FRACTURE MECHANISM
MAPS BASED ON EXPERIMENTAL DATA
4.1 The Fracture
Diagram
Metals
can fracture during monotonic loading in one of a number of ways: by cleavage,
by transgranular ductile fracture, by intergranular fracture, and so forth.
The aim here is to assemble, for metals, the data on the failure of found bar
specimens in tension, identifying the regime of stress and temperature over
which a given mechanism is dominant.
The
procedure is as follows: First, we tabulate, for each experimental investigation,
the homologous temperature (T/TM) and the normalised tensile stress
(sn/E ; where sn is the nominal stress in a
creep test and E is Youngís Modulus adjusted in the temperature of the test).
Together with the time-to-fracture (tf is secs) and fracture strain
(ef, or the reduction in area at failure). Second, we attempt to
assign a mode of failure to each. Some investigators specifically describe how
the specimen failed; by cup-and-cone fracture, by rupture, by intergranular
fracture and so on. Others show micrographs with sufficient detail for the
fracture mode to be identified. Still others (mostly those studying the
material from an engineering point of view) do neither; but change of mechanism
can sometimes be inferred from the data because of a sudden change in fracture
strain, because the time-to-fracture versus stress curve has an abrupt change
in slope, because the ratio of fracture strain to reduction in area changes, or
because there is a change in the temperature dependence of time-to-fracture.
This
information is assembled into the diagrams that follow. Most of them show four
mechanism-fields: ductile fracture, transgranular creep failure, intergranular
creep failure and rupture. But there exist other distinct mechanisms (cleavage,
for example) and subdivisions of the mechanisms listed already ñ though for
three out of the four materials studied here, the data is not sufficiently
detailed to allow these distinctions to be drawn at present. For the fourth,
iron, evidence for these extra fields exists. There is a cleavage field
characterized by a catastrophic reduction in ductility at the
ductile-to-brittle transition temperature (for round bars) and by fractographic
appearance. In addition, it is possible to subdivide the intergranular
creep-fracture field, at least in an approximate way, to show a regime in which
near-spherical voids grow on boundaries; a regime at higher stress in which the
voids become finger-like; and, at still higher stresses, a regime in which they
become wedge-like.
There
are difficulties and ambiguities in a study of this sort. There is the
influence of purity: a diagram applies to one purity of metal, or composition
of alloy, with one grain size, and in one state of heat-treatment. Specimen
shape is also important. Flat specimens tend to fail by rupture more readily
than round bars because unconstrained necking leads to rapid thinning through
the thickness of the flat plate. The reduced constraint also means that the
fracture stress will be lower. The data used here are from round bar tests.
This still does not eliminate the ambiguity arising from the variation in
fracture strains between specimens having different ratios of cross-sectional
area to gauge length. The total fracture strain is composed of homogeneous
elongation and elongation due to necking. The elongation due to necking is
proportional to the diameter of the specimen at the onset of necking. The
strain due to this elongation will depend on the length of the specimen. Wires
have virtually no component of fracture strain owing to necking, while short
fat specimens have a great deal. All experimental work carried out here used
specimens having a diameter to gauge length ratio of about 1:10. This ratio
usually was not considered of particular importance in collected data provided
that the specimens were not wires. In general, fracture data from wires have
been excluded from the maps for this reason. However, these shape
considerations do not apply for modes of fracture that involve little or no
necking. For example, it is not always important in grain boundary creep
cavitation.
The
data reported here arise form creep tests carried out under constant load
conditions so that the stress on the specimen actually increases throughout the
test. Constant stress creep tests introduce ambiguity into the determination
of fracture life and mechanisms as functions of stress and temperature.
Nevertheless, the stress is not constant in such tests if inhomogeneous strain
or necking occurs, or if there is a great deal of cavitation. In a constant
load test, the load is definitely constant regardless of how the specimen
deforms or fractures. When constant stress fracture lives are put on the maps,
they invariably appear to be longer than the constant load fracture lives at
the same initial stress. Since creep fracture data are almost exclusively from
constant load tests, this mode was chosen as the standard loading for the data
used in the maps ñ while keeping the above qualifications in mind. In Appendix
Four, an approximate method has been derived for converting time-to-fracture
determined under constant stress to time-to-fracture under constant load.
