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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 51(1985) 31-49 NORTH-HOLLAND COUPLED ELASTO-PLASTICI~ AND DAMAGE CONSTITUTIVE EQUATIONS Jean LEMAITRE ~~or~toir~ & ~~~an~~ et Techn~i5g~e, ENSET- Univ. Paris 6-CNRS, 94230 Cachan, France Received 26 November 1984 The basic equations to mode1 the coupling between strain and damage behaviors are written within the framework of the thermodynamics of irreversible processes. The damage is represented by a scalar internal variable which expresses the loss of strength of materials during such processes as fatigue, ductile or creep strains. Applications are given in elasticity coupled with damage for brittle failure of concrete or high-cycle fatigue of metals, in plasticity coupled with damage for ductile fracture or low-cycle fatigue, and in visco-plasticity coupled with creep and fatigue damages. 1. Uncoupled and coupled structures calculations The usual way to predict the conditions of failure in structural calculation is to proceed in three steps, as indicated on the scheme in Fig. 1. (1) The stress and strain fields are first calculated with elastic, plastic or visco-plastic constitutive equations by a method of calculation adapted to the problem, the finite element method for example [l]_ WMAGE CRACK ~UTI~ REAGAN LAWS LAWS STRESS AND STRAIN FIELDS Fig. 1. Prediction of failure with classical constitutive equations. ~~78~/85/$3.~ @ 1985, Elsevier Science Publishers B.V. (North-Holland)

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 51(1985) 31-49 NORTH-HOLLAND

COUPLED ELASTO-PLASTICI~ AND DAMAGE CONSTITUTIVE EQUATIONS

Jean LEMAITRE ~~or~toir~ & ~~~an~~ et Techn~i5g~e, ENSET- Univ. Paris 6-CNRS, 94230 Cachan, France

Received 26 November 1984

The basic equations to mode1 the coupling between strain and damage behaviors are written within the framework of the thermodynamics of irreversible processes. The damage is represented by a scalar internal variable which expresses the loss of strength of materials during such processes as fatigue, ductile or creep strains. Applications are given in elasticity coupled with damage for brittle failure of concrete or high-cycle fatigue of metals, in plasticity coupled with damage for ductile fracture or low-cycle fatigue, and in visco-plasticity coupled with creep and fatigue damages.

1. Uncoupled and coupled structures calculations

The usual way to predict the conditions of failure in structural calculation is to proceed in three steps, as indicated on the scheme in Fig. 1.

(1) The stress and strain fields are first calculated with elastic, plastic or visco-plastic constitutive equations by a method of calculation adapted to the problem, the finite element method for example [l]_

WMAGE CRACK

~UTI~ REAGAN

LAWS LAWS

STRESS AND

STRAIN FIELDS

Fig. 1. Prediction of failure with classical constitutive equations.

~~78~/85/$3.~ @ 1985, Elsevier Science Publishers B.V. (North-Holland)

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32 J. Lemaitre, Coupled elasto-plasticity and damage

(2) In a second step, these results are used to calculate the condition of a macro-crack initiation with the help of a damage constitutive equation [2], for example, the number of cycles to fatigue failure due to a loading variable with time.

(3) Finally, the growth and the propagation of the crack through the structure is analysed by means of fracture mechanics, using concepts of stress intensity factors or strain energy release rate [3].

This procedure implicitly assumes that the state of damage of the structure does not influence the state of stress or strain, that is there is no coupling between constitutive equations of strain and damage. This assumption was not too bad as far as the structure calculations were not too accurate. With the progress of constitutive equations, taking into account anisotropic strain hardening in plasticity or visco-plasticity, and the progress of numerical calculations in relation with faster computers, this non-coupling hypothesis must be released in order to have an accuracy of the same order of magnitude in the representation of physics and in the numerical procedure.

