Steel Properties at Low Temperatures

Aircraft and chemical processing equipment are now required to work at subzero temperatures and the behavior of metals at temperatures down to -150°C needs consideration, especially from the point of view of welded design where changes in section and undercutting at welds may occur.

An increase in tensile and yield strength at low temperature is characteristic of metals and alloys in general. Copper, nickel, aluminium and austenitic alloys retain much or all of their tensile ductility and resistance to shock at low temperatures in spite of the increase in strength.

In the case of unnotched mild steel, the elongation and reduction of area is satisfactory down to -130°C and then falls off seriously. It is found almost exclusively in ferritic steels, however, that a sharp drop in Izo-d value occurs at temperatures around 0°C (see Figs. 1 and 2).

The transition temperature at which brittle fracture occurs is lowered by:

• a decrease in carbon content, less than 0,15% is desirable
• a decrease in velocity of deformation
• a decrease in depth of `notch`
• an increase in radius of `notch`, e.g. 6 mm minimum
• an increase in nickel content, e.g. 9%
• a decrease in grain size; it is desirable, therefore, to use steel deoxidized with aluminium normalized to give fine pearlitic structure and to avoid the presence of bainite even if tempered subsequently
• an increase in manganese content; Mn/C ratio should be greater than 21, preferably 8.

Figure 1.	(a) Yield and cohesive stress curves
	(b) Slow notch bend test
	(c) Effect of temperature on the Izod value of mild steel

Figure 2. Effect of low temperatures on the mechanical
properties of steel in plain and notched conditions

Surface grinding with grit coarser than 180 and shot-blasting causes embrittlement at -100°C due to surface work-hardening, which, however, is corrected by annealing at 650-700°C for 1 h. This heat-treatment also provides a safeguard against the initiation of brittle fracture of welded structures by removing residual stresses.

Where temperatures lower than -100°C or where notch-impact stresses are involved in equipment operating below zero, it is preferable to use an 18/8 austenitic or a non-ferrous metal.

The 9% Ni steel provides an attractive combination of properties at a moderate price. Its excellent toughness is due to a fine-grained structure of tough nickel-ferrite devoid of embrittling carbide networks, which are taken into solution during tempering at 570°C to form stable austenite islands. This tempering is particularly important because of the low ferrite-austenite transformation temperatures.

A 4% Mn Ni (rest iron) is suitable for castings for use down to -196°C. Care should be taken to select plates without surface defects and to ensure freedom from notches in design and fabrication. Fig. 3 shows tensile and impact strengths for various alloys.

Figure 3. Tensile and impact strengths of various alloys at subzero temperatures

Steel Properties at High Temperatures

Creep is the slow plastic deformation of metals under a constant stress, which becomes important in:

1. The soft metals used at about room temperature, such as lead pipes and white metal bearings.
2. Steam and chemical plant operating at 450-550°C.
3. Gas turbines working at high temperatures.

Creep can take place and lead to fracture at static stresses much smaller than those which will break the specimen when loaded quickly in the temperature range 0,5-0,7 of the melting point T_m.

The Variation with time of the extension of a metal under different stresses is shown in Fig. 4a. Three conditions can be recognized:

• The primary stage, when relatively rapid extension takes place but at a decreasing rate. This is of interest to a designer since it forms part of the total extension reached in a given time, and may affect clearances.
• The secondary period during which creep occurs at a more or less constant rate, sometimes referred to as the minimum creep rate. This is the important part of the curve for most applications.
• The tertiary creep stage when the rate of extension accelerates and finally leads to rupture. The use of alloys in this stage should be avoided; but the change from the secondary to the tertiary stage is not always easy to determine from creep curves for some materials.

Figure 4.	a) Family of creep curves at stresses increasing from A to C
	b) Stress-time curves at different creep strain and repture

The limited nature of the information available from the creep curve is clearer when a family of curves is considered covering a range of operating stresses.

