Continuum damage mechanics: Present state and future trends

Nuclear Engineering and Design 105 (1987) 19-33 North-Holland, Amsterdam

19

CONTINUUM DAMAGE MECHANICS: PRESENT STATE AND FUTURE TRENDS J.L. C H A B O C H E Office National d'Etudes et de Recherches A~rospatiales, BP 72, 92322 Ch~tillon Cedex, France

Received 4 November 1986

Continuum Damage Mechanics (CDM) has developed since the initial works of Kachanov and Rabotnov. The paper gives a review of its main features, of the present possibilities and of further developments. Several aspects are considered successively: - damage definitions and measures, - damage growth equations and anisotropy effects, - use of CDM for local approaches of fracture. Various materials, loading conditions and damaging processes are incorporated in the same general framework. Particular attention is given to the possible connections between different definitions of damage, especially between the CDM definition and the information obtained from material science. such equations, which can be used for crack initiation problems, as well as crack growth problems, in the context of local approaches (section 6).

1. I n t r o d u c t i o n

During the past two decades many researches have been developed around the concept of a macroscopic damage variable initially proposed by Kachanov [1] and Rabotnov [2] for creep. The distributed defects in materials, not only lead to the crack initiation and the final fracture, but also induce the material deterioration (material damage), such as decrease of strength, rigidity, toughness, stability and of residual life. The new concepts initiated the development of Continuum Damage Mechanics (CDM) [3]. They are supported by the general framework of thermodynamics of irreversible processes and offer complementary possibilities to the Fracture Mechanics. The aim of the present paper is to review some general features of CDM and to summarize its main possibilities, considering successively the following aspects: - definition and damage measures, including the possibility to describe the microstructural damaged state in terms of appropriate mechanical variables, - formulation of equations governing the evolution of these damage variables (section 3), -description of the mechanical behaviour of the damaged material. This can be studied in the framework of the thermodynamics and can include the influence of damage anisotropy (see section 5), - solution of boundary value problems by means of

2. Damage

measures

and definitions

The first step in developing a damage theory concerns the definition of the damage variable. Obviously damage is not directly measurable as strain or plastic strain (already with the hypothesis of the unloaded configuration). In the present section we consider different possibilities to define the damage internal variable through indirect measurement procedures. In fact such measurements are not always practicable but furnish conceptual definitions. Let us note that interpretation of each measure in terms of a damage variable needs a corresponding model. Before any damage theory, it is necessary to precise what we mean by the ultimate state of the damage processes: under the present development of CDM, this final state corresponds generally to the macroscopic crack initiation, that is the presence of a material discontinuity, sufficiently large as regards the microscopic heterogeneities (grains, subgrains.... ): in such a case, the main macroscopic crack is assumed to be developed through several grains, in order to show a sufficient macroscopic homogeneity in size, geometry and direction, leading to a possible treatment through the Frac-

0 0 2 9 - 5 4 9 3 / 8 7 / $ 0 3 . 5 0 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics P u b l i s h i n g Division)

J.L. Chaboche / Continuum damage mechanics

20

!Cavities ! Micro- I ! Macro- ] MacroDislocations{slip t crack j Micro! crack I propagation I bands i initiation j propagation] initiationi mm_

o.bl Classical crack initiation

'D

!D

(a)

(~)

011 I

Presentdefinition of crack initiation D1 --

~ II

]1

Damagemechanics

Fracture

Ol" N1/N~I1 :

i

\ /

Micro*o0c

initiations

-,,-

\ ~ /"---"

idealization

\ ~.L ~ a

specimen) is NF2. If the measured remaining life is N2 the damage after the initial damaging process writes:

I

D = 1 - N2/NF.

macroscoDiccrack //

Fig. 1. Schematic illustration of a macroscopic crack initiation concept.

ture Mechanics concepts (see schematic illustration in fig. 1). 2.1. Measures through the remaining life

As an engineer point of view, the main objective of a damage theory is to allow prediction of lifetime of the structure. Then the remaining life concept is a natural way to define damage. The most conventional definition for such damage parameter is the life ratio N / N v in fatigue, where N and N v represent (for a given loading condition) the present number of cycles already applied and the total number of cycles to crack initiation (or failure), respectively. In that case the damage theory limits to the linear Miner's rule. For instance, the remaining life at the second level, N2, after damaging to N 1 / N v , at a first level writes: N2 - l - D = 1 Nv2

N--! Nv, "

11

Fig. 2. Schematic of damage evolution curves as deduced from remaining life measures; (a) linear accumulation, (b) non-linear accumulation.

Fracture ains ~ mechanics

~

~2/NF2

mechanics

Surface

\?~r

~F

(1)

More generally the remaining life concept does not necessarily lead to the linear rule: after a certain damaging process, the present damage is measured by performing a "measure test", with a fixed loading under which the nominal life (for an initially undamaged

(2)

The remaining life measurements make evidence of interesting properties for damage. For instance in fatigue there is not an unique damage evolution curve as function of the life ratio N / N F, but a dependence on the applied loading (fig. 2) [4,5]. This leads to the conclusion that damage growth equations have to show unseparability, of the damage and loading variables [6,7]. However, such measurements are not sufficient to completely fix the values for damage. As shown in [4], a one-to-one mapping changes the damage value without changing any remaining life result. Let us note that the remaining life measures are also practicable for creep damage [8]. 2.2. Damage measures from the microstructure

A second natural way to define a damage variable is to observe and quantify the irreversible defects: intergranular cavities in creep, surface microcracks in fatigue, dimensions of cavities in ductile fracture. Such measurements have been already used in many situations (see [9,10,11] for instance). Some difficulties arise when dealing with the interpretation: such measures are destructive, which limit their use to observe the development of damage, - defects are difficult to observe during the first phase of damaging processes. Moreover, the initial state is not easily characterized, the quantification has to be done in terms of macroscopic variables (usable in structure computation). Then in each case, convenient hypotheses have to be considered and a particular model developed. -

J.L. Chaboche / Continuum damage mechanics O

0 i i

,

s

",.,,;.',(h)o=(fi)u

i/__ _ J /

21

;..,,. b ," i°i,

ii O*

(a)

0

o

(b)

Fig. 4. (a) Net stress tensor for damage growth, (b) effective stress tensor for deformation.

state of material damage may be described by a second rank symmetric tensor [13] (fig. 3b), generalizing eq. (3) as:

V -b-

X3

o=

X3

x" - -~' a ~ --

C

1,,/

Interpretation in terms of mechanical parameters can be made by means of a net area reduction (fig. 3a). By definition, the damage variable /9, associated with direction n is: SD

S -

S - ~

f/,k, ®,,,, dSW,

(4)

--

3. Net area reductions - (a) the damaged element, (b) volume element with grain boundary cavities, (c) net area reduction on principal planes of damage tensor. Fig.