Data
from constant displacement-rate tensile tests are included on the maps. At low
temperatures, the small amount of thermal activation available causes the
time-to-fracture contours to bunch up. At 0.2 TM and lower, the
difference in stress between that required for failure in 1 second and that
required for failure in 106 seconds is less than the variation in
tensile test data between nominally identical specimens. Since the tensile
test at low temperatures has hardly any region of steady-state deformation, the
true stress increases monotonically until fracture occurs. The true stress at fracture
might be thought of as some measure of the inherent strength of the material,
deformed and work-hardened to its absolute limit. Should one use this true
stress to represent the fracture strength? Perhaps, but if this true stress
were applied at the beginning of a test the load required would considerably
exceed the load required to break the specimen. In order to be consistent with
the constant load creep data, the U.T.S. from tensile tests were chosen for the
fracture strength. The U.T.S. is a good measure of the fracture strength of a
material at low temperatures because, when multiplied by the specimenís initial
area, it gives the load that will just break the specimen whether applied over
a long or short period.
At
a given low temperature, the strain to fracture is nearly constant regardless
of strain-rate. Therefore the time-to-fracture is simply this strain divided
by the strain-rate in a tensile test. In most tensile tests, the test is
carried out in a reasonable length of time, so that all tensile test fracture
times lie in the range 102 to 104 seconds. The time is
really irrelevant in such a situation and is not recorded on the maps. At
higher temperatures, the stain-rate sensitivity of materials increases so that
the strain-rate determines the stress which is applied and, hence, the
mechanism of fracture. The fracture strain is, then, a function of
strain-rate, so that times-to-fracture are relevant variable. Whether or not
the data from a constant strain or displacement-rate test is compatible with
that from a constant load test really depends on the temperature and the
strain-rate. If the temperature is high enough and the strain-rate low enough
to attain a steady state in a tensile test then the nearly constant stress at
which this occurs ought to be equivalent to the observed. A steady state is
obtained when the rate of work hardening is matched by the rate of recovery, or
when the defect structure responsible for deformation does not change with
strain. Then, the stress for continued deformation does not change either.
Such a situation is recognized by a leveling off of the stress-strain curve at
some stress for some arbitrary amount of strain. If this leveling off is
insufficient then the U.T.S. is recorded but no time-to-failure is ascribed to
it in the maps.
In
spite of the above difficulties, it does appear that the general form of the
diagrams are reproducible and that, because of the normalized axes, diagrams
for materials with the same structure and comparable purity are very similar.
To support this point, one may note the similarities in the diagram constructed
here for 99.8% copper (Figure 4.1) with that prepared without any collaboration
by C. V. Gandhi on 99.8% nickel (Figure 4.2).
4.2 Copper
Pure
copper has neither enough strength nor oxidation resistance to make it suitable
for engineering applications at elevated temperatures. However, its physical
properties are well known and it is readily accessible in a pure state. For
this reason, many scientific investigators have studied the creep and fracture
of copper as a representative of the F.C.C> metals. Their work, being for
scientific rather than for applied purposes, usually identified mechanisms of
deformation and fracture as well as tabulating the time-to-failure as a
function of the stress, temperature, and microstructural variables.
The
normalized data for the fracture of found bars of O.F.H.C. copper, together
with some data on purer copper, is tabulated in Table 4.1 and plotted in
Figure. 4.1. The value of Youngís Modulus as a MN/m2 was obtained
from Frost and Ashby (1973). This was also the source for the melting
temperature of pure copper: 1356 Kelvin. The map itself shows the fields of
dominance of four mechanisms. At high stresses and low temperatures, the metal
fails by ductile fracture, that is, by the formation of a fibrous ìcupî,
surrounded by a shear-lip or ìconeî, forming in the necked region of the
tensile specimen. This plastic deformation of the specimen until it reaches
the U.T.S., when necking occurs. As the neck curvature increases, triaxial
stresses assist the growth of holes nucleated at inclusions until these holes
link up to form a transverse crack which propagates to the surface by maximum
normal stress. Nucleation and growth of the holes are observed to occur only
in the neck in pure copper. This is probably due to the small size and
sparcity of inclusions. The above mechanism results in a fractional reduction
in area that is always greater than the fractional elongation to fracture.