As the applied stress decreases the primary stage decreases and the secondary stage is extended and the extension during the tertiary stage tends to decrease. Modifying the temperature of the test has a somewhat similar effect on the shape of the curves.

Design data are usually given as series of curves for constant creep strain (0,01-0,03%, etc.), relating stress and time for a given temperature. It is important to know whether the data used are for the secondary stage only or whether it also includes the primary stage (Fig. 4b).
In designing plants that work at temperatures well above atmospheric temperatures, the designer must consider carefully what possible maximum strains he can allow and what the final life of the plant is likely to be. The permissible amounts of creep depend largely on the article and service conditions. Examples for steel are:

	Rate of Creep mm/min	Time, h	Maximum Permissible Strain, mm
Turbine rotor wheels, shrunk on shafts	10-11	100000	0,0025
Steam piping, welded joints, boiler tubes	10-9	100000	0,075
Superheated tubes	10-8	20000	0,5

In designing missiles data are needed at higher temperatures and stresses and shorter time (5-60 min) than are determined for creep tests. This data is often plotted as isochronous stress-strain curves.

Creep tests

For long-time applications it is necessary to carry out lengthy tests to get the design data. It is dangerous to extrapolate from short time tests, which may not produce all the structural changes, e.g. spheroidation of carbide. For alloy development and production control short time tests are used.

Long time creep tests

A uniaxial tensile stress is applied by the means of a lever system to a specimen (similar to that used in tensile testing) situated in a tubular furnace and the temperature is very accurately controlled. A very sensitive mirror extensometer (of Martens type) is used to measure creep rate of 1×10^-8 strain/h. From a series of tests at a single temperature, a limiting creep stress is estimated for a certain arbitrary small rate of creep, and a factor of safety is used in design.

Short time tests

The rupture test is used to determine time-to-rupture under specified conditions of temperature and stress with only approximate measurement of strain by dial gauge during the course of the experiments because total strain may be around 50%. It is a useful test for sorting out new alloys and has direct application to design where creep deformation can be tolerated but fracture must be prevented.

The engineering tension test is widely used to provide basic design information on images/the strength of materials and as an acceptance test for the specification of materials. In the tension test a specimen is subjected to a continually increasing uniaxial tensile force while simultaneous observations are made of the elongation of the specimen. An engineering stress-strain curve is constructed from the load elongation measurements (Fig. 1).

Figure 1. The engineering stress-strain curve

It is obtained by dividing the load by the original area of the cross section of the specimen.

(1)

The strain used for the engineering stress-strain curve is the average linear strain, which is obtained by dividing the elongation of the gage length of the specimen, d, by its original length.

(2)

Since both the stress and the strain are obtained by dividing the load and elongation by constant factors, the load-elongation curve will have the same shape as the engineering stress-strain curve. The two curves are frequently used interchangeably.

The shape and magnitude of the stress-strain curve of a metal will depend on its composition, heat treatment, prior history of plastic deformation, and the strain rate, temperature, and state of stress imposed during the testing. The parameters, which are used to describe the stress-strain curve of a metal, are the tensile strength, yield strength or yield point, percent elongation, and reduction of area. The first two are strength parameters; the last two indicate ductility.

The general shape of the engineering stress-strain curve (Fig. 1) requires further explanation. In the elastic region stress is linearly proportional to strain. When the load exceeds a value corresponding to the yield strength, the specimen undergoes gross plastic deformation. It is permanently deformed if the load is released to zero. The stress to produce continued plastic deformation increases with increasing plastic strain, i.e., the metal strain-hardens. The volume of the specimen remains constant during plastic deformation, A·L = A₀·L₀ and as the specimen elongates, it decreases uniformly along the gage length in cross-sectional area.

