Fatigue life predictions for discrete strain Markov processes

IntJ FatiguelO No 4 (1988) pp 227-236

Fatigue life predictions for discrete strain Markov processes* D.J. D o w d e l l , H . H . E . Leipholz and T . H . T o p p e r

A method for life prediction is investigated for a strain history composed of discrete strain levels. Life prediction is based on the choice of the hysteresis loop under ideal Masing material response as the definition of a damaging event. The Markov transition matrix is chosen as the basis for strain history description. Relationships between the level crossing, peak and valley, Markov matrix and the rainflow counting techniques are discussed. The solution for the probabilities of hysteresis loops is developed based on a stochastic matrix that implicitly describes Masing material response. The probabilities of hysteresis loops can then be used in the linear damage rule for life prediction. Key words: fatigue; fatigue life prediction; Masing material; Markov matrix; hysteresis loop

Fatigue is a process whereby an environment of fluctuating stress causes irreversible changes in the structure of the material of a component progressing towards failure. Failure occurs when the component can no longer perform its intended purpose. In many cases, fatigue results in sudden and unexpected collapses endangering peoples lives. During the design process, engineers endeavour to avoid such failures through the application of their knowledge of material properties and the fluctuating loads in order to evaluate the expected fatigue life and ensure adequately safety during service. Fatigue life is quantified in terms of damage to failure. Where a crack can be detected and sustained by a component for a large portion of its service life, the crack length is a convenient measure of damage. In structural components, where cracks cannot be detected, it is assumed that a hypothetical damage measure exists. If the measure of damage is a monotonically increasing function of the applied loads, then the damage function has an inverse and can be linearized. Under such assumptions, the fatigue life for a stationary random load history can be predicted with the use of the linear damage rule proposed by Palmgren and Miner. 1 Difficulty in the application of the linear damage rule results from improper characterization of damaging events and from not taking the interaction effect of high load levels into consideration. 2 It has been noted that the best fatigue life predictions result from the choice of the hysteresis loop as the definition of a damaging event. 3"4"s The interaction of load levels can be taken into consideration through the use of the modified life law concept. 2'6 Determination of the probability distribution of the designated damaging events (hysteresis loops) is the critical step in the prediction of fatigue lives. Given a stochastic load history, fatigue life can be predicted using the linear damage rule if the probability values of closed hysteresis loops can be calculated.

"This issueof Int J Fatigue is dedicatedto the memoryof Dr Horst Leipholz

H y s t e r e s i s loops u n d e r d i s c r e t e s t r a i n histories Similar in philosophy to the finite element approach, a process in continuous strain and time can be modelled by a process in discrete strain and time, as illustrated in Fig. 1. Accurate representation of any load history is gained when a large number of discrete strain levels can be used. A common measure of material behaviour under cyclic loading is the cyclic stress-strain curve. Ideal Masing materials are materials that display the property that the shape of the stress-strain reponse after a stress reversal is exactly equal to the cyclic stress-strain curve magnified by a factor of two. The stress-strain response of many engineering materials can be modelled as ideal Masing materials with good accuracy. 13.14 Assuming ideal Masing material response, the resulting hysteresis loop patterns for 2, 3, 4 and 6 discrete strain levels are illustrated in Fig. 2. The number of possible hysteresis loops, x, for any number of discrete strain levels, n, may be found from the formula x = 2 "-1 -

1

(1)

The number of possible hysteresis loops is a base-2 exponential function of the number of discrete strain levels.

Load h i s t o r y c h a r a c t e r i z a t i o n Any characterization of a load history must be based on a counting method. Various counting methods, adapted from Dowling, 7 are summarized in Table 1. The methods given in Table 1 can be classified into three categories in order of increased complexity: • • •

level crossing counts peak, reversal or range counts hysteresis loop counts.

Depending on the character of the strain history (narrow band, wide band, duty cycle, sequenced etc) and the method

0142-1123/88/040227 - 10 $3.00 © 1988 Butterworth & Co (Publishers) Ltd Int J Fatigue October 1988

227

,

,./

/" r

v

/I

/~, , a (1~ ivl II

I,

L~ "1.

