zzu-u~~E q a r_ ha raIl i II III

MICROMECHANICS OF FATIGUE

GRANT No AFOSR-89-0329 D IFinal scientific report ELECTE

June 1992 N

Jean LEMAlTRE, Rend BILLARDON and Marie-Pierre VIDONNE

Rapport Interne no 134

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In this final report, the results given in the first two annual reports are recalled.

Application of the derived tools to Apha-Two-Titanium Aluminide Aliov is made with a

first series of strain controlled fatigue tests the locally coupled model is firstidentified and then checked on a secLond series of stress controlled fatigue tests.The good agreement allows to use this locally coupled method to predict fatigue crackinitiation on brittle or quasibrittle materials.

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MICROMECHANICS OF FATIGUE

GRANT No AFOSR-89-0329

Final scientific report

June 1992

Jean LEMA1TRE, Ren6 BILLARDON and Marie-Pierre VIDONNE

Rapport Interne n' 134

Aceassloa Por

N'% 4 Tty

30 rDT1C QUJAI4TY 1N3FCTE!1TFT) 31 P !

1A

ABSTRACT

In section I., brittle and ductile isotropic damage mechanisms are studied from a

meso-mechanical viewpoint. Relationships between crack density and void volume

fraction defined at meso-scale on one hand, and a scalar internal variable characterizing

damage on the other hand, are given.

In section 2., a general form for the evolution law for this damage variable is

derived. A threshold which defines the onset of this evolution is derived from

thermodvnamical considerations.

In section 3.. it is proposed to relate the ultimate stage of continuum damage

evolution, i.e. the local failure, to localization phenomena. The corresponding criteria are

studied in details.

In Section 5, a post processor is fully described which allows the calculation of the

crack initiation conditions from the history of strain components taken as the output of a finite

element calculation. It is based upon damage mechanics using coupled strain damage

constitutive equations for linear isotropic elasticity, perfect plasticity and a unified kinetic law of

damage evolution. The localization of damage allows this coupling to be considered only for

the damaging point for which the input strain history is taken from a classical structure

calculation in elasticity or elastoplasticity. The listing of the code, a "friendly" code, with less

than 6(X) FORTRAN instructions is given and some examples show its ability to model ductile

failure in one or multi dimension, brittle failure, low and high cycle fatigue with the n(..ilinear

accumulation, and, multiaxial fatigue.

In section 6, application is performed on the Alpha-Two-Titanium Alum:Aide Alloy.With

a first serie of strain controlled fatigue test the locally coupled model is first identified and then

checked on a second serie of stress controlled fatigue tests. The good agreement allows to use

this locally coupled method to predict fatigue crack initiation on biAttle or quasibrittle materials.

TABLE OF CONTENTS

1. MICROMECHANICS OF DAMAGE

1.1. Brittle isotropic damage

1.1.1. Microcracks and zcalar damage variable

1.1.2. Brittle damage growth

1.2. Ductile isotropic damage

1.2.1. Microcavities and scalar damage variable

1.2.2. Ductile damage growth

2. GENERAL PROPERTIES AND FORMULATION

2.1. A general form for continuum damage evolution law

2.2. Damage threshold

3. LOCAL FAILURE CRITERIA

3.1. Introduction

3.2. Rate problem analysis : the linear case

3.3. The non-linear case : some results

4. A CRACK INITIATION POST PROCESSOR

4.1. Introduction

4.2. Locally coupled analysis of damage in structure

4.3. Constitutive equations

4.4. Numerical procedure

4.5. Examples

4.6. Listing of DAMAGE 90

5. APPLICATION TO ALPHA-TWO TITANIUM ALUMINIDE ALLOY

5.1. Material

5.2. Tensile specimens

5.3. Identification tests

5.4. Verification tests

1. MIICROMIECHAN'ICS OF DAMAGE

Micromechanics consists in deriving the behaviour of materials at meso-scale from

the study of specific mechanisms at micro-scale. The micro-mechanisms must be

precisely defined from physical observations for both the geometries and the kinematics.

Their mechanical modelling is performed with elementary usual constituutve equations

established at meso- or macro-scale for strain, crack growth and fracture. When

compared to the direct analysis of the macroscopic properties, the additional power of this

approach comes from a better modelling of the possible interactions between different

mechanisms and from the homogenization that bridges the gap between micro- and meso-

scales.

When using classical continuum thermodynamics concepts, the effects at meso-

scale of material degradation at micro-scale are characterized by an internal variable called

damage. Hereaft-,: the effects of the material degradation are assumed to be isotropic at

meso-scale and the damage variable to be a scalar denoted by D. In this section,

definitions for this scalar variable are derived from the study of two different micro-

mechanisms.

1.1. Brittle isotropic damage

1.1.1. Microcracks and scalar damage variable

The main mechanism of brittlc damage is the nucleation, growth and coalescence

of microcracks up to the initiation at meso-scale of a crack. Hereafter, a relationship

bctwkeen the micro-crack pattern and the damage variable D is established.

Let us consider a Representative Volume Element at meso-scale as a cube of

dimiension ( * ll). This RVE is assumed to be constituted at microscale of cubic cells

of d:mension (d - d - d) in which may lie a microcrack of any area s, and any orientation

(,ce Fin. I). The number of cells is m-1/ld 3 and the number of cracks n 5 m./ /

/ /

--- - mesoscale R V F

-d

SmicrocrackS~area si

Fig. 1. Micro- and meso-models for brittle damage

5

The e-eometrv heine defined. thet. modelling consists in writine :he balance of the

usiated cnricr, caculated by MlAsNc Orcture mechanics one one banid, and calculated

by continuuim damace mechanics on to other hand.

I-or a cracked cell i, subjected to a Llven state of stress, if G, denotes the strain

energy release rate cor-responding to a crack of area q. Di the equivalent damage of the

cell and Yi the strain energy density release rate. the balance of dissipateJ1 energy can be

wntten as:

G, SI- Y it d-"

For thie n cr-acked calls of the meso-cuba. the previous relation becomesn * n

Assun'-.iing that britle growth of microcrawks occurs at (i = (77, consain:.cres-dn

to 'I== constant, it can be xr-ittenTnatn . -n

-r:.c it- it iý -sn med thlat S' Corc SpondLN to IDi .ilert) of the pre vious

rc1.::o 1 P c..~

T fl

~~cnlab:a C% 1!n: 1'! i1,1 1c~nn F :heII' crta ýn: c I)a thc i tr)-sc.al nv

r Y, I 'A~~ ~ ~ 71:"" of 1)ý f tc ll1

In Ph: if a in a (nt that t 011 w wcrck int iats, 2 )I1~~ Pn D~nl'Iý [ it MIS can be

'Ahen

n> kI

th).l

Hence,

n1si

D Dc12 k

nIn this case, the damage variable D appears as the micro-cracks surface density (" s, / 12)

corrected by a factor (here De/k).

If the following simplest fracture criterion is considerednSsi

nV si = 2-k= 1 -4 Dc = 1 then, D-1 12

By the way, this calculation gives an order of magnitude of a characteristic length

which allows for the matching between fracture mechanics and damage mechanics namely

1, the size of the Representation Volume Element. Since,

D Gc D,n -

I k ~GC

I lence, for the simple fracture criterion k = 1, ) = 1,

Gc

For most metallic materials

li eht alloys Steel and high alloys

.0005 < C < 0.05 MPa m

"< Yc < 10. MPa

0.0 025 < 1 < 0.005 n

and for concrete in tension Gc = 3 10-5 Mpa.m, Yc - 1.5.10-4 NMPa, so that 1 2. .10-1.

This shows that the size of the physical RVE must be of the order of the millimeter for

metals and of the order of the decimeter for concrete.

7

1.1.2. Brittle da --er()kkthl

Aýs a specific exxample. let us derive the kinetic damnace evolution law at meso-scale

which corresponds to the fatiguLe microcrack-s grow.th at micro-scale of Fig 1.. For

simplicity sake, the analysis is restricted to a two dimensional problem, for which e

denotes the uniform thickness of the whole RVE.

With Do: = k., it has been established thatn n .

SSi

D~-- anid D= 1

and.L! for eaich cracketd cell, if 2a denotes the crack leneth, the followinu relations hold(EE I, F" ti. ý .

The surfac T wroth raite s, of each crack can be expressed as a function of the sri

-i rIeas e raite Cl, of t~he corresponodln e cell. by mleansý otf the Paris, Ilaw of ft ucrckeox t.If N denTotes the numnber of cycles of loadn Tn moe0 n

%1KNItl\ th alplitudeC of the Sustrs rintensity factor (%ith Knlj = O). then

6 N

"t and TI arT~ aenlcutu.~ Ih £4 for many metallic materials.

~2i'aN~me\tn-Ijt tI > ariS law corresponds'- to the inte Cration over one cycle ot,

C K Tk

:\ r~I it on Nh (p hewe nGan Y, canT be fouind thronu~h their definition from the

e ci ncruv. I, w decnotes the elastic strain energYv density and Wl thle elastic strain

v. nerev, of the elementary cell, then

owl i)w 1ox \heroas )l D

8

Since

Wi= wi d2 e,

G(wi d2 e) dDal) -ds

Ifsi 2 ai e , Gi=Yidi , Ge=YidiDi=d,2 -die i=Ydi i i

then

si = rl C e E'1 21dT12 Y2 y,1

Hence, the damage rate isx'_ si I C Eq/- d /2 n .r1-1

f2 1e Y121

A-ssuming that all the n cracked cells have the same strain energy density release rate

"Y= Y,. the homogenized strain energyV density release rate for the meso-RVE is

Y = n Yn and Y= n Y•

'o2 sequentlyn T- .. II T I

e,•Y•' Yi=neYne Yn en 2 y 2 Y

Tr C ET"/ dl/ e yI

12 n,

1• :his example. the damage rate is an increasing function of the strain energy density

rcease ratet for most materials since r1 = 4. the damage rate is quasi-proportional to the

•'.:>v. enrergy densitv release rate. The damage rate is also proportional to the rate Y.

