PREDICTION OF ENDURANCE ON THE BASIS OF CHARACTERISTIC

LOADING PARAMETERS

I. G. Zavalich and L. A. Shefer UDC 539.4

A considerable amount of experimental data has now been accumulated on the endurance of materials under different types of endurance for random loads is calculated from test results for harmonic loading using various hypotheses of fatigue damage summation which involve schematization of the loading process. Such an approach can lead to substantial errors [I].

The present article proposes a method which permits the conversion of fatigue character- istics from one process to another without schematization and use of the fatigue damage sum- mation hypotheses. The method is based on the equivalence of random and programmed loading processes [2].

Analysis of results of tests of random narrow-band and programmed loading of specimens of different types of materials shows that, given equal RMS stress deviations S, endurance is significantly lower in the first case. It was shown in [2] that this difference is due to the presence of relatively low-amplitude stresses in the program blocks. Due to the con- siderable duration of the continuous loading, these low stresses allow the specimen material to recover, thus increasing endurance. It turned out that eliminating stresses below the leve'l Omi n = (0.8-1.2)S from the blocks led to a situation whereby the endurance under random and programmed loading were equal.

Tests were conducted in cantilever bending on flat specimens of materials differing in both chemical composition and strength properties (aluminum alloy AMg6, heat'resistant steel D152, glass-fiber-reinforced plastic PPN), making it possible to determine Omi n as a quantity independent of the test material.

Based on the results, the loading process can be tentatively divided into two parts: damaging and nondamaging. The boundary is the mean value of Omin from the above range, i.e., the rms stress level Omin = S. Here, whereas maxima lying below the rms level in programmed loading allow the material to recover due to the duration of the loading in the program block, this effect is absent in random loading because the probability of long continous load- ing with nondamaging stress maxima is nearly zero. Thus, the nondamaging part of the ran- dom process is regarded as a component part of the loading, not causing fatigue damage, but not preventing such damage either. Thus, attention should be focused on the damaging part of the process and its description.

The following quantities can be used as parameters characterizing the damaging part of the loading process: the rms of the damaging maxima; the number of such maximums; the maxi- mum spread of the process.

Let us examine the dimensionless parameter G, which encompasses these parameters:

S n s p G = m s ~max __ Sms ns

S m n S S m n

where Sms is the rms of the damaging maxima; Sm, rms of the positive maximum; ns, mean number of damaging maxima per unit of time; n, mean number of positive maxima per unit of time; Omax, maximum spread of the process; and P = Omax/S , peak factor.

With a known maximum probability density fm(o), Eq. (I) can be represented as fol- lows:

( i )

Chelyabinsk Polytechnic Institute. Translated from Problemy Prochnosti, No. 10, pp. 25-30, October, 1982. Original article submitted October 28, ]980.

1314 0039-2316/82/1410-1314507.50 @ 1983 Plenum Publishing Corporation

80

;0 2

2

k.5 5.0 5,5 a

S, MPa

100! 801

60 . c

40 8o .a.

20

&O 5,5 t~ No

I

I G h 2 3 ~ G b

Fig. 1. Fatigue characteristics of alloy AMg6: 1) harmonic load- ing (o); 2-5) random loading

[ m) i = 0.28, Gi = 1.97; • i = 0.39, G i = 2.06; o) i = 0.6, Gi = 2.23; A) i = I, G i = 2.53].

i~ ~max UNax ml] I12 ~Vm (oi eo I G (~ do I

G = ~ ~ P.. ( 2 ) ~ ~

For harmonic loading, the parameter Gh, in accordance with Eq. (I), is equal to 4.

