CN108280256B - Modeling method based on nonlinear fatigue accumulation damage mechanism degradation-impact model - Google Patents

Modeling method based on nonlinear fatigue accumulation damage mechanism degradation-impact model Download PDF

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CN108280256B
CN108280256B CN201711406456.8A CN201711406456A CN108280256B CN 108280256 B CN108280256 B CN 108280256B CN 201711406456 A CN201711406456 A CN 201711406456A CN 108280256 B CN108280256 B CN 108280256B
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许丹
何娇兰
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Abstract

The invention provides a modeling method based on a nonlinear fatigue accumulation damage mechanism degradation-impact model, which comprises the following steps: firstly, the method comprises the following steps: a fatigue degradation model under constant amplitude load; II, secondly: an impact load model; thirdly, the method comprises the following steps: a hysteresis effect model under an impact overload condition; fourthly, the method comprises the following steps: a hysteresis effect correlation model based on a Piecewise Deterministic Markov Process (PDMP); through the steps, the method simulates the actual use environment, also considers the accidental impact process caused under the accidental condition, and has authenticity; the method has new knowledge and understanding on crack expansion in practical engineering application, and has practical significance; the invention combines the fatigue degradation process and the random impact process by applying a scientific method, gives clear mathematical expression for calculating the reliability of the sample, has academic value, provides a way for solving the problem for engineering, and has popularization and application values.

Description

Modeling method based on nonlinear fatigue accumulation damage mechanism degradation-impact model
Technical Field
The invention provides a modeling method of a degradation-impact model based on a nonlinear fatigue cumulative damage mechanism, which relates to a fatigue cumulative damage calculation method when a random degradation process and a random impact process are acted together, is a nonlinear fatigue damage superposition model, and provides a nonlinear fatigue damage modeling method of degradation and impact interaction aiming at the problems that the degradation process is delayed when the impact is too large and the impact damage is influenced by the degradation process. This is a reliability modeling problem in the field of reliability technology.
Background
The fuselage is one of the most important components of an aircraft and is usually designed using the concept of damage tolerance. During service, the fuselage is often subjected to fatigue loads caused by the air flow and to occasional impacts caused by turbulence, which cause cracks. This real load condition can be translated into a scientific problem combining a fatigue loading sequence and some spike overloads. Researchers often simulate this actual physical process with fatigue degradation processes and random impact processes. For such realistic problems, in the existing research at home and abroad, two kinds of directions appear:
1. the fatigue degradation process and the random impact process are considered to be two mutually independent processes, and the correlation between the two processes is not considered.
2. Some researchers consider the correlation between the two processes of degradation and shock to some extent, but mainly model studies aiming at that the shock is too large to cause the system to directly crash and the shock is not large to cause damage increment. The interaction between both impact and degradation is not taken into account.
Therefore, considering the interaction between the random degradation process and the random impact process, and providing a modeling method capable of calculating the interaction is an urgent problem to be solved.
Disclosure of Invention
The invention aims to provide a modeling method based on a fatigue damage mechanism random degradation and random impact model, which is used for solving the problem of crack propagation under the effects of degradation and impact, establishing a nonlinear fatigue damage model by considering the degradation process delay caused by the overload of an impact load and the impact damage size influenced by the degradation process, and establishing an interaction model between the nonlinear fatigue damage model and the degradation process so as to solve the algorithm problem of the whole crack propagation process.
The invention is realized by adopting the following technical scheme, and the invention discloses a modeling method based on a fatigue damage mechanism random degradation and random impact model, which comprises the following steps:
the method comprises the following steps: fatigue degradation model under constant amplitude load
In the fatigue damage stage, a Paris (Paris) model is adopted to describe crack propagation under constant amplitude load; the Paris model is widely used in current crack propagation research, and a differential expression of the crack length a of the Paris model
Figure BDA0001520431920000021
The contents are as follows:
Figure BDA0001520431920000022
Figure BDA0001520431920000023
Δσ＝σmax-σmin(3)
Figure BDA0001520431920000024
the correction factor C is influenced by the dispersibility of the material, which is assumed to be a random variable following a normal distribution, i.e.