Ill/I

\( \ / V

,/'1 AI 3 //I ,./v ~ IV' 'I/l,/I I

'

IIvii t/

v

v k/I/

IIIlll

lllllllll

I

II

IIIlll

IIIIlllll

I

11

IIlIRII

IIIIIIIAI

I

I~

INlll IIlIIIIII~AI I lllll IJ 114 III/~INIVI~4111 AIIIII III I II Ill]llllll I~1111 I/IVtL II I IIIMIIIIIII IIIIII II~/NI T=ime II I lIIlllIIIlllll Ill/IllI/lll/I

II I II I! IIII

II/IVI/llI/I

III

IlIIIVlfl

llIl

Discrete strain-time history

Continuous strain-time history

Fig. 1 Modelling of a continuous strain-time history by a discrete strain-time history

Stress

2 Levels 1 Loop

i / Strain

e2

3 Levels 3 Loops

'-

e3 e2

e,1

4 Levels 7 Loops

e2

.

may be underconservative and resulting designs dangerous. For example, in the case of a wide band random process, representation of such a history in terms of a mean crossing peak count neglects the majority of small cycles which may, in fact, contribute most of the fatigue damage. For the case of a narrow band random process, a fatigue life calculation based on the level crossing count has a simple solution. An analytical solution for the fatigue life of a component subjected to a random history that is normally distributed is given by Crandall and Mark. s Practical application of this technique is restricted to resonant systems with a single dominant vibration mode. Since many fatigue problems are not the result of resonant vibrations, a more general approach is sought. A previous attempt to describe the probability of occurrence of hysteresis loops was made by Philippin, Leipholz and Topper. 9'1°'11In these references, the probability of closed hysteresis loops resulting from a modified version of a mean crossing peak count is determined as a function of the probabilities of reversals at each of four discrete strain levels under ideal elasto-plastic material response. For application to a general service history, several limitations are evident in this work: •

•

I

6 Levels 31 Loops

e,

•

7/'~

~

e'6

e6

Fig. 2 Stress-strain response under a discrete strain history

of description, the reproduction of that character may have varied success. Subsquently, life predictions based on data collected from any one counting method will be influenced by the counting method. In cases where damaging events are misrepresented or overlooked, fatigue life predictions

228

narrow band histories can only be considered because small cycles are excluded by the mean crossing peak count; results are only applicable to ideal elasto-plastic materials; the formulation was limited to four discrete strain levels plus a mean strain level.

It is the objective of this work to improve on the work of Philippin et a/by: • • •

extending the analysis to include wide band processes through the inclusion of small cycles; using ideal Masing material response so that fatigue near the endurance limit can be considered; increasing the number of discrete strain levels that can be used.*

*The original intent of the research was to be able to handle 6 discrete strain levels, however, the analysis presented in this paper can be used with any number of discrete strain levels

Int J Fatigue O c t o b e r 1 9 8 8

Table 1. Summary of load h i s t o r y description methods adapted from Dowling = Method

Description

Peak count

Reversals that are maximums above the mean, and reversals that are minimums below the mean are counted.

Illustration I

¢.-

2

Mean crossing peak count

t

The single greatest strain peak is counted between successive crossings of the mean strain level. .E 2 +a t

Peak and valley count

All maximum reversals are counted and all minimum reversals are counted. Two distributions are produced, one for maximums and one for minimums. ¢,+., C/3

Level crossing count

Each level crossing is counted for increased strain above the mean, and for decreasing strain below the mean.

I

.=::

f

Fatigue meter count

This counting method is similar to the level crossing count, except that before a level crossing can be counted, an assigned lower level must be crossed.

Int J Fatigue October 1988

41

"V

"

"v

\

22g

Method

Description

Range count

Each range, defined as the difference between two successive reversal points, is counted.

Illustration

r2

r4

i-

0 ~D

Range mean count

This method adds the information of the mean strain level to the informaton gathered in a range count.

m2 m4

~0 CO

Markov matrix count

Successive reversal points are counted as transitions from one strain level to another, and stored in a two dimensional distribution.