As
the temperature is raised, the metal starts to creep, and in the range of
temperature and stress shown in the figure, it fails by a transgranular creep
failure. The fracture mechanism is identical with that of ductile fracture:
nucleation, growth and linkage of internal voids or holes. And, because the
longitudinal strain in the specimen causes the growth of the holes, the
fractional reduction in area should again be greater than or equal to the
fractional reduction in area should again be greater than or equal to the
fractional elongation of the specimen at fracture. However, the dominant mode
of deformation causing this growth and linkage has changed: it is power-law
creep, not glide-plasticity (see, for example, Frost and Ashby, 1973). This
change in flow mechanism influences the onset of macroscopic instability
(necking), it changes the local stresses at inclusions which cause voids to
nucleate there, and may change the condition for local instability which makes
the voids link up. For all these reasons, we draw a boundary between these two
fields, which simply shows where power-law creep becomes the dominant mode of
flow. The data required to do this for a number of metals is given by Frost
and Ashby (1973, 1976). The time-to-fracture contours begin to spread out with
increasing temperature in the transgranular creep failure field because thermal
activation has an appreciable effect over the lifetime of the specimen.
Below
this transgranular creep failure field lies a field of intergranular creep
fracture. Specimens stressed in this regime fail because holes nulceate and
grow along grain boundaries and in a direction perpendicular to the largest
normal traction. These cavities grow by the stress-directed diffusion of
vacancies to the cavities. Since grain boundaries and the specimen surface are
the only sinks for atoms that will produce specimen extension, only cavities
very near the boundaries, or connected by a quick diffusion path to them, will
grow. Finally, these holes link up, reducing the internal cross-sectional area
of the specimen, until plasticity causes the remaining ligaments to fail
abruptly. Such samples usually show little or no necking, and may fail after
very small strains. Since the cavities contribute to the elongation of the
specimen and their growth does not depend on the amount of strain occurring the
specimen, it is possible for the fractional reduction in area to be less than
the fractional elongation of the specimen. Fleck et al. (1970) present nominal
longitudinal strain and area strain as a function of temperature for copper.
Around 693 Kelvin the logarithmic area strain becomes about equal to the
logarithmic longitudinal strain. This implies a change from transgranular
creep fracture to intergranular creep fracture. In general,, the transition
from a transgranular to an intergraular failure is either noted in, or can be
inferred from, most of the papers listed on Figure 4.1. Where a data point can
be unambiguously identified as an intergranular failure, it is shown as a full
symbol on the Figure; this information together with direct observation of the
transition itself allows the field boundary to be drawn. It is shown here as a
straight line because the data is not sufficiently detailed to distinguish a
curvature.
As
the temperature is raised further, strain-induced grain growth and dynamic
recrystallization accompany the creep test. This seems to have two effects:
by detaching the grain boundaries from voids which have nucleated on them, the
recrystallization inhibits growth of voids and, thus, suppresses grain-boundary
fracture. This, in turn, permits the continued deformation which is itself
enhanced by the softening effect of recrystallization until rupture by necking
to a point or chisel-edge intervenes. There are very few data in this regime,
so that its characteristics are not well known.
The
data points on Figure 4.1 are labeled, as far as possible, with numbers which
show the logarithm, to the base 10, of the time-to-fracture in seconds. This
gives an approximate picture of how contours of constant time-to-fracture might
lie on the diagram. For copper, almost all the data lie between 101.8
secs (about a minute) and 105.5 secs (about 100 hours). These are
short compared with data for engineering materials, for which creep data for up
to 108 seconds is available.
The
purity of the copper used in each investigation is listed on Figure 4.1. Four
of the sever investigations used OFHC copper, and their results are broadly
self-consistent; allowing for normal experimental scatter, field boundaries
deduced from them superimpose, and times-to-fractures, under a given set of
conditions, are the same. The purer coppers tend to show longer
times-to-fracture. This is probably because the inclusion density is lower, or
because (in the case of Ruchwiedís (1972) work) directional solidification led
to a rather special microstructure. This high-purity data is included in the
figure but was given little weight when assigning positions for the field
boundaries. The diagram can therefore be regarded as describing OFHC copper.
