Efficient integration technique for generalized viscoplasticity coupled to damage

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING

Int. J. Numer. Meth. Engng. 44, 1727–1747 (1999)

EFFICIENT INTEGRATION TECHNIQUE FOR GENERALIZEDVISCOPLASTICITY COUPLED TO DAMAGE

MAGNUS JOHANSSON, ROLF MAHNKEN AND KENNETH RUNESSON∗Division of Solid Mechanics, Chalmers University of Technology, S-41296 G�oteborg, Sweden

SUMMARY

A generalized viscoplasticity theory, with kinetic coupling to damage, was presented by Johansson andRunesson.1 This theory, which is based on the Duvaut–Lions’ concept of viscoplastic reguralization, in-cludes the (unconventional) concept of dynamic yield surface that is approached asymptotically at in�niteloading rate. In this paper, we extend the model concept to include Microcrack-Closure-Reopening (MCR)e�ects. Primarily, we deal with the issue of e�cient integration and iteration for computing the stress (andother state variables) within an strain-driven format. In particular, we demonstrate the e�ciency of a novel‘multi-level’ Newton-like iteration algorithm for the model problem involving von Mises quasistatic yieldsurface with non-linear mixed hardening. The Algorithmic Tangent Sti�ness (ATS) tensor is derived and themodel is implemented in the commercial FE code ABAQUS. Copyright ? 1999 John Wiley & Sons, Ltd.

KEY WORDS: viscoplasticity; damage; dynamic yield surface; integration

1. INTRODUCTION

Realistic models for the description of the rate-dependent macroscopic characteristics of manyengineering materials are obtained within the framework of viscoplasticity, by which a thresholdvalue of stress must be exceeded before rate-dependent inelastic strain can develop, see e.g. thework of Perzyna.2 For example, a quite versatile model for metals and alloys was developed atONERA in France.3–5 A comprehensive summary and evaluation of the model features, as wellas comparison with other models, were given by Chaboche.6

In the classical viscoplasticity formulations, the response becomes completely elastic withoutlimit on the stress at in�nite loading rate, which is not realistic. Therefore, in a previous paperby Johansson and Runesson,1 a dynamic yield surface was introduced, and it is approached atin�nite loading rate. This is accomplished by a special choice of ‘overstress’ function, which hasthe property that the apparent uidity becomes in�nite when the stress approaches (from inside)the dynamic yield surface. Although rather simple in concept, this feature is essential for thepossibility to describe the in uence of large strain rate on the total stress–strain relation.

∗ Correspondence to: Kenneth Runesson, Division of Solid Mechanics, Chalmers University of Technology, S-41296G�oteborg, Sweden. E-mail: [email protected]

Contract=grant sponsor: NFFPContract=grant sponsor: ABB Stal Inc.Contract=grant sponsor: Saab Military Aircraft Inc.Contract=grant sponsor: Volvo Aero Corporation

CCC 0029–5981/99/111727–21$17.50 Received 15 August 1997Copyright ? 1999 John Wiley & Sons, Ltd. Revised 19 June 1998

1728 M. JOHANSSON, R. MAHNKEN AND K. RUNESSON

In this paper, we consider (in particular) the numerical aspects of the above-mentioned modelconcept. The integrated constitutive equations can in general be seen as a system of non-linearequations, from which the unknowns must be solved in an iterative fashion. An additional di�cultyis that some of the unknowns have upper and lower bounds. For this kind of problems thepossibility and the advantage of combining the good global convergence properties of, for example,the Bisecting method and the local convergence properties of the Newton method are discussed.The paper is organized as follows: Section 2 contains a brief summary of the basic relations

of generalized viscoplasticity based on the Duvaut–Lions concept,7 including the dynamic yieldsurface. As compared to Johansson and Runesson,1 we extend the model concept to includeMicrocrack-Closure-Reopening (MCR) e�ects. For de�niteness, a model for metal and alloys isgiven in Section 3. The incremental relations obtained upon implicit integration are reviewed inSection 4, whereas the various iterative strategies to solve for the updated stress (and other statevariables) at each Gauss point are discussed in Section 5. The proper ATS-tensor, to be usedin equilibrium iterations, is derived in Section 6. Computational results, which demonstrate thee�ectiveness of the proposed procedures, are given in Section 7.

