FOUNDATION OF A MECHANO-CHEMICAL FATIGUE THEORY (MCFT)
by
E.
Altus
February, 1989
t.,
1
.UG. 1989
.
UTIAS Report No .330
.
CNISSN
0082-5255
.
FOUNDATION OF A MECHANO-CHEMICAL FATIGUE THEORY (MCFT)
by
E. Altus*
Submitted September 1988
©Institute for Aerospace Studies 1989
*On sabbatical leave from the Faculty of Mechanical Engineering,
Technion, Israel Institute of Technology, Haifa, Israel.
.
-Acknowledgement
This work was supported by the Ontari
0
Centre for Material Research,
Grant No. TP2-325, and acknowledged with tnanks. The author wishes to thank
Dr. R. C. Tennyson, Director of UTIAS, and the Technion, Israel Institute of
Technology, for giving the opportun1ty to spend a fru1tful sabbatical year
at the Institute •
..
Abstract
A theory which identifies the chemica1
reaction between a broken
molecule and its neighbour as the main micro-sca1e source of fatigue fai1ure
is deve10ped.
Sy app1ying statistica1 1aws, the macro-behaviour of the
material is revea1ed.
Three types of macro damage accumulation functions
emerge,
as a direct resu1t
of different va1ues of the micro-scale
parameters. The reference type shows secondary and tertiary stages, the
second type inc1udes an additiona1 primary stage, and the third type,
1eading to a material with a specific endurance limit, has on1y a primary
stage.
A well-known empirical power-law re1ationship between the stress
level and the number of cyc1es to fallure is obtained analytically by the
theory
for the
reference type of damage,
when
a Wei bull
strengt h
distribution function for the molecules is used.
Three microscopic material parameters control the macro-behaviour:
(1)
a statistica1 shape parameter, describing the strength distribution of the
molecular chains, (2) a stress concentration factor, which describes the
amount of weakening a broken chain can cause to its neighbour, and (3) a
probability function for achemical reaction to occur.
The micro-scale roots of the theory enables the use of physical
internal variables for the damage evolution equations.
Thus, a clear
interpretation of the macro-fatigue response, inc1uding the existence of an
endurance limit is achieved.
Experimental
·
correlations with
av~ilable
fatigue data for different materials, incllJding metals, plastics and
comopsites, show the general va1idity of the theory, not only for polymeric
interfaces.
This is due to the mechano-chemica1 mechanism, on which the
theory is based.
The mathematical simi1arity of the basic equations to the ones found in
chaotic theories is demonstrated and discussed briefly •
Contents
Abstract
1.
INTROOUCTION ANO MOTIVATION
2.
STATISTICAl STRENGTH PARAMETERS
3.
REVERSE BREAKAGE BY A MECHANICAl PROCESS
4.
MECHANO-CHEMtCAl FATIGUE THEORY (MCFT)
5.
FATIGUE MOOElS
(a) Materials With No Fatigue Resistance (Rf
=
0)
(b) Materials with Fatigue Resistance Through k
(c) Materials with Fatigue
~esistance
Through 9
6.
FATIGUE AHO CHAOS
7.
DISCUSSION
8.
CONClUSIONS
REFERENCES
FIGURES
APPENDIX
1
2
6
6
14
14
18
19
20
21
23
24
1.
INTROOUCTION ANO MOTIVATION
The fatigue and damage mechanics of structural materials can be
approached from two distinct points of view.
In the macro (or
phenomeno
1
ogi ca
1)
approach, the materi al i s taken as a cont i nuum, and the
fatigue or damage process is characterized by functions of a tensorial
nature, to be fitted to the general continuum theories.
A good example is
in Composite Materials (Taljera 1985), where the real non-homogeneous,
cracked structure is modell ed as continuous, with damage properties taken
experimentally.
J!"
the second approach, the analysis admits some internal
structure, which gives an insight into the real physical process but makes
the engineering application less immediate. Many examples can be taken from
molecular models as in polymers (Flory 1975), where the engineering
application comes only af ter intermediate additional studies are performed.
An outstanding case, where a direct relationship between the micro model and
the engineering use is beautifully demonstrated in Zhurkov (1965), where the
macro behaviour, so many orders of magnitude away from the rnolecular level,
is predicted.
In advanced structural materials, such as ceramics, or fibre
composite
materials,
the
importance
of
the micro
properties grows
drastically, since the damage region which may eau se total failure is
extremely smal
1 ,
as in the cases of delamination and interface debonding. A
recent review on the micro-macro approaches as related to mechanical failure
is found in Lemaitre (1986).
Recent technological advances in microscopie measurements allow a
better understanding of these sensitive interphase regions (Hatsuo 1984),
and opens new possibilities to model the strength and fatigue properties of
the material • Most macro theories and models ofdeformation are categorized
as elastic, viscoelastic, viscoplastic, etc.
While the elastic behaviour
can describe the response of the material to a monotonie, quasi statie
loading up to failure, the other theories are rate dependent, based on the
observation that the mechanisms of damage are time dependent.
The plastic
behaviour, although designed for non-monotonie loading, assumes an elastic
behaviour for the unloading stage, and therefore cannot predict fatigue
progress •
The related micro models in polymers are based on the spring-like
behaviour of the atoms and molecules in the elastic case.
The plastic
deformation is usually takèn as some kind of unraveling of the chains
(Kardomafeas
&
Yannas 1985), and the rate dependent model s are based on the
kinetic (and statistical) nature of molecular bonds (Zhurkov 1965, Tenmonia
et al 1985, OeVries et al 1971, Zhurkov
&
Korsukov 1974).