Ti::< %ill be used in section 2.1 as a -uideline to derive a gencral kinetic law for dama-e.

9

1.2. Ductile isotropic damage

1.2.1. NIicrocavities and ;calar damage variable

The main mechanism of ductile damage is the nucleation, the growth and the

coalescence of microcavities by large local plastic deformations. Hereafter, a relationship

between the density of micro voids and the damage variable D is established.

Let us consider again a Representative Volume Element at meso-scale as a cube ofdimension (I * I * 1). This RVE is assumed to be constituted at microscale of cubic cells

of dimension (d * d * d) in which may lie a void of volume d3 (Fig.2.)./ /

/ /

d

void

1 //

Fig. 2. Micro-meso element for ductile damaue

On this very simple geometr,. the modelling consists in writing the balance of the

dissipated energy calculated from the growth of the cavities on one hand, and calculated

by continuum damage mechanics on the other hand.

For the geometrical model under consideration, the porosity P can be defined as

P= I P- _ nd3

Po 13

where p and P0 are the current and initial porosity respectively, and n the number of

cavities.

According to Gurson's model, the porosity P at meso-scale is equal to the

hydrostatic part c = c of the plastic strain due to the growth Of Voids.

I lere, this assumption leads to the following equality written at meso scale

ýP1

At meso-scale, given an homogenized stress a-- and a plastic strain rate c the total

power dissipated is

This can be split in two parts by means of the deviatoric and hydrostatic quantities i.e.

D *pD p

= (YiJJ + 3 i5H i + 0

The first term is the power dissipated in pure plasticity by slips. The second term which

corresponds to the irreversible change of volume may be interpreted as the power

dissipated in the RVE to increase the material discontinuities by growth of the cavities.

This latter part must balance the damage dissipation i.e.3 13, =y613

C~H ~H P Dlso that

6 ~P-Y

Assuming for simplicity, proportional loading, perfect plasticitv,-- const, and the

initial condition P = 0 -- D = 0, the integration yields

3 OH d3

As for brittle damage, because of the localization of the damage phenomenon, it can be

assumed that the meso-crack initiation occurs when a set of cavities occupies only part ofthe flat volume (I * I * d), the other cavities in the RVE being neglected.

In other words, the critical value of the porosity corresponding to D = I is assumed as(3 12 d d

Pc= n - = k W- k

3 GHThis allows for the calculation of the term in the damage equation

3 (H d

I lence,

D nd2

11

1.2.2. Ductile damige growth

As a spccif, c example, let us derive the kinetic damage evolution law at meso-scale

which corresponds to the voids eTowth at micro-scale. For simplicity sake, the analysis

is restricted to the pa-rtcular case k = 1.

The kinetic law for damage evolution can be directly derived from the expression

for D established in the previous section, so that

d2 ddD= •-:- n2+ n 1d

The first teM,- accounts for the increase of the number of cavities and n denotes the

number of cavities nucleated per second. The second term accounts for the cavity

growth. In the Gurson model, the porosity rate is also the sum of two terms accounting

:,r nucleation and ,rowth.

a) Dama,,e (grOVth by n-UceLmriot of cavities

To modecl nucleatlon, Tvereaard proposed the following kinetic law for porosity,P)= ,-\ Gc -- B OG

%k nere -\A and B are m-atenal parameters.

Assumini `or simplicitry sake a sudden nucleation of cavities of a fixed size d

aaP n -•-

anqd

: [i : 2 (A Ceq -B t•

so that.

d ~(A +B

Damage can be expressed as a function of the accumulated plastic strain rate p \vitten in

terms of the plastic tangent modulus ET. Assuming proportional loading, i.e.

G~{_CHOHa GH

• 2 .p .p C3p= 3 - ET

so that,

-1 ET N + B p

12

b) Damage growth by enlargement of afixed number n of cavities.

The problem of void growth has received much attention i- 4h- nmist 20 years.

Essential results are the McClintock and Rice & Tracey analyses which derive the rate of

growth of a cylindrical or spherical cavity of volume V in a perfectly plastic infinite body

as a function of the accumulated plastic strain rate p and triaxiality ratio c5H/(eq, i.e.

=0.85 V p exp(3 -O)

Taking

V = d3

leads to3 G3H

3d 2 d=0.85 d3 p exp(3 ;__ )

Since

d d d2 d=2 n n, -_, D=2D d

D0.57 D ;exp 1 j

c) Conclusions

In this example, the damage rate by nucleation and growth of voids appears as

- proportional to the accumulated plastic strain rate,

- an increasing function of the triaxiality ratio-

- through ET or D, a function of the current state of the material.

This will be used in section 2.1 as a guideline to derive a general kinetic law for damage

evolution.

13

2. GENERAL PROPERTIES AND FORMULATION

14

2.1. A general form for continuum damage evolution law

a) General formalism

Within the framework of classical continuum thermodynamics, the thermo-

mechanical state of a material is described by the following set of independent state

variables:

V (-, T, V) with V - (D, VP, ,

where S, re denote the total and plastic strain tensors respectively, T the temperature, V

the set of the internal variables, and D the damage variable. For simplicity sake, hereafter

only isothermal situations will be considered.

The reversible behaviour is described by' the Helmholtz specific free energv

T = , V),

chosen as

T' Y41 -P, D) + yP(.P N'P)v, It hp

with

T e 1 . P) 0 -

where E denotes the elasticity matrix of the undamaged material. The thermodynamic

forces

A (,A)

are defined from the following state laws

z. -= , A=(-Y, -LAP) = P

where p denotes the mass density, z the stress tensor, and Y the strain energy density

release rate or damage energy release rate.

The irreversible behaviour is described by a dissipation potential

> = ((z, A -,)),

from which the following evolution laws are derived:

a)z r)A

In particular, the damage evolution law can be chosen such that

@FD

15

b) dcvnaie vs (micro-)phrstiitv

In section 1.. it has beet established that damage is alwavs related to some

irreversiblc strain eiUher at micro- or meso-level. This property can be taken into account

in the evolution law for the darma-c variable by assuming that the factor X is proportional

to the accumulated plastic strain so that

DF D

T:,w irreversible na:are of daimace is directly taken into account by the fact that the variable

p is alvk avs positive or null.

In most materials, a certain amount of plasticity must be accumulated before

U,,ige at meso-level appears. In metals, this corresponds to the accumulation of micro-

stresses in the vicinitv of initial defects, of dislocations. prior to the nucleation of

m.,ro-cracks or mncr,• voids. To model this phenomenon, since the damage evolution is

C,• c-.d b% the accumulated plas: e strain rate, it is natural to introduce a threshold pD o(V

t?:: \ rable p. sucn trat

- Pit PpD

=() if p < p)

1= -•)- p -p-pD)

v' h-re (I-b is the -icavvyside step function.

In monotonic loadin e. PD can be identified as the uniaxial damage threshold whereas

for fatigue or creep processes PD is a function of the applied stress, as it will be discussed

in section 12.2

c) driving force for damag, c

From the thermodynamical analysis, it has been deduced that the driving force for

damage is the strain energy density release rate Y. Hence, FD must be a function of Y

FD = FD (Y,...)

16

d) influence of 'he triaxialirv ratio JAnother important feature of fracture mechanisms is the influence of the triaxiality

ratio (O'H/(Geq), where it is recalled that cH denotes the hydrostatic stress and GYeq the Von

Mises' equivalent stress. This effect is directly taken into account through the damage

energy release rate Y which is a function of the triaxiality factor Rv"y _ Oeq2 Rv

2 E (1 - D) 2

with

Rv (I+v) + 3(1-2v 2 Y)

e) a generalform

In order to choose the proper and simplest expression for FD, let us recall the

kinetic damage laws obtained by micromechanics for particular mechanisms in section 1.

6 Ti C E~'12 cIT12 e Ti1Brittle damage bvfatigueg rowth of micro-cracks D = y

12 n

rT being of the order of 4

= (const) * Y

Although no plasticity has been introduced in the analysis, it always exists at

micro-scale at the crack tips of the micro-cracks and it is possible, at least formally

to relate Yý to p through a plasticity constitutive equation

Y = Y(Gcq) --* Y@•(cq)

and

Ductile damage by nucleation of micro-cavities = El- A + B p

Ductile damage by enlargement of micro-cavities: D = 0.57 D p exp 3 OH

17

The qualitative conclusion which can be drawn from these three results is that the

damage rate D can be considered as proportional to Y, which is a function of (GH/'c;q),

and p

6- Y p

or

FD - y2

As in any realistic constitutive equation, a material dependent scale factor, such asI

(const), a ET, or 0.57, must be introduced. Let us denote by S this material constant so

that

FD- -S

Finally, according to the qualitative properties listed above, the damage potential is

natural'v ritten as :

FDY'. ýpD)= -S t11p-pf)

where the factor 2 has been introduced to compensate for the factor (1/2) coming from the

derivation. Hence, the proposed general continuum damage evolution law is the

followtin

•where t'•ko material dependent parameters are introduced, viz. S and Pi which characterize

the energetic resistance against the damage process and the damage threshold,

relpect:, !.v. The effects of the temperature T are taken into account through the variation

of these coefficients with T and through the accumulated plastic strain rate p which is also

a function of T.