2. SUMMARY OF CONSTITUTIVE RELATIONS

2.1. Thermodynamic basis

The thermodynamic state is assumed to be determined by the elastic part of the strain Ue, thehardening variables �i;† and the (scalar) damage �. Decomposing the total strain into elastic andinelastic parts, U= Ue + Up, we propose the following uncoupled form of Helmholtz’ free energy(per unit volume),8 in a generic model:

(Ue; �i; �) :=e(Ue; �) + p(�i) (1)

From the Clausius–Duhem-inequality, the following relations for the stress b and the dissipationD are obtained:

b= @=@U; D := bp : Up +∑iKi�i + A�¿0 (2)

where we have introduced the dissipative stresses

bp := − @@Up = b; Ki := − @

@�i; A := − @

@�(3)

According to the ‘principle of equivalent strain’,9 we de�ne the e�ective stress b as

b := @e

@Ue with e(Ue) :=e(Ue; 0) (4)

If we assume that Ue can be expressed as Ue(b; �), then (4) implies that b can be given as b(b; �).This relationship will be employed subsequently.

† For simplicity, the hardening variables �i are treated as scalars. In practice, they may be tensors of even order

Copyright ? 1999 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng. 44, 1727–1747 (1999)

GENERALIZED VISCOPLASTICITY COUPLED TO DAMAGE 1729

Figure 1. Image point on the dynamic yield surface bd. (For simplicity, the projection bs is depicted as the closest pointin the Euclidean norm of b onto �s = 0, and restricted to stress space)

2.2. Viscoplastic admissibility—static and dynamic yield surfaces

In classical viscoplastic models the behaviour is either elastic or viscoplastic. The apparent uidity is then assumed to increase monotonically with increasing stress level, in such a fashionthat it becomes unbounded only when the stress is unbounded. In order to obtain unlimited uidityfor bounded stresses, we introduce the concept of a dynamic yield surface.1 Hence, the elastic andviscoplastic regions are de�ned by the convex sets Bs and Bd, de�ned by

Bs(�) := {(b; Ki) |�s(b; Ki)60}; Bd(�) := {(b; Ki) |�d(b; Ki)60} (5)

where �s(b; Ki) and �d(b; Ki) are the static and dynamic yield functions (see Figure 1), respec-tively. These sets are de�ned as to satisfy Bs(�)⊂Bd(�), which is accomplished by ensuring that�s(b; Ki)¡�d(b; Ki) for all states (b; Ki).

2.3. Flow, hardening and damage rules

The viscoplastic ow and hardening rules are formulated in the spirit of Duvaut and Lions7

Up = 1t∗�(�)(E e)−1 : (b − bs) (6)

�i=1t∗�(�)

∑jH−1ij (Kj − K sj ) (7)

where t∗ is the relaxation time, �(�)¿0 is the ‘overstress’ function, Ee := @2e=@Ue⊗ @Ue is thetensor of elastic tangent moduli, and Hij := @2p=@�i@�j is the matrix of hardening moduli. The



static solution (bs; K si ) is the ‘projection’ of the current state (b; Ki) onto Bs in a certain metric,and it is de�ned by the following Kuhn–Tucker conditions‡ (for smooth �s(bs; K si )= 0)

bs = b − �sEe : @�s

@bs :M; M :=@b@b

K si= Ki − �s∑jHij@�s∗

@K si�s¿0; �s60; �s�s = 0

(8)

where we have introduced the plastic potential �s∗(bs; K si ), in order to admit non-associativehardening rules.

Remark. The 4th order tensor M is used to capture the MCR-e�ect, which is described inSection 3.

The overstress function �(�) is introduced in order to account for the e�ect of the dynamic yieldsurface, whereby �∈ (0; 1) de�nes the actual position of the (stress) state relative to the quasistaticand dynamic yield surfaces. If �=0, then the state satis�es �s = 0, and if �=1, then the statesatis�es �d =0. Two conditions should be placed on the properties of �(�):

(1) The rate-independent solution must be retrieved when t∗ → 0, which is the case when �(0)¿0.For convenience we choose �(0)= 1.

(2) The apparent uidity must grow in an unlimited fashion when the state approaches the dy-namic yield surface, i.e. �(1)=∞.

A typical choice is �(�) := 1=(1− �)p, where the exponent p is a material parameter. Moreover, �is de�ned from the image point of the current stress state onto �d =0 as follows (for 0¡�61):

bd := 1�(b − bs) + bs (9)

Kdi :=1�(Ki − K si ) + K si (10)

which is illustrated in Figure 1. A quite general class of damage rules may be formulated as:

�=1t∗�(�)

@�@A

(11)

where �(A; �; �pa) is a positive scalar function that is monotonically increasing in its arguments.The argument �pa is the accumulated plastic strain, obtained from

� pa =

√23|Up|; Up = U− (Ee)−1 : b (12)

‡ These are not Kuhn–Tucker conditions in the strict sense, since the principle of maximum dissipation has been abandonedwhen �∗ 6=�



3. A MODEL FOR METALS AND ALLOYS

In order to model e�ects of closure and reopening of microcracks (MCR-e�ects) in a simplefashion,10; 11 we introduce the elastic part of the free energy as

e :=G|Uedev|2 +Kb2(�evol)

2 (13)

where G is the shear modulus and Kb is the bulk modulus, whereas Ue is de�ned as

Uedev := (1− �)1=2Uedev; �evol := (1− g(�evol)�)1=2�evol with �evol = Ue : T (14)

The MCR-function g(x) is chosen,8; 12 as follows:

g(x) := g0 + (1− g0)H(x) (15)

where H(x) is the Heaviside step function: H(x)= 1 if x¿0, H(x)= 0 if x¡0. With (13)–(15),we may express b as

b= @@Ue = 2G(1− �)U

edev + Kb(1− g(�evol)�)�evolT (16)

Combining this expression with the e�ective stress in (4), we obtain

b=(1− �)bdev + (1− g(�evol)�)�mT with �m = 13 b : T (17)

whereby the (4th order) projection tensor M in (8) becomes

M=I

1− � +T⊗ T

1− g(�evol)�(18)

For the sake of de�niteness, we choose the von Mises yield function with the (initial) quasistaticand dynamic yield stress �sy and �

dy, respectively. We thus de�ne

�s := �e − �sy − K with �e :=

√32|cdev|; c := b − B (19)

�d := �e − �dy − K (20)

where K is the isotropic hardening (drag-stress) and B the kinematic hardening (back-stress).

Remark. For this choice of yield functions it follows that Bs(�)⊂Bd(�) if �sy¡�dy, see

Appendix. Moreover, the actual state will always be located inside the dynamic yield surface,i.e. (b; K;B)∈Bd(�).

The stresses K and B are obtained from the ‘hardening’ part p of the free energy, which ischosen as

p :=12rH�2 +

12(1− r)H�2e with �e =

√23|R| (21)

where � and R are the isotropic and the kinematic hardening variable, respectively. The parameterr ∈ [0; 1] controls the relation between isotropic and kinematic hardening, and H is the initial valuehardening modulus of both the static and dynamic yield surface (for simplicity).



Moreover, the damage stress A becomes

A= − @@�=�2e6G

+g(�evol)�

2m

2Kbwith �e :=

√32|bdev| (22)

The potential functions �s∗ and � are chosen as

�s∗ := �s +K2

2K∞+

B2e2B∞

with Be :=

√32|Bdev| (23)

� :=A2

2S(1− �)m T (�pa) (24)

where T (�pa)∈C∞ is a threshold function. In order to obtain a smooth initial damage development,we choose T as the Hermitian polynomial

T (�pa)=

0 if �pa6�0(�pa − �0)2(�1 − �0)2

(3− 2 �

pa − �0�1 − �0

)if �0¡�pa¡�1

1 if �16�pa

(25)

In this model, K∞ and B∞ are saturation values of the isotropic and kinematic hardening, respec-tively, whereas S and m are parameters that de�ne the ductility and the rate of damage development.The choice m 6=1 represents a trivial, but quite important, generalization of the classical damagelaw suggested originally by Lemaitre.8

In summary, the constitutive equations for b; K and B are obtained upon using the rate equations(6) and (7) as

˙bdev +1t∗�(�)(bdev − bsdev) = 2GUdev; ˙�m =Kb�vol (26)

K +1t∗�(�)(K − K s) = 0; K(0)= 0 (27)

Bdev +1t∗�(�)(Bdev − Bsdev)= 0; Bm = 0; Bdev(0)= 0 (28)

By combining (11) and (24), the damage rule becomes

�=1t∗�(�)�s

AS(1− �)m T (�

pa); �(0)= 0 (29)

The statically admissible state (bs; K s;Bs) is determined as the solution of the ‘projection’ problem

bsdev= bdev − 3G�sbsdev − Bsdev

�se; �sm = �m ; �se =

√32|bsdev − Bsdev|

K s= K + rH�s(1− K s

K∞

)

Bsdev= Bdev + (1− r)H�s(bsdev − Bsdev

�se− BsdevB∞

); Bsm =Bm

�s¿0; �s60; �s�s = 0

(30)



Remark. The image point on the dynamic yield surface is de�ned only if �s¿0, i.e. when theactual state is outside the elastic region Bs.

4. NUMERICAL INTEGRATION OF THE CONSTITUTIVE EQUATIONS

4.1. Strain-driven core algorithm—preliminaries

In this section we derive explicit incremental relations for the chosen model problem, fromwhich updated values (at the new time tn+1) of the various state variables can be computed. Fromthese it becomes clear that three scalar equations (in the case of viscoplastic loading) can beestablished, from which the unknowns ��s (de�ned below), � and � can be solved for a giventime increment �t. These equations are derived upon establishing the appropiate conditions forthe (quasi) static yield criterion, the dynamic yield criterion, and the integrated damage law. Theywill be considered subsequently in turn.

