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Fatigue Lifetime Estimation based on Rain ow Counted Data using the Local Strain Approach

Klaus Dre ler, TecMath GmbH, Sauerwiesen 2, D-67661 Kaiserslautern Michael Hack, Univ. Kaiserslautern, P.O.Box 3049, D-67653 Kaiserslautern Abstract

In the automotive industry both the local strain approach and rain ow counting are well known and approved tools in the numerical estimation of the lifetime of a new developed part especially in the automotive industry. This paper is devoted to the combination of both tools and a new algorithm is given that takes advantage of the inner structure of the most used damage parameters.

1 Lifetime estimation and mathematical tools

Whenever new parts are developed in the automotive industry one has to estimate the lifetime of these parts to verify the security of the whole car. This process has to include both physical tests and application of numerical tools. On one hand the results of calculations just based on the theory and algorithms which are known today are not reliable enough if they are used as the only test. On the other hand physical tests for any part in any stage of development are much too expensive (see 5]). What is the typical reason for the failure of a part in a car? The parts are usually not destroyed by one large load, but by the accumulation of the 1

2 OPERATIONS IN THE LOCAL STRAIN APPROACH

2

energy dissipated by many (typically several millions) hysteresis loops. It is assumed that the damage of a loop is rate{independent. This means the frequency of the oscillations is not taken into account. The damage induced by the individual loops is accumulated using Miner's rule. (see 9]) If during a test track drive one has measured the strain values at a point one uses the tool of rain ow counting (see 8]), which means classifying and counting of the corresponding hysteresis loops. The damage induced by these loops is used to estimate the lifetime (for a mathematical description see 3] and 1]). But often one has no information about the local values, since to get real local values you have to built up a specimen put it into a car and drive it around a test track. The way out is to measure outer loadings the part is subjected to during a test track drive. These measurements can be used for any stage of development of the part, especially if one does shape optimization. Of course in this cases the method used to do the estimation of lifetime must not depend on data you need to measure on specimens rst. Here the local strain approach is used.

2 Operations in the local strain approach

The local strain approach is used as a mathematical tool to estimate the time to failure of a given part when applying a given unidirectional load. The necessary information are the local maxima and minima of the load. This leads to a string of numbers, we call the stream of turning points. (The mathematical reason for the restriction to this stream is that the damage as a functional on the loading is assumed to be rate{independent. For a mathematical description see 3].) Using information about the geometry and the material this stream of extremal values is transformed into a stream of hysteresis loops in the stress{ strain{plane at a point of interest (see 11]) . (We will call the operator transforming streams of turning points into the stream of hysteresis loops H.) These points are chosen as the points where the failure of the part is expected. One can nd these points especially in notches. Here the stress and strain induced by the unidirectional load are one-dimensional hence we can speak about the stress{strain{plane and about hysteresis loops.

3 DAMAGE ESTIMATION USING RAINFLOW COUNTED DATA

3

Having the loop stream a damage is evaluated using material properties, experimental data and a so{called damage parameter. This damage parameter induces a functional D, that maps the stream of hysteresis loops into its damage value indicating how much the stream of loops has damaged the part. Usually a value of zero means not damaged at all and the value one indicates that the part is broken. Schematically we get: Load stream

H Stream of D ?! loops in the hysteresis ?! {"{plane

Damage

Because of the division into two independent steps (the operators H and D), it is obvious that the stream of hysteresis loops in the stress{strain{plane gives the whole information to estimate the damage. The following questions arise: (1) Let a damage parameter be given (i.e. an operator D). Which information about the ordering of the loops in the stream is necessary to estimate the damage? (2) Let us assume we do not know the stream of extremal load data but the information about the load is given via a rain ow count of the load. Which information about the stream of hysteresis loops can we gain?

3 Damage estimation using rain ow counted data

Let us at rst consider the second question. What means rain ow counting the stream of loading values? The rst de nition is due to Endo 8], we want to sketch the hysteresis type of approach. We follow the trajectory in the loading{strain{plane, nd all closed hysteresis loops, delete them from the loading signal (i.e. replace the original signal with a signal leading to the same trajectory but the closed loop), classify the loops by the turning points and count the loops having similar

3 DAMAGE ESTIMATION USING RAINFLOW COUNTED DATA

4

loading values, where the loop opens and turns around. The classi cation is done by subdividing the set of possible loading values into a nite number of intervals. We nish up in an integer matrix RFM where the entries denote the number of loops and the part of the signal where no more loops occur which we name the residual RES. (For a mathematical treatment see 3].)

