Nuclear Engineering and Design 80 (1984) 233-245
North-Holland, Amsterdam
HOW TO USE DAMAGE
MECHANICS
233
*
(This paper is dedicated to the memory of A. Sawczuk (1927-1984))
Jean LEMAITRE
Professor at University Paris 6, Laboratoire de Mbcanique et Technologie, E.N.S.E. T/Universitb PARIS 6/CNRS,
61, Avenue du Prbsident Wilson, 94230 Cachan, France
Received 22 August 1983
The background of continuum damage mechanics is first presented in the framework of thermodynamics with some
examples of constitutive equations for ductile damage, creep damage and fatigue damage. After the general scheme of
structural calculations for macro-crack initiation, through non-coupled or coupled strain damage equations, some examples of
"simple applications" are given: fracture limits of metal forming, surface initial damage in fatigue, creep fatigue interaction,
and bifurcation of cracks.
1. Introduction
Damage mechanics has now reached a state of development which allows engineering applications. This part
of solid mechanics based on metallurgy gives a better
understanding of rupture problems in structures by the
definition of a variable which represents the deterioration of the materials before the initiation of a macrocrack. This variable has to be local in the sense of
continuum mechanics in order to be introduced in
structural calculations.
Damage in metals is mainly the process of the initiation and growth of micro-cracks and cavities. At that
scale the phenomenon is discontinuous. Kachanov in
1958 [1] was the first to introduce a continuous variable
related to the density of such defects. This variable has
constitutive equations for evolution, written in terms of
stress or strain which may be used in structural calculations in order to predict the initiation of macro-cracks.
These constitutive equations have been formulated in
the framework of thermodynamics and identified for
many phenomena: dissipation and low cycle fatigue in
metals (Lemaitre [2] 1971), coupling between damage
and creep (Leckie [3]-Hult [4] 1974), high cycle fatigue
(Chaboche [5] 1974), creep-fatigue interaction [6],
ductile plastic damage [7]; that is the three main kinds
of damage: fatigue damage, ductile damage and creep
damage.
The assumption of isotropic damage, is often sufficient to give a good prediction of the carrying capacity,
the number of cycles or the time to local failure in
structural components. The calculations are not too
difficult because of the scalar nature of the damage
variable in this case. For anisotropic damage the variable is of tensorial nature (Chaboche [8] 1978, Murakami
[9], Krajcinovic [10]) and the work to be done for
identification of the models and for applications is
much more complicated [10,11]. Nevertheless, since
1975, damage mechanics has been applied with success
in several fields to evaluate the integrity of structural
components and it is the conviction of the present
writer that it will bec. ome in the near future one of the
main tools for analyzing the strength of materials as a
complement to fracture mechanics. Before explaining
" h o w to use damage mechanics" with some examples, it
is necessary to give a short background on the subject.
2. Elements of damage mechanics
2.1. Damage variable
* Invited lecture of Division L of the 7th International Conference on Structural Mechanics in Reactor Technology
(SMiRT-7), Chicago, Illinois, USA, August 22-26, 1983.
In a damaged body, let us consider a volume element
at macro-scale, that is of a size large enough to contain
0 0 2 9 - 5 4 9 3 / 8 4 / $ 0 3 . 0 0 © E l s e v i e r S c i e n c e P u b l i s h e r s B.V.
( N o r t h - H o l l a n d Physics P u b l i s h i n g D i v i s i o n )
234
~<~
J. Lemaitre / How to use damage mechanics
-~
I/
\S
Fig. 1. Damaged element.
m a n y defects, a n d small enough to be considered as a
material point of the mechanics of c o n t i n u a (fig. 1). Let
S the overall section area of this element, defined by the
n o r m a l n, a n d S the effective resisting area, that is S
diminished from the surface intersections of the microcracks a n d cavities a n d corrected for the microstress
concentrations in the vicinity of discontinuities a n d for
the interactions between closed defects. By definition,
the d a m a g e variable D associated with the direction of
the normal n is:
D. S s ~ ,
D. = 0 corresponds to the u n d a m a g e d state,
D. = De, a critical value, corresponds to the rupture of
the element in two parts (0.2 ~< De ~< 0.8 for metals).
2.1.1. Hypothesis o f isotropy
W h e n cracks a n d cavities are oriented D, is a function of n which leads to an intrinsic variable of tensorial
n a t u r e [8,9]. If cracks a n d cavities are equally distributed in all d i r e c t i o n s , / 9 , does n o t depend u p o n n a n d
the isotropic intrinsic variable is a scalar: D.
2.1.2. Concept o f effective stress
The stress vector taken as the density of forces with
regard to the effective area S is called effective stress
vector ~'. It is related to the usual stress vector T by:
S
T
1"= T-~ or ]?= 1 _ D
which leads to the effective Cauchy stress tensor o
related to the usual stress tensor o by:
2.1.3. Hypothesis o f strain equivalence
The effective resisting area S can be calculated
through m a t h e m a t i c a l homogenization techniques [12]
but the shape a n d the size of the defects must be
known, which is somewhat difficult, even with a good
electron microscope! To avoid this difficulty the following hypothesis is m a d e [13]:
Every strain behavior of a d a m a g e d material is represented by constitutive equations of the u n d a m a g e d
material in the potential of which the stress is simply
replaced by the effective stress.