In contrast to the above models, the
-
fatigue phenomena cannot fit into
any of the categories.
To a large extent it is observed as a rate
independent process. However, the damage progression in each cycle suggests
that the unloading process cannot be elastic.
In the macro approach, the crack growth rate under fatigue 10ading is
found to be rel ated to the energy rel ease rates (Ma11 et al
1987),
known as
"Paris 1aw".
It is 1imited to very large cracks and gives no c1ue as to the
physica1 micro mechanisms.
Such a micro mechanism is proposed and discussed
here.
Finally, the fatigue process is usua11y related to a comparative1y
sma11 plastic deformation part and a dominant "bond breakage" part.
For
simp1icity, on1y the latter mechanism is examined here.
Sliding processes
causing non-recoverable deformations are neg1ected.
2.0 STATISTICAl STRENGTH PARAMETERS
In this section, the inf1uence of the statistica1 strength and
non-homogeneity parameters of the chain molecules (macromolecules) on the
overall macro behaviour is demonstrated.
A one-dimensiona1 model
is
examined for convenience.
Consider a po1ymeric bond (interface) between two materials as shown in
Fig.
1.
In rea1ity, the macromolecules are statistically oriented, each
may have different properties.
In the model, all chains are represented by
straight 1inear springs with a statistica1 density of strength distribution
f(f1) or
f'(a).
f1 and
a
are the re1ative displacement (strains) and
unidirectiona1 stresses of the chain, respective1y.
Assuming displacement
controlled process (parts land II are rigid), it is simp1er to use f
instead of f'.
First, consider the case where the strength of the chains is the on1y
non-uni form parameter of the materi al.
The probabi 1 ity of breakage of a
chain 10aded from zero to f1 wi11 be
6
F (f1)
=
J
f (f1' ) df1
I(l)
o
so that the re1ative number of unbroken chains is
( 2)
and the outer force wi11 be governed by
~(f1):
where N, E, A are the total number of chains, their modulus and area,
respecti vely.
The ave rage outer stress is simply
(4)
Equat i on
(4)
expresses the macro stress/strai n rel ati on of the interface.
lt is inherently nonlinear, created solely by the statistical distribution
F(6) •
Under monotonie loading conditions, the tangent macro modulus is:
Ë{+)
t
=
ä
(6)
=
E[l -
F(6) - 6f(6)]
,6
(5 )
which is monotonically decreasing. Here (-) denotes macro parameters, (+)
is for loading, ( t) is for the tangent property and (
6)
is pa..rttal
differentiation. Therefore, the maximum macro stress will ocèur when Ef+)iS
zero, or when
(6)
and
(7)
Under unloading conditions, the macro modulus Ë(-)wi1l be governed by the
number of molecules remained unbroken:
(a)
where 6J1l is the max i mum 6 experi enced through the hi story of 1 oadi ng. The
RHS of
~tS)
shows that the fully unloaded specimen will always return to the
initial zero strain (no residual strains).
Next
we
consider the case where the initial length of the chains has a
statistical distribution too. Therefore, even at a zero outer loading, each
chain has already been elongated out of its zero load position. This causes
internal stresses which may come as a result of cooling, phase
tranfonnation, or any other inhomogeneity.
Since there is no priority to
tension or compression, a symmetrie internal strain distribution density
funct i on h( fI) is assumed.
Note that the symmetry provi des zero net force
wh en no outer load is present.
h(fI) can be considered as a characteristic
mater i al function reflecting its heterogeneity.
Buckling phenomena during
compressive loading is neglected.
Two strain parameters have to be addressed here:
the true strain fI in
eaeh unbroken chain, and the external strain fle.
For each fle' the true
strain distribution will be h(fI-6e ), meaning that in the interval [fI, Mdfl]
there are h(fI-6e)d6 ehains (relative number). Their probability to survive
is
~(6),
from
Eq. (2).
Then, the relative number of surviving chains from 6
to 6+dfl at fle is:
S(6)dfl
=
h(6 - fleH
1 -
F(6)]dfl,
(9)
where S(fI) is the survival density.
The total number of surviving ehains
is
Cl>
( 10)
-Cl>
The outer fa ree wi
11
be:
Cl>
(11)
-Cl>
sa that the average outer stress is:
Cl>
( 12)
_Cl>
It is seen that the initial 6e , for which no external force is applied, is
not zero, but the solution of
(13)
If
where 6 is the Oirac
·
function, then (12) converges to (4) as in the first
case.
The compressive strength of the chains is usually much higher than for
tension. lf no compression failure is allowed,
F(6 (
·
0)
=
0,
( 15)
and Eq. (12) is separated to
~
0
a(6e )
=
E[f 6h(6 - 6e)r1 - F(6)]d6
+
f 6h(6 - 6e)d6].
(16)
o
-CDThe tangent modulus under monotonie 10ading is
( 17)
and the unloading roodulus is derived from the relative number of surviving
chains at each stage:
(18)
Since, during unloading, new chains are not breaking and the process is
linear, the recovered strain will be
~ ~
E
(-)
=
a(6e)/Ë(-)
=
f
6h(6 - 6e)[1 - F(6)]d6 /
f
h(6 - 6e)[1 - F(6)]d6
-~ ~
(19)
Thus, total unloading will not bring the displacements to the same original
position. The residual strain wil1 be:
Figure 2 shows a schematic globa1 stress-strain behaviour for the two
cases discussed.
