CN105301113A - Metal ultrasonic fatigue testing method of uniform cross section segment-containing plate-shaped sample - Google Patents

Abstract

The invention discloses a metal ultrasonic fatigue testing method of a uniform cross section segment-containing plate-shaped sample. The method comprises the following parts: a first part, firstly, according to density and elastic modulus of a test material, and a dimension designing formula of the uniform cross section segment-containing plate-shaped ultrasonic fatigue sample under a axial tension-compression loading is calculated by analysis; a second part, a stress transformation formula between the uniform cross section segment-containing plate-shaped sample and an arc-shaped sample is derived, a stress of the uniform cross section segment-containing plate-shaped ultrasonic fatigue sample is transformed into a stress corresponding to the arc-shaped sample on a system control software, and a ultrasonic fatigue test control of the uniform cross section segment-containing plate-shaped sample is completed by using existing equipment and software; a third part, a stress correction formula is derived, when a deviation exists between an actual size and a design size of the uniform cross section segment-containing plate-shaped sample, so that more accurate test control is guaranteed. The method is suitable different metal materials, and the cover range is wide; the method can be used for completing an ultra-high cycle fatigue test of the uniform cross section segment-containing plate-shaped sample, upgrade is not needed, and the cost is saved.

Description

Metal ultrasonic fatigue test method for plate-shaped sample containing uniform section

Technical Field

The invention relates to the field of ultrasonic fatigue performance testing of metal materials, in particular to a metal ultrasonic fatigue testing method of a plate-shaped sample containing a uniform section.

Background

The fatigue phenomenon is a phenomenon that a structural material forms cracks under cyclic load and after a certain number of cycles, or the cracks are further propagated until the structural material is completely broken. Statistics show that more than about 80% of structural strength failure is caused by fatigue, so the fatigue performance test and research of the material are significant. Conventional fatigue performance studies are generally at 10⁷As a limit cycle. In fact, some high-strength steels applied to key parts or structural steels subjected to high-frequency load often have practical service lives of 10⁸～10¹⁰Throughout the week, increasing results have also demonstrated that many engineering steels are at 10⁷After secondary stress cycle, fatigue fracture still occurs, and the fracture generally originates from the interior of the sample and is in a cracking formFor concealment, it is more important to study the ultra-high cycle fatigue performance of the material.

The ultrasonic fatigue test is a new technology for testing the fatigue performance of materials, has obvious advantages compared with the traditional fatigue test technology, has extremely high working frequency, and can greatly improve the efficiency of the fatigue test, the working frequency of the traditional high-frequency fatigue test is 100-200 Hz, and the working frequency of the ultrasonic fatigue test can reach 2.0 × 10⁴Hz, test a fatigue life of 1.0 × 10⁹The conventional high-frequency fatigue test needs about 100 days, and the ultrasonic fatigue test can be completed only about one day. The ultrasonic fatigue test technology is an innovation in the field of fatigue test, greatly improves the test efficiency and the research and development efficiency, and is suitable for testing the ultrahigh-cycle fatigue performance of metal materials.

The ultrasonic fatigue test technology is to make a sample excited to resonate so as to generate a resonance phenomenon. The ultrasonic signal is sent out by a piezoelectric ceramic transducer, and the electric signal supplied by a high-frequency power supply is converted into mechanical vibration with the same frequency and then amplified by a vibration displacement amplifier. One end of the sample is connected with the displacement amplifier, and the other end is free. The resonance occurs under the excitation of the displacement amplifier, the resonance wave is generated in the sample, the tensile-compression symmetric cyclic load is formed along the axial direction of the sample, and the longitudinal displacement and stress (strain) field is established, so that the geometric shape of the ultrasonic fatigue sample must meet the resonance condition under the ultrasonic frequency when the ultrasonic fatigue sample is designed.

The ultrasonic fatigue testing machine introduced by the martial Steel is a USF-2000 ultrasonic fatigue testing machine produced by Shimadzu corporation of Japan, and system control software on the equipment is only provided with two sample types: circular arc shaped specimens and notch shaped specimens as shown in figures 1 and 2. The lack of specimen type greatly limits the application of ultrasonic fatigue testing machines. The main problems are as follows:

1. when ultrasonic fatigue is required for a plate-like specimen, for example, for automotive sheet steel, only two specimen types in the existing control software cannot be completed.

2. For materials containing obvious defects (such as inclusion and shrinkage cavity), a sample is required to have an equal stress area, so that the material defects can be searched for in a larger area in the process of carrying out a super-high cycle fatigue test to obtain a safer fatigue test result, and meanwhile, when the influence of the surface treatment of the material on the fatigue performance needs to be researched, the sample is also required to have an equal stress area to facilitate the surface treatment. The existing control software only has two sample types without equal stress areas when working, and cannot be completed.

3. The displacement amplitude interval of the stable work of the common ultrasonic fatigue testing machine is 10-50 mu m. If some materials with lower strength level are designed into arc-shaped samples for testing, the situation that the displacement amplitude is too small to cause the test to be unstable or even the vibration cannot be started can occur.

4. After the sample is processed, the actual size and the design size of the sample inevitably have deviation due to processing errors, and in addition, since the ultrasonic fatigue sample has high requirements on the surface smoothness, after the sample is processed, the middle part of the sample is usually required to be polished by sand paper, which also causes the deviation of the size of the sample from the pre-designed size. When the deviation between the actual size and the design size of the sample reaches a certain degree, the actual vibration frequency and the stress amplitude of the sample are correspondingly deviated from the design value, so that the precision of the test result is influenced.

Disclosure of Invention

In order to solve the problems in the background art, the invention provides an ultrasonic fatigue test method for a plate-shaped test sample containing a section. The invention mainly solves the following three problems:

1. the method for designing the test sample for the ultrasonic tension-compression fatigue test of the plate-shaped test sample with the uniform section is provided, so that the resonance can be effectively generated on an ultrasonic fatigue testing machine.

2. A control method for ultrasonic fatigue test of a plate-shaped sample containing a uniform section is provided. The stress of the plate-shaped sample with the equal section is converted into the stress of the circular-arc-shaped sample on system control software through a stress conversion formula between the plate-shaped sample with the equal section and the circular-arc-shaped sample, and the ultrasonic fatigue test control of the plate-shaped sample with the equal section is completed by utilizing the existing equipment and software.

3. In order to eliminate the influence of the size deviation of the sample on the test result and obtain a more accurate ultrasonic fatigue test result, the invention provides a stress correction method when the actual size and the design size of the plate-shaped sample with the equal section have the deviation.

To achieve the above object, the present invention is realized by:

a metal ultrasonic fatigue test method of a plate-shaped sample containing a uniform section comprises the following three parts:

firstly, according to the density and the elastic modulus of a test material, obtaining a size design formula of a plate-shaped axial tension-compression ultrasonic fatigue sample containing a constant section by adopting analytical calculation;

a second part is used for deducing a stress conversion formula between the plate-shaped test sample containing the equal section and the circular arc-shaped test sample, converting the stress of the plate-shaped ultrasonic fatigue test sample containing the equal section into the stress corresponding to the circular arc-shaped test sample on system control software, and using the existing equipment and software to complete the ultrasonic fatigue test control of the plate-shaped test sample containing the equal section;

and the third part is used for deducing a stress correction formula when the actual size and the design size of the plate-shaped sample containing the uniform section have deviation, so that the test control is more accurate.