Further
support for the general shape of Figure 4.1 is obtained form the work of Gandhi
(1977) who has constructed a fracture map for 99.8% nickel (Figure 4.2).
Though based on a completely different set of studies, the data and field
boundaries, when plotted on normalized axes like those of Figure 4.1, almost
superimpose on the map for copper. The tests performed here on copper are also
presented in Figure 4.1. The present results are consistent with the data of
others and the mechanisms observed are in accord with those on the map.
4.3 Austenitic Stainless
Steels (18-8 series)
The
18-8 austenitic stainless steels are among the most widely used alloys because
of their ductility at cryogenic temperatures, their corrosion resistance, and
their high temperature strength and oxidation resistance. 304 stainless steel
is fundamentally typical of the 18-8 series, being essentially an alloy of iron
with 18 weight percent chromium and 8 weight percent nickel. It exhibits all
of the above desirable properties though not to the extent shown by more
advanced members of this series. Type 316 is one of the most creep and creep
fracture resistant steels of the 18-8 series. The main compositional
difference between 304 and 316 is the addition of molybdenum. The nominal
compositions of these two are given in Table 4.2 on the next page.
There
is some variation in the mechanical properties of different heats of 304, as
there is also for 316. Compositional variations are one reason for this; the
permitted range of compositions for these alloys is wide, allowing a range of
18 to 20 percent chromium and 8 to 12 percent nickel in 304 (16 to 18% Cr and
10 to 14% Ni in 316). In additional, the properties of the 18-8 series depend
on the thermal and mechanical history. Strengthening is usually accomplished
by work-hardening. In 316, Garofalo et al. (1961a) found that homogenizing and
quenching followed by deformation and an aging treatment at some intermediate
temperature, led to higher strengths at room temperature and, for short times,
at high temperature. Finally, there is the problem of sensitization in these
steels if they are cooled slowly through the range 800 to 400oC.
This leads to corrosion or oxidation of grain boundaries and premature
intergranular failure. For all these reasons, only data for stainless steel
which had been homogenized and quenched in oil or water, has been used in this
study. Homogenization is usually carried out between 1850 and 2050oF
for the 180-8 series. Such material should contain a reproducible
microstructure, and data obtained from it by different investigators should be
comparable.
There
is a great deal of engineering data on the mechanical properties of 304 and 316
steels. Some has been tabulated in Table 4.3 and used to construct fracture
mechanism maps (Figures 4.3 and 4.4). The fracture stresses recorded here are
all normalized by the Youngís Modulus corresponding to the temperature at which
the test was performed. Appropriate values of E were obtained from frost and
Ashby (1976) who expressed E as a linear function of temperature:
E(T)
= 21.6 x 104 [1 ñ 4.7 x 10-4(T ñ 300)] MN/n2
where the temperature, T, is in degrees Kelvin. The moduli
of 304 and 316 are virtually identical and differ only slightly from austenite
(Garofalo et al., 1960). The equation derived for E(T) from Garofaloís data
is, for both 304 and 316, well represented by
E(T)
= 20l.0 x 104 [1 ñ 4.3 x 10-4 (T ñ 300)] MN/m2
The melting point, TM, is also used only as a
normalizing parameter. For simplicity we have used the melting point of pure
iron, 1810oK, for both steels.
Unfortunately,
few statements or implications concerning the mechanisms of fracture were reported
in the engineering studies. One scientific study by Wray (1969) produced a map
for 316, with axes of strain-rate and temperature, on which he delineated a
region of grain boundary cavitation. he also observed ductile fracture,
transgranular creep fracture, intergranular creep fracture and rupture. Like
copper, one would expect the mechanism at low temperatures to be ductile
fracture. As the temperature is increased, creep deformation should become
dominant and transgranular creep fracture would appear while, at lower
stresses, intergranular creep fracture would be observed. Maps for 304 and 316
should look roughly similar, though 316 will be more creep fracture resistant.