Steady-state loading processes are characterized by a structure (irregularity) parameter i. This parameter is defined as the ratio of the mean number of crossings of the level of the expectation value per unit of time to the mean number of extrema per unit of time. The structure parameter of random processes i varies from unity (narrow-band process) to zero. Meanwhile, given a constant peak factor, a reduction in i leads to a reduction in the char- acteristic parameter G determined from Eq. (2). Calculations show that for Gaussian steady- state random process with maximum probability densities which can be represented by Rice's

relation [3], the upper interpolation limit in Eq. (2) can be taken as Omax = ~ when the peak factor P ~ 3.2. The error of the calculation of G here is no greater than 0.4%.

Let us examine the fatigue resistance characteristics under harmonic and random centered loadings, using flat specimens of AMg6 [4] as an example. The tests were conducted in canti- lever bending until a crack appeared (Fig. la). The measure of damage chosen was 0.5% change in the natural frequency of the specimens, which corresponds to the appearance of a crack with an area equal to ]-1.5% of the initial area of the specimen working section. Endurance in this case was taken as the number of intersections of the expectation value, with the sign of the derivative assigned, until the chosen damage criterion No was satisfied (the frequency of the harmonic signal and the effective frequencies of the random processes were the same and were equal to 30 Hz).

It is apparent from Fig. ] that, given equal rms's for the stresses, the lowest endurance corresponds to random loading with a Rayleigh distribution of the maxima (i = i), while the greatest endurance corresponds to harmonic loading. A reduction in the structure parameter (i < 1) leads to an increase in endurance compared to that seen with narrow-band random load- ing.

Analysis of the fatigue resistance characteristics under harmonic and random loadings in the plane S--G, where each process corresponds to a specific value of the characteristic parameter Gi, allowed us to obtain a bundle of rays which converged at one point G0, So (Fig. Ib). Each ray in the figure characterizes a certain endurance. Meanwhile, the less the slope of the ray with respect to the horizontal, the greater the endurance to which it corresponds.

If we assume that the above law remains in force at endurances in excess of the empirical data (No > 107), then there may be a ray that occupies a certain limiting position correspond- ing to infinite endurance. In the general case, such a ray may differ from the horizontal So

1315

Sh-$o ~i - So

1,5,

b~

1 0

/

u - - 3 - - �9 --~. {

0,2 O L+ 0,5 0,8 i

2 I ! l

200 ~00 500 [~00 tOCO ~U, MPa

Fig. 2 Fig. 3

So~ ~kiPa

720

80

Fig. 2. Ratio of stress rms under harmonic and random loadings for different materials: I) AMg6; 2) ~Mg61; 3) 2024; 4) 01420.

Fig. 3. Dependence of fatigue-characteristic parameters on the strength properties of materials: I) B = f1(o u) -- according to the appearance of a crack; 2) B = f2(o u) -- according to fracture; 3) S0 = f3(Ou).

(ray C in Fig. lb), thereby establishing different values of the stress rms Soi in re!aLion to the structure of the loading process. However, it is not presently possible to experi- mentally establish this limiting straight line. Thus, for this line we will take the ray in the horizontal position, which defines So as the maximum possible stress rms, independent of the structure of the loading process, at which the endurance No approaches infinity. Such an assumption epters into the safety factor, since the adopted value So is the minimum pos- sible value.

The following relation may be written on the basis of the data in Fig. Ib for two load- ing processes, differing in structure, with the characteristic parameters G i and Gj at a con- stant endurance (No = const):

Sy--So Go--G j Si--%o =' Oo--O i (3)

Since, as confirmed by the results in [5], the ratio of endurances for two loading pro- cesses of different structures is constant given constant stress rms's, then, allowing for Eq. (3), we should represent the fatigue curves for these processes with parallel straight lines in the plane log (S -- S0)--logN0:

lg Noj = Aj - - B lg (S~ - - So); ( 4 )

lg N0~ = A~ - - B lg (Si - - S ~ .