To

° AIM

°AV /l -~4 '/~° m3

V

2 ]

Range pair count

Ranges and means are counted through a process of continually editing reversals from the history. The first figure to the right shows three reversals removed from the history, and the order in which they are removed. Reversals are removed from the smallest to largest. The second figure shows the history after some editing.

1 2 3 4 5 6 , 0 0 o , 0 20 01 01 =E300 1 1 1 o

V ~4011 5011 0 61 0111

01 1

._c 0

._c '~ 0 ~3

230

Int J Fatigue October 1988

Method

Description

Illustration

Rainflow counting

Rainflow counting takes its name from an analogy to the flow of water off multiple roof peaks. A strain range is counted when a stream of water meets a lower roof. After a range is counted, the peaks are edited from the history. The Rainflow counting method retains the same information as the Range Pair counting method, however, the strain history may be processed sequentially.

Strain

Tirnl

Stress

Z It was found that the Markov matrix description of load histories satisfies the first requirement above and is able to reproduce a wide variety of load histories with reasonable

for every transition between reversal levels. For a history containing a number of reversals, s, the sum of all the entries in the Markov reversal count will be s:

accuracy.15,16,17 ~ Bo = s

(2)

i=lj=l

Level crossing a n d p e a k and v a l l e y histories A level crossing count is the simplest method of strain history description. Assume that vector c contains, in each element, the number of level crossings for a particular history. The distribution of peaks contained in the ranges between the levels, p, can be found by taking differences between successive elements of vector c. The elements of the new vector p represent the difference in the distribution of peaks which are maxima, p+, and the distribution of peaks which are minima, p - . Since only the difference in the numbers of maxima and minima in a certain range are specified, elements of the vectors p+ and p - can take on a range of values. In order to have a more precise description of a load history, a description in terms of the peak and valley distributions may be used. In reverse, given distributions for the peaks and valleys, one can always calculate the distribution of peaks p and by integration determine the level crossing count c. The peak and valley load history description is clearly superior to the level crossing count.

Markov transition matrix Given a sequence of reversals at a set of discrete strain levels, a Markov reversal count matrix, B, can be generated by incrementing the appropriate element within the matrix

Int J Fatigue October 1988

To reconstruct or analyse a history, the Markov reversal count, [B], may be represented in terms of a Markov transition matrix, [~], defined in terms of stochastic matrices. The elements of the transition matrix, [~], are defined such that each element, ~#, describes the probability of transition from strain level i t o j given that a reversal has just occurred at level i. Transitions from one level to the same level do not contribute to fatigue damage and therefore are not considered. The values on the main diagonal, ~ii are therefore undefined. The load history causing fatigue damage is assumed to be a string of transitions with alternating direction of travel. Reversals occur at the end of each transition. A transition with increasing strain is described by ~# w h e r e j > i for all i. All reversals in increasing strain lie in the upper triangular portion of the transition matrix. For transitions with decreasing strain, j < i for all i, and the transition probabilities lie in the lower triangular portion of the transition matrix. The matrix, [~], can therefore be split into two individual transition matrices identified by [~+] and [~-], where [~+] is upper triangular and [~-] is lower triangular. Both [l~+] and [1~-] are stochastic matrices. The Markov transition matrix [~] serves to store in compact form the non-zero elements of both directional transition matrices and can be expressed as the sum

[P] = [P+] + [p-]

(3) 231

As stochastic matrices, [p÷] and [p-] must satisfy the property that the sum of the rows must be unity and all non-zero elements are positive. Since only elements of [~+] in the part of the row to the right of the main diagonal are non-zero, it can be stated that

._c ro

~13u=

1

j=

1,2,3 ..... n-

1

Time

(4)

i=j+ 1

For [~-], only elements to the left o f the main diagonal are non-zero, therefore j-t