Remark. Due to the particularly simple structure of the model problem it is possible to reducethe problem setting to only three scalar equations. In a more general situation, it is necessary toinvolve the Cartesian components of bs, K s and Bs, as well as of b, K and B, as unknowns inaddition to ��s; � and �, cf. the discussion in Johansson et al.13

4.2. Quasistatic solution

The backward Euler rule is used to integrate the evolution equations (26)–(28), which gives

n+1bdev =1

1 + n+1��(n+1bedev + n+1��n+1bsdev); n+1�m = n+1�em (31)

n+1K =1

1 + n+1��(nK + n+1��n+1K s) (32)

n+1Bdev =1

1 + n+1��(nBdev + n+1��n+1Bsdev);

n+1Bm =0 (33)

where the scalar ��(�) and the e�ective elastic trial stress n+1be are de�ned as

��(�)=�tt∗�(�); n+1bedev = nbdev + 2G�Udev; ��em =Kb��v (34)

Upon introducing (34) into (31)–(33), we obtain the following expression for n+1�s (after certainalgebraic manipulations):

n+1�s( ��s; �)= n+1�e( ��s)− n+1h( ��s; �) ��s (35)

where we have introduced the trial value of n+1� as

n+1�e( ��s)= n+1 ˜�ee( ��s)− �sy − aK ( ��s)nK; n+1 ˜�ee( ��

s)=

√32|n+1bedev − aB( ��s)nB| (36)

and also the hardening function h and the plastic multiplier ��s

n+1h( ��s; �)=3G1− � + aB(

��s)(1− r)H + aK ( ��s)rH; ��s := �s(1 + ��)¿0 (37)



The coe�cients aK and aB are given as

aK ( ��s)=

(1 +

rH ��s

K∞

)−1; aB( ��s)=

(1 +

(1− r)H ��sB∞

)−1(38)

with the properties aK (0)= aB(0)= 1, and aK ≡ 1 when K∞=∞ (linear isotropic hardening),whereas aB≡ 1 when B∞=∞ (linear kinematic hardening).

Remark. It appears that the quasistatic yield function �s is dependent on the variable � onlyindirectly via the plastic multiplier ��s.

When the plastic multiplier ��s and the damage � are known, the quasistatic stresses can becalculated by

n+1bsdev = c1( ��s; �)n+1bedev + (1− c1( ��s; �))aB( ��s)nBdev (39)n+1K s = aK ( ��s)(nK + rH ��s) (40)

n+1Bsdev = c2( ��s)n+1bedev + (1− c2( ��s))aB( ��s)nBdev (41)

where the coe�cients c1 and c2 are given as

c1( ��s; �)= 1− 3G ��s

n+1 ˜�ee( ��s)(1− �)

; c2( ��s)= aB( ��s)(1− r)H ��sn+1 ˜�ee( ��s)

(42)

4.3. Dynamic solution

By combining (9),(10), (31)–(33) and (39)–(41), we derive the following expressions for thedynamic stresses:

n+1bddev( ��s; �; �) =(1− c1( ��s; �)�(1 + ��(�))

+ c1( ��s; �)

)n+1bedev

−aB( ��s)(1− c1( ��s; �)�(1 + ��(�))

+ c1( ��s; �)− 1)nBdev (43)

n+1Kd( ��s; �) =

(1− aK ( ��s)�(1 + ��(�))

+ aK ( ��s)

)nK +

(1− 1

�(1 + ��(�))

)aK ( ��s)rH ��s (44)

n+1Bddev( ��s; �) =

(1− (1− c2( ��s))aB( ��s)

�(1 + ��(�))+ (1− c2( ��s)aB( ��s))

)nBdev

+(1− 1

�(1 + ��(�))

)c2( ��s)n+1bedev (45)

The dynamic yield function can be written asn+1�d( ��s; �; �)= n+1�de ( ��

s; �; �)− �dy − n+1Kd( ��s; �) (46)

where we have introduced

n+1�de ( ��s; �; �)=

√32|n+1bddev( ��s; �; �)− n+1Bddev( ��

s; �)| (47)



4.4. Damage solution

The third incremental relation is obtained from the integrated damage law. Upon combining(31) and (39), we �rst obtain the e�ective stress as

n+1bdev( ��s; �; �) =1

1 +��(�)((1 + c1( ��s; �)��(�))n+1bedev

+��(�)(1− c1( ��s; �))aB( ��s) nBdev); ��m =Kb��v (48)