Initially the strain hardening more than compensates for this decrease in area and the engineering stress (proportional to load P) continues to rise with increasing strain. Eventually a point is reached where the decrease in specimen cross-sectional area is greater than the increase in deformation load arising from strain hardening. This condition will be reached first at some point in the specimen that is slightly weaker than the rest. All further plastic deformation is concentrated in this region, and the specimen begins to neck or thin down locally. Because the cross-sectional area now is decreasing far more rapidly than strain hardening increases the deformation load, the actual load required to deform the specimen falls off and the engineering stress likewise continues to decrease until fracture occurs.

Tensile Strength

The tensile strength, or ultimate tensile strength (UTS), is the maximum load divided by the original cross-sectional area of the specimen.

(3)

The tensile strength is the value most often quoted from the results of a tension test; yet in reality it is a value of little fundamental significance with regard to the strength of a metal. For ductile metals the tensile strength should be regarded as a measure of the maximum load, which a metal can withstand under the very restrictive conditions of uniaxial loading. It will be shown that this value bears little relation to the useful strength of the metal under the more complex conditions of stress, which are usually encountered.

For many years it was customary to base the strength of members on the tensile strength, suitably reduced by a factor of safety. The current trend is to the more rational approach of basing the static design of ductile metals on the yield strength.

However, because of the long practice of using the tensile strength to determine the strength of materials, it has become a very familiar property, and as such it is a very useful identification of a material in the same sense that the chemical composition serves to identify a metal or alloy.

Further, because the tensile strength is easy to determine and is a quite reproducible property, it is useful for the purposes of specifications and for quality control of a product. Extensive empirical correlations between tensile strength and properties such as hardness and fatigue strength are often quite useful. For brittle materials, the tensile strength is a valid criterion for design.

Measures of Yielding

The stress at which plastic deformation or yielding is observed to begin depends on the sensitivity of the strain measurements. With most materials there is a gradual transition from elastic to plastic behavior, and the point at which plastic deformation begins is hard to define with precision. Various criteria for the initiation of yielding are used depending on the sensitivity of the strain measurements and the intended use of the data.

1. True elastic limit based on micro strain measurements at strains on order of 2 x 10^-6 in | in. This elastic limit is a very low value and is related to the motion of a few hundred dislocations.
2. Proportional limit is the highest stress at which stress is directly proportional to strain. It is obtained by observing the deviation from the straight-line portion of the stress-strain curve.
3. Elastic limit is the greatest stress the material can withstand without any measurable permanent strain remaining on the complete release of load. With increasing sensitivity of strain measurement, the value of the elastic limit is decreased until at the limit it equals the true elastic limit determined from micro strain measurements. With the sensitivity of strain usually employed in engineering studies (10^-4in | in), the elastic limit is greater than the proportional limit. Determination of the elastic limit requires a tedious incremental loading-unloading test procedure.
4. The yield strength is the stress required to produce a small-specified amount of plastic deformation. The usual definition of this property is the offset yield strength determined by the stress corresponding to the intersection of the stress-strain curve and a line parallel to the elastic part of the curve offset by a specified strain (Fig. 1). In the United States the offset is usually specified as a strain of 0.2 or 0.1 percent (e = 0.002 or 0.001).

   (4)

A good way of looking at offset yield strength is that after a specimen has been loaded to its 0.2 percent offset yield strength and then unloaded it will be 0.2 percent longer than before the test. The offset yield strength is often referred to in Great Britain as the proof stress, where offset values are either 0.1 or 0.5 percent. The yield strength obtained by an offset method is commonly used for design and specification purposes because it avoids the practical difficulties of measuring the elastic limit or proportional limit.

Some materials have essentially no linear portion to their stress-strain curve, for example, soft copper or gray cast iron. For these materials the offset method cannot be used and the usual practice is to define the yield strength as the stress to produce some total strain, for example, e = 0.005.