It is seen that the strength statistica1 distribution
causes a non1inear macro response, with a monotonie stiffness reduction.
Thenna1 stress distribution or other non-homogeneous effects inc1uding a
distribution of modulus E(t.) wi11 cause a residua1 strain upon un1oading,
gi vi ng a simil ar stress-strai n behavi our as in Fi g. 2b.
However, as seen
from Fi g. 2, the fat i gue process cannot be produced si nee the un1 oadi ng
stage is always elastic.
For a non-e1astic process to occur during
unloading, the geometrica1 re1ationship and forces between neighbouring
chains must be taken into account.
3.
REVERSE BREAKAGE BY A MECHANICAL PROCESS
--Once it is realized that the triggering mechanism of fatigue lies in
the non-e1astic part of the un10ading process, it is fruitful to examine a
simple structure which may produce reverse breakage and exp10re its
limitations. Consider an elastic configuration as shown in Fig. 3, which is
a typical unit of a chain, having a skeleton A-B-C- ••• , and secondary bonds
designated here by four different springs.
Two springs (2, 3) will be
stretched when t. and F are increased while the other two wil1 be compressed.
Thi s property of the structure may lead to a breakage of springs 1 or 4
while t.F is negative (i.e., reverse loading).
Examini~g
the above structure
for different spring stiffnesses and initia1 configurations, a general force
displacement behaviour can be shown as in Fig. 3b.
The first force drop
(a-b) is related to the breakage of springs 2 or 3, while (c-d) is due to
the breakage of springs 1 or 4. However, in all cases checked, It.al
<
lt.cl,
since springs 1 or 4 are in compression when the first breakage occurs.
This phenomenon seems to characterize all such structures, so that a
tension-tension (and even a great part of tension-compression) fatigue
cannot be achieved by this model.
A1though not a proof, this example serves to support
·
the argument, that
a purely mechanica1 mechanism, based on elastic (and even non1inear)
springs, with a statistica1 strength distribution or residua1 stresses,
cannot predict fatigue behaviour.
4.
MECHANO-CHEMICAL FATIGUE THEORY
(MC~Tl
Let a unidirectional set of chains of molecules represent the polymeric
interface material as shown in Fig. 4a.
An externa1 load is presented here
through a globa1 displacement, equal to all chains (Fig. 4b). Due to the
statistica1 strength distribution, the weakest chain breaks, springs back
its two halves towards bath sides, realigning with the unbroken neighbours
(Fig. 4c). A1though the strength of the principal bonds in polymers is very
high compared to the average macro value, the statistica1 stress
distribution is such that local stresses as high as 100 times the average
stress, exi st (DeVri es et al 1971, Zhurkov
&
Korsukov 1974).
They are the
main source of fatigue nuc1eation.
The alignment mayalso involve relative
s 1 i di n 9 of the cha ins.
The two edges of the broken cha in (us ua 11 y carbon
atoms)
are very active compared
to their surrounding,
and
unstable.
Therefore, they tend to react with other
.
atoms in their vicinity.
Upon
unloading (Fig. 4d), the two edges are further compressed towards some of
the particles of the unbroken chain, raising the possibility of a chemical
react i on even further, whi ch decreases the stabil ity of the nei ghbouri ng
chains (Fig. 4d).
This
wil
1 result in a lower strength value when the
mater i al is re10aded again.
For convenience, this effect will be regarded
as a chemical stress concentration factor.
The end result is a successive
breakage (Figs. 4e, f).
This process continues in each cyc1e from new
sources of open edges.
It is asslJmed that one breakage in a chain reduces
its 10ading capacity to zero.
The above mechanism is the basis for the
MCFT.
Let the strength characteristic distribution of the chains be given by
a density
f{t.),
wnefe t. is the strain as in Eq.
(I). When the material is
10aded first to 6(+ , the re1ative number of broken chains will be
t.{+)
BI
=
J
f{t.)d6
=
F{6+).
o
(21)
and the number survived is 1-F(t.+).
Bn denotes the additiona1 relative
number of chains broken in cyc1e n.
Each cycle extends between two adjacent
peaks of t., meaning that the first cyc1e ends at the first peak (6t), etc.
As wi11 be seen, this definition is compatible with the order of the fatigue
damage progression. The compression state at the vicinity of the free edges
(Fig. 4c) will be sensitive to the breaking load or displacement.
The
larger the displacement, the higher the
compres~iQn
and(the probabi1ity of a
chemical reaction.
During unloading (from
t.~+)
to 6 -»), the compression
wil 1 increase
fur~h1r,
so
t~at
the total probabil ity of reaction g, is a
function of both 6 + and 6(- •
In the second cycle some of the "reacted" unbroken chains behave like
having a stress concentration factor k, so that their probability of
breakage changes from F(t.) to F(kt.).( The
nu~ber
of additional broken fibres
in the second loading process from 6 -) to 6
t
+} is therefore:
B
2
=
[g(t.(+), 6(-»e F(t.(+»]e[F(kt.(+) - F(6(+»]
1 - F{t.+)
where 9 is the probabi1ity function of chemica1 reaction. The first part of
Eq.
(22)
denotes the number of participant chains and the second is their
probabi1ity of breakage.
Equation
(22)
can a1so be written as
(23)
where , crn be regarded as a fatigue rate intensity function.