Simultaneous solution of Eqs. (3) and (4) under the condition that one of the loading processes is harmonic, i.e., that Aj = A h and Gj = Gh, allows us to obtain a generalized fatigue-curve equation for any of the loading processes examined:

lgNo~ = A h - - B l g l Oo-- c~ (S~ -- So)]. (5) Go 6

E x p r e s s i o n (5 ) c o i n c i d e s i n f o r m w i t h t h e f a m i l i a r f o r m of t h e W e i b u l l e q u a t i o n f o r fatigue curves in the case of harmonic loading. It was shown in [6] that representation of the results of fatigue tests in the form (5) best corresponds to the empirical data for light alloys. In the form proposed, Eq. (5) is based on statistical parameters of the loading pro- cess and makes it possible to determine endurance characteristics in the case of different random loads with allowance for the criterion Gi, which reflects the structure of the loading process.

The relation obtained was used to approximate the results of tests, in cantilever bend- ing, of the following different materials: light alloys MATS, ~Mg6, IMV2, 01420, AMg61, and 2024T3; heat-resistant steel EP678; and a composite material -- glass-fiber-reinforced plastic

1316

TABLE ]. Numerical Values of the Parameters of Eq. (5) for Different Materials

Material, au, MPa

IMV2; 240 MAI5; 280

AMg6; ~u

AMg61; 35O

Or#anic- fiber- re~nforeed plastics; 440

2024T3; 450

01420; 500

fp678 1300 N o t e .

S o n roe Type of specimen

Our Data , Without concentrator

[4] Without concentrator With hoIe (O=1 turn)

[4] Without concentrator

>> })

>> )>

Our data Without concentrator

Same Without concentrator

[7J Circular with notch Without concentrator

[4]

Our data

G~"

)> ))

With hole (~ =1 ram) With hole ( O =2 ram)

Without concentrator

Crack appearance

Bcr S~ MPa 6~

1 , 0 2 , 5 3 5,0

1,0 2 : ~ ] 4,8 1,0 4,6

1,0 2,53 3,9 0,6 2,23 4,2 0,3! 2,06 4,0 0,2~ 1,97 4,3 1,0 2,53 * 0,8'. 2,43 [ -~ 1,0 2,15 ] 3.5

I 0,91 2,47 I * 1,0 2,63 3,3 0,8~ 2,39 3,I

23 13,92 4,7

15 4,06 4,6 19 3,88 4,2

29 I 3,92 * * 27 4,11 * * 30 3 86 [ . -~ 26 4,13 * . ~ 3,6 -~ 3,8

-~13 .! 4~1 t 3.2~'*

44 4,05 ] 3,1 46 3,87 [ 2,9 43 4,08 / 3,1 41 4,19] 2,9 44 3,92 I 3,2

/

45 i3,87 2,8 i

3,2 3,4 3,2

0,7, 2,33 0,61 2,27 1,0 2,53

1,0 2,53

1,0 2,53

3,0

3,0 142 ! 3,97 I * I

Fracture

Gof

4,07

4,16 3,83

:# ,l~

3,89 4,10

4,08 4,13 3,90 4,04 3,87 4,30

3,86

x- 4r

Mean for the material

B c r Bf &f' MPa

25

16 18

.a

%-

5 4 4 4 4 4

4

3,2

t

3,0

Bf So, MPa

4,7 I 24

4,4 17

- - 28

3,7 34

- - 13

3,2 } 56

3 , 0 44

- - 142

** denotes specimens of the glven material tested only to cracking; * denotes specimens tested to fracture without intermediate checking of the natural frequency.

The tensile-strength range of the materials was 240-1300 MPa, while the empirical endurances No ranged from 104 to 107 .

We analyzed test results from harmonic and steady-state random loadings with different values of the structure parameter (i = 0.28-I). Sometimes the stress spread exceeded the yield point in the random loadings with a high rms, i.e., caused elastoplastic strains. How- ever, due to their abrupt appearance, these strains did not cause the empirical data to de- viate from the proposed approximate relation (5). No special studies have been done yet on the effect of abrupt high-intensity overloads on endurance.