D. -

N

where dSg~k) and I, (k) are the area of a given boundary occupied by the k th cavity in volume V and the unit normal vector to it, respectively. Fig. 4a indicates schematically that some equivalence is supposed between the present damaged material and the undamaged one (in terms of damage rate) to define the net stress a*. The problem of damage anisotropy is considered in section 5.

Xa

x,..K a~

s,(v3

S* (3)

D n = 0 corresponds to the undamaged state, D, = D c, a critical value, corresponds to the rupture of the element (0.2 < D c < 0.8 for metals). The corresponding net stress concept has been developed by Murakami and Ohno [12,13] in the case of creep damage of polycristalline metals. The microstructural changes can be characterized mainly by nucleation and growth of various cavities on grain boundaries. Assuming that the principal effect consists in the net area reduction due to the distribution of cavities, the

2.3. Damage measures through physical parameters and the effective stress concept The influence of damage on physical quantities can be measured and used to define properly the damage variable: - density change [14], which can be interpreted as a damage variable in ductile failure (see section 3.3), - resistivity change [15] which, through a convenient model, leads to very similar damage measures compared to mechanical parameters (see below), acoustic emission, change in sound velocity . . . . change in the fatigue limit [16] which can he also interpreted in terms of the remaining life, change in the mechanical behaviour of the material, interpreted through the effective stress concept: " a damaged volume of material under the applied stress o shows the same strain response than the undamaged one submitted to the effective stress 8 ". If the damage D represents the loss of effective area taking into account decohesions and local stress concentrations,

22

J.L. Chaboche / Continuum damage mechanics

one can write: S

undamaged state:

o =

.

S

(5)

This definition through the gross behaviour of the material is supported by the results of homogenization techniques [17], considering periodic arrays of defects. Damage measures through the effective stress concept have been performed in several situations. Let us mention: - the case of ductile rupture [18], measuring the change of elastic modulus. From elasticity equations for both damaged and undamaged material o=/~%,

is = ( o / X ) N

(7)

1-D

6=E%,

one obtains: E

o

3=~o= E

~

1-D

D=I---

/~

(6)

E'

Fig. 5a illustrates the case of copper [18]. - the brittle creep damage [19]: using the power law to describe the secondary creep, considered as an

Several tests give the exponent N. Damage follows easily from the measurement of strain rate during tertiary creep and the effective stress concept (5): D = 1 - ( i l i a ) '/N.

(8)

Fig. 5b shows the case of the superalloy IN 100 [19]. The damage in composites develops continuously and shows various mechanisms at the microscale: fiber debonding, matrix microcracking, delamination... The case of fatigue is illustrated in fig. 6 for graphite/epoxy laminates (from [20]). One observes simultaneously the change in the elastic modulus on the first cycle (due to partial debonding), its progressive decrease during fatigue (fig. 6b), accompanying microcracking of the matrix with some stabilization, and the final delamination giving rise to rupture. Let us note some similarities with the case of metals, with much more pronounced effects. Ref. [20] shows clearly the correspondence between the decrease of elastic modulus, the decrease of residual

0

MPo)

IN I00 IO00*C

.5(

500400-

0--

300-

:239

2OO-

I00- ~,'

~t.t,,i

0

o

~5

t

9b 10o l'lO-

zaU2GN 180"C

O . _0_~

=__.~ .

. . . . .

.0-

5(;

.6.4-

21Z MP o --~ 226 ~ \

.2-

2 6 8 ~ 2b

~

14b "CpI ) I. $5

'

0~

'

8b

'

160

'

r

"CpR : 1,07

, .~_,~-,-i-,-; 0,5

,

-'~-Z]k''

|

t/tc

Fig. 5. (a) ductile plastic damage for 99.9% copper, T = 20°C, (b) creep damage measures - Superalloy IN 100 at 1.000°C and light alloy AU2GN at 180°C.

J.L. Chaboche/ Continuumdamagemechanics Table 1

60 50

12-

1 cycle 40000 cycles

Observable variables

1 2 3

_8oooooyc,e,

////74

40-

$,

23

Internal variables

Associated variables

Elastic strain

Stress tensor

tensor

o

ce

30

Entropy s

Temperature T

20 10

2

4

6

8

10

12

14

Accumulated plastic strain P

Radius of yield surface: Ry + R R

Damage

Damage strain energy release rate Y

D

0.9

the total strain tensor by Cp = ( - % and the accumulated plastic strain by p = (~/p : ~p)1/2.

~, 0.8

3.1. Thermodynamicpotential

0.7 /

0.6 ~ 0

Normalized cycles (n/N)

0'.2

014

0'.6

, 0.8

~--, 1

Fig. 6. Fatigue of [0/90] graphite/epoxy specimens, O~n = 345 MPa (50 ksi), R = 0.1, from ref. [20]; (a) stress-strain curves at various stages of the fatigue life, (b) normalized residual elastic modulus.

strength and the remaining life in fatigue. This last example shows the adequacy of Damage Mechanics to describe completely different kinds of materials. Ref. [21] gives further theoretical developments on the case of composites. Let us note that damage can be active or passive [22]. When the defects, especially microcracks, are closed, they do not affect the macroscopic behaviour. Damage state is not eliminated but has to be considered as passive: this can be the case under compression for instance. Ref. [23] gives some developments around that point.

3. Thermodynamic aspects The present developments are based on a thermodynamic theory of irreversible processes with internal state variables [24,25,26]. Here the presentation is limited to the simple case of isotropic hardening within the small strain hypothesis and to isotropic damage evolution. Extension to kinematic hardening is well known [27] and theories with anisotropic damage evolution have been developed in [26] and [28]. Generalization to finite strain can be found in [29]. The chosen state variables are given in table 1 for the present case. The plastic strain tensor is defined from

The specific free energy 'it', taken as the thermodynamic potential in which elasticity and plasticity are uncoupled, gives the law of thermoelasticity coupled with damage and the definition of the associated variables related to internal variables [26]: g, = '/,,(%, T, D) + ~p(T, p ) .

(9)

As proposed in [30], the damaged elastic behaviour is described through a strain equivalence and, referring to the effective stress concept:

X/Ie= ~1( 1 - -

D)A : % : %.

(10)

The stress writes: 3~

a=p~=(1-D)A:%

or

0=/~:%.