If the strain
1imits
6
+),
6t-)
are kept constant at each cyc1e, the candidate chains for
failure are taken on1y from the new1y generated sites near the previous
breakage poi nts. Iherefore
and for the n-th cyc1e
The tota1 re1ative number of broken chains af ter n cycles is
n-1
On
=
B1
l •
i ,
i=O
(24 )
( 25)
(26)
where On is the Oamage function. Equation
(26)
can be used to find
B1
~nd
•
by performing few fatigue experiments to failure at different
6'+)
and
6
l - J•
The simp1e geometric nature of the accumulative damage allows an
immediate ca1cu1ation of the condition for the fatigue limit state (the
condition for which the fatigue process will be stopped, and no further
damage wi11 occur):
T
=
1
i
m
On
=
B l' (
1-.)
<
1.
(27)
n+cD
The above ana1ysis assumes a displacement controlled fatigue cycle.
It
predicts new chain breakage near the sites of the first cycle breakage only.
However, more of ten the fat i gue process i s stress controll ed.
Th is wi 11
cause new sites of broken chains at
ever~
cyc1e, in addition to the old
sites, since at each 10ading, a higher 6 l
+)
wouldhave tri be achieved, to
keep the outer stress range
(a~+), a~-))
constant.
For the stress
controlled fatigue, which is the condition assumed from now on, the maximum
outer stress during the first loading is (Eq. 4),
( 28)
The above stress level
(o~+))
will remain for all consecutive cycles.
Sim11ar to Eq. (22), during the second loading, the number of additional
broken chains at
t~e
initial sites for the second cycle is
(29)
Not i ce that the 1 eft part i s rel ated to cycl e 1.
Here 61-) and
6~
+)
are st 111 unknowns.
The fi rst i s cal cul ated from the lower 1 imit of the
outer stress:
( 30)
No
.
fibers are broken dring unloading.
In addition to the breakage at the
0ld) sites, new sites( Qf broken chains are created since the displacement
6~+
is higher than 6t+'. Their relative number is,
(31)
B2a expresses the contribution of the chemical reaction to the damage
progression, while B2b is due to the statistical strength distribution of
the unbroken chains.
One more rel.at.ionship is found by equating the
external load of the s
.
econd cycle to
0ci+):
Substituting Eqs. (29),
(30), (31)
in (32),
6~+)
can be determined through a
fatigue, the parameters of each cycle have to be found separately and
successively.
The analysis of the third cycle and the following ones is more involved
and needs additional structuring of the failure process. A group of two or
more adjacent broken chains can be called a crack. Only the chains at the
edge of the crack are active and can cause further breakage at each cycle by
the MCFT.
The souree of new chain breakage in the third cycle can come not
merely from B2b and B2a' but also from those sites where the chemical
reaction which took place in the first cycle, but no breakage occurred in
the secon,d )CYCl
er...u.
Th; sis due to the
1
arger di spl acements in the thi rd
cycle (63+
>
6f-').
!f,
in the second cycle, B2a new edge sites were
created by breakage (Eq. 29), then the number of unbroken edges whi ch went
through chemical reaction is,
(33)
61-)
where gl
=
g6t+) and 62
=
6~+)
for convenience. From the above amount of
candidates, the number of breakages in the third cycle will be:
(34)
This type of additional breakage continues in every cycle, from any souree
where chemical reaction takes place.
Since each new breakage creates
several possible responses, it is fruitful to characterize the different
types of edges and treat their propagation systematically.
Consider the state at
6~+),
where the material is loaded to
a~+)
the
n-th time. Three types of edge broken chains exist: The first type with no
chemical reaction, meaning that it was created prior to the n-th cycle and
did not react chemically with the nearby unbroken chain.
The second type
has had chemical reaction in previous cycles but caused no breakage.
(The
neighbouring chain is strong enough to survive the stress concentration.)
The third type has been created during the n-th cycle, either at new sites
or by expansion of an "old" crack. Notice also that the size of the crack
does not enter the damage process.
The additional chain breakage from the n-th to the (n+1) cycle is
demonstrated sChematically in Fig. 5.
N~n)
(i
=
1, 2,
3)
are the number of
edges related to the th ree types at the n-th cycle. Each edge can create a
new type according to the statistica1 nature of the MCFT.
a,~,
y
show the
proportions by which each type is separated during the (n+l) cyc1e. In the
case of no chemical reaction, it is assumed that
if
an edge did not react
chemically in previous cycles, it wil1 continue to be passive.
The
motivation for this
assumrt~on
is that the chemical probability function 9
depends
Ol)
the strain II + at the time of breakage (Fig. 4c) and on the
minimum 4 l -) (Fig. 4d). Since the ana1ysis is confined to the case where
lll-)
>
ll~-) when m
>
n, then if the chemical reaction did not occur during
tWe first cycle af ter breakage, it wil1 not occur at all. This yields a
=
1
for all cyc1 es.
In the second type (N 2), part of the
·
se edges wil1 break
(~lN~n)),
creating near-breaking sites in the (n+l) cyc1e, while the others will
rema in unbroken
(~2N~n)).
In the third type (N 3), which consists of the new1y broken edges (or
cracks of a single broken chain, created in the previous cyc1e
n), all
three types are expected to deve10p af ter an additiona1 cyc1e
(YiN~n)J.
Finally, the remaining source of chain breakage for the (n+l) cycle
inc1udes the chains which were not broken and are not
rh~mically
reacted or
re1ated to any edge. Tbey wi11 produce two types: ö
1
N
4
n) new cracks of one
chai n wi dth, and
ö2N~n)
unbroken chains. These new cracks are added to the
potential number of edges for the (n+2) chemica1 reaction.
