JP2001324432A - Method for predicting creep life of equipment under narrow-band random stress variation and narrow-band random temperature variation - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide a method for predicting a creep life, when both narrow- band random stress variations and narrow-band random temperature variations are applied on equipment. SOLUTION: In the method for predicting the creep life of equipment under random stress variations and random temperature variations for obtaining the probability density function of the amount of damage accumulation, on the basis of a Robinson damage sum and to predict the life of the equipment on the basis of the probability density function, (1) a damage coefficient which expresses the amount of damage per unit time is approximated by a linear expression, when the random stress variations and random temperature variations lie within narrow bands, and (2) the random stress variations σ(t)(instant) are expressed by the sum of a time mean value σ(t)(mean) and probability amount of variations σ', and the random temperature variations θ(t)(instant) are expressed by the sum of a time mean value θ(t)(mean) and a probability amount of variations θ'(t).

Description

DETAILED DESCRIPTION OF THE INVENTION

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for predicting the life of an industrial gas appliance or the like, and more particularly, to a gas appliance by treating a damage process of a gas appliance component or the like as a stochastic process. And other methods for predicting creep life.

[0002]

2. Description of the Related Art High-temperature gas equipment materials such as industrial furnaces do not have a common standard for inspection timing and method, and countermeasures are taken according to the use of the equipment. These devices are often used in thermally and chemically harsh environments such as high temperatures and corrosion.Even if the devices have exactly the same specifications, the load on the devices differs depending on the user, and the accumulation speed of damage to the devices and the life of the devices A relatively large variation occurs. To address this problem, a method of monitoring the state of the components of the device in detail is conceivable.However, there are problems such as limitations on the operating environment of the sensor and installation location, and an increase in the cost required for monitoring. At present there is little technology.

[0003] In particular, in gas equipment under actual operating conditions, the starting and stopping of the equipment are repeated according to the operation schedule of the equipment, and the amount of heat transfer with an object to be heated is varied. Narrow band random stress fluctuation in which the peak value of the load stress fluctuates randomly, and narrow band random temperature fluctuation in which the peak value of the temperature of the load fluctuates randomly. Here, the narrow band means that the variation in the peak value of the applied stress of the thermal stress or the applied temperature of the temperature is relatively narrow. In high temperature gas appliances,
It is considered that damage due to creep deformation occurs. Creep deformation is a deformation caused by an increase in strain with time when a certain amount of stress is applied to a material at a temperature equal to or more than half the melting point at an absolute temperature. For this reason, in the development of high-temperature gas appliances, it is necessary to develop a damage evaluation technique capable of evaluating damage accumulation due to creep deformation caused by load fluctuation under actual operating conditions.

[0004] As such a damage evaluation technique, a method of treating a damage process of a material as a stochastic process is known. The following two types of methods are known.

[0005] The first method treats crack propagation in a material as a stochastic process, and furthermore, studies using crack propagation resistance as a factor of irregularity in a model of damage development, and research on load application. It can be categorized into studies employing stress irregularities. These are damaged by introducing a random term, which is the source of the irregularity, into a part of the Pais rule, which is basically a deterministic equation that expresses crack propagation regardless of the cause of the irregularity, and forming it as a stochastic differential equation. Build a model of development.

[0006] The second method is based on the concept of continuum damage mechanics, in which the temporal and spatial variations of microscopic material properties due to the occurrence of fluctuating loads and microcracks are caused by macroscopic changes in material strength. It is a method of formulating the effect on the change of various characteristics and describing the evolution of damage. This method is one of practical methods because it deals with damage parameters that can be defined from macroscopic characteristics.

As a typical example of the above method, Silberschm
There is a study by idt. In this study, the nonlinear Langevin equation for the damage accumulation of randomly varying short-axis tensile loads (I-mode) was

(Equation 1) Gave. Where f (p) is the right-hand side of the deterministic equation for mode I damage

(Equation 2) L (t) is a probability term, A, B, C, and D are experimental values, and g (p) is modeled assuming that the strength of the probability term is proportional to the degree of damage accumulation at a given time . Also, Silberschmidt
The numerical analysis of the nonlinear Langevin equation shows the qualitative change of the PDF with respect to the change of the stress fluctuation strength, and the calculation shows the empirical facts about the shortening of the material life caused by the stress fluctuation. ing.