We examined experimental results obtained on both smooth specimens of different configur- ation and on specimens with stress concentrators. Calculations were performed simultaneously for endurance characteristics under harmonic and random loadings obtained on specimens of a given type. The parameters of Eq. (5) were calculated by the least-squares method. Numeri- cal values of Gi were found from known maximum probability densities for the processes ana- lyzed (in the random loadings, Oma x corresponded to a probability P = 0.997).

The minimum of the mean square error of the empirical data approximation was used to establish the parameters Ah, B, So, and Go (Table I). The stress rms ratio for harmonic and

random loading (S h -- S0)/(S i -- So) is shown in Fig. 2 as a function of the structure param- eter i for the different materials. Also shown is relation (5), which satisfactorily agrees with the test data.

Table I shows numerical values of the parameters of Eq. (5), describing the empirical endurances according to both the moment of appearance of a crack and the moment of complete fracture of the working cross section of the specimen. The appearance of a crack was deter- mined from a 0.5% change in the natural vibration frequency of the specimen.

Analysis of the theoretical data shows that parameter B in Eq. (5) depends on the degree of damage to the specimens (the appearance of a crack or complete fracture of the working cross section) but is independent of the structure of the loading and the degree of stress concentration. A constant B was also obtained in [6] for light alloys in the construction of generalized fatigue curves for smooth specimens, notched specimens (s o = I-2.28), and full- size parts under harmonic loading in the case where metallurgical and processing factors were identical. The mean values of B for these specimens corresponding to the fatigue curves (5) in accordance with the appearance of a crack Bcr and fracture Bf show that Bcr > Bf for a given material. On the other hand, an increase in the ultimate strength o u of a material leads to a reduction in B within the limits of the same damage criterion.

1317

s. Mp

20O

150

' < t h

50

cj

u

~5 5,0

jm(~)s

0,,~ 0,4 F h

0,3 .,~

-I I 2 3 t -t 0 1 2 a e

j:~(tls l 0,7)

a(t) 5 0,~

0,~ O,q

I 2 5 L -I i 2 t b d

Fig. 5

5.5 5.0 ~.5 jg N~ F i g . 4

-I

_ Fq

k 02

0,[ [

Fig. 4. Fatigue characteristics of specimens of alloy AMg6: I) harmonic

loading; 2) total effect of two harmonics; 3, 4) total effect of random

and harmonic loadings [*) Sh/S i = 2, Gi = 2.03; x) Sh/S i = 5, G i = 2.03; u) Sh/S i = I, G i = 2.2]; 5) random loading (i = 0.82, G t = 2.43).

Fig. 5. Histograms of probability densities of stress maxima under the combined action of harmonic and random loadings: a) purely random pro-

cess (i = 0.82, G i = 2.43); b) Sh/Si = I; c) Sh/S i = 2; d) Sh/Si = 5.

Figure 3 shows mean values of B in relation to Ou, corresponding to the fatigue curves (5) in accordance with cracking and fracture. The resulting relations are approximated well by the analytic expression

B = ~ . cth (0,0026%), ( 6 )

where ~ = 2.91 (moment of crack appearance) and 2.65 (moment of fracturel.

The numerical values of So found from cracking and fracture are abouE the same for spec- imens of a given material (see Table I), i.e., are independent of specimen design features and damage level. There is a general tendency for So to increase with @u (curve 3 in Fig. 3), al- though So is evidently a multiparameter function of the properties of the material.

The parameter Go, determined in accordance with crack appearance and fracture, is a stable characteristic which is independent of the specimen material, specimen design~ or de- gree of damage to the specimen. Numerical values of Go lie within the range 3.8-4~ for all of the materials. Calculations showed that use of a mean value Go = 4 in Eq. (5) leads Eo an endurance error of no more than 8% within the range of empirical values cited.