Figure BDA0001520431920000025
Y is a geometric factor which can be simplified under special conditions, and when the cracks of the central holes of one aluminum plate and one iron plate are calculated, the geometric factor Y is equal to 1.12; when calculating the surface cracks of an aluminum plate and an iron plate, the value of the geometric factor Y is approximate to
Figure BDA0001520431920000026
Step two: impact load model
In fact, a constant amplitude load is accompanied by random impacts; the crack growth under constant amplitude and impact loading is schematically shown in FIG. 2; in FIG. 2, TNIs the occurrence time of the Nth impact, KNIs the impact strength; in the present invention, all impacts used to calculate the hysteresis effect are called overloads; the number of times of occurrence of the impact and the size of the impact are both required to be given certain random distribution;
before the first impact is reached, the crack propagation is only dependent on the constant amplitude load; when the Nth impact appears, the crack grows in the plastic zone caused by overload, and the growth rate is very low until the plastic zone disappears; later, the crack growth rate returned to normal before the (N +1) th impact occurred; the impact process is equivalent to a jump process; the degradation process under the constant amplitude load is a determination process which is updated continuously along with time; x (n) is assumed to be the state of crack propagation during degradation and impact, and is expressed as follows:
X(n)＝Xn,n∈[TN,TN+1) (5)
step three: hysteresis effect model under impact overload condition
In practice, the impact load does not have an ideal constant amplitude load, but has random environmental impact; for certain materials, such as aluminum and iron, overloading can delay surface crack propagation rather than accelerate its growth; this delay is a hysteresis effect caused by shock overload;
the weilinberg model (i.e., Willenborg model) describes this overload hysteresis; under the action of impact load, a large single plastic area appears near the crack tip, and the crack propagation rate reaches the minimum value; in the next load cycle, when the plastic region exceeds the upper boundary value, the hysteresis effect gradually decreases and then completely disappears; in the hysteresis region, fatigue crack propagation can be described by the following equation:
Figure BDA0001520431920000031
Figure BDA0001520431920000032
σcomp＝σreq-σmax(8)
σreqstress without hysteresis effect, σreqThe expression of (c) can be derived by the euclidean (Irwin) function and geometric criteria:
ρreq＝ρres(9)
ρres＝ρol-(a(n)-aol) (10)
Figure BDA0001520431920000033
Figure BDA0001520431920000034
based on the formulae (9) to (12), σreqThe final derivation of (c) is:
Figure BDA0001520431920000035
ρreqis formed byreqα is the coefficient of the Irwin function, which is 1 for the crack propagation problem of flat panels;
the duration of the hysteresis effect depends on the respective size of the plastic zone; from the equations (7) and (8), the duration can be found when the stress thereof satisfies the following relationship,
σmin＞σreq-σmax(14)
that is, when the condition of equation (13) is satisfied, the overload hysteresis effect ends; the determination of the size of the hysteresis region by the stress intensity in equations (12) and (13) may be converted into the determination of the size of the hysteresis region by the crack length, where the converted equation is expressed as:
Figure BDA0001520431920000041
as can be seen from equations (1) and (2) of the fatigue model, the future crack propagation rate depends on the current crack length; therefore, the conventional linear cumulative damage rule is no longer suitable for calculating the crack length of the hysteresis region; wherein r ═ σmin/σmax；
The derived equation (15) can determine the following two cases:
Figure BDA0001520431920000042
whether a hysteresis effect occurs or not can be judged when each impact arrives;
Figure BDA0001520431920000046
when in use
Figure BDA0001520431920000043
A hysteresis effect occurs;
Figure BDA0001520431920000044
the end of the hysteresis effect can be judged; when the accumulated crack length satisfies the formula (15), the hysteresis effect ends;
step four: hysteresis effect correlation model based on segmented deterministic Markov process (PDMP)
This step establishes a delayed crack propagation model due to overload, in which the path of crack propagation is influenced by many random factors in the impact process, such as impact arrival time, impact size, accumulated crack length, and material properties, the Willenborg model is used to describe the relationship between random impact load and crack propagation rate, the Paris model describes the deterministic process in the absence of hysteresis, which is crack propagation under constant amplitude load, the piecewise deterministic markov process is a process that can link the stochastic impact process with the deterministic crack propagation model, and in general, the piecewise deterministic markov process is well suited to describe the deterministic process plus the stochastic time, deterministic process refers to fatigue degradation, and stochastic event is random impact