Measures of Ductility

At our present degree of understanding, ductility is a qualitative, subjective property of a material. In general, measurements of ductility are of interest in three ways:

1. To indicate the extent to which a metal can be deformed without fracture in metalworking operations such as rolling and extrusion.
2. To indicate to the designer, in a general way, the ability of the metal to flow plastically before fracture. A high ductility indicates that the material is "forgiving" and likely to deform locally without fracture should the designer err in the stress calculation or the prediction of severe loads.
3. To serve as an indicator of changes in impurity level or processing conditions. Ductility measurements may be specified to assess material quality even though no direct relationship exists between the ductility measurement and performance in service.

The conventional measures of ductility that are obtained from the tension test are the engineering strain at fracture ef (usually called the elongation) and the reduction of area at fracture q. Both of these properties are obtained after fracture by putting the specimen back together and taking measurements of L_f and A_f .

	(5)
	(6)

Because an appreciable fraction of the plastic deformation will be concentrated in the necked region of the tension specimen, the value of e_f will depend on the gage length L₀ over which the measurement was taken. The smaller the gage length the greater will be the contribution to the overall elongation from the necked region and the higher will be the value of e_f. Therefore, when reporting values of percentage elongation, the gage length L₀ always should be given.

The reduction of area does not suffer from this difficulty. Reduction of area values can be converted into an equivalent zero-gage-length elongation e₀. From the constancy of volume relationship for plastic deformation A*L = A₀*L₀, we obtain

(7)

This represents the elongation based on a very short gage length near the fracture.

Another way to avoid the complication from necking is to base the percentage elongation on the uniform strain out to the point at which necking begins. The uniform elongation e_u correlates well with stretch-forming operations. Since the engineering stress-strain curve often is quite flat in the vicinity of necking, it may be difficult to establish the strain at maximum load without ambiguity. In this case the method suggested by Nelson and Winlock is useful.

The ability of a material to absorb energy when deformed elastically and to return it when unloaded is called resilience. This is usually measured by the modulus of resilience, which is the strain energy per unit volume required to stress the material from, zero stress to the yield stress s. The strain energy per unit volume for uniaxial tension is

(1)

From the above definition the modulus of resilience is

(2)

This equation indicates that the ideal material for resisting energy loads in applications where the material must not undergo permanent distortion, such as mechanical springs, is having a high yield stress and a low modulus of elasticity. Table 1 gives some values of modulus of resilience for different materials.

Table 1. Modulus of resilience for various materials

Material	E, psi	s0, psi	Modulus of resilience, Ur
Medium-carbon steel	30×106	45000	33,7
High-carbon spring steel	30×106	140000	320
Duraluminium	10,5×106	18000	17,0
Cooper	16×106	4000	5,3
Rubber	150	300	300
Acrylic polymer	0,5×106	2000	4,0

Toughness

The toughness of a material is its ability to absorb energy in the plastic range. The ability to withstand occasional, stresses above the yield stress without fracturing is particularly desirable in parts such as freight-car couplings, gears, chains, and crane hooks. Toughness is a commonly used concept, which is difficult to pin down and define. One way of looking at toughness is to consider that it is the total area under the stress-strain curve. This area is an indication of the amount of work per unit volume, which can be done, on the material without causing it to rupture. Figure 1.2 shows the stress-strain curves for high- and low-toughness materials. The high-carbon spring steel has a higher yield strength and tensile strength than the medium-carbon structural steel. However, the structural steel is more ductile and has a greater total elongation. The total area under the stresstrain curve is greater for the structural steel, and therefore it is a tougher material. This illustrates that toughness is a parameter that comprises both strength and ductility. The crosshatched regions in Fig. 1 indicate the modulus of resilience for each steel. Because of its higher yield strength, the spring steel has the greater resilience.
Several mathematical approximations for the area under the stress-strain curve have been suggested. For ductile metals that have a stress-strain curve like that of the structural steel, the area under the curve can be approximated by either of the following equations:

(3)

(4)

For brittle materials the stress-strain curve is sometimes assumed to be a parabola, and the area under the curve is given by

(5)

Figure 1. Comparison of stress-strain curves

All these relations are only approximations to the area under the stress-strain curves. Further, the curves do not represent the true behavior in the plastic range, since they are all based on the original area of the specimen.