The corresponding equivalent stress n+1�e( ��s; �; �) is needed in the expression for the damagestress

n+1A( ��s; �; �)=(n+1�e( ��s; �; �))2

6G+g(n+1�m)(n+1�m)2

2Kb(49)

Finally, from the integrated form of the damage law, the following expression is obtained:

n+1��( ��s; �; �)= �− n�−��s��(�)1 + ��(�)

n+1A( ��s; �; �)S(1− �)m T (n+1�pa( ��

s; �; �))= 0 (50)

4.5. Summary of algorithm

The numerical algorithm is summarized in Box 1. Whenever viscoplastic loading is detected,the set of equations �(Y)= 0 must be solved iteratively, which is discussed next.

Box 1. Summary of numerical algorithm

1. For a given state (nU; nb; nK; nB; n�), compute

n+1U= nU+�U; n+1be = nb + Ee : �U

2. Check loading =unloading of updated quasistatic solution (n+1bs; n+1K s; n+1Bs)

Set ��s = 0 n+1�= n+1�s = n+1�e

• if n+1�e60, then elastic unloading ( ��s = 0)n+1b= n+1be; n+1K = nK; n+1B= nB; n+1�= n�

Exit.• elseif n+1�e¿0, then viscoplastic loading ( ��s¿0)Compute Y= [ ��s; �; �]t from �= [�s;�d ;��]t = 0.Compute (n+1b; n+1K; n+1B) and, �nally, n+1b.



5. LOCAL ITERATION PROCEDURE

5.1. Preliminaries

At viscoplastic loading, the solution Y= [ ��s; �; �]t of the problem

�(Y)= 0; ai6Yi6bi; i=1; 2; 3 (51)

must be found iteratively, where ai and bi are lower and upper bounds, respectively. Several con-ceptually di�erent iteration schemes (possibly in combination) can be envisioned for this problem.Below, we discuss two schemes.

5.2. Classical Newton iterations

The most straightforward approach is to use Newton iterations

Y(k+1) =Y(k) − (J(k))−1�(k) k =0; 1; 2; : : : (52)

where the Jacobian matrix J is de�ned as

J=@�@Y

=

@�s

@ ��s@�s

@�@�s

@�

@�d

@ ��s@�d

@�@�d

@�

@��

@ ��s@��

@�@��

@�

(53)

The complete expressions of the Jacobian matrix elements are rather lengthy and are thus omittedhere for brevity. Quasi–Newton methods with some approximation of the Jacobian J can also beused. These methods show very good convergence properties close to the solution point. However,it is easy to envision situations for which convergence is very slow or no convergence is achieved,as shown by the one-dimensional problem in Figure 2. In case, the method fails a possible remedyis a line-search technique, where ||�(Y)|| is used as a merit function, e.g. Reference 14. Adrawback is that the step size may become small, thus reducing the overall e�ciency of thealgorithm quite signi�cantly.

5.3. Multi-level iterations

In order to gain control and robustness of the iteration process (but not necessarily highere�ciency), it has been considered advantageous to adopt a multi-level iteration strategy, wherebythe multi-dimensional problem (51) is tackled by solving a sequence of one-dimensional problems.To �x ideas, consider the case with two unknowns

�=[�1(Y)�2(Y)

]where Y=

[Y1Y2

](54)

Firstly, for given (�xed) Y2 we use the iteration Y(k+1)1 =Y (k)1 + �Y (k)1 ; k =0; 1; 2; : : : until the

condition �1(Y(k)1 ; Y2)= 0 is satis�ed (within the given tolerance). However, since the solution

Y1(Y2)= arg {�1(Y1(Y2); Y2)= 0} (55)



Figure 2. One-dimensional non-linear problems with bounds, where a Newton method, in: (a) shows slow convergence;and in (b) fails to converge

(if it exists) does not in general satisfy the second condition �2(Y1(Y2); Y2)= 0, an additional(outer) iteration Y (k+1)2 =Y (k)2 +�Y (k)2 becomes necessary. The basic idea is to determine a solutionY1(Y2) according to (55) whenever Y2 is modi�ed. In this manner the two-dimensional problem(54) is converted into a sequence of one-dimensional problems. This is regarded as the mainadvantage of the approach, since it allows us to combine the advantages of various one-dimensionalsolution schemes in order to achieve optimal robustness and e�ciency. For example, the goodglobal convergence properties of a Bisecting method or a Pegasus method,15 may be combinedwith the good local convergence property of a Newton method.