(11)

The thermodynamic force Y associated to D is defined by: 0q" Y=0~=-½A:%:%

(12)

Let us note that D includes all the damaging effects, the density p being considered as constant. In the finite strain case [29], the change of p (due to the growth of cavities) can be used as a damage variable. Moreover, in the present case, complete separability of the hardening and damage processes is assumed, '/'p does not depend on D. being the density of elastic strain energy defined by dW~ = a : %, the expression for Y shows that [30]: -y

We l-D-2

1 dive dD

at constant a and temperature.

(13)

J.L. Chaboche / Continuum damage mechanics

24 C~ J

Hardening and damage being uncoupled, it is sufficient to assume:

/

II

/F

,'E-6E

I '

- YD > O.

A

:. ')/ "

,'

?

As - Y is a quadratic function, this leads to /} > 0. The rupture criterion " - Y = [ Y[ = Yc ~ crack initiation" corresponds to an elastic energy criterion. It may be written in terms of D through the one-dimensional rupture stress o R [28]:

:/i .

~i":5:-::, '/i~;;

(16)

e

02

]_

o

" 8~p-:-See Fig. 7. Schematic of the dissipation during plastic flow and damage growth.

Then the variable - Y can be considered as the elastic strain energy release rate associated to a unit damage growth. Analogy with Fracture Mechanics concepts is clear. - Y may be calculated as a function of the hydrostatic stress o n = ½Tr(o) and the Von Mises equivalent stress O~q = (2-0 3 , : o,)a/2 where o' is the stress deviator [31]: °eq

[

(-tl

-}(1 + ~,) + 3(1 - 2z,) O'eq

(14)

The relation (13) defining - Y as the elastic strain energy release rate and the above dissipation aspects can be illustrated in fig. 7, showing the different parts of dissipation during the plasticity and rupture process [26]. The curve OA'B' represents the evolution of hardening during plastic flow OAB. Parts AB and BC correspond to the plastic flow and the elastic strain increase during the damaging process, respectively (schematized at constant stress). The total dissipated energy separates into (1) the energy stored in the system (hardening), (~) the heat dissipated energy, (~) the energy released by the system during the damaging process - Y3D, eventually converted into heat.

3.2. Dissipation The second principle of thermodynamics imposes that the intrinsic dissipation has to be positive:

o:ip-Rp-

Y/) > 0.

OR

Y~- 2E(1 - Oc) ~ Dc= 1

(15)

(2EYc)l/2

.

(17)

Many experiments have shown that: 0.2 < Dc < 0.8, which allows (1 - De) x to be neglected with regard to 1 when x is much greater than 1. The potential of dissipation is a scalar convex function of flux variables (ip, p, D, and the heat flux q) or their dual variables (by means of the Legendre-Fenchel transform), the state variables acting as parameters [25]: qb*(O, R , Y; Ce, T , p , D ) .

It gives the constitutive equations for the evolution of dissipative variables [26]. Here: ip

=

3q,* 30'

3,#* 3R'

P-

b -

3~* 3Y"

(18)

If q~* is a convex function of - Y, the damage dissipation (16) is automatically positive [30]. In the case of time independent plasticity and isotropic hardening the plastic flow can be particularized with the Von Mises plastic potential [26]:

f(o, R, O ) = 8 ~ q - k - k
(19)

where Gq is the equivalent effective stress (here %a/(1 - D ) ) and /~ = R/(1- D). It follows then from the normality rule ( o ' is the stress deviator):

ip=~ 3f~

3

X

o'

2 1 - D Oeq'

3f P= -X3R

X 1-D

~.

(20) . xl/2

-- ( ~ l [ p : Cp)

.

The plastic multiplier is determined from the consistency condition f ' = 0. In fact the above normality rule is a sufficient but not a necessary condition [32]. In some cases it appears as too restrictive, especially for materials or rupture conditions where the energetic rupture criterion (17) does not allow a correct description, as it is the case in creep. The only condition to be verified is b > 0 (because - Y is always positive).

J.L. Chaboche / Continuum damage mechanics 3.3. A slightly different theory for large strain damage [29] In that case the density changes constitute a damaging effect in the mechanical sense. The above approach can be modified into (in the isothermal case): " / ( % , p , 13) = ½Z : % : I[e + " / l ( P , T ) + "/2(fl, T ) , (21) where L is now independent of the damage variable/3. The isotropic part of damage is taken into account in the elastic moduli by: o = p-~% = p L : % = p 0 ( 1 - D ) L :

%.

(22)

This theory corresponds to a particular case of the preceding one in that concerns the elastic behaviour. The plastic flow rule and the damage evolution are written in the framework of generalized standard materials. The plastic potential is:

25

law, especially a form suggested by the Rice and Tracey formula for the growth of spherical voids under high triaxiality [33]. The main differences with the above theory (sections 3.1, 3.2) are: - the damaged elastic behaviour (only through the density change), - the additional term "/2 in the free energy, which leads to a completely different sense to the associated thermodynamic force and to the dissipation, - the hydrostatic term in the plastic strain rate [26]. In the first theory it comes only through the anisotropic form of the damage variable [34], - the form of the plastic potential. Comparing (23) and (19) shows that damage reduces the hardening R in the second theory though it does not in the first one. With o H = 0 for instance: Oeq = p(R + k) instead of

aeq = R + k(1 - D ).

4. Damage

growth

equations

4.1. Creep damage where J ( a / p ) = Oeq/P. The generafized force B is conjugated to the damage variable/3: B=o

0"/2 0,8 "

(24)

Continuum Damage Mechanics was developed first for the case of creep damage [1,2]. The RabotnovKachanov equation can be considered as very classical. D varying between 0 for the undamaged material and 1 for the rupture, it writes, under pure tensile stress:

The flow rule now writes [29]: (1 - D) -k dt,

dD =

~p=X

Of a(o/0)'

Decomposing it into deviatoric and spherical parts: •,

=2

'

.

(28)

(25)

=-

Xn

,[°hi

(26)

where r, k, A are material and temperature dependent coefficients. Their determination is made from constant stress creep tests, for which the integration from 0 to 1 gives the rupture time:

Oeq to= k +1l

and for the internal variables:

Of 0f

/3=-X-~-~ = +)tg(~).

(A)-r

(29)

while evolution of damage is given by:

)1/2

D= l- (l-

~)l/(k

+ l).