To check the feasibility of extending the proposed meEhod to polyharmonic processes, a unit described in [5] was used to test flat specimens of AMg61 to fracture. ?he specimens were acted on by two harmonic processes with close frequencies and equal sEress amplitudes o a :

We took the fundamental loading frequency ~ = 20 Hz; the frequency difference s = 0.5 Hz. The stress rms for such loading was determined as S = o3. The probability density of the stress maxima for this process (7) is written in the form

2 (8 )

1318

The parameter Gi, characterizing the loading process, was determined with Eq. (2) and is equal to ].585 in the present case.

Figure 4 shows experimental values of endurance under the biharmonic loadile~ngin_eJ_n point corresponds to a mean value from the results of tests of six specimens. The resulting test data agrees satisfactorily with the theoretical relation (5) -- curve 2 in Fig. 4.

A topic of considerable practical interest is loading in the form of the sum of random and harmonic processes (e.g., a random external force and a harmonic load from an enginer) Specimens of AMg61 were tested to fracture with the combined action of a steady-state random loading with structure parameter i = 0.81 and effective frequency We = 28 Hz and a harmonic loading with frequency ~ = 8 Hz. The ratio of the rms's of the harmonic and random stresses in the total loading was taken as Sh/S i = 1, 2, and 5. Histograms of the probability densi- ties of the maxima for the above loadings, shown in Fig. 5 as a function of a quantile of the distribution t, were constructed from scans of records of vibrations at least 50 sec long for each realization. Also shown is a histogram of the initial random process (see Fig. 5a), whiqh agrees well with the theoretical distribution [3]. The introduction of harmonics into the random process distorts the maximum probability density. Meanwhile, the greater S h com- pared to Si, the stronger the expressed predominance of the maxima corresponding to the ampli- tude of the harmonic loading (Fig. 5d).

The histograms make it possible to determine the characteristic parameters G i of the loading processes (1) and, with Eq. (5), to determine the corresponding fatigue curves. Fig- ure 4 shows experimental endurances for combined loadings. The data agrees satisfactorily with the theoretical relations (curves 3, 4, 5). Each point corresponds to a mean value for six specimens.

Equation (5) and approximate relation (6) were checked against the results of tests of specimens of the earlier-noted materials in cantilever bending. Although the type of stress state does not determine the given expressions, for more thorough substantiation of the meth- od it would be desirable to experimentally check the relations for other types of stress state.

The following sequence may be established for determining endurance for the class of loading processes examined in bending, based on the results of the present study:

1. Given known strength properties and an assigned degree of fatigue damage, parameter B of generalized fatigue equation (5) is determined from Eq. (6).

2. Fatigue tests under some type of loading (simplest under harmonic) are conducted un- til an assigned degree of fatigue damage is reached in order to obtain a reference fatigue curve for structural elements for which fatigue characteristics are desired.

3. The least squares method is used with Eq. (5) to approximate test data with known B, Go, and Gio This makes it possible to determine the parameters Ah and So for the given elements.

4. Fatigue characteristics for a different type of loading are found by determining the characteristic parameter G i of the loading and by subsequent use of Eq. (5).

I �9

2.

3.

4.

5.

6.

LITERATURE CITED

V. P. Kogaev, Strength Design for Stresses Varying with Time [in Russian], Mashino- stroenie, Moscow (1977). L. A. Shefer, V. G. Ezhov, and I. G. Zavalich, "Study of equivalence between random and programmed loadings," Probl. Prochn., No. 8, 93-96 (1980). B. R. Levin, Theoretical Principles of Statistical Radio Engineering [in Russian], Sovet- skoe Radio, Moscow (]969). N. I. Grinenko, L. A. Shefer, V. G. Ezhov, and I. G. Zavalich, "Prediction of the Fa- tigue properties of materials under random loading," in: Prediction of the Strength of Materials and Structural Elements of Machines with Long Save Lives [in Russian], Naukova Dumka, Kiev (1977), pp. 152-161. L. A. Shefer and V. G. Ezhov, "Effect of the character of the structure of random pro- cesses on endurance," Probl. Prochn., No. 7, 38-42 (1978). M. N. Stepnov and E. V. Giatsintov, Fatigue of Light Structural Alloys [in Russian], Mashinostroenie, Moscow (1973).