Example of one-dimensional linear elasticity:
~ = o/E
¢e = 6 / E
u n d a m a g e d state,
o
(1 - D ) E
damaged state,
c ~ being the elastic strain and E the Young's modulus.
2.2. Thermodynamics
In order to model the isotropic p h e n o m e n a of elasticity, thermal effects, plasticity a n d damage within the
framework of the t h e r m o d y n a m i c s of irreversible
processes the state variables given in table 1 must be
introduced [13]. The plastic strain tensor is defined from
the total strain tensor by c p = c - c * and the scalar p by:
p = (2iP: iP)1/2.
2.2.1. Thermodynamic potential
The free energy ~b, taken as the t h e r m o d y n a m i c
potential in which elasticity a n d plasticity are uncoupled, gives the law of thermoelasticity coupled with
d a m a g e and the definition of the associated variables
Table 1
Observable
variables
Internal
variables
Associated
variables
Elastic strain
tensor
Stress tensor
o
Temperature
T
Entropy
s
~e
Accumulated
plastic
strain p
Increment of
the yield
surface: R
Damage D
Damage strain
energy release
rate y
J. Lemaitre / How to use damage mechanics
As ( - y ) is a quadratic function: b >/0.
The rupture criterion:
related to internal variables.
= ~ e ( , e, T, D ) + % ( T , p ) .
In the case of linear elasticity where a is the fourth
order tensor of elasticity and p the density considered as
constant:
- Y =Yc ~ crack initiation,
may be written in terms of D through the one-dimensional rupture stress oR:
1
I/%= ~ p a : c ¢ : ¢ ¢ ( 1 - D ) .
02
Yc
The damage elasticity law obeys the hypothesis of strain
equivalence:
a~
o=p~ce=a:~.c(1-D)
or
O = a : ¢ e.
- ½ a : ce: c~"
We, being the density of elastic strain energy defined by
dW¢ = a : dc ¢, the expression for y shows that:
-y =
1-D
-
1 dW¢
-2 dD
By analogy with " G " in fracture mechanics, - y is
called " d a m a g e strain energy release rate".
- y may be calculated as a function of the hydrostatic stress o H = ½tr(o), the Von Mises equivalent stress
(related to the second invariant of the stress deviator)
and the Poisson ratio v:
Many experiments have shown that:
which allows (1 - De) x to be neglected with regard to 1
when x is much greater than 1.
The potential of dissipation is a scalar convex function of flux variables (iP, p, 19, and the heat flux q ) or
their dual variables (by means of the Legendre-Fenchel
transform), the state variables acting as parameters:
It gives the constitutive equations for the evolution of
dissipative variables, in particular for D:
D= -O~*/Oy ].
2.3. Damage models
O'¢q ~ [ 2~(0' -- OH'I) : ( O -- OH1)] 1/2
2]
Oeq ] J"
By analogy with the one-dimensional case, taken as a
1 . , --y =
stress reference o*, Oeq = O*, OH = 3o
o ' 2 / 2 E ( 1 - D) 2, it is possible to define a " d a m a g e
equivalent stress" [14]:
Many models may be derived from this formalism,
the difficulty being in the choice of the analytic expressions for the function 9~* and in its identification from
experiments in each particular case of damage. Thus a
measure of damage is needed. A m o n g several different
methods [13] it may come from the variation of Young's
modulus through elasticity law coupled with damage:
ti=E%
o* = Ooq z3(1 + v) + 3 ( 1 - 2 v ) ( ~ - ~
which may be used to write any three-dimensional
damage constitutive equation (isotropic) in the same
way as for the one-dimensional case.
2.2.2. Dissipation
The second principle of thermodynamics imposes
that damage dissipation must be non-negative:
- - y D >/O.
(2Eyo) 1/2"
q 0 . ( ~ p , p , y , q; ¢¢, T,p, D).
at constant a and T.
°e2
[](l+v)+3(l_2v)(°.]
°Y = 2 E ( 1 - D ) 2
OR
2 E ( 1 - De) --* De = 1
0.2 ~< De ~< 0.8
The variable y associated with D is defined by:
a~ =
y=p-ff-~
235
or
o=E(1-D)~e.
The quantity E ( 1 - D ) = E can be considered as the
elasticity modulus of the damaged material from which:
D=I-~
Young's modulus E being known, careful measurement
of the elasticity modulus E on specimens damaged by
ductile plasticity, creep or fatigue leads to a quantitative
evaluation of the damage variable D as shown in fig. 2.
Some examples of damage models are now listed below.
236
J. Lemaitre
/
H o w to use damage mechanics
MPo)
400-
In the case of radial loading, when the principal directions of stresses do not vary, the triaxiality ratio oH/%q
is constant and this expression may be integrated using
the conditions:
300
P < PD (damage threshold) ~ D = 0,
200
P = PR (strain to rupture) ~ D = D~.
0
500.