Figure BDA0001520431920000045
Figure BDA0001520431920000045
② jump procedure (X)N,TNAnd N) is not less than 0, ③ state transition, state transition between fatigue degradation process and random impact process;
the implementation of the specific algorithm of the segmental deterministic Markov process can be carried out by the following three steps:
1. selecting a behavior vector
A multidimensional behavior vector z (n) with environmental and stress variations was constructed to describe the crack state of the specimen:
Figure BDA0001520431920000051
2. push to flow function
Figure BDA0001520431920000052
The purpose of this step is to find a definite path between two random impacts; suppose that the occurrence time of two random impacts is TNAnd TN+1Of the current function
Figure BDA0001520431920000053
Expressed as:
Figure BDA0001520431920000054
wherein, η (T)N) Representing the accumulated crack length at the occurrence of the nth impact, the boundary conditions for the occurrence of the hysteresis effect are:
Figure BDA0001520431920000055
3. deriving a hopping procedure
This step includes the derivation of the jump time and the magnitude of the impact load. The magnitude of the impact follows a normal distribution
Figure BDA0001520431920000056
The mean and standard deviation thereof are muolAnd deltaol(ii) a The time interval between two adjacent random impacts follows an exponential distribution with an intensity lambda,
TN+1-TN～E(λ) (19)
if the system does not fail, its maximum stress intensity factor K must be metmaxCannot exceed the threshold KthresholdAnd the cumulative crack length a (n) fails to sing the maximum threshold athreshold. The reliability expression is:
R(n)＝P{Kmax(n)≤Kthreshold∩a(n)≤athreshold} (20)
through the steps, firstly, the fatigue degradation and the sudden impact load under the banner load are considered by the model, and the whole life cycle process is comprehensively analyzed; secondly, by considering the hysteresis effect of crack propagation under the condition of impact overload, the propagation rate is slowed down, and the boundary conditions of the hysteresis effect and the ending hysteresis effect are deduced; finally, a piecewise deterministic Markov process is provided to describe the process of combining fatigue degradation and random impact load, a coupling model is established by considering the interaction between the fatigue degradation and the random impact load, and a calculation method of reliability is provided; based on the three obvious advantages, the method has great practical significance for practical engineering problems related to crack propagation.
The constant amplitude load in the step one refers to that the amplitude of the load borne by the constant amplitude load is a determined constant and does not change along with the change of time; the "Paris (Paris) model" refers to a mathematical model that describes the fatigue fracture process of certain metallic materials.
The "hysteresis effect" in step two refers to a slower change in the state of the system than when the hysteresis effect is not present.
Wherein, the "WeilinBorg (Willenborg) model" in step III refers to a mathematical model used for describing crack propagation when a hysteresis effect occurs; the "Irwin (Irwin) function" is the basic theory in fatigue and fracture mechanics.
The Markov process in the fourth step is a random process and is an important method for researching the state space of the discrete event dynamic system; the invention relates to a 'segmental deterministic Markov process (PDMP)', which refers to a degradation process and an impact process, wherein the degradation process is a deterministic process with random parameters, and the random impact process is an uncertain random process.
The invention has the following advantages:
1. the invention simulates the actual use environment, considers the process of fatigue degradation which is continuously generated along with the time, also considers the accidental impact process caused under the accidental condition, and has authenticity.
2. The invention considers that the crack propagation has a hysteresis effect when the impact overload occurs, and the crack propagation rate is not generally considered to be increased continuously. The method has new knowledge and understanding on crack propagation in practical engineering application, and has practical significance.
3. The invention combines the fatigue degradation process and the random impact process by applying a scientific method, gives a definite mathematical expression for calculating the reliability of the sample, which has not been found in the previous research and engineering. Therefore, the method not only has academic value, but also provides a way for solving the problem for engineering.