The quantities of the above are:
For the lino chemically reacted" edges:
For the chemica11y reacted but unbroken:
For the new edges:
=
F{klln+l) - F{kll n)
1 -
F{klln)
,,(n) -_
1
"
~2-
~l(35)
(36)
(37)
(38)
(39)
y~n)
=
1 - gn
(40)
Notice that the sum of all elements of a,
~,
y, or
6
must be
1.
From the
above, the number of new breaks expanding from old cracks is
--
(a)
(n)
(n)
(n)
(n)
8n+1 =
13
1
N2 + Y1
N3 •
(41)
Thus,
and
N(n)
=
1 - 0
4
n
(a)
=
1 -
On - 8n
+
1
(n)
=
1 - On - N2
6(n) _ F(6n
+
1) - F(6n)
1
-
1 - F
(6
n
)
(n+1 )
N2
6~n)
=
1 _
6~n)
•
The number of new cracks created in the (n+1) cycle is therefore:
(42a)
( 42b)
( 43)
(44)
Sjnce)new
rr~çks
are behaving like edge type
3,
their number is added to
N3
n
+ l . N
4
n ) is an intermediate value showing the potential number of
chains which can create new cracks.
From Fig.
5
and the above relationships, the number of sites for each
type
in
the (n+1) cycle is deduced:
(45a)
(n+1)
(n)
(n)
(n)
(n)
N2
=
~2
N2
+
Y2
N3
'
(45b)
(n+1)
(a)
(b)
N3
=
8n+1 + 8n+1,
(45e)
The damage progression can now be written as:
( 46)
where 0n+1 are the relative number of chains broken through the whole
fatigue life till the (n+1) cycle. The equilibrium equation is then
( 47)
For further ealculations,
;t
is convenient to introduee the stress ratio
between the outer stress and the re al chain stress:
(48)
whieh can be considered as the survival ratio, or the residual stiffness
ratio. It is also useful to introduee the fatigue resistanee parameter Rf;
(49)
whieh is the ratio between the maximum stress for whieh no failure occurs at
any number of cycles (called the enduranee limit), and the statie strength.
For any mater; al, 0 ( Rf
(1.
A non-fati gue resi stant or
l
fully resi stant
So far, no specific distribution function or any material parameters
were inserted in the analysis. In the next section, such fatigue models are
studied, especially the extreme cases, in order to understand the capability
and limits of the MCFT.
For simplicity, the loading profile is restricted
to the case where the outer stresses range between zero and a constant
maximum val ue in tension
(a
o ).
.
5.
FATIGUE MODELS
(a) Materials Wit" No Fatigue Resistance (Rf
=
0)
Consider first the simplest case, where in each cycle, from zero to a
constant stress, the unbroken chains near the edge of every crack breaks.
It means that the probabil i ty of chemi cal reacti on is 100% and the stress
concentration caused by it is infinite, i.e.,
9n
=
1;
k
+
m.(50)
Then, From Eqs. (35)-(42)
~1
<
1;
~2
<
1
(Sla)
and
Y1
=
1;
Y2
=
Y3
=
O.
(Slb)
It reflects the most extreme case of the HeFT,
where the material
deteriorates at the fastest rate.
Notice that at the other extreme, if
Ic =
1.
in (50), there will be no fat i gue damage accumul at i on at any stress
level, since no edges 'l«)uld break af ter the first cycle.
This behaviour
corresponds to a fully fat i gue resi stant behavi our and can be accompl i shed
al ternatively by setting 9
=
0 or
~
=
m.Although
~1
and
~2
are not
defined explicitly, they are "t,ounded between 0 and 1.
As
wi1l be seen, the
actual number has no effect on the results in this case.
For the first
three cycles, the results are:
( 52a)
(52e)
ete. Thus the n-th eyele will yield
n
= n
[ m i
]
. 1 2m. 1 - m· 2
1=
1-
1-(53a)
where
( 53b,
el
Equations (53) represent n implieit equations (using Eq. 48) whieh are
solved sueeessively, starting with (S2a) for
t.1.
The Weibull distribution
funetion is used here for the strength:
(54 )
where
a
and
~
are material parameters.
The distribution density is
( 55)
The statie strength (outer stress and displaeement at failure) under
monotonie loading ean be found analytieally for the above distribution,
using Eqs. (6), (7),
(54), (55):
t.
s is a eonvenient referenee statie failure parameter. From (4), the statie
strength is:
Substituting (56) in (54), the normalized probability function is written as
a function of
~
alone:
=
1 - exp[ - } (
is)
~]
.
(58)
a is thus identified with the statie strength while
~
is directly related to
the shape of the strength distribution function.
Equation
demonstration.
!::.
=
!::.s·
(Se)
is plotted for different
~
values in Fig. 6 for
Not i ce that the poi nt of maximum f
'
does not coi nc i de wi th
The solution of (53), using (58), is solved numerically.
In Fig. 7,
the effect of
~
on the damage progression to failure is shown, normalized to
the number of cycles to failure, Nf • It is seen that the damage rate dO/dN
is increasing monotonically for any
~,
which is a direct result of allowing
an additional edge chain to break af ter each cycle (Eq. 50).
Larger
~
values cause a smaller damage accumulation during most of the fatigue life
until the end stage, where the deterioration is more catastrophic.
~
=
CDcorresponds to the case when all chains are stronger than the fatigue
stress, so that no damage is accumulated.
Two distinct regions can be
identified.
In the first, which includes most of the cycles
(",60~),
the
stiffness reduction is approximately linear with the number of cycles, but
wi th rel atively low damage accumul ati on.
