Review of fatigue research at the Institute of Aerophysics (March 1959- March 1962)

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17 jan 63 REVIEW OF FATIGUE RESEARCH AT INSTITUTE OF AEROPHYSICS

TEU!t!;SUlE HOGESCHOOL DELFT

(March 1959 - March 1962 ) VUCCTUIGCOUWKUNDE

BlCLlOilllEK

by

E. D. Poppleton

REVIEW OF FA TIG UE RESEARCH AT INSTITUTE OF AEROPHYSICS (March 1959 - March 1962)

by

E. D. Poppleton

AUGUST 1962 UTIA REVIEW NO. 21

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FOREWORD

This review constitutes the Final Report on work completed

with the financial support of the United States Air Force Office of Scientific Research of the Air Research and Development Command under Contract No. AF 49(638)-548 and Supplemental Agreement No. 1 (61-392).

SUMMARY

This review gives a brief description of an investigation of the fatigue of aluminium aHoy specimens under random axial loading. This re-search was carried out at Institute of Aerophysics under contract to AFOSR.

TABLE OF CONTENTS

Page

I. INTRODUCTION 1

ll. PRELIMINARY CONSTANT AMPLITUDE F ATIGUE TESTS 1

(Refs. 1 and 2)

IIl. DESIGN OF A RANDOM LOAD FA TIGUE MACHINE 1

(E,efs. 3, 5, and 6)

IV. DERIVATION OF A FATIGUE DAMAGE LAW.(Ref. 4) 3

V. INVESTIGATION OF FATIGUE OF ALUMIWUM ~LLOY DUE 4

TO RANDOM AXIAL LOADING (Ref. 5)

REFERENCES 6

FIGURES

1. INTRODUCTION

This review covers the research conducted at the Institute of Aerophysics, during the period March'1959 to March 1962, under contract to the United States Air Force Office of Scientific Research. The work included the development of a machine capable of testing specimens under random axial

loads, the testing of a large number of specimens under various types ofaxial

fatigue loading and studies of methods of analysing the damage suffered by

specimens during fatigue. A drawing of the specimen used throughout the

ex-perimental programme is shown in Fig. 1.

The review of the work is presented, in chronological order, in the four following sections, reference to the publications in which the work was

first described being given in the title of the section. A few figures are given,

which illustrate important features of the work.

Il. PRELIMINARY CONSTANT AMPLITUDE FATIGUE TESTS (Refs. 1 and 2)

An essential preliminary to the programme of random load

testing was the determination of the static strength characteristics and of the

constant amplitude S-N curve for the material. The latter was accomplished

using an Amsler Vibrophore fatigue machine, since this phase of the work pro

-ceeded concurrently with the design and assembly of the UTIA facility, which

was consequently not available until later. Nine specimens were used to

in-vestigate each stress amplitude a~d it was found that, for amplitudes greater

than about 30 ksi, the distribution of the logarithms of the fatigue lives was

approximately normal. The median logN was ther~fore estimated from the

best straight line drawn through the data plotted on normal probability paper,

although it was recognised that this distribution would not be an accurate

des-cription of the process at low probability levels.

At stress amplitudes less than 30 ksi, however, it was found that

there was a considerable difference between the actual distribution of failures and

the log-normal distribution. A larger number of specimens was tested at 20 and 22 ksi, and it was found that,in each case, the failures could be split into two groups, both of which had an approximately log-normal distribution, but with

considerably different medians and variances. This bimodal behaviour is

con-sistent with the idea that fatigue failures in this stress range occur as aresult

of the action of one of two different fatigue mechanisms. One of these (designated Short Term Fatigue, STF, in Ref. 1) predominates at high stress levels and

leads to short fatigue lives while the other mechanism (LTF) gives rise to the

long lives typical of stress amplitudes weU below the "knee" of the S-N curve. The S-N curve is shown in Fig. 2, and the distributions of the logarithms of the lives are indicated pictorially.

In addition to the above constant amplitude tests, a number of "partial damage" tests were made, in which a known number of cycles (n1) at

a stress level SI were applied to a specimen and the number of cycles (n2) required to break the specimen at a second standard level (S2 = 34 ksi) was measured. A total of fourteen values of SI were investigated with an average of about four values of n1 at each level, and in all cases the distribution of the failures was reasonably represented by a single log-normal distribution.