From the physical point of view, this coupling is due to the nature of damage. The brittle or fatigue or ductile damages are mainly the development of micro-cracks and micro-cavities; that is, creation of surfaces of separation in the materials which reduce the rigidity of the solids. We observe the following:

(1) The elasticity modulus decreases as the damage progresses. (2) The plastic strain-hardening decreases also and may become negative just before failure. (3) It is to be recognized now that the increase of plastic strain rate in tertiary creep is

mainly due to damage by intercristalline micro-cracking. Roughly speaking, using uncoupled constitutive equations gives rise to small errors on

stresses or strains if the loading is far from the rupture conditions, but the errors may be of the order of 10 to 50% close to the failure. This means that to predict the rupture of the structures with a good accuracy, it is necessary to use coupled constitutive equations. Then, the procedure of prediction is reduced to only one step in which stress, strain and damage fields histories are obtained in the same time as shown by the scheme in Fig. 2. A macro-crack is then considered as the set of points for which the local damage has reached its critical value at failure.

Fig. 2. Prediction of failure with coupled constitutive equations.

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J. Lemaitre, Coupled e~~sto-p~a~tici~ and damage 33

2. General coustitutive equations

2.1. Thermodynamic variables

In order to model elasticity, plasticity and damage in view of structure calculations, the following variables are introduced: -the total strain tensor E, -the elastic strain tensor Ed, -the plastic strain tensor ~~ = F - Ed,

- anisotropic strain hardening is modelled by the kinematic tensorial variable X which represents the translation of the yield surface in the stress space, associated with the isotropic hardening scalar variable R which represents the increase of the ‘radius’ of that yield surface. Their corresponding state variables are respectively (Y and p. Associated with X and R, they define the power dissipated by micro-internal stresses in the body (X : ci) and the power dissipated to increase the yield stress (RJ~). (The symbol : denotes the contracted tensorial double product.)

If the phenomenon of damage is regarded as isotropic, it can be defined by a scalar variable D which represents the surface density of intersection of micro-cracks and micro-cavities with any plane in the body. If SS is the area of the section of a finite volume element, (1 - D)SS is the area which effectively resists to the stress. This proves that

(virgin element + ) 0 d D S 1 (+-broken element) . (2.1)

In fact, the volume element breaks by atomic decohesion before that value of 1; the criterion for macro-crack initiation is D = DC (DC being of the order of 0.2 to 0.8 depending upon materials). Let Y be the variable associated with D as Yfi is the power dissipated in the creation of surfaces of separation.

If thermal effects are considered, the variable temperature T is introduced together with its associated variable entropy s.

All these variables are summarised in Table 1.

Table 1 ~e~odyn~ic variables

State variables

Observable Internal Associated variables

;

(T

s

Ee o-

EP --Ei

a X P R D Y

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34 _I. Lemaitre, Coupled ~lasto-~~ast~&~ty and damage

2.2. Thermodynamic potential

The free energy is taken as the thermodynamic potential: a convex function of ail state variables from which the state laws derive as well as the associated variables corresponding to the dissipative phenomena. For classical elasto-plasticity, + depends on strains by means of ~~

only,

(2.2)

A very useful concept is that of an effective stress, that is, the stress relative to the effective area [4],

cr

a=l-D’ (2.3)

Associated to the hypothesis of strain equivalence [5], it gives the key to write an analytical expression for 4. That hypothesis states that any strain constitutive equation of a damaged body derives from the same function as for a virgin material, except that the stress is replaced by the effective stress.

For example, the one-dimensional elastic strain energy of a damaged body is we = i&s, = ~E.E~(I - D), from which follows law of elasticity, namely,

(2.4)

where E is Young’s modulus. Now, the thermodynamic potential for isothermal linear elasticity, isotropic damage and

plasticity, not coupled with elasticity is

1 ICi=-a:r”:ee(l-D)+tClp(tY,P),

2P (2.5)

where a is the fourth-order tensor of elastic coefficients, functions of temperature and p the density.

The coupling between elasticity and damage is introduced through the factor (1 - D) in the first term of (2.5).