The
field boundaries on the maps for 304 and 316 were arrived at in the following
way. Smith et al. (1950) state that a prominent kink in their plot of rupture
time against stress separates transgranular fracture (which occurs at shorter
times and higher stresses) from intgergranular fracture. The locus of these
kinks provides the main basis for the transition from transgranular to
intergranular fracture shown on the maps. Whenever the fracture was stated or
implied to be transgranular or intergranular these points were identified. For
instance, Garofalo et al. (1961a) observed that all their fractures were
intgergranular under their testing conditions. These points together with the
lines deduced from Sith et al. (1950) established the intergranular fracture
field. Above this field, the failures were all transgranular. This
transgranular region was further separated into two fields corresponding to
ductile fracture (i.e. hole growth by dislocation glide) and transgranular
creep fracture (i.e. hole growth by dislocation creep). The separation was
taken to occur on the line along which the U.T.S. levels off. This was found
to be in good agreement with the line separating dislocation glide and creep in
the deformation mechanism map for 316 and 304 stainless steel (Frost and Ashby,
1976).
Wray (1969) observed rupture or
complete necking to occur above 760oC virtually independent of stress in 316.
Aborn and Bain (1930) reported that the carbides in 18-8 stainless steels
dissolve somewhere around 760oC. We associate this dissolution with
the onset of rupture behavior described by Wray above 760oC. As the
carbides dissolve, the number of possible nucleation sites becomes less and
cavity growth is decreased due to the migration of unpinned grain boundaries
away from existing cavities. This permits continued deformation leading to
much greater strains and reduction-in-area.
These
maps may be presented in another format as shown in Figures 4.5 and 4.6. Here
the homologous temperature axis is replaced by a logarithmic time scale. This
format is a common way of displaying creep fracture data in the literature.
Data obtained at one temperature is connected by a line and labeled with that
temperature. The field boundaries from the fracture mechanism maps may be
transposed on to these new diagrams. In this way, old data may be seen with a
new perspective and a change in slope, whether gradual or sharp, may be
associated with a change in fracture mechanism.
At
this point, it is necessary to point out that intergranular cavitation in
stainless steel is very sensitive to the presence of boron, the cavitation
being suppressed by extremely small quantities of this element. The steels
tested here or used to construct the maps were not purposely inoculated with
boron and so show cavitation after only a few hundred hours. Some boron-containing
stainless steels do not show cavitation even after many thousand hours under
load at high temperatures.
Also,
we considered only solution-treated steels with no traces of cold work left in
them. The presence of cold work would enhance recrystallization or accelerated
grain growth. This is trre even at relatively low temperature (~ 500 oC)
if the fracture life is several years or more. Grain movement will suppress
the growth of intergranular cavities and so favor a transgranular fracture
mode. Therefore, stainless steels which have not been solution-treated after
forming which contain boron will probably show more transgranular fractures and
longer times-to-fracture than would be expected from the maps presented here.
4.4 Iron
The
fracture behavior of iron is more varied than that of the F.C.C. metals
examined so far. This is due partly to its susceptibility to cleavage at low
temperatures and to its transformation to austenite and s-iron at higher temperatures. Also, studies here and elsewhere
indicate that subdivisions of the intergranular creep fracture field exist.
Chapter Two showed how the shape of grain boundary cavities changes as a
function of stress and temperature. This may lead to different dependencies of
the time-to-failure on the controlling parameters of stress and temperature.
Iron
itself is rarely used for its mechanical properties because it is relatively
weak in the unalloyed state and because it is liable to corrode, especially by
oxidation. However, as iron is the main constituent of all steels, it is the
object of many fundamental scientific studies. Like copper, the data for iron
tends to be scattered over temperature and stress without the thoroughness
given to standard engineering materials. Also like copper the mechanism of
fracture is usually reported with the data and, because of this, the fracture
mechanism map for iron is well based on actual observations.