[0008]

However, the above-described method for estimating the creep life of equipment has the following problems. That is, when estimating the creep life of a gas appliance by the first method, since the calculation is basically based on the progress of the crack, it is necessary to determine a place where the crack is easily formed in the appliance. At this time, generally,
A location where cracks are likely to occur is determined based on a location where stress is likely to be concentrated in the equipment. However, since the components constituting the gas equipment operating at the production site have complicated shapes, it is often difficult to predict a place where cracks are likely to occur in the equipment. Further, since the components constituting the gas appliance are complicated in shape, the process up to destruction may greatly differ depending on the place where the crack is formed. Furthermore, when a crack is formed, the situation must be measured in detail, but it is difficult to measure the situation in detail because the components constituting the gas appliance have a complicated shape.

[0009] From the above, when predicting the creep life of gas equipment operating at a production site with high accuracy, determinism basically representing crack propagation by directly calculating as a clear crack having a clear size and position. It is often difficult to construct a model of damage evolution by incorporating a random term, which is a source of irregularity, into a part of the Paris rule, which is a statistical equation, and forming a stochastic differential equation.

In the second method, the method of predicting the creep life of gas equipment by the method of Silberschmidt has the advantage that it is not necessary to consider the progress of cracks. However, there is no mention of the case where the temperature fluctuates. No, its impact cannot be evaluated. Therefore, the creep life cannot be accurately predicted when there is a temperature fluctuation. However, in gas appliances, usually, not only the stress but also the temperature often fluctuates, and in such a case, there is a problem that the gas appliance cannot be applied. Therefore, it is difficult to accurately predict the life of gas appliances by this method.

The present invention has been made to solve the above problems. That is, the subject is
When treating the damage process of a material as a stochastic process, the influence of the temporal and spatial fluctuations of microscopic material properties due to the occurrence of fluctuating loads and microcracks on the macroscopic changes in material strength A method for predicting creep life of equipment by formulating and describing the evolution of damage, providing a method for predicting creep life when both narrow-band random stress fluctuations and narrow-band random temperature fluctuations are applied to the equipment. That is.

[0012]

In order to solve the above problems, a method for predicting creep life according to the present invention obtains a probability density function of a damage accumulation amount based on Robinson damage sum, and obtains a device based on the probability density function. In the method for predicting the creep life of equipment under random stress fluctuations and random temperature fluctuations for predicting the service life of a device, (1) when the random stress fluctuations and the random temperature fluctuations are in a narrow band, the damage coefficient representing the amount of damage per unit time is calculated. (2) expressing the random stress fluctuation σ (t) (instantaneous) by the sum of a time average σ (t) (average) and a stochastic fluctuation σ ′; It is characterized in that θ (t) (instantaneous) is represented by the sum of the time average value θ (t) (average) and the stochastic variation θ ′ (t).

According to the creep life prediction method having such characteristics, the probability density function of the amount of accumulated damage is obtained using the damage coefficient based on the sum of Robinson's damages. Robinson's damage sum is a method of calculating the damage accumulation amount by using the Larson and Miller parameters on the horizontal axis and accumulating the life determined by the damage degree curve using stress on the vertical axis. Larson and Miller parameters are empirical functions where stress is represented by temperature and lifetime in the case of creep rupture. Therefore, both stress and temperature can be considered when estimating life. Furthermore, the random stress amplitude fluctuation σ (t) (instantaneous) is represented by the sum of a time average value σ (t) (average) and a stochastic fluctuation σ ′ (t),
The random temperature fluctuation θ (t) (instantaneous) is the time average value θ (t)
(Average) and the stochastic variation θ ′ (t).
By approximating the damage coefficient representing the damage amount per one time with a linear expression, a Langevin equation of the damage accumulation amount is derived. Here, the Langevin equation of damage accumulation is a function that includes a stochastic process due to stress fluctuations and temperature fluctuations in a dynamic equation that shows the damage evolution indicated by Robinson's damage sum when the stress and temperature are constant. Stochastic differential equation. Thus, Robinson's sum of damages is extended when the load stress and load temperature fluctuate randomly in a narrow band. Accordingly, a model of the development of damage accumulation due to creep deformation when both the applied stress and the applied temperature randomly fluctuate in a narrow band by solving the Langevin equation can be shown. That is, it is possible to accurately predict the life of a gas appliance in which both the stress and the temperature fluctuate.

Further, the damage accumulation process uses a Langevin equation and an equivalent Fokker-Planck equation.

