CN110455919B - Method for evaluating solid-solid interface contact characteristics by utilizing nonlinear effect - Google Patents

Method for evaluating solid-solid interface contact characteristics by utilizing nonlinear effect Download PDF

Info

Publication number
CN110455919B
CN110455919B CN201910812907.0A CN201910812907A CN110455919B CN 110455919 B CN110455919 B CN 110455919B CN 201910812907 A CN201910812907 A CN 201910812907A CN 110455919 B CN110455919 B CN 110455919B
Authority
CN
China
Prior art keywords
solid
wave
interface
contact
nonlinear
Prior art date
Legal status (The legal status is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the status listed.)
Active
Application number
CN201910812907.0A
Other languages
Chinese (zh)
Other versions
CN110455919A (en
Inventor
韩庆邦
钱昕彤
仲晓敏
殷澄
姜学平
Current Assignee (The listed assignees may be inaccurate. Google has not performed a legal analysis and makes no representation or warranty as to the accuracy of the list.)
Nanjing Youqiao Electronic Technology Co ltd
Original Assignee
Changzhou Campus of Hohai University
Priority date (The priority date is an assumption and is not a legal conclusion. Google has not performed a legal analysis and makes no representation as to the accuracy of the date listed.)
Filing date
Publication date
Application filed by Changzhou Campus of Hohai University filed Critical Changzhou Campus of Hohai University
Priority to CN201910812907.0A priority Critical patent/CN110455919B/en
Publication of CN110455919A publication Critical patent/CN110455919A/en
Application granted granted Critical
Publication of CN110455919B publication Critical patent/CN110455919B/en
Active legal-status Critical Current
Anticipated expiration legal-status Critical

Links

Images

Classifications

    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N29/00—Investigating or analysing materials by the use of ultrasonic, sonic or infrasonic waves; Visualisation of the interior of objects by transmitting ultrasonic or sonic waves through the object
    • G01N29/04—Analysing solids
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N29/00—Investigating or analysing materials by the use of ultrasonic, sonic or infrasonic waves; Visualisation of the interior of objects by transmitting ultrasonic or sonic waves through the object
    • G01N29/04—Analysing solids
    • G01N29/12—Analysing solids by measuring frequency or resonance of acoustic waves
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N29/00—Investigating or analysing materials by the use of ultrasonic, sonic or infrasonic waves; Visualisation of the interior of objects by transmitting ultrasonic or sonic waves through the object
    • G01N29/44—Processing the detected response signal, e.g. electronic circuits specially adapted therefor
    • G01N29/4409—Processing the detected response signal, e.g. electronic circuits specially adapted therefor by comparison
    • G01N29/4418—Processing the detected response signal, e.g. electronic circuits specially adapted therefor by comparison with a model, e.g. best-fit, regression analysis
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N29/00—Investigating or analysing materials by the use of ultrasonic, sonic or infrasonic waves; Visualisation of the interior of objects by transmitting ultrasonic or sonic waves through the object
    • G01N29/44—Processing the detected response signal, e.g. electronic circuits specially adapted therefor
    • G01N29/4409—Processing the detected response signal, e.g. electronic circuits specially adapted therefor by comparison
    • G01N29/4436—Processing the detected response signal, e.g. electronic circuits specially adapted therefor by comparison with a reference signal
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N29/00—Investigating or analysing materials by the use of ultrasonic, sonic or infrasonic waves; Visualisation of the interior of objects by transmitting ultrasonic or sonic waves through the object
    • G01N29/44—Processing the detected response signal, e.g. electronic circuits specially adapted therefor
    • G01N29/4472—Mathematical theories or simulation
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N29/00—Investigating or analysing materials by the use of ultrasonic, sonic or infrasonic waves; Visualisation of the interior of objects by transmitting ultrasonic or sonic waves through the object
    • G01N29/44—Processing the detected response signal, e.g. electronic circuits specially adapted therefor
    • G01N29/46—Processing the detected response signal, e.g. electronic circuits specially adapted therefor by spectral analysis, e.g. Fourier analysis or wavelet analysis
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N29/00—Investigating or analysing materials by the use of ultrasonic, sonic or infrasonic waves; Visualisation of the interior of objects by transmitting ultrasonic or sonic waves through the object
    • G01N29/44—Processing the detected response signal, e.g. electronic circuits specially adapted therefor
    • G01N29/48—Processing the detected response signal, e.g. electronic circuits specially adapted therefor by amplitude comparison
    • G—PHYSICS
    • G06—COMPUTING; CALCULATING OR COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G06F17/11—Complex mathematical operations for solving equations, e.g. nonlinear equations, general mathematical optimization problems
    • G06F17/13—Differential equations
    • G—PHYSICS
    • G01—MEASURING; TESTING
    • G01N—INVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N2291/00—Indexing codes associated with group G01N29/00
    • G01N2291/02—Indexing codes associated with the analysed material
    • G01N2291/023—Solids
    • G01N2291/0231—Composite or layered materials

