CN101122560A - Mechanical structure crack expansion rate and crack expansion life span predication method - Google Patents
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Abstract
A crack growth rate and crack growth life prediction method for a mechanical structure belongs to a calculation method for fatigue crack growth rate and crack growth length of the mechanical structure. At first, a relationship between S-N and P-S-N curve expressions (power function form) and a crack growth rate curve expression (Paris formula) is built up in the method. A determination method of the crack growth rate under the conditions of block spectrum load, overload hysteresis load and random load is proposed. And a determination method of Paris curve parameters is provided when the S-N and P-S-N curves cannot be expressed in power function form. And a method for improving the prediction precision of the crack growth rate and the crack growth life is proposed. The S-N and P-S-N curves are used by the method to predict the crack growth rate and the crack growth life. Therefore, the method has the advantages of little dependence on tests and high precision and can shorten the product development period and save much manpower, material resources and financial resources.
Description
Technical Field
The invention relates to a method for calculating the fatigue crack propagation rate and crack propagation length of a mechanical structure that is required to withstand repeated stress levels over its lifetime.
Background
The crack growth rate curve is the basic data for crack growth analysis, however, it is affected by many factors, such as structure form, processing technique, load spectrum type, stress level, etc., and is difficult to determine, especially the crack growth rate under variable amplitude load. Generally, accurate crack growth rate can be obtained only through a large number of crack growth tests, and the method has high requirements on test technology, is time-consuming and labor-consuming, and increases the cost and the period of product development.
Conventionally, attempts have been made to link the conventional fatigue design method (including the static strength design method) and the durability design method based on fracture mechanics to establish the correlation therebetween, such as establishing the fatigue limit σ W And crack propagation threshold value delta K th The relationship between the two, so that other performance parameters are determined by certain fatigue performance parameters, and the design and test process are simplified, but the relationship cannot be established with the crack propagation rate, and the crack propagation rate cannot be predicted.
The S-N curve is the basis of the traditional fatigue design method. During the course of the past decades, extensive experience and experimental data have accumulated. If the functional relation between the S-N curve and the crack expansion rate curve can be established, the crack expansion rate of the structure can be predicted by utilizing abundant data in the traditional fatigue design, so that the dependence of the crack expansion rate on a test is reduced, and the cost and time of product development are greatly saved.
Disclosure of Invention
Aiming at the background technology, the invention provides a mechanical structure crack propagation rate and crack propagation life prediction method which has small test dependence and high precision, can shorten the product development period and can save manpower, material resources and financial resources.
A method for predicting crack propagation rate and crack propagation life of a mechanical structure is characterized by comprising the following steps:
(1) Establishing the relation between S-N, P-S-N curve expressions and crack expansion rate curve expressions;
the relation between the S-N curve expression and the crack propagation rate curve expression is
Wherein the expression of the S-N curve is C 1 ＝Δσ m N f The expression of the crack propagation rate curve is
In the formula C 1 M, C, N are material constants, Δ σ represents the stress level to which the structure is subjected, N f Denotes the structural fatigue life, a 0 、a f Respectively representing the initial crack size and the ultimate crack size, a and N respectively representing the crack length and the number of load cycles,represents the rate of change of the crack length with the number of load cycles, i.e., the crack propagation rate, and β (a) represents a stress intensity factor correction coefficient, which is related to the structural form, the crack length, and the like.
If the expression of the P-S-N curve is
The relationship between the P-S-N curve and the crack growth rate curve is
In the formula C 1p 、m p Is a material constant, N p Representing fatigue life with a certain degree of reliability p;
(2) A block spectrum load, an overload hysteresis load and a crack propagation rate and crack propagation life prediction method under random load;
when the mechanical structure works under the block spectrum load, the crack propagation rate parameter under each stage of load is
Wherein the expression of the S-N curve of the structure under each stage of load isA crack propagation rate of
In the formula, delta sigma i Representing the i-th order stress level, N fi Is shown andΔσ i corresponding fatigue life; c 1i 、m i 、C i 、n i Respectively is a material constant under the ith level load; a is a fi Denotes the ultimate crack size, Δ K, at i-th order load i Is expressed by Δ σ i The corresponding stress intensity factor.