That is, a Fokker-Planck equation corresponding to the Langevin equation is derived. Here, the Fokker-Planck equation is a second-order partial differential equation in a probability density function which is derived assuming that a third or higher order moment of a transition amount can be ignored in a continuous Markov process. In addition, the Markov process is a process in which the future information time t ₂ on the probability variable is fully described by the information at the present time t _1. By solving this Fokker-Planck equation, the probability density function of the damage accumulation amount at an arbitrary time from the start of the experiment to the destruction can be expressed in a normal distribution form. Furthermore, based on this Fokker-Planck equation, it is possible to obtain an equation for predicting the remaining life from an arbitrary damage accumulation amount in a material that has already been damaged. Therefore, it is possible to obtain a probability density function of damage and a predictive expression of the remaining creep life when both stress and temperature fluctuate randomly in a narrow band.

[0016]

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS An embodiment embodying a method for predicting the creep life of equipment under narrow-band random stress fluctuation and narrow-band random temperature fluctuation according to the present invention will be described in detail with reference to the accompanying drawings and mathematical expressions. Will be described. In addition, FIG. 1 briefly describes the symbols used in the mathematical expressions of the present embodiment.

(Thought of Damage Model by Stochastic Differential Equation) A model of damage development of a material by a stochastic differential equation is as follows.
It is a method of expressing random time evolution in the state space, considering changes in state quantities such as the length of cracks in the material and the amount of damage accumulated in the material as a stochastic process.

The curves (1) to (3) 2, the start of the experiment at t = t _b
When random stress fluctuation and random temperature fluctuation are applied to a material having initial damage p = p _b , the damage p (t) accumulated in the material is
Fig. 3 schematically shows a path taken on a pt plane. A stochastic differential equation is used to describe such a path. In the present embodiment, the following Langevin equation is used.

(Equation 3) Where a (p, t) is the right side of the deterministic differential equation for damage development, b (p, t) is the effect of random fluctuating stress on damage development, and dW is the Wiener process Represents the increment of. This equation does not represent the path of development of the damage obtained in a single experimental result, but rather the entire path drawn by multiple experimental results.

The two distributions g (p, t | p _b , t _b ) in FIG. 2, t = t
_g1 and t _g2 are the probability density functions (PDF) estimated from the frequency of the path drawn on the pt plane as a result of repeating the experiment under the same initial conditions (p _b , t _b ). It represents the time change. The PDF is delta-like immediately after the start of the experiment, but as the damage develops thereafter, the peak attenuates as shown by the dashed line C in the figure, and the width of the distribution expands with it. It is the Fokker-Planck equation that represents the time change of such PDF

(Equation 4) It is. This equation can be derived from the equation (Equation 3). By solving this equation, the damage probability distribution and the average damage accumulation amount at any time after the start of the experiment (two-dot chain line E in the figure), deviation Etc. can be estimated. Also PDF
It is also possible to calculate the remaining life distribution based on the concept of First Passage Time described later.

(Application to Creep Life Evaluation) The following analysis shows Robinson's life ratio which is a linear damage law based on a creep damage curve under a constant stress and temperature condition indicated by Larson and Miller parameters. The rule is extended to the case of narrow-band random stress amplitude fluctuation and narrow-band random temperature fluctuation using Langevin equation and Fokker-Planck equation.

That is, consider a case where a material is in a stress / temperature range where creep is a problem, and a random fluctuating stress and fluctuating temperature are applied. Now, it is assumed that these fluctuation values can be approximated by a step-like function that jumps at equal time intervals Δt and keeps constant stress σ _i and temperature θ _i until the next jump. Here, the subscript i represents the number of jumps for each Δt from the start of the experiment until the time of interest. Damage accumulated in the material at a certain number of repetitions n after the start of the experiment (hereinafter referred to as damage accumulation)
_pn is the sum of the damage accumulated in the material for each square wave according to Robinson's damage sum

(Equation 5) Can be expressed as Where p _n delta _t is damage accumulation amount after nΔt seconds, T _i
Represents the creep rupture time of the material when a constant stress and a constant temperature are applied to the undamaged material. In practice, T _i is assumed to be a function of stress and temperature T _i = T _i (σ, θ), and Larson,
It can be estimated from a damage degree curve using the mirror parameter σ = θ (k + logT _i ). Here, k is a constant determined by experiment. In the equation (Equation 5),
Formally, 1 / T _i represents the amount of damage the material suffers per unit time. Therefore, a function representing the amount of damage accumulated per unit time when a creep test is performed at a certain stress σ and temperature θ

(Equation 6) And this is called the creep damage coefficient, which is a basic quantity that determines the creep damage accumulation process. By using the creep damage coefficient, the damage amount dp accumulated by the material at a certain time interval dt can be expressed by the following equation.

