30079_18d.pdf

hypothesis may be written as follows: If a design limit of creep strain 8 D is specified, it is predicted

that the creep strain 8 D will be reached when

ST-=!

(!8-78)

i=l L 1

where t t = time of exposure at the rth combination of stress level and temperature

L 1 = time required to produce creep strain 8 D if entire exposure were held constant at the /th

combination of stress level and temperature

Stress rupture may also be predicted by (18.78) if the L 1 values correspond to stress rupture. This

prediction technique gives relatively accurate results if the creep deformation is dominated by stage

II steady-state creep behavior. Under other circumstances the method may yield predictions that are

seriously in error.

Other cumulative creep prediction techniques that have been proposed include the time-hardening

rule, the strain-hardening rule, and the life-fraction rule. The time-hardening rule is based on the

assumption that the major factor governing the creep rate is the length of exposure at a given tem-

perature and stress level, no matter what the past history of exposure has been. The strain-hardening

rule is based on the assumption that the major factor governing the creep rate is the amount of prior

strain, no matter what the past history of exposure has been. The life-fraction rule is a compromise

between the time-hardening rule and the strain-hardening rule which accounts for influence of both

time history and strain history. The life-fraction rule is probably the most accurate of these prediction

techniques.

18.7 COMBINED CREEP AND FATIGUE

There are several important high-performance applications of current interest in which conditions

persist that lead to combined creep and fatigue. For example, aircraft gas turbines and nuclear power

reactors are subjected to this combination of failure modes. To make matters worse, the duty cycle

in these applications might include a sequence of events including fluctuating stress levels at constant

temperature, fluctuating temperature levels at constant stress, and periods during which both stress

and temperature are simultaneously fluctuating. Furthermore, there is evidence to indicate that the

fatigue and creep processes interact to produce a synergistic response.

It has been observed that interrupted stressing may accelerate, retard, or leave unaffected the time

under stress required to produce stress rupture. The same observation has also been made with respect

to creep rate. Temperature cycling at constant stress level may also produce a variety of responses,

depending on material properties and the details of the temperature cycle.

No general law has been found by which cumulative creep and stress rupture response under

temperature cycling at constant stress or stress cycling at constant temperature in the creep range can

be accurately predicted. However, some recent progress has been made in developing life prediction

techniques for combined creep and fatigue. For example, a procedure sometimes used to predict

failure under combined creep and fatigue conditions for isothermal cyclic stressing is to assume that

the creep behavior is controlled by the mean stress cr m and that the fatigue behavior is controlled by

the stress amplitude cr a , with the two processes combining linearly to produce failure. This approach

is similar to the development of the Goodman diagram described in Section 18.5.4 except that instead

of an intercept of cr u on the cr m axis, as shown in Fig. 18.38, the intercept used is the creep-limited

static stress o~ cr , as shown in Fig. 18.64. The creep-limited static stress corresponds either to the

design limit on creep strain at the design life or to creep rupture at the design life, depending on

which failure mode governs. The linear prediction rule then may be stated as

Failure is predicted to occur under combined isothermal creep and fatigue if

&„ <r m

— + — > 1

(18.79)

(T N

0- cr

An elliptic relationship is also shown in Fig. 18.64, which may be written as

Failure is predicted to occur under combined isothermal creep and fatigue if

/<r a \ 2 /o- m y

M + M ^ 1

(1880 )

\(T N / \cr c j

The linear rule is usually (but not always) conservative. In the higher-temperature portion of the

creep range the elliptic relationship usually gives better agreement with data. For example, in Fig.

18.65fl actual data for combined isothermal creep and fatigue tests are shown for several different

Fig. 18.64 Failure prediction diagram for combined creep and fatigue under

constant-temperature conditions.

temperatures using a cobalt-base S-816 alloy. The elliptic approximation is clearly better at higher

temperatures for this alloy. Similar data are shown in Fig. 18.65& for 2024 aluminum alloy. Detailed

studies of the relationships among creep strain, strain at rupture, mean stress, and alternating stress

amplitude over a range of stresses and constant temperatures involve extensive, complex testing

programs. The results of one study of this type 8 2 are shown in Fig. 18.66 for S-816 alloy at two

different temperatures.