Box 2. Multi-level algorithm for the iterative solution of a nonlinear multi-dimensional problem

Algorithm FIND-ZERO(l):Input: Yl+1Output: Yl−1, Yl satisfying �l−1 = 0, �l=0Notation:

Y=

Yl−1YlYl+1

; �=

�l−1(Y)�l(Y)�l+1(Y)

where Yl−1 :=

Y1...

Yl−1

; Yl+1 :=

Yl+1...Yn

Iteration steps:1. Initialize Y (k = 0)l

2. If l¿1 then FIND-ZERO(l− 1) Y(k)l−1 =Yl−1(Y(k)l ) satisfying �l−1 = 0

3. Calculate the residual �(k)l =�l(Y(k)l−1; Y

(k)l ;Yl+1)

4. If |�(k)l |¡ tol then RETURN5. Compute �Y (k)l Y (k+1)l =Y (k)l +�Y (k)l6. k = k + 1, GOTO 2.

For the case of n unknowns, Yl; l=1; : : : ; n, a sequence of subproblems are solved, startingfrom the highest level (l= n). For given values of Yl+1, the proposed algorithm for solving such a



subproblem on the lth level is summarized in Box 2. It is emphasized that the convergence on eachlevel crucially depends on how e�ciently the increment �Y (k)l is computed. Two possibilities are:• Bisecting method:

�Y (k)l =

{ 12 (Y

minl − Y (k)l ); Ymaxl =Y (k)l ; if �l(Y

(k)l ) · �l(Ymaxl )¿0

12 (Y

maxl − Y (k)l ); Yminl =Y (k)l ; else

(56)

• Newton Method:

�Y (k)l =−[d�(k)ldYl

]−1�(k)l (57)

where the total derivative of the residual �l is given as

d�ldYl

=@�l@Yl

+@�l@Yl−1

dYl−1dYl

For the evaluation of dYl−1=dYl, we exploit Step 2 of the algorithm in Box 2. This gives thecondition

�l−1(Yl−1(Yl); Yl;Yl+1)= 0 d�l−1dYl

=@�l−1@Yl

+@�l−1@Yl−1

dYl−1dYl

= 0

The total derivative of �l thus becomes

d�ldYl

=@�l@Yl

− @�l@Yl−1

[@�l−1@Yl−1

]−1 @�l−1@Yl

(58)

which is simply the result of a static-condensation of the Jacobian J.

Examples of alternative iteration methods for one-dimensional problems are Regula Falsi and thePegasus method. We refer to Engelin-M�ullges and Reuter15 for the speci�c iteration schemes.On certain occasions it was observed that the subproblem in Step 2 of Box 2 did not have a

solution. Such a situation should not be interpreted such that the full problem �=0 does not havea solution. Rather, it means that Yl must be updated more carefully. For example, we may give�Y (k)l the heuristic value �Y (k)l = fac(bl − al), where fac ∈ [0; 1] is a given factor, e.g. fac = 0·1.Only if all possible trial-steps fail to give a solution, the algorithm returns with the message thatno solution has been found.Finally, for the applications discussed in Section 7, the multi-level algorithm is designed such

that on the levels l=2; 3 (for � and �) bisecting is used until the residuals |�d| and |��| are lesserthan a prescribed value (e.g. 100) or if the Newton iteration fails to converge, whereas Newtoniterations are carried out on level l=1 (for ��s). There are, of course, many other possibilities tocombine di�erent iteration techniques, which might increase the e�ciency and the robustness ofthe algorithm even more.

6. EQUILIBRIUM ITERATIONS—ALGORITHMIC TANGENT STIFFNESS

To obtain quadratic convergence at equilibrium iterations, it is necessary to derive the AlgorithmicTangent Sti�ness (ATS) tensor. In fact, this tensor can be obtained in closed form for the adopted



model problem, as shown subsequently. First we note that the ATS-tensor Ea is de�ned (for anymaterial response) by the linearized incremental relation

�(�b)=Ea(�U) : �(�U) (59)

for variations �(�U) of the current strain increment �U.Assuming viscoplastic response, we recall from (48) that n+1b can be expressed as

n+1bdev = h1 (�U;Y(�U)) n+1bedev + h2 (�U;Y(�U)) nB; n+1�m =Kb��v (60)

where h1 and h2 are de�ned as

h1 (�U;Y) =1 + c1(�U;Y)��(Y)

1 + ��(Y)

h2 (�U;Y) =��(Y)(1− c1(�U;Y)) aB(Y)

1 + ��(Y)

(61)

From the de�nitions

�(�bdev)= Ea;dev : �(�U); �(��mT)= Ea;m : �(�U) (62)

we obtain

Ea;dev =

((@h1@Y

)t ( dYd (�U)

)+

@h1@(�U)

)n+1bedev

+

((@h2@Y

)t ( dYd(�U)

)+

@h2@(�U)

)nB+ h1Ee (63)

and

Ea;m =KbT⊗ T (64)

From �= 0, we conclude that ��=0, i.e.