In order to explain the technical scheme of the invention in more detail, the inventor makes further detail as follows:

first, it should be clear that the term "plate-like sample with uniform cross section" as used herein means that the sample itself is plate-like and includes a portion with the same cross section;

for the plate-shaped ultrasonic fatigue test sample containing the equal section, the calculation of the test sample size design formula comprises the following specific steps:

measuring the density rho and the elastic modulus E of a test material;

step two, analyzing and calculating the plate-shaped axial tension-compression ultrasonic fatigue sample containing the uniform section, and the method comprises the following steps:

2.1 miming b₁，b₂，L₁，L₂Data, b₁Is the thickness of a constant cross-sectional section of the plate, b₂Is the thickness at both ends of the plate, L₁Is half the length of the equal section of the plate, L₂Is the variable cross-sectional length of the plate; in the following calculation, for the convenience of calculation, the dimensions of mm, g and ms are adopted, i.e. the dimension unit is mm, the mass unit is g, and the time unit is ms. Other parameters are unified into units of mm, g and ms through calculation; for example, the density ρ 7850kg/m³＝7850×10³g/(10³mm³)＝7.85×10^-3g/mm³；

2.2 calculating the length L at both ends of the plate₃(ii) a According to the continuous system vibration theory, the material meets the ideal elastomer condition, the origin of coordinates is assumed to be the axial center of the sample, and the axial direction of the sample is taken to be the x axis. U (x, t) is the longitudinal vibratory displacement of the section at the coordinate x at the time t, S (x) is the area of the cross section of the sample at the coordinate x, S' (x) is the first derivative of a function S (x); the longitudinal wave equation of the sample at resonance is:

$\frac{{\∂}^{2}U(x,t)}{\∂t^2}=\frac{E}{\mathrm{\ρ}}(\frac{{\∂}^{2}U\left(x\right)}{\∂x^2}+\frac{S^{\′}\left(x\right)}{S\left(x\right)}\frac{\∂U(x,t)}{\∂x})---\left(1\right)$

assuming that the sample satisfies the resonance condition, the separation variable of U (x, t) is U (x, t) ═ U (x) e^iωtSubstituted into the formula (1) to obtain

$\frac{{\∂}^{2}U\left(x\right)}{\∂x^2}+\frac{S^{\′}\left(x\right)}{S\left(x\right)}\frac{\∂U\left(x\right)}{\∂x}+k^{2}U\left(x\right)=0---\left(2\right)$

In the formula,in order to derive the function u (x),in order to obtain a first-order derivative,in order to obtain a second-order derivative,rho is the density of the sample material, E is the elastic modulus of the sample material, and f is the vibration frequency of the sample;

for the plate-shaped sample containing the equal section, the cross section area equation of the sample is calculated

$\left\{\begin{array}{cc}S\left(x\right)=b_{1}w& \left|x\right|\≤L_1\\ S\left(x\right)=b_{1}w{\cosh }^2\[\mathrm{\α}(x-L_1)\]& L_1\≤\left|x\right|\≤L_1+L_2\\ S\left(x\right)=b_{2}w& L_1+L_2\≤\left|x\right|\≤L\end{array}\right.$

Wherein, b₁Is the thickness of a constant cross-sectional section of the plate, b₂Is the thickness at both ends of the plate, w is the width of the plate;

calculating a boundary condition U according to a cross-sectional area equation of the sample_x＝L＝U_max，U|_x＝00 and continuity Condition $U{|}_{x\→L_1-}=U{|}_{x\→L+},U^{\′}{|}_{x\→L_1-}=U^{\′}{|}_{x\→L_1+},U{|}_{x\→(L+L_2)-}=U{|}_{x\→(L_1+L_2)+},U^{\′}{|}_{x\→(L_1+L_2)-}=U^{\′}{|}_{x\→(L_1+L_2)+}$ The equation of vibration in the longitudinal direction of the sample U (x) is obtained as:

$\left\{\begin{array}{cc}U\left(x\right)=C_1\sin \left(kx\right)& \left|x\right|<L_1\\ U\left(x\right)=C_2\exp \&lsqb;({\mathrm{\&beta;}}_1-{\mathrm{\&alpha;}}_1)(x-L_1)\&rsqb;+C_3\exp \&lsqb;-({\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1)(x-L_1)\&rsqb;& L_1<\left|x\right|<L_1+L_2\\ U\left(x\right)=U_{\max }\cos \&lsqb;k(L-x)\&rsqb;& L_1+L_2<\left|x\right|<L\end{array}\right.---\left(3\right)$

in the formula:

U_max＝U|_x＝Li.e. the amplitude of displacement at the free end of the sample α₁，β₁，Comprises the following steps:

determining the length L at both ends of a plate-shaped test specimen₃Comprises the following steps:

$L_3=\frac{1}{k}\arctan \left\{\frac{cos\left({\mathrm{kL}}_1\right)\&lsqb;{\mathrm{\&beta;}}_1\cosh \left({\mathrm{\&beta;}}_1{L}_2\right)-{\mathrm{\&alpha;}}_1\sinh \left({\mathrm{\&beta;}}_1{L}_2\right)\&rsqb;-ksin\left({\mathrm{kL}}_1\right)\sinh \left({\mathrm{\&beta;}}_1{L}_2\right)}{\&lsqb;{\mathrm{\&alpha;}}_{1}sin\left({\mathrm{kL}}_1\right)+k\cos \left({\mathrm{kL}}_1\right)\&rsqb;\sinh \left({\mathrm{\&beta;}}_1{L}_2\right)+{\mathrm{\&beta;}}_{1}sin\left({\mathrm{kL}}_1\right)\cosh \left({\mathrm{\&beta;}}_1{L}_2\right)}\right\}---\left(4\right)$

the variable cross-section curve of the plate-shaped sample containing the equal cross-section during analysis and calculation of the sample is a catenary, and the catenary is replaced by an arc curve during actual processing due to difficulty in finishing machining; by varying the length L of the section₂Minimum thickness b₁Thickness b at both ends₂To obtain the radius of the transition arc of the variable section

The design formula of the equal-section-plate-shaped ultrasonic fatigue test sample is suitable for an axial tension-compression ultrasonic fatigue test with the stress ratio of-1, and can ensure that the equal-section-containing plate-shaped axial tension-compression test sample can effectively generate resonance on an ultrasonic fatigue testing machine.

The stress conversion formula of the plate-shaped sample with the equal section is given below, and the stress of the plate-shaped ultrasonic fatigue sample with the equal section is converted into the stress corresponding to the arc-shaped sample on the system control software, so that the aim of completing the ultrasonic fatigue test control of the plate-shaped sample with the equal section by adopting the conventional equipment and software is fulfilled.