When the mechanical structure works under the block spectrum load, the crack propagation length calculation formula is
a i ＝G -1 (a i-1 ，a i ，m i )
Wherein a is i Denotes the crack length, G, of the structure after the i-th order load 1 The inverse function of the function G is represented,
the calculation formulas of the crack propagation rate and the crack propagation length of the mechanical structure under the overload hysteresis load are respectively
a i ＝G -1 (a i-1 ，a i ，m i )
Wherein
n i ＝m i ，
N i And N i-1 Representing the number of cycles of the i-th and i-1-th stage loads, respectively, phi i An overload hysteresis factor for the i-th order load level.
When the mechanical structure works under random load, the whole random load spectrum is divided into a plurality of sections, rain flow counting is carried out on each section, each section is approximate to a block spectrum load, and then the crack expansion rate and the crack expansion service life are approximately predicted according to the crack expansion rate determining method under the block spectrum load.
When the whole S-N curve is used as C 1 ＝Δσ m N f When the expression has larger error, the partial formula C of S-N curve is used 1 ＝Δσ m N f Fitting to obtain more accurate S-N curve parameters, thereby effectively improving the crack propagation rate and the prediction precision of the crack propagation life; similarly, for other S-N curve expression forms, it is converted to C 1 ＝Δσ m N f Form, then the crack propagation rate and crack propagation life were calculated.
The method is suitable for predicting the fatigue extension life of the mechanical structure, does not need to carry out a crack extension test, only needs to utilize the S-N curve measured by the test, can save test cost and time, shortens the development period of products, saves manpower, material resources and financial resources, and is feasible, reliable and high in precision in engineering.
Drawings
FIG. 1 is an S-N curve of a mechanical structure; FIG. 2 is a crack growth rate curve for a mechanical structure; FIG. 3 is a partial S-N curve fit; FIG. 4 is a crack propagation curve under multi-level loading; fig. 5 is a graph showing random loading spectra and segmentation.
Detailed Description
The linear portion of the curve in FIG. 1 can be generally expressed as
C 1 ＝Δσ m N f (1)
In the formula C 1 M is a material constant which can be determined by experiments or a mechanical design manual; Δ σ represents the stress level to which the structure is subjected, N f Indicating the fatigue life of the structure.
The linear portion of the curve in FIG. 2 can be generally expressed as
Wherein C and n are material constants and are generally determined by experiments; a. n represents the crack length and the number of load cycles respectively;
the change rate of the crack length along with the number of load cycles, namely the crack expansion rate is shown; β (a) represents a stress intensity factor correction coefficient, which is related to the structural form, the crack length, and the like, and the expression form thereof can be determined according to the stress intensity factor manual.
The variables are separated from (0,a) for formula (2) 0 ) Integrate to (N) f ，a f ) To obtain
In the formula a 0 The initial crack size can be determined from the equivalent initial defect size of the structure, or given using empirical formulas in machine design manuals, a for a given structure 0 Is a constant; a is f Indicates the ultimate crack size and can be determined by the formula (4)
In the formula K IC Denotes the fracture toughness of the material, which can be determined by a mechanical design Manual,. Sigma. max The maximum stress level is indicated.
The left side of the easy-to-know formula (3) is a constant, let
Then equation (3) can be further written as
C′＝Δσ n N f (6)
In formula (6), Δ σ is the stress applied to the structure, and N f Is a structural life corresponding to Δ σ, and therefore the equation reflects the relationship between stress and fatigue lifeI.e. the S-N curve in the conventional fatigue design. It can be seen that there is a certain relationship between the crack growth rate curve and the S-N curve, and from the crack growth rate curve, it can be deduced that the S-N is obtained by comparing the formula (6) with the formula (1)
The relationship between the crack growth rate curve and the S-N curve expression can thus be summarized as
The P-S-N curve is generally described as
Wherein C is 1p 、m p For experimental constants, for common materials, it can be queried by machine design manuals, or can be determined by experiments, N p Representing the fatigue life with a reliability p.
The formula (9) represents the utilization parameter C 1p And m p The fatigue life with a certain degree of reliability p can be calculated. C is determined because the S-N curve equation and the crack growth rate equation have an inherent relationship 1p And m p The fatigue crack growth rate calculated with the expression (8) also necessarily has a certain reliability p, and the predicted fatigue crack growth life reliability is also p. Therefore, crack propagation rate with a certain reliability P can also be predicted using the P-S-N curve.