100'
~
0
D
•
~ ~'
I00
C 10 2
liO -
Neglecting elastic strain in the calculation of p [ p
(2~: ~)] and using PD/PR =~O/CR, the equation for
damage evolution D may be written in terms of the
one-dimensional threshold Cr~ and one-dimensional
strain to rupture OR:
_~ =.G5
io °/[
//
'R--'~
. . . . . . .
2b
*
[,~b
'
~
8'0
'
-'e
'
~-'p n =' 35
16oI
'
~-
"CpR =
1,07
Fig. 2. Ductile plastic damage of copper 99.9%.
2.3.1. Ductile plastic damage
Ductile damage in metals is essentially the initiation
and growth of cavities due to large deformations ( p = 0.2
to 2). Experiments of ductile rupture show that a starting expression for the potential ~0" can be:
S
p ~(1 + ~ ) + 3 ( 1
-- 2~)
t ;l /
--~
Identification of such a model consists in the quantitative evolution of the coefficients co, c R and D¢ (Poisson's
coefficient being known from elasticity) which can be
done from experimental results such as those shown on
fig. 2.
2.3.2. Creep damage
Metals loaded at high temperature (above J of the
absolute melting temperature) are subjected to another
kind of damage mainly due to intercrystalline microcracking. The early Kachanov's model [1] extended to
the three-dimensional case may be derived from a
potential written as:
_r+l
2
2
from which:
from which:
b
D= ~sYp.
S is a coefficient, material and temperature dependent. If in the expression for ( - y ) , O c q / ( 1 - D ) is
replaced by its value as a function of p taken from the
hardening law of plasticity coupled with damage: O~q/(1
- D) = Kp 1/M. ( K and M being coefficients which are
material dependent) one obtains the following differential constitutive equation for ductile plastic damage
[151:
b = -ff~
~(1 + ~) + 3(1 - 2~)
%
2 E ] r/2
-YTJ
Ocp*
Oy
o.
1)=((1-D)A
A
) • or
)r
'
where o* is the damage equivalent stress, A and r being
coefficients which depend upon the material and the
temperature.
F r o m this expression it is easy to calculate the time
t c to the initiation of a creep damage macro-crack at a
point in a component where the state of stress is constant in time with the conditions:
t=0---~D=0,
t = t c ~ - - D = D c.
J. Lemaitre / How to use damage mechanics
0 (MPo)
600
~~~x-,_._..._,
^;""~"
400200"
X
T
mX
""-- X
760
81 5
850
~.~..x~
~ x
~ x
~
"x~x-..
~x_"""~X~x
~
x---~'Omx..~o ~ a ( - . . . . - ' ' ~
"~x~,
~x--..tx
"--x.~
x~
~-~
~"X X~.,,
~
"~x
o-o~ ~.
x-.~
~
So and s o being material and temperature dependent
coefficients. It is more convenient to write this constitutive equation in terms of stress, which can be done by
replacing p by its value taken from the hardening law of
micro-plasticity, in a similar way as for ductile plastic
damage. The result [with ( M / K M ) ( 2 E S o ) ~o = B0, 2s 0 +
M - 1 = fl0] is:
900
oO-o.
/ a
dD=Bo
x ~
60"
40"
~
~ 1000
x
237
~
[
1040
i f 0 ~ ~(1 + 7,) + 3(1
20-
x
10
,
102
~
,
lO 3
,,2]so
2(l+p)+3(1-2v)(~)
- -
J
(;]
> oj;
21,) °H
-
6~°ldO~q,,
-
%q
1100 *C
~ tc(h)
dD=0
Fig. 3. Creep rupture curves for IN 100 refractory alloy [6].
Neglecting (1 - O c ) r+' with regard to 1 gives:
tc= ~
This equation allows the coefficients A and r to be
identified from one-dimensionai creep rupture tests
curves such as shown in fig. 3 for several temperatures.
2. 3. 3. Fatigue damage
Fatigue damage occurs when metals are subjected to
periodic or variable loadings. The repetition of maximal
values of stress induces transcrystalline micro-cracks
which may yield failure within 10 to 10 4 cycles (low
cycle fatigue in the plastic range) or 105 to 107 cycles
(high cycle fatigue in the elastic range).
The mathematical treatment to model fatigue damage
is more complicated than for the two other kinds of
damage because several effects such as nonlinear cumulation, effect of mean stress etc., are important. Assuming that damage fatigue is always associated with a
micro-plasticity phenomenon represented by a variable
of micro-plastic strain p, some (but not all) principal
features of the fatigue phenomenon may be represented
by the potential:
if
#~q[2(1 + , ) +
3 ( 1 - 2 ~ , ) ( OH
( o mbeing the fatigue l i m i t ) .
To obtain a classical constitutive equation for fatigue,
which gives the variation of damage per cycle 8 D / / S N
the model has to be integrated over one cycle. Assuming
a "positive" and proportional loading ( o H / % q = cte)
defined by maximum OcqM and m i n i m u m %qm values of
%q and if (1 - D ) #o÷ 1 is considered as a constant over
one cycle:
[ OH \2]So
-
8D
aN
-
=
(fl0 + 1)(1 - D ) ~°+l
The number of cycles to failure for a periodic loading
T
800 9~,C~-.~00 * C
600.
