Su Chun Qu Zhongzhou Hao Huibing

(School of Mechanical Engineering， Southeast University， Nanjing 211189， China)

F or mechanical parts，failures usually result from the competition between soft failure(degradation)and hard failure，and shock processes can speed up both of the failures.In recent years， studies on competing failures have been extensively explored;but most of them assume that the two kinds of failure processes are independent of each other［1-4］.

In real circumstances， however， there exist correlations between the degradation process and random shocks.For example，the degradation process makes the system more vulnerable to random shocks，and random shocks can accelerate the degradation process.Up to now， different categories of random shock assumptions have been constructed， including the cumulative shock model［5］， the extreme shock model［6］， the mixed model［7］and the δshock model［8］.Moreover， effects of correlations， competing processes between shocks and degradation have also been studied.Su et al.［9］proposed a reliability assessment method considering competing failure based on the Wiener process.Peng et al.［10］developed reliability models for systems that undergo multiple dependent competing failure processes，in which two kinds of failure processes are dependent upon each other due to the impact from the same shock processes.Investigating both the degradation process and the shock process，Wang et al.［11］constructed a system reliability model on competitive failure processes with fuzzy degradation data，which was evaluated with a multi-state system reliability theory.In this paper，the reliability analysis is conducted on the basis of the dependent competing failure process.In which， hard failure is caused by the shock process， while soft failure is the result of continuous degradation and abrupt damage from the same process.The cumulative shock model is applied for the sake of establishing two probabilistic models，and the correlation between hard failure and soft failure is considered.Based on that， reliability is consequently estimated including the situation of the shifting-threshold value.The case study with sensitivity analysis implies that the proposed models are in line with the actual situation，which also demonstrates that the proposed models can be applied to the components that endure dependent failure processes.

1 Dependent Competing Failure Process(DCFP)

As shown in Fig.1， letX(t)be the wear volume of the continuous degradation by timet，and it is monotonically increased with time.Shock loads will cause additional abrupt damageYi(i=1，2，…)and speed up the degradation process.Soft failure will occur when the overall degradation，Xs(t)， is beyond the critical strength levelH.

Fig.1 Soft failure process

Jiang et al.［12］pointed out that when sustaining shocks，components become more susceptible to hard failures.Thus，the same random shock process can also result in hard failure.As seen in Fig.2(a)， letWidenote the magnitude of thei-th shock(i=1，2，…)， hard failure occurs when the cumulative shock load magnitude exceeds the threshold valueD1.The system will fail when either of the two failures occurs.For most materials， the strength will gradually decrease with time.Fig.2(b)shows that the critical strength value decreases fromD1toD2after the arrival of a run ofmshock loads.

Fig.2 Hard failure process.(a)Fixed threshold;(b)Shifting threshold

2 Reliability Modeling for DCFP

2.1 Shock process analysis

It is assumed that random shocks arrive according to a homogeneous Poisson process with rate λ.LetN(t)denote the number of shocks until timet，then

As shown in Fig.2(a)， in the cumulative shock model， letTbe the time that the system incurs hard failure，and the system will not fail until the cumulative shock damage exceeds the threshold valueD1.Thus， the reliability function of the hard failure process can take the form as

Specifically，when the magnitudes of the shocks are independent identically distributed(i.i.d)random variables following a normal distribution asWi～N(μW，σ2W)， the reliability function can be obtained as

where Φ(·)is the cumulative distribution function of a standard normally distributed variable.

As shown in Fig.2(b)，the hard failure threshold value decreases fromD1toD2right after the arrival of a run ofmshocks.In such a shifting-threshold situation，the equivalent reliability function is

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2.2 Soft failure model due to degradation and shocks

Descriptions for gradual degradation，X(t)，make a great difference in various studies［4，7-8］.Some researchers apply the linear degradation path model［6，12］，and a stochastic process can also be employed［13］.

Each random shock can cause abrupt damage.The abrupt damage in the overall degradation are measured by the shock damage sizes as{Y1，Y2，…}.The cumulative damage size due to random shocks until timetis given as［10］

whereN(t)is the total number of shocks to the system until timet.