6 5 4 3 2 1 1 1 2

backward residue

2 5 3 6 4 1 7

2

3 1

4 1

5 6 1

3 4 5 6

1 1 1 1

forward residue

is subdivided into six parts. The loops are indicated by the numbered triangles. Each occurrence of a loop starting in class i and turning around in class j increases the entry of RFM(i,j) by one. E.g. loop 6 is counted in RFM(4,2). After the deletion of the loops we end up with the residue.

Figure 1: An example for the rain ow count. The loading space

But why are we interested in the damage of a rain ow counted loading, since one needs the loading signal to calculate this count. Why don't we use the original data? Several reasons have to be considered: (1) The load can be rain ow counted on{line, i.e. during the drive on the test{track to reduce the arising data, such that much longer drives and much ner measurement can be applied. (2) There exist mathematical tools to manipulate the rain ow count, e.g. to extrapolate the data in a way such that the resulting rain ow count has the same distribution of hysteresis loops.

3 DAMAGE ESTIMATION USING RAINFLOW COUNTED DATA

5

This gives a much better extrapolation of track data than simply to merge several load histories. As well the rain ow data can be totally arti cial to examine special situations. (3) We can manipulate the rain ow matrix to simulate special situations. (4) We can generate arti cial rain ow matrices. respect to time of the hysteresis loops (in the plane load vs. local strain) is lost, when the data is rain ow counted. We get a set of loops instead of a stream. The problem is, that the counting is done on the loading stream. Hence we only can get information on the location of the loops with respect to the loading. We can not read the exact location of the local strains from the rain ow count. For each class of hysteresis loops we can get the upper and lower loading value Lu; Ll and by using formulae from geometry (see 11]) and material properties we can calculate the amplitude of strain "a. By the transformation from hysteresis loops in the loading{strain{plane into such in the stress{strain{plane, we also loose the knowledge on the exact location in the stress{strain{plane. We just have the amplitudes ( a and "a). Most of the damage parameters do also encounter for the location of the hysteresis loops in the stress{strain{plane. The reason is that the damage induced by a hysteresis loop of a given stress and strain amplitude is larger under tension than under pressure. The crucial value hereby is the mean stress ( m). Therefore one can not speak of the damage of a rain ow count with respect to such an mean stress dependent damage parameter. But it is possible to come back from a rain ow count to a stream of turning points. A rain ow count consists of the tuple (RFM; RES), where RFM denotes the rain ow matrix with entries RFM(i; j ) giving the number of hysteresis loops from the level i to the level j and RES is the residual of the count.

Observation: It is clear that all information on the original order with

3 DAMAGE ESTIMATION USING RAINFLOW COUNTED DATA

6

L

Lo

Lu

ε

σ

ε

Figure 2: Let us assume that Lmin and Lmax are the extremal

loading values. If a hysteresis loop is classi ed in the loading space to occur between Lu and Lo then its strain amplitude is xed, but its location in the strain space can vary between the two indicated extremal locations. Since hysteresis loops can be nested it can be situated anywhere between, too. By the transformation into the stress{strain{plane the di erent strain locations lead to di erent mean stresses.

3 DAMAGE ESTIMATION USING RAINFLOW COUNTED DATA

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We consider the set of streams of loading turning points which lead to the given rain ow count (RFM; RES). If RC denotes the rain ow count operator, we write formally

L : = RC? (RFM; RES)

1

For each stream L 2 L we can calculate the damage D(H(L)). Hence we can canonically de ne the damage of L and for (RFM; RES) by the expectation value of the damage of the stream of turning points in L, where we assume that every stream has the same probability. The assumption makes sense, whenever we have not got any further information on the loading. 1 X D(RFM; RES) : = jLj D(H(L))

L2L

If we use the tool of rain ow reconstruction on (RFM; RES) we get a \stochastically optimal" stream L (see 6]). This means that we get each L 2 L with the same probability, to avoid a systematics. Of course one won't score all streams in L, hence one will use a sample mean-value as an estimate of the expectation value. One chooses L ; : : : ; LN 2 L and considers

1

1 D(RFM; RES) = N

N X i=1

D(H(Li))

6], which distributes all hysteresis loops stochastically and there usually are lots of hysteresis loops of the same kind, there is an intrinsic averaging even if we only reconstruct one stream of turning points. Therefore often one does only one reconstruction, but we can not exclude that the damage of a reconstructed stream is a exception value with respect to the set L. This method does not depend on H or on D, since we always construct a full stream of loading turning points, which is used as input to the usual local strain approach. If we examine the most used damage parameters, we nd additional properties not used by the latter procedure 9]:

Remark: Because of the special structure of the reconstruction algorithm

3 DAMAGE ESTIMATION USING RAINFLOW COUNTED DATA

(i) The damage is linearly accumulated. (ii) The rating of the damage is done hysteresis loop by hysteresis loop.

