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# FATIGUE CURVE APPROXIMATION USING DANIELS’ SEQUENCE AND MARKOV CHAIN

Viacheslavs Cimanis, докторант

Yuri Paramonov, профессор, доктор технических наук, профессор

Рижский технический университет, Латвия

Участник первенства: Национальное первенство по научной аналитике - "Латвия";

The possibility of using the model based on Daniels’ sequence and Markov chain theory for approximation of S-N fatigue curve of composite material is studied. The model allows to see the connection between static strength distribution parameters and parameter of S-N fatigue curve. Although the model is too simple and does not provide numerical coincidence with experimental fatigue test data but it can explain existence of fatigue limit and it can be used as nonlinear regression model of S-N fatigue curve with and without fatigue limit. By the use of this model we can predict fatigue curve changes as consequence of static strength parameter changes. Numerical example is given.

Key words: strength, fatigue life, Markov chain, Daniels’ sequence

Introduction. Every year the use of composite material in aircraft structure increases. To provide reliability of flight we should study the fatigue phenomenon of this material. A lot of papers and books are devoted to this problem (see for example [1]). One of the main quantitative characteristics of this phenomenon is fatigue curve. There are many offers for its description.For example, the Weibull’ equation is used very often:, where S-1, C, B, and aare some parameters,  S is the stress amplitude and N is the corresponding average number of cycles. Seven equations of the quantile fatigue curve are given in [2]. Parameters of these and similar equations have no connections with the parameters of cumulative distribution function (cdf) of tensile strength of composite material component. Our paper is devoted manly to development of some idea already studied in [3,4]: to find the connection of tensile strength distribution parameters and parameters of fatigue curve, S-N, for unidirectional composite using model, based on the Markov chain theory, with state space defined by  Daniel sequence [3]. The successful fitting of experimental fatigue curves can be considered as proof of “likelihood” of the studied model.

Daniels’ sequence model. In Daniels’ papers [5,6], a relationship between the distribution functions of fiber strength and strength of an aggregate of parallel fibers at a uniform distribution of load between them was determined. “Developing” this model in time, we come to a sequence of local stresses {s0,s1,s2,...} which are called Daniels’ sequence (DS) [3]: si+1=S/(1-F(si)), где i=1,2,..., where s0=S is the initial rated stress in the undamaged specimen, F(s) is cdf of tensile strength. DS can be considered as sequence of stresses in the cross section where the failure proceeds, during fatigue loading at the constant mode of loading. It has following specific feature. If initial stress, S, is upper than some value (DS-fatigue-limit (DSFLm)), then stress-sequence grows up to infinity. DSFLm is defined as the maximum value of S for which there is solution of equation s=S/(1-F(s)). This value is equal to SD=max x(1-F(x)).Growth of local stress corresponds to decreasing of local cross section. Let us define that the failure of specimen takes place if local cross section area become less than some value pC (initial cross section area is equal to one). Then critical local stress corresponding to this event, S*UT, is defined from equation Fs(S*UT)=1-pC. The number kmax {sS*UT}, where km is some scale coefficient, can be called as Daniels’  fatigue life  (DSFLf) at stress S.

Here we consider the data of fatigue test of carbon-fiber composite [7]. In accordance with [7] it was supposed that tensile strength of carbon fiber strands has cdf of lognormal distribution, Fs(x)=Ф(log(x)-0)/1), where Ф(.)  is cdf of standard normal distribution, with parameters 0=6.44  and 1=0.1816.  These carbon fiber strands are longitudinal items of specimens which was used for fatigue test. But if we try to calculate DS for corresponding maximum cycle  stresses : (S1,S2,S3)=(323.7  309.7  290.1)  we see that these stresses are under DSFLm, which is equal in this case to 446.85 MPa.  CorrespondingDSFLfs are equal to infinity!

Fig.1. Daniels sequence of local stresses (a) and the corresponding D-fatigue curve for S =323.7, 309.7 and 290.1 MPa (b) for for ks=1.6, km=1, pC=0.1, S*UT=494 MPa.

So in framework of DS-model using cdf of strength of strands the failure of specimens can be explained only by existence of significant local stress concentration. Results of calculations of DSs for the same set of initial stress (S1,S2,S3) taking into account the stress concentration coefficient, ks=1.6, are given in Fig. 2a. For illustration of explanation by DS-model the existence of limit fatigue life phenomenon the results of calculation for S=270 MPa are given also. In last case  DSFLf is equal to infinity.

In Fig. 1a we see that DFLf (the order number of DS up to failure of specimen) for  km=1 is very small:3,4,7. So although DS allows to make quality explanation of fatigue failure of material, can explain phenomenon of fatigue limit, but the quantity prediction is very poor.  And it does not explain the scatter of fatigue life. But the possibility of explanation of phenomenon of fatigue limit is very attractive. So in following we study the model based on use the theory of Markov chain with space of states based on DS.