Then the overall degradation of the system including gradual degradation and shock damage can be expressed asXs(t)=X(t)+S(t).Soft failure will occur when the overall degradation is beyond the threshold valueH.Thus，the probability that the component survives is

In this paper，it is assumed that the magnitude of gradual degradation，X(t)，at timetfollows a normal distribution asN(μ(t)，σ2(t)).And the shock damage sizes are also i.i.d variables taking the form asYi～N(μY，σ2Y).Considering the independence of the two random variables，Xs(t)takes the distribution asN(μ(t)+N(t)μY，σ2(t)+N(t)σ2Y).Reliability is equal to the probability that the total damage to the system has not exceeded the failure threshold.Using Eqs.(1)and(6)，the reliability function for the soft failure process can be obtained as

2.3 System reliability analysis

2.3.1 Reliability analysis under a cumulative shock model

Fig.2(a)shows a cumulative shock model;that is to say，a system is considered to be failed only when the cumulative magnitudes exceed the threshold valueD1.Although the two failure processes are dependent for being affected by the same shock process，it is still reasonable to assume that they are physically independent of each other［10］.Thus，the probability that the component survives from failure can be expressed as

It is also assumed that the shock process follows the homogeneous Poisson process described in Section 2.1.Based on the specific assumptions ofX(t)，Wi，Yi(i=1，2，…)，and using Eqs.(3)and(7)，the reliability function of the dependent competing failure process can be obtained as

2.3.2 Reliability analysis due to a shifting-thresholdvalue

In Fig.2(b)，the hard failure threshold decreases from D1toD2right after the arrival ofmshocks.This kind of problem was considered by Jiang et al［12］.In their research，a generalized run shock model was given.In this paper，we propose a different approach for reliability analysis based on the cumulative shock model.Similarly，by using Eqs.(4)and(7)，we can obtain the reliability that the component survives from failure as

When the assumptions made in the previous section are taken into consideration，the specific reliability function take the form as

3 Numerical Example

In this section，a case study is provided to illustrate the proposed models in Section 2.This example is based on the fatigue crack growth data of an alloy［14］.The loading cycles are considered to be the loading timet.Information about the shocking process is used to demonstrate the proposed model.

Fig.3 shows the fatigue crack growth path.We select 13 samples out of a total of 21 samples since the remaining samples are not completed.The least squares method is employed to evaluate the parameters，and the results are

whereKis the number of cycles.

Fig.3 Fatigue crack growth paths of samples

Sizes of random shock loads，Wi(i=1，2，…)，which are measured in units of component life［3］，are assumed to follow a normal distribution，Wi～N(2，0.5);the shock damage sizes take the form asYi～N(0.02，0.01)fori=1，2，…，and the threshold value of soft failureH=2.0 inches(1 inch=25.4 mm);the threshold value of hard failureD1=35 units;and the arrival rate λ =0.5 × 10-4.In addition，we assume thatm=3，and the lower level of hard failure threshold valueD2=25 units.

According to Eqs.(9)and(11)，the reliability curves of the two models constructed in Section 2.3 are plotted in Fig.4，respectively.For Case 1，reliability almost remains at 1 whenK＜5×104cycles.This is because the effect of random shocks is not significant and the gradual wear degradation amount is not large enough to cause any failure.In the next time period，with the gradual degra-dation and random shocks，reliability drops quickly，and the effect of shocks becomes a significant factor of the reliability performance.Eventually，reliability drops to a quite low level whenK＞2.5 ×105cycles.Analysis of Case 2 can be obtained similarly.

Fig.4 Reliability function of different models

In addition，the reliability curve based on degradation without shocks is also provided in comparison with the two shock-considering models in Fig.4.According to the results，it can be concluded that shocks will accelerate the failure process，which validates the effectiveness of models constructed in Section 2.Moreover，when the hard failure threshold value decreases fromD1toD2，failure will occur sooner.

Fig.5 shows the reliability distributions with different rates of random shocks in Case 2.A higher arrival rate will make the reliability drop more quickly.It is reasonable because more intensively frequent random shocks，larger sizes of shock loads and shock damages will resultantly make a component more vulnerable.

Fig.5 Sensitivity analysis for λ

4 Conclusion

This paper focuses on the reliability analysis of components with dependent competing failure processes due to hard failure and soft failure.Generalized probabilistic methods based on the cumulative shock model are proposed and two specific models with normal distribution are obtained.Compared with the degradation process without shocks，models established in this paper indicate that random shocks have significant effects on the failure process.When considering the shifting-threshold situation，reliability decreases even faster.

The two proposed models only consider one degradation path.In real situations，components may have multiple degradation measures，and this will be the focus of our future research.

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