Several important properties, though not directly introduced in the formulation,

are also naturally exhibited by' this general evolution law, i.e.

- the non linear accumulation of damage,

- the effect of mean stress in fatigue.

- the non linear interaction of different kinds of damage.

18

2.2. Damage threshold

Under monotonic loading, the damage threshold PD can be identified with theuniaxial damage threshold Ec, whereas in the case of fatigue or creep loadings it is a

function of the applied stress. It corresponds to the critical level of plasticity which

induces the nucleation of microciacks without any consequence on the mechanical

properties and can be related to the energy stored in the material.

Experiments in fatigue have shown that the total plastic strain energy dissipated

may reach tremendous values before failure althou2h the stored energy remains constant

at microcrack initiation. This stored energy is the result of microstress concentrations

which develop in the neighbouring of dislocation networks in metals and of

inhomogeneities in other materials. For a unit volume, it is equal to the differencet

between the total plastic strain energy ( (5ij J dt ) and the energy dissipated in heat as0

given by the Clai;sius-Duhem inequality of the second principle of thermodynamics.

For instance, in the case of a material exhibiting kinematic X and isotropic R

strain hardenings and no damage, under an isothermal transformation the rate of energy

dissipated in heat is

0 = ij lrj - Rp - Xij oj - 0

This expression may be calculated from

- the potential of dissipation e.g.

(ZD .Z)cq - R - (5y - 3 Xij Xii

- its associated normality flow rule*p dF - al a

Iij

- and the yield criterion

f (DZ-)q - R -cy =0

so that

+ ay+ 2X•

Hence, the stored energy Ws as a function of time t is

W ,(t t - t( 3 ° .Ws(t) 0 9 oif dt - + X- XiXjP dt

0 0

19

L*

This formula can be simplified if the following assumptions are made

- the effect of the kinematic hardening is neglectedp'Ws (c,,eq - oy) dp

- the variation of •eq is neglected as for a quasi perfectly plastic material

s = [Sup ((eq)- oy] P

If this stored energy is considered as a constant for the damage threshold PD, its

value can be identified in the one-dimensional monotonic case used as a reference withPD = CPD"

In the particular case considered above, and with the crude approximations made,

the onset for the damage process, or damage threshold, corresponds in the case of amonotonic loading to the ultimate stress cu :

[Sup (c~cq' - 5y] PD = (Ou - ) FpD

so that

PL p SLIp(Ge(l) _- Y

As a sinmnar%, in the particular case considered in this section. the whole set of

equations that governs the damage evolution is

• ) y • (Ys - 0",

s P 'f ' P - PD S Sup(G e& ) -

20

3. LOCAL FAILURE CRITERIA

21

The ultimate stage of continuum damage evolution corresponds :o meso-crack

initiation- As a first approximation, this critical damage state can be characterized by a

critical value Dc of the damag- - iriable, for instance Dc = 1.In fact, the value of Dc is not solely material dependent. As discussed below, it

also depends on the local stress state.

It is proposed to relate the conditions for the meso-crack initiation to theconditions for the localization of the deformation in the material. In this section, these

latter conditions are studied in details, inside, at the boundaries or at the interfaces of rate-

independent solids for both the linear and the non-linear case. Physical interrretations of

these conditions are also given.

3.1. Introduction

Since the pioneering works of Hadamard, Hill, Mandel and Rice, the localization

of the deformation in rate-independent materials is treated as the bifurcation of the rate

probklem.

tlere, wýe give a general view of the various bifurcation and localizationphenomena for possibly heterogeneous solids made of rate-independent materials. Under

the small sLrain assumption, the behaviour of these materials is described by the following

piece-v'e ilinear rate constitutive laws

when f <0, or f =0and 3:;Z(v)<0=1.. .= - (l*) with

- when f =0 andb 3'(v) Ž0

where and "Vv) = £ respectively denote the stress and strain rates, v the velocity. andf

the vied unction.

T',-e general class of materials modelled by relations (1) includes for instance the

elasto-plastic damageable solids the behaviour of which is described by the constitutive

equations discussed in section 2.1.a.

22

It is shown that, in general, different types of localization phenomena may occur,

depending on the failure of one of the three conditions which are described in section 3.2.

Their physical interpretation is the following

the ellipticity condition is very classical. Its failure is the condition for

localization given by Rice and linked to the appearance of deformation modes involving

discontinuities of the velocity gradient. It has also been related to stationary acceleration

waves ;- the boundary complementing condition governs instabilities at the boundary of

the solid. Its failure leads to deformation mode localized at the boundary and is related

to stationary surface waves (for i"stance Rayleigh waves)

- the interfacial complementing condition governs instabilities at interfaces. Its

failure leads to deformation modes localized at each side of the interface and is related to

stationary interfacial waves (Stonely waves).

3.2. Rate problem analysis : the linear case

Let us consider for instance the body sketched in Fi,.3.

ineracia! mode

S(in ular

sigua s"-- 'urf ace surfaceS

"ssuinarl surface/SQ crossing I

inside• •:

surface inl•'Nle

n

P nQ Q

nin ni

(F () 1

Fig. 3. Different types of localization modes

23

Qualitative results can be exhibited from the analysis of the rate problem for the

so-called linear comparison solid (see Hill). In this case, this linear problem is well-

posed if and only if the following conditions are met

- the ellipticit.' condition : the rate equilibrium equations must be elliptic in theclosure of the body 02, i.e.

det(n. HE. n) # 0 for any vector n # 0, and any point M E ".

- the boundary complementing condition : this relation between the coefficients ofthe field and boundary operators must be satisfied at every point P belonging to theboundary F where the boundary conditions are formally written as O(n) = . This

condition is easily phrased in terms of an associated problem on a half space defined by

z > 0. It requires for every vector k = (kj, k-, 0) # 0, that the only solution to the rateequilibrium equations with constant coefficients (equal to those of the operator at point

P). in the form

v(x, v, z) = w(z) exp[i (kj x + k- y)]

with bounded w and satisfvin g the homogeneous boundary conditions .(•) = 0, is the

identically zero solution v 0.

- the interfacial complementing condition : this relation between the coefficients ofthe field operators in Q1 and Q 2 must be satisfied at every point Q of the interface I

between Q 1 and Q,. This condition is again easily phrased in terms of an associated

problem on the whole space divided by the plane interface z = 0. It requires for everyvector k = (kl, k,, 0) # 0, that the only solution to the rate equilibrium equations withconstant coefficients (equal to those of the operators at point Q, in 0-1 for z < 0 and in Q,_

for z > 0), in the form

(vN(x, y, z), v,(x, y, z)) = (w%(z), w2(z)) exp[i (kI x + k v)J

with bounded w1 ,w2) and satisfying the continuity requirements (continuity of the

velocity and the traction rates) across the interface z = 0, is the identically zero solution

(v (z), v'2(z)) - (0, 0) ; (where vj and v2 are the solutions, respectively for z < 0 and forz > 0).

When these three conditions are fulfilled, the rate boundary problem admits a

finite number of linearly independent solutions, which depend continuously on the data,and which constitute diffuse modes of deformation.

24

Remarks

- these three conditions are local, and this is particulally important when

considering their numerical implementation;

- the above-given results remain valid for an arbitrary number of non-intersecting

interfaces, an interfacial condition being written for each interface;

- the failure of these conditions can be interpreted as localization criteria as recalled

in section 3.1. These localization criteria can also be used as indicators of the local failure

of the material ;

- both boundary and interfacial complementing conditions fail in the elliptic regime

of the equilibrium equations, or at the latest, when the ellipticity condition fails. Thus,localized modes of deformation at the boundary or at the interface generally occur before

the onset of so-called shear banding modes.

3.3. The non-linear case : some results

Although the complete analysis of the non-linear problem is not yet available,

some results can be given for the possibility of emergence of deformation modes

involving jumps of the velocity gradient for the bi-linear rate constitutive laws (I).

The necessary and sufficient conditions for the onset of such modes inside the

body have been given by Borrd & Maier who extended the results given by Rice, andRudnicki & Rice for so-called continuous and discontinuous localizations. We have

amplified these results by seeking necessary and sufficient conditions for which a

discontinuity surface for the velocity gradient appears at, or reaches the boundary of the

solid. These conditions are given below for the constitutive laws (1) with

hwhere it is assumed that h > 0, and i3 is strictly positive definite.

At a point P of the boundary, F where only surface traction rates F are applied, the

necessary and sufficient conditions for continuous localization [i.e. the material is in

loading (L = H) on each side of the singular surface] are

i) there exists • such that m . i0

(2a) ii det (n.H.n) =0

iii) (m . H . n) . (n . .n)-I . (n . z a) m . E : a

25

4. A CRACK INITIATION POST PROCESSOR

26

4. 1. Introd[cion~L

The constituti\e equations for elasto-[pliticitv and damage at microscale developed

during the first %ear of the grant (see the tir,,t annual report, LEMAITRE-BILLARDON,

Nlai 1990) %\ere used to write a post processor allowing the determination of crack initiation

even when the damage is highly localized as in high cycle fatigue.

4.2. Locally Coupled Analysis of Damagr in Structures

In most applications, the damage is very localized in such a way that the damaged

material occupies a volume small in comparison to the macro-scale of the structural

component and even to the mesoscale of the RVE. This is due to the high sensitivity of

damage to stress concentrations at the macroscale and to defects at the microscale. This

allows us to consider that the effect of the damage on the state of stress and strain occurs

only in very small damaged regions. In other words, the coupling between damage and

strains may be neglected everywhere in the structure except in the RVE(s) where the

damage develops. This is the principle of the locally coupled analysis [Lienard, 1989;

Lemaitre, 1990] where the procedure may be split into the following two steps as shown in

Figure 4.