Landscapes

  • Physics & Mathematics (AREA)
  • General Physics & Mathematics (AREA)
  • Engineering & Computer Science (AREA)
  • Pathology (AREA)
  • Immunology (AREA)
  • Health & Medical Sciences (AREA)
  • General Health & Medical Sciences (AREA)
  • Biochemistry (AREA)
  • Analytical Chemistry (AREA)
  • Chemical & Material Sciences (AREA)
  • Life Sciences & Earth Sciences (AREA)
  • Mathematical Physics (AREA)
  • Signal Processing (AREA)
  • Pure & Applied Mathematics (AREA)
  • Mathematical Optimization (AREA)
  • Mathematical Analysis (AREA)
  • Computational Mathematics (AREA)
  • Algebra (AREA)
  • Data Mining & Analysis (AREA)
  • Theoretical Computer Science (AREA)
  • Acoustics & Sound (AREA)
  • Databases & Information Systems (AREA)
  • Operations Research (AREA)
  • Software Systems (AREA)
  • General Engineering & Computer Science (AREA)
  • Spectroscopy & Molecular Physics (AREA)
  • Investigating Or Analyzing Materials By The Use Of Ultrasonic Waves (AREA)

Abstract

The invention discloses a method for evaluating contact characteristics of a solid-solid interface by utilizing a nonlinear effect, which comprises the following steps of: s01, establishing an isotropic solid-solid bonding interface model; s02, deducing the expressions of the reflected wave and the transmitted wave by using a perturbation method; s03, defining four nonlinear parameters to evaluate the contact characteristics of the solid-solid bonding interface; s04, drawing a curve of the four nonlinear parameters changing along with the contact stress; s05, selecting excitation signal aliasing with proper frequency by using an ultrasonic signal generator; s06, performing fast Fourier transform on the acquired signals to obtain corresponding frequency spectrum images, and measuring to obtain nonlinear parameters of the test piece under different pressures; and S07, comparing the theoretical value with the actual value, and analyzing to obtain the monotonic relation between the nonlinear parameter and the bonding strength of the test piece. The method for evaluating the contact characteristic of the solid-solid interface by utilizing the nonlinear effect can better evaluate the bonding strength and the contact condition of the interface of the bonding piece and more effectively monitor and maintain the quality of the bonding piece.