The majority of the above tests we re made with a mean stress of 16 ksi and a total of about 1000 specimens was used in the programme.

lIl. DESIGN OF A RANDOM LOAD FATIGUE MACHINE (Refs. 3, 5 and 6) The specification for the machine was that, in addition to usual sinusoidal testing. it should be capable of applying to a specimen of reasonable size a fluctuating axialload representative of aircraft and missile service load-ing. These service loadings arise from the response o~ an elastic structure to random excitation. and the power spectrum of such a response is commonly concentrated around a small number of natural frequencies. It was desirable, therefore, to have a means of varying the shape of the power spectrum of the stress history of the specimen and an obvious prime mover for the machine was thus an electromagnetic shaker excited by a random noise generator. By introducing .suitable filters. the shape of the power spectrum could be readily changed although the direct use of the shaker to apply axial loads to anything but a very small specimen would have required a very large system. Previou5 random load tests by other workers have been made by direct use of a shaker to apply bending loads to small notched cantilevers, but the interpretation of such tests is quite difficult. since the stress distribution in the specimen is not easily determined.

In the present machine (Fig. 3), the uniformity of stress in the specimen and a reasonable specimen size are retained. at the expense of some flexibility in the choice of stress power spectrum, by interposing an elastic system between the shaker and the specimen. It is possible to give this system either one or two degrees of freedom and control of the dynamics of the system is obtained by the use of adjustable springs and masses. By applying a filtered noise signal to the shaker, a variety of power spectrum shapes is obtainable and the machine is thus able to simulate quite accurately the stress history

re-sulting from the response of any two degree of freedom structure to a random input having any power spectrum. In principle. there is no difficulty in incor

-porating further degrees of freedom but it was felt that the extra complication would be unwarranted, since, as far as a large amount of fatigue testing is con-cerned. the frequency content of the stress history is probably irrelevant, per se. Of considerably more importance is the simulation of the correct am-plitude distribution and, in particular, the correct representation of the

"irregularity" of the history. Areasonabie meaSure of this latter property is the ratio (R) of the frequency of zero crossings with positive slope to the fre-quency of peaks and since this depends only on certain moments of the power

spectrum. representative values (in the range R = O. 6 to 1. 0) are readily obtained on this machine. As far as the distribution of peak stresses is concerned, this

is fixed by the dynamics of the machine and the fact th at the noise generator

provides a stationary Gaussian signal. However, considerable control on the

amplitude distribution can be had by varying the RMS output from the noise

generator, in a random manner. The distribution of peaks of the resulting

quasi-stationary process may be calcu ated from the known probability density of the RMS output of the noise generator.

A detailed description of the UTIA fatigue machine and Hs

per-formance is given in Refs. 3 and 5, and a photograph of the facility is shown

in Fig. 3.

IV. DERIVATION OF A FATIGUE DAMAGE LAW (Ref. 4)

For the analysis of fatigue damage under random loading it is

desirabie to use a formulation which does not require that the load history be

idealised as a succession of half sine waves. Torbe has made a significant

contribution in this regard by expressing the rate of damage accumulation in

terms of the instantaneous values of the damage itself, the stress (8) and

first time derivative of the stress (8). The parameters in Torbe's damage

equation may be determined from conventional sinusoidal tests and then the

equation may be used for the analysis of general non-sinusoidal loading.

A disadvantage of Torbe's original work is that it is possible to

incorporate only smaH effects due to past stress history on the current rate

of damage accumulation. Moreover, the tests required to determine the

necessary parameters may not be very weU conditioned. and the inher,ent

scatter would probably make the variation in the results rather large.

It is suggested in Ref. 4 that the general ideas of Torbe should

be combined with the technique of Corten and Dolan. which has proved very

successful in the analysis of fatigue tests with programmed loading. In this

way, it is, hoped that the parameters involved would have an acceptable

statistical behaviour, and their number, and the experimental programme

required to determine them, would be considerably reduced. Expressing all

stresses .as fractions of a reference stress SR' and using the Corten and

Dolan parameter d. and a paraIlleter ,P , which is related to their "damage

nucleus parameter" m, it is shown that the instantaneous rate of damage

accumulation may be written as a series of functions of the form:-• 1. ..

2.)/'(

k .. \0{ , I

"

(s

+ S .. .

:: (J..-

2.k'>(~~

')

4

N~

I

si

I~I

" <... <:<:B (s+s+s"\.)(~ .. f 1. -r S ",,-,,'L ) (1)

f' 'l f'l

where ",M~,

N

Rare respectively the values of p. and the fatigue life in a

constant amplitude test at level SR'

J1

is the instantaneous value of jJ-. ,

and k is an integer. The double summation is to allow for the effect of

possible variations in ,"mean" .stress in the random stress history.

As in the work of Corten and Dolan, the value of

A

will be fixed by a peak stress

S

which has occurred prior to the instant considered, but this may not be a constant under random loading where continually higher loads appear in the stress history. However, it is likely that)1 will remain constant for fairly long periods and it is shown that, during one such time interval, T/ , the increment in damage is approximately given by

.>..

-=

~ , l

J (

"5) (2) I L\ , L

where w,.. is the circular frequency of peaks

.

..

=

joint probability density of S, S,

s

.

