The present invention relates to a system and method for estimating fatigue life of components operating under cyclic stress, particularly, components subject to low-cycle fatigue (LCF).
In a plurality of applications in technical systems parts or components can be subject to stresses which alternate or vary over time, of a mechanical or thermal nature for example. In such cases individual parts can for example be subject to direct mechanical stresses through the occurrence of compressive or tensile forces. A time-varying thermal stress of this type arises on the other hand for example for the parts or components in a turbine system, especially in a gas turbine, when the gas turbine is started up or shut down.
Extreme cyclic loading, both mechanical and thermal, results in material fatigue, also referred to as low cycle fatigue (LCF), which, in many cases limits the life of the component. The design of the components with respect to LCF life is done based on corresponding material curves which are determined in most cases by performing curve fits to experimental LCF test data. To avoid material failures during service life, a series of safety factors are taken into account while designing the component. These should, in particular, contain the uncertainties resulting from the determination of the material curves in estimating the fatigue life span of the component.
Now it is seen that the individual methods of fitting, for which a large number of degrees of freedom are available, differ vastly in teens of their robustness with respect to changes in the basic underlying measurement data. But since high uncertainties in the material curves would in turn lead to high uncertainties in estimated fatigue life spans, such a fit method that is not very robust would lead to high trade-offs in the calculated, allowable fatigue life spans, and hence indirectly to high costs.
A common way to determine LCF material curves is to use a simple approach based on linear regression and the principle of least squares (LS). However, as mentioned above, such a method is not optimal in terms of robustness. The uncertainties in the resulting material curves are taken as given and accounted for within the safety-factor concept.
The object of the present invention is to provide a method and system for robust and reliable determination of LCF material curves from experimental test data. These curves can then be used in order to estimate average LCF life times of components operating under cyclic loading.
The above object is achieved by the method according to the claims and the system according to the claims.
The underlying idea of the present invention is to customize or fit the LCF material curves to experimental data from strain controlled fatigue (LCF) tests in the framework of the statistical “Maximum Likelihood” theory. The (mid-life) stress amplitude as well as the number of cycles to crack initiation are considered as random variables. Depending on the underlying probability distribution functions as well as the curve parameters that are to be determined, the “curve fit” here changes into a problem of non-linear optimization that differs from a conventional least squares (LS) approach. The choice of stress amplitude and LCF life time as dependent variables (in contrast to, e.g., elastic, plastic, or total strain amplitudes) has proved to be beneficial with regard to the robustness of the method, which is thus more reliable and obviates the need for providing high factors of safety in fatigue life estimation.
Additional advantages are realized by embodiments according to the dependent claims.
In an exemplary embodiment, for improved reliability in obtaining LCF characteristics, said probability distribution functions fσ and fN represent log normal distributions.
In a preferred embodiment to aid computation, the computing of said first set and second set of curve parameters comprises determining those values of said parameters for which a negative logarithm of said likelihood functional ‘L’ assumes a minimum value, such that said likelihood functional ‘L’ is maximized.
In an exemplary embodiment, said first material curve (10) is defined by a Romberg-Osgood equation based relationship between stress ‘σa’ and strain ‘εa’, wherein
and wherein E, K′ and n′ form said first set of parameters θC1 whose values are determined such that said likelihood functional ‘L’ is maximized.
In an exemplary embodiment, said second LCF material curve (20) is defined by a Coffin-Manson-Basquin equation based relationship between strain ‘εa’ and number of cycles to crack initiation wherein
and wherein ε′f, σ′f, E, b and c form said second set of parameters θC2 whose values are determined such that said likelihood functional ‘L’ is maximized.
In one embodiment, at least one of said parameters in said first and second set of parameters has predetermined fixed value from known material characteristics of said component (6). By fixing the value of one or more parameters, for example, by incorporating prior knowledge of material properties, the computational burden on the system may be reduced.
In accordance with another aspect of the present invention, a method is provided for operating a component under cyclic stress, said method comprising scheduling a downtime or maintenance interval of said component taking into account an estimated fatigue life of said component, said estimated fatigue life being determined by a method according to any of the above-mentioned embodiments.
In an exemplary embodiment, said component is a gas turbine component. The present invention is particularly useful for gas turbine components which operate under high cyclic stress (both mechanical and thermal) and hence prone to low cycle fatigue.
The present invention is further described hereinafter with reference to illustrated embodiments shown in the accompanying drawings, in which:
Referring no to
The testing means 2 may comprise, for example, a servo-controlled closed loop testing machine, a portion (length) of component 6 or the representative specimen having a uniform gage section is subject to axial straining. An extensometer may be attached to the uniform gage length to control and measure the strain over the gauge section. In the illustrated embodiment, a first strain-controlled test performed on the component/specimen involves applying a completely reversed cyclic straining to the component/specimen and measuring the corresponding stress amplitudes for various test strain (amplitude) levels. A second strain-controlled test performed on the component/specimen involves, for different test strain (amplitude) levels, applying a completely reversed cyclic straining on the component/specimen with constant strain amplitude till fatigue failure (i.e., crack initiation) of the component/specimen occurs, and measuring the number of cycles to crack initiation for each test strain (amplitude) level.