In order to realize the ultrasonic fatigue test control of the plate-shaped test sample with the equal section, the working principle of the ultrasonic fatigue test technology is firstly known. The ultrasonic fatigue test technology realizes the control of the stress amplitude of the sample by controlling the displacement amplitude of the end part of the sample. The voltage of the energy transducer and the vibration displacement amplitude of the output end are in a linear relation, and after the displacement amplitude of the end part of the sample is given, the ultrasonic fatigue testing machine adjusts the displacement amplitude by changing the voltage of the energy transducer. Thus, for a given stress amplitude σ_maxFirst, the corresponding vibration displacement amplitude U is determined_max。

The stress distribution function sigma (x) of the plate-shaped ultrasonic fatigue test sample with the uniform section can be obtained by deriving a displacement amplitude function U (x):maximum stress amplitude sigma thereof_maxAt the specimen mid-section (x ═ 0); obtaining the maximum stress amplitude sigma of the plate-shaped sample containing the uniform section_maxComprises the following steps:

wherein E is the modulus of elasticity of the sample material,U_max＝U|_x＝Lis the maximum displacement amplitude of the sample,

drawing up the dimension parameter L₁，L₂，b₁，b₂Wherein b is₁Is the thickness of a constant cross-sectional section of the plate, b₂Is the thickness at both ends of the plate, L₁Is half the length of the equal section of the plate, L₂Is the variable cross-sectional length of the plate;

firstly, the length L of the two ends of the plate is obtained according to a size design formula (4)₃The stress amplitude needing to be loaded is set to be sigma_maxThe stress amplitude σ of the plate-like sample containing the uniform cross-section can be obtained from the equation (5)_maxCorresponding displacement amplitude

PreparingThe data of the data is transmitted to the data receiver,the radius of the thinnest point in the circular arc shaped sample,the radius of the cylinder at both ends of the sample,half the length of the middle section of the sample. Then the length at both ends of the circular arc shaped specimenThe length of the arc-shaped sample can be automatically given by system control software, and can also be given by an analytical formula of the lengths of two ends of the arc-shaped sample:

$L_2^0=\frac{1}{k}\arctan \{\frac{1}{k}\&lsqb;\frac{\mathrm{\&beta;}}{\tanh \left({\mathrm{\&beta;L}}_1^0\right)}-\mathrm{\&alpha;}\tanh \left({\mathrm{\&alpha;L}}_1^0\right)\&rsqb;\}---\left(6\right)$

in the formula, $\mathrm{\&alpha;}=\frac{1}{L_1^0}\arccos h\left(\frac{R_2^0}{R_1^0}\right),\mathrm{\&beta;}=\sqrt{{\mathrm{\&alpha;}}^2-k^2},k=2\mathrm{\&pi;}f/\sqrt{\frac{E}{\mathrm{\&rho;}}},$ ρ is the density of the sample material, E is the modulus of elasticity of the sample material, and f is the sample vibration frequency.

Stress distribution function sigma of circular arc-shaped test piece⁰(x) The arc-shaped sample is obtained by derivation of a displacement functionLarge amplitude of stressAt the specimen middle section (x ═ 0):

in the formula, $\mathrm{\&alpha;}=\frac{1}{L_1^0}\arccos h\left(\frac{R_2^0}{R_1^0}\right),\mathrm{\&beta;}=\sqrt{{\mathrm{\&alpha;}}^2-k^2},k=2\mathrm{\&pi;}f/\sqrt{\frac{E}{\mathrm{\&rho;}}}.$

as can be seen from the equation (7), in order to make the displacement amplitude of the plate-like sample including the uniform cross-section be U_maxThe displacement amplitude U of the circular arc-shaped sample at the same level as that of the plate-shaped sample with the equal section is determined_maxLower corresponding stress amplitudeComprises the following steps:

the formula (8) is a stress conversion formula between a plate-shaped sample containing equal section sections and a circular-arc-shaped sample, wherein sigma_maxThe stress amplitude of the plate-shaped sample containing the equal section is obtained,the stress amplitude of the corresponding arc-shaped sample under the same displacement amplitude is obtained.

The displacement amplitude range of the USF-2000 ultrasonic fatigue testing machine is 10-50 mu m, so the dimension parameter L is measured₁，L₂，L₃，R₁，R₂The test stress amplitude range of the plate-shaped test sample containing the equal section is

The method for controlling the ultrasonic fatigue test of the plate-shaped test sample with the uniform section is provided.

The stress correction method when the actual dimension of the plate-shaped sample with the equal section is deviated from the designed dimension is given below, and the accurate control of the ultrasonic fatigue test of the plate-shaped sample with the equal section is further realized.

It is assumed that the dimension of the actually processed plate-like test piece containing the uniform cross-section segments is L 'measured by accurate measurement'₁，L'₂，L'₃，b′₁，b'₂，b′₁Is the measured thickness of the section of the plate with equal section, b'₂Is measured thickness, L 'at both ends of the sheet'₁Is half, L 'of the actual measurement length of the equal section of the plate'₂Is the actual measurement length of the plate variable section, L'₃Measured length at both ends of the boardThe actual resonant frequency of the sample may not be 2.0 × 10 due to the deviation of the measured dimensions from the design dimensions⁴Hz. At this time, the resonance frequency f 'of the sample can be inverted from the sample size according to the formula (4), and L'₁，L'₂，b′₁，b'₂Substituting formula (4) and setting the vibration frequency in the vibration frequency range of the ultrasonic fatigue testing machine, namely 19.50 × 10³Hz～20.50×10³The values of Hz are calculated one by one at intervals of 10Hz to obtain the lengths L 'at two ends of the plate'₃And the corresponding relation between the resonant frequency f ', namely L'₃The value of (f) yields the actual resonance frequency f'.

If f' is less than 19.50 × 10³Hz or greater than 20.50 × 10³Hz indicates that the dimensional deviation of the sample is too large, so that the sample cannot start to vibrate and needs to be processed again.

According to the obtained f', the stress amplitude sigma of the plate-shaped sample containing the uniform section can be obtained_maxActual vibration displacement amplitude of

Wherein $k^{\&prime;}=2{\mathrm{\&pi;f}}^{\&prime;}/\sqrt{\frac{E}{\mathrm{\&rho;}}};{\mathrm{\&beta;}}_1^{\&prime;}=\sqrt{{\mathrm{\&alpha;}}_1^{\&prime;2}-k^{\&prime;2}},{\mathrm{\&alpha;}}_1^{\&prime;}=\frac{1}{2L_2^{\&prime;}}ln\left(\frac{b_2^{\&prime;}}{b_1^{\&prime;}}\right);$

According to U'_maxFrom the equation (8), the amplitude U 'of the circular arc sample of a predetermined size can be obtained'_maxLower corresponding stress amplitudeWherein,for given dimension values of the circular arc shaped test specimen, f⁰＝2.0×10⁴Hz。The expression of (a) is as follows:

the formula (9) is a stress correction formula of the plate-shaped sample containing the equal section. Wherein,for given dimension values of the circular arc shaped test specimen, f⁰＝2.0×10⁴Hz，L'₁，L'₂，L'₃，b′₁，b'₂Is the actual size value of a plate-shaped sample containing equal section segments, and f ' is the actual size L ' of the plate-shaped sample '₁，L'₂，L'₃，b′₁，b'₂And (5) inversely calculating the obtained actual resonant frequency.

The invention is implemented on the basis of the USF-2000 ultrasonic fatigue testing machine which is mainstream in the market at present. However, the invention is also applicable to other ultrasonic fatigue testing machines, and the USF-2000 ultrasonic fatigue testing machine control software only comprises a circular arc-shaped test sample and a notch-shaped test sample, so the second part of the technical scheme of the invention is specially used for the USF-2000 ultrasonic fatigue testing machine. Other ultrasonic fatigue testing machines can adopt the first part and the third part.