500.
1000"C
4oo.
~°x
300] 11oo°c
x
"o% ~
~ ~
a~o
~ _
so / _y~O+,.
• *(y,/,;r)=;-o-~t~o
) p
from which:
ay
10
102
103
104
105
106
N F
Fig. 4. Woehler rupture fatigue curves for IN100 refractory
alloy f -- 5 Hz [5].
238
J. L e m a i t r e / H o w to use damage mechanics
defined by (%qM, Oeqm) is obtained from integration of
this cyclic model for N = 0 --* D = 0 and N = N R *-- D =
1:
(/3o + 1)~0~l
- x / Bo+l
M
#0+iX)
-- O'~]m
(x) =xifx>O
(x)
0 if x..< 0.
-l
N R
o H is the mean value of the hydrostatic stress o H,
/ 0
\2]s° "
2(/30 + 2 ) B o :~(1 + u) + 3(1 - 2 v ) [ ~H ) I
~%qJ J
M ( O H ) = O , ( 1 --bb-HH).
Applied to the one-dimensional case (%q = o, oH/ocq =
1 / 3 ) this is the equation of the classical Woehler curve
(fig. 4) from which it is possible to identify the two
coefficients B 0 and fl0 for each material at each temperature considered. A more realistic model, which is more
complicated and which does not correspond to an explicit form of the potential of dissipation, is [5]:
(
where All = ½sup, ° sup, J2[o(t ) - o(t0) ] over one cycle,
corresponds to the Sines fatigue limit criterion:
[(.,-02)2+(02-0,)2+(03-0,)211/~
o~ being the principal stresses.
Identification of the 5 coefficients of the model for each
material: /3, a, oj( fatigue limit) b, % (ultimate stress)
needs the Woehler curve and some two stress levels
fatigue tests.
3. Principle of structural calculations for macro-crack
initiation
The damage models give the possibility to calculate
the damage evolution with time in each point of a
structure up to macro-crack initiation as function of the
stress or strain fields. The first thing to do is to calculate
these fields.
8D = [1 _ (1 _ D),e+ 1] ,,(A,,. ~.,J2M,
All
6-N
M~.)(-I -- D )
J2(") = ~1
where
3.1. Uncoupled calculations
The scheme of the calculation is given in fig. 5.
The geometry of the structure is known and the
loads are given as functions of time.
The constitutive equations identified for the material
of the structure are also known. Let us consider, as
an example, Prandtl-Reuss elasto-plastic constitutive equations using the isotropic hardening hypothesis:
't
st-rgin .
consLtLuLive
equation
damage
evoluLion
laws
crock ]
propogoLion
ows
sLress and
sLroin fields
hisLories
methode of]
calculaLion J
damage
mechanics ]
imechanics
fracture[
Fig. 5. Scheme of non-coupled calculation of strains and macro-crack initiation and propagation.
J. Lemaitre/ How to use damagemechanics
(=(e+(p,
l+v
C= ~ a -
lence gives a simple way to introduce the damage in the
strain constitutive equations. The previous equations of
elasto-plasticity become:
v
~ tr(o)l,
do~q
239
(=~e+(p,
aD
d(P =
(e
Ky t Oeq-k I I-My Oeq'
My
l+v
E
where k is the yield stress, Ky and My are hardening
coefficients, o ° is the stress deviator.
The stress and strain fields corresponding to the
loading are obtained by finite dement techniques,
for example:
o(M,t),,P(M,t).
It is these quantities that have to be introduced in
damage models. Special techniques may be used in
order to save time in the integrations of differential
constitutive equations for damage, when the loading
varies with time [16].
3.2. Coupled calculations
In fact, the damage field has an influence on the
state of stress or strain mainly due to its effect on
Young's modulus of the material. This third cause of
redistribution of stress has to be taken into account in
the strain structure calculation. The concept of effective
stress associated with the hypothesis of strain equiva-
d(P
=
o
1' _--+tr(°)1
1-D
:I-D
3
--
daeq
/c,
o I3
- kz
'(p)
to which the damage model corresponding to the damage
involved has to be added, in fatigue for example (see
section 2.3)
AII-D) )~8.
8NS-D-D[ 1 - ( 1 - D ) a + l ] "( M(1
The scheme of the calculation given in fig. 6 is simpler
but the method of calculation is more complicated due
to the fact that the step by step calculation of the
stresses, the strains and the damage must cover the
whole lifetime of the structure [17]. It is interesting to
note that, with this formulation, there is no need for
fracture mechanics to calculate the crack behavior. D ~De at a point M 0 defines a macro-crack in M 0 and
corresponds to a zero stiffness at that point. Then,
further calculation gives the conditions at which the
strain _ damageI
constitutive ]
equations I
l llst'
es';t,ooII
It and damage I1
'
I crack initiation 1
;
end propogationI' k_~condition
..... I
methode of coupledJ
calculations
Fig. 6. Scheme of structural calculation for coupled strain and damage.