(30)

(27)

Let us note that g is an increasing function of ort/p but '/2 is a decreasing function of/3, so that B < 0 and the hydrostatic strain rate is positive. As in the first theory the growth of damage decreases the free energy. In [29], the function g is chosen as an exponential and different forms are considered for the relation between fl and p, by taking into account the mass conservation

The concept of effective stress, introduced in the secondary creep law (Norton's equation), allows the calculation of tertiary creep curves as well as the prediction of the change in creep ductility [28,34]. Fig. 8 gives an example: = ~p

o_ /¢(1

(31) D)

"

J.L. Chaboche/ Continuumdamage mechanics

26

which can describe adequately the isochronous surfaces in a large domain [38,39]. However, the form (34) automatically gives no damage under pure compression, which is not perfectly true in polycristalline metals, as shown by two level creep tests [40]. Let us mention a possible different form for the damage growth equation, using the creep strain instead of time [41]:

epl%) I 5-

- • x

+

Predictions o Tests

\ Failure

4-

"' '~"'~ INIO0

Z = 1000°C

3"

a B

dD=A[Jo(o)] %q de.q,

(35)

2' such form was deduced from cavity measurements on notched specimens. It is possible to show that (28) and (31) are equivalent to (35) through the following one-toone mapping [42] (in tension):

1. ~x~.-X.-~

o

10

X--X- : ~ ( ' - x

20

40

,

t (h)

60

Fig. 8. Calculated and measured creep curves on superalloy IN 100. Prediction of creep ductility.

D ~ [1 - (1 - D ) k - " + ' ] a + '

(36)

4.2. Fatigue damage Under multiaxial stress conditions the damage equations can be generalized by describing isodamage surfaces (or isochronous surfaces) defined with proper stress invariants: - the octahedral shear stress ./2(o) related to the effects of shear, - the hydrostatic stress ./1(o) = Tr(o) = a : 1, which greatly affects the growth of the cavities, - the maximum principal stress Jo(a) = 01, which opens the microcracks and causes them to grow. Following Hayhurst's method [35], the equivalent stress can be defined through a linear combination:

X(o)=aJo(o)+flJl(a)+(1-a-fl)J2(o),

(32)

where a, /3 are coefficients dependent on the material and temperature. The time to failure under a fixed multiaxial stress is expressed as:

The creep damage equations can be considered as taking into account the evolution of microstructural defects in an indirect manner. Works by the material scientists have shown the combination of two mechanisms; the nucleation and growth of cavities [36,37]. It is possible to make some connections between the equations obtained from material science and the more macroscopic ones developed in the framework of C D M [381. In that concerns the multiaxial stress criterion, the physical interpretations lead often to a product form:

x( o) = [ Jo(o)]"[ J2(,)] '-~,

(34)

In the case of fatigue several aspects have to be considered: the existence of micro-initiation and nficro-propagation stages, - the non-linear-cumulative effects for two-level tests or block-program loading conditions, the existence of a fatigue limit, but its marked decrease after prior damage, the effect of mean-stress either for the fatigue limit and for the S - N curves. Fatigue damage accumulation models have been considered in [43,5,7]. A common general form is obtained with: -

-

-

- [OM--ff]

dD=D,~( . . . . )[ M ( ~ ) ]

'8

dN,

(37)

where a M and 8 are the maximum and mean stress, respectively. Several choices [44] for a lead to the rules considered in [5,45,46]. The key of the non-linear effect is the dependency in o M and ff which, after integrating for a two-level test, gives:

NF 2N2-I-(N1)O-'~2)/t'-'~') .

(38)

The function M ( 8 ) is deduced from a linear dependency between 8 and the fatigue limit. This cumulative damage equation allows a very good description of multi-level fatigue tests [7,44]. In a certain sense it includes in a continuous way the micro-initiation and micro-propagation phases as discussed in [5,11,47]. Moreover, by a convenient variable change, eq. (37) can be incorporated into Continuum Damage

27

J.L Chaboche / Continuum damage mechanics Mechanics, with the effective stress concept, as shown by damage measurements described in [19,26]. The microcrack measurements made in [11,48] for instance then show the possible equivalence between: the definition by the effective stress concept, the definition in terms of the remaining life concepts (subjacent in eq. (37)), the quantification of physical damage, in terms of microcracking. In the case of Low-cycle Fatigue, the conventional parametrization of the life is written in terms of the plastic (or total) strain range. Provided the existence of a one-to-one relation between o M and Z~Cp (the cyclic curve) eq. (37) can still be used and contains independently the influence of mean-stress. The generalization to multiaxial loading conditions is a difficult problem. At least two parameters have to be considered: an equivalent shear-stress amplitude, a mean (or maximum) hydrostatic stress. Experiments near the fatigue limit show the independency in the mean shear-stress and in the hydrostatic range [49]. A possible form to generalize (37) has been proposed in [28].

4.4. Ductile plastic damage Ductile damage in metals is essentially the initiation and growth of cavities due to large deformations. Experiments of ductile rupture show that the dissipative potential q~* can be expressed, in the framework of time-independent plasticity [26]:

~b* = Oeq--/~ -- k + 2-1 S --~

ip =~3q~* - A Oq'* = (Zip: 0o ' P = " ' ~ R 0q,* = b = -• ~S-Yp.

One advantage of the CDM approach in creep and fatigue is to allow a natural way to predict creep-fatigue interaction. The simplest hypothesis consists in a direct summation of creep and fatigue damages, which leads to [4,19]: d D =fc(O, D ) dt +fF(OM, o, D ) d N ,

(39)

where fc is for instance deduced from eq. (28) and fF from eq. (37). These two functions can be determined from pure tensile creep tests and pure fatigue tests (high frequency). The conditions at low-frequency or with hold times are then predicted by integrating numerically (39). This approach has given good results for several materials [19,50,51,52]. Let us note that the additive hypothesis does not correspond to the direct addition of physical damages of different natures. For instance microcracks and cavities are not added. Only their mechanical effects add, in the framework of the effective stress concept. The mechanical effects are obtained by the above mentioned one-to-one mappings between the physical damage (crackgor cavities) and the corresponding macroscopic variable D (see sections 4.1 and 4.2).

(40) • % )

~1/2

(41) '

(42)

Using the relation (14) between Oeq and - Y, and a particular choice for hardening:

Oeq=/~ + k = gp 1/M.