The
fracture mechanism diagram for iron, shown in Figure 4.7, was constructed from
the data in Table 4.4. The melting point was taken as 1810 Kelvin. Youngís
Modulus for ferrite from 0 to 1184 Kelvin was calculated from the single
crystal elastic constants of Dever (1972) and by taking the geometric mean of
E¢ = C11 ( 2C122
/ (C11 + C12))
and
E"
= (C44 (3C21 - 2C44) / (C44 + C12)
)
The result is shown as a function of temperature in Figure
4,.8. The value of E thus obtained at room temperature is 1.98 x 10 5 MN/m2.
This compares favorably with the average value from several investigations of
1.97 (+ 0.04) x 10 5 MN/m2 (ASM Metals Handbook, Vol. 1, 1961). Such a
computed value should be appropriate for an untextured polycrystalline material
containing anisotropic grains. Youngís Modulus for austenite (1184 to 1663
Kelvin) is from Koster (1948) and is given by
E(T
= 2.16 x 10 \5 [1 ñ (T ñ300)4.7 x 10ñ4] MN/m2
where T is in Kelvin.
At
very low temperatures the fracture behavior is characterized by cleavage.
Although cleavage is not subdivided on the diagram, it ought to separate into
one region where microcrack nucleation determines the fracture stress, and
another in which microcrack propagation should require very little energy.
Once a critical microcrack is nucleated, it should propagate to fracture using
only the stored elastic strain energy remaining after nucleation and not
require any further loading of the specimen. Microcrack nucleation has been
theoretically attributed to (1) cracking of inclusions or inclusion-matrix
interfaces due to micro-yielding near the particle, (2) cracking at the grain
boundary due to a dislocation pile-up there, or (3) the inter-section of twins.
A thorough study by McMahon and Cohen (1965) revealed that nearly every crack
in an extra-pure Ferrovac E iron had nucleated at a cracked carbide due to
micro-yielding around the particle. At higher temperatures a larger amount of
plastic work is required at the crack tip for propagation than for nucleation
and must be supplied by increasing the stress and the strain energy. If we
accept that crack nucleation occurs at particles, the temperature of the
transition between cleavage control mechanisms is dependent on purity and
particle size. In zone-refined, electrolytic iron, cleavage did not occur at
77 Kelvin (Simonsen and Dossin, 1965). As iron of this purity is not readily
available and as cleavage is common in nominally pure iron and steels, the
absence of cleavage in such pure material is interesting mainly because it
demonstrates that cleavage, like most fracture mechanisms, is dependent on the
presence of particles. Regardless of what controls the fracture process, the
mechanism of fracture at these low temperatures in nominally pure iron is
cleavage.
As
the temperature is raised still further, the fracture surface appearance
changes from 100% cleavage facets (solid squares) to a mixture of facets and
fibrous fracture dimples (half-filled squares) and, finally, to 100% fibrous
fracture. The reduction in area, and the strain to fracture increase
simultaneously. This is the ductile-brittle transition and it indicates the
change in fracture mechanism from cleavage to ductile fracture. This mechanism
field boundary is the most thoroughly studied of all such transitions. Its
position in stress-temperature space has been well documented for many
materials, grain sizes and inclusion contents. Nevertheless, it forms only one
small part of the total fracture picture and other mechanism boundaries may
prove equally important when considering the high-temperature life of
materials.
The
transgraular ductile fracture and transgranular creep fracture of iron should
be much like that of copper or stainless steel. The boundary between these two
fracture mechanisms has been drawn from the deformation mechanism maps. It is
that line which separates dislocation glide deformation from power-law creep
deformation.
The
intergranular creep fracture field of iron is not necessarily different from
those of the metals examined so far, but more is known about it. Since
previously crept iron specimens can be broken open at low temperatures along
the grain boundaries, the shape of the cavities there may be examined in plan
view. This has allowed cavity shape to be studied as a function of stress and
temperature as detailed in Chapter Two. At very low stresses, the cavities are
round or faceted, their exact shape being determined mainly by the equilibrium
of surface tensions. At higher stresses, the smooth perimeter of the cavities
becomes unstable and finger-like projections shoot out ahead of the main crack.
At still higher stresses, just below the transition to transgranular creep
fracture, groups of irregularly shaped voids and wedge cracks appear. The
boundaries between these sub-mechanisms is indicated by dashed lines and was
determined from work done here and by Taplin and Wingrove (1967).