240
J. Lemaitre / How to use damage mechanics
n e i g h b o u r point M 1 also loses its rigidity to give a
bigger crack together with M 0. As a consequence, the
crack growth is the succession of points for which
D = D~. A n example is given in section 4.3.
4. Specific applicaEons
After the general schemes of calculation some specific
applications are given in order to prove that the d a m a g e
business may be simple!
4.1. Fracture limits o f metal f o r m i n g
R e t u r n i n g to the ductile plastic damage model, it
allows the possibility to calculate the accumulated strain
(elastic strain n e g l e c t e d ) p = P R corresponding to the
local rupture D = Dc;
2 + v) + 3(1 -- 2~') _ _
PR = c R ~-~
the considered metal, the Poisson's ratio u and the strain
to rupture in one-dimensional tension ~R. Of course
some plastic analysis of the process is necessary to
determine o . / o ~ . K n o w i n g PR, one has to check if the
m a x i m u m accumulated strain calculated, from the
kinematic strain field of the process c~M~: p = (~c: c)~/2
is less than pR or not.
11/2~
?
(M)
If the process is far from a radial loading, OH/Oeq is a
function of time a n d one must use the differential
model D( p, ovi/ooa ) which has to be integrated for each
particular case together with p = ( ~ : ~)1/2 the fracture
limit for p (or c) being defined by D = D c.
A n interesting particular case is the one of plane
stress which corresponds to the deep drawing of sheets.
In that case the triaxiality ratio may be calculated as a
function of the principal strains ~l a n d c2, ( ( 3 = - - ( ( 1 +
~2)) from the incompressibility condition of plasticity
using the integrated Hencky law of plasticity:
c2
This expression contains the large influence of the triaxiality ratio on fracture; numerical values are in agreem e n t with those given by the models for the growth of
voids of Mac-Clintock [18] and Rice and Tracey [19],
a n d with experiments on the fracture of n o t c h e d specimens [20].
R e m e m b e r that the analytical integration of the
model is only possible for radial loading. T h e n for any
metal-forming process corresponding to quasi-radial
loading, the curves g i v i n g p R / c R as a function of OH/Oeq
are the master curves to determine the limiting value p R
in order to avoid local fracture. They are d r a w n o n fig.
7. To use it for practical purposes one must known, for
--+1
OH
Cl
Oeq
~F3
~1
-}-(
~I-{-I
The accumulated plastic strain may also be calculated
as a function of ~1 a n d c2:
=
2
2
xl/2
-
I n t r o d u c i n g these formulas in the expression of p R leads
to the condition of local rupture expressed in terms of ~l
P~
C1
CR
C
]5
/
o,5
~\
./~ = 0,3
/y
0
I
I
I
1
2
3
Ueq
Fig. 7. Accumulated strain to local rupture in metals.
_.;5
_.'1
_.&
0
.o'5
Fig. 8. Fracture limits of deep drawing.
.~
.~5
C2
CR
J. Lemaitre / How to use damage mechanics
a n d c 2 [21]:
d a m a g e does not affect the stress field, we use the
simple pure fatigue model m e n t i o n e d in section 2.3.
+','2+
e~
241
\OR/
4
[
2 B 0 2(1 + p) + 3(1 - 2 p )
\ Oeq ]
1-:
J
0D
3N
=o.
(fl0 + 1)(1 - D )
[ 0o+1
The corresponding curves are the well-known fracture
limits curves of deep drawing (fig. 8). They are quasistraight lines in the useful-range as shown by m a n y
experiments [22] a n d by more sophisticated analysis
taking into account necking instability [23]. It is interesting to observe that, here again, c l / c R expressed as a
function of ~2/~R, are the master carves for any metal
loaded in radial a n d plane stress conditions, the material
d e p e n d e n t p a r a m e t e r (added to ER) being Poisson's
ratio 1,.
x~o,~M
#°+l
_#0+ 1"~
-Oeq,~ 1.
T h e loading being periodic O~qM, o ~ m, OH/O,,q are constant
- within the d o m a i n ~ s N = 0 --* D = Do, a simple integration of the model gives the n u m b e r of cycles to
initiation N i for which N = N i ~ D = D~ = ,
N~=
(1 -
+
+
/~o+ I'~ - I
m )
: O
\2Is°
(/30 +2)2Bo ~(1+")+3(I-2~)(~-mH) J
4.2. Initial value o f damage
-
The p r o b l e m of initial conditions in structural calculations is a difficult one because o n e knows neither the
exact mechanical conditions of the m a c h i n i n g a n d forming process n o r the states of internal stress, h a r d e n i n g or
damage. T h e initial value of d a m a g e is especially imp o r t a n t in the fatigue p h e n o m e n o n where cracks more
often initiate o n free surfaces of structures even when
the stress field is uniform. Let us show how it is possible
to o b t a i n an a p p r o x i m a t e estimation of a n equivalent
initial value of " s u r f a c i c " d a m a g e D O for fatigue from
classical tests. Consider a structure c o m p o n e n t in which
a d o m a i n ~ is subjected to a u n i f o r m stress field,
periodic in time (fig. 9). A t the initial value of time t o ,
the damage D is assumed to be zero everywhere except
in a s u b d o m a i n ~ c o n t a i n i n g the free surface O ~ of
a n d its n e i g h b o u r h o o d of small d e p t h where D = Do, is
a n u n k n o w n value. One way to estimate D O is to calculate the ratio of the n u m b e r of cycles N~ to initiation of
a macro-crack in the s u b d o m a i n .@s to the n u m b e r of
cycles to failure in the d o m a i n ( ~ - ~ ) . A s s u m i n g that
•"~--- D ( ~ . ) -_ 0
D(t.):
D.