~[3ij = 1

j=

2,3,4 ..... n

(5)

i=1

Peak: k, i < j

a

must be satisfied. The parts of the rows of the Markov transition matrix, [~], over which the above sums are defined, are termed part-rows. The distribution of peaks, p +, must satisfy the equation p+ [13-] = p -

(6)

=_

Similarly, the distribution of valleys, p - , must satisfy p - [p+] = p+

co

Time

(7)

subject to the condition that

f p:= i=l

= 1/2

(8)

i=1

Through the substitution of one equation in the other, the following two identities must be satisifed: p - [[3+] [[3-] = p -

Valley: k, i ~>1

b

Fig. 3 Peaks and valleys in a strain history

(9) L Bji = L Bkj

and

i=j+ I

p+ [p-] [p+] = p+

(lO)

subject to the condition of Equation (8). If the Markov matrix generates a valid strain history, then the two equations above will be satisfied for unique peak and valley vectors.

j = 1, 2, 3 . . . . , n - 1

(13)

k=j+ I

The row and column sums described in the above equations are significant. For maximum at j , a vector, q, is defined such that its elements are the part-row sums: j-l j-i

qi = .~__ . . . Byk = ~= Bkj

j=2,3,4

.... n

(14)

For a minimum j , a vector r can be defined having elements

Relationship between transition and reversal count matrices

rj = ~ Bjk = ~ Bki k =j+ 1

It is assumed that a strain can be modelled based on the data collected in terms of a Markov reversal count. In this section, the method of conversion from the reversal count matrix to the stochastic matrix will be investigated. Assume that a Markov reversal count, [B], is collected over a time, t, or a number of events, x. The elements of [B], B,), are functions of time or the number of events:

8=

BO(t)

(11)

Bo~x)

The sum of the elements in the reversal count is given as s in Equation (2). Any sequence of reversals obeys the simple property that for any peak at j in a sequence of reversals at i, j and k, j must be a level greater than both i and k for a maximum, and for a minimum j must be less than both i and k, as illustrated in Fig. 3. Since i and k are both arbitrary it can be stated that: j-i

i=1

j = 2,3,4 ..... n

k=l

for a maximum a t j . For a minimum at j :

232

(15)

The distributions along the rows of the Markov reversal count matrix is described by the distribution along the rows of the transition matrix. It can be stated that:

BJt=L~=Bjk ~T,J-1

1

13yt

j=

2,3,4 ..... n

(16)

for I < j . For l > j , it can be stated that:

Bj, =

Bik

[~jt

j=

1,2,3 ..... n-I

(17)

Lk=j+ t Now, for m = 1, 2 , . . . , n - 1, the components of the vector q [ ~ - ] can be found as: n

i=m+l q'f3mi "~ =

Im- I

'

/

Bkl

Bmi =i=m+ Bmi = r~ (18) ,

Therefore, it can be concluded that

q[B-] = r

(19)

Similarly, the vector r[[~ +] can be found to be:

j-i

~Bj~ = ~Bkj

j = 1,2,3 ..... n--1

k =j+ 1

(12)

r[13 +] = ¢

(20)

The vectors q and r must satisfy the relationships

Int J Fatigue O c t o b e r 1 9 8 8

q[p-] [p+] = q

(21)

r[p+l[p -] = r

(22)

Stress

and:

w

Path: 1,5, 3, 4

q =

sp +

(23)

r =

sp-

(24)

//

/

The vectors q and r can therefore be expressed as multiples of the peak and valley distribution vectors p+ and p - . In equation form

// #

Strain

and:

The above derivation shows that the simple division of the elements in the part-rows of the reversal count by the appropriate row or column sums of that matrix yields the Markov transition matrix. Simlation based on this transition matrix for a number of reversals, s, will statistically reproduce the original count matrix. In reverse, the expected Markov reversal count may be calculated using the formula [B]

= s[diag(p +) [~+] + diag(p-) [p-l].