* A classical structure calculation in elasticity or elastoplasticity by the Finite

Element Method (FEM) or any other method to obtain the fields of strain and stress.

* A local analysis at the critical point(s) only dealing with the elasto-plastic

constitutive equations coupled with the kinetic law of damage evolution, that is a set of

differential equations.

This method is much more simple and saves a lot of computer time in comparison to

the fully coupled analysis which takes into account the coupling between damage and strain

in the whole structui-'e. The fully coupled method must be used when the damage is not

localized but diffused in a large region as in some areas of ductile and creep failures

27

wamum • mt n ra - -- -- -

(BENALLAL, 19891

S~ POST PROCESOR

The ltocall comuupled nlssi osialo thos caases cofbritle o aiu

falu e whn the stru tur relm ai nseasi everwh

r excewp t hecitial - poines)

E II

alsobeauw ak nt esat th macrosae uc o

Cmnmcnuum mdl ecfaRV •a m c lermad eqofa mLzad~ngFEIM Calculado~ns

Fig. 4 Locally coupled analysis of crack initiation. c

uhe locally coupled analysis is most suitable for those cases of brittle or fatigue

failures when the structure remains elastic everywhere except in the critical point(s).

If the damare localizes, this is of course because stress concentrations may occur but

•so, because some weakness in strength always exists at the microscale. Let us cotnsider a

muc-ro-mechanics model of a RVE at mesoscale made of a matrix containing a mi-cro-

element weaker by its yield stress, all other material characteristics being the same in the

inclusion and in the matr• (Figure 5).

• Te behavior of the matrix is elastic or elasto-plastic with the following,

characteristics of the material: yield stress oy, ultimate stress ou and fatigue himit of

(of < oy).

28

=F

Fig. 5 Two scale Representative Volume Element

The behavior of the irclusion is elastic-perfectly plastic and can suffer damage.

Its weakness is due to a plastic threshold which is taken to be lower than oy and equal to

the fatig-ue limit of if not known by another consideration

ý.t ý,o.t = (7f

Below the fatigue limit of no damage should occur. The fatigue limit of the inclusion is

supposed to be reduced in the same proportion:

G4f =CYf-(3 y

The second assumption which makes the calculation simple is the Lin-Taylor's strain

compatibility hypothesis [Taylor, 193S] which states that the state of strain at micro-scale is

equal to the state of strain imposed at macroscale.

Then, there is no boundary value problem to be considered. Only the set of coupled

constitutive equations must be solved for the given history of the mesostrains at the critical

29

point(s).

The determination of this critical point is the bridge between the F.E.M. calculation

and the post processor. If the loading is proportional, that is constant principal directions

of the stresses, this is the point M* where at any time the damage equivalent stress is

maximum

CF *(NI) = Max(a *(M)) -- M *

If the loading is non-proportional, the damage equivalent stress may vary differently at

different points as functions of the time like parameter defining the history. A small &* for

a long time may be more damaging than a large o* for a short time! There is no rigorous

way to select the critical point(s) but an "intelligent" look at the evolution of &* as function

of the space and the time may restrict the number of the dangerous areas to a few points at

which the calculation may be performed.

Another point of interest is the mesocrack initiation criterion. The result of the

calculation is a crack initiation at micro-scale Lhat is a crack of the size of the inclusion; its

area is 8A = d2 . This corresponds to a value of the strain energy release rate at mesoscale

G that may be calculated from the variation of the strain energy W at constan-t stress

2 8A

5W.But --- W is also the energy dissipated in the inclusion by the damaging process at constant

stress

d3'J YdD

G(8A) 08A

Assuming a constant strain energy density release rate Y = Yc yields

"3f0

*t

G(5A) = Yc Dc d

Another interesting relation comes from the matching between damage mechanics and

fracture mechanics. Writing the equality between the energy dissipated by damage at

micro-crack initiation and the energy dissipated to create the same crack through a brittle

fracture mechanics process:

Yc Dc d 3 = Gc d2 , where Gc is the toughness of the material. It yields

Yc Dc d = Gc

Comparing the twko relations for Yc Dc d shows that

G(SA) = Gc --* the instability of the meso RVE

"l'hen, the criterion of micro-crack initiation is also the criterion for a brittle crack instability

at mesoscale.

4.3. Contitutive Equatos

4.3.1. Formulation

The following hypotheses are made:

* small deformation

* uncoupled partitioning of the total strain in elastic strain and plastic strain

Eij= E+EPi

* linear isotropic elasticity

31

perfect plasticity. The latter hypothesis is justified by the fact that in most cases

damage develops when the accumulated plastic smain is large enough to consider the strain4

hardening as saturated. Nevertheless, the code described in section 4.3 allows one to

consider stepwise perfect plasticity in order to take into consideration high values of strain

hardening and the cyclic stress strain curve for multilevel fatigue processes.

The threshold of plasticity cs is bounded by the fatigue limit and the ultimate stres

of •_ as _< au

The plastic constitutive equations are derived in the most classical way from the yield

function in which the kinetic coupling with damage obeys to the strain equivalence principle

applied on the plastic yield function:

f = cs = 0 with •Cq7

"Thf plas:ic strain rate derives from f through the normality rule of standard materials

[Lelaitre and Chaboche, 1990].

~z2GD; =0a• f 3. 3GD if=

U @(5j 2 Geq -D jf= 0

where r . is the plastic multiplier. This shows that

- DP J )I 1 where p is the accumulated plastic strain.

The kinetic law of damage evolution, valid for any kind of ductile, brittle, low or high

cycle fatigue damage derives from the potential of dissipation F in the framework of the

State Kinetic Coupling theory [Marquis and Lemaitre, 1989]

32

y2G 2a Rv G •2R,F=f+ Y= _ _

2S(1 - D) 2E(1 - D) 2 2E

S is a damage strength, a constant characteristic of each material

=F Y ,o Y- , or I=-D - if P>p1PD

aY S I-D Smesocrack initiation if D = Dc as demonstrated in section 2.

PD is a damage threshold corresponding to microcrack nucleation, it is related to

the energy stored in the material. Consider the material of the inclusion having a plastic

tir-eshold stress a and a fatigue limit o f considered as the "physical yield stress"(Y

(Figure 6).

7--

/ - /111 5tbr~zjenersy

Fig. 6. Modelling a stress-strain curve.

The second principle of thermodynamics states that in the absence of any damage, the

energy dissipated in heat is alp. Then the stored energy density is 0

Ws f JijdE:P- ap

From the perfectly plastic constitutive equation

W = (,s- Of

Considering that the threshold PD is governed by a critical value of this energy identified

from the unidimensional reference tension test having a damage threshold EPD, a plastic

tlhreshold equal to the ultimate stress ca and a fatigue limit c3f

(as- G'f)PD = (G, - 4f)EPD

ard PD EPD- with au =w-as -- G~t a"

If piecewise perfect plasticity of several thresholds cysi is considered

Ws= Gsi(P-Pi -p ) - fP

and PD is defined by

PDx. asi(pi - Pi-1) - OfPD = (au - Cf)FPDp>O

The critical value of damage at mesocrack iniliation Dc corresponds to an instability

[Billardon and Doghri, 19891. Nevertheless, it can be related to a certain amount of energy

dissipated in the damaging process

34

f YdD = constant at failure

0

Assuming for that calculation a proportional loading for which Rv = const

fI YdD=Df ciRdD-= RVDc

0 0 2E 2E

Taking again the pure tension test as a reference which gives the critical value of the

damage at mesocrack initiation DIc related to the stress to rupture aR and the tensile

decohesive ultimate stress Gu together with Rv = 1 gives the rupture criterion:

EL D -D Dc ,i.e., Dc = DI,2E 2E G 2 R

Collectin g all the equations gives the set of constitutive equations to be solved by a

numerical analvsis

El.- C' + Ep

e 1+ V C71j V OSkk iE 1 -1 D I - D

3-piD, .-- iff 0O2 Gs Geq

IjP I. - D y... i1-D

0 -- if f<0

RV ... ifppD2ES

PD =PD a 2

0 ... if P< PD"1¶

Mesocrack initiation if D =Djc G1 1asRv

As the plastic strain rate tP is governed by I, and I related to the damage rate D, in

this model it is the damage which governs the plastic strain.

Note also that the perfect plasticity hypothesis allows one to wrTite Rv c nly as a

function of the total strain, since

E(1-D) e e EP aa(d sPa 0G---12-2v E H EH as EIj3 -k

ceqj = cs 01 - D)

C7H E E -Ithen

Geq I - 2v as

4.3.2. Identification of the Material Parameters

In order to be able to perform numerical calculations, it is an important task to

dctermine the material coefficients in each case of application. This is always difficult

because one is never sure that the handbook or the-test results used correspond exactly to

the material considered in the application. Those coefficients are

E and v for elasticity

Of, Cy, Ou and the choice of as for plasticity

S, EpD, DIc for damage

36i

1 2

The da-maged parameters are deduced from the measurement of Eas the law. of elasticiz,

allows to write [Lemaia-e and Dufailly, 19S7]

E

Then, the damage D is plotted versus the accumulated plastic strain p as in Figure 8 from

whiuch it is easy to dedace.:

the damage threshold cpD, defined by Ep < CpD -*D=0

t-he damace strength S as S -- - '2E 6D

s£nce in one dimension R,, I and D ps2 ES

the critical value of the damage D Ic def-ined by D Ic Nla.-(D).

0

Figure 8. Identification of damaged parameters.