Simple Markov chain model. We consider a Markov chain, the first r states of which are related with items of Daniels’ sequence {s0,s1,...,sr-1}, (r+1)-th is absorbing state (local stress is equal or more than S*UT). We assume that the only transitions to the nearest ‘senior’ states can take place and we have the following matrix of transition probabilities:

, qi=1-pi, i=1,...,r.

The main characteristics of this type of Markov chain are well known. Time to failure (time to absorption) T=X1+X2+...+Xr, where Xi (time the process spends in i-th state), i = 1,…,r, are independent random variables. Random variable Xi has geometric distribution with probability mass function P=(Xi=n)=(1-pi)n-1pi, i=1,2,... Expectation value and variance are equal to E(Xi)=1/pi and V(Xi)=(1-pi)/p2i. Probability generating function for random variable T  is equal to .  The cumulative distribution function of the number of steps up to specimen failure (number of steps of Markov chain   up to absorption in absorbing state),TA, is defined by equation  ,  t=1,2,3,...., where a row vector =(1,0,...,0) , the b is vector column (0, … 0,1)´. All these formulae are well known. A new step which we offer to do is 1) the connection of probabilities pi, i=1,...,r, with parameter of composite material component strength distribution and parameters of cycles of fatigue loading and 2) the connection of Markov chain state space with DS .

In what follows, for definiteness, loading by a pulsing cycle is assumed; S is the maximum (nominal) stress of the cycle, and his the vector-parameter (its components are parameters of the distribution functions of strength,...).  It is assumed that one step of Markov chain in general case corresponds to kM cycles (the kM is also a component of the vector ). Then fatigue life (the fatigue cycle number up to specimen failure), T, is equal to kmTA. The p-quantile fatigue curve which defines the fatigue life tp(S) (the number of cycles) corresponding to the probability of failure p under an initial normal stress S and the corresponding mean fatigue curve are defined by equations  .  By fitting experimental data we can get the estimate of the parameter  (first of all, the values km and ks),  by using either the nonlinear method of least squares or the method of maximum likelihood.

In Fig. 2 we see example of   fitting of the date of [7] using Markov chain model and the same cdf of tensile strength of strands as for example in the Fig.1, additionally assuming that ks=1.6 and km=12.2847.  The items of matrix P are defined in following way: p1=Ф((log(S)-0)/1); s2=S/(1-p1); pic=Ф((log(si)-0)/1),  pi=(pic-p(i-1)c)/(1-p(i-1)c),   Si+1=S/(1-pic), i=1,2,...,r.

Fig. 3. Fatigue test data (+) and Markov model mean fatigue curve for  ks=1.6 and km=12.2847; simbols (??)show two standard deviation intervals.

Conclusion

Using of Daniels’ sequence  and cdf of strength of longitudinal items (strands) allows to explain the existence of fatigue limit but its value is to large and fatigue failure under loading at stress level lower than its value can be explain  by the local stress concentration (or local decreasing of strength). But in this case “predicted DS-fatigue life” is too small. Reasonable fitting of fatigue test data of carbon-fiber composite specimen was obtained using Markov chain model with states of space based on Daniels’ sequence taking into account the local stress concentration and some scale factor. Although the model is too simple and does not provide too precise numerical coincidence with experimental fatigue test data but it can explain existence of fatigue limit and it can be used as nonlinear regression model of S-N fatigue curve with and without fatigue limit. By the use of this model we can try to predict fatigue curve changes as consequence of tensile strength parameter changes.

References:

 1. Harris B. Fatigue in Composites.  Cambridge, England: Woodhead publishing limited, 2003.   2. Pascual F.G., Meeker W.Q. Estimating Fatigue Curves with the Random Fatigue-limit Model. Technometrics. Vol. 41, 1999, pp. 277-302.3. Paramonov Yu., Kuznetsov A., Kleinhofs M. Reliabilty of fatigue-prone airframes and composite materials. Riga: RTU, 2011.4. Paramonov Yu., Kleinhofs M., Paramonova A. Markov model of connection between the distribution of static strength and fatigue life of a fibrous composite .// Springer Science+Business Media, Inc, Translated from Mekhanika Kompozitnykh Materialov, September-October, 2006, Vol. 42, No.5, pp. 615-630,5. Kleinhofs M. Investigation of static strength and fatigue of composite material used in aircraft structure. Candidate degree thesis, Riga, 1983.6. Daniels H.E. The statistical theory of the strength of bundles of threads. I. Proceedings of the Royal Society of London, Series A, 1945; 183(995): 405-435.7. Daniels H.E. The maximum of a Gaussian process whose mean path has a maximum, with an application to the strength of bundles of fibers. Advances in Applied Probability, 1989; 21(2): 315-33.
Комментарии: 6

## Деревянкин Павел Андреевич

True: the most significant parameters of quality in terms of design and technology are reliability and durability. These parameters are largely determined by performance level details, such as fatigue resistance, corrosion resistance, wear resistance, the contact stiffness, etc.