Due to different material properties, the S-N curve of some materials may not have a significant straight line in the log-log coordinate, or may have only a partial straight line, and if the whole S-N curve is described by equation (1), a large error may be caused, so that the prediction result is unreliable. In order to adapt the conclusions of the above derivation to all S-N curve forms, local fitting methods can be used to obtain the S-N curve parameters.
Suppose A in FIG. 3 1 ～A 7 Seven sets of (Δ σ, N) data points. If one wants to obtain the structure at delta sigma 2 The crack growth rate parameter under load can be determined by using formula (1) to point A 1 ～A 3 Fitting to obtain local S-N curve parameters C 1 And m. The crack growth rate parameter is then determined according to equation (8). The structure is at delta sigma 6 The crack propagation rate parameter under the action of load can be similarly measured by the comparison of A 5 ～A 7 And fitting to obtain.
For a load spectrum with n stages (FIG. 4), let Δ σ be i 、N i 、N fi And a fi For the ith level load amplitude, cycle number, fatigue life and ultimate crack size, the S-N curve and crack propagation rate curve of the structure under the ith level load can be written as
In the formula, delta sigma i Representing the i-th order stress level, N fi Representing the fatigue life corresponding to the i-th order stress level; c 1i 、 m i 、C i 、n i Respectively is a material constant under the ith level load; a is a fi Denotes the ultimate crack size, Δ k, at i-th order load l Is represented by a and Δ σ i The corresponding stress intensity factor.
According to formula (8), there are
Order to
Then C i Can be further written as
Under the action of the ith stage load (from N time) i-1 To N i ) Cracks from a i-1 Extend to a i (as shown in FIG. 4), equation (11) is integrated
Bringing formula (14) into (15)
Bringing formula (10) into formula (16)
a i ＝G -1 (a i-1 ，a i ，m i ) (18)
From the above derivation, for a given multilevel load spectrum, the fatigue damage (N) at each level of load can be calculated according to the conventional fatigue analysis method i -N i-1 )/N fi And calculating the ultimate crack size a at each level of load through the fracture toughness of the material fi Then calculating the corresponding crack length a of the structure after each stage of cyclic load action through the formulas (17) to (18) i Thereby obtaining a crack propagation curve.
Studies have shown that after a high load has occurred in the load spectrum, there is a greater effect on the subsequent crack propagation rate, i.e. a significant reduction in the crack propagation rate, a phenomenon known as the retardation of crack propagation by overload.
For a load spectrum with n-order loads, overload hysteresis effects must be taken into account if there is a large overload stress level. Applying the Wheeler overload closure model, rewrite equation (11) to
Wherein phi i The overload hysteresis factor can be determined according to a Wheeler model.
The crack propagation length under overload hysteresis load can be obtained as
α i ＝G -1 (a i-1 ，a i ，m i ) (21)
Because the amplitude of the random load is irregularly changed, the crack propagation rate of the random load cannot be determined according to the S-N curve like the load block spectrum. If the random loads are counted by rain flow counting, the crack propagation rate and crack propagation size cannot be considered in terms of load order. As an approximation, the entire on-board load spectrum can be divided into several segments (as shown in FIG. 5, the load spectrum is divided into T < T 1 、T 1 ＜t＜t 2 、t 2 ＜t＜T 2 And T > T 2 Four segments) and rain flow counting is performed for each segment so that each segment approximates a block spectrum. On the basis of the above-mentioned technical scheme,and calculating the crack propagation rate and the crack propagation length section by section according to the sequence of the load sections, thereby achieving the purposes of approximately predicting the crack propagation rate and prolonging the service life.
Best Mode for Carrying Out The Invention
The following describes an embodiment of the structural crack propagation analysis under block spectral load according to the present invention (constant amplitude loading is a specific example of block spectral load, and random loading can be approximated as block spectral load).
(1) Determination of the Property parameters of the Material used for the construction, such as the S-N Curve, the fracture toughness K of the Material IC Stress intensity correction factor beta (a), fatigue notch coefficient, and the like.
(2) Calculating the S-N curve of the structure at each stress level according to the S-N curve and the fatigue gap coefficient of the material, and determining the parameter C 1i And m i The corresponding S-N curve can also be determined by experiment. If the reliability requirement exists, the P-S-N curve parameter is used for replacing the corresponding S-N curve parameter.