1319

7. J. Kovalevskii, "Relations between endurance under pulsating loading with random ampli- tude ordering and corresponding programmed loadings," in: Fatigue, Strength, and Durabil- ity of Aircraft Structures [in Russian], Mashinostroenie, Moscow (1965), pp~ 164-181.

FATIGUE DAMAGE TO A PSEUDO-e-TITANIUM ALLOY IN THE

I0-33-Hz FREQUENCY RANGE

A. Ya. Krasovskii, Yu. N. Petrov, G. N. Nadezhdin, L. E. Matokhnyuk, V. L. Svechnikov, and T. Ya. Yakovleva

UDC 669.295

The frequency of cyclic loading is an extremely important factor affecting the fatigue fracture of metals. Frequency exerts an effect because sign-changing deformation may be ac- companied by several phenomena, especially heating of the material. However, relatively few works have examined the role of frequency in structural changes in the sonic and ultrasonic ranges [I-3]. Due to their low level of energy dissipation, titanium alloys do not undergo appreciable heating even when there is a significant increase in frequency.

The present work uses the results of tests of a pseudo-m-titanium alloy VT18U at fre- quencies of 30 and 300 Hz and I0 kHz to study the evolution of its dislocation structure during fatigue damage accumulation.

The fatigue tests were conducted on smooth specimens with a working diameter of 7 mm.

The test results provide evidence of an increase in endurance and endurance limit with an increase in loading frequency (Fig. I). This effect is connected with the fact that given identical stress amplitudes the level of inelastic strain on which fatigue damage accumula- tion depends will be greater in the case of low-frequency loading. At high frequencies the process of plastic strain cannot be completed to the same degree as at low frequencies and damage to the material is less [4].

The structure of the metal was studied with an electron microscope for specimens tested at different frequencies but the same stress level (450 MPa). Structural changes in alloy VT18U during cyclic loading were studied step-by-step for each test frequency (30 and 300 Hz and 10 kHz) after numbers of cycles corresponding to 0.1Nfrac, 0.5Nfrac, 0.8-0.9Nfrac, and Nfrac , where Nfrac is the life of the specimen at the chosen stress level.

The structure was studied by the method of transmission electron microscopy on foils. The main alloying element in the alloy studied is aluminum. The aluminum may be found either in solid solution in the e phase or in the form of the intermetallide TisAl. It is well known that the strain properties of titanium alloys at room temperature are related to diffusion of dissolved atoms and nonconservative dislocation movement [5, 6]. Hexagonal dislocation net- works may also form as a result of cross slip, along with high-energy dislocation ]oops, etc. Fatigue fracture is controlled mainly by plastic strain mechanisms (cross slip and twinning), the intensive occurrence of which is due to the stacking-fault energy (SFE) of the metal [7, 8]. The vacancy mechanism also contributes to some degree to fracture under cyclic loading~

In connection with the above, it is of immediate interest to study the effect of alu- minum on the stacking-fault energy of cyclically deformed a-titanium. The stacking-fault energy determines the capacity of the dislocations to complete cross slip. The available literature data indicates that aluminum lowers the SFE of m-titanium by one order [9, 10] and thereby impedes cross slip. Other substitutional elements, such as tin [9] in solid solution in m-phase titanium, probably have the same effect.

Proceeding on the basis of the stated goal of this article, we broke the investigation down as follows:

Institute of Strength Problems, Academy of Sciences of the Ukrainian SSR, Kiev. Insti- tute of the Physics of Metals, Academy of Sciences of the Ukrainian SSR. Translated from Problemy Prochnosti, No. 10, pp. 30-34, October, 1982. Original article submitted October 5, 1981.

1320 0039-2316/82/1410-1320507.50 �9 1983 Plenum Publishing Corporation