The engineering stress-strain curve does not give a true indication of the deformation characteristics of a metal because it is based entirely on the original dimensions of the specimen, and these dimensions change continuously during the test. Also, ductile metal which is pulled in tension becomes unstable and necks down during the course of the test. Because the cross-sectional area of the specimen is decreasing rapidly at this stage in the test, the load required continuing deformation falls off. The average stress based on original area likewise decreases, and this produces the fall-off in the stress-strain curve beyond the point of maximum load. Actually, the metal continues to strain-harden all the way up to fracture, so that the stress required to produce further deformation should also increase. If the true stress, based on the actual cross-sectional area of the specimen, is used, it is found that the stress-strain curve increases continuously up to fracture. If the strain measurement is also based on instantaneous measurements, the curve, which is obtained, is known as a true-stress-true-strain curve. This is also known as a flow curve since it represents the basic plastic-flow characteristics of the material. Any point on the flow curve can be considered the yield stress for a metal strained in tension by the amount shown on the curve. Thus, if the load is removed at this point and then reapplied, the material will behave elastically throughout the entire range of reloading.
The true stress s is expressed in terms of engineering stress s by

(1)

The derivation of Eq. (1) assumes both constancy of volume and a homogenous distribution of strain along the gage length of the tension specimen. Thus, Eq. (1) should only be used until the onset of necking. Beyond maximum load the true stress should be determined from actual measurements of load and cross-sectional area.

(2)

The true strain emay be determined from the engineering or conventional strain e by

(3)

Figure 1. Comparison of engineering and true stress-strain curves

This equation is applicable only to the onset of necking for the reasons discussed above. Beyond maximum load the true strain should be based on actual area or diameter measurements.

(4)

Figure 1 compares the true-stress-true-strain curve with its corresponding engineering stress-strain curve. Note that because of the relatively large plastic strains, the elastic region has been compressed into the y-axis. In agreement with Eqs. (1) and (3), the true-stress-true-strain curve is always to the left of the engineering curve until the maximum load is reached. However, beyond maximum load the high-localized strains in the necked region that are used in Eq. (4) far exceed the engineering strain calculated from Eq. (1). Frequently the flow curve is linear from maximum load to fracture, while in other cases its slope continuously decreases up to fracture. The formation of a necked region or mild notch introduces triaxial stresses, which make it difficult to determine accurately the longitudinal tensile stress on out to fracture.
The following parameters usually are determined from the true-stress-true-strain curve.

True Stress at Maximum Load

The true stress at maximum load corresponds to the true tensile strength. For most materials necking begins at maximum load at a value of strain where the true stress equals the slope of the flow curve. Let s_u and e_u denote the true stress and true strain at maximum load when the cross-sectional area of the specimen is Au. The ultimate tensile strength is given by

Eliminating P_max yields

(5)

True Fracture Stress

The true fracture stress is the load at fracture divided by the cross-sectional area at fracture. This stress should be corrected for the, triaxial state of stress existing in the tensile specimen at fracture. Since the data required for this correction are often not available, true-fracture-stress values are frequently in error.

True Fracture Strain

The true fracture strain e_f is the true strain based on the original area A₀ and the area after fracture A_f

(6)

This parameter represents the maximum true strain that the material can withstand before fracture and is analogous to the total strain to fracture of the engineering stress-strain curve. Since Eq. (3) is not valid beyond the onset of necking, it is not possible to calculate e_f from measured values of e_f. However, for cylindrical tensile specimens the reduction of area q is related to the true fracture strain by the relationship

(7)

True Uniform Strain

The true uniform strain e_u is the true strain based only on the strain up to maximum load. It may be calculated from either the specimen cross-sectional area A_u or the gage length L_u at maximum load. Equation (3) may be used to convert conventional uniform strain to true uniform strain. The uniform strain is often useful in estimating the formability of metals from the results of a tension test.