As shown in
Referring back to
The first LCF material curve, namely, stress-strain curve to be fitted on the first set of data samples, is defined by a first set of curve parameters, while the second curve, namely, strain-life curve to be fitted on the second set of data samples, is defined by a first set of curve parameters. The objective of the proposed “curve “fit” method is to numerically determine the values of the above-mentioned curve parameters for which a likelihood function ‘L’ as defined in equation (1) is maximized.
where
fσ and fN are probability distribution functions,
εa,j and εa,k represent test strain levels in the first and second sets of data samples respectively,
σa,j represents measured stress amplitude values in the first set of data samples
Ni,k represents the measured number of cycles to crack initiation in the second set of data samples, and
θC1 and θC2 represent the first set and second set of curve parameters respectively whose values have to be determined such that the likelihood functional ‘L’ is maximized.
Depending on the probability distribution functions used as well as the curve parameters that are to be determined, the “curve fit” here changes into a problem of non-linear optimization that differs from a conventional least squares (LS) fit method. In the embodiment illustrated herein, the probability distribution functions fσ and fN represent log normal distribution. However, alternate embodiments may incorporate other types of probability distribution, such as Weibull distribution. As illustrated below, the computation involved herein comprises determining values of the sets of parameters θC1 and θC2 for which a negative logarithm of the likelihood functional ‘L’ is minimized.
As an example, the first LCF material curve may defined by a Romberg-Osgood equation based relationship between stress ‘σa’ and strains ‘εa’ as expressed in equation (2) below:
wherein E, K′ and n′ form the first set of parameters θC1, referred to subsequently herein as Romberg-Osgood parameters, or θRO.
Again, as an example, the second LCF material curve may be defined by a Coffin-Manson-Basquin equation based relationship between strain ‘εa’ and number of cycles to crack initiation ‘Ni’ as expressed in equation (3) below:
wherein ε′f, σ′f, E, b and c form the second set of parameters θC2, referred to subsequently herein as Coffin-Manson-Basquin parameters, or θCMB.
Thus, in this example, the problem involves determining θRO and θCMB such that the negative logarithm of the likelihood functional ‘L’ is minimized. As mentioned above, the probability distribution functions fσ and fN in this case represent log normal distributions. For the probability distribution functions fσ and fN, the corresponding median values for σa and Ni are given respectively by equations (2) and equations (3) mentioned above. That is to say the median value for σa is RO−1(εa,j|θRO) and the median value of Ni is CMB−1(εa,k|θCMB). The variances for log σa and log Ni are assumed to depend on εa. Based on the above considerations, the problem may be considered to be that of minimizing the expression given by equation (4):
As shown to one skilled in the art, the parameters in θRO and θCMB may be constrained to fulfill equations (5a) and (5b) below:
σ′f=K′·(ε′f)n′ (5a)
b=n′·c (5b)
Based on the interrelation mentioned above, and assuming that the variances σRO and σCMB are constant, the problem may be finally considered to be reduced to minimizing the expression given by equation (6)
The above non-linear functional is minimized using numerical methods to yield the values of the parameter sets θRO and θCMB. If one or more individual parameters are known in advance, then their values can be fixed in equation (6). The dimension of the parameter space that is to be examined is reduced as a result. As an example, the elastic modulus E may be determined from prior experimentation and the value thus obtained may be fixed in equation (6). This would greatly reduce computational burden on the system.
Referring to
As can be seen, the “curve fit” method proposed herein involves a problem of non-linear optimization that differs vastly from a conventional least squares (LS) fit method. Unlike in the conventional procedure with regard to the residues that are to be minimized, (i.e., amplitudes of stress and/or elastic and plastic strain), the dependent variables herein are identified as the stress amplitudes and the fatigue life span. This has proved to be extremely beneficial with regard to the robustness of the method, which is thus more reliable and obviates the need for providing high factors of safety in fatigue life estimation.
Referring back to
The output of the design means 4 may comprise, for example, a prescribed number of cycles of operation for different levels of operational cyclic stress. Based on the output of the design means 4, the operation of the component 6 may be controlled by the control means 5. In particular, the control means 5 may be comprise prognosis means for scheduling and implementing appropriate downtimes or maintenance intervals for the component 6 taking into account the estimated life-span and operating stress on the component 6.
While this invention has been described in detail with reference to certain preferred embodiments, it should be appreciated that the present invention is not limited to those precise embodiments. Rather, in view of the present disclosure which describes the current best mode for practicing the invention, many modifications and variations would present themselves, to those of skill in the art without departing from the scope and spirit of this invention. The scope of the invention is, therefore, indicated by the following claims rather than by the foregoing description. All changes, modifications, and variations coming within the meaning and range of equivalency of the claims are to be considered within their scope.