8

Especially the latter point shows, that the full reconstruction of the stream is too much:

RFM RES

reconstr. ?! of turningstream ?! points

loop-wise rating

Since the information we use to construct the stream of turning points is completely contained in (RFM; RES), it has to be possible to do a damage evaluation without reconstructing a complete stream of turning points:

RFM RES

?!

loop{wise rating

Using the latter assumptions (i.e. linear accumulation and loop-wise rating) we get the damage of the whole stream of turning point as follows: Let the stream of turning points consist of the hysteresis loops fh j = 1; : : : ; T g, the total damage is evaluated as T X Dtot = d(h )

=1

where d(h ) denotes the damage induced by the hysteresis loop h . If we have a rain ow count instead, we can not any more speak of the total damage but of the expectation value of it. Again we use the individual damages of each loop. For each class of loops hij , we have RFM(i; j ) representatives which we name hij ; = 1; : : : ; RFM(i; j ). We get the expectation value of the damage as: 1 0 nKl RFM i;j X X d(hij )A D(RFM; RES) = E @

( ) ( ) ( )

i;j =1

=1

where nKl denotes the number of classes used for the rain ow count.

4 DAMAGE PARAMETERS WITH HYSTERESIS MEMORY

9

We assume the damage of the single hysteresis loops to be stochastically independent. Hence the expectation value coincides with the linear superposition of the single expectation values:

D(RFM; RES) =

= =:

nKl RFM(i;j ) X X i;j =1 nKl X i;j =1 nKl X i;j =1

=1

E d(hij )

( )

RFM(i; j )E d(hij ) RFM(i; j )E dij

Hence we need for the calculation of the damage of a rain ow count (RFM; RES) the expectation values of the damage of the di erent hysteresis loop types that occur in the matrix. In the following we will see depending on the damage parameter whether and how we can calculate E dij .

4 Damage parameters with hysteresis memory | Example of the PSW T -parameter

In physical tests one can observe that a part which is under tension can not sustain as many hysteresis loops of a given stress-amplitude as the same part under tension. Hence damage parameters were introduced that deal with the in uence of the mean stress One of the most used is the so{called PSWT {parameter (see 7]), which is given by the formula q PSWT : = ( a + m) "a E where : amplitude of stress a "a : amplitude of strain : mean stress m E : modulus of elasticity () : : positive function branch

+ +

4 DAMAGE PARAMETERS WITH HYSTERESIS MEMORY

10

Observations:

(1) We only need to know the set of hysteresis loops in the stress{strain{ plane: we do not need to know the ordering of these loops. (2) We need to know the location of the hysteresis loops in the stress space (we need the mean stress m). The location of the hysteresis loops does not only depend on the given values of the loading but the history of the loading has an in uence, too. If we use the reconstruction algorithm, we construct globally for all hysteresis loops one and only one ordering (history). The idea of the method of hysteresis loop reconstruction is to build up an individual history for each hysteresis loop. To estimate E dij , we do the following (i) For each class of hysteresis loops from class i to j construct some histories. (ii) Calculate the location of the hysteresis loop in the stress space, depending on the constructed history. (iii) Rate the hysteresis loop using the PSWT {parameter. (iv) Build the mean of all damages corresponding to the histories constructed in (i). The result is an estimate of E dij . At rst glance this procedure seems not to be able to compete with the reconstruction algorithm, since we construct a lot of histories instead of one respectively a few. The key point is that the histories here are local histories | they only need to re ect the hysteresis history at the point where the given hysteresis loop is situated. Therefore we examine, what is the minimal information about the history to be able to calculate E dij .

4 DAMAGE PARAMETERS WITH HYSTERESIS MEMORY

11

)

Figure 3: Example: We are interested in the dotted loop in the right picture. To this loop corresponds the dotted part of the time signal in the left half of the picture

Notation: We use the notation

interval

a; b]

if a

b

and the

a; b] for the convex interval b; a] else.

hull of a and b, i.e. the

Observations:

(1) Obviously the location of the hysteresis loop can not depend on the future: Let (L ) denote the loading stream and Lt the point where our given hysteresis loop ends its opening branch. Then the location of the hysteresis loop can only depend on (L )t .