Compared with the prior art, the invention has the following advantages and effects:

1. the method for designing the size of the plate-shaped ultrasonic fatigue test sample containing the uniform section is suitable for different metal materials, and the coverage range is wide;

2. the designed plate-shaped axial tension-compression test sample containing the uniform section can effectively generate resonance on an ultrasonic fatigue testing machine, and the test frequency is as high as 2.0 × 10⁴Hz, can greatly accelerate the fatigue test, and can be used for completing the ultra-high cycle fatigue test of the plate-shaped sample with the equal section;

3. the ultrasonic fatigue test control of the plate-shaped sample is indirectly realized by utilizing the existing equipment and software through the ultrasonic fatigue test control of the arc-shaped sample, the software and the equipment are not required to be upgraded, and the test cost is saved;

4. in the resonance process of the plate-shaped sample containing the equal section, the stress of the equal section area of the sample is equal; before the test, the area can be subjected to surface treatment to study the influence of different surface treatments on the fatigue performance of the material; or for the plate-shaped material with obvious defects (such as inclusion and shrinkage), the plate-shaped material can be designed into a plate-shaped sample with equal section sections, so that the material defects can be conveniently found in a larger area in the test process, and a safer fatigue test result can be obtained;

5. the plate-shaped test sample containing the uniform section can be used for the ultra-high cycle fatigue performance test of materials with lower strength level; if a material with a lower strength level is designed into an arc-shaped sample for testing, the situation that the test is unstable or even cannot start vibration due to too small displacement amplitude can occur; compared with arc-shaped samples and other samples with the same shape, the displacement amplitude of the plate-shaped sample with the equal section is relatively larger under the same stress level, so that the stable completion of the test can be ensured;

6. the ultrasonic fatigue test method of the plate-shaped sample with the equal section considers the stress correction method when the actual size and the design size of the sample have deviation, corrects the stress error caused by the size error, and can enable the ultrasonic fatigue test of the plate-shaped sample with the equal section to be controlled more completely and accurately.

Drawings

FIG. 1 is a circular arc specimen provided by USF-2000 ultrasonic fatigue tester apparatus;

FIG. 2 is a sample of a circular arc notch provided by USF-2000 ultrasonic fatigue tester equipment;

FIG. 3 is a schematic diagram of a plate-shaped ultrasonic fatigue test sample containing a section and the stress distribution thereof;

FIG. 4 is a schematic view of a circular arc-shaped sample and its stress distribution;

FIG. 5 is a schematic diagram of inverse calculation of resonant frequency;

FIG. 6 is a schematic diagram of an embodiment of the present invention.

In the figure, 1-transducer: converting an electrical signal provided by a power source into a mechanical vibration signal, 2-displacement amplifier: amplifying the vibration displacement amplitude from the transducer to obtain the required displacement amplitude for the sample, 3-the plate-shaped ultrasonic fatigue sample with the equal section, and 4-the cooling air nozzle.

Detailed Description

The invention is further described with reference to the following figures and specific examples.

First, it should be clear that the term "plate-like sample having an equal cross section" as used herein means that the sample itself has a plate shape and the sample includes a portion having the same cross section, such as the sample shown in fig. 3 of the present invention. The following description of the invention is mostly based on the sample shown in fig. 3, but the technical solution of the invention is not limited to the sample exactly the same as that of fig. 3, and the technical solution of the invention can be applied to a plate-like sample having an equal cross section similar to that of fig. 3.

A metal ultrasonic fatigue test method of a plate-shaped sample containing a uniform section comprises the following three parts:

firstly, according to the density and the elastic modulus of a test material, obtaining a size design formula of a plate-shaped axial tension-compression ultrasonic fatigue sample containing a constant section by adopting analytical calculation;

a second part is used for deducing a stress conversion formula between the plate-shaped test sample containing the equal section and the circular arc-shaped test sample, converting the stress of the plate-shaped ultrasonic fatigue test sample containing the equal section into the stress corresponding to the circular arc-shaped test sample on system control software, and using the existing equipment and software to complete the ultrasonic fatigue test control of the plate-shaped test sample containing the equal section;

and the third part is used for deducing a stress correction formula when the actual size and the design size of the plate-shaped sample containing the uniform section have deviation, so that the test control is more accurate.

In the first section, the calculation of the sizing formula includes the following specific steps:

measuring the density rho and the elastic modulus E of a test material;

step two, analyzing and calculating the plate-shaped axial tension-compression ultrasonic fatigue sample containing the uniform section, and the method comprises the following steps:

2.1 miming b₁，b₂，L₁，L₂Data, b₁Is the thickness of a constant cross-sectional section of the plate, b₂Is the thickness at both ends of the plate, L₁Is half the length of the equal section of the plate, L₂Is the variable cross-sectional length of the plate;

2.2 calculating the length L at both ends of the plate₃(ii) a According to the continuous system vibration theory, the material meets the ideal elastomer condition, the origin of coordinates is assumed as the axial center of the sample, and the axial direction of the sample is taken as the x axis; u (x, t) is the longitudinal vibratory displacement of the cross section at coordinate x at time t; s (x) is the area of the cross-section of the sample at coordinate x, and S' (x) is the first derivative of the function S (x), then the longitudinal wave equation for the sample at resonance is:

$\frac{{\&part;}^{2}U(x,t)}{\&part;t^2}=\frac{E}{\mathrm{\&rho;}}(\frac{{\&part;}^{2}U\left(x\right)}{\&part;x^2}+\frac{S^{\&prime;}\left(x\right)}{S\left(x\right)}\frac{\&part;U(x,t)}{\&part;x})---\left(1\right)$

assuming that the sample satisfies the resonance condition, the separation variable of U (x, t) is U (x, t) ═ U (x) e^iωtSubstituting into formula (1) to obtain

$\frac{{\&part;}^{2}U\left(x\right)}{\&part;x^2}+\frac{S^{\&prime;}\left(x\right)}{S\left(x\right)}\frac{\&part;U\left(x\right)}{\&part;x}+k^{2}U\left(x\right)=0---\left(2\right)$

In the formula,in order to derive the function u (x),in order to obtain a first-order derivative,is a second order derivative. k is a mixing parameter of the sample material,ρ is the density of the sample material, E is the modulus of elasticity of the sample material, and f is the sample vibration frequency.

According to the cross-sectional area equation of the plate-like specimen

$\left\{\begin{array}{cc}S\left(x\right)=b_{1}w& \left|x\right|\&le;L_1\\ S\left(x\right)=b_{1}w\cos h^2\&lsqb;\mathrm{\&alpha;}(x-L_1)\&rsqb;& L_1\&le;\left|x\right|\&le;L_1+L_2\\ S\left(x\right)=b_{2}w& L_1+L_2\&le;\left|x\right|\&le;L\end{array}\right.$

Wherein, b₁Is the thickness of a constant cross-sectional section of the plate, b₂Is the thickness at both ends of the plate and w is the width of the plate.

