each load increment, and after the convergence has been reached, the over all elements are tested in order to detect the fully damaged elements (elements where the damage variable has reach its critical value in all the Gauss points). If so, the fully damaged element is killed from the structure and a new adaptive meshing of the part is worked out. For the mesh adaptation, two kinds of error estimators are used (George and Borouchaki, 1997; Borouchaki et al, 2002). The first one is based on the tools curvature and called the geometrical error estimator; while the second one is based on the gradient of the damage field inside the deformed part and is called the physical error estimator. After each remeshing, the mechanical fields have to be transferred from the old to the new mesh and the next loading step is worked out using ABAQUS solver.

I n p u t A B A Q U S d a t a n e w lo a d i n c r e m e n t B .C .

S T A R T

In i t i a l m e s h

g e n e r a t o r ( B L 2 D )

A B A Q U S A n a l y s e s

U M A T o r V U M A T

f u l l y d a m a g e d

e l e m e n t

A d a p t i v e R e - m e s h in g ( B L 2 D )

( g e o m . a n d p h y s ic . e r r o r e s t im a t o r )

S t o r e v a r ia b l e s

n e x t s t e pA n a l y s i s c o m p le t e d

N O

K i l l e le m e n t s

Y E S

F i e l d s t r a n s f e r n e w m e s h

N O

S T O P

Y E S

Figure 1: Flowchart of the used Shell Script


The first 2D example concerns the fracture prediction in a deep drawing of a 0.98 mm thickness sheet. Figure 2 shows the ductile damage distribution inside the sheet for three different values of the punch displacement d. Note the clear yield localization observed for d=6.4 mm (Figure 2b) and the final fracture obtained for d=9.6 mm (Figure 2c).

(a) d=3.6 mm (b) d=6.4 mm (c) d=9.6 mm

Figure 2: Axisymmetric deep drawing. Damage distribution inside the sheet for different configurations

The second example deals with the blanking of a 1.0 mm thickness sheet. Figure 3 shows four different configurations corresponding to different punch displacement. One can observe that a macroscopic crack initiates close to the die edge (Figure 3d) and propagates in the direction of the punch edge. The blanking operation is completed for d=0.54 mm.

(a) d=0.05 mm (b) d=0.2 mm


(c) d=0.39 mm (d) d= 0.54 mm

Figure 3: Numerical simulation of an axisymmetric blanking operation

The third example concerns the orthogonal cutting process by chip formation using the same material constants. Figure 4 shows the capability of the proposed methodology to simulate correctly the cutting operation by chip formation due to the damage coupling effect. This gives the opportunity to study (virtually) the effect of the process parameters (tool velocity, friction, heat exchange, chip thickness, material ductility) of the chip formation and fragmentation. This aspect will be shown during the oral presentation together with more 2D and 3D metal forming process simulations.

(a) d=0.1 mm (b) d=15.0 mm

( c ) d=28.0 mm (d) d=41.2 mm

Figure 4: Simulation of the blanking operation with the fully coupled procedure


4.2 3D Hydraulic deep drawing

This example deals with the hydraulic deep drawing of 3.00 mm thickness and 245.0 mm diameter circular sheet using the same material constants as for 2D cases above. Linear isoparametric hexahedric solid element have been used (Saanouni et al, 2001a). Figure 5 shows the damage distribution inside the sheet at three different moments of the process. Four macroscopic cracks initiate as indicated in Figure 5b and propagate until the final fracture shown in Figure 5c. This latter numerical result compares well with the experimental result taken from (Saanouni et al, 2001a).

(a) (b)

(c) (d)

Figure 5: Fracture prediction during the hydraulic deep drawing of a circular sheet

5. Some conclusions

In this work a fully coupled thermoelastoplastic-damage constitutive equations at finite strains have been implemented into ABAQUS using both Umat and Vumat user’s subroutines. An incremental procedure combining ABAQUS together with adaptive remeshing tools has been developed in order to improve metal forming processes with respect to the ductile damage occurrence. The proposed procedure has been shown


to be very helpful in order to improve “virtually” various sheet and bulk metal working processes. However, further extension of the proposed model to include the induced plastic and the damage anisotropy and the higher gradient effect (non-local formulation) are under progress.

Acknowledgments

The financial support of the FEDER (contract N° 99-2-50-059) and CRCA via the Pôle Mécanique et Matériaux Charpardanais (PMMC) is gratefully acknowledged.

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numerical simulation of damage in metal forming processes using abaqus

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