Several other empirical methods have recently been proposed for the purpose of making life

predictions under more general conditions of combined creep and low-cycle fatigue. These methods

include:

1. Frequency-modified stress and strain-range method. 8 3

2. Total time to fracture versus time-of-one-cycle method. 8 4

3. Total time to fracture versus number of cycles to fracture method. 8 5

4. Summation of damage fractions using interspersed fatigue with creep method. 8 6

5. Strain-range partitioning method. 8 7

The frequency-modified strain-range approach of Coffin was developed by including frequency-

dependent terms in the basic Manson-Coffin-Morrow equation, cited earlier as (18.54). The resulting

equation can be expressed as

Ae - AN a f v b + BN c f v d

(18.81)

where the first term on the right-hand side of the equation represents the elastic component of strain

range, and the second term represents the plastic component. The constants A and B are the intercepts,

respectively, of the elastic and plastic strain components at N f = 1 cycle and v — \ cycle/min. The

exponents a, b, c, and d are constants for a particular material at a given temperature. When the

constants are experimentally evaluated, this expression provides a relationship between total strain

range Ae and cycles to failure N f .

The total time to fracture versus time-of-one-cycle method is based on the expression

t f = — = CrJ

(18.82)

Fig. 18.65 Combined isothermal creep and fatigue data plotted on coordinates suggested in

Figure 18.64. (a) Data for S-816 alloy for 100-hr life, where cr N is fatigue strength for 100-hr life

and (T cr is creep rupture stress for 100-hr life. (From Refs. 80 and 81.) (b) Data for 2024 alumi-

num alloy, where o- N is fatigue strength for life indicated on curves and o- cr is creep stress for

corresponding time to rupture. (From Refs. 80 and 82.)

Fig. 18.66 Strain at fracture for various combinations of mean and alternating stresses in unnotched specimens of S-816 alloy, (a) Data taken at 816 0 C.

(b) Data taken at 90O 0 C. (From Refs. 80 and 81.)

where t f is the total time to fracture in minutes, v is frequency expressed in cycles per minute, N f is

total cycles to failure, t c — 1 / v is the time for one cycle in minutes, and C and k are constants for

a particular material at a particular temperature for a particular total strain range.

The total time to fracture versus number-of-cycles method characterizes the fatigue-creep inter-

action as

t f = DNj m (18.83)

which is identical to (18.82) if D = C ll(l ~ k} and m = k/(l - K). However, it has been postulated

that there are three different sets of constants D and m: one set for continuous cycling at varying

strain rates, a second set for cyclic relaxation, and a third set for cyclic creep.

The interspersed fatigue and creep analysis proposed by the Metal Properties Council involves

the use of a specified combined test cycle on unnotched bars. The test cycle consists of a specified

period at constant tensile load followed by various numbers of fully reversed strain-controlled fatigue

cycles. The specified test cycle is repeated until failure occurs. For example, in one investigation the

specified combined test cycle consisted of 23 hr at constant tensile load followed by either 1.5, 2.5,

5.5, or 22.5 fully reversed strain-controlled fatigue cycles. The failure data are then plotted as fatigue

damage fraction versus creep damage fraction, as illustrated in Fig. 18.67.

The fatigue damage fraction is the ratio of total number of fatigue cycles N' f included in the

combined test cycle divided by the number of fatigue cycles N f to cause failure if no creep time

were interspersed. The creep damage fraction is the ratio of total creep time t cr included in the

combined test cycle divided by the total creep life to failure t f if no fatigue cycles were interspersed.

A "best-fit" curve through the data provides the basis for making a graphical estimate of life under

combined creep and fatigue conditions, as shown in Fig. 18.67.

The strain-range partitioning method is based on the concept that any cycle of completely reversed

inelastic strain may be partitioned into the following strain-range components: completely reversed

plasticity, Ae^; tensile plasticity reversed by compressive creep, Ae pc ; tensile creep reversed by

compressive plasticity, Ae cp ; and completely reversed creep, Ae cc . The first letter of each subscript

Fig. 18.67 Plot of fatigue damage fraction versus creep damage fraction for 1 Cr-1 Mo- 1 A V

rotor steel at 100O 0 F in air, using the method of the Metal Properties Council. (After Ref. 88,

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