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/T_M) and the normalised tensile stress (s_n/E ; where s_n 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 (t_f is secs) and fracture strain (e_f, 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 T_M 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 10^1.8 secs (about a minute) and 10^5.5 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 400^oC. 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 2050^oF 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, T_M, 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 760^oC. We associate this dissolution with the onset of rupture behavior described by Wray above 760^oC. 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 10^ñ4] 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 927^oC the austenite transformation occurs. Wray (1975) found that, in the temperature range 950 to 1350^oC, 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 1350^oC and in the strain rate range 2.8 x 10^ñ5 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) = Ks^N                                                                          (2)

 

This may be integrated to

 

            e = Ks^Nt + 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 / a_o)( 1(t) / 1_o) = s_o exp e                            (4)

 

where s_o 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) = Ks_o^N exp (Ne) = Ks_o^N exp (Ne) NT (de/ dt)           (6)

 

In a constant stress test, a steady-state strain rate is obtained which is given by

 

            e_ss = Ks_o^N

 

Therefore, Equation 6 becomes

 

            (de/ dt) = e_ss exp (Ne) (1 + Nt (de /dt))

 

The solution to this differential equation is

 

            = exp (-Ne) ((e/e_ss) 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/e_ss) ñ (e_p exp(nNe_p) / e_ss) )                           (8)

 

where e_p 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 e_p (t), i.e. the time dependent, primary creep strain. Then for each value of t and e_p (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 - e_p) / e_ss )                                                    (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

 

            e_ss = exp (-Ne_CL) ( (e_CL - e_p) / t)                                               (10)

 

where e_CL 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:

 

            e_sst + e_p = e_CS = exp (-Ne_CL)(e_CL - e_p) + e_p                               (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 e_p. 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:

 

            e_sst_f = 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:

 

            e_CL = e* + e_p = e_sst_f^CS + e_p

 

Substituting this into Equation 9 one obtains

 

            t_f^CL = t_f^CS exp { -N(e_sst_f^CS + e_p) }                                              (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 t_f^CL 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 e_ss is evaluated using Equation 10, then Equation 13 is an implicit solution for t_f^CS.

 

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.