(8)

True Local Necking Strain

The local necking strain e_n is the strain required to deform the specimen from maximum load to fracture.

(9)

The flow curve of many metals in the region of uniform plastic deformation can be expressed by the simple power curve relation

(10)

where n is the strain-hardening exponent and K is the strength coefficient. A log-log plot of true stress and true strain up to maximum load will result in a straight-line if Eq. (10) is satisfied by the data (Fig. 1). The linear slope of this line is n and K is the true stress at e = 1.0 (corresponds to q = 0.63). The strain-hardening exponent may have values from n = 0 (perfectly plastic solid) to n = 1 (elastic solid) (see Fig. 2). For most metals n has values between 0.10 and 0.50 (see Table 1.).
It is important to note that the rate of strain hardening ds /de, is not identical with the strain-hardening exponent.
From the definition of n

or

(11)

Figure 2. Log/log plot of true stress-strain curve

Figure 3. Various forms of power curve s=K* e ⁿ

Table 1. Values for n and K for metals at room temperature

Metal	Condition	n	K, psi
0,05% C steel	Annealed	0,26	77000
SAE 4340 steel	Annealed	0,15	93000
0,60% C steel	Quenched and tempered 1000oF	0,10	228000
0,60% C steel	Quenched and tempered 1300oF	0,19	178000
Copper	Annealed	0,54	46400
70/30 brass	Annealed	0,49	130000

There is nothing basic about Eq. (10) and deviations from this relationship frequently are observed, often at low strains (10^-3) or high strains (e»1,0). One common type of deviation is for a log-log plot of Eq. (10) to result in two straight lines with different slopes. Sometimes data which do not plot according to Eq. (10) will yield a straight line according to the relationship

(12)

Datsko has shown how e₀, can be considered to be the amount of strain hardening that the material received prior to the tension test.
Another common variation on Eq. (10) is the Ludwig equation

(13)

where s₀ is the yield stress and K and n are the same constants as in Eq. (10). This equation may be more satisfying than Eq. (10) since the latter implies that at zero true strain the stress is zero. Morrison has shown that s₀ can be obtained from the intercept of the strain-hardening point of the stress-strain curve and the elastic modulus line by

The true-stress-true-strain curve of metals such as austenitic stainless steel, which deviate markedly from Eq. (10) at low strains, can be expressed by

where e^K is approximately equal to the proportional limit and n1 is the slope of the deviation of stress from Eq. (10) plotted against e. Still other expressions for the flow curve have been discussed in the literature.
The true strain term in Eqs.(10) to (13) properly should be the plastic strain

e_p= e_total- e_E= e_total- s/E

Studies of the basic structural changes that occur when a metal is subjected to cyclic stress have found it convenient to divide the fatigue process into the following stages:

• Crack initiation includes the early development of fatigue damage which can be removed by a suitable thermal anneal.
• Slip-band crack growth involves the deepening of the initial crack on planes of high shear stress. This frequently is called stage I crack growth.
• Crack growth on planes of high tensile stress involves growth of well-defined crack in direction normal to maximum tensile stress. Usually called stage II crack growth.
• Ultimate ductile failure occurs when the crack reaches sufficient length so that the remaining cross section cannot support the applied load.

The relative proportion of the total cycles to failure that are involved with each stage depends on the test conditions and the material. However, it is well established that a fatigue crack can be formed before 10 percent of the total life of the specimen has elapsed. There is, of course, considerable ambiguity in deciding when a deepened slip band should be called a crack.

In general, larger proportions of the total cycles to failure are involved with the propagation of stage II cracks in low-cycle fatigue than in long-life fatigue, while stage I crack growth comprises the largest segment for low-stress, high-cycle fatigue. If the tensile stress is high, as in the fatigue of sharply notched specimens, stage I crack growth may not be observed at all.

An overpowering structural consideration in fatigue is the fact that fatigue cracks usually are initiated at a free surface. In those rare instances where fatigue cracks initiate in the interior there is always an interface involved, such as the interface of a carburized surface layer and the base metal.