At
927oC the austenite transformation occurs. Wray (1975) found that,
in the temperature range 950 to 1350oC, failure in zone-refined iron
(99.997 at % pure) was determined only by plastic deformation leading to
separation at a point in the neck. Concurrent recrystallization as also noted.
This is the same mechanism that led to rupture in copper and stainless steel.
However, rupture in zone-refined materials may be the dominant mode of fracture
at most temperatures and stresses because of the lack of nucleation sites for
cavities. When Wray (1975) examined electrolytic iron (99.97 at % pure) in the
temperature range 950 to 1350oC and in the strain rate range 2.8 x
10ñ5 to 2.3 x 10 2 sec ñ1, he found regions of
transgranular creep fracture and intergranular creep fracture. This
electrolytic iron is of a purity compatible with that of the irons used to construct
the map. The transgranular creep fracture field for nominally pure iron
extends into the austenite region until it is cut off by the rupture field.
The intergranular creep fracture field will probably persist at very low
stresses because the strain occurring during the life other specimen is not
enough to cause recrystallization. This behavior should extend into s-iron where only the two mechanisms, rupture
and intergranular creep fracture, will be observed.
APPENDIX FOUR
ON RELATING CONSTANT STRESS CREEP TESTS AND CONSTANT LOAD
CREEP TESTS
The strain produced in a uniaxial creep specimen by a stress
acting over time can be represented in stress-strain-time (s-e-t)
space by a surface. Analytically, this is described by
e
= e(t,s)
In a constant stress (CS) test, all deformation is carried
out in one e-t plane that is
perpendicular to the stress axis. The strain-rate in this case is simply given
by the partial derivative of strain with respect to time, i.e.
(de/dt) = (de/dt)
In a constant load (CL) test, the stress varies as well and the
total derivative of strain with respect to time is given by
(de/ dt) = (de/dt) + (de/ds)(ds/dt) (1)
Let us assume for simplicity that all of the primary creep
strain occurs initially and the material deforms from t = 0 according to
(de/dt)
= KsN (2)
This may be integrated to
e = KsNt
+ c
and differentiated with respect to stress to obtain
(de/ds)
= KNs(N-1)t (3)
By assuming constant volume and homogeneous deformation, we
see that the stress varies with strain in the following way:
s = (L / a(t)) = (L / ao)( 1(t) / 1o)
= so exp e (4)
where so
is the initial stress and e is the
logarithmic longitudinal strain. Differentiating this with respect to time
results in
(ds / dt) = s
o exp (e) (de/ dt) (5)
Substituting Equations 2,3,4 and 5 into Equation 1 gives
(de/ dt) = KsoN
exp (Ne) = KsoN exp (Ne)
NT (de/ dt) (6)
In a constant stress test, a steady-state strain rate is
obtained which is given by
ess = KsoN
Therefore, Equation 6 becomes
(de/ dt) = ess
exp (Ne) (1 + Nt (de /dt))
The solution to this differential equation is
=
exp (-Ne) ((e/ess) ñ C) (7)
where C is some constant which may be determined by
recalling the assumption that all the primary creep occurs immediately upon
loading. Then
t
= exp (-Ne) ( (e/ess) ñ (ep exp(nNep) / ess)
) (8)
where ep
is the primary creep strain observed in a constant stress test. Equation 8
will generate the constant load curve that would be obtained from a specimen
loaded by an initial stress that causes a known primary creep strain and a
known steady-state strain rate in a constant stress test. Equation 8 could be
improved by using ep (t),
i.e. the time dependent, primary creep strain. Then for each value of t and ep (t) the strain could be
determined in order to generate the creep curve.