'38

The plastic yield threshold ys is a matter of choice regarding the type of accuracy

needed in the calculation.

* take as cf for the highest bound on the time-like loading parameter to crack

initation.

* take as.= cu for the lowest bound on the time-like loading parameter to crack

initiaton.

4.4. Numerical Procedure4.4.1. Method of Integration

The method used is a strain driven algorithm: at each "time" (or time-like parameter)

increment, and for a given total strain increment, one knows the values of the stresses and

the other material model variables at the beginning of the increment (tn) and they have to beupdated at the end of the increment (tn+l). We show here that the general numerical

procedure [Benallal, Billardon, Doghri, 1988] becomes particularly simple and efficient

because analytically derived in a closed form.

In the following, the subscript (n+l) will be omitted and all the variables which do

not contain the subscript (n+l) are computed at tn+1. Tensorial notations will be used for

convenience.

We first assume that the increment is entirely elastic. So, we have

G = .trc 1 + 2.i(c - c)

where ?. and J.1 are the Larne coefficients

Ev EEv =and i - E(I - 2v)(1 + v) 2(1 + v)

and 1 is the second order identity tensor.

All the other "plastic" variables are equal ;o their values at tn. If this "elastic

39

predictor" satisfies the yield condition f < 0, then the assumption is valid and the

computation for this time increment ends. If f > 0, this elastic state is "corrected" as

explained in the following in order to find the plastic solution.

The rate relations of Section 3 are discretized in an incremental form corresponding to

a fully implicit integration scheme which presents the advantage of unconditional stabilit,

[Ortiz and Popov, 1985]. Then the solution at tn+1 has to satisfy the following relations:

f = &eq - (Ts=

Xtr~ 1 + 2p.(E - EP- AS-P)

AFP =NAp

AD= YAps

where A() = ()n+- (-)n and N 3

2 aeq

If we replace AEP by its expression in the second equation, we see that the problem is

reduced to the first 2 equations for the 2 unknowns: C5 and p.

Let's find 8 and p which satisfy:

f = G-eq - cOs = 0

h = - Xtre 1- 2t(- EP) + 2jNAp = 0

This nonlinear system is solved iteratively by a Newton method. For each iteration (s) we

have

f + a---:C = 0

h + a- :Ca + p Cp = 0

40

b

where f, h and their partial derivatives are taken at tn+l and at the iteration (s). The

"corrections" Ca and Cp are defined by

C =()s+1- (6)S and Cp =(P)s+1 - (P)s

The starting iteration (s--O) corresponds to the elastic predictor.

The set of equations for f and h may be rewritten as

f +N:Ca =0

h +[Hl + 2ýAp -N-]: Ca + 2 NCp = 0

'Ahere II is the fourth order identity tensor and

ON lV3(II l®I)_N® N]a 8 6eqL2 3

Since N:N =3 and N- aN = 0, the system Diyes explicitly the correction for p:2 ad

Cp= f-N:h3ýt

we shall prove that the correction for 8 can be found explicitly also. First let's prove that

C- is a deviatoric tensor.

From the second equation of the system and using the expression of -- one obtains

tr(Ca) -trh

Using the definitions of h and C6 ,

/, S

tr[(a)s+] =(3X + 2p)tr E = tr[(a)0] = constant

These relations being true for any iteration (s), then tr(Ca) = 0 and the proof is achieved.

Using this last result together with the expressions of D and Cp yields for the

second equation of the system:

3

where A= 1+- 3--Ap I--2-ApN®NGeq ) (eq

One can also check the following result: the tensor A is invertable if, and only if,

Geq • 0, and in this case:

A-1 3•1 11I+ 2t' AP N 0N]1 + ± Ap N Neq

0eq

Using this expression gives explicitly the correction'for •

Ca =2(f_ -N: h)N - -lL h +A .Ap(N:h) N3 1 + 3p. Ap aeq

Teq

To summarize, this method is a fully implicit scheme with the advantage of an explicit

scheme: no linear system to be solved and the unknown are updated explictly.

Once p and C are found, EP and D are calculated from their discretized constitutive

equations and the stress components are given by yij = (1 - D)aij.

Let us remark that the method described above can be applied to the case of nonlinear

42

isotropic and k-inematic hardenings, with or without damage.

4.4.2. Jump in Cycles Procedure

For periodic loadings of fatigue with large number of cycles, the computation step by

step in time becomes prohibitive and the complete computation is out of a reasonable

occupation of any computer. For that reason, a simplified method has been proposed

[Billardon, 1989] and a slightly different version is used in this work. It allows one to

"jump" large numbers of cycles for a given approximation to the final solution.

(a) Before any damage growth (i.e., when p < PD)

(i) the calculation is performed until a stabilized cycle Ns is reached. Let 8p be the

increment of p over tis cycle. We assume that during a number AN of cycles, p remains

liner- versus N with tue slope _p. This number AN is gven by8SN

Thi nubrANiAivnb

w4,here Ap is a given value which determines the accuracy on accumulated plastic strain.

(ii) a iým AN of cycles is performed and p is updated as

p(Ns + AN) = p(Ns) + AIN - 6p.

Go to (i)

(b) Following damage growth (i.e. when p > PD)

(i) the calculation is performed with a constant value of the damage (1 0) until a

stabilized cycle Ns is reached. Then a fully coupled elasto-plastic and damage computation

is performed for the next cycle. Let 5p and 5D be resp. the increments of p and D for this

cycle. We assume that during a number AN of cycles, p and D remain linear versus N

43

with resp. the slopes _._p and 8D. This number AN is -ivenby8N 5N

where AD is a given value which determines the accuracy on the damage.

(ii) A Jump AN of cycles is performed and p and D are updated as

P(Ns+ 1 + AN) p(Ns+ 1)+ ,AN-5p

D(NS+ 1 + AN) = D(NS + 1) + AN-D

Go to (i)

The choice of Ap and AD is of a first importance. For this wxork we chose

AD 2 D- and Ap - AD , whereA-E is the strain amplitude.50 Y(AE)

This method gives good results but as any heuristic method, sometimes it fails and

the values of AD or Ap must be changed.

4.4.3. The Post Proessr DAMAGE 90

A computer program called DAMAGE 90 has been written as a "friendly" code on the

basis of the method developed. The input data are the material parameters and the history

of the total strain components rij. DAMAGE 90 gives as output the evolution of the

damage D, the accumulated plastic strain p, and the stresses oij up to crack initiation.

DAMAGE 90 distinguishes between two loading cases:

(a) General Loading History. In this case,'the history is defined by the values of

4 A

cj at given "time" values. DAMAGE 90 assumes a linear history between two consecutive

given values.

(b) Piecewise Periodic History. In this case, the lowiiny is defined by blocks of

cycles. In each block, each total strain qj varies linearly between two given "peak" values.

DAMAGE 90 asks the user if he wishes to perform a complete calculation, or a simplified

one using the jump in cycles procedure described in Section 4.2 if the number of cycles is

large.

DAMtAGE 90 is written in FORTRAN 77 as available on a CONVEX computer, and

contains near 600 FORTRAN instructions. It is designed to be used as a post processol

after a FEM code. However, it may be used in an interactive way. In all the examples of

Section 5, the average computing time on the CONVEX machine was between 1 and 15 sel

for each run.

The questions asked b.' DAMAGE 90 for a case corresponding to an example of

section 5.4 are listed in Ap-endix 1 together with the results. Appendix 2 is the complete

Fort-ran listing of the prog-am DAMAGE 90.

4.5. F

The following examples were perfonned with the "DAMA,\GE 90" code in order to

point out the main properties of the constitutive equations used together with the locally

coupled method.

4.5. 1. Case of Ductile Damage

The material data are those of a stainless steel

E = 200,000 MPa, v = 0.3

Cf =200 MPa, 3y = 300 MPa, ou= 1 500 MPa

S 0.06 MPa, EPD = 10%, Dic = 0.99

45

Pure tension at microscaie V

The pure tension case at micro-scale is obtained by giving PD = EPD = 10%,

Dc = DIc = 0.99 and E1 1 as the main inpuL DANMAGE 90 computes the other strains

le lylvEii= E3=-2E1+ 2 E(I- D)

which represents the elasto-plastic contraction resulting in the incompressible plastic flow

hyPothesis (EP = E5'

For a better representadon of the large strain hard-ning of that material, the piecewise

perfect plasticity procedure is used with the following data:

c1 1% 0 0.25 1.5 5

CYSMPa I 200 I 300 I 400 I 500

The results are those of Figure 9 where the strain softening and the damage are linear

functions of the strain as it is introduced in the constitutive equations

$00

400 -

300 0

200

.2

0/.10 is so

Fig. 9. Elastoplasticity and ductile damage in pure tension at microscale.

46

Plane Strain a: meso (or micro) scale

Those cases of two dimensional loadings wxere performed in pure perfect plasticity

%kith a plasticity threshold as = cu= 500 MNPa.

The influence of the triaxiality (ratio (TH/GTeq) upon the accumiuiattxa plastic strain to

ruptu-re PR was obtained by calculations for which in each of them CYHj/(Yq is fixed to a

certain value. As from the constitutive equations, c¶7!i_ E EH ,this fixes the value ofa-eq 1-2v as

El I+ E- £- = 3EH. Then, for each point £11 was chosen and E-22 taken as E22 = 3EH - El I.