(43)

K and M being coefficients which are material dependent, one obtains the following differential constitutive equation for ductile plastic damage [18]: b =

4.3. Creep-fatigue interaction

( -S y)2,

+

+ 3(1 -

In the case of radial loading, when the principal directions of stresses do not vary, the triaxiality ratio on/O~q is constant and this expression may be integrated using the conditions: P < P i~ (damage threshold) = D = 0, p = p ~ (strain to rupture) = D = D c. Neglecting elastic strain in the calculation of p and using Pr)/PR = Er)/CR, the equation for damage evolution may be written in terms of the one-dimensional threshold c D and one-dimensional strain to rupture ~R [181:

o

,.-

(I p

+ . ) + 3(1 - 2 . )

t /21 / - ,D

(45)

Identification of such a model consists in the quantitative evaluation of the coefficients ~r), ~R and D c (Poisson's coefficient being known from elasticity), which can be done from experimental results such as those shown on fig. 5a. As shown in fig. 9, the present model gives the influence of triaxiality on the strain to failure [18]. It compares fairly well with the MacClintock [53] or Rice and Tracey [33] models and can be used to predict the

J.L. Chaboche / Continuum damage mechanics

28 p, ,U..

t 05

o

,,

i

~'

-"-

3

Ocq

Fig. 9. Influence of triaxiality on fracture strain: • A 508 steel, ® H steel, - . . . . . domain covered by Mc-Clintock RiceTracey models, - domain covered by the present model.

fracture limits of metal forming [31]. Let us note that the modified thermodynamic framework developed by Rousselier [29] (see section 3.3) leads to similar results. It fits directly the Rice and Tracey model for a special choice of g'2(fl)- As shown in [29] this theory allows a correct prediction of plastic instabilities and can be used to predict ductile fracture by means of local approaches [54,55]. Let us mention also some works based on homogenization concepts which, in the case of ductile fracture, allow informations about the damage evolution equations [56,57].

5. Damage anisotropy The directional nature of damage is clear in many situations. In creep for instance, microstructure observations have shown two classes of metallic materials [38,58]: - for copper, cavitation takes place on grain boundaries more or less perpendicular to the maximum principal stress, - for aluminium alloys, grain boundary cavitation is much more isotropically distributed. Clearly these observations relate to the anisotropy of the damage rate equation. For a situation with varying directions of principal stresses the anisotropy of the present damage state could be reduced (even for copper type materials). In Continuum Damage Mechanics the problem of anisotropy concerns also the effect of damage on the constitutive behaviour. Several levels of theories can then be considered, where the damage variable is

considered as a scalar function [59], vectors [22], second [12] or fourth order tensors [60]. See also [61] for a more complete review. The scalar measure of damage anisotropy is defined (in the creep range) by considering the grain boundaries orthogonal to the direction n and the fraction of boundaries which are cavitated. Then to each direction n is associated a scalar valued function V(n). In ref. [59], it is shown that a tensorial decomposition of such elementary damage measure leads to even order tensors. However, if one considers the damage as produced by small parallel cracks, it is possible to associate a vector to each crack direction [22]. Such kind of theory, similar to the slip theory of plasticity can be developed with a similar thermodynamic framework as the one presented in sections 3.1 and 3.2, introducing for instance a vector as the dual variable to the present damage variable (instead of the scalar Y or of the fourth order tensor Y defined in [26]). This kind of theory is attractive to give directly the distinction between active damage (open microcracks) and passive damage (closed microcracks). The main difficulty is the number of independent systems which have to be considered in a general case, but several useful applications have been done for concrete under special loading cases [22]. Two kinds of theories are summarized below, with a little more details. In the first one damage acts as a second order tensor and its evolution is directed by the so-called net stress tensor. Influence of damage on the constitutive behaviour is described by a fourth order transformation. In the second one damage is defined directly as a fourth order tensor.

5.1. The anisotropic theory of Murakarni and Ohno [12,13] As shown in section 2.2, the damaged state of a material can be interpreted by a second order damage tensor. In its principal directions it writes 12 = I2jvi ® J,j. Considering the vector force acting on a surface element and on the net surface element, the net stress is introduced in [12] as: o* = ½ ( o . ~ +

~.o),

~ = ( 1 - 12) 1.

(46)

The creep damage evolution is expressed in the form: r

where X(o*) is the invariant describing the isochronous surface (see eq. (32)). Coefficient ~, adjusts the degree of anisotropy of the damage evolution. On the other hand, the effect of the present damage

J.L Chaboche / Continuum damage mechanics on the strain behaviour (elastic, plastic, creep) can only be introduced through a fourth-rank tensor. Murakami and Ohno use an effective stress of the form [12]: S = ½IF: o + ( F : o)T],

(48)

where the fourth-rank tensor F is deduced from the tensor ~2 or rather from (I)= ( 1 - $2) -1 by a general expression deduced from the representation theory. This effective stress is introduced in the creep constitutive law. As a particular case, under tension and neglecting hardening it writes:

[

4p=

o K(1-c~2)

[

'

I)=

o A(1-/2)

]r

(49)

29

state: D = 1 - A : ,'~-x.

(53)

such an operator D can represent any observed behaviour for the damaged elastic material. We have: o =A:

ce = (1-/)):

A: 'e

(54)

and the symmetry of A and A is verified. The thermodynamic framework presented in section 3.1 can be generalized in the present case. Note that slightly different formulations have been developed, using a second order damage tensor [65]. The creep damage evolution is expressed here as: b= 0(6)b,

(55)

Integrating (49b) between 0 and 1 leads to: I2 = 1 - (1 - t /_t c ).x/(r+l) .

(50)

Experiments on perforated plates have shown the adequacy of the present theory for both the anisotropic effects on the elastic-plastic behaviour and on the rupture conditions [62]. Also experiments on slotted plates show similar possibilities [63]. The theory is capable to predict fairly correctly creep rupture for sequence tension-torsion experiments [12,13], especially longer lives when changing the direction of the maximum principal stress.

5.2. The theory developed by the author [60,64]

where Q ( 6 ) gives the directionality of damage evolution, in the principal effective stress system. It is introduced by rotating a fourth order tensor Q* which is fixed for a given material. Its identification is made by linearly superposing the isotropic case and the one corresponding to planar microcracks, all parallel, developing perpendicular to the direction of maximum principal stress: Q* = vl + (1 - y ) G .

(56)

The anisotropy tensor G is defined from elastic analysis results [64]. The scalar damage rate b in relation (55) (D is a scalar measure of damage density) obeys the Rabotnov-Kachanov equation. The damage law writes: k

The damage variable is defined directly through the effective stress concept, that is by means of an equivalence made between the damaged and undamaged materials in terms of their stress-strain behaviours. In the case of linear elasticity, one writes o = -4 : c and 6 = A : %, respectively, where A represents the damaged elasticity tensor. The effective stress follows: 6=A:%=A:fx

1:o.

where invariants 2 and X are defined properly [34]. Eq. (57) has to be compared to eq. (47) of the first theory. Using the effective stress concept in the thermodynamic framework leads to the creep strain rate equation, applied here under pure tension [28,34]:

(51) 4p= 1 - D

The fourth-rank tensor M = A : A-1 can be set in a form that generalizes the initial theories of Kachanov and Rabotnov, such that the effective stress can be expressed: = (t-D)-'

:°,

K(12D)

the damage equation (57) expresses in that case: JDm(i)r(1-n)-k.