The law of elasticity is derived from (2.5) by

cr=p$=a:FC(l-D).

The thermodynamic potential also defines the variable Y associated with D,

P-6)

(2.7)

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which gives to (-Y) the sense of an elastic strain energy. If (-Y) is calculated within the hypothesis of isotropic elasticity, the result is [6]

2

-y= tTeq 2E(l - 0)’

[5(1+ v) + 3(1- 2v) CZJ = 2E(;aD)* R” , (2.8)

where v is Poisson’s ratio, uH is the hydrostatic stress, uH = 4tr(a), a,, is the Von Mises equivalent stress, ueq = (8~” : u*)“~,

CT* being the stress deviator, CF* = CT - ~~1, and R, = $(l + v)+ 3(1- 2~)(a~/cr,,)~ for short.

This quantity may be used as a damage criterion by the definition of an equivalent one- dimensional stress ET*, giving the same value of Y as the equivalent one-dimensional case for which R, = 1 (in (2.8)),

-y= (T

*2

2E(l- 0)” (2.9)

or u” = u,,Rjf2. (2.10)

This stress damage criterion may be used for damage theories as the Von Mises equivalent stress is used for plasticity theories.

The dissipation, which must be positive in order for the second principle of thermodynamics to be satisfied, is written through the Clausius-Duhem inequality as follows:

a:iP-X:CC-R@-Yl%q.F. , PdT,O, (2.11)

.Gp, h, e, D are the mechanical dissipative flux variables; it is assumed that their laws of evolution derive from a potential of dissipation, a convex function of all dual variables a, X, R, Y, 0, grad T/T (the state variables acting as parameters),

(b”(u, X, R, Y, y; E, T, .ce, czp, a,p, II).

The hypothesis of generalised normality is the following:

w* . NJ* i'P=- au ’ ff=-dX*

a+* a+* P”:_-g’ a=____

8Y *

(2.12)

(2.13)

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36 J. Lemaitre, Coupled elasto-plasticity and damage

For phenomena which do not depend explicitly upon time, such as plasticity, $* becomes the indicatrice function of the convex and the plasticity constitutive equations are written by means of a plasticity multiplicator A calculated through a condition of irreversibility or consistency,

(2.14)

The coupling between plastic strain and damage appears to be D and Y in the potential +*. These constitutive equations have to be identified from experiments for each material and for each kind of damage.

2.4. Identification of damage constitutive equation

The dependence of #* on a, X, R is found through one-dimensional strain hardening cyclic tests for plasticity and also from creep or relaxation tests for visco-plasticity. The dependence of #* on Y or D needs measurement of the damage variable. A way to do it is to evaluate D through its influence on the elasticity modulus.

From the one-dimensional law of elasticity (2.4), calling (1 - D)E = E the damaged elasti- city modulus, it is easy to obtain

D=l-g-. (2.15)

Then, careful measurements of the Young modulus E on the virgin material and the elasticity modulus I? on the damaged material yield D [7]. An example of the evolution of the damage during a low cycle creep fatigue test is given in Fig. 3.

Fig. 3. Variation of damage during a one-dimensional low-cycle creep fatigue test. A316 stainless steel; T = SSO“C, number of cycles to failure NC = 218 c.

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J. Lemaitre, Coupled elasto-plasticity and damage

Knowing the evolution of damage in particular cases of brittle damage, ductile damage, creep damage, many empirical models have been proposed. synthetic view on these models is to use the potential of dissipation written functions as follows:

f$* = qfqq x, R; T, D)+ 4;(Y; T, D).

37

fatigue damage, A way to have a as a sum of two

(2.16)

Most of these models may be obtained, at least for their main properties, taking 4; as a power function of Y,

4; = so (qso+l(I _ 0))’ ) so+ 1 So

(2.17)

from which follows

fi=_K=_a@:= -y s”(l_D)_l ( > aY aY so (2.18)

or

a=- (2.19)

for phenomena which depend explicitly, or not, upon time. So and so are the only two material constants, temperature dependent, which must be identified for each type of damage and each material. If D = D, failure of the volume element occurs.