## Исаева Людмила Евгеньевна

В работе исследованы, казалось бы, простые модели, позволяющие объяснить существование предела выносливости. Анализ механических свойств углеродного волокна, с помощью тонких испытаний на выносливость дает возможность предсказать изменение кривой усталости, как следствие растяжения и оценить параметры прочности. Работа заслуживает положительной оценки.

## Таратин Вячеслав Викторович

Статья содержательная и хорошо аргументиролванная. Графический матерал. а также математические выкладки хорошо её дополняют. Безусловна актуальность работы, так как она связана с безопасностью эксплуатации авиационной техники. Заслуживает высокой оценки. Докторанту хотелось бы пожелать успехов в его научных делах и на творческом поприще.

## Хлопков Юрий Иванович

In materials science, fatigue is the progressive and localized structural damage that occurs when a material is subjected to cyclic loading and is the most important reason of catastrophe in aviation, railways, sea and space. One the first catastrophe was Versailles train crash in 1843. At least 55 passengers were killed trapped in the carriages. Importance of this problems is more increasing for complex modern and futures technical devices. Investigations of authors about the possibility of using the model based on Daniels’ sequence and Markov chain theory for approximation of S-N fatigue curve of composite material on my outlook is very interesting. Consider, that this report deserves good mark. - Zay Yar Myo Myint (Зай Яр Мьо Мьинт)

## Игнатова Анна Михайловна

очень хорошая работа, но есть вопрос, учитывается ли в Вашем расчете возможные дефекты в структуре материала? ведь они неизбежны, хотя вы и делает оговорку что расчет простой, тем не менее интересно может ли он быть дополнен какими либо коэффициентами? или иным способом для повышения точности?

## Cimanis Viacheslavs Janovich

В статье представленная модель сравнивается с данными эксперимента (образцы также могут иметь дефекты в структуре композита). В статье на графике 3 отображена модель и ее стандартное отклонение, как мы видим, большенство данных эксперимента находятся в данном интервале. Мы можем заключить, что возможные дефекты композита, также учтены предлагаемой моделью.
Комментарии: 6

## Деревянкин Павел Андреевич

True: the most significant parameters of quality in terms of design and technology are reliability and durability. These parameters are largely determined by performance level details, such as fatigue resistance, corrosion resistance, wear resistance, the contact stiffness, etc.

## Исаева Людмила Евгеньевна

В работе исследованы, казалось бы, простые модели, позволяющие объяснить существование предела выносливости. Анализ механических свойств углеродного волокна, с помощью тонких испытаний на выносливость дает возможность предсказать изменение кривой усталости, как следствие растяжения и оценить параметры прочности. Работа заслуживает положительной оценки.

## Таратин Вячеслав Викторович

Статья содержательная и хорошо аргументиролванная. Графический матерал. а также математические выкладки хорошо её дополняют. Безусловна актуальность работы, так как она связана с безопасностью эксплуатации авиационной техники. Заслуживает высокой оценки. Докторанту хотелось бы пожелать успехов в его научных делах и на творческом поприще.

## Хлопков Юрий Иванович

In materials science, fatigue is the progressive and localized structural damage that occurs when a material is subjected to cyclic loading and is the most important reason of catastrophe in aviation, railways, sea and space. One the first catastrophe was Versailles train crash in 1843. At least 55 passengers were killed trapped in the carriages. Importance of this problems is more increasing for complex modern and futures technical devices. Investigations of authors about the possibility of using the model based on Daniels’ sequence and Markov chain theory for approximation of S-N fatigue curve of composite material on my outlook is very interesting. Consider, that this report deserves good mark. - Zay Yar Myo Myint (Зай Яр Мьо Мьинт)

## Игнатова Анна Михайловна

очень хорошая работа, но есть вопрос, учитывается ли в Вашем расчете возможные дефекты в структуре материала? ведь они неизбежны, хотя вы и делает оговорку что расчет простой, тем не менее интересно может ли он быть дополнен какими либо коэффициентами? или иным способом для повышения точности?

## Cimanis Viacheslavs Janovich

В статье представленная модель сравнивается с данными эксперимента (образцы также могут иметь дефекты в структуре композита). В статье на графике 3 отображена модель и ее стандартное отклонение, как мы видим, большенство данных эксперимента находятся в данном интервале. Мы можем заключить, что возможные дефекты композита, также учтены предлагаемой моделью.
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