~t
the
·
second stage, the
deterioration becomes
rapid until total failure.
Oue to the similar
characteri st i cs found in creep, the above are regarded as "secondary" and
"terti ary" fatigue stages.
It can be shown
analytically (see Appendix) that the approximate
solution of Eq. (53) yields a power law relationship between the ratio of
the st res s
1
eve
1
and the number of cyc 1 es needed to cause the same damage,
i
.e. ,
(59)
For mN
=
0
we
get the fatigue life relationship
From ((59) and (60) we get:
NI
N
Nf = Nf;
(61)
Meaning th at the same amount of damage is áccumulated for the same relative
number of cycles at
~
stress level. This yields a unique damage curve for
all stress levels, once
~
is given. Therefore, the stress level a/as
=
0.28
written in Fig. 7 can be omitted. This property leads directly to a "linear
damage law" , as exhibited in Minerls rule.
The subject will be discussed
extensively in a subsequent study.
Furthermore, it can be shown that the expl icit stress-fatigue 1 ife
relationship (the S-N curve) normalized to the static strength value is:
ao _
- -
(62)
The approximate value holds for (Nf) 10,
~
)
5), which covers all practical
cases.
Figure 8 is a regular S-N curve based on Eq. (62) for different
~.
It is seen that a drastic difference in fatigue life can be caused by
~
for
materials which have the same static strength •
.
Moreover, some parts of the
graphs exhibit an almost linear behaviour, especially for high
~,
which is a
known experimental result for many materials.
0
1t
may explain also the large
scatter in fatigue test results, where
~
may be different from one specimen
to the other, while the statie strength and initial modulus values have
almost no scatter.
It shows that a good quality control during
manufacturing processes can improve the fatigue life by orders of magnitude,
without having to increase significantly the statie strength values.
lnterestingly, Eq. (62) yields for
ao
=
as'
(63)
which for
~
large enough is approximated by
(64)
Although (64) has no practical importance, its value being between
1
and 2
is physically acceptable, since for any fatigue stress value which is lower
(recall that every cycle ends
~t
the maximum stress, therefore the first
loading and fully unloading is more than 1 cycle).
A material represented
by
Eq. (62)
has no fatigue resistance, since it does not have an endurance
limit. This behaviour is typical of some aluminum alloys.
Notice that the
whole macro fatigue characteristic for the model is dependent on one micro
scale parameter,
~.
A
further insight into the damage progression can be seen on the chain
strength distribution space, as shown in Fig. 9.
The first cycle creates
damage similar to a statie loading (area
Al
in the figure). However, all
Al
sites, which represent the number of chains brok en , will expand in the
second cycle (k
+
èo).
Since the near edge chains ar.e homogeneous in their
strength distribution, their breakage in the second cycle is represented by
the area
B2'
Then
82
=
Al
and additional new cracks
A2
are created during
the second cycle.
In the thi rd cycl e, all ol d crack s
(Al
+
A2)
will expand,
50
that
B3
=
Al
+
A2•
Continuing this process,
we
get for the n-th cycle
Therefore
n
n-l
Bn
=
~
Ai'
1
~
Bi
=
Al (n-1)
+
A2(n-2)
+ •••
An-l
and the total relative number of broken chains can be expressed as
n
n
n
On
=
~
Ai
+
~
Bi
=
~
(n
+
1 -
1)
Ai •
i =1
i =2
i =3
(b) Materials with Fatigue Resistance Through k
In the case where
9
=
1;
k is finite,
(65)
(66)
( 67)
(65)
not all cracks expand af ter each cycl e si nee some edge chai ns may be st rong
enough to sustain the finite stress concentration. A finite k can be termed
as "mec hanical resistance" to fatigue.
No
analytical solution could be
found here, and the sti ffness reduction as a function of the number of
cycles is solved numerically and shown in Fig. 10 for different kiS and the
same
~.Three types of fatigue responses are identified.
For large enough k
values (k
>
20), the behaviour is identical to the no-resistance behaviour,
showi ng secondary and terti ary stages only
.
.
For 8.85
<
I<
<
20, all three
stages of fatigue failure are present. Experimental evidence for such macro
behaviour can be seen even for non-homogeneous materials (Oaniel
&
Charewicz
1986).
Notice the strong nonlinear effect of k on the fatigue life in this
region.
In the third type, for small enough kiS (here, k
<
8.85), the
damage accumulation stops completely, showing primary and secondary stages
onl y.
The absence of a tert i ary stage corresponds to the exi stence of an
endurance limit.
The above three response types can be clearly seen in Fig. 11, where
the S-N curves are plotted for different kvalues. The major effect of k is
changing the endurance limit stress value.
The above fatigue response is more realistic and reflects the behaviour
of lOOSt materials having a "knee" point on the $-N curve.
Notice again the
"almost" linear behaviour in region I, also exhibited by many materials
(Hertzberg
&
Manson 1980, Weiss 1963).
(c) Materials with Fatigue Resistance through
9
As defined,
9
reflects the probability that the active edge of a broken
chain will have a chemical reaction with a member of a neighbouring chain,
thus causing a "stress concentration" (k).
The specific value of 9 will
depend on
the type of chemical
bonds existing in each material
(Van der Waals, etc.), and on the configuration of the molecules, i.e., on
the di stances between the broken edge and lts nei ghbouri ng atoms.
Assumi ng
that the two properties are independent, we can write
(66a)
where ( )c and ( )g are for chemical and geometry related probabi1ities,
respectively.
Intuitively, gc is a more fundamental function of each
material , independent of loading and other
'
external conditions.