During
primary creep the strain-rate may not depend on stress in the same way as it
does during steady-state creep. If the primary creep strain is small an
approximation to Equation 8 is
t
= exp (-Ne) ( (e - ep) / ess ) (9)
which implies that the primary creep strains under both
loading modes are equal at the beginning of loading. This is reasonable as the
primary creep strain is frequently independent of stress. The above equation
may be rewritten as
ess = exp (-NeCL) ( (eCL - ep)
/ t) (10)
where eCL
is the strain at time t in a constant load creep test. Using Equation 10 the
steady-state strain rate may be obtained from the constant load curve resulting
from an equal initial stress ñ providing that primary creep has ended and that
the deformation is still homogeneous. Indeed, the entire constant stress curve
may be generated by rearranging Equation 10 and adding on the primary creep
strain:
esst + ep = eCS
= exp (-NeCL)(eCL - ep) + ep (11)
This maps constant load strain to constant stress strain
and, implicitly, vice-versa. Again, a more accurate mapping would be obtained
if the time-dependent primary creep strain were used for ep. This mapping is only valid
while the deformation is homogeneous and before tertiary creep has begun in the
constant stress test.
To
find the relation between the two times-to-failure, we sue the Monkman-Grant
relation:
esstf = constant = e* (12)
which is reasonably valid for many materials, especially
pure metals. Feltham and Meaking (1959) found this equation to be well obeyed
by their experimental constant stress tests performed on copper over a wide
range of stresses and temperatures. Equation 12 implies that there is a
constant, extrapolated, steady-state creep strain, e*, at which failure occurs (Figure 1). As seen in the figure,
failure will be imminent in a constant load test when the specimen has strained
an amount:
eCL = e* + ep = esstfCS + ep
Substituting this into Equation 9 one obtains
tfCL
= tfCS exp { -N(esstfCS + ep) } (13)
The time-to-failure for a constant load test may thus be
approximately predicted from a constant stress test. From Figure 1 one sees
that tfCL determined from Equation 13 is a conservative
estimate of the actual time-to-failure. In Figure 2 the ratio of the two
times-to-failure versus strain may be seen for various values of N. The
dependence on N is exponential so that N must be known accurately in order to
predict the time-to-failure under constant load. Indeed, if anything happens
that alters N as the stress increases during a constant load test, then the
mapping from constant stress to constant load curves (Equation 11) and the
relation between the times-to-failure (Equation 13) are invalid. To handle the
case of a changing N, the above analysis would have to be carried out using an
N that was a function of the stress, or using the deformation law
e - K sinh (Ns)
This lead to mathematical complications that make a
graphical solution to the problem the simpler approach. Given a series of
constant stress curves determined at stresses greater than or equal to the
initial stress in a constant load test, one determines the creep-rate for
various strains when the decreasing cross-sectional area causes the stress to
rise under constant load to a value used in one of the constant stress tests.
A smooth, monotonic curve having the correct slopes at the various strains is
the constant load curve. This idea is present in greater detail in a thesis by
A. J. F. Paterson (1973).
It
is not possible to invert Equation 13 in order to obtain explicitly the
time-to-failure under constant stress conditions. If ess is evaluated using Equation 10, then Equation 13
is an implicit solution for tfCS.
Finally, all of the above analysis ignores the possibility
of necking. If necking occurs, the cross-sectional area is related only in a
complicated way to the over-all longitudinal strain. The stress not only
increases more rapidly at the neck, but varies over the specimenís gauge
length. as a result, the strain rate varies from point to point in the
specimen. The analysis presented here is not valid once pronounced necking
occurs in a specimen.
Hart
(1969) showed that in a fisco-plastic material, homogeneous deformation becomes
unstable when
m
+ (1/N) < 1 (14)
where m is a work-hardening parameter defined by
m
= (s1ns
/ se) | e
As
Burke and Nix (1976) point out, Equation 14 means that some constant stress
tests become unstable before a steady-state is ever reached because values of m
+ 1/N exceed unity after only a few percent strain. However, some diametral
perturbation is required before an unstable specimen can begin necking. Thus,
specimens with a smooth gauge section of constant diameter, tested in a furnace
with a good constant temperature zone, may deform homogeneously to a greater
strain than specimens not tested under these conditions. Of greater importance
is the rate of growth of a neck once it is initiated. If this rate is slow
with respect to the total rate of deformation, a very diffuse neck will form
and the deformation will be nearly homogeneous. This is most likely to occur
when N is small. Thus, the analysis presented here should apply best where the
mechanism of d3eformation is diffusional flow, though it should still work in
power-law creep providing the region of power-law breakdown is not near.