@�@(�U) + J

dYd(�U) = 0

dYd(�U) = − J−1 @�

@(�U) (65)

Remark. The Jacobian J := @�=@Y, as well as @hi=@Y; i=1; 2, are established as part of thelocal iteration procedure for the constitutive problem.It remains to use the relation (17), with g(x) de�ned in (15), to obtain

Ea = (1− �)Ea;dev + (1− g(n+1�em) �)Ea;m − (n+1bdev + g(n+1�em) n+1�m T)⊗d�

d(�U) (66)

where it is noted that d�=d(�U) is part of dY=d(�U) given in (65).

7. COMPUTATIONAL RESULTS

7.1. Uniaxial strain-driven problem

The classical Newton and the multilevel algorithms were compared with respect to e�ciencyand robustness for a tension-bar (in uniaxial stress) subjected to monotonic loading with constant



Figure 3. Uniaxial strain-driven problem: scaled stress versus scaled total strain for di�erent t∗ (�y := �sy=E)

strain rate �=4× 10−7 (s−1) for 332 timesteps. This problem corresponds to a situation of mixedstress and strain control on the constitutive (Gausspoint) level for multiaxial stress.The following values of the loading and material parameters were used:

E�sy= 400;

H�sy= 40;

K∞�sy

=B∞�sy

= 0·2; �=0·3; r=0·5

S�sy=

1150

; m=2; �0 = 0; �1 = 10−6; g0 = 1

t∗=1 or 104 (s) ;�dy�sy= 4; p=2; �=4·10−7 (s−1)

(67)

Whereas t∗=104 (s) represents truly viscoplastic response, the choice t∗=1 (s) corresponds (inpractice) to rate-independent response. The response curves are shown in Figure 3, and it can beseen that the ductility decreases with increasing t∗.§

The total CPU time for the classical Newton and the multi-level combined Bisecting-Newtonalgorithms were obtained as shown in Table I. In Table II the number of iterations for di�erentloadsteps are shown for both methods (the numbers in brackets refer to the levels 1, 2, 3). Theresults in the tables demonstrate the e�ciency of the multi-level algorithm. It should be notedthat a two-level procedure in each loadstep is adopted, since equilibrium iterations are required tosatisfy the uniaxial stress condition. The number of iterations in Table II refers to the total numberof local iterations in one loadstep.

§The fracture strain is unrealistically large, which is of no concern for the present investigation focussed on numericale�ciency. In reality, softening due to necking can be expected, which cannot be simulated with the small strain theoryadopted in the present paper



Table I. Uniaxial strain-driven problem: CPU-time (s) for the classical Newton and for themulti-level algorithms for di�erent t∗

t∗(s) Newton Multi-level

1 17·3 11·0104 8·1 7·1

Table II. Uniaxial strain-driven problem: numberof iterations for the classical Newton and for themulti-level algorithms for timesteps (t∗ = 1 (s))

Timestep Newton Multi-level

1 42 23 (7, 13, 3)10 43 26 (7, 16, 3)247 43 23 (4, 16, 3)

7.2. Metal sheet with a hole (plane strain and plane stress)

The constitutive model described above was implemented in the commercial FE-code Abaqus.16

The classical problem of a thin metal sheet in Figure 4(a) with a central cylindrical hole, subjectedto prescribed end displacement in the x2-direction along the edges x2 =±L, was considered next.Due to the double symmetry, only one quarter is analysed. The used �nite element mesh, asshown in Figure 4(b), consists of 1728 biquadratic elements. Geometric data were chosen asfollows:

wL=12;

RL=16;

tL=

1250

(68)

where t is the thickness.For the same displacement rate u=L=4× 10−7 (s−1), two di�erent magnitudes of the displace-

ment increment have been used in the computations: �u=L=8× 10−4 and �u=L=2× 10−5. Thelatter was used in order to get an accurate ‘reference’ solution to the problem.The load–displacement and the damage–displacement response curves (for the most damaged

Gauss point) are shown for plane strain in Figures 5 and 7, and for plane stress in Figures 6and 8, respectively. The spread of the damage zone at local failure¶ is shown in Figure 9 for thecase of plane strain and in Figure 10 for the case of plane stress.In Table III the largest residual forces16 are shown for di�erent load steps and iterations. From

this table the e�ciency of the ATS-tensor is shown.

¶Local failure is achieved when � ≈ 1 in the most damaged Gauss point.