Description

Method for evaluating solid-solid interface contact characteristics by utilizing nonlinear effect
Technical Field
The invention relates to a method for evaluating contact characteristics of a solid-solid interface by utilizing a nonlinear effect, belonging to the technical field of ultrasonic detection and analysis.
Background
The bonding is a method for connecting materials together to form an assembly, and the bonded assembly has the advantages of no stress concentration, good fatigue resistance, light structure, capability of avoiding electrochemical reaction of metal connection, capability of connecting objects with complex shapes and the like, and is widely and massively applied in the industries of aerospace, electronics, medical instruments, wood, buildings and the like.
However, various micro defects are inevitably formed inside the adhesive or at the bonding interface during the production of the adhesive. Under the influence of the forming process and the working environment, the metal material and the bonding structure thereof are easy to have various defects of early mechanical property degradation. In some very important situations, irreparable consequences can result if the defective bonded assembly cannot be located and replaced in a timely manner. Therefore, nondestructive testing of the adhesive and prediction of the adhesive strength are actual problems to be solved urgently.
Ultrasonic waves are an important means for solving the problem of interfacial adhesion evaluation. The traditional linear ultrasonic flaw detection technology is mainly based on linear parameters generated in the sound wave transmission process to detect flaws in materials. However, in practical application, the detection signal needs to reduce the nonlinear interference of noise, and the ultrasonic wave is severely attenuated when passing through the defect, so that the linear parameter changes insignificantly, and the flaw detection effect is not very ideal.
Disclosure of Invention
The invention aims to overcome the defects of the prior art and provide a method for evaluating the contact characteristic of a solid-solid interface by utilizing a nonlinear effect, which can better evaluate the bonding strength and the contact condition of a bonding piece interface and more effectively monitor and maintain the quality of the bonding piece.
In order to solve the technical problems, the technical scheme adopted by the invention is as follows:
a method for evaluating solid-solid interface contact characteristics by utilizing a nonlinear effect comprises the following steps:
s01, establishing an isotropic solid-solid bonding interface model, and giving a longitudinal wave motion equation and a stress displacement relation;
s02, deriving expressions of reflected waves and transmitted waves by a perturbation method according to a nonlinear wave equation of aliasing incident beams and boundary conditions of a solid-solid interface;
s03, analyzing to obtain a nonlinear effect of an aliasing beam incident on a solid-solid bonding interface, and defining four nonlinear parameters to evaluate the contact characteristics of the solid-solid bonding interface by combining the power law relation of the interface linear rigidity and the contact stress;
s04, drawing a curve of four nonlinear parameters changing along with contact stress under the condition of aliasing of beams with different frequencies according to the numerical calculation result;
s05, selecting excitation signal aliasing with proper frequency by using an ultrasonic signal generator, exciting on the tested piece through an ultrasonic probe, receiving an acquisition signal by using another ultrasonic probe on the other side of the tested piece, and visually acquiring the signal through a signal acquisition device;
s06, carrying out fast Fourier transform on the acquired signals to obtain corresponding frequency spectrum images, and measuring to obtain nonlinear parameters of the test piece under different pressures based on the theoretical basis of S03;
and S07, comparing the theoretical value with the actual value, and analyzing to obtain the monotonic relation between the nonlinear parameter and the bonding strength of the test piece.
In S01, the one-dimensional elastic longitudinal wave propagates in the X-axis direction, and the reference plane positions of the contact surfaces are set to X ═ X, respectively-,x＝X+In the region X < X-Incident one-dimensional longitudinal wave fincTo obtain a reflected wave frefIn the region X > X+Obtaining a transmitted wave ftra；
The one-dimensional longitudinal wave is positively propagated along the x axis, and the relation between the motion equation and the stress displacement is as follows:
Figure BDA0002185532210000021
Figure BDA0002185532210000022
wherein u (x, t) represents the displacement of the acoustic wave propagating along the x direction; σ (x, t) represents a stress variation; sigma0Two solid contact static stresses; t represents time; ρ represents a mediumThe density of (c).
In S02, the expression of the aliased beam nonlinear wave equation is as follows:
Figure BDA0002185532210000031
where phi is the velocity potential and the velocity expression of the acoustic wave is
Figure BDA0002185532210000034
Wherein, c0The propagation velocity of small-amplitude sound waves in ideal gas is constant and depends on a medium; gamma is the ratio of specific heat at constant pressure to specific heat at constant volume; p is pressure, P0Is the initial pressure; rho0Is the initial density.