(59)

(52) 5.3. Comparison between the two theories

where D is an asymmetrical fourth-rank damage tensor. It can be measured through the elastic behaviour measured in the present state and in the initial (undamaged)

In the particular case of tension, the comparison with relation (49) in the theory of Murakami-Ohno

,L L. Chaboche / Continuum damage mechanics

30

D=I -'E/E

o.,

i[? '" y

~ 0.5

=

1

2a/d I

Fig. 10. Damages defined by the net area reduction and by the effective cross section. Results of elastic calculations on perforated (O) or cracked ( + ) plates. Plain lines are calculated by eq. (61).

easily shows the main difference between I2 and D, difference corresponding to the two definitions used. Integrating (59) between 0 and 1 leads to: D=I

-

(1

-

._ c ).1/(k+1)

t/t

(60)

and combining with (50) gives [34]: O = 1 -- (1 -- ff~)(r+l)/(k+l)

(61)

Fig. 10 shows this correspondence together with elasticity results for perforated and cracked plates. It proves that the influence of damage on the constitutive behaviour is less sensitive than given by the net area definition. This is described in the two theories by means of the influence factor c in the first one and the use of exponent k instead of r in relation (59) for the second one. Moreover the predicted creep ductilities with the two theories are identical if the coefficient c in the theory of Murakami (damage effect on the constitutive equations) is c = (n + 1 ) / ( k + 1). In that concerns anisotropic effects several similarities between the two theories have been pointed out [34] either for the rupture conditions (damaging effect) and for the constitutive behaviour.

worthiness in the past and in the future. However in some situations this global approach presents some difficulties, even inconsistencies, especially when material non-linearities play an important role, for example in ductile fracture or creep crack growth. An alternate method, presently under development is the so-called "local approach of fracture". Some attention is given here to the additional possibilities offered by Damage Mechanics to this development. See refs. [66,67] for more detailed discussions. In fact, two levels can be considered for local approaches: - Numerical methods using discrete crack increments: one uses the technique of node release to produce the crack growth when some critical value of a physical quantity is reached at some critical distance ahead of the crack tip [68-71]. This physical quantity can be equivalent stress or strain, energy or some measure of local damage. In such situations, the prediction of crack growth using this method will be mesh dependent [69], but the local mesh size is fixed by means of statistical considerations [55,72]. Simplified approaches can be developed, using analytical stress-strain fields near the crack tip [73]. - Use of the Continuum Damage Mechanics, including the progressive deterioration and the corresponding stiffness reduction. The crack (damaged zone) is then taken as the locus of material points with no rigidity where damage has reached its critical value (fig. 11). Conceptually, due to the stress redistribution associated with viscoplasticity and strength decrease, there is no need to introduce a critical distance nor node release techniques for crack growth simulation. After some

/

~

~

~

/ H/gh grad/enfof

6. Local a p p r o a c h e s of fracture

The Linear Elastic and Non-Linear Fracture Mechanics are very useful to describe the crack propagation and the fracture of different structures. In many cases L E F M constitutes a practical tool to predict crack growth, especially in situation of non dissipative "small scale yielding" materials. Not doubt on its utility and

Fig. 11. Crack as a damaged zone.

J.L. Chaboche / Continuum damage mechanics maximum, the stresses in the plastic zone decrease continuously as damage grows, approaching the crack tip. This approach was studied first by Hayhurst et al. [74] in the creep domain and is now under a large development in ductile fracture [54,55], creep [75,76,77] and fatigue [78], rupture of concrete [79,80]... In some cases, analytical solutions can be obtained, considering a discontinuous growth for damage (sudden change from 0 to 1) [81,82]. The practical use of such local approaches needs further developments in finite element technique and the systematic checking of the discretization procedures. Also simplified methods have to be studied, including for instance the use of analytically defined stress-strain fields.

[4]

[5]

[6]

[7]

[8] 7.

C o n c l u s i o n

The main features of Damage Mechanics have been reviewed together with its present use and developments, considering successively: the measures and definitions of damage, its incorporation into thermodynamic theories, the damage growth equations, the influence of damage on the mechanical behaviour, especially in the case of anisotropy, the use of damage mechanics, as an additional tool, to develop local approaches of fracture. This rational approach of damage proceeds as an extension of Continuum Mechanics. It allows many possibilities, introducing for instance some connections between microstructure measurements and mechanical parameters. Moreover this theory finds applications in many different situations (various loading conditions, various materials, various physical processes... ). In that concerns life predictions in structures, Damage Mechanics allows to take into account the coupling effects between deterioration processes and mechanical behaviour. It will certainly be developed further an extensively used in the future as a complementary tool between Continuum Mechanics and Fracture Mechanics.

[9] [10]

[11]

[12]

[13]

[14]

[15]

[16]

[17] R e f e r e n c e s

[1] L.M. Kachanov, Time of the rupture process under creep conditions, Izv. Akad. Nauk. SSR, Otd Tekh. Nauk. 8 (1958) 26-31. [2] Y.N. Rabotnov, Creep problems in structural members (North-Holland, Amsterdam, 1969. [3] J. Hult, Continuum Damage Mechanics - Capabilities,

[18]

[19]

[20]