/

/

4

6

Fig. 4 Description of a stress-strain path in a discrete strain history

Stress

II I

Loop: 1, 5, 3, 4, 3

As illustrated in Fig. 2 and Table 2, the number of hysteresis loops, x, grows base-2 exponentially with the number of discrete strain levels used to characterize a history. The number of stress-strain path segments, .y, forming the network of hysteresis loop can be found from the formula:

7///]

Strain

(26)

Because of the complexity of the hystersis response for more than 6 strain levels, it was found necessary to devise a system for the identification of stress-strain paths, reversal points and hysteresis loops. The system that was devised was based on the algorithm for modelling of an ideal Masing material. Since the stress-strain path followed during any excursion in strain is influenced by the material's memory of previously unclosed reversal points, the list of strain levels of yet unclosed loops forms a description from which the actual stress-strain path, reversal point and any closed loops can be reconstructed. The number lists that form the names of valid stress-strain paths, reversal points and loops can be easily generated by a recursive computer algorithm for any number of discrete strain levels. Fig. 4 illustrates the name of a particular stress-strain path. If that stress-strain path is assumed to end in a reversal then the reversal point or path-reversal would be identified by the same list of numbers. A loop closes when a stress-strain path meets or crosses a previous path-reversal. In order to identify a closed loop, as illustrated in Fig. 5, the last and third last entries in the number list forming the name must be equal.

Path-reversal

5

(25)

Including hysteresis response

y = 2" - 2

#

V

transition

2

3

4

5

6

Fig. 5 Fig. 5 Description of a closed hysteresis loop in a stress-strain history

of load direction is taken into account implicitly in the structure of the matrix. Although the [y] matrix can be very large, as illustrated in column 4, Table 2, the matrix is also quite sparse. Columns 5 and 6 in Table 2 show the number of elements and the percentage of the full matrix populated for up to 10 discrete strain levels.

Solution for path-reversal probabilities The path-reversal probability vector, 7r, is defined as the vector containing the probability of reversal at each of the path-reversal points in any one time step. The path-reversal probability vector must satisfy the equation

[7]

matrix

The level transition matrix can be expanded to include the ideal Masing material response through the generation of the path-reversal transition matrix, [7]. The indices of the path-reversal transition matrix correspond to each possible reversal point in the network of stress-strain paths. Computer algorithms, based on the natural naming system for pathreversals, were used to automatically generate the [7] matrix. The [7] matrix is a true stochastic matrix. The alteration

Int J Fatigue October 1 9 8 8

1

=

(27)

~

under the condition that the sum of the reversal probabilities satisfies y

Z ~, = 1

(28)

i=l

The above equations can be solved directly by solving the non-trivial system of equations: ([7] - / )

=

0

(29)

233

Table 2 Numbers of hysteresis loops, stress-strain paths and corresponding numbers of entries in level transition and path reversal transition matrices Number of discrete strain levels

Number of closed hysteresis loops

Number of stress-strain paths or path-reversal

Number of entries in expanded path-reversal transition matrix

Number of non-zero entries in path-reversal matrix

Percentage of path reversal matrix populated

2 3 4 5 6 7 8 9 10

1 3 7 15 31 63 127 255 511

2 6 14 30 62 126 254 510 1022

4 36 196 900 3 844 15 876 64 516 260 100 1 044484

2 10 32 86 212 498 1 136 2 524 5612

50.0 27.8 16.3 9.56 5.52 3.14 1.76 0.97 0.54

n

2n-1 - 1

(2"-2) =

(n+ 1)2"-1 + 2 n

100((n + 1)2"-1 + 2n~ ~, } (- 2) 2 ="

2"-2

subject to the condition of Equation (28). For large problems, however, the solution of Equation (29) is difficult. Large amounts of computer memory are required, and, due to the size of the [7] matrix and ill-conditioning of some problems, a solution is not always possible using direct means. Since the [7] matrix is very sparse, there is potential for savings in memory space if iterative techniques can be used. Iterative solution techniques were investigated.

Iterative solution techniques Starting with the initial estimate %, repeated multiplication of this vector by the [7] matrix gives the sequence r~o, re1, ~2 . . . . . ~,, ~,+1 . . . .