In S02, the solid-solid interface boundary condition satisfies:
when the elastic wave has interaction across the boundary, the displacement and stress conditions are as follows:
Figure BDA0002185532210000032
when the elastic waves do not interact across the boundary, i.e. there is no contact between the two solids at this time, the displacement and stress conditions are as follows:
Figure BDA0002185532210000033
wherein u (+0, t) is acoustic wave in region X > X+Displacement of (2); u (-0, t) is acoustic wave in region X < X-Displacement of (2); σ (+0, t) is that sound wave is in area X > X+The stress of (2); sigma (-0, t) is that the sound wave is in the region X < X-Of the stress of (c).
The specific steps of deriving the expressions of the reflected wave and the transmitted wave by using the perturbation method are as follows:
the solutions of step a, formula (1) and formula (2) are as follows:
u(x,t)＝finc(x-ct)+fref(x+ct)x＜X- (7)
u(x,t)＝ftra(x-ct)x＞X+ (8)
in the formula, function finc(x-ct),fref(x + ct) and ftra(x-ct) represents an expression of an incident wave propagating right along the x-axis on the left side of the interface, a reflected wave propagating left along the x-axis, and a transmitted wave propagating right along the x-axis on the right side of the interface, respectively, and the wave velocity c of the acoustic wave is (λ/ρ)1/2λ is the elastic constant;
step b, in order to solve the equations of the formula (1) and the formula (2), introducing the following variables:
Z(t)＝[u(X+,t)+u(X-,t)]/2 (9)
Y(t)＝u(X+,t)-u(X-,t)＝h(t)-h0 (10)
one-dimensional elastic longitudinal wave propagates along the direction of the X axis, and the reference plane positions of the contact surfaces are respectively set as X ═ X-,x＝X+(ii) a Wherein u (+0, t) is an acoustic wave at the interface X ═ X+A displacement of (a); u (-0, t) is acoustic wave at interface X ═ X-A displacement of (a); h (t) is the gap distance between the contact surfaces when the sound waves are transmitted; h is0The gap distance between the contact surfaces is constant when the contact surface is silent, and is related to the roughness of the solid medium and the contact surface; z represents the half of the displacement sum of the two ends of the contact interface, and Y represents the displacement difference of the two ends of the contact interface;
the derivation of equations (10) and (11) yields:
Figure BDA0002185532210000041
Figure BDA0002185532210000042
wherein, σ (h)0+ Y) is the stress at the contact interface when the acoustic wave propagates;
and solving to obtain an expression of the reflected wave and the transmitted wave:
Figure BDA0002185532210000043
Figure BDA0002185532210000044
wherein Y represents the displacement difference between the two ends of the interface;
c, exciting to obtain a frequency f1，f2Two ultrasonic waves will generate nf due to the nonlinear interaction between the two ultrasonic waves at the interface1，nf2，f1+f2Sum frequency wave sum f1-f2The difference frequency wave of (a) is,
regardless of the acoustic wave attenuation and the initial phase difference, the introduced excitation signal expression is as follows:
Figure BDA0002185532210000045
in the formula, A1And A2，ω1And omega2Respectively the amplitude and angular frequency of the two excited sound waves,
solving a reflected wave equation and a transmitted wave equation by using a perturbation method according to a wave equation and an incident wave expression; considering that the gap distance changes very little when the displacement of the incident wave is small, the function σ (h) may be derived from this in this case when h is h0The nearby taylor expansion is replaced, taking the second order term, the expression is as follows:
σ(h)＝σ(h0+Y)＝σ0-K1Y+K2Y2 (16)
wherein, the first and the second end of the pipe are connected with each other,
Figure BDA0002185532210000051
K1represents linear stiffness;
Figure BDA0002185532210000052
K2representing the nonlinear rigidity of the contact surface, namely the second-order rigidity; h is the gap distance of the contact interface;
and d, substituting the formula (15) and the formula (16) into the formula (12) to obtain a first-order nonlinear ordinary differential equation related to Y, wherein the expression form is as follows:
Figure BDA0002185532210000053
e, solving an approximate solution by using a perturbation method as follows:
consider an approximate solution of the equation Y ═ Y1+Y2Wherein Y is1Is a first order trace, Y2Is a second order trace, is an approximate solution of the equation, and, at the same time, Y1,Y2The following equation is satisfied:
Figure BDA0002185532210000054
Figure BDA0002185532210000055
obtaining by solution:
Figure BDA0002185532210000056
wherein the content of the first and second substances,
Figure BDA0002185532210000057
δ1＝arctan(ω1/a)，δ2＝arctan(ω2/a)
Figure BDA0002185532210000061
wherein, take Δ ω ═ ω2-ω1，Σω＝ω1+ω2，
Figure BDA0002185532210000062
ψ1＝θ1+δ1-δ2，ψ2＝θ2-δ1-δ2，θ1＝arctan[a/(Δω)]，θ2＝arctan(a/Σω)；
Step f, finally obtaining an expression of the reflected wave and the transmitted wave:
Figure BDA0002185532210000063
Figure BDA0002185532210000071
in S03, the expression of the power law relationship between interface linear stiffness and contact stress is as follows:
K1＝Cσ0 m (24)
wherein C and m are normal numbers, will
Figure BDA0002185532210000072
And
Figure BDA0002185532210000073
and substituting to obtain a relation function of second-order rigidity and contact pressure:
Figure BDA0002185532210000074
defining a non-linearity parameter beta1,β2The nonlinear parameter gamma is the ratio of the amplitude of the difference frequency component and sum frequency component to the amplitude of the fundamental frequency in the transmitted wave1,γ2The ratio of the amplitude of the difference frequency component and the sum frequency component in the reflected wave to the amplitude of the fundamental frequency is:
Figure BDA0002185532210000075
Figure BDA0002185532210000076
Figure BDA0002185532210000077
Figure BDA0002185532210000081
the invention has the beneficial effects that: the invention provides a method for evaluating the contact characteristic of a solid-solid interface by utilizing a nonlinear effect, which is used for evaluating the contact characteristic of the solid-solid bonding interface by comparing a measured value of a nonlinear parameter with a reference value to analyze the bonding strength at the solid-solid bonding interface. In practical application, the detection signal needs to reduce the nonlinear interference of noise, and meanwhile, the ultrasonic wave can be seriously attenuated when passing through the defect, so that the linear parameter change is not obvious, and the linear ultrasonic flaw detection effect is not ideal directly. Compared with the single beam serving as an excitation source, the nonlinear ultrasonic detection technology can better make up the defect, and compared with the single beam serving as an excitation source, the aliasing beam with any frequency serving as an excitation source can more easily see different frequency components when receiving response signals, and can more easily measure nonlinear parameters, the defect of linear ultrasonic detection is well made up, so that the sensitivity of nonlinear ultrasonic detection using the aliasing beam as the excitation source is higher. Meanwhile, the method for evaluating the contact characteristic of the solid-solid bonding interface based on the nonlinear parameters has stronger practicability.
Drawings
FIG. 1 is a schematic representation of a model of the present invention;
FIG. 2 shows the nonlinear coefficient β of the present invention at different difference frequencies1Graph (theoretical value) against stress;
FIG. 3 shows the nonlinear coefficient β of the present invention at different sum frequencies2Graph (theoretical value) against stress;
FIG. 4 shows the nonlinear coefficient γ of the present invention at different difference frequencies1Graph of relationship with pressure(theoretical value);
FIG. 5 shows the nonlinear coefficient γ of the present invention at different sum frequencies2Pressure dependence (theoretical);
FIG. 6 is a schematic view of the experimental set-up of the present invention;
FIG. 7 is a graph of the spectrum of a transmitted signal without applied pressure in an example of the invention;
FIG. 8 is a graph of the spectrum of the transmission signal when pressure is applied in an example of the present invention;
FIG. 9 is a graph comparing the curves of the theoretical values and the actual measured values of the present invention.
Detailed Description
The present invention is further described with reference to the accompanying drawings, and the following examples are only for clearly illustrating the technical solutions of the present invention, and should not be taken as limiting the scope of the present invention.
Nonlinear ultrasonic detection techniques can clearly see different frequency components when receiving a response signal. In addition to the fundamental frequency signal in the incident wave, higher harmonics are present in the received acoustic wave. Compared with the fundamental wave signal, the attenuation of harmonic wave is very small, which well makes up the defect of linear ultrasonic detection, so the sensitivity of nonlinear ultrasonic detection is higher.
The method of beam aliasing is controllable, so that any space can be selected, different waveforms can be converted, sound waves with different frequencies and different propagation directions can be selected, and the effect of avoiding nonlinear interference on a system can be achieved. The method for measuring the nonlinear effect of the system by adopting one path of incident sound wave response summation not only can remove the nonlinearity of the system, but also has higher sensitivity to the tiny change of the internal structure of the detection material and better application prospect. Therefore, compared with the single beam serving as the excitation source, harmonic signals of different frequencies can be more easily received by using the aliasing beam of any frequency as the excitation source, the nonlinear parameter can be more easily measured, and the method has higher accuracy and higher practicability.
The invention discloses a method for evaluating solid-solid interface contact characteristics by utilizing nonlinear effect, which is characterized in that two signals with different frequencies are input, the generated nonlinear effect of difference frequency and sum frequency signals is utilized, and the solid-solid interface contact characteristics are evaluated by introducing nonlinear coefficients, and the method mainly comprises the following steps:
step one, establishing a corresponding isotropic solid-solid bonding interface model according to actual conditions. As shown in fig. 1, which is a schematic diagram of a model of the present invention, a one-dimensional longitudinal elastic wave propagates along an X-axis direction, and reference plane positions of contact surfaces are set to X ═ X, respectively-,x＝X+In the region X < X-Incident one-dimensional longitudinal wave fincTo obtain a reflected wave frefIn the region X > X+Obtaining a transmitted wave ftra。