31

limitations and promises, in: Mechanisms of Deformation and Fracture (Pergamon, Oxford, 1979) pp. 233-347. J.L. Chaboche, Lifetime predictions and cumulative damage under high temperature conditions, Int. Symp. on Low Cycle Fatigue and Life Prediction, Firminy, France, 1980 (ASTM STP 770, 1982). S.S. Manson, Some useful concepts for the designer in treating cumulative damage at elevated temperature, I.C.M. 3, Cambridge, 1979, Vol. 1, pp. 13-45. E. Krempl, On phenomenological failure laws for metals under repeated and sustained loading (fatigue and creep), Conf. on Environmental Degradation of Engineering Materials, Blacksburg, Virginia, 1977. J.L. Chaboche, Une loi diffrrentielle d'endommagement de fatigue avec cumulation non linraire, Revue Fran~aise de M6canique 50-51 (1974); English translation in "Annales de I'ITBTP", HS 39 (1977). D.A. Woodford, A critical assessment of the life fraction rule for creep rupture under nonsteady stress or temperature, Int. Conf. on Creep Fatigue elevated Temperature Applications, Philadelphia, Sept. 1973, Sheffield, April 1974. V.D. Dyson and D. McLean, Creep of Nimonic 80A in torsion and in tension, Met. Sci. 2 (1977) 37. C. Levaillant and A. Pineau, Assessment of high temperature LCF and life of austenitic stainless steels by using intergranular damage as a correlating parameter, LCF and Life Prediction, ASTM-STP 770, Eds.: Amzallag, Leis, Rabbe (1982) pp. 169-193. G. Cailletaud and C. Levaillant, Creep-fatigue life prediction: What about initiation?, Nucl. Engrg. Des. 83 (1984) 279-292. S. Murakami and N. Ohno, A continuum theory of creep and creep damage, 3rd IUTAM Symp. on Creep in Structures, Leicester, 1980. S. Murakami, Notion of continuum damage mechanics and its application to anisotropic creep damage theory, J. Engrg. Mater. and Technol. 105 (1983) 99. J.J. Jonas and B. Baudelet, Effect of crack and cavity generation on tensile stability, Acta Metallurgica 25 (Pergamon Press, 1977) 43-50. G. Cailletaud, H. Policella and G. Baudin, Mesure de drformation et d'endommagement par mrthode 61ectrique, La Recherche Arrospatiale, No. 1980-1. T. Bui-Quoc, J. Dubuc, A. Bazergui and A. Biron, Cumulative fatigue damage under stress controlled conditions, J. of Basic Engrg. Trans. ASME (1971) 691-698. C. Duvaut, Analyse fonctionelle - M~canique des millieux continus-homogrnrisation, Theoretical and Applied Mechanics (North-Holland, Amsterdam, 1976). J. LemMtre, A Continuum Damage Mechanics model for ductile fracture, J. Engrg. Mater. and Technol. 107 (1985) 83-89. J. Lemaitre and J.L. Chaboche, Aspect phrnomrnologique de la rupture par endommagement, J. de Mrcanique Appliqure 2, No. 3 (1978) 317-365. A. Charewicz and I.M. Daniel, Fatigue damage mecha-

32

J.L. Chaboche / Continuum damage mechanics

nisms and residual properties of graphite/epoxy laminates, Symp. IUTAM on Mechanics of Damage and Fatigue, Ha~'fa, 1985. [21] R. Talreja, A continuum mechanics characterization of damage in composite materials", Proc. Royal Soc. Lond. A 399 (1985) 195-216. [22] D. Krajcinovic and G.U. Fonseka, The Continuous Damage Theory of brittle materials, Parts 1 and 2, J. of Applied Mechanics, Trans. ASME 48 (1981) 809-824. [23] P. Ladeveze and J. Lema~tre, Damage effective stress in quasi-unilateral condition, Congr+s IUTAM, Lyngby, Denmark, 1984. [24] F. Sidoroff, On the formulation of plasticity and viscoplasticity with internal variables, Arch. Mech., Poland, Vol. 27, 5-7 (1975) pp. 807-819. [25] P. Germain, Q.S. Nguyen and P. Suquet, Continuum thermodynamics, J. of Applied Mechanics, ASME 50 (1983) 1010-1020. [26] J. Lema~tre and J.L. Chaboche, Mrcanique des Matrriaux So[ides (Dunod, Paris, 1985). [27] B. Halphen and S. Nguyen, Sur les matrriaux standard grn6ralisrs, J. Mrcanique 14, No. 1 (1975) 39-63. [28] J.L. Chaboche, Description thermodynamique et phrnomrnologique de la viscoplasticit6 cyclique avec endommagement, Th+se Univ. Paris VI et publication ONERA no 1978-3. [29] G. Rousselier, Finite deformation constitutive relations including ductile fracture damage, IUTAM Syrup. on Three-Dimensional Constitutive Relations and Ductile Fracture, Dourdan 1980, Eds. Nemat-Nasser (North-Holland, Amsterdam, 1981) pp. 331-355. [30] J.L. Chaboche, Sur l'utilisation des variables d'rtat interne pour la description du comportement viscoplastique et de la rupture par endommagement, Symp. Franco-Polonais de Rhrologie et Mrcanique, Cravovie, 1977. [31] J. Lema~tre, How to use Damage Mechanics, Nucl. Engrg. Des. 80 (1984) 233-245. [32] E.T. Onat, Representation of mechanical behaviour in the presence of internal damage, IUTAM Symp. on Mechanics of Damage and Fatigue, Haifa, 1985. [33] J.R. Rice and D.M. Tracey, On the ductile enlargement of voids in triaxial stress fields, J. Mech. Phys. Solids 17 (1969) 201. [34] J.L. Chaboche, Constitutive equations in creep-fracture damage, Engineering Approaches to High Temperature Design, Eds. Wishire, Owen (Pineridge Press, Swansea, 1983). [35] D.R. Hayhurst, Creep rupture under multi-axial state of stress, J. Mech. Phys. Solids 20, No. 6 (1972) 381-390. [36] G. Greenwood, Creep life and ductility, Int. Congress on Metals, Cambridge, 1973; Microstructure and the Design of Alloys 2 (1973) 91. [37] B.F. Dyson, Constrained cavity growth, its use in quantifying recent creep fracture experiments, Canadian Met. Quart. 18 (1979) 31. [38] D.R. Hayhurst, On the role of the creep continuum damage in structural mechanics, Engineering Approaches to High

[39]

[40]

[41]

[42]

[43] [44]

[45]

[46]

[47]

[48]

[49]

[50]

[51]

[52]

[53] [54]

Temperature Design, Eds. Wilshire, Owen (Pineridge Press, Swansea, 1983). P. Delobelle, Etude en contraintes biaxiales des lois de comportement d'un acier inoxydable du type 17-12 SPH Modrlisation et identification - Introduction de l'endommagement, cas de l'Inconel 718. Th~se de Doctorat d'Etat, Besan~on, 1985. H. Policella, P. Paulmier and S. Savalle, Effet de la contrainte moyenne en fatigue vibratorie h haute temprrature, Rrsultats exp~rimentaux sur IN 100 h 900°C, R.T. Onera, no 49/1765. J. Beziat, A. Diboine, C. Levaillant and A. Pineau, Creep damage in a 316 S.S. under triaxial stresses, 4th Int. Seminar on Inelastic analysis and Life Prediction in High Temperature Environment, Chicago, Illinois, 1983. E. Contesti, Endommagement en fluage: Exprrience et modrlisation, Rrunion GRECO Grandes Drformations et endommagement - GIS Rupture ~t chaud, Aussois (France), 30 janvier 1986. S.M. Marco and W.L. Starkey, A concept of fatigue damage, Trans. ASME 76, no 4 (1954) 627-632. M. Chaudonneret and J.L. Chaboche, Fatigue life prediction of notched specimens, Int. Conf. on Fatigue of Engineering Materials and Structures, Sheffield, Sept. 1986. S. Subramanyan, A cumulative damage rule based on the knee point of the S-N curve, Trans. ASME, J. of Mater. Technol. (1976) 316-321. Z. Hashin and C. Cairo, Cumulative damage under two level cycling: Some theoretical predictions and test data, Fatigue of Engineering Materials and Structures 2 (1979) 345-350. K.J. Miller and K.P. Zachariah, Cumulative damage laws for fatigue crack initiation and stage 1 propagation, Journal of Strain Analysis 12, No. 4 (1977). D.F. Socie, J.W. Fash and F.A. Leckie, A continuum damage model for fatigue analysis of cast iron, ASME Conf. in Life Prediction, Albany, New York, 1983. K. Dang Van, Sur la rrsistance h la fatigue des mrtaux, Sciences et Technique de rArmement, 47, 36me Fasc. (1973). A. Plumtree and J. Lema~tre, Apphcation of damage concepts to predict creep-fatigue failures, ASME. Pressure Vessels and Piping Conf. Montreal, June 1979. A. Del Puglia and E. Vitale, Damage concept in creep fatigue: Current theories and applications, Proc. ASMEPVP Conf. Orlando, Florida, 1982 in: Inelastic Analysis and Life Prediction at Elevated Temperature, Ed. G. Baylac, ASME-PVP 59, Book No. 216. G. Cailletaud and J.L. Chaboche, Lifetime prediction in 304 S.S. by damage approach, ASME-PVP Conf. Orlando, Florida, 1982, ASME Paper no 82-PVP 79. F. McClintock, A criterion for ductile fracture by the growth of holes, ASME J. of Applied Mechanics (1968). G. Rousselier, J.C. Devaux and G. Mottet, Ductile initiation and crack growth in tensile specimens - Application of Continuum Damage Mechanics, SMiRT 8, Brussels, 1985.

J.L. Chaboche / Continuum damage mechanics [55] G. Rousselier, Les modrles de rupture ductile et leurs possibilitrs actuelles dans le cadre de l'approache locale de la rupture, Srminaire Approaches Locales de la Rupture, Moret-sur-Loing, France, 1986. [56] P. Suquet, Plasticit6 et homogrnrisation, Thbse Doctorat d'Etat, Univ. Paris 6 (1982). [57] A. Dragon and A. Chihab, Quantifying of ductile fracture damage evolution by homogenization approach, SMirt 8, Brussel, 1985. [58] F.A. Leckie and D.R. Hayhurst, Creep rupture of structures, Proc. Royal Soc. London 340 (1974) 323-347. [59] F.A. Leckie and E.T. Onat, Tensorial nature of damage measuring internal variables, IUTAM Symp. on Physical Non-Linearities in Structural Analysis, Senlis, France, 1980 (Springer). [60] J.L. Chaboche, Continuous Damage Mechanics: a tool to describe phenomena before crack initiation, Nucl. Engrg. Des. 64 (1981) 233-247. [61] D. Krajcinovic, Continuum Damage Mechanics, Applied Mechanics Reviews 37 No. 1 (1984). [62] S. Murakami and T. Imaizumi, Mechanical description of creep damage state and its experimental verification, J. de Mrchanique Throrique et Appliqure 1, No. 5 (1982) 743-761. [63] A. Litewka, Experimental simulation of anisotropic damage, Mechanika Teoretyczna I Stosowana 2/3, 21 (1983) 361-370. [64] J.L. Chaboche, Le concept de contrainte effective appliqu6 h l'61asticit6 et h la viscoplasticit6 en prrsence d'un endommagment anisotrope, Col. EUROMECH 115, Grenoble 1979 (CNRS, 1982). [65] J.P. Cordebois and F. Sidoroff, Anisotropie 61astique induite par endommagement. Col. EUROMECH 115, Comportement Mrcanique des Sohdes Anisotropes, Grenoble, 1979 (CNRS, 1982). [66] J. Janson and J. Hult, Fracture mechanics and damage mechanics, a combined approach, J. Mrcanique Appliqure 1, No. 1 (1977). [67] J. Lema~tre, Local approach of fracture, Symp. on Mechanics of Damage and Fatigue, 1-4 July 1985, Ha'ifa, Israel. [68] L. Anquez, Elastoplastic crack propagation (fatigue and failure), La Recherche Arrospatiale 1983-2. [69] J.C. Newman Jr., Finite element analysis of fatigue crack propagation including the effects of crack closure, Ph. D Thesis, V.P.I., Blacksburg (1974).

33

[70] T.D. Hinnerirhs, Viscoplastic and creep crack growth analysis by the finite element method, Ph. D Thesis, Air Force Institute of Technology (1980). [71] J.C. Devaux and G. Mottet, Drchirure ductile des aciers faiblement allirs: ModUles numrriques, Rapport no 79-057, Framatome (1984). [72] F. Mudry, A local approach to cleavage fracture, Srminaire International "Approches locales de la Rupture", Moretsur-Loing, France, 1986. [73] E. Mass and A. Pineau, Creep crack growth behaviour of type 316 L steel, Engrg. Fracture Mechanics 22, No. 2 (1985) 307-325. [74] D.R. Hayhurst, et al., Estimates of the creep rupture lifetime of structures using the finite element method, J. Mech. Phys. of Solids 23 (1975) 335. [75] K. Saanouni, J.L. Chaboche and C. Bathias, On the creep crack growth prediction by a local approach, Symp. IUTAM on Mechanics of Damage and Fatigue, Haffa, 1985, J. of Engineering Fracture Mechanics 25, No. 5 / 6 (1986) 677-691. [76] D.R. Hayhurst, On the use of creep damage constitutive equations to predict the behaviour of cracked specimens under constant applied loads, 5th Int. Seminar "Inelastic Analysis and Life Prediction in High Temperature Environment", Paris, 1985. [77] K.P. Walker and D.A. Wilson, Constitutive modeling of engine materials, PWA report FR17911 (AFWAL-TR-844073) (1984). [78] A. Benallal, R. Billardon and D. Marquis, Prrvision de l'amorsage et de la propagation des fissures par la mrcanique de l'endommagment, Sprciale Publication SFM, Paris (1984). [79] J. Mazars, A description of micro and macro scale damage of concrete structures, Symp. on Mechanics of Damage and Fatigue, 1-4 July 1985, Ha'ifa, Israel. [80] Bazant, Cedolin, Blunt crack band propagation in finite element analysis, J. Engrg. Mech. Div. (April 1979) p. 297. [81] H.D. Bui and A. Ehrlacher, Propagation of damage in elastic and plastic solids, ICF 6, Vol. 2, Cannes, 1981. [82] H.D. Bui, K. Dang Van and E. De Langre, A simplified analysis of creep crack growth using local approach, S~minaire International "Approaches locales de la Rupture, Moret-sur-Loing, France, 1986.