30079_18b.pdf

Fig. 18.12 Surface flaw shape parameter. (From Ref. 22. Adapted by permission of

Prentice-Hall, Inc., Englewood Cliffs, New Jersey.)

To approximate the effects of strain hardening, a flow stress cr 0 , taken to be an average of the

yield and ultimate strengths, is often used when computing the plastic collapse stress. The plastic

collapse stress a c is that applied stress which produces cr 0 across the remaining uncracked ligament,

and is the maximum applied stress that a perfectly plastic material can sustain. This stress may be

determined using a limit load analysis. In general, the plastic collapse stress is a function of geometry,

type of loading, type of support (boundary conditions), and through-thickness constraint (plane stress

or plane strain). 6 ' 2 5 For a single through-thickness crack of length a in a strip with width b loaded

in tension (see Fig. 18.9), if end rotations are restrained, the plastic collapse stress under plane stress

conditions may be approximated by 2 5

a c = a 0 (\ - alb}

(18.39)

18.5 FATIGUE AND STRESS CONCENTRATION

Static or quasistatic loading is rarely observed in modern engineering practice, making it essential

for the designer to address himself or herself to the implications of repeated loads, fluctuating loads,

and rapidly applied loads. By far, the majority of engineering design projects involve machine parts

Fig. 18.13 Failure assessment diagram.

subjected to fluctuating or cyclic loads. Such loading induces fluctuating or cyclic stresses that often

result in failure by fatigue.

Fatigue failure investigations over the years have led to the observation that the fatigue process

actually embraces two domains of cyclic stressing or straining that are significantly different in

character, and in each of which failure is probably produced by different physical mechanisms. One

domain of cyclic loading is that for which significant plastic strain occurs during each cycle. This

domain is associated with high loads and short lives, or low numbers of cycles to produce fatigue

failure, and is commonly referred to as low-cycle fatigue. The other domain of cyclic loading is that

for which the strain cycles are largely confined to the elastic range. This domain is associated with

lower loads and long lives, or high numbers of cycles to produce fatigue failure, and is commonly

referred to as high-cycle fatigue. Low-cycle fatigue is typically associated with cycle lives from 1 up

to about 10 4 or 10 5 cycles. Fatigue may be characterized as a progressive failure phenomenon that

proceeds by the initiation and propagation of cracks to an unstable size. Although there is not

complete agreement on the microscopic details of the initiation and propagation of the cracks, pro-

cesses of reversed slip and dislocation interaction appear to produce fatigue nuclei from which cracks

may grow. Finally, the crack length reaches a critical dimension and one additional cycle then causes

complete failure. The final failure region will typically show evidence of plastic deformation produced

just prior to final separation. For ductile materials the final fracture area often appears as a shear lip

produced by crack propagation along the planes of maximum shear.

Although designers find these basic observations of great interest, they must be even more inter-

ested in the macroscopic phenomenological aspects of fatigue failure and in avoiding fatigue failure

during the design life. Some of the macroscopic effects and basic data requiring consideration in

designing under fatigue loading include:

1. The effects of a simple, completely reversed alternating stress on the strength and properties

of engineering materials.

2. The effects of a steady stress with superposed alternating component, that is, the effects of

cyclic stresses with a nonzero mean.

3. The effects of alternating stresses in a multiaxial state of stress.

4. The effects of stress gradients and residual stresses, such as imposed by shot peening or

cold rolling, for example.

5. The effects of stress raisers, such as notches, fillets, holes, threads, riveted joints, and welds.

6. The effects of surface finish, including the effects of machining, cladding, electroplating,

and coating.

7. The effects of temperature on fatigue behavior of engineering materials.

8. The effects of size of the structural element.

9. The effects of accumulating cycles at various stress levels and the permanence of the effect.

10. The extent of the variation in fatigue properties to be expected for a given material.

11. The effects of humidity, corrosive media, and other environmental factors.

12. The effects of interaction between fatigue and other modes of failure, such as creep, cor-

rosion, and fretting.