Drawings
FIG. 1 is a flow chart of the method of the present invention
FIG. 2 is a schematic representation of crack growth under constant amplitude and impact loads in accordance with the present invention;
FIG. 3 is a piecewise deterministic Markov modeling process of the present invention;
FIG. 4 is a flow chart of a crack propagation algorithm implementation of the present invention;
FIG. 5 is a reliability graph of the present invention
FIG. 6 is a first-pass probability density function graph of the present invention
The symbol numbers in the figures are illustrated as follows: in fig. 1, "PDMP" refers to a piecewise deterministic markov process; in FIG. 2, a (n) is the crack length under the current load, athA failure threshold indicative of the length of the crack, the system being considered to fail when the threshold is reached; t is1Indicating the arrival of the first burst, T2Indicating the arrival of a second burst of impacts, T3Indicating the arrival of a third burst impact; n represents the number of constant amplitude load cycles; in fig. 3, phi denotes a flow function; in fig. 4, the definitions of the symbols are given in table 2 below.
TABLE 2 description of symbols in the present invention
Figure BDA0001520431920000071
Figure BDA0001520431920000081
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and examples.
The following embodiment is implemented by taking crack length calculation as a research object according to the process shown in fig. 1, firstly, a crack propagation model under fatigue degradation is determined as a Paris model, and a mathematical explicit expression of crack length calculation is deduced according to the Paris model; determining an incidental impact model accompanied under fatigue degradation, wherein the occurrence frequency of impact is determined to obey Poisson distribution, the impact size obeys normal distribution, the time of impact occurrence obeys exponential distribution according to the occurrence frequency of impact obeys Poisson distribution, and the influence of each impact arrival is calculated according to the occurrence time and the occurrence size; then considering the hysteresis effect caused by the impact under the overload condition, the crack propagation rate is slowed down in the plastic area, judging whether the coming impact causes hysteresis or not at the moment of arrival of each impact through conditions, and the schematic diagram of the crack growth under the constant-amplitude load and the impact load is shown in FIG. 2; finally, fatigue degradation and an impact model are combined through a piecewise deterministic Markov process, and a mathematical model and a description in the piecewise deterministic Markov process are shown in figure 3, wherein the deterministic process refers to a fatigue degradation process, the Markov process refers to an expansion state of a crack under an impact load, and a crack expansion length caused by nonlinear superposition degradation and impact, a curve of crack expansion reliability changing along with time is calculated by using a model Carlo simulation algorithm, so that the purpose of reliability evaluation is achieved, and the implementation flow of the algorithm in the invention is shown in figure 4.
The invention discloses a modeling method based on a nonlinear fatigue accumulation damage mechanism degradation-impact model, which is shown in figure 1 and comprises the following specific implementation steps:
the method comprises the following steps: fatigue degradation model under banner load
Table 1 gives the values of the parameters in the model, which are verified by the markov model karo simulation algorithm.
Parameters in the model of Table 1
Figure BDA0001520431920000091
Calculated using the Paris model in the following cases:
(1) before the first impact occurs;
(2) after the ith impact comes, if the hysteresis effect occurs and the time for ending the hysteresis effect is TenAt TenAnd (i +1) th impact arrival, crack propagation was calculated using the Paris model.
(3) After the ith impact arrives, no hysteresis effect occurs, and crack propagation in the time period is directly calculated by using a Paris model
In the above three cases, the calculation formula of crack propagation is as follows:
Figure BDA0001520431920000092
T0to the onset of fatigue-only degradation, TiThe end time is the time when only fatigue degradation is effected. In case (1), T0＝0，Ti＝T1(ii) a In case (2), T0＝Ten，Ti＝Ti+1(ii) a For case (3), T0＝Ti，Ti＝Ti+1。
Step two: impact load model
The number of times of occurrence of the impact follows a poisson distribution, and the magnitude of the impact assumes a normal distribution.