Calculating a boundary condition U according to a cross-sectional area equation of the sample_x＝L＝U_max，U|_x＝00 and continuity Condition $U{|}_{x\&RightArrow;L_1-}=U{|}_{x\&RightArrow;L+},U^{\&prime;}{|}_{x\&RightArrow;L_1-}=U^{\&prime;}{|}_{x\&RightArrow;L_1+},U{|}_{x\&RightArrow;(L+L_2)-}=U{|}_{x\&RightArrow;(L_1+L_2)+},U^{\&prime;}{|}_{x\&RightArrow;(L_1+L_2)-}=U^{\&prime;}{|}_{x\&RightArrow;(L_1+L_2)+}$ The equation of vibration in the longitudinal direction of the sample U (x) is obtained as:

$\left\{\begin{array}{cc}U\left(x\right)=C_1\sin \left(kx\right)& \left|x\right|<L_1\\ U\left(x\right)=C_2\exp \&lsqb;({\mathrm{\&beta;}}_1-{\mathrm{\&alpha;}}_1)(x-L_1)\&rsqb;+C_3\exp \&lsqb;-({\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1)(x-L_1)\&rsqb;& L_1<\left|x\right|<L_1+L_2\\ U\left(x\right)=U_{\max }\cos \&lsqb;k(L-x)\&rsqb;& L_1+L_2<\left|x\right|<L\end{array}\right.---\left(3\right)$

in the formula:

U_max＝U|_x＝Lα, the maximum displacement amplitude of the sample₁，β₁，Comprises the following steps:

determining the length L at both ends of the plate₃Comprises the following steps:

$L_3=\frac{1}{k}\arctan \left\{\frac{cos\left({\mathrm{kL}}_1\right)\&lsqb;{\mathrm{\&beta;}}_1\cosh \left({\mathrm{\&beta;}}_1{L}_2\right)-{\mathrm{\&alpha;}}_1\sinh \left({\mathrm{\&beta;}}_1{L}_2\right)\&rsqb;-ksin\left({\mathrm{kL}}_1\right)\sinh \left({\mathrm{\&beta;}}_1{L}_2\right)}{\&lsqb;{\mathrm{\&alpha;}}_{1}sin\left({\mathrm{kL}}_1\right)+k\cos \left({\mathrm{kL}}_1\right)\&rsqb;\sinh \left({\mathrm{\&beta;}}_1{L}_2\right)+{\mathrm{\&beta;}}_{1}sin\left({\mathrm{kL}}_1\right)\cosh \left({\mathrm{\&beta;}}_1{L}_2\right)}\right\}---\left(4\right)$

in the analysis and calculation of the sample, the variable cross-section curve of the plate-shaped sample containing the equal cross-section sections in the figure 3 is assumed to be a catenary, and the catenary is replaced by a circular arc curve in the actual processing because the mechanical processing is difficult to complete; by varying the length L of the section₂Minimum thickness b₁Thickness b at both ends₂To obtain the radius of the transition arc of the variable section

The second part is as follows:

the stress distribution function sigma (x) of the plate-shaped ultrasonic fatigue test sample with the uniform section is obtained by deriving a displacement function U (x):maximum stress amplitude sigma thereof_maxThe maximum stress amplitude σ of a plate-like specimen containing equal cross-section sections was determined at the intermediate cross-section (x ═ 0) of the specimen_maxComprises the following steps:

wherein E is the modulus of elasticity of the sample material,U_max＝U|_x＝Lis the maximum displacement amplitude of the sample,

for a plate-shaped sample containing equal section sections, a dimension parameter L is set₁，L₂，b₁，b₂First, the length L of the plate at both ends is obtained according to the formula (4)₃The stress amplitude needing to be loaded is set to be sigma_maxThe stress amplitude σ of the plate-like sample containing the uniform cross-section is obtained from the equation (5)_maxCorresponding displacement amplitude

For the self-contained arc ultrasonic fatigue test sample in the system control software, as shown in FIG. 4, a drawing is drawn upThe data of the data is transmitted to the data receiver,the radius of the thinnest point in the circular arc shaped sample,the radius of the cylinder at both ends of the sample,is half of the length of the middle variable section of the sample; then the length at both ends of the circular arc shaped specimenThe length of the arc-shaped sample can be automatically given by system control software, or can be given by an analytical formula of the lengths of two ends of the arc-shaped sample:

$L_2^0=\frac{1}{k}\arctan \{\frac{1}{k}\&lsqb;\frac{\mathrm{\&beta;}}{\tanh \left({\mathrm{\&beta;L}}_1^0\right)}-\mathrm{\&alpha;}\tanh \left({\mathrm{\&alpha;L}}_1^0\right)\&rsqb;\};---\left(6\right)$

in the formula, $\mathrm{\&alpha;}=\frac{1}{L_1^0}\arccos h\left(\frac{R_2^0}{R_1^0}\right),\mathrm{\&beta;}=\sqrt{{\mathrm{\&alpha;}}^2-k^2},k=2\mathrm{\&pi;}f/\sqrt{\frac{E}{\mathrm{\&rho;}}},$ ρ is the density of the sample material, E is the modulus of elasticity of the sample material, and f isThe sample vibration frequency.

Stress distribution function sigma of circular arc-shaped test piece⁰(x) The maximum stress amplitude of the arc-shaped sample is obtained by derivation of the displacement functionAt the specimen middle section (x ═ 0):

in the formula, $\mathrm{\&alpha;}=\frac{1}{L_1^0}\arccos h\left(\frac{R_2^0}{R_1^0}\right),\mathrm{\&beta;}=\sqrt{{\mathrm{\&alpha;}}^2-k^2},k=2\mathrm{\&pi;}f/\sqrt{\frac{E}{\mathrm{\&rho;}}}.$

as can be seen from the equation (7), in order to make the displacement amplitude of the plate-like sample including the uniform cross-section be U_maxThe circular arc sample is drawn up on the section plate with equal sectionSame displacement amplitude U of sample_maxLower corresponding stress amplitudeComprises the following steps:

the formula (8) is a stress conversion formula between a plate-shaped sample containing equal section sections and a circular-arc-shaped sample, wherein sigma_maxThe stress amplitude of the plate-shaped sample containing the equal section is obtained,the stress amplitude of the corresponding arc-shaped sample under the same displacement amplitude is obtained.

The third part is as follows:

it is assumed that the dimension of the actually processed plate-like test piece containing the uniform cross-section segments is L 'measured by accurate measurement'₁，L'₂，L'₃，b′₁，b'₂B 'as shown in FIG. 3'₁Is the measured thickness of the section of the plate with equal section, b'₂Is measured thickness, L 'at both ends of the sheet'₁Is half, L 'of the actual measurement length of the equal section of the plate'₂Is the actual measurement length of the plate variable section, L'₃The actual resonant frequency of the test sample may not be 2.0 × 10 due to the deviation of the measured dimensions from the design dimensions of the test sample⁴Hz. In this case, the actual resonance frequency f 'of the sample can be inversely calculated from the measured dimensions of the sample according to equation (4), and L'₁，L'₂，b′₁，b'₂Substituting formula (4) and setting the vibration frequency in the vibration frequency range of the ultrasonic fatigue testing machine, namely 19.50 × 10³Hz～20.50×10³Calculating value one by one at intervals of 10Hz between Hz to obtain the actual measurement length L 'at two ends of the plate'₃And the corresponding relation between the resonant frequency f ', namely L'₃The value of (d) yields the actual resonant frequency f';

if f' is less than 19.50 × 10³Hz or greater than 20.50 × 10³Hz indicates that the size deviation of the sample is too large, so that the sample cannot start to vibrate and needs to be processed again;

according to the obtained f', the stress amplitude sigma of the plate-shaped sample containing the uniform section can be obtained_maxActual vibration displacement amplitude of