18.5.1 Fatigue Loading and Laboratory Testing

Faced with the design of a fatigue-sensitive element in a machine or structure, a designer is very

interested in the fatigue response of engineering materials to various loadings that might occur

throughout the design life of the machine under consideration. That is, the designer is interested in

the effects of various loading spectra and associated stress spectra, which will in general be a function

of the design configuration and the operational use of the machine.

Perhaps the simplest fatigue stress spectrum to which an element may be subjected is a zero-

mean sinusoidal stress-time pattern of constant amplitude and fixed frequency, applied for a specified

number of cycles. Such a stress-time pattern, often referred to as a completely reversed cyclic stress,

is illustrated in Fig. 18.14«. Utilizing the sketch of Fig. 18.14, we can conveniently define several

useful terms and symbols; these include:

cr ma x = maximum stress in the cycle

cr m = mean stress = (o- ma x + cr min )/2

cr mi n = minimum stress in the cycle

a- a = alternating stress amplitude = (cr ma x - cr min )/2

Ao- = range of stress - o- ma x - <7 mi n

R = stress ratio = a- min /cr ma x

A = amplitude ratio = <r a l<r m = (1 - R)/(I + R)

Fig. 18.14 Several constant-amplitude stress-time patterns of interest: (a) completely reversed,

R= -1; Ob) nonzero mean stress; (c) released tension, R = O.

Any two of the quantities just defined, except the combinations cr a and ACT or the combination A and

R, are sufficient to describe completely the stress-time pattern above.

More complicated stress-time patterns are produced when the mean stress, or stress amplitude,

or both mean and stress amplitude change during the operational cycle, as illustrated in Fig. 18.15.

It may be noted that this stress-time spectrum is beginning to approach a degree of realism. Finally,

in Fig. 18.16 a sketch of a realistic stress spectrum is given. This type of quasirandom stress-time

pattern might be encountered in an airframe structural member during a typical mission including

refueling, taxi, takeoff, gusts, maneuvers, and landing. The obtaining of useful, realistic data is a

challenging task in itself. Instrumentation of existing machines, such as operational aircraft, provide

some useful information to the designer if his or her mission is similar to the one performed by the

instrumented machine. Recorded data from accelerometers, strain gauges, and other transducers may

in any event provide a basis from which a statistical representation can be developed and extrapolated

to future needs if the fatigue processes are understood.

Basic data for evaluating the response of materials, parts, or structures are obtained from carefully

controlled laboratory tests. Various types of testing machines and systems commonly used include:

1. Rotating-bending machines:

a. Constant bending moment type

b. Cantilever bending type

2. Reciprocating-bending machines.

Fig. 18.15 Stress-time pattern In which both mean and amplitude change to produce a more

complicated stress spectrum.

3. Axial direct-stress machines:

a. Brute-force type

b. Resonant type

4. Vibrating shaker machines:

a. Mechanical type

b. Electromagnetic type

5. Repeated torsion machines.

6. Multiaxial stress machines.

Fig. 18.16 A quasirandom stress-time pattern that might be typical of an operational aircraft

during any given mission.

7. Computer-controlled closed-loop machines.

8. Component testing machines for special applications.

9. Full-scale or prototype fatigue testing systems.

Computer-controlled fatigue testing machines are widely used in all modern fatigue testing lab-

oratories. Usually such machines take the form of precisely controlled hydraulic systems with feed-

back to electronic controlling devices capable of producing and controlling virtually any strain-time,

load-time, or displacement-time pattern desired. A schematic diagram of such a system is shown in

Fig. 18.17.

Special testing machines for component testing and full-scale prototype testing systems are not

found in the general fatigue testing laboratory. These systems are built up especially to suit a particular

need, for example, to perform a full-scale fatigue test of a commercial jet aircraft.

It may be observed that fatigue testing machines range from very simple to very complex.

The very complex testing systems, used, for example, to test a full-scale prototype, produce very

specialized data applicable only to the particular prototype and test conditions used; thus, for the

particular prototype and test conditions the results are very accurate, but extrapolation to other test

Fig. 18.17 Schematic diagram of a computer-controlled closed-loop fatigue testing machine.

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