(Equation 7) This is the dynamic equation that represents the evolution of creep damage over time.

Next, the influence of randomly fluctuating stress and temperature on the damage dynamic equation (Equation 7) will be examined. Or later,
The stress and temperature that fluctuate randomly with time are referred to as fluctuating stress and fluctuating temperature, respectively. The instantaneous values are represented by σ (instantaneous) for the fluctuating stress and θ for the fluctuating temperature.
(Instantaneous). Here, assuming the steady operation of the working machine, it is assumed that the fluctuating stress and the fluctuating temperature fluctuate randomly around a certain time average value, and these are changed to the time average value σ (average) (t) as follows. , Θ (mean (t) and stochastic fluctuations σ ′, θ ′.

(Equation 8)

(Equation 9) Each term in these equations is a function of time. Further, here, among the components of the equations (8) and (9), the narrow-band fluctuation in which the magnitude of the stochastic fluctuation is sufficiently smaller than the average value.

(Equation 10)

[Equation 11] think of. The stochastic variation on the right side of Equations (Equation 12) and (Equation 13) is calculated using parameters Qσ and Qθ representing the magnitude of the variation and noise ξσ (t) and ξθ (t) for expressing the probabilistic variation. It expresses like.

(Equation 12)

(Equation 13) Here, ξ _i (t), i = σ, θ is a mathematical expression of a fast-changing irregular function having a Gaussian distribution, and its ensemble average is <ξ _i (t)> = 0, different time t ≠ The values ξ _i (t) and ξ _i (t ′) at t ′ are statistically independent, and the autocorrelation function is <ξ _i (t) ξ _i using the Dirac delta function δ (t).
(t ′)> = δ (tt ′). Also, ξσ
(t) and ξθ (s) are independent of each other <ξσ (t) ξθ (s)> =
Set to 0. Therefore, σ ′ and θ ′ have the following properties. (a) The ensemble average of σ´ is

[Equation 14]

(Equation 15) (b) The auto-correlation and cross-correlation are

(Equation 16)

[Equation 17]

(Equation 18) (c) σ ′ (t) and θ ′ (t) show a Gaussian distribution. In order to estimate the amount of damage accumulation, the instantaneous value σ (instantaneous) of the fluctuating stress and the instantaneous value θ (instantaneous) of the fluctuating temperature are used to _calculate φ _c (σ (instant),
θ (instantaneous)), but it is practically difficult to directly use the fluctuating stress and fluctuating temperature. Therefore,
As shown below, the damage coefficient φ _c (σ (instantaneous), θ (instantaneous)) is Taylor-expanded around σ (average) and θ (average), and the damage coefficient is estimated from the average value and the intensity of fluctuation. .

[Equation 19] Here, equation (8) and equation (9) were used. However, under the conditional expressions (Equation 10) and (Equation 11) of the narrow band fluctuation, the second-order and higher-order terms of the equation (Equation 19) become very small to other terms. Therefore, ignoring second-order or higher minute terms in equation (Equation 19),

(Equation 20) To approximate the damage factor. By substituting equation (20) into equation (7), the following equation is obtained.

(Equation 21) Get. This equation is a Langevin equation representing Robinson's damage law in the case of narrow-band random stress / temperature fluctuation. Φ _c (average) in the above equation represents φ _c (σ (average), θ (average)), and dWσ (t) and dWθ (t) are σ ′,
This is the increment of the Wiener process for θ ′. Note that there is a relationship dW _i = ξ _i dt between dW _i and ξ _i , i = σ, θ.
Since the coefficient of each term on the right side of the equation (Equation 21) is a constant, it can be easily integrated, and a developed equation of p (t) is obtained as in the following equation.

(Equation 22) Here, t _b is the start time of the test, p _b represents the initial amount of damage that has already existed in the material at time t _b. This equation represents the results of countless creep tests starting from the initial state (p _b , t _b ). However, since what is needed in practical use is the stochastic expected value of damage accumulated at time t, it is possible to estimate the evolution of the average value by taking the ensemble average <p> of the above equation. it can.

(Equation 23) As is evident from this equation, the evolution of the mean value of the damage in this model is consistent with the evolution of the damage calculated by Robinson's damage law in the usual way in the absence of variation. Furthermore, the square deviation of the change in damage accumulation is

(Equation 24) Becomes Where α = (∂φ _c (mean) / ∂σ) Qσ,
β = (∂φ _c (average) / ∂θ) Qθ. Thus, the distribution of damage at any time between the time the material begins to fail and the time it breaks has a slope proportional to the slope of the damage curve due to creep and the magnitude of stress and temperature fluctuations, and a spread proportional to the square root of the elapsed time. Will be.