Fatigue has certain things in common with plastic flow and fracture under static or unidirectional deformation. The work of Gough has shown that a metal deforms under cyclic strain by slip on the same atomic planes and in the same crystallographic directions as in unidirectional strain. Whereas with unidirectional deformation slip is usually widespread throughout all the grains, in fatigue some grains will show slip lines while other grains will give no evidence of slip.

Slip lines are generally formed during the first few thousand cycles of stress. Successive cycles produce additional slip bands, but the number of slip bands is not directly proportional to the number of cycles of stress. In many metals the increase in visible slip soon reaches a saturation value, which is observed as distorted regions of heavy slip. Cracks are usually found to occur in the regions of heavy deformation parallel to what was originally a slip band. Slip bands have been observed at stresses below the fatigue limit of ferrous materials. Therefore, the occurrence of slip during fatigue does not in itself mean that a crack will form.

A study of crack formation in fatigue can be facilitated by interrupting the fatigue test to remove the deformed surface by electro polishing. There will generally be several slip bands which are more persistent than the rest and which will remain visible when the other slip lines have been polished away. Such slip bands have been observed after only 5 percent of the total life of the specimen. These persistent slip bands are embryonic fatigue cracks, since they open into wide cracks on the application of small tensile strains. Once formed, fatigue cracks tend to propagate initially along slip planes, although they later take a direction normal to the maximum applied tensile stress. Fatigue-crack propagation is ordinarily transgranular.

An important structural feature, which appears to be unique to fatigue deformation, is the formation on the surface of ridges and grooves called slip-band extrusions and slip-band intrusions. Extremely careful metallography on tapered sections through the surface of the specimen has shown that fatigue cracks initiate at intrusions and extrusions.

W. A. Wood, who made many basic contributions to the understanding of the mechanism of fatigue, suggested a mechanism for producing slip-band extrusions and intrusions. He interpreted microscopic observations of slip produced by fatigue as indicating that the slip bands are the result of a systematic buildup of fine slip movements, corresponding to movements of the order of 10^-7cm rather than steps of 10^-5 to 10^-4 cm, which are observed for static slip hands.

Such a mechanism is believed to allow for the accommodation of the large total strain (summation of the micro strain in each cycle) without causing appreciable strain hardening. The notch would be a stress raiser with a notch root of atomic dimensions. Such a situation might well be the start of a fatigue crack. This mechanism for the initiation of a fatigue crack is in agreement with the facts that fatigue cracks start at surfaces and that cracks have been found lo initiate at slip-band intrusions and extrusions.

Extensive structural studies of dislocation arrangements in persistent slip bands have brought much basic understanding to the fatigue fracture process.

The stage I crack propagates initially along the persistent slip bands. In a polycrystalline metal the crack may extend for only a few grain diameters before the crack propagation changes to stage II. The rate of crack propagation in stage I is generally very low, on the order of angstroms per cycle, compared with crack propagation rates of microns per cycle for stage II. The fracture surface of stage I fractures is practically featureless.

Failure to observe striations on a fatigue surface may be due to a very small spacing that cannot be resolved with the observational method used, insufficient ductility at the crack tip to produce a ripple by plastic deformation that is large enough to be observed or obliteration of the striations by some sort of damage lo the surface. Since stage II cracking does not occur for the entire fatigue life, it does not follow that counting striations will give the complete history of cycles to failure.

Stage II crack propagation occurs by a plastic blunting process that is illustrated in Fig. 1.

Fig.1 Plastic blunting process for growth of stage II fatigue crack

At the start of the loading cycle the crack tip is sharp (Fig. 1 a). As the tensile load is applied the small double notch at the crack tip concentrates the slip along planes at 45° to the plane of the crack (Fig. 1b). As the crack widens to its maximum extension (Fig. 1c) it grows longer by plastic shearing and at the same time its tip becomes blunter. When the load is changed lo compression the slip direction in the end zones is reversed. The crack faces are crushed together and the new crack surface created in tension is forced into the plane of the crack

The major share of the fatigue life of the component may be taken up in the propagation of crack. By applying fracture mechanics principles it is possible to predict the number of cycles spent in growing a crack to some specified length or to final failure.