The one-dimensional longitudinal wave is positively propagated along the x axis, and the relation between the motion equation and the stress displacement is as follows:
Figure BDA0002185532210000091
Figure BDA0002185532210000092
wherein u (x, t) represents the displacement of the acoustic wave propagating along the x direction; σ (x, t) represents the stress variation; sigma0Two solid contact static stresses; t represents time; ρ represents the density of the medium.
And step two, deriving expressions of reflected waves and transmitted waves by using a perturbation method according to a nonlinear wave equation of aliasing incident beams and the boundary conditions of a solid-solid interface.
Wherein, the expression of the aliasing beam nonlinear wave equation is as follows:
Figure BDA0002185532210000101
where phi is the velocity potential and the velocity expression of the acoustic wave is
Figure BDA0002185532210000104
Wherein, c0The propagation velocity of small-amplitude sound waves in ideal gas is constant and depends on a medium; gamma is the ratio of specific heat at constant pressure to specific heat at constant volume; p is pressure, P0 is initial pressure; ρ is a unit of a gradient0Is the initial density.
Wherein, the boundary condition of the solid-solid interface satisfies:
when the elastic wave has interaction across the boundary, the displacement and stress conditions are as follows:
Figure BDA0002185532210000102
when the elastic waves do not interact across the boundary, i.e., the two solid phases are not in contact at this time, the displacement and stress conditions are as follows:
Figure BDA0002185532210000103
one-dimensional elastic longitudinal wave propagates along the direction of the X axis, and the reference plane positions of the contact surfaces are respectively set as X ═ X-,x＝X+. Wherein u (+0, t) is acoustic wave in region X > X+Displacement of (2); u (-0, t) is acoustic wave in region X < X-Displacement of (2); σ (+0, t) is that sound wave is in area X > X+The stress of (2); sigma (-0, t) is that sound wave is in region X < X-Of the stress of (c). The specific steps of deriving the expressions of the reflected wave and the transmitted wave by using the perturbation method are as follows:
the solutions of step a, formula (1) and formula (2) are as follows:
u(x,t)＝finc(x-ct)+fref(x+ct)x＜X- (7)
u(x,t)＝ftra(x-ct)x＞X+ (8)
in the formula, function finc(x-ct),fref(x + ct) and ftra(x-ct) represents an expression of an incident wave propagating right in the x-axis direction on the left side of the interface, an expression of a reflected wave propagating left in the x-axis direction, and an expression of a transmitted wave propagating right in the x-axis direction on the right side of the interface, respectively, and the acoustic wave velocity c ═ λ/ρ1/2And λ is the elastic constant.
Step b, in order to solve the equations of the formula (1) and the formula (2), introducing the following variables:
Z(t)＝[u(X+,t)+u(X-,t)]/2 (9)
Y(t)＝u(X+,t)-u(X-,t)＝h(t)-h0 (10)
one-dimensional elastic longitudinal wave propagates along the direction of the X axis, and the reference plane positions of the contact surfaces are respectively set as X ═ X-,x＝X+. Wherein u (+0, t) is the sound wave at the interface X ═ X+A displacement of (a); u (-0, t) is acoustic wave at the interface X ═ X-A displacement of (a); h (t) is the gap distance between the contact surfaces when the sound waves are transmitted; h is0The gap distance between the contact surfaces in the case of a non-acoustic wave is constant and depends on the roughness of the solid medium and the contact surfaces. Z represents half of the sum of displacements at both ends of the contact interface, and Y represents the difference in displacement between both ends of the interface.
The derivation of equations (10) and (11) yields:
Figure BDA0002185532210000111
Figure BDA0002185532210000112
wherein, σ (h)0+ Y) is the stress at the contact interface as the acoustic wave propagates.
And solving to obtain an expression of the reflected wave and the transmitted wave:
Figure BDA0002185532210000113
where Y represents the difference in displacement between the two ends of the interface.
Step c, exciting the frequency f1，f2Two ultrasonic waves will generate nf due to the nonlinear interaction between the two ultrasonic waves at the interface1，nf2，f1+f2Sum frequency ofSum of waves f1-f2The difference frequency wave of (1).
Regardless of the acoustic wave attenuation and the initial phase difference, the introduced excitation signal expression is as follows:
Figure BDA0002185532210000121
in the formula, A1And A2，ω1And omega2The amplitude and angular frequency of the two excited sound waves respectively.
And solving the equations of the reflected wave and the transmitted wave by using a perturbation method according to the wave equation and the incident wave expression. Considering that the gap distance changes very little when the displacement of the incident wave is small, the function σ (h) may be derived from this in this case when h is h0The nearby taylor expansion is replaced, taking the second order term, the expression is as follows:
σ(h)＝σ(h0+Y)＝σ0-K1Y+K2Y2 (16)
wherein the content of the first and second substances,
Figure BDA0002185532210000122
K