(Foker-Planck Equation) In the damage evaluation and life evaluation of a material, not only the average value and deviation of damage accumulated in the material at a certain point in time, but also the PDF and probability distribution of damage play an important role. The commonly used normal distribution, lognormal distribution, and Weibull distribution are intended for the probability of failure.However, by solving the Fokker-Planck equation, the time change of the PDF of the amount of damage accumulated in the material can be calculated. Can be caught.

The Fokker-Planck equation can be derived from the Langevin equation. In this analysis,
The following partial differential equation is obtained from the equation (Equation 21).

(Equation 25) This equation is the Fokker-Planck equation of the fatigue damage accumulation process for the narrow band random stress amplitude fluctuation and the narrow band random temperature fluctuation. Here, g (p, t | p _b , t _b ) is a conditional PDF conditioned by an initial value (p, t) = (p _b , t _b ). In the above equation, since the coefficient of each term is a constant, g
(p, t | p _b , t _b ) can be solved analytically. The final solution is the following normal distribution:

(Equation 26) From this equation, under the condition that the initial damage (p _b , t _b ) exists,
It is possible to estimate the PDF probability density distribution of the damage accumulation amount at an arbitrary time from the start of damage to the destruction, or the PDF of the time to reach the arbitrary damage accumulation amount.

Further, the remaining life distribution of the material can be estimated based on the concept of remaining life evaluation First Passage Time. In this analysis, the First Passage Time is the undestructed state. The damage value in 0 ≤ p <1 is the destruction state in the shortest time.
It means the average time required to reach p = 1. This time is obtained from the Fokker-Planck equation and its solution as follows.

[Equation 27] Here, T (p) represents the average remaining life expected from the damage accumulation amount p at a certain point in time. The first term on the right-hand side represents the remaining life value given by the existing Robinson's damage rule when there is no change in the stress amplitude and temperature for each iteration, and the second and subsequent terms show the effect of the change on the remaining life.

As described above in detail, in the method for predicting the life of a device under a narrow-band random stress fluctuation according to the present embodiment, the damage coefficient φ _c (σ (instant), θ (instant))
Is subjected to Taylor expansion around σ (average) and θ (average), ignoring second-order and higher-order small terms in the equation (Equation 19) in which the damage coefficient is estimated from the average value of fluctuation stress and the intensity of fluctuation. , Equation (20) is obtained. Further, by substituting this equation into Equation (Equation 7), it is possible to obtain a Langevin equation (Equation 21) representing the sum of Robinson's damages in the case of narrow-band random stress fluctuation and narrow-band random temperature fluctuation. Equation (Equation 2)
Since the coefficient of each term on the right side of 1) is a constant, it can be easily integrated, and a development formula of the normalized damage accumulation amount p (t) is obtained as in Expression (22). Thereby, the average value and deviation of damage accumulated in the material at a certain point in time when both stress and temperature fluctuate randomly in a narrow band can be obtained. Therefore, it is possible to accurately predict the life of a device in which both the stress and the temperature fluctuate randomly.

Further, a Fokker-Planck equation (Equation 25) representing the evolution of a conditional probability density function relating to Robinson's damage sum corresponding to the Langevin equation is derived. Since each coefficient in the equation (Equation 25) is a constant, By solving it, a conditional probability density function g (p, t | p _b , t _b ) of the normal distribution type shown in the equation (Formula 26) is finally obtained. By solving this Fokker-Planck equation, a normally distributed conditional probability density function in the case where both stress and temperature fluctuate randomly is obtained,
Furthermore, the probability density function calculates the probability density distribution of the damage accumulation amount at any time from the start of damage to the destruction under the condition where the initial damage (p _b , t _b ) exists, or It is possible to estimate the probability density distribution of the time to reach the damage accumulation amount. Furthermore, based on the Fokker-Planck equation, a prediction formula of the remaining life of a material that has already been damaged from an arbitrary amount of damage accumulation can be obtained. Therefore, it is possible to accurately predict the life of a gas appliance in which both the stress and the temperature fluctuate.

It should be noted that the present embodiment is merely an example and does not limit the present invention in any way. Therefore, naturally, the present invention can be variously modified and improved without departing from the gist thereof.

