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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
LABORATOIRE DE MECANIQUE ET TECHNOLOGIE
(E.N.S. CACHAN / C.N.R.S. / Universit6 Paris 6)
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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,,