3. Elasticity coupled with damage

The kinds of damage which may occur in the elastic range are essentially: (1) The brittle fracture of concrete, that is, micro-cracking by decohesion between cement

and sand, and cracking of the sand; (2) The high-cycle fatigue of metals for which the loading corresponding to number of

cycles to failure higher than 10’ do not produce any significant plastic strain. This damage is essentially transcristalline micro-cracking.

3.1. Formulation

Damage is always associated with an irreversible strain p. In the case of fatigue in the ‘elastic’ range, the values of that strain are negligible as compared to elastic strain. Neverthe- less it exists, and it is responsible for internal damping for example. Let us call 7r that micro-plasticity strain (V =p) and let r be its associated variable.

As far as isothermal processes are concerned, the three state variables are P, D and 7~. The thermodynamic potential for linear isotropic elasticity coupled with damage is +(E~, D, T), and its dual function +*(a, D, 7~) is

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38 J. Lemaitre, Coupled elasto-plasticity and damage

tr(a2) -- 1-D

V

E M41’ .~ 1-D >

from which follows

a** l-Iv u v tr(a) l Ee=P-=-----

dCr E 1-D El-D ’

av 2

y=p-= - oeq

R”, w*

aD 2E(l- D) r=p=.

(3.1)

(3.2a)

(3.2b)

The constitutive equation for elastic strain is coupled with damage through the factor (1 - D) with the constitutive equation (2.19) as high-cycle fatigue does not depend explicitly, upon time.

The different damage behaviors are taken into account by different expressions for h’ related to i.

3.2. Application to high-cycle fatigue of metals

The high-cycle fatigue of metals is difficult to model by means of continuum mechanics because at an early stage, a single crack develops which is not exactly in agreement with the hypothesis of damage being equally distributed in the volume element!

Nevertheless, many continuous damage models have been proposed [8]. One of the simplest, valid for three-dimensional fatigue, is the following:

A possible constitutive equation for micro-plasticity is

(3.3)

with h’ = 7j(l- D) (as demonstrated in Section 4.1); in (3.3),

Aa,, = [$(u; - a?)@; - ,;)]“2 , (3.4)

where 6: are the components of an internal stress, which may be identified with the mean values of the components of the stress deviator over a time period of one cycle. a: Max and u: min being the maximum and the minimum values of the stress deviator during the cycle,

6y = $(u? Max - a: min) ; (3.5)

k and p are material constants characteristic of internal friction. With (2.9) and an evident change of coefficient for B, we have

d = +c)‘““(+.$’ g$. (3.6)

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J. Lemaitre, Coupled elasto-plasticity and damage 39

This constitutive equation for high-cycle fatigue has the following properties: (i) it is valid for three-dimensional loading;

(ii) nonlinear evolution of damage; (iii) nonlinear relation between number of cycles to failure and stress; (iv) effect of hydrostatic stress given by (T* ; (v) linear cumulation if /? = const., but nonlinear cumulation if p is taken as a function of

(T* : p(cr*); (vi) effect of mean stress given by a:, but equal damage for positive (tension) or negative stress

(compression). To take into account a different behavior in these two cases a concept of ‘effective stress in quasi unilateral conditions’ must be introduced [9].

The three coefficients to be identified for each material are B, so and p but most often so = 1. This can be done from the classical Woehler curve which expresses the number of cycles to failure as a function of the maximum stress, or its amplitude, from the results of tension-compression fatigue tests. Taking the case of mean stress equal to zero, so = 1, (3.6) for a one-dimensional loading reduces to

(3.7)

Integrating over one cycle N defined by the maximum value (gM) and the minimum value (a,,, = -uM) of the stress, and neglecting the variation of (1 - D)p+l during the cycle gives

SD uM 4

I

4BaK’ -= SN o (/I + l)(l - D)p+’ dD(a) . (3.8)

To obtain the corresponding number of cycles to failure NR, we must integrate once more with initial and final conditions

N=O + D=O, N=N, t D=D,-1. (3.9)

We then obtain

N = p + 1 Cr$+l) -___ R /3+2 4B ’

(3.10)

Equation (3.10) is the equation of the Woehler curve given by the model when the mean stress ti is zero. Then, (p + 1) may be calculated from the slope of the corresponding experimental curve in a log-log plot. The nonlinearity of the phenomenon corresponds to values of p of the order of 2 to 10.

The final constitutive equations for elasticity coupled with high-cycle fatigue damage are (3.2a) and (3.6), with u* and Au,, given by (2.10) and (3.4), resp., and geeq and R, as defined in Section 2.2.

As an example, Fig. 4 shows the evolution of the damage zone of a tulip specimen using these constitutive equations implemented in a finite element code [lo] using a step-by-step

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40 J. Lemaitre, Coupled elast~-plasticity and damage

II

G ._ 2 ._ .z .E

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J. Lemaitre, Coupled elasto-plasticity and damage 41

procedure. The difference with classical FEM calculations is that the rigidity matrix varies as it is a function of the damage elasticity modulus E = E(1 - 0). The external loading is a periodic ‘0, +’ force and the state of stress is axisymmetrical.

3.3. Application to the behavior of concrete

From a phenomenological point of view, it is better to write the damage as a function of strain rather than stress. Then, Y and ti may be expressed as a function of strain through the stress and the elasticity law.

Furthermore, to take into account the large difference of strength in tension and com- pression, an equivalent strain E * is introduced. It contains the unilateral conditions of opening and closure of micro-cracks:

E * = ((&J2 + (&2)2 + (&3)*y2 )

where Ed, Ed, &3 are the principal strains and the symbol ( ) denotes the positive part,

w= ( X ifX>O,

0 ifXa0.

3E FISSURE

NOTCHED SPECIMEN

(3.11)

LOADING

Q (kN)

Fig. 5(a). Crack growth in a concrete notch specimen loaded in mode 1.

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A damage model integrated with respect to time, proposed by Mazars, is the following [ll]:

l_ Q&-4+ L a

D= I &* exp(G * - EDI> I if e*>.zD,

0 if .5*<ED,

(3.12)

where EP (a damage threshold), a and b are material coefhcients which may be identified from tension~ompression tests. (D = DC means rupture of the volume element.)

This model is restricted to monotonic loading, i* >O, but it is valid for any three- dimensional case.

In order to predict the failure of concrete structures it must be used together with the elasticity law (3.2a). Using the finite element method a step-by-step procedure on displacement is used.

An example is given in Fig. 5: it is a notch plate specimen for which the calculated damage zone (D = DC) is compared with the path of the experimental crack [ll]. For that example, the material coefficients are E = 3~~ MPa, Y = 0.2, ED = 10V4, a = 0.8, b = 2 - 104.

Fig. 5(b). Calculated damage zone (black elements).

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43 J. Lmzaitre, Coupled elasto-plasticity and damage

Elasto-plasticity coupled with damage

For simplicity, we restrict ourselves to isotropic hardening characterised by the only internal variable p. The classical hypothesis of generalised standard materials (normality rule) and associated plasticity (plasticity criterion used as plastic potential) are adopted.

The kinds of damage which may occur together with plasticity is ductile damage due to large strains or low cycle fatigue (NR < 10” cycles) when temperature is below about $ of the absolute melting temperature.

(1) Ductile damage is mainly initiation and growth of cavities generally at the boundary of inclusions with plastic instabilities, this phenomenon is one of the limitations in metal foxing processes.

(2) Low-cycle fatigue is associated with service cycles of parts of nuclear plants, airplane engines, pressure vessels, etc. At room temperature, this is generally initiation and growth of transgranular crack.

4.1. Formulation

Considering isothermal processes, the state variables are E, se, cp, p and D. The laws of evolution of the three dissipative variables derive from the potential of dissipation taken as the function of the plasticity criterion,

f(a,R;D)=O+ 39f0,

f<O or f<O3 ip=O;

~*=f(o;R;D)+~;E(Y,d;D),

(4.1)

(4.2)

(4.3)

If the Von Mises plasticity criterion is considered the coupling between plasticity and damage is introduced as follows:

+ o;q-R ----ay=O, 1-D (4.4)

uy being the initial yield stress of the material. Then

Lj_pP3= -y so i ( > aY so 1-D'

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44 .I. Lemaitre, Coupled eiasto-plasticity and damage

The multiplicator h’ is derived from the consistency condition f = 0 which ensures that the actual yield state of stress follows the loading stress. Furthermore, we must have

h’=O iff<Oorf<O; now,

or

af af + af . f=a,:ti+z~+aD~

R’(p) . ez,-R . p+(1_D)2D=O?

or, with /i -R --

p-l-D and T_D)=~Y,

/i = H(f) k& - D> R’(p) - CTY(- Y/S,,)“o ’

with ( > as in (3.12) and

0 iff<O, H(f)={l iff =O;

R(p) characterises the nonlinear strain strain hardening curve,

R=$$ h(p) = KptJM,

(4.6)

(4.7)

(4.8)

hardening; let us take the simple expression of the

(4.9)

where K and M are material coefficients which may be identified from tension strain hardening test results. In the range of eP where the damage is zero or negligible, the equation of the strain hardening curve is

a=ayfK&y, (4.10)

which allows to determine (TV and M from the slope of a plot of log(a - fly) against log sB.

4.2. Application to ductile damage

A ductile damage must be added to the equations of Section 4.1, In the general damage model

_yso i _yso fi= - ( > so 1_D= & (3 ri; (4.11)

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p is now the classical accumulated plastic strain. Substituting (2.81,

( ~k~Rv . >

so

Ij= 2E&(l- D)z P.

Usually ductile damage occurs for strain large enough to plastic ~strain-h~dening saturated);, then we may consider

a-l -= const. = G”, 1-D

(4.12)

consider the material as perfectly

a; being the stress corresponding to the threshold. Now we have

(4.13)

(4.14)

Multi-dimensional experiments show that so = 1. Then, the only one material-dependent coefficient is (a~/2ESo) and it may again be

identified from uniaxial monotonic experiment. In this case (aH/aeq = 4, rj = .Q the model (4.14) reduces to

(4.15)

Write an evident integration with an initial condition defined by a threshold ED and a rupture condition defined by the strain to rupture &R,

&s&ED+ D=O, &=ER -+ D=Dc;

then,

2

I)=?-

2ESo cp + const. or D=D,c,

ER-CD

which means that [12]

2 U” DC

-=-

2ESo and fi=

DC -----R,b.

ER - ED ER- ED

(4.16)

Then, the final constitutive equations for isotropic elasto-plasticity coupled with ductile damage are

i=p+i’p, (4.17)

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46 J. Lemaitre, Coupled elasto-plasticity and damage

&e_l+v 6 Y tr(a) ---___ E 1-D El-&

ip = $H(f) K kLl> uD

EP

D;R, *;’ (l-W/M _ uy

&R: ED

(4.18)

with

Lj, DC -R@, &R- ED

p = (S&P : &P)1/2 )

PC&D --) D=O, D = D, + crack initiation .

This set of constitutive equations may be used to predict the limits in metal forming (D + DC) [13] or to predict the influence of a forming process on the behavior of materials by the final values of cP or p or R and D [14].

The identification of the material constants is easy to perform. Example of steel 35NCD16 at room temperature:

E = 210,000 MPa , Y = 0.3 )

oy = 1200 MPa , K = 3340 MPa = , M 3.1,

& ,, = 0.02, &R = 0.37, D, = 0.24.

4.3. Application to low-cycle fatigue of metal

Low-cycle fatigue is considered when the loading is high enough to yield plastic strain and a number of cycles to failure less than lo4 cycles. A fatigue model, coherent with the isotropic plasticity developed, derives from the general damage model (2.19); substituting (4.7) into (2.19) yields

(4.19)

Neglecting the second term in the denominator of (4.19) and with an evident change of notations for C, and so = 1, we obtain

d& d = C (1 _ D>2 (a,, - uy(l - D))“-lb,, . (4.20)

Identification of coefficients C and M come from Woehler curve as explained in Section 3.2, and a calculation of cyclic plasticity, coupled with low-cycle fatigue damage, uses the elasto-plastic constitutive equations of Section 4.2 associated with the last one.

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Elasto-visco-plasticity coupled with damage

Visco-plasticity in metals occurs when the temperature is above approximately one third of

5.

the absolute melting temperature. Two kinds of damage may occur: the creep damage and a low-cycle fatigue damage if the loading is cyclic.

The coupled constitutive equations may be derived from the same formalism as previously. The potential of dissipation is written as the sum of two terms, one for visco-plastic strain,

one for creep fatigue damage, both phenomena being time-dependent,

then N p-NJM

1-D’

or with R = h(p) as in plasticity,

p= ((l_u;)K)N~,

- D)-’ ,

with (2.9) and an evident change of notations, we get

(5.2)

(5.3)

This damage model generalises the one-dimensional Kachanov’s law proposed in 1958 [4]. The final constitutive equations for isotropic visco-plasticity coupled with creep and fatigue

damages are (5.3) and

(5.4)

with u* and e as in (2.10) and (4.5), resp., and (TH, uD, ueq and R, as defined in Section 2. K, A4, N are visco-plastic temperature-dependent material coefficients which can be identified from one-dimensional creep tests.

A and r, creep-fatigue damage coefficients in (5.3), are identified from one-dimensional creep fatigue tests by means of Woehler (or Manson-Coffin curves) corresponding to different frequencies.

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in in

J. Lemaitre, Coupled el~st~-~~~st~c~t~ and damage

1x1 DAMAGE VERSUs TIME

TIME : WOURS DAMAGE: 1x10-’

ROTbTlON

Fig. 6. Calculation of a turbine disc model, INCO 718 alloy temperature T = 550 “C.

Equations (5.3) and (5.4) have been used for the calculation of the time for a crack initiation a model of a turbine disc [15]. Fig. 6 shows the result for the evolution of the creep damage the critical zone due to the centrifugal force. [16] and 1171 contains some other examples.

6. Conclusion

The sets of constitutive equations for elasticity, plasticity and visco-plasticity coupled with brittle, fatigue, ductile or creep damage are given ‘ready for use’ in step-by-step incremental methods of structure calculations. Some problems of convergence may arise, but not more difficult as those for classical plasticity.

The difficulty which remains is to obtain the material-dependent coefficients for each application. A handbook of such coefficients does not yet exist, and this situation may last another ten years or more! If you want to make numerical calculations with the models described, you have to identify the coefficients, by yourself, from tests results which may be found in the literature for most of them. It is not difficult, the methods are given in this paper, it is a good exercise, it gives a better understanding of the domain of applicability of the constitutive equations, and it helps to make a good criticism of the validity of the numerical results obtained!

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References

[l] D.R.H. Owen and E. Hinton, Finite Elements in Plasticity (Pineridge, Swansea, 1980). [2] J. Lemaitre and J.L. Chaboche, Aspects phenomenologiques de la rupture par endommagement, J. Met.

Appl. (3) (1978). [3] G. Sih, Mechanics of Fracture, Vols. I-V (Noordhoff, Leiden, 1973-1977).

[4] L.M. Kachanov, Time to the rupture process under creep conditions, Izv. Akad. Nauk SSR Otdel. Tehn. Nauk

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