The points of the limit curve in strains correspondingr also to rupture wvere obained

,Aihproportional loading in strain histories: E-2 ( X EC 3

P, -P 7,

1 ý0 S 1* 25

Fig. 10. Effect of the triaxiality Fig. 11. Crack initiation limitupon the strain to rupture. curve in plane ctrisncz

rPR is the strain to rupture in

pure tension all/Cy = 1/3,

C-PR =19.6%

the results are plotted in Figures 10 and 11I where are obtained the classical strong effect of the

triaxiality which makes the rupture more and more brittle (PR - PD) -4 0 [McClintock, 1968]

and the classical "S" shape of the limit curve of metal forming [Cordebois, 19831 here in plane

strains.

47

4.5.2 Case of Brittle DamageThe material data are those of a ceramic

E = 400,000 MPa v = 0.2

of = 200 MPa, Gy = 250 MPa, ou = 300 MPa

s = 0.00012 MPa, EPD = 0, Dlc = 0.05

An elastic tension case is considered at meso-scale, ElI is the main input and the two

other principal strain components are imposed as -22 = £33 = -V 1-

At microscale this induces a pure tension in the elastic range but a three-dimensional

state of stress occurs as soon as the plastic threshold is reached due to the difference of the

elasto-plastic contraction of the inclusion and the elastic contraction imposed by the matrix.

The plastic threshold is taken as os = 200 MPa.

The result are those of Figure 12 for the behavior at mesoscale where no appreciable

plastic strain appears and of Figure 13 at microscale where the effect of damage is sensitive.

4.5.3. Low and High Cycle Fatigue

In the damage model, there is no difference in the equation between low and high

cycle fatigue, which obey, this is the hypothesis, to the same energetic mechanisms.

Furthermore, vw-ritten as a damage rate equation it is valid even when a cycle cannot be

defined.

The material data are those of an aluminum alloy

E = 72,000 MPa, v = 0.32

of = 303 MPa, oy = 306 MPa, ou = 500 MPa

s = 6 MPa, EpD = 10%, DIc = 0.9 9

48

0, b

200

IQ / ,00

.15

X 2 )4 )I ? 0

Fig. 12. Stress strain and brittle Fig. 13. Stress-strain and damagedamage at mesoscale in pure at microscale when pure tensiontension is applied at mesoscale

As for the example of Section 5.1, pure tension cases at microscale are considered by

giving as input data c 11, the other strains being computed by DAMNIAGE 90.

In order to take into account the cyclic strain hardening the following plastic

thresholds taken from a cyclic stress strain curve are considered for the different constant

strain amplitudes imposed

Ell% + 0.425 + 0.43 + 0.47 +1 + 3.5 + 4.5

as MPa 303 305 308 370 440 460

The number of cycles to rupture obtained are given as a function of the strain amplitude

imposed in Figure 14 on which the number of cycles to damage initiation No (p =PD) is also

reported. As shown by experiments, the ratio No/NR increases as NR increases.

4 q

%

\

4

2

1

.5! I i I

Fig. 14. Fatigue rupture curve in pure tension at microscale.

The details of the results corresponding to the case Az1 1 = 7% are given in the graphs

of Figure 15 where are reported the evolution of the strain E11 and the stress-strain loops

(a11, C11) as a function of the number of cycles, the equivalent stress (Teq and the

accumulated plastic strain p together with the damage D.

It is interesting to compare with the case of a pure elastic tension case at mesoscale

inducing a three-dimensional state of stress at microscale. The same input are used as for

the case of Figure 12 except that Lhe strains applied are

EI1 = ± 3.5%, E22 = -33 = -V Ell.

The four similar graphs are shown in Figure 16.

90

"i Ik

Fig. 15. Results of a very low cycle fatigue in pure tension at microscale AE1 7%, NR

= 40 cycles.

Fig. 1 6. Results of a very low cycle fatigue in pure tension at mesoscale inducing three-

dimensional fatigue at microscale AE1 7%, NR 8

[51

-i111 I I

4.5.4. Nonlinear Accumulation in Fatigue

An important feature of fatigue damage is the nonlinearity of the accumulation of

damages due to loadings of different amplitudes. If a two level loading history is

considered wui .'14 cycles at the strain amplitude Az1 and N2 cycles at the strain amplitude

AE2- with N1 + N2 = NR the number of cycles to rupture, in opposition with the

Palmgreen-Miner rule, generally

N1 N2N, + N2 < 1 if AF1 > AE2NRI(Azl) NR2(AF2)

N1 N2+ >lifA£l <AE2NR 1(Az£l) NR 2(A£-2)

NR1 and NR2 being the number of cycles to failure corresponding to the constant

amplitude cases of fatigue respectively at A-1 and AE2 .

This effect was studied with the same material data as those of the example 5.3 on a

one-dimensional case at mesoscale E22 = -33 = - v E11.

Ell ± 0.47% for the highest level NR = 7720 cycles

E1 1 ± 0.425% for the lowest level NR = 109570 cycles

The results are given in the accumulation diagram of Figure 17, thcy are in good

qualitative agreement with experimental results

4.5.5. Mlitiaxial FatieIe

Two cases of multiaxial fatigue were studied: biaxial fatigue and fatigue in tension

and shear. Both were performed with the material data of examples 5.3 and 5.4. The

game played with the code, DAMAGE 90 consisted of finding the state of strains which

produces a given number of cycles to failure. This was done by repeated trial from many

calculations.

52

N2

1.

.8_Av vAv AV AvAvV AA N

N, N2

.6

.4 Linear

AAAAA A acc vva,cn

ýýVV V V N.2 N1I N2

0

_ N ,0 .2 .4 .6 .8 1

Fig. 17. Accumulation diagram for two level tests in tension at mesoscale.

Biaxial Fatigue in Plane strain at meso (or micro) scale

A plane state of strain is considered with the following in phase strains

E1= +±x E22=±y E3 3 = 0

Figure 18 shows contours of cycles to failure corresponding to NR = 7800 cycles.

" ' -N . : 7 3 0 0 , . 1,•,,

Fig. 18. Biaxial fatigue in plane strain.

53

Fatigue in Tension and Shear strains imposed at meso (or micro) scale

This is the same type of calculation with

11=+±x E12=±y all other components = O

Figure 19 summarizes the results for the same number of cycles to failure.

.1r %

3 -I .Z .3 .3 .3 .3 .1cSI

Fig. 19 Fatigue in tension and shear strains imposed.

';4

4.6. Listing of DAMAGE 90

L_* DkAIGZ 90 STP- )*** URA GOOD WES.C 3.lstiV-p01rf0Ctly Plastic la.. COWI9d to A bUI dAmA90 Model. BI7I

C-1. ullyimplcit ntegatio schme. UTCin.1) zha .1... of,,10 cp.- 2* - A £s i cycles procedur. vt*zm. 1~)* 1 . value of P. :

L C.. Written by Issasa DObri &=fl1*S.M~0

L-C~ Version : My 199 3DflM*)DOO

ZPVD--= RE.AL- (A-U,O-Z) "=i1qi,*)' Is the stress Stat. un±a.ia1 3 I' or .VCHA.AC'I=*l MrXI ,A.UPDAS2IDC. * ClC*3L'?. Wr.S-.a .Q2L REDMf(UIIOTAXXCidASAC.-M20 XrILEI.FL12 1y-A.NA.X n XD

WAP-kxTLR(WrMT-dlt-PV1) GDO BYE..V�iNIAL*8 1r5N~),~Vl5AV S~V)55J ( (UNZXC!.t. y P.A ='-) l

* ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ RT M.))~~4NS.S1()S~() tizI*)' 1 . t2.. atr~z bistory cyclic? 'IT"' or'''INXCtmft50),5"(00)UtS(6,00)17BI650jPS(6,0)MU 1 "~I() SYTEL(So) x7)CT=C.H.E. 'y' ..L¶.CY-C.S.t. U.* .~

Dr1-CEZ ISLOPE(6) ,tEU)6) ,fl0CX)6) ,NSC1YI50) * CLI.NE. IT, .A1.CTCC.Ut. '3')srv'**

~o~N/rI02,v4E .T. .~XZII(CyC C.EQ. y 3 .03. (~CTCLI. 'V

C- MADf MAXA *)i03 . TVim LOAZZ- IS CZ`C.C.).S. 6 1

P2.-S Uaflri's. Do yvuf w~ish Ut. js.p in cycles procedure for0A.,=.6 arge 3 7 1*71 or

S?'2D1- 1.10- IF0Q*MNE. -y LA:~jP-.n1. ''.1

WA 1 U2()S, *)' I .Giv th nus-b0: a' blocks of cccs.3-zt cp2.ctzda

WRL= (?.S , Give material cc.oslonts and tŽ.. straln.. history' Do ThL.1.PLOCK1*l r ?. WRITE(P'.S. )' I ive tI. nvc'. ocycles Io, block:'I:

Int(S,*)I wifl S.vo you th. daS grow-,%Ž u7 to crock irrit'at'o DO LS..32,%AX-O.nS*) Give for 9-1% xtraL.ý coz;onaot P.2.5 and ain values

.........................................................3 ?.( lit peak , 2=d p..' o!f I2-trir

1~(P.S. 5' 3 :0y . Give LI' -=WR31rrt)3'S,.) 1cOr.-" a odu.4c WRIT t (2fIPISI)I - Give 12.. value of th. plastic threshold stress

*vn n )E a 3IC. for this block :1 ~(?IS. )' I POISSCU''a0 ratio :' LAD(P.3, )SY-rL(IBLj

ITZM* )x.-uWRT (?z 1 ' ) 1 Suggest a ciccb~r !iacrasnrts per cycle11571Z-- PtAST C~7 plastic thrashold S.Gs fai..n'~ 4)

given w-it- ioad~in C ive ED.3-)'UEIL).'= (?-L, -) Fatigue. limit SIof :' LI D

READ-M, tS!G ME ) r((YC.IC.?Q. n'o').01. (Cý=C.Q. 'N') )tJ,.n1aS.t)P.5,')' Yield stress 3:G7 :', GT(S)