Wherein $k^{\&prime;}=2{\mathrm{\&pi;f}}^{\&prime;}/\sqrt{\frac{E}{\mathrm{\&rho;}}},{\mathrm{\&beta;}}_1^{\&prime;}=\sqrt{{\mathrm{\&alpha;}}_1^{\&prime;2}-k^{\&prime;2}},{\mathrm{\&alpha;}}_1^{\&prime;}=\frac{1}{2L_2^{\&prime;}}ln\left(\frac{b_2^{\&prime;}}{b_1^{\&prime;}}\right);$

According to U'_maxFrom the equation (8), the amplitude U 'of the circular arc sample of a predetermined size can be obtained'_maxLower corresponding stress amplitudeWherein,for given dimension values of the circular arc shaped test specimen, f⁰＝2.0×10⁴Hz；The expression of (a) is as follows:

the formula (9) is a stress correction formula of the plate-shaped sample containing the uniform section; wherein,for given dimension values of the circular arc shaped test specimen, f⁰＝2.0×10⁴Hz，L'₁，L'₂，L'₃，b′₁，b'₂Is a measured dimension value of a plate-shaped sample containing equal section segments, and f ' is a measured dimension L ' of the plate-shaped sample '₁，L'₂，L'₃，b′₁，b'₂And (5) inversely calculating the obtained actual resonant frequency.

Example 1

Taking a steel material for automobile sheets as an example, it is assumed that dimensional accuracy control is very good during sample processing.

1. The elastic modulus E of the steel material for the automobile plate is 206GPa, and the density rho is 7850kg/m³。

2. The sample size was designed. The preset axial tension and compression test sample is a plate-shaped ultrasonic fatigue test sample containing a uniform section, and is shown in figure 3. First, the size parameter, L, is set₁＝12.5mm，L₂＝20mm，b₁＝3mm，b₂＝10mm，b₁Is the thickness of a constant cross-sectional section of the plate, b₂Is the thickness at both ends of the plate, L₁Is half the length of the equal section of the plate, L₂Is the variable cross-sectional length of the plate. When calculating, all known parameters are converted into dimension units of mm, g and ms, and the lengths L at two ends of the plate-shaped sample are calculated by a formula (4) which is a calculation formula of the lengths at two ends of the plate-shaped sample₃The radius R of the transition arc of the variable section segment is 58.89mm when the radius R is 11.13 mm. Then according to L₁，L₂，L₃，b₁，b₂R and other parameters are processed into ultrasonic fatigue test samples, and the working areas in the middle of the test samples, namely the surfaces of the equal section sections and the variable section sections, are polished to reach the finish degree R_a0.32. Tapping one side of the sample, wherein the diameter of the screw hole is 5 mm.

3. Fixing the processed sample in the displacement amplifier of the ultrasonic fatigue testing machine, adjusting the cooling air nozzle to align with the stress concentration part of the sample, opening the valve, and cooling the sample, as shown in fig. 6.

4. Opening the control software of the ultrasonic fatigue system, selecting the type of the circular arc sample, and drawing up the size parameter of the circular arc sample The radius of the thinnest point in the circular arc shaped sample,the radius of the cylinder at both ends of the sample,half the length of the middle section of the sample. The lengths of the two ends of the circular arc sample can be obtained by an analytical formula (6) of the lengths of the two ends of the circular arc sample or by ultrasonic fatigue testing machine system control software

5. Drawing up stress amplitude sigma to be loaded for plate-shaped test sample containing equal section_max200MPa according to the size parameters of the circular arc-shaped sample $L_1^0=20mm,L_2^0=8.53mm,R_1^0=1.5mm,R_2^0=5mm$ And the size parameter L of the plate-shaped sample containing the equal section₁＝12.5mm，L₂＝20mm，L₃＝11.13mm，b₁＝3mm，b₂10 mm. The stress amplitude of the corresponding arc-shaped sample under the same displacement amplitude is obtained by the stress conversion formula, namely the formula (8)Namely, the stress amplitude of the circular arc-shaped sample needs to be input in the system control software, and the vibration displacement amplitude is 22.1 μm at this time. Sigma_maxWhen the pressure is 200MPa, if the test is directly carried out by using the circular arc sample, the size parameter $L_1^0=20mm,L_2^0=8.53mm,R_1^0=1.5mm,R_2^0=5mm$ The displacement amplitude of the arc-shaped sample is 9.85 micrometers, and the test cannot be completed when the displacement amplitude exceeds the vibration range of the ultrasonic fatigue testing machine.

6. After other test parameters are set in control system software, an axial tension-compression ultrasonic fatigue test of a plate-shaped test sample with a uniform section is started, and the vibration frequency is 2.0 × 10⁴Hz, 1.38 × 10⁷After weekly cycles, the specimens broke.

Example 2

Taking the same steel material for automobile sheets as in example 1 as an example, it is assumed that dimensional accuracy control is not good during sample processing, and deviation between the actual dimension and the design dimension of the sample is large due to surface grinding after sample processing is completed.

1. The elastic modulus E of the steel material for automobile plates is 206GPa, and the density rho is 7850kg/m³。

2. The sample size was designed. The preset axial tension and compression test sample is a plate-shaped ultrasonic fatigue test sample containing a uniform section, and is shown in figure 3. First, the size parameter, L, is set₁＝12.5mm，L₂＝20mm，b₁＝3mm，b₂＝10mm，b₁Is the thickness of a constant cross-sectional section of the plate, b₂Is the thickness at both ends of the plate, L₁Is half the length of the equal section of the plate, L₂Is the variable cross-sectional length of the plate. All known parameters are calculatedConverting into dimension units of mm, g and ms, and calculating the length L at two ends of the plate-shaped sample by using a formula (4) which is a calculation formula of the length at two ends of the plate-shaped sample₃The radius R of the transition arc of the variable section segment is 58.89mm when the radius R is 11.13 mm. Then according to L₁，L₂，L₃，b₁，b₂R and other parameters are processed into ultrasonic fatigue test samples, and the working areas in the middle of the test samples, namely the surfaces of the equal section sections and the variable section sections, are polished to reach the finish degree R_a0.32. Tapping one side of the sample, wherein the diameter of the screw hole is 5 mm.

3. The dimensions of the processed sample were measured, and the actual dimensions thus measured were L'₁＝12.5mm，L'₂＝20mm，L'₃＝11.00mm，b′₁＝2.86mm，b'₂9.90mm, the corresponding actual resonance frequency f' 19.88 × 10 can be calculated from fig. 5³Hz。

4. Fixing the processed sample in the displacement amplifier of the ultrasonic fatigue testing machine, adjusting the cooling air nozzle, aligning the sample with the equal section part, opening the valve, and cooling the sample, as shown in fig. 6.