The aircraft industry has leaded the effort to understand and predict fatigue crack growth. They have developed the safe-life or fail-safe design approach. In this method, a component is designed in a way that if a crack forms, it will not grow to a critical size between specified inspection intervals. Thus, by knowing the material growth rate characteristics and with regular inspections, a cracked component may be kept in service for an extended useful life. This concept is shown schematically in Fig. 1.

Figure 1. Extended service life of a cracked component

Fatigue Crack Growth Curves

Typical constant amplitude crack propagation data are shown in Fig. 2. The crack length, a, is plotted versus the corresponding number of cycles, N, at which the crack was measured. Figure 2. Constant amplitude crack growth data

As shown, most of the life of the component is spent while the crack length is relatively small. In addition, the crack growth rate increases with increased applied stress.

The crack growth rate, da/dN, is obtained by taking the derivative of the above crack length, a, versus cycles, N, curve. Two generally accepted numerical approaches for obtaining this derivative are the spline fitting method and the incremental polynomial method. These methods are explained in detail in many numerical methods textbooks. Values of log da/dN can then be plotted versus log DK, for a given crack length, using the equation

(1)

where Ds is the remote stress applied to the component as shown in Fig. 3.

Figure 3. Remote stress range

A plot of log da/dN versus log DK, a sigmoidal curve, is shown in Fig. 4. This curve may be divided into three regions. At low stress intensities, Region I, cracking behavior is associated with threshold, DK_th, effects. In the mid-region, Region II, the curve is essentially linear. Many structures operate in this region. Finally, in the Region III, at high DK values, crack growth rates are extremely high and little fatigue life is involved.

Figure 4. Three regions of crack growth rate curve

Region II

Most of the current applications of LEFM concepts to describe crack growth behavior are associated with Region II. In this region the slope of the log da/dN versus log DK curve is approximately linear and lies roughly between 10^-6 and 10^-3 in/cycle. Many curve fits to this region have been suggested. The Paris equation, which was proposed in the early 1960s, is the most widely accepted. In this equation

(2)

where C and m are material constants and DK is the stress intensity range Kmax - Kmin.

Values of the exponent, m, are usually between 3 and 4. These range from 2,3 to 6,7 with a sample average of m = 3,5. In addition, tests may be performed. ASTM E647 sets guidelines for these tests.

The crack growth life, in terms of cycles to failure, may be calculated using Eq. (2). The relation may be generally described by

Thus, cycles to failure, Nf, may be calculated as

(3)

where a_i is the initial crack length and af is the final (critical) crack length. Using the Paris formulation,

	(4)

Because DK is a function of the crack length and a correction factor that is dependent on crack length [see Eq. (1)], the integration above must often be solved numerically. As a first approximation, the correction factor can be calculated at the initial crack length and Eq. (4) can be evaluated in closed form.

As an example of closed form integration, fatigue life calculations for a small edge-crack in a large plate are performed below. In this case the correction factor, f(g) does not vary with crack length. The stress intensity factor range is

(5)

Substituting into the Paris equation yields

(6)

Separating variables and integrating (for m<>2) gives

(7)

Before this equation may be solved, the final crack size, af, must be evaluated. This may be done using as follows:

(8)

For more complicated formulations of DK, where the correction factor varies with the crack length, a, iterative procedures may be required to solve for a_f in Eq. (8).

It is important to note that the fatigue-life estimation is strongly dependent on a_i, and generally not sensitive to af (when a_i«a_f). Large changes in a_f result in small changes of N_f as shown schematically in Fig. 5.

Figure 5. Effect of final crack size on life