(30)

where each estimate, 7t., is defined by:

~. = Go ([7])"

(3])

in Fig. 6. In this paper, the hypothesis will be adopted that the two probabilities are equal. The probability of travelling on a path segment can be determined as the signed sum of the probabilities shown in the hatched area in Fig. 6, so that the reversals preceding movement towards the path segment carry a positive sign and reversals proceding movement away from the path segment carry a negative sign. If the hysteresis loop has end points at levels a and b, and the loop closes at b, then the hatched area is defined as the area between levels a and b + 1 contained within the hysteresis loop. To this sum is added the probability of entering the network, indicated in Fig. 6 by the small circle.

E x a m p l e f a t i g u e life c a l c u l a t i o n In this example, the ideas developed in this paper will be applied using a four strain level example.

For the class of stochastic matrices of which [7] is a member, it is known that the limit of the sequence exists and is equal to the vector lim ~, = n

Stress[ Loop closing

(32)

n~00

Iteration has the advantage that as many steps as necessary of the iterative process can be applied so that the required accuracy is reached. Operations involve only multiplication and addition with the non-zero elements of the [7] matrix, therefore, only non-zero elements need to be stored. Iterative sequences using [7] matrices were found to converge very slowly. Slow convergence prompted the study of convergence acceleration techniques. Two convergence acceleration techniques were studied in reference [12].

Strain

Sum for hysteresis loop probabilities Once the path reversal probabilities, ~i, are established through the solution techniques described above, the hysteresis loop probabilities must be interpreted from that data. It can safely be assumed that the probability of closure of a hysteresis loop is related to the probability of an excursion in stress and strain travelling along the stress-strain segment that just closes the hysteresis loop, as illustrated

234

1

2

aa+la+

b-lb

Fig. 6 Closure of a hysteresis loop. The probability of closure is assumedto be the probabilityof travelling along the path segment that just closesthe loop Int J Fatigue O~.tober 1988

Given a Markov level transition matrix: I

[~]=

0

0.3

1 0.3 0.2

0 0.7 0.2

0.4 0.6 0 0.6

0.3 1 0.4 1 0

(33) [7] =

1.

calculate the peak and valley histograms

2.

construct the path-reversal transition matrix

3.

calculate the path-reversal probability vector

4.

find the probability of the hystersis loop designated by: loop: 1, 4, 2, 4

5.

given life data for all possible hysteresis loops, determine the expected life of a component subject to the strain history described by the Markov matrix.

Peak and valley histograms First, split the transition matrix into forward and reverse direction transition matrices: 0

0.3 0.4 0 0.6 0.4 0 0 0 0

[3+=

0

p- =

0 0 0 3 O.7 L O.2 0.2

0 0 0 0.6

0 (34)

Since both the forward direction and reverse direction level transition probabilities, contained in vectors p+ a n d / ) - , satisfy Equations (9) and (10), the vector p+ may be solved with the linearly dependent set of equations: ~+

[13-13 + -

+

0

P

-/l

[--0.52 _l 0.26 P / 0.2 L o

0 0 0 0.6 0 0 0 0 0 0 0 0 0 0

0 0 0.3 0 0.3 0 0 0 0 0 0 0 0 0

0 0.3 0 0.4 0 0.3 0 0 0 0 0 0 0 0.3

0 0 0 0 0 0 0.6 0 0.6 0 0 0 0 0

0 0 0 0 0 00 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.2 0 0 0 0.2 1 0 0 0 0 0 0 0.2 0 0 0 0.2 0 0 0.6 0 0.4 0 0 0 0 0.7 0 0.3 0 0 0.6 0 0.4 0 0 0.2 0 0 0 0.2 0 0 0 0 0 0

(39)