Fig. 9. Initial surfacic damage on a domain ~.
within the domain (~ - ~s): N = 0 --*D = 0, the same
integration gives:
1~{,,#o+1
NR=
-x
/ O
\2Is°
from which N i / N R = (1 - D o ) ao+2 a n d
N i / 1/(B° + 2)
Fig. 10 gives a m e a n for the d e t e r m i n a t i o n of this
initial surfacic d a m a g e D O as a function of N i / N R a n d
flo, material dependent, acting parameter:
T h e model being identified for the material considered,
fl0 is known. T o o b t a i n the ratio N i / N R some fatigue
tests are performed on specimens m a d e from the same
material a n d with the same m a c h i n i n g as those of the
structure from which the n u m b e r of cycles to crack
initiation (a crack of = 0.1 m m depth) N i is observed
together with the n u m b e r of cycles to complete failure
N R. In fact, N i / N R depends u p o n the level of the stress
[24,25] which m e a n s that these tests must be performed
within the range of the stress assumed for the structure
u n d e r consideration. D O being d e t e r m i n e d from the
abacus, this is the value that has to b e introduced in the
surfacic d o m a i n ~ s , of the structure for structural a n d
d a m a g e calculations as explained in section 3.
N o t e that the measures of fig. 10 d e p e n d u p o n the
fatigue model chosen, which m e a n s that the same model
242
J. Lemaitre / How to use damage mechanics
1_ Do
Do
d a m a g e is related to the strain amplitude, and the
strain-range partitioning m e t h o d [28] give better accuracy but they are difficult to apply for a three-dimensional state of stress a n d when the stress or strain are
not periodic.
The models described in section 2.3 may be used
assuming that creep d a m a g e i n c r e m e n t d D c and fatigue
d a m a g e increment d D v are functions of the two damages
added:
°iS1
O05
~9s
dDc=fc(a, Dc + DF),
d O v = f v ( O , D v + De),
0,15
-
- • 0,85
a n d by addition:
dO =dD c + dO v = fc(o,
,2 -~/
:
o2s I/
01
02
0/8
W i t h the n o t a t i o n of section 2.3 the damage per cycle of
period At: 3D/3N is [6]:
8D
.~,(
o*
):
, ~ = fo ( 1 - D ) A d t
I
05
NR
fl+l
Fig.
D) + f v ( o , D).
a
All
,8
10. Initial surfacic damage.
a n d by integration for N = 0 --* D = 0 a n d N = N R ~ D
---1:
m u s t be used for crack initiation a n d growth in the
structure. F o r example if the second fatigue model of
section 3 is used, the rule also depends o n the a coefficient.
NR=fo
4.3. Creep fatigue interaction
where N F a n d N¢ are functions of the local loading by:
W h e n metallic c o m p o n e n t s are subjected to periodic
or varying loading at high t e m p e r a t u r e ( T > J T ~ = K
melting point) b o t h the p h e n o m e n a of creep d a m a g e
a n d fatigue d a m a g e occur simultaneously. The difficult
p r o b l e m is that these two p h e n o m e n a interact together
in a n o n l i n e a r manner. Most often the linear interaction
rule of Taira [26]:
A tN R
- - +
NR
=1
leads to errors by a factor of 2 to 10 or more:
- A t is the period of a square wave periodic loading,
- t e is the time to crack initiation as if the creep
damage would act alone,
N F is the n u m b e r of cycles to crack initiation as if the
fatigue d a m a g e would exist alone,
- N a is the n u m b e r of cycles to crack initiation in the
real case.
The M a n s o n - C o f f i n law [27] in which the c r e e p - f a t i g u e
dD
~
1 (l-D)-:
N¢
r+]
1
[1-(1-D)'+1]
NF ( f l + l ) ( 1 - a ) ( 1 - D )
"
~
T h e creep a n d fatigue models being identified for the
material considered ( A, r; B, a, o t, b a n d % coefficients)
if the loading is not periodic, N R is o b t a i n e d by a
numerical integration [16] in which the input is the
stress history represented by o~) a n d A n ( N ). If the
loading is periodic, it is possible, here again, to work
with master curves: N F a n d N~ are constant, it is possible
to write with the variable change:
(1 - D )
¢+1 = u
a n d with:
r+l
B+l
J. Lemaitre / How to use damage mechanics
243
NR
1
NF
X= q 5
0,.
0,8
~.=1
,
•\
\
\',
\,
\
-,.... '-. .'....--~.
\
....
.....
xaj~"
-....-~...