In the invention, the number of the impacts is assumed to obey Poisson distribution { N (n), n is more than or equal to 0} and Poisson (lambda), and the size sigma of the impacts isolSubject to a normal distribution of the signals,
Figure BDA0001520431920000093
step three: hysteresis effect model under impact overload condition
When the impact load is too large, a hysteresis phenomenon may be caused to crack propagation. According to the Willenborg model, the conditions for judging whether the hysteresis effect occurs at the time of impact are as follows:
Figure BDA0001520431920000101
step four: hysteresis effect correlation model based on segmented deterministic Markov process (PDMP)
According to the Willenborg model, it is very difficult to obtain an explicit mathematical expression of crack length by its formula, so by obtaining an expression of crack propagation using the load interval Δ n of 1 by the forward euler formula, crack propagation can be described by the multidimensional variable z (n):
Figure BDA0001520431920000102
considering that the same batch of products also has the dispersibility of the material, the geometric parameter C of the material is randomized, C-N (mu)c,δc). The markov monte carlo simulation algorithm substitutes the frequency for the probability and estimates the general idea with the sample, and the change of the reliability curve of crack propagation along with the time is calculated as shown in fig. 5. FIG. 6 is a graph of a probability density function for a first crossing of a corruption threshold corresponding to a reliability curve.
As can be seen from fig. 5, when the reliability is R ═ 0.9, the duty cycle is 10000 cycles.

Claims (1)

1. The modeling method based on the nonlinear fatigue accumulation damage mechanism degradation-impact model is characterized by comprising the following steps: the method comprises the following steps:
the method comprises the following steps: fatigue degradation model under constant amplitude load
In the fatigue damage stage, a Paris model is adopted to describe crack propagation under constant amplitude load; the Paris model is widely used in current crack propagation research, and a differential expression of the crack length a of the Paris model
Figure FDA0002185772620000011
The contents are as follows:
Figure FDA0002185772620000012
Figure FDA0002185772620000013
Δσ＝σmax-σmin(3)
Figure FDA0002185772620000014
the correction factor C is influenced by the dispersibility of the material, which is assumed to be a random variable following a normal distribution, i.e.
Figure FDA0002185772620000015
Y is a geometric factor which can be simplified under special conditions, and when the cracks of the central holes of one aluminum plate and one iron plate are calculated, the geometric factor Y is equal to 1.12; when calculating the surface cracks of an aluminum plate and an iron plate, the value of the geometric factor Y is approximate to
Figure FDA0002185772620000016
Step two: impact load model
In fact, a constant amplitude load is accompanied by random impacts; all impacts used to calculate hysteresis effects are called overloads; the number of times of occurrence of the impact and the size of the impact are both required to be given a random distribution;
before the first impact is reached, the crack propagation is only dependent on the constant amplitude load; when the Nth impact appears, the crack grows in the plastic zone caused by overload, and the growth rate is very low until the plastic zone disappears; later, the crack growth rate returned to normal before the (N +1) th impact occurred; the impact process is equivalent to a jump process; the degradation process under the constant amplitude load is a determination process which is updated continuously along with time; x (n) is assumed to be the state of crack propagation during degradation and impact, and is expressed as follows:
X(n)＝Xn,n∈[TN,TN+1)
step three: hysteresis effect model under impact overload condition
In practice, the impact load does not have an ideal constant amplitude load, but has random environmental impact; overloading delays the surface crack propagation rather than accelerating its growth; this delay is a hysteresis effect caused by shock overload;
the weilinberg model, the Willenborg model, describes this overload hysteresis; under the action of impact load, a large single plastic area appears near the crack tip, and the crack propagation rate reaches the minimum value; in the next load cycle, when the plastic region exceeds the upper boundary value, the hysteresis effect gradually decreases and then completely disappears; in the hysteresis region, fatigue crack propagation can be described by the following equation:
Figure FDA0002185772620000021
Figure FDA0002185772620000022
σcomp＝σreq-σmax(8)
σreqstress without hysteresis effect, σreqThe expression of (c) is derived by the euclidean, i.e. Irwin function and geometric criteria:
ρreq＝ρres(9)
ρres＝ρol-(a(n)-aol) (10)
Figure FDA0002185772620000023
Figure FDA0002185772620000024
based on the formulae (9) to (12), σreqThe final derivation of (c) is:
Figure FDA0002185772620000025