* ISIGY 0Rr(P.S,1. . IlYOU t177Ct~ IS ON= CYCLIC.-0~r~vS.' ltijt. stress SIC.. :1

XLAfl(.'3, )SIGV ImITtEl'.s,)' Give th. o.-2.r of point. which d.fin.. th isahctocWOLTt (?IS,*)'I" MnAGto rMVTLUCV dD - (T/S) 4, GiCve S- .y (200 ma.) :PLOD(J20. ')S 2LA30)10, *)InMMAT..(A) WRITE rS.) Give the. -1u..s of ti-. at these pocist,

N31~t).S,' C.'90?TS03SSH01.3 d=- if prpfl. DO you k.o. pDO32f)3)?"4Gl.*~V

0?,U)ArSPt.y'.R.XAS0t0'. _4 2 135)11 Cive th. values. of -. 12.' -strain at these'"S31 I ?-.t,' CG- tho Vaec. of PD ' ti..s :1)READ txa3, * j S.- ZDopoK T(IT GI-1I r EQ~.?0. )-1 SI2.0- IND DO

ELSE? IlI'S7.Qo'XR(-t?.E.N)~ MG5,tS, Give th. -. ).-, of Ls. plastic th'r.shold str.ss S)~7xrt('.1,1) IG. at these. Li..

2 G ive &I'D in tension EO?3 (YLIC.C11V73 WRDCI.-f/)lsCf2ooyJ =feS* Suggest an Ls-:t~.a1 tin. increment (to

REA J-Z,- E DLit.".aii. between first 3 2 n.

STOP"" 50310 LAZA. COOD BYE.' C... INTIl31LZZ. STAlO.V r~p.D.d2.(pstts~dptII IF DO 101.1 R.s'AIJ

WPrtUsCK* 052.011 LNIOISl tnc D-O O you k.0o. DC 7 "Y"I''J7IT)Q* P ?''N'ND DOD

ktAD -M, X4.'SDC STAMTV)2 )=5?'o

"SI1E.t rs,)' Give th. Value. of Dr C .a0 0 Dc (1) STATIr'fl(M-0.

bSITCe(1'S.*) Give DIC In tension DS7J.N(TST).O.D- - Dlc [ (SI~u/SIG%)--2j / Me CI DREDLF% DI IF3* nit to(3113. 1. I'y'I) .OR. 131S PD. EQ. IT )LPDrSOp0

2CL~.l.5Go TO 301

DP.s$4 C. Ff50 pp

V=TLZ-'dir.c-t.aut' ELSE 7 MP.Q y).R A0DE.' a~

F=rf.-' smr..rut- STXEV 2) GQE. SPO)'frd=

am fr5)V nýF NK2o.T 2s tatUýlW QUO) COKQ5

W--=(rr0 .310 )CQMf. z57.lr~

WRz't (r.J7.314) m

313 7'R.-9,(19X. '11',9X.2V *O9X.'33',10X, '12' X.29X.'13,X33'. V~rTZ

. 115. '0' 23X. 'S:G Gq.5 s0' O TO 303

315 ?"3--%c ZAS, lox. -AE.5 p',x ?a. sr-S EQ.s 2t-5s rm rT

C- IT LCAOLQ3 ILS PICT CrC-IC mW iC- 37 M LCAM~gr IS CQ.C=

O~c~O~L"I/1000. ~c-..7=V.40000.2C'7-1ICTC.L.1

*S'-53s...R) C5Tfl'IO .

I /)TZ(IrF'4.)?"~oV) ]DO -1,6Do Isý''mxxx imam0 a

~ES DO.

S:Gs-sYfEt)1Grv) Wttgm rr....................

I?)S.T.1OTORSXS.O~.IL)S~' LTASE THEJ' 'l W.ITE)30.*)'aEGf~YflC OF THE eCYCE.

& 7ALUýtS or 50 SIGf 2,u d Sic.. W1pa=EIo,*,'C'IC± IRIAL) -,NRCIC.-

C.. 1? 30 ~ 0V17-tl,'-: LIC. BY 3 MC IF

MtICONVW .C. 'y J.3I. s'AL'EV~l) 1.0 .0. )THLN CX-LL Ztt HJ 10 ? S5.' 75 5t.5~

rF(tS2.Lo3.(S')Cs-sIG?)SCrCSa/ssoY))'s-2)rJ(1 SERSI,?A

Vt ... JD.7X7. I .0 5-E.lG1. ORS?'b. 0' t I IS1ST T R

* -f--l ( .c3 1 -CT - 1 .0 5oct5n.v ELSE l(Z)F~4

I-PASS-1PASS-1CmtrvL'IL'0X/2. V p73151.0

C** 1! cVE2smcrý VC.X1LE )00tC'c.Lr..NCY=X)Ir' voST-I .151,5

lr))A3sD.EQ..'o OA3.E.0' T .0WEpLf2)S)lI

IPASS-0 ELZE

STA55VI(XS7)-7A~tV( 17) ).( ST)-ISLCft)IST) .. xu-2. VrlrZ/17tRElm to v IF

1)0 1=,i-& =M5Ito

TUr.T-LX.D'TrX StATEY) ZS?)3551EV () I?CALL 0UMPV (NA.U ?r SRAW*SR '12EVOl 1SIX,0

56

D~~5IGS/E0/1O000 ?-I. +=U) o-S'.s.:c/3 ./toIrt32GS.Z2.S0C?.aa.SCS .0T.5013S)VP'-- PLESZ REVIEW MEED~/1 2 336VAL= or S0Gf SIC. and SIC. .' IT-~l

& MWf.ST.1 .StATEVSTRSS .SOA0R.IWfU3 Cvgz-0-so rqAX0/Tyc.. 37 00 =xE== l~rv g Tm IC. By 2 DWYgX.Cpll.xo

* LZT-DLCCt(SC=0/SIGS3*2)^R10 wato -) KMCa.0. ;'SFDW

I? AOS0PAS.z . ll.I3(QPLZ2X2)CTsx/2s.l 3mu-7

17ASS-.0 %lR.rO(80,*3 S~p,8= ~cm2

S 7.q&I t( I0 S S'RkI t D3 .ST R. A4 ( S T IDLTxiwysm4*imcyc.3 I3L)-mncC

3 ('.0, *3 * 8m.0033 80. 3 'Czt.0 p= 0 0 FOft N=.~ cZC

GO TO 308 t=0 I?12- IF &lm IT

w C**S .L1.6OF C 0? ES Arom CA(.A= G7O

vr10 X IF - C.i0 (B

SIGHTS 00 M -slmstf (I) O - I3 a. O 0D SIc .3

*cv=I I 0TOO SMALL.0E00EITDPIm 310Et=-L0 .07.053) SIM...0 33t 0nEGX- 0030 P(./ý7

IF(I0CK3.t6 0-~ Do J-1.S0G~~O 31 3SOP.S 303[1ZF3 S I0.70?A3G~'7t3S

lD Lrol(NAMIS 000 4) Z= D

SeD-v 3)krw (2 -0'. IS 000-P 30'-f L'0'IcL-.c~ 0?)Dc.n7crn7Ip..( .000 .7 fP(. 3000

17315003.03.0 a307vP.**~ PPr=.1RD TSOEJ.~3C 007C ?ZE0 p0

073LDK IF.O0~ UlE8001,cnr-Go mcc-r-m 3~0 if0-SG 3507/0073 * S0I 333-0cm3l

C--t- : 00t7P VROS=LR UI-E1.*'YL -00001513 )O -00073 1.7SO.03m .R.ANPOEI. .uE3 0'I-WFD00)RIE7D-X~r0a.s

1,11E~s"),ro f MSCIAErY (2IF 0'

I 000 U Q - O.W70a. Q T--i"I 8.0 3703LOS 07 -IG/2/C I

3CM-RTIZ(033~ l0( 7b 10

LLS 0 974.0TE380.3C-0E(01.3 -Rra.C 1''900570..0 PSn-r.7 &ND83)C~5. IF ~ ~ a -S T.,

033to0.t~ 400 3 .Oo. (jop4 ol .13 .roTRO000380,3SEt0 07 030 epss(sTa.E 0mx

38.40 Do0 .7033 Lv :-, 7CCT.E KrL.I3.c3Za)3d0L0~~.AX0-~.07/00. LOr(-m~r. 3 CS IElS)PIvss0p S0 STo1330.MP 3Cn.-)0T=

5733A'400~.'n3.0.tXS0.E~ '') ~0bN0Su.6e~57SL

Vr='-i.( I1~ PER (I) - FZDI(t 13T) ) -T3DM

lm rr L--- rr nm C.ISTC PR=IC--Z DOLS wE YT1.Z? 72 Tl X==or-CC= 130 n -I.IMgSS

fl3(E17"s.-z ,1. I3 U--E . j x U(13) -M II 115 (I ) t5TRSHlI AS () 3/051153Z...M... NO.......=ZZT D

wu-.EKS. P"S-.rPAS L- Corr.cticoae. ver the elastic predictorGO TO 304 C. ... Correction Over p

33D 17 CDug2m~t3.-Z2= DOwýýl

Z___ C.... .Correction Over eff.ct-Iv. stress308 mmX0)~ JOB Is =DM TO IMU- y--t AR Do I-I.UZSS

xx* CETS(Z3-2.F!'33:/3.113 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ D DO'3..35~T.133mxE?~ 70.L

IF(~~~c! (JMA.'') . CP ann FLAX=Z 1zRRC-ICn5~ )F~.1 .rLI2* fa1~p..oue3761-0

_-C CRTRE.=

a W 605DIS 3rE-v

D)LZ=1 -. 118 (A-H.O-Z3 C-- UPAT M 76A0e*~c* ~.... Update inc. at p

&~~~~~~~~~~ 31.3301s6.s!s3)~~~3~s(~ .... Updatet effecLives stress' 336.P5736).6S~5.36Do .I-.KISS

LmCA EcL7L.VN.A CAL DOA35.0S.S!'S7

C 0C

TICLpz-ýS:61I.- 6 DrV2-ES-.3LS3

E~ I~c-o ~E~m IFDO2 II.97SS-1 C. ... 1.0 ve con-Srt too Slos'iy~or ve -n~;..i....o

DO C-..~ C... .pro-gr.. vill propose a 3=11or ýtiýo increment.