(3) Calculating the ultimate crack size a at each stress level according to the fracture toughness and the stress intensity correction factor of the material fi 。
(4) Calculating the crack expansion rate curve parameter C under each level of stress by using the relation between the S-N curve and the crack expansion rate curve i And n i . And calculating the corresponding crack length according to the formulas (17) to (18). Taking min { a) as the ultimate crack size of the structure under the multilevel load fi When a is i ＝min{a fi And when the structure is broken, the structure is invalid.
(5) If the load level difference is large, that is, the influence of the load hysteresis effect is large, the influence of the hysteresis effect on the crack propagation should be considered, and at this time, the corresponding crack length is calculated by equations (20) to (21).
Possibility of industrial utilization
The mechanical structure crack expansion rate and crack length prediction method is suitable for fatigue-resistant design of mechanical structures. The method is mainly characterized in that a relation between an S-N curve and a crack expansion rate curve is established, and fatigue performance parameters such as crack expansion rate and crack length under block spectrum load, overload hysteresis load and random load are calculated based on the relation. As the S-N curve data is rich, the new method can save a large amount of test cost and time and shorten the product development period. In addition, if the S-N curve of the structure does not exist or the precision is not enough, the novel method only needs to use the S-N curve of the structure measured by the test without carrying out a crack propagation test, and the requirements on the test technology and the like are not high. Therefore, the new method is feasible and reliable in engineering.
Claims (2)
1. A method for predicting crack growth rate and crack growth life of a mechanical structure is characterized by comprising the following steps: (1) Establishing the relation between S-N, P-S-N curve expressions and crack expansion rate curve expressions; the relation between the S-N curve expression and the crack propagation rate curve expression is
Wherein the expression of the S-N curve is C 1 ＝Δσ m N f The expression of the crack propagation rate curve is
In the formula C 1 M, C, N are material constants, Δ σ represents the stress level to which the structure is subjected, N f Denotes the fatigue life of the structure, a 0 、a f Respectively representing the initial crack size and the ultimate crack size, a and N respectively representing the crack length and the number of load cycles,represents the rate of change of the crack length with the number of load cycles, i.e., the crack propagation rate, and β (a) represents a stress intensity factor correction coefficient, which is related to the structural form, the crack length, and the like:
if the expression of the P-S-N curve is
The relationship between the P-S-N curve and the crack growth rate curve is
In the formula C 1p 、m p Is a material constant, N p Representing a fatigue life with a certain degree of reliability p;
(2) A block spectrum load, an overload hysteresis load and a crack propagation rate and crack propagation life prediction method under random load;
when the mechanical structure works under the block spectrum load, the crack propagation rate parameter under each stage of load is
Wherein the expression of the S-N curve of the structure under each stage of load is
A crack propagation rate of
In the formula, delta sigma i Representing the i-th order stress level, N fi Is expressed by Δ σ i Corresponding fatigue life; c 1i 、m i 、C i 、n i Respectively, the material constant under the i-th level load, a fi Denotes the ultimate crack size, Δ K, at i-th order load i Is expressed by Δ σ i The corresponding stress intensity factor is given by the corresponding stress intensity factor,
when the mechanical structure works under the block spectrum load, the crack propagation length calculation formula is
a i ＝G -1 (a i-1 ，a i ，m i )
Wherein a is i Denotes the crack length, G, of the structure after the i-th order load 1 The inverse function of the function G is represented,
the calculation formulas of the crack propagation rate and the crack propagation length of the mechanical structure under overload hysteresis load are respectively
a i ＝G -1 (a i-1 ，a i ，m i )
Wherein
n i ＝m i ，
N i And N i-1 Representing the number of cycles, phi, of the i-th and i-1-th stage loads, respectively i An overload hysteresis factor for the i-th order load level;
when the mechanical structure works under random load, the whole random load spectrum is divided into a plurality of sections, rain flow counting is carried out on each section, each section is approximate to a block spectrum load, and then the crack expansion rate and the crack expansion service life are approximately predicted according to the crack expansion rate determining method under the block spectrum load.
2. The method of predicting crack growth rate and crack growth life of a mechanical structure according to claim 1, wherein C is used for the entire S-N curve 1 ＝Δσ m N f When the expression has large error, the local formula C of S-N curve 1 ＝Δσ m N f Fitting to obtain more accurate S-N curve parameters, thereby effectively improving the crack propagation rate and the prediction precision of the crack propagation life; similarly, for other S-N curve expression forms, it is converted to C 1 ＝Δσ m N f Form, then the crack propagation rate and crack propagation life were calculated.
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