However,
the smallest
di~tance
from a broken edge to the nearest unbroken neighbour
depends on 6(+
and 6(-), i.e., on
~he
stress levels at the breaking stage
and the amount of unloading af ter breakage so that (66a) is interpreted as:
Finding the explicit functions for 9 needs further study. Hevertheless,
we
can explore the material behaviour for the simplest case:
gc
=
1;
gg
=
const.
<
1
(67)
meani ng that a constant percentage of the broken edges
wi
11
create a new
bond vii th one of the unb roken cha ins.
The i nfl uence of
9
<
1
on the S-H
response is shown in Fig.
12
for the same
~
=
3
as in Fig.
11,
and k
+
GO.It is seen that
9
<
1
is also a source for an endurance limit, as k, but
tends to shift thë
-
whole S-N curve, for all stress levels. Notice also that
region
11
is much larger than in Fig.
11,
while region
1
diminishes.
Figures
11
and
12
show that although the influence of constant k and
9
is similar, the way they affect the damage progression is different.
To
examine this difference further, two cases, leading to the same number of
cycles to failure under the same fatigue stress level of loading, are
compared in Fig.
13.
This is done by taking an intersection point between
two lines in Figs.
11
and
12.
The secondary fatigue stage for
9
<
1
is more
dominant than for k
<
GO.Hotice also that the higher stiffness values in
the primary and secondary stages do not guarantee a longer fat i gue
1
i fee
6.
FATIGUE AHO CHAOS
The form of the damage evolution equations, especially for the basic
case (Eqs.
52, 53)
deserves additional attention from a more general
perspective. The general shape of
(53),
using
(48),
is
F(O(n))
=
x(O(n))-O(n)
(68)
F is the basic strength characteristic function of the material ,
0
is the
damage function and
X
is a number based on all previous cycles, i.e., a
functional of the damage through its evolution. This structure shows a kind
of lIself similarity" of
0
towards F.
A functional self-slmilarity can also
be found in Chaotic phenomena.
A
known example
;s the lIuniversal equation"
(J.
M. Feigenbaum, 1979):
g(x)
=
-ag(
g(x/a)).
(69)
Although the self-similarity in
(68)
is more complex, its geometrical
interpretation is direct.
Figure 14 shows three damage curves for a basic
case for three stress levels. From Eqs. 60,61, any two of the curves are
contractions (expansions) of the third by an appropriate factor, found from
(60).
Moreover, for any segment of one of the curves (say, AB in Fig. 14)
there is an identical curve in each of the other two (AIBI, A"B") •
All
three segments differ by a linear transformation only (Le., rotation and
translation).
In other words, the original curve, af ter two successive
transformations, stretching and linear, remains identical.
If
(70)
represents the orfg
-
i nal curve, then
(71 )
is any other curve, dependi ng on the expans i on factor c.
tiow, a
1
i near
transformation yields,
( 72)
The new vector,
<Tl,
f 1
(,,»
has the same functional relationship as the
original one.
Therefore:
Comparing (73) and (69)
we
see that the universal equation is a private case
of the damage equation.
7.
OISCUSSION
Although not representing the general case, the S-N power relationship
in Eqs. (60), (62) is a natural reference equation to the fatigue response
for the following reasons.
First, it is an analytical solution to the
damage evolution equations in (S3a) when a Weibull distribution function is
used.
Second, it is graphically (Figs. 10, 11) and physically (non-fatigue
res i stant) a 1 imi t 1 i ne to whi ch other cases can be rel ated.
Thi rd, i t
reflects the extreme case when the fatigue resistivity (Rf
=
0) vanishes,
and fourth, it exhibits an identical formula tothe
we"
known empirical
"Basquinls Law" for high cycle fatigue (Ashby
&
Jones 1986):
a
(amax - amin)Nf
=
C,
(74)
where a and Care found experimentally (a is usua11y between 1/8 and 1/15
for most materials). Comparing (74) with (62) we get
a
=
-,
1
~
(75)
where amin
= 0 ts
-
assumed throughout this study.
,
tobreover, Eq. (74) is
found for a wide
range
of materials, including metals, and is not confined
only to a polymeric interface.
It is therefore useful to compare Eq. (62) with experimental data of
unnotched specimens under fatigue loading, as in Fig. 15.
Taking into
account the variety of sources (Daniel .\ Charewicz 1986, Hertzberg
&
Manson
1980, Weiss 1963) from which the data originate, the different experimental
conditions, size effects, surface roughness, etc., it is hard to overlook
the validity of Eq. (62) and, consequently, the theory itself. Although not
a complete experimental evidence, three groups can be identified in Fig. 15.
Fiber reinforced materials (like Graphite-Epoxy laminates) are the most
fatigueresi stant, whi ch is due to thei r crack arrest mechani sms.
Steel s
form a second group and plastics seem to be theweakest; despite their large
structural diversity.
The
wi
de range of static strength for the material s
plotted, and the possibility to control Rf by substituting a finite stress
concentration k (Fig.
11),
can only expand the generality of the basis on
which the HeFT lies.
Oespite the extreme differences between metals
(crystalline) and polymers, the similarity between their fatigue response
may be due to similar strength distribution shapes of their micro-components
(grains vs. macromolecules) , and the basic common phenomena of chemical
reaction during unloading.
The inconsistency of the MCFT with experimental results at high stress
levels, seen in Fig. 13, points to the limits of the theory.