The entry in row 1, column 2, 7t2, describes the probability of reversal at reversal point 2 immediately following a reversal at reversal point 1 as shown in Fig. 7. Since a reversal at point 2 follows a reversal at point 1, and since a reversal at level 1 must certainly follow a reversal at level 2 in a stress-strain excursion in decreasing strain, the entry T12 takes the value of unity just as 13-zl. For a reversal at reversal point 4 in row 4 of matrix (39), Fig. 7 shows that only two reversal points are possible: reversals at either point 5 or 7. Therefore, only entries 745 and T47 in row 4, columns 5 and 7, are non-zero. The probability of reversal at point 5 after a reversal at point 4 is equal to the probability of reversal at level 3 after a reversal at level 2. This probability value can be found in 13+z3. Similarly, reversal at point 7 after reversal at point 4 can be found as the probability of reversal at level 4 following level 2 and can be taken from 13+z4

Path-reversal probability vector Solution of [7] for the eigenvector corresponding to the eigenvalue ~. = 1 subject to the condition of Equation (28) yields the solution for the path-reversal probability vector: = [0.0460 0.0460 0.0613 0.0740 0.0444 0.0317 0.0756 0.1731 0.1731 0.0577 0.0597 0.0418 0.0398 0.0756] (40)

z] =

o

o

-0.7 0.09 0.06

0.4 - 0.46 0.2

o

0.3 0.37 -0.263

=

o

(35)

and p - may be solved with the linearly dependent set of equations:

t,-[P-P+

12 -- 0 1 0 0 0.3 0 0.4 0 121 13 0 0 0 0.7 132 0 0 0 0 1323 0 0 0 0.7 131 0.3 0 0.4 0 0 0 0 0 14 0 0 0 0 143 1434 0 0 0 0 0 0 0 0 142 0 0 0 0 1423 14232 0 0 0 0 1424 0 0 0 0 141 .0.3 0 0.4 0

Probability of loop: 1 , 4 , 2, 4 The loop closing probability P1,4¢4can be calculated as the signed sum of all the reversals in the shaded area of Fig. Stress

=

0.34 -0.50 0.2 o

0.18 0.25 -0.4 o

0 "] 0 0 --1

J

= 0

(36)

)

Subject to the condition of equation (8), solution of (35) above yields: p+ = [ 0 0.0460 0.1655 0.2885 ]

0 7)

and the solution of (36) gives: p - = [ 0.1533 0.1735 0.1732 0 ]

(38)

Strain

Path-reversal transition matrix By inspection of the stress-strain pattern shown in Fig. 7, the [7] matrix is found to be:

Int J Fatigue October 1988

1

2

3

4

Fig. 7 Example calculation of hysteresis loop closure probability

7. The probability of loop closure is equal to the signed sum of the probabilities: P1,4,2,4

=

~lA,2 -- ~1A,2.,3 + 711,4,2,3,2

= 0.0577 -- 0.0597 + 0.0418 = 0.0398 Fatigue

(41)

life calculation

After calculation of the remainder of hysteresis loop probabilities and given the fatigue life data for each possible hysteresis loop, the expected fatigue life for the strain history described by the Markov matrix above can be calculated as shown in Table 3. The probability of hysteresis loop closure is divided by the number of cycles to failure for that loop, giving the expected damage of that cycle caused in one reversal of the strain history. Summing the expected damage gives an expected damage per reversal for the total strain history. Inverting this number gives an estimate of the fatigue life, as shown at the bottom of Table 3.

Table 3. example

Fatigue

life

prediction

for

4-level

Loop name

Closure probability

Cycles to failure

Damage per reversal ( x 10 -e)

121 1323 131 1434 14232 1424 141

0.0460 0.0740 0.0137 0.1731 0.0597 0.0398 0.0756

5 000 000 3 000 000 500 000 2 000 000 4 000 000 300 000 1O0 000

0.0092 0.0247 0.0634 0.0866 0.0149 O.1327 0.7560

Total

0.4999 Life =

1.0875 x 10 -e 1

1.0875 x 10 -6

= 919 540 reversals

5.

Sandor, B. I. "Strength of Materials" (Prentice Hall, New York, 1978) pp 315-324

6.

Dowdell D. J., Leipholz, H. H. E. and Topper T. H. "The modified life law applied to SAE-1045 steel" Int J Frecture31 (1986) pp 29-36

7.