1031.,13-CwE - - GO TO 30000 DO CmO IF

*- ItCqf-Tl VTC=6R F.ESTRSE.-SIC.1

C- 1.520 MEO. IND0 IF3 C.~.~.j /~Do I-1ISISS

~SREST3SI)L575.'S3 *A.C.0.S (1)-

2C C.... .F... resid-l.1t5.SE 333 RE.5115-0.

C TýI1X. ST 31A1! C ... O)1l-VE TEST Oil 03M rorLo CONfli101O3

IF (1) -R=. FAESS.

001.0 C... .correct ion -- ~ pIT3 3L6A 052mý . *y')i OR. (LN1Axl -5 EC I I )Cl I E0- 3 - M

5131243 2 ) -=Tu *5-131131 3-0 . - i U'pS-R243 (I I 3S0 C -~F5(33 1131puft2) 2 C .... C.Cr-etion over *f!-rt-iv. stress

C .... .Tr-c of str.L.a t-.,~ DO1- Ir*Ns

C .... EOf-civ. str..ssMU( )S31 323--0. AS (I I 3/VE:KDo01 -1.W!51 x:m Do

31)C' ... OF THE MXA02UIVE TL.50 ON4 THE0 Y1003 CONDITIONLIM DO lam Xi'CA:- cO ..(.. S.5134r158wm C 0or THE Loci 05 THE1 ILASOIxC cmR45060003

1731.511.031 I.1T.C. )SICi~..x-. C' ... C OF TEST0 ON4 THE0 YIELD0 COMMONC-~ ITtS I= 1202. 610460101 nm IF

P..LsOse'-91GS S00 CouTrintIrly.LO.0.)fl411

S98

~~m -S-.MN 2 -S..¶2

O1.AB(3)-O5T7AI(2) & STA1-vI3).3TA=(z I I STR53 I STA

END r.k7r-nr 2.22O 30) LI. sfu).X4S fSTRS.)Z).1-4,6),*C... Plastic Str.a.n Lmr m & STA-l-V( 3) .S--P-7t 2 ) SMSBS--xR

Do X-t.NU.S om-(90.444 )7---,STA-- 5.- ~ ).s~as,

DP5TXVW)I)-Xf)'=I 1,0 rvWC1Z(LE10.4.* ,3(~.,)~3LO4

cmD IX.Z1O.4,1 . 0 /,2Z04.

C ... .Damaq. icrn 444 7rC3¶-A(3Z.E1O .4.SX.E'0.4.3X.L10.4.3XEIO.4,4%r.E3.41CDo. zr.rtmT

DO.1-Ds/s0

C .... .Cos5t*c mut?*S5*S zmca

SrksLS MRSM )CZ-CCZD-

s7A1--7().2)

END I r

C ... 1,1 a~d 2nd Str.., 1nvarýc-s

- vI ): .Lr 2 C7.C~S() -V C-X 3 ~I -

1? Cr - ) CCN 11 .7IEN~D DO

C.DC E~ X .. ...... . '.......... S........ .T... ..

C............. .........

C ... lrnv r.aat .! 2 s-,-.tric 2ndorrt.ss

DL2Lc:CC LEAL's IA)1.C,-.I)

REAL'sa V)G),VI())

to 1-1.w-.ss

DEPLICT REAS a IA-3,.O-Z)

59

5. APPLICATION TO THE ALPHA TWO TITANIUMALUMINIDE ALLOY

60

The material was provide by AFOSR from the Materials Laboratory WRDCIMLLN.

Wright Patterson AFB in order to apply the micromechanics model to a practical case

interesting for applications.

5.1. Material

- Composition (Ti3 Al) Ti -24 Al- 11 Nb

Ti Al Nb Fe 0 N

bal 13.9 21.7 0.073 0.065 0.012

- Heat treatment

Solution 2100'F/ I HRIVACUM Forced fan. Cool with argon

Aging 1400'F/IHR/VACUM forced fan air cool

- Mechanical properties.

Figure 20 gives the yield stress and the ultimate stress as a function of the tempe rature.

800

700

600

500

400a300400 a

200JYS, MPaJ

100 UTS, MPa

0 -I0 200 400 600 800

Temperature, C

Fig. 20 Yield stress and ultimate stiess of the Alpha-Two. Titanium Aluminide Alloy.

Figure 21 gives theYoung Modulus also as a function of the temperature

61

100

90

0

•o 800.

(3lcj) o::• 70 o

0O¢E 60

50

400 200 400 600 800

Temperature, CFig. 21 Young Modulus of the Alpha-Two Titanium Alumide Alloy

5.2. Tensile specimens

The location of the tensile specimens as taken in the piece of material is indicated in

figure 22.

) : J> N /'"

.. ' .'! •"• "-, " K

"--t 0-1 6\ý "Ib "1 e.--, r

Fig. 22 Localization of the tensile specimens

The drawing of specimens is incicated in figure 23.

62

The drawing of specimens is indicated in figure 23 below

A A

EPROUVETTE CYLINDRIQUE

63

5.3. Identification tests

The 3 basic tests needed for the identification of the model are described in the following

figures.

Test n'l is a classical tensile test strain controlled 1 = 0-5 s-1 on specimen n' 41

Young modulus 90 000 MPa

Failure stress 523

Failure strain 1.59 %

The figure 24 shows a typical non ductile type behavior without any softening prior to failure.

Test n°2 is a strain controlled cyclic test ( 1 = 0-5 s-1, = ± 0.96 %) on specimen n' 40

Number of cycles to failure 52.

The figures 25 a, b, c and d show a large cyclic hardening without any softening due to

damage.

Test n°3 is a cyclic test on specimen n' 20 ( 10-5 s-1, -= 1.35 %) for wich the failure occurs

after a single cycle (Figure 26).

The conclusion of these tests from the point of view of damage is a typical brittle behavior for

which the damage mechanics must be applied only through the localy coupled approach as explained

in section 4 of this report.

64

The crack initiation occurs at microlevel and then prapagate very iast to induce

macroscopic failure

identification of the model

Using the DAMAGE 90 post processor in an iterative process of identification the

following set of materials parameters has been obtained from the tree basic test results.

E P if P > PDa

D =Dc -crack initiation

E =90 000 MPa

v=.3

S=3

EPD = 0.01

af= 315 MPa

cy = 320 MPa

au = 525 MPa

Dc = 0,2

65

CDJ

z r4

CZ u

Cl)

C) C) ) C) C

(0o C~j

wzoliZWWmZO U)HQUJwCI)(n

66

c:)~0

0-

0

C:)C:0~

G O C) CD C

00

CCD

Hz - w m z n - :- U (

H _ _ _ U67

0

U')

-zC)z

0 DCNJ 4 co

LC)

C0

0ý

____ 0

WZo t 3.zwwmrzo nýawcoc,)6,8

Lt*)

C)

z tCA

0) 0 0D C00 C ((0C,'j C\j o

0

cu*

69

IC)

CC)C)C)~

Lflr

C)D

a)-z w m z C/D c- IU CO U

70

LO

___ ___ C-

C0

CD

0 ___

C)~

6

H) CD C\J CD I

LC)

71

5.4. VWrification tests

In order to check the applicability of the model to the Alpha-Two Titanium Aluminide

Alloy, some stress controlled tests have been performed.

The main features of these tests are reported in the table fig. 27 and the figures 28, 29,

30, 31, give the shapes of the stress-strain loops for different number of cycles.

Figures 32 indicates the type of brittle failure by pictures taken with an electron beam

microscope.

Those stress controlled tests have been also calculated using the Post-Processor

DAMAGE 90. The comparison between the calculated number of cycles to failure and the test

results is given in the chart of figure 33.

The agrement is within in order of magnitude of the usual discrepency.

72

r-CO ~ (-r-0LOCV)(D0)C~l -

C) a C: C\j

0)

IC ) - YC- C)0

Ctr

U)U

+1I

U-) CD tacoo

r-)U) -

C)

Lo.

C: - CC

CXC"

0u)0

CCD

.-. U~c ocn L.-

=3 um

CL*

.- 0

C o

C\I

0.0

4))

a-~

C) C) C) C)- C) C ý C

CC'JC;

El

CC

474

0)'I

- C)

00

C)3

C)C

CLf

C~jC0

LO0

+1,

00

0)

It,3

+75

CDC

00

00

o U

C/ (0 E>C)CoD C)C

c~cm

C'C)

I.76

N LO.

00l -

(NjW

L)

0n

C)C

40~

4\ \

W\

07

I.6.

(0

(0 C) C 00G ~ ~ ~ C C)00

1..~cl (0(NVNI

0Q

(0

77

jr'

Fig. 32 Micrograph pictures of fracture surface

78

00

00

+1

cf) tn

E' 00E=: =

7P Z,,