ρreqis formed byreqα is the coefficient of the Irwin function, which takes the value of 1 for the crack propagation problem of the flat panel;
the duration of the hysteresis effect depends on the respective size of the plastic zone; from the equations (7) and (8), the duration can be found when the stress thereof satisfies the following relationship,
σmin＞σreq-σmax(14)
that is, when the condition of equation (13) is satisfied, the overload hysteresis effect ends; the determination of the size of the hysteresis region by the stress intensity in equations (12) and (13) can be converted into the determination of the size of the hysteresis region by the crack length, and the converted equation is expressed as:
Figure FDA0002185772620000031
as can be seen from equations (1) and (2) of the fatigue model, the future crack propagation rate depends on the current crack length; therefore, the conventional linear cumulative damage rule is no longer suitable for calculating the crack length of the hysteresis region; wherein r ═ σmin/σmax；
The derived equation (15) can determine the following two cases:
Figure FDA0002185772620000032
whether a hysteresis effect occurs or not can be judged when each impact arrives;
Figure FDA0002185772620000033
when in use
Figure FDA0002185772620000034
A hysteresis effect occurs;
Figure FDA0002185772620000035
the end of the hysteresis effect can be judged; when the accumulated crack length satisfies the formula (15), the hysteresis effect ends;
step four: hysteresis effect correlation model based on segmented deterministic Markov process (PDMP)
This step establishes a delayed crack propagation model caused by overload, in which the path of crack propagation is influenced by many random factors in the impact process, such as impact arrival time, impact size, accumulated crack length and material properties, the Willenborg model is used to describe the relationship between random impact load and crack propagation rate, the Paris model describes the deterministic process in the absence of hysteresis effect, which is crack propagation under constant amplitude load, the piecewise deterministic markov process is able to link the stochastic impact process with the deterministic crack propagation model, the piecewise deterministic markov process is well suited to describe the deterministic process plus random time, the deterministic process is fatigue degradation and the stochastic event is random impact, and thus the piecewise deterministic markov process is used to describe the interaction between fatigue degradation and stochastic impact, the piecewise deterministic markov process has three characteristics, ① deterministic process plus random time
Figure FDA0002185772620000036
② jump procedure (X)N,TNAnd N) is not less than 0, ③ state transition, state transition between fatigue degradation process and random impact process;
the implementation of the specific algorithm of the segmental deterministic Markov process can be carried out by the following three steps:
1. selecting a behavior vector
A multidimensional behavior vector z (n) with environmental and stress variations was constructed to describe the crack state of the specimen:
Figure FDA0002185772620000041
2. push to flow function
Figure FDA0002185772620000042
The purpose of this step is to find a definite path between two random impacts; suppose that the occurrence time of two random impacts is TNAnd TN+1Of the current function
Figure FDA0002185772620000043
Expressed as:
Figure FDA0002185772620000044
wherein, η (T)N) Representing the accumulated crack length at the occurrence of the nth impact, the boundary conditions for the occurrence of the hysteresis effect are:
Figure FDA0002185772620000045
3. deriving a hopping procedure
This step includes the derivation of the jump time and the impact load magnitude; the magnitude of the impact follows a normal distribution
Figure FDA0002185772620000046
The mean and standard deviation thereof are muolAnd deltaol(ii) a The time interval between two adjacent random impacts follows an exponential distribution with an intensity lambda,
TN+1-TN～E(λ) (19)
if the system does not fail, its maximum stress intensity factor K must be metmaxCannot exceed the threshold KthresholdAnd the cumulative crack length a (n) fails to sing the maximum threshold athreshold(ii) a The reliability expression is:
R(n)＝P{Kmax(n)≤Kthreshold∩a(n)≤athreshold} (20)
through the steps, firstly, the fatigue degradation and the sudden impact load under the banner load are considered by the model, and the whole life cycle experience process is comprehensively analyzed; secondly, by considering the hysteresis effect of crack propagation under the condition of impact overload, the propagation rate is slowed down, and the boundary conditions of the hysteresis effect and the ending hysteresis effect are deduced; finally, a piecewise deterministic Markov process is provided to describe the process of combining fatigue degradation and random impact load, a coupling model is established by considering the interaction between the fatigue degradation and the random impact load, and a reliability calculation method is provided.
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