Figure 4. Metal sheet with a hole: (a) Geometry and loading; (b) FE-discretization

Figure 5. Metal sheet with a hole: Force versus end displacement (plane strain)



Figure 6. Metal sheet with a hole: Force versus end displacement (plane stress)

Figure 7. Metal sheet with a hole: Damage development in the most damaged Gauss point (plane strain)



Figure 8. Metal sheet with a hole: Damage development in the most damaged Gauss point (plane stress)

Figure 9. Metal sheet with a hole: Distribution of damage at local failure for t∗ =1 (s) (plane strain)

8. CONCLUDING REMARKS AND OUTLOOK

In this paper we have discussed numerical issues related to a newly developed viscoplasticitymodel with a dynamic yield surface and with coupling to damage. In particular, we have com-pared computationally two conceptually di�erent iteration strategies for solving the incremental



Figure 10. Metal sheet with a hole: Distribution of damage at local failure for t∗ =1 (s) (plane stress)

Table III. Metal sheet with a hole: Largest residual force in the plane strain FE-model for �u=L=8× 10−4and t∗=1 (s)

Iteration numberDisplacement

u=L× 10−4 1 2 3 4 5 6

8 −1·817E−08 1·039E−1316 0·241 −7·627E−03 −1·651E−04 −3·326E−09 8·934E−1424 1·44 0·307 7·968E−03 8·998E−04 −1·377E−07 −9·688E−1432 2·25 0·601 −2·161E−02 3·700E−03 −2·851E−06 −1·723E−12...

...72 −8·095E−02 8·999E−03 −2·155E−05 −1·751E−1080 0·145 1·407E-02 −2·154E−03 −1·639E−04 −5·395E−09

constitutive problem obtained upon implicit integration of the pertinent evolution equations forviscoplastic ow, hardening and damage. The proposed multi-level iteration strategy, which isbased on bisecting and Newton iterations, gives increased control and robustness compared tothe classical (simultaneous) Newton iteration scheme. In the uniaxial strain-driven problem theproposed multi-level iteration strategy also shows higher numerical e�ciency.As to the global equilibrium iterations, the ATS-tensor was derived in closed form for the

considered constitutive model for metals, whereby it appears that the Jacobian of the incrementalconstitutive problem will be employed quite conveniently. The computational results con�rm thequadratic convergence that should be expected.Future work will primarly be directed to veri�cation and testing of the model based on experi-

mental data. To this end the material parameters have to be estimated. The general procedure forminimizing a least-square functional with a gradient based optimizations scheme is presented inaccompanying paper.17



APPENDIX I

Theorem. For the chosen von Mises-based model in Section 3; we obtain Bs(�)⊂Bd(�) if�sy6�

dy .

Proof. That (b; K;B)∈Bs(�) means that �s(b; K;B)60, whereas (b; K;B)∈Bd(�) means that�d(b; K;B)60. Hence, for �s60 we obtain

�d(b; K;B) =√32|bdev − Bdev| − �dy − K

=�s(b; K;B)− (�dy − �sy)6(�sy − �dy)60 (69)

where the last inequality follows from �sy6�dy.

Theorem. The actual state (b; K;B) is then always (viscoplastically) admissible (if �sy6�dy) inthe sense that it is located inside the dynamic yield surface; i.e. �d(b; K;B)60.

Proof. From (9) and (10) we �rst obtain

b= bs + �(bd − bs); K =K s + �(Kd − K s); B=Bs + �(Bd − Bs) (70)

This gives

�d(b; K;B) =√32|bdev − Bdev| − �dy − K

=

√32|(1− �)(bsdev − Bsdev) + �(bddev − Bddev)| − (1− �)K s − �Kd − �dy

6 (1− �)(√

32|bsdev − Bsdev| − K s

)+ �

(√32|bddev − Bddev| − Kd

)− �dy

= (1− �)(�sy − �dy)60 (71)

where the triangle inequality was used to obtain the �rst inequality in (71).

ACKNOWLEDGEMENTS

We wish to acknowledge support from the Swedish National Research Program in Aeronautics(NFFP), ABB Stal Inc., Saab Military Aircraft Inc., and Volvo Aero Corporation.

REFERENCES

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7. G. Duvaut and J. L. Lions, Les Inequations en Mecanique et en Physique, Dunod, Paris, 1972.8. J. Lemaitre, ‘Micromechanics of crack initiation’, Int. J. Fracture, 42, 87–99 (1990).9. J. Lemaitre, A Course on Damage Mechanics, Springer, Berlin, 1992.10. S. Yazdani and H. L. Schreyer, ‘An anisotropic damage model with dilatation for concrete’, Mech. Mater., 7, 231–244

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Efficient integration technique for generalized viscoplasticity coupled to damage

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