At higher
stresses, micro-plastic deformations take place (sliding), which can extend
the fatigue life due to its stress relaxation effect. Another phenomenon is
the dependence of the stress concentrati on factor k on the si ze of the
crack.
At low stress levels, relatively more "big" cracks wi,., be created
for the same amount of damage 0, si nce the number of cycl es to reach 0 wi 11
be higher, giving more chances for the critical cracks to grow.
Thus, the
theory may predict shorter fati gue 1 He than expected for higher stress
levels, as the experimental results indicate.
The stat i c stress
as'
whi ch i staken so far as the static strength,
deserves more attent
i
on.
Si nce no pl ast ic deformat i on i s i nc 1 uded in the
theory,
as
can be defi ned more accurately as the static strength in the
absence
Of
plastic deformation.
In other words,
if
in the higher stress
levels, no plastic deformation is allowed, as would be the statie strength.
as is therefore always greater than the actual ultimate tensile strength,
and represents a limit.
Different heat treatments, chemical additives, etc., are used to
improve the statie strength of metals.
Mos~
methods are based on preventing
the original structure to undergo plastic deformation by slippage. Since as
represents the limit at which statie failure occurs without slipping, this
1 imit may be used as a useful upper bound to the improvement one can get
from these methods.
8.
CONCLUSIONS
The characteristics of the MCFT found in this stlJdy can be sUlnl11arized
in the following:
1.
It explains macro response by physical parameters
~,
k, g.
2. Gives logical reasoning to damage evolution processes.
3. Includes inner structure of the damaged material.
4. Appears to be relevant to metals, polymers and composites,
and not merely to interfaces.
5. Explains naturally the endurance limit phenomena.
6. Eliminates the need for empirical equations.
7. Shows relation to chaotic behaviour.
..
REFERENCES
=
Ashby, F. M.
&
Jones, R.H.O. 1986, In Engineering Materials 1, Pergamon
Press, 135-139.
Daniel, I. M.
&
Charewicz, A. 1986, Eng. Fracture Mechanics,
~,
793-808.
DeVries, K. l., loyd, B. A.
&
Williams, M. l. 1971, J. App. Physics, 42,
No.
12, 4644-4653 •
Feigenbaum, J.
M.~
1979, J. Stat. Phys.
~,
669-706.
Flory, J. P. 1975, Science, 188, 1268-1276.
Hatsuo, Ishida 1984, Polymer Composites,
~,
101-122.
Hertzberg, R. W.
&
Manson, A. J. 1980, Fatigue of Engineering Plastics,
Academic Press.
Kardomateas, A. G.
&
Vannas, I. V. 1985, Phil. Mag. A, 52, 39-50.
lemaitre, J. 1986, J. Eng. Fract. Mechanics,
~,
523-527.
Mall, S., Razaizadeh, M. A.
&
Gurumurthy, R. 1987, J. Eng. Mat. Tech., 109,
17-21.
Taljera, R. 1985, J. Composite Materials,
~,
355-375.
Termonia, V., Meakin, P.
&
Smith, P. 1985, Macromolecules, 18 (11),
2246-2252.
Weiss, V., Ed. 1963, Aerospace Structural Metals Handbook, Syracuse
University Press.
Zhurkov, S. N. 1965, Int. J. Fracture Mechanics, 1, 311-323.
Zhurkov, S. N.
&
Korsukov, A. F. 1974 J. Polym. Sci., polym. Phys. Edition,
APPENDIX
The power law (Eq. 59), relating the number of cycles to the outer
stress level for the case 9
=
1, k
+
CDis proven in the following. Let a
o
and
o~
denote two outer fat i gue stress
1
evel s imposed separatelyon the
material such that
K =
al/o
o
0
(A.l)
The evolutionary form of the governing equations (52), (53) demand
an
induction type of proof. For the first cycle (52a) and
0
0
,
we
get:
-ao~m~
-cml '"
~
~
m1
=
e
0
=
e
=
1 -
cm1
(A.2)
where
~
a
«
1,
c
=
ao
o
«
1
(A.3)
The damage af ter one cycle is small, so that
(A.4)
Using
(A.4), (A.2)
(A.5)
or
(A.6)
For
o~,
by the same procedure,
(A.7)
so that
It means that
~Ml
-
K~ ~ml
~NI
N=1 -
~n
n=1
(A.9)
where M, mand N, nare the stiffnesses and cycle numbers for the
o~
and
°
0
cases, respectively.
Denoting
.t
(
mi
)
'1n(..t)
=
TI
2
i=1
mi_l - mi-2
( A.lO)
etc., the lemma for the general cycle nUlTlber can be stated as follows.
Prove that i f
~
=
Eo
o
(A.11a)
and
(A.11b)
for
(A.11c)
Then
(A.12a)
and
(A.12b)
for
(A.l2c)
Proof:
First, find the ratio between
1t
m
(.t+l)
and
1tM(L+K~).
Recall that from
(A.13)
and
Moreover, the change of m between two sueeessive eyeles is small relative to
rn itself, i.e.,
(A.15 )
Therefore
l<i<K~
( A.16)
Using (A.14), (A.15) to show that
(A.1l)
and
1<
i
<
K~,
( A.1S)
the following is derived through (A.12), (A.13), (A.11e), (A.12e):
[~m(l+l)]~
=
[~m(l)]~
( A.19)
~M(L+K~)
~M(L)
Now, from the Weibull distribut10n (54) and (A.11e),
SUbstituting (A.20) in (A.19) and using (A.lla). (A.llb). we get
['ltm(1+1)]~
=
1
(A. 21)
'It