=1 =1

)

Figure 4:We truncate the future of the loop.

(2) In (L )t we can apply the following deletion rules, such that the stream after applying one of the rules leads to the same location of the hysteresis loop:

=1

(D1) We can delete Li if Li 2

Li?1 ; Li+1 ]

(monotonous deletion)

4 DAMAGE PARAMETERS WITH HYSTERESIS MEMORY

(D2) We can delete Li and Li if deletion)

+1

12 (Madelung

Li; Li+1 ]

Li?1 ; Li+2 ]

)

Figure 5: Madelung deletions correspond to the deletion of closed hysteresis loops. (D3) We can delete L1 if jL1j < jL2j

)

metals and alloys, that maxima in the absolute values delete all the memory up to this point (see 10].

Figure 6: This last rule corresponds to the behavior of

(3) To any given stream of turning-points leading to a location of our hysteresis loop there exists one of minimal length leading to the same location. This minimal stream is of the form (L ; : : : ; Ln), where (a) jL j > jL j (b) Li 2 Li ; Li ] for i < n ? 2 = (it follows that jLi j > jLi j for i < n ? 1) (4) If the hysteresis loop we are looking for has its opening branch from class i to class j , then Ln? = i; Ln = j , since we have truncated the future and hence the opening branch of our loop is the last part of the signal.

1 1 2 +1 +2 +1 1

5 CONSTRUCTION OF A MINIMAL STREAM

13

If we follow the trajectory in the loading{strain-plane induced by this minimal stream, we get just the nesting of the hysteresis loops around the hysteresis loop, we are interested in.

5 Construction of a minimal stream

We construct a minimal stream by building up the nesting of the loops from inside to outside. We assume that we have an asymmetric rain ow count, i.e. we can distinguish between hanging hysteresis loops, which are classi ed in RFM(i; j ) where i > j and standing ones (i < j ) We have to verify that the probability distribution of our minimal streams corresponds to the inverse image of the rain ow count. Let us rst consider the case to construct the whole stream. We know (see 6]) that the number of possibilities to insert a loop (i; j ) into a stream where all larger loops have been randomly inserted is

i>j P (i; j )

= #fpossible hysteresis loops which hij can hang ing i? N N j XX X X = RFM(k; `) + RFM(k; `)

1

+ #f 2 f1; : : : ; M ? 1gjRES

i<j P (i; j )

k=1 `=j

k=i+1 `=1

j; RES

+1

i

g

= #fpossible hysteresis loops which hij can stand ing i? N N i XX X X = RFM(k; `) + RFM(k; `)

1

+ #f 2 f1; : : : ; M ? 1gjRES

k=1 `=j

k=j +1 `=1

j; RES

+1

i

g

have been inserted. Therefore we can rst choose randomly a position for our smaller loop. If this position is again a loop, we can choose the position of this larger loop

Remark: We see that this number does not depend on how the larger loops

5 CONSTRUCTION OF A MINIMAL STREAM

14

now independently. Hence we can do the same process like with the small loop. The only xed positions are those in the residue. If we nd a position in the residue we x all the nested loops. We can formulate the algorithm in pseudo{code:

Method:

given: RFM : rain ow matrix RES : residue (i,j) : a hysteresis loop classi ed in RFM(i; j ) randomly chosen nesting of hysteresis loops Goal: where (i; j ) is the innermost hysteresis loop. for any RFM(i; j ) 6= 0 (if equal zero we need no expectation value of the damage of these hysteresis loops). nesting : = (i; j ) Calculate the number P(i; j ) of all possible hysteresis loops where (i; j ) can be nested in. Choose randomly one of these positions. if the position is a hysteresis loop (k; `) in the rain ow matrix then add (k; `) to nesting (i; j ) : = (k; `) until position is part of the residue.

Construction of a nesting

repeat

Since we used the probabilities P (i; j ) and using the fact that these probabilities do not depend on the outer loops we get the result:

Theorem 5.1 The above algorithm constructs each possible nesting with the

same probability.

How to transform the nesting into a minimal stream? We now start with the residue and add the largest loop. Since we do not need to know anything of the future of the innermost loop and the closure of the larger loops will occur in the future, we only need the resulting string of turning points until the turning point of the inserted largest loop.

5 CONSTRUCTION OF A MINIMAL STREAM

15

Now we can apply deletion rule (D3) and delete all points before the maximal amplitude of the resulting string. Clearly we have to add the nested loops from outside to inside. We have to distinguish, whether the loop is inserted in the rst half of the outer loop or the closing part:

in the closing branch of hanging loops. The opposite holds for sitting loops.