Dowling N. E. 'Fatigue failure predictions for complicated stress-strain histories' TAM Report No. 337 (Department of Theoretical and Applied Mechanics, University of Illinois, USA, 1971 )

8.

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9.

Philippin G. and Leipholz H. "On the Distribution of the Stress States of a Material With Ideal Elasto-Plastic Behaviour Subjected to a Finite Number of Strain Levels" Mech. Res. Commun 2 (Pergamon Press, 1975) pp 215-220

10.

Philippin G., Topper T. H. and Leipholz H. H. E. 'Mean life evaluation for a stochastic loading programme with a finite number of strain levels using Miner's rule' Bulletin 46 (Shock and Vibration Center, Washington, DC, USA, 1976)

11.

Philippin G. A., Topper T. H. and Leipholz H. H. E. 'On the mean life evaluation of a material with ideal elastoplastic behaviour subjected to a loading programme with a finite number of strain levels" Bulletin 47 (Shock and Vibration Information Center, Washington, DC, USA 1977)

12.

Dowdell, D. J. and Leipholz H. H. E. 'On solution convergence in a fatigue life calculation" 11th Canadian Congress of Applied Mechanics, Edmonton, Alberta, Canada, 1987

13.

Masing G. "Eigenspannungen und Verfestigung bei Messing' Proc 2nd International Congress for Applied Mechanics, Zurich, Switzerland, 1926

14.

Williams, D. P. and Topper, T. H. "A generalized model of structural reversed plasticity" Exp Mechanics (April 1981 ) pp 145-154

To demonstrate the application of the above theory, an experimental programme was undertaken in the lab using strain programs containing six discrete strain levels. The experimental results are presented in a paper to follow.

15.

Huck i . and Schutz W. 'Generating the FALSTAFF load history by digital mini computers' Proc 8th ICAF Symposium, "Problems with Fatigue in Aircraft, Lausanne, Switzerland, June, 1975,/CAF Doc No 801, Sec 3.62 pp 1-23

16.

Noble B. and Daniel J. W. "Applied Linear Algebra" (Prentice Hall, Englewood Cliffs, NJ, USA, 1977)

References

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Dowdell, D. J. 'A strain based approach to fatigue life prediction for complex load histories' M.Sc. thesis (University of Waterloo, Canda, 1977)

Summary and conclusions (1)

(2)

(3)

(4)

The objective of this paper was to describe a method of determining the probability of closed hysteresis loops resulting from the appfication of a stochastic load history to an ideal Masing material. The probabilities of closed hysteresis loops can be used in the linear damage rule for life prediction. The Markov matrix was selected as the load history description that provides an improvement to the modified mean crossing peak count used in previous work. The part-row sums of the Markov reversal count matrix represent the peak and valley distributions. The peak and valley distributions relate the Markov reversal count to the Markov transition matrix. The peak and valley distributions can also be related to the level crossing count. A general method of solution for the probability of dosed hysteresis loops is presented based on the expansion of the Markov level transition matrix into a large matrix called the path-reversal transition matrix. The path-reversal transition matrix can be solved either directly or by iteration to give the path reversal probabilities. The hysteresis loop closing probabilities can be deduced from the path-reversal data.

Experimental programme

1. 2.

3.

4.

236

Miner, M . A. "Cumulative damage in fatigue" J Appl Mech 12 (1945) A159-A154 Leipholz, H. H. E., Topper, T. and El Menoufy, M. 'Lifetime prediction for metallic components subjected to stochastic loading' J Computers & Structures 16 1-4 (1983) pp 499-507 Okemura, H., Sakai, S. and leusuki I. 'Cumulative fatigue damage under random loads" Fatigue Engng Mater Struct 1 (1979) pp 409-419 Socie D. F. "Fatigue life prediction using local stress-strain concepts" Exp Mechanics 17 2 (February 1977)

Authors

D. J. Dowdell is now at Kayaba Kohgyo Shataku A211, Shimokuzawa 1475, Sagamibara-Shi, Kanagawa 229, Japan. T. H. Topper is and H. H. E. Leipholz, now deceased, was with the Civil Engineering Department, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1.

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