5. Opening the control software of the ultrasonic fatigue system, selecting the type of the circular arc sample, and drawing up the size parameter of the circular arc sample The radius of the thinnest point in the circular arc shaped sample,the radius of the cylinder at both ends of the sample,half the length of the middle section of the sample. The lengths of the two ends of the circular arc sample can be obtained by an analytical formula (6) of the lengths of the two ends of the circular arc sample or by ultrasonic fatigue testing machine system control software

6. Drawing up stress amplitude sigma to be loaded for plate-shaped test sample containing equal section_max200MPa according to the size parameters of the circular arc-shaped sample $L_1^0=20mm,L_2^0=8.53mm,R_1^0=1.5mm,R_2^0=5mm$ And the measured dimension parameter L 'of the plate-shaped sample containing the equal section segment'₁＝12.5mm，L'₂＝20mm，L'₃＝11.00mm，b′₁＝2.86mm，b'₂9.90mm and the calculated actual resonance frequency f' 19.88 × 10³Hz. The stress amplitude of the corresponding arc-shaped sample under the same displacement amplitude is obtained by the stress correction formula, namely the formula (9)Namely, the stress amplitude of the circular arc-shaped sample needs to be input in the system control software, and the vibration displacement amplitude is 22.0 μm at this time. As can be seen from comparison example 1, the corrected stress amplitude 447MPa is 2MPa less than the uncorrected stress amplitude 449MPa, the displacement amplitude is 0.1 μm less, and if the uncorrected stress amplitude 449MPa is directly loaded, the actual life will probably be shortened.

7. After other test parameters are set in control system software, an axial tension-compression ultrasonic fatigue test of a plate-shaped test sample with a uniform section is started, and the vibration frequency is 19.88 × 10³Hz, through 1.84 × 10⁷After weekly cycles, the specimens broke.

The above-mentioned embodiments are merely preferred embodiments of the present invention, and the scope of the present invention is not limited thereto, so that variations based on the shape and principle of the present invention should be covered within the scope of the present invention.

Claims (4)

1. A metal ultrasonic fatigue test method of a plate-shaped sample containing a uniform section is characterized by comprising the following three parts:

firstly, according to the density and the elastic modulus of a test material, obtaining a size design formula of a plate-shaped axial tension-compression ultrasonic fatigue sample containing a constant section by adopting analytical calculation;

a second part is used for deducing a stress conversion formula between the plate-shaped test sample containing the equal section and the circular arc-shaped test sample, converting the stress of the plate-shaped ultrasonic fatigue test sample containing the equal section into the stress corresponding to the circular arc-shaped test sample on system control software, and using the existing equipment and software to complete the ultrasonic fatigue test control of the plate-shaped test sample containing the equal section;

and the third part is used for deducing a stress correction formula when the actual size and the design size of the plate-shaped sample containing the uniform section have deviation, so that the test control is more accurate.

2. The metal ultrasonic fatigue test method of the plate-like test piece with the constant section segment according to claim 1, characterized in that:

in the first section, the calculation of the sizing formula includes the following specific steps:

measuring the density rho and the elastic modulus E of a test material;

step two, analyzing and calculating the plate-shaped axial tension-compression ultrasonic fatigue sample containing the uniform section, and the method comprises the following steps:

2.1 miming b₁，b₂，L₁，L₂Data, b₁Is the thickness of a constant cross-sectional section of the plate, b₂Is the thickness at both ends of the plate, L₁Is half the length of the equal section of the plate, L₂Is the variable cross-sectional length of the plate;

2.2 calculating the length L at both ends of the plate₃(ii) a According to the continuous system vibration theory, the material meets the ideal elastomer condition, the origin of coordinates is assumed as the axial center of the sample, and the axial direction of the sample is taken as the x axis; u (x, t) is the longitudinal vibratory displacement of the cross section at coordinate x at time t; s (x) is the area of the cross-section of the sample at coordinate x, and S' (x) is the first derivative of the function S (x), then the longitudinal wave equation for the sample at resonance is:

$\frac{{\&part;}^{2}U(x,t)}{\&part;t^2}=\frac{E}{\mathrm{\&rho;}}(\frac{{\&part;}^{2}U\left(x\right)}{\&part;x^2}+\frac{S^{\&prime;}\left(x\right)}{S\left(x\right)}\frac{\&part;U(x,t)}{\&part;x})---\left(1\right)$

assuming that the sample satisfies the resonance condition, the separation variable of U (x, t) is U (x, t) ═ U (x) e^iωtSubstituting into formula (1) to obtain

$\frac{{\&part;}^{2}U\left(x\right)}{\&part;x^2}+\frac{S^{\&prime;}\left(x\right)}{S\left(x\right)}\frac{\&part;U\left(x\right)}{\&part;x}+k^{2}U\left(x\right)=0---\left(2\right)$

In the formula,in order to derive the function u (x),in order to obtain a first-order derivative,for second order derivation, k is a mixing parameter of the sample material,rho is the density of the sample material, E is the elastic modulus of the sample material, and f is the vibration frequency of the sample;

according to the cross-sectional area equation of the plate-like specimen

$\left\{\begin{array}{cc}S\left(x\right)=b_{1}w& \left|x\right|\&le;L_1\\ S\left(x\right)=b_{1}w\cos h^2\&lsqb;\mathrm{\&alpha;}(x-L_1)\&rsqb;& L_1\&le;\left|x\right|\&le;L_1+L_2\\ S\left(x\right)=b_{2}w& L_1+L_2\&le;\left|x\right|\&le;L\end{array}\right.$

Wherein, b₁Is the thickness of a constant cross-sectional section of the plate, b₂Is the thickness at both ends of the plate, w is the width of the plate;

calculating a boundary condition U according to a cross-sectional area equation of the sample_x＝L＝U_max，U|′_x＝L＝0，U|_x＝00 and continuity Condition $U{|}_{x\&RightArrow;L_1-}=U{|}_{x\&RightArrow;L_1+},U^{\&prime;}{|}_{x\&RightArrow;L-}=U^{\&prime;}{|}_{x\&RightArrow;L+},U{|}_{x\&RightArrow;(L+L_2)-}=U{|}_{x\&RightArrow;(L+L_2)+},U^{\&prime;}{|}_{x\&RightArrow;(L_1+L_2)-},U^{\&prime;}{|}_{x\&RightArrow;(L_1+L_2)+}$ The equation of vibration in the longitudinal direction of the sample U (x) is obtained as:

$\left\{\begin{array}{cc}U\left(x\right)=C_{1}sin\left(kx\right)& \left|x\right|<L_1\\ U\left(x\right)=C_2\exp \&lsqb;({\mathrm{\&beta;}}_1-{\mathrm{\&alpha;}}_1)(x-L_1)\&rsqb;+C_3\exp \&lsqb;-({\mathrm{\&alpha;}}_1+{\mathrm{\&beta;}}_1)(x-L_1)\&rsqb;& L_1<\left|x\right|<L_1+L_2\\ U\left(x\right)=U_{max}cos\&lsqb;k(L-x)\&rsqb;& L_1+L_2<\left|x\right|<L\end{array}\right.---\left(3\right)$

in the formula:

U_max＝U|_x＝Lα, the maximum displacement amplitude of the sample₁，β₁，Comprises the following steps:

determining the length L at both ends of the plate₃Comprises the following steps:

$L_3=\frac{1}{k}\arctan \left\{\frac{cos\left({\mathrm{kL}}_1\right)\&lsqb;{\mathrm{\&beta;}}_1\cosh \left({\mathrm{\&beta;}}_1{L}_2\right)-{\mathrm{\&alpha;}}_1\sinh \left({\mathrm{\&beta;}}_1{L}_2\right)\&rsqb;-ksin\left({\mathrm{kL}}_1\right)\sinh \left({\mathrm{\&beta;}}_1{L}_2\right)}{\&lsqb;{\mathrm{\&alpha;}}_{1}sin\left({\mathrm{kL}}_1\right)+k\cos \left({\mathrm{kL}}_1\right)\&rsqb;\sinh \left({\mathrm{\&beta;}}_1{L}_2\right)+{\mathrm{\&beta;}}_{1}sin\left({\mathrm{kL}}_1\right)\cosh \left({\mathrm{\&beta;}}_1{L}_2\right)}\right\}---\left(4\right)$

during analysis and calculation of the sample, a variable cross-section curve of a plate-shaped sample containing a uniform cross-section is assumed to be a catenary, and the catenary is replaced by an arc curve during actual machining due to difficulty in completion of machining; by varying the length L of the section₂Minimum thickness b₁Thickness b at both ends₂To obtain the radius of the transition arc of the variable section

3. The metal ultrasonic fatigue test method of the plate-like test piece with the constant section segment according to claim 1, characterized in that:

the second part is as follows:

the stress distribution function sigma (x) of the plate-shaped ultrasonic fatigue test sample with the uniform section is obtained by deriving a displacement function U (x):maximum stress amplitude sigma thereof_maxA plate-like sample containing equal cross-section segments was obtained at the middle cross-section (x ═ 0) of the sampleMaximum stress amplitude σ of_maxComprises the following steps:

wherein E is the modulus of elasticity of the sample material,U_max＝U|_x＝Lis the maximum displacement amplitude of the sample,

for a plate-shaped sample containing equal section sections, a dimension parameter L is set₁，L₂，b₁，b₂First, the length L of the plate at both ends is obtained according to the formula (4)₃The stress amplitude needing to be loaded is set to be sigma_maxThe stress amplitude σ of the plate-like sample containing the uniform cross-section is obtained from the equation (5)_maxCorresponding displacement amplitude

For the self-contained arc ultrasonic fatigue test sample in the system control software, the method plansThe data of the data is transmitted to the data receiver,the radius of the thinnest point in the circular arc shaped sample,the radius of the cylinder at both ends of the sample,is half of the length of the middle variable section of the sample; then the length at both ends of the circular arc shaped specimenThe length of the arc-shaped sample is automatically given by system control software or given by an analytical formula of the lengths of two ends of the arc-shaped sample:

$L_2^0=\frac{1}{k}\arctan \{\frac{1}{k}\&lsqb;\frac{\mathrm{\&beta;}}{\tanh \left({\mathrm{\&beta;L}}_1^0\right)}-\mathrm{\&alpha;}\tanh \left({\mathrm{\&alpha;L}}_1^0\right)\&rsqb;\};---\left(6\right)$

in the formula, $\mathrm{\&alpha;}=\frac{1}{L_1^0}\arccos h\left(\frac{R_2^0}{R_1^0}\right),$ $\mathrm{\&beta;}=\sqrt{{\mathrm{\&alpha;}}^2-k^2},$ $k=2\mathrm{\&pi;}f/\sqrt{\frac{E}{\mathrm{\&rho;}}},$ rho is the density of the sample material, E is the elastic modulus of the sample material, and f is the vibration frequency of the sample;

stress distribution function sigma of circular arc-shaped test piece⁰(x) The maximum stress amplitude of the arc-shaped sample is obtained by derivation of the displacement functionAt the specimen middle section (x ═ 0):

in the formula, $\mathrm{\&alpha;}=\frac{1}{L_1^0}\arccos h\left(\frac{R_2^0}{R_1^0}\right),$ $\mathrm{\&beta;}=\sqrt{{\mathrm{\&alpha;}}^2-k^2},$ $k=2\mathrm{\&pi;}f/\sqrt{\frac{E}{\mathrm{\&rho;}}};$

as can be seen from the equation (7), in order to make the displacement amplitude of the plate-like sample including the uniform cross-section be U_maxThe displacement amplitude U of the circular arc-shaped sample at the same level as that of the plate-shaped sample with the equal section is determined_maxLower corresponding stress amplitudeComprises the following steps:

the formula (8) is a stress conversion formula between a plate-shaped sample containing equal section sections and a circular-arc-shaped sample, wherein sigma_maxThe stress amplitude of the plate-shaped sample containing the equal section is obtained,the stress amplitude of the corresponding arc-shaped sample under the same displacement amplitude is obtained.

4. The method for metal ultrasonic fatigue test of plate-like test piece with constant section according to claim 1, wherein the test piece is prepared by mixing a plurality of materials

The third part is as follows:

it is assumed that the dimension of the actually processed plate-like test piece containing the uniform cross-section segments is L 'measured by accurate measurement'₁，L'₂，L'₃，b₁'，b'₂，b′₁Is the measured thickness of the section of the plate with equal section, b'₂Is measured thickness, L 'at both ends of the sheet'₁Is half, L 'of the actual measurement length of the equal section of the plate'₂Is the actual measurement length of the plate variable section, L'₃The actual resonant frequency of the sample may not be 2.0 × 10 due to the deviation of the measured size of the sample from the design size⁴Hz; in this case, the actual resonance frequency f 'of the sample can be inversely calculated from the measured dimensions of the sample according to equation (4), and L'₁，L'₂，b₁'，b'₂Substituting formula (4) and setting the vibration frequency in the vibration frequency range of the ultrasonic fatigue testing machine, namely 19.50 × 10³Hz～20.50×10³Calculating value one by one at intervals of 10Hz between Hz to obtain the actual measurement length L 'at two ends of the plate'₃And the corresponding relation between the resonant frequency f ', namely L'₃The value of (d) yields the actual resonant frequency f';

if f' is less than 19.50 × 10³Hz or greater than 20.50 × 10³Hz indicates that the size deviation of the sample is too large, so that the sample cannot start to vibrate and needs to be processed again;

according to the obtained f', the stress amplitude sigma of the plate-shaped sample containing the uniform section can be obtained_maxActual vibration displacement amplitude of

Wherein $k^{\&prime;}=2{\mathrm{\&pi;f}}^{\&prime;}/\sqrt{\frac{E}{\mathrm{\&rho;}}},$ ${\mathrm{\&beta;}}_1^{\&prime;}=\sqrt{{\mathrm{\&alpha;}}_1^{\&prime;2}-k^{\&prime;2}},$ ${\mathrm{\&alpha;}}_1^{\&prime;}=\frac{1}{2L_2^{\&prime;}}ln\left(\frac{b_2^{\&prime;}}{b_1^{\&prime;}}\right);$

According to U'_maxFrom the equation (8), the amplitude U 'of the circular arc sample of a predetermined size can be obtained'_maxLower corresponding stress amplitudeWherein,for given dimension values of the circular arc shaped test specimen, f⁰＝2.0×10⁴Hz；The expression of (a) is as follows:

the formula (9) is a stress correction formula of the plate-shaped sample containing the uniform section; wherein,for given dimension values of the circular arc shaped test specimen, f⁰＝2.0×10⁴Hz，L'₁，L'₂，L'₃，b₁'，b'₂Is a measured dimension value of a plate-shaped sample containing equal section segments, and f ' is a measured dimension L ' of the plate-shaped sample '₁，L'₂，L'₃，b₁'，b'₂And (5) inversely calculating the obtained actual resonant frequency.