Lemma 5.2 Hanging loops start in the opening branch of sitting loops and Proof:

The start of a loop corresponds to a turning point in the string, i.e. the direction has to change direction. Since the rst half of a hanging loop is decreasing it must start on the increasing part of the outer loop. The same argument holds for sitting loops. 2 Again since we can neglect the future we need only the starting and turning point of the loop. If the loop is situated in the opening branch of the outer loop we can delete the turning point of the outer loop. We can summarize

Method:

Insert outermost loop into residue Delete everything after turning point of the loop Apply deletion rule (D3) until rst point is point of maximal amplitude while there are still loops to insert Insert loop using the latter lemma If necessary delete turning point of outer loop

Construction of the minimal stream

end while

Now we have just to construct the trajectory to this minimal stream and we get a sample value of dij .

6 RESULT

16

Remarks:

(a) The complexity of the algorithm to estimate D(RFM,RES) does only depend on the number of classes of the RFM and the sparsity of the matrix. It does not depend on the overall number of loops counted in the rain ow process. This implies that this method is especially useful for data of long drives or for extrapolated data. With slight changes we can use it with matrices that represent just the distribution of the loops rather than real rain ow counts. (b) Since the individual E dij are independent, we can calculate them in parallel and use modern computer architecture. (c) We see that we can calculate all the P(i; j ) in advance. (d) If we use the global construction method, the expense that is necessary for the individual loop classes is xed by the number of their occurrence. But we know that for the damage the large loops are much more important that the small loops which is just opposite to the usual distribution. In our method we can handle the sample size for each class of loops individually and can give the larger loops their importance.

Hence we get an e ective method to estimate E dij . Hereby we can estimate D(RFM,RES). We can apply the algorithm to any damage parameter that uses besides the amplitudes of stress and strain only the position of the hysteresis loop with respect to the stress to evaluate the damage of this hysteresis loop.

6 Result

We see, that we can use the damage estimation based on rain ow counted data, if damage parameters with hysteresis memory are involved. The calculation of the total damage can be done independently and in parallel for each class of hysteresis loops. By the independence we can give higher importance to the larger loops and less importance (less computational e ort) to the smaller ones as those do not contribute too much to the damage.

REFERENCES

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References

1] Beste, A.; Brokate, M.; Dre ler, K.: Kann man berechnen, wie lange ein Auto halt? (in german). In: Bachem, A.; Junger,M.; Schrader, R.: Mathematik in der Praxis , Springer 1995. 2] Brokate, M. : Hysteresis Operators . In: Visintin A. (ed.): Phase Transitions and Hysteresis , Lecture Notes in Mathematics 1584, Springer 1994, pp. 1{38. 3] Brokate M.; Dre ler K.; Krejc P. : Rain ow counting and energy dissipation for hysteresis models in elastoplasticity , submitted. 4] Brokate M.; Visintin A. : Properties of the Preisach model for hysteresis , J. Reine Angew. Math. 402 (1989), pp. 1{40. 5] Dre ler K.; Kottgen V.B.; Beste A.; Kotzle H. : Feasibilities of numerical procedures in fatigue analysis , VDI Berichte Nr. 1153 (1994), pp. 43{59. 6] Dre ler K.; Hack M.; Kruger W. : Stochastic reconstruction of loading histories from a rain ow matrix, submitted. (AGTM-Bericht Nr. 134) 7] Smith K.N.; Watson P.; Topper T.H.: A stress-strain function for the fatigue of metals, J. of Materials 5 (4) (1970), pp. 767{778. 8] Matsuishi M.; Endo T.: Fatigue of metals subjected to varying stresses. In: Proc. Kyushu Branch of Japan. Soc. of Mech. Eng. (1968), pp. 37{40 9] Miner, M.A. : Cumulative damage in fatigue, J.Appl.Mech. 12 (1945), pp. A159{A164. 10] Masing, G.: Eigenspannungen und Verfestigung beim Messing. In: Proc. 2nd Int. Congress of Appl. Mech., (1926), pp. 332{335. 11] Neuber H. : Theory of stress concentration for shear{strained prismatical bodies with arbitrary nonlinear stress{strain law. Trans. ASME, J.Appl.Mech. 28 (1961), pp. 544{550.

REFERENCES

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12] Preisach F.: Uber die magnetische Nachwirkung , Z. Physik 94, 1935, pp. 277{302.