,
q8
1
"--
0 -I
0
0
0
0,2
"\.."'\. \<,I;%
\\ \~
'
0,,,-'. '.,, '~('("'.\
_",. \-..
0,4
"¢.
0,6
0,2
~
. . . . .
0,4
.
-
...
• ',:
N~
- - -":"~- ~--~-.---'-Z'~ '''~
0,6
0,8
I
Ng
Ng
I,~.,
\
X _-1j5
qs. "~'"5.
X= 2
_/o~
06-
",..
aP-'"~--,~
~
~
"---4z~ _ - ~
0
Fig. ll.
0,2
0,4
"-.'~
. ? . . ~ x . N..
0,6
0,8
0
I
q2
6,4
6,6
q'8
I
Creep fatigue interaction diagrams.
we obtain:
NR
o
f [ N~ u1-x
-+
Idu
a n expression in which two p a r a m e t e r s A a n d a appear.
By numerical calculation it is possible to draw the
well-known interaction diagram (Nrt/NF, NR/N¢) for
any material represented b y one value of A a n d the
value of a which corresponds to the local stress (fig. 11).
T o use these curves one m u s t first k n o w the values of r
a n d fl of the material, to select the a p p r o p r i a t e value of
x = (r + 1)/(fl + 1).
K n o w i n g the local stress, a is calculated by:
°t=l--a(AII-OI(I-~-~H)
(see section 2.3). This allows the a p p r o p r i a t e curve to be
selected. N c a n d N F are calculated from the previous
formulas from which the ratio N c / N r is determined.
T h e n N R / N F is the o r d i n a t e of the intersection p o i n t of
the curve determined a n d of the straight line: N R / N F =
( Nc/NFX NR/Nc).
K n o w i n g N R / N F a n d N F any child knows h o w to
calculate N R.
4. 4. Bifurcations of cracks
T h e last application deals with the b e h a v i o r of
macro-cracks. Classical fracture mechanics, with the
concepts of stress-intensity factors, strain energy release
rate or c o n t o u r integrals allows the solution of classical
p r o b l e m s of elasticity in two dimensions a n d elastoplasticity if the plastic energy involved at the crack tip is
small in c o m p a r i s o n to the elastic energy. F o r high
244
J. Lemaitre / How to use damage mechanics
1.
Fig. 12. Bifurcation of cracks.
nonlinear and three-dimensional problems involving the
effects of history of loading it is the conviction of the
author that a local approach by means of damage
mechanics is more appropriate.
D = De is the condition for a macro-crack initiation
at the point considered. Remember from section 3.2
that a crack in the sense of fracture mechanics may be
considered as the set of points for which the damage has
reached its critical value De. Then, a structural calculation performed with strain constitutive equations coupled with damage as explained in section 3.2 gives the
points (or the element if the F E M technique is used) for
which D = De, as a function of the history of loading.
This represents the behavior of the crack(s).
As an example let us describe the problem of the
bifurcation of a crack in a sheet loaded in its plane in a
non-proportional condition, that is the calculation of
the angle of an increment of crack with its previous
shape due to a variation of the direction of the loading
[29]. Consider a thin sheet (fig. 12) submitted to biaxial
loading, F 1 and F 2, low enough to have high cycle
fatigue for which plasticity can be ignored even at the
crack tip. The material is 2024 aluminum alloy. The
initial crack is inclined at an angle ~t to the direction of
the loading which corresponds to the direction of crack
growth for F 1 / F E = r (with Fimin=0 and Fimax=
constant). When the ratio r is modified the phenomenon
of bifurcation occurs and it is characterized by the angle
/~ (fig. 12). The damaged elasticity constitutive equa-
tions are:
¢=
1 + v
E
o
1-D
I, t r ( a )
- - I
E1-D
/ o
3N
\21s°
(rio + 1)
[( UeqM//3°+1__( Oeqm~fl°+l].
× [\-(7~1
\l-D}
They are implemented in a plane stress finite element
code using three-nodes triangles which allows a decrease
of the stiffness due to the damaged elasticity modulus
E=(1-D).
The numerical strategy adopted is the
standard Euler algorithm and the integration over time
is made for a sufficiently small damage increment A D
for which the stiffness matrix is re-evaluated. For one
value of the ratio F 1 / F 2 this procedure is applied up to
D = De in some elements above the crack tip which
defines the /} angle. The results are in good agreement
with those of ref. [30]. An example is given in fig. 12.
5. C o n c l u s i o n s
- The formalism of continuous damage mechanics is
simple.
J. Lemaitre / How to use damage mechanics
Hypothesis of isotropic damage makes equations sim-
-
ple.
Anisotropic damage is complicated.
Modelling damage constitutive equations needs good
experience and good experiments.
- Identification of models is complicated.
Damage evaluation needs good stress and strain
structural calculations.
- Crack initiation and crack growth are obtained from
structural calculation by a simple procedure.
- Ductile fracture, initial value of damage, c r e e p - f a tigue interaction, in three-dimensional conditions may
be obtained from simple master curves.