Failure is given by

Z

ó _é

=

1, as usual.

The integral in the above equation is evaluated for the case of a Gaussian stress history and the result is found to be independent of k, so for this case, the details of damage accumulation within a cycle appear to be irrelevant. In addition, experimental programmes are suggested for the determination of the time variation of

j).

and the behaviour of the parameters

SF! .

V. INVESTIGATION OF FATIGUE OF ALUMINUM ALLOY DUE TO RANDOM AXIAL LOADING (Rei. 5)

In order to investigate the behaviour of the damage nucleus parameter

jl

(see Section IV), a number of partial damage tests were made. In these the specimen was subjected to a random loading having a single degree-of-freedom power spectrum of known RMS stress, for a given time, and then broken under sinusoidalloading. Analysis of these tests, using the ideas of Corten and Dolan, indicated that the values of.J1 were considerably higher than the results of programmed loading tests would suggest.

This result was confirmed by analysing the results of tests in which specimens were broken under a random loading having,a single-degree-of-freedom power spectrum. These results are shown in Fig. 4, where the root mean square stress is plotted as a function of the median fatigue life, in hours. The results have been analysed in the manner indicated in Section IV, and the integral in Eq. (2) has been evaluated for the Gaussian stress history, assuming that)l has some weighted .mean value throughout the life. This gives the following result for the failure time T:

where

cr

is the non dimensional RMS stress and

r

is the complete gamma function, NR is the fatigue life at the reference stress amplitude and given mean stress, and it has been assumed that the random loading is in a narrow band near

w

p • Putting in the measured fatigue lives, T, showed that in all cases the value';;" was quite large, and in addition, it was found that Miner' s rule gave extremely unconservative estimates of the fatigue life. The appro-priate form of Miner's rule is found by putting ;;'::./AI<, and d

=

$ (the slope of the log S - log N curve) in the above equation.

The effect of the shape of the power spectrum on the fatigue life was investigated briefly and the results are shown in Fig. 4. For these tests the machine was used in the two degree of freedom configuration, with au irregularity factor of about 0. 6. This gives a stress history having the appear-ance of a random vibration superimposed on a "mean" stress which itself

fluctuates in·a random manner. The results are shown in Fig. 3 and it is clear that, for a given RMS stress, this stress history causes damage similar to the single degree of freedom case, where there is little variation in mean stress, and the probability density of peak stresses approximates the Rayleigh distri-bution.

In order to investigate the effect of stress distribution func-tion (or "load spectrum") on the fatigue life, in the single-degree-of-freedom configuration, tests were made in which the RMS output from the noise generator was varied in a random manner. This was accomplished by changing the dial setting on the generator every minute according to a set of readings derived from a table of random numbers. Several sets of these random numbers were

generated from different distribution functions and the resulting fatigue lives are shown in Fig. 4. The majority of tests were made with a rectangular distribution of RMS stress, in which all values of cf" between zero and a given maximum are equally probable. This distribution has many more low level stress peaks than occurred in the stationary one-degree of freedom tests, where the probability density of peaks was a good approximation to the Ray-leight distribution. Other distributions of RMS stress were also investigated including a truncated Gaussian, which results in an approximately exponential distribution function or "load spectrum" and three special distributions (A,

Band C) in which only values of c::i'" between two truncation points were retained.

All of the test results were analysed in terms of a number of damage theories including those of Corten and Dolan and Freudenthal and Heller.

1. Swanson, S. R. 2. Swanson, S. R. 3. Carr, J. B. 4. Poppleton, E. D. 5. Swanson, S. R. 6. REFERENCES

Systematic Axial Load Fatigue Tests Using

Un-notched Aluminum AHoy 2024-T4-Extruded

Bar Specimens.

UTIA Technical Note No. 35. AFOSR 344, May

1960.

A Two-Distribution Interpretation of Fatigue

S-N Data.

Canadian Aeronautical Journal, Vol. 6, No. 6,

June 1960.

Analysis of the Performance of the UTIA

Random Load Fatigue Facility.

UTIA Technical Note No. 54 AFOSR 1965,

October 1961.

On the Prediction of Fatigue Life Under Random Loading.

UTIA Report No. 82, Feb. 1962.

An Investigation of the Fatigue of Aluminum

AHoy due to Random Axial Loading.

UTIA Report No. 84 (to be published shortly).

Annual Progress Report 1960, Institutue of Aerophysics. University of Toronto, October,

1960.

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using

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Stationary with Axial Loading

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Test Results

Statio nary test points

are m ean values from

six 0 r more tests

a) Two Degrees of Freedom

Quasi Stati onary

Test Resu lts: t-t-f - I -l:=~ b) Single degree of freedom

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FIGU RE 4

-TIME TO F AlL UR E - HOURS