(Estimation of Life of Gas Appliances in Ceramic) Next, estimation of the life of gas appliances in ceramic is performed based on the ceramic crack growth rule. Behavior of SCG typically relationship stress intensity factor K _I and crack propagation velocity v

[Equation 28] Is represented by Where a is the crack length, K _I denotes a stress intensity factor of I mode. For most structural ceramic materials, the power law crack propagation velocity

(Equation 29) Represented by Where K _IC is the critical stress intensity factor, A
And n are material constants. The stress intensity factor is related to the applied stress σ and a by the following equation.

[Equation 30] Here, Y is a parameter related to the shape of the crack.
In this section, the crack propagation law of ceramics by equations (28) to (30) is used.

(Equation 31) Consider the evolution of the crack length and the probability density function of the crack length when the applied stress fluctuates randomly. In this analysis, it is assumed that the stress particularly indicates a narrow-band random fluctuation.

(Lange-Vang Equation of Crack Growth Rate) The influence of stress fluctuation on the crack growth rate da / dt is expressed by an additional term to the equation (Equation 31) as follows.

(Equation 32) Here, the first term on the right-hand side indicates the crack propagation speed when the stress σ is constant, and corresponds to the crack propagation speed when the stress used normally in the crack propagation equation does not change. The second term on the right side represents the effect on the crack growth rate due to the random fluctuation of the applied stress.The coefficient α is a coefficient related to the strength of the fluctuation, and ξ (t) is the ensemble average of which <ξ
(t)〉 = 0, and the autocorrelation function is <ξ (t) ξ (t-τ)〉 =
δ (τ) is a random function having the characteristic of τ = 0. Now, as one attempt, assume a state in which the stress fluctuates randomly over time with respect to some average value.

[Equation 33] Here, σ (instant) is the instantaneous value of the fluctuating stress, σ (average)
Indicates a time average value, and 'indicates a variation. Further, it is assumed that the variation of the stress exhibits the following properties. That is, (1) the ensemble average of σ ′ is

(Equation 34) Becomes (2) σ ′ is determined by the random variable ξ (t) and the constant Q related to the intensity of fluctuation.

(Equation 35) And its autocorrelation function is

[Equation 36] Becomes (3) σ ′ indicates a Gaussian distribution. (4) Also, since we are considering random fluctuation in a narrow band,

(37) The crack growth rate resulting from the application of the fluctuating stress having the above properties to the material is a random variable. In order to obtain the crack propagation speed at this time, Expression (Equation 33) is substituted into Expression (Equation 31). However, the fluctuating stress has the above property (I
In consideration of having V), the equation (Equation 31) is subjected to Taylor expansion for σ (average) as follows.

(38) Using equation (33),

[Equation 39] This is the Langevin equation relating to crack propagation when a variable stress is applied. Γ in the above equation
= A (Y σ (average) / K _IC ) ⁿ The equation (39) corresponds to the equation (32), and the coefficient of the second term on the right-hand side with respect to the stress fluctuation strength is

(Equation 40) Can be determined as follows. The equation (Equation 39) becomes a linear equation when n = 0,2, but when n = 0, it becomes a commonly used deterministic equation and is not the subject of this analysis. Also common n>
When 0 and n ≠ 0.2, equation (39) is a nonlinear equation. In this analysis, the latter general case is targeted, but as it is, it is necessary to rely on a numerical analysis method. However, the following variable transformation

[Equation 41] Can be solved analytically. At this time

(Equation 42) Therefore, equation (39) can be converted into an Ito-type stochastic differential equation as shown in the following equation.

[Equation 43] Here, dW (t) represents the increment of the one-dimensional Wiener process. In this equation, the coefficient of the first term on the right side, which is the advection term (2-n / n) γ
And the coefficient [n (2-n) / 2] (γ / σ (average)) Q of the second term, which is the diffusion term, can be treated as a constant, so that it can be easily integrated,

[Equation 44] Get. Here, z (t _b ) is the initial value of z (t), and t _b is the start time of the present stochastic process.

(Life Prediction in Ceramics for Miner's Rule) Finally, the influence of random stress fluctuation in a narrow band on ceramics for Miner's rule will be considered. In this analysis, the lifetime value of silicon nitride given by Ohji et al. Is used. Relationship lifetime t _L of the stress σ and the material loaded on the material is given by Ohji et al.

[Equation 45] This equation represents the remaining life when the stress σ is applied to the undamaged. Here, a function that has the following dimension of [1 / hour] and expresses the damage that material suffers per unit time

[Equation 46] And this is called the damage factor. Now, to start the fatigue experiment in time t _b, the material is broken to the time t _e after N _f times repeatedly. This section of time [t _b , t _e ] is divided into N _f minute time intervals Δt of equal length, and numbers are assigned in chronological order.