-
-
-
R
e
f
e
r
e
n
c
e
s
[1] L.M. Kachanov, Time of the rupture process under creep
conditions, TVZ Akad Nauk S.S.R. Otd Tech. Nauk 8
(1958).
[2] J. Lemaitre, Evaluation of dissipation and damage in
metals, submitted to dynamic loading, Proc. I.C.M. 1
Kyoto Japan (1971).
[3] F. Leckie and D. Hayhurst, Creep rupture of structures,
Proc. R. Soc. London A240 (1974) 323.
[4] J. Hult, Creep in continua and structures, Topics in Applied Continuum Mechanics (Springer Verlag, Vienna,
1974).
[5] J.L. Chaboche, Une loi diff6rentielle d'endommagement
de fatigue avec cumulation non lin6aire, Rev. Franqaise de
M6canique 50-51 (1974).
[6] J. Lemaitre and J.L. Chaboche, A nonlinear model of
creep-fatigue damage cumulation and interaction, Proc.
IUTAM Symp. of Mechanics of visco-elastic media and
bodies (Springer-Verlag, Gothenburg, 1974).
[7] J. Lemaitre and J. Dufailly, Mod61isation et identification
de l'endommagement plastique des m~tanx, 36me Congr6s
Franc,sais de M6canique Grenoble (1977).
[8] J.L. Chaboche, Le concept de contrainte effective appliqu6
l'61asticit6 et h la viscoplasticit6 en pr6sence d'un endommagement anisotrope. Colloque EUROMECH 115,
Grenoble Edition du CNRS 1982 (1979).
[9] S. Murakami and N. Ohno, A continuum theory of creep
and creep damage, 3rd IUTAM Symp. on Creep in Structures, Leicester (1980).
[10] D. Krajcinovic and G.U. Fonseka, The continuous damage
theory of brittle materials, Trans. ASME J. Appl. Mechanics 48 (1981) 809-824.
[11] E. Krempl, On the identification problem in materials
deformation modelling - EUROMECH on Damage
Mechanics - Cachan (France) (1981).
245
[12] P. Suquet, Piasticit6 et homog6n6isation, Th6se d'Etat,
Universit6 Paris 6 (1982).
[13] J. Lemaitre and J.L. Chaboche, Aspect ph6nom6nologie de
la rupture par endommagement, J. M6canique Appl. 2
(1978) 317-365.
[14] J. Lemaitre and D. Baptiste, On damage criteria. Workshop
NSF on Mechanics of damage and fracture, Atlanta USA
(1982).
[15] J. Lemaitre, A continuous damage mechanics model for
ductile fracture, Trans ASME J. Eng. Mater. Technol.
1984.
[16] A. Benallal, Prevision de l'amorgage en fatigue ~ chaud,
Universit6 Paris 6, Th~se de Docteur IngOaieur (1981).
[17] R. Billardon, D. Marquis, A. Benallal and D. Grolade,
Prediction of crack initiation by damage theory, Euromeeh
on Damage Mechanics Cachan, France (1981).
[18] F. Mac Clintock, A criterion for ductile fracture by the
growth of holes, J. Appl. Mechanics (June 1968).
[19] J. Rice and D. Tracey, On ductile enlargment of voids in
triaxial stress fields, J. Mechanics Phys. Sol. 17 (1969).
[20] F. Mudry, Etude de la rupture ductile et de la rupture par
clivage d'aciers faiblement alli6s, Th6se, Ecole des Mines,
Universit6 de Technologic de Compiegne, (1982).
[21] J. Lemaitre, A three dimensional ductile damage model
applied to deep drawing forming limits, ICM 4 Stockholm
(1983).
[22] M. Grumbach and G. Sanz, Influence de quelques
param6tres sur les courbes limites d'emboutissage, Rev.
M6tallurgie (April 1972).
[23] J.P. Cordebois, Crit6res d'instabilit6 plastique et endommagement ductile en grandes d6formations (Application
l'emboutissage), Th6se de Docteur 6s Sciences, Universit6
Paris 6 (1983).
[24] B. Jacquelin, F. Hourlier and A. Pineau, Crack initiation
under low cycle multiaxial fatigue in 316 L stainless steel,
Pressure vessels and piping Conf. Orlando (USA) Vol. 59,
ASME (1982).
[25] G. Cailletaud, Microamorcsage, micropropagation, La Recherche A6rospatiale 6 (1982).
[26] S. Taira, Lifetime of structure subjected to varying load
and Temperature, Creep in structures, ed. N.J. Hoff
(Academic Press, New York, 1962).
[27] S.S. Manson, Behavior of materials under conditions of
thermal stress (NACA Tec. note 2933) (1954).
[28] L.F. Coffin, A study of the effects of cyclic thermal stress
in a ductile metal, Trans. ASME 76.931 (1954).
[29] R. BiUardon and J. Lemaitre, Numerical prediction of
crack growth under mixed-mode loading by continuum
damage theory, Congr~ A.U.M. LYON (1983).
[30] H. Policella and M. Chaudonneret, Installation d'essais de
traction biaxiale sous chargement monotone ou cyclique,
Werkstoff Techniche Verlagsgesellshaft M.B.H Karslmhe
(1981).