[Equation 47]

[Equation 48] Assuming that the stress value applied to the material at time t _i is σ (t _i ) = σ _i , damage Δ p _i to the material between t _i and t _i + Δt
Is

[Equation 49] It can be expressed as. Therefore, the damage p (t _N ) accumulated in the material from the time t _b to the time t _N is calculated by taking the sum of the damage Δ p _i incurred at each minute time interval.

[Equation 50] Can be given by Here, Δ in equation (50)
If we take the limit of t → 0,

(Equation 51) Becomes Next, the effect of the fluctuating stress on the equation (Equation 51) will be examined. Here, the fluctuating stress is defined as a deterministic term σ as follows:
(Average) (t) and a stochastic variation σ ′.

(Equation 52) Each term in these equations is a function of time.

Now, when the fluctuating temperature and fluctuating stress are decomposed as in equation (52), the magnitude of the stochastic variation is sufficiently smaller than the size of the deterministic term.

(Equation 53) Consider a narrow band variation that can be expressed as: It is also assumed that the stochastic variation σ ′ has the following properties. (a) About the ensemble average of σ´

(Equation 54) (b) The autocorrelation function of σ´ is

[Equation 55] Becomes (c) σ ′ (t) indicates a Gaussian distribution. Under these conditions, the equation (Equation 51) is Taylor-expanded around σ (average).

[Equation 56] Since the second-order or higher-order minute terms in the above equation are small compared to the other terms in the small-band fluctuation, this is ignored.
Using

[Equation 57] Becomes And a stochastic differential equation for stochastic damage accumulation

[Equation 58] Get. Note that dWσ (t) in the above equation is Wie for σ ′
Indicates the increment of the ner process. Since the coefficient of each term on the right side of the equation (Equation 58) is a constant, the development of p (t) can be obtained by simple integration.

[Equation 59] Here, p _b represents the initial damage already existing in the material at the time t _b . Therefore, the stochastic expected value of damage accumulated at a certain time t can be calculated by taking the ensemble average of the above equation.

[Equation 60] Which is consistent with the development when there is no change. Furthermore, the squared deviation of the accumulated damage variation is:

[Equation 61] Becomes

(Fokker-Plank equation) In the damage evaluation and life evaluation of a material, not only the average value and deviation of damage accumulated in a material at a certain point in time, but also the probability density distribution and probability distribution of damage play an important role. Fulfill. The probability density distribution of damage generally shows a normal distribution or a lognormal distribution, but the distribution when the stress fluctuates randomly is not clear at present. Here, the Langevin equation

(Equation 62) Derives the Fokker-Plank equation equivalent to and the solution to the probability density distribution function of damage, and the probability distribution of damage accumulated in the material at a certain time under the condition that random stress fluctuation is applied to the material. Attempt to determine the parameters that characterize the shape and distribution shape.

Now, a function f (p (t)) of the random variable p (t) is introduced. The change of the function f during the minute time interval dt

[Equation 63] It expresses like. Expand to the second power of dp to take into account the contribution of the higher order derivative in proportion to the small time interval dt. Further, the following equation is obtained by substituting the equation (Equation 58) and organizing it.

[Equation 64] Here, (dt) ² → 0, dt; dWσ → 0, (dWσ) ² = dt were used. Taking the ensemble average of both sides of this equation,

[Equation 65] Becomes Here, <dWσ> = 0. The function f (p
(t)) is a conditional probability density function (hereinafter referred to as a conditional PDF) conditioned at t = t _{b with} an initial value p = p _b g (p, t | p _b ,
t _b ) and g (p, t | p _b , t _b )
5) is expressed again as follows.

[Equation 66] g (∞, t | p _b , t _b ) = g (-∞, t | p _b , t _b ) = 0, ∂g (∞,
t | p _b , t _b ) / ∂p = ∂g (-∞, t | p _b , t _b ) / ∂p = 0
By integrating this equation as follows, the following partial differential equation is obtained.

[Equation 67] This equation gives Fo, which represents the evolution of conditional PDF for creep strain.
This is the kker-Plank equation.

[0035]

As is apparent from the above description, according to the present invention, the probability density function of the damage accumulation amount is obtained from the damage accumulation process based on the sum of Robinson's damage, and the life of the device is determined based on the probability density function. In the method for predicting the creep life of a device under random stress fluctuation and random temperature fluctuation to be predicted, (1) when the random stress fluctuation and the random temperature fluctuation are in a narrow band, a damage coefficient representing a damage amount per unit time is calculated. (2) The random stress fluctuation σ (t) (instantaneous) is expressed by the sum of a time average σ (t) (average) and a stochastic fluctuation σ ′ (t), θ (t)
(Instantaneous) is converted to the time average value θ (t) (average) and the stochastic variation θ ′ (t)
Thus, the Langevin equation including the stochastic process can be derived from the dynamic equation showing the damage evolution indicated by Robinson's damage sum when the stress and the temperature are constant. This Langevin equation includes both a stochastic process due to stress fluctuation and a stochastic process due to temperature fluctuation. This can provide a model for the evolution of damage accumulation for both stress and temperature.
Therefore, it is possible to accurately predict the life of a device in which both the stress and the temperature fluctuate. That is, Silberschmidt
In this study, a nonlinear Langevin equation (Equation 1) was given for damage accumulation of randomly varying short-axis tensile loads (I-mode). F (p) in (Equation 1) is the right side of the deterministic equation for mode I damage as shown in Eq. (2), L (t) is a probability term, and A, B, C, and D are G (p)
Is unsatisfactory because it does not give an undefined and clear functional form. On the other hand, in the present invention, the influence of the stress and the temperature fluctuation on the damage accumulation amount can be clearly determined from the stress and the temperature derivative of the damage degree curve. That is, Silberschm
Although idt's study failed to provide an accurate model of damage development when both stress and temperature fluctuate, the present invention provides that stress and temperature derivatives can be used to determine the relationship between stress and temperature. A model of damage development when both fluctuate can be clearly shown.

Further, the present invention uses the Langevin equation and the equivalent Fokker-Planck equation as the damage accumulating process so that the stress amplitude and temperature fluctuate randomly from the start of the experiment to the fracture. The probability density function of the damage accumulation amount at the time of can be expressed in a normal distribution form. Furthermore, based on the Fokker-Planck equation, a prediction formula of the remaining life of a material that has already been damaged from an arbitrary amount of damage accumulation can be obtained. This makes it possible to obtain a conditional probability density function of damage and a prediction equation of the remaining life when both stress and temperature fluctuate randomly. Therefore, it is possible to accurately predict the life of a device in which both the stress and the temperature fluctuate. In other words, in the analysis of Silberschmidt, the qualitative change of the PDF with respect to the change of the stress fluctuation strength is shown by numerically solving the nonlinear Langevin equation, and the empirical facts regarding the shortening of the material life caused by the stress fluctuation are shown. As shown by calculation, PDF
The effect on the functional form and the remaining life is not considered. In addition, although there is no mention of the presence or absence of temperature fluctuations, the effects cannot be evaluated, but in this analysis, the damage PD considering both the effects of random stress fluctuations and temperature fluctuations was not considered.
F was derived and the change over time in the distribution shape was clearly shown, and it was clarified that the remaining life could be evaluated from any amount of damage accumulation in the presence of both changes. This allows Silb
In the analysis of erschmidt, it is not possible to accurately predict the life of a device in which both stress and temperature fluctuate, but in the present invention, it is necessary to accurately predict the life of a device in which both stress and temperature fluctuate. Can be.

[Brief description of the drawings]

FIG. 1 is a table showing symbols of mathematical formulas used in the present embodiment.

FIG. 2 is a graph showing a temporal change of a probability density function (PDF) estimated from the frequency of passing through a specific area on a pt plane.

Claims (2)

[Claims] 1. A method of predicting a creep life of a device under random stress fluctuation and random temperature fluctuation, wherein a probability density function of a damage accumulation amount based on a Robinson damage sum is obtained, and a life of the device is predicted based on the probability density function. (1) When the random stress fluctuation and the random temperature fluctuation are in a narrow band, a damage coefficient representing a damage amount per unit time is approximated by a linear expression, (2) The random stress fluctuation σ (t) (Instantaneous) is represented by the sum of the time average value σ (t) (average) and the stochastic variation σ ′, and the random temperature variation θ (t) (instantaneous) is represented by the time average value θ (t) (average). A method for predicting creep life of a device under a narrow-band random stress fluctuation and a narrow-band random temperature fluctuation, which is expressed by the sum of the stochastic fluctuation θ ′ (t). 2. The method for predicting creep life of equipment under a narrow-band random stress fluctuation and a narrow-band random temperature fluctuation according to claim 1, wherein the damage accumulation process includes a Langevin equation and an equivalent Fokker-Planck equation. A method for predicting the creep life of a device under a narrow-band random stress fluctuation and a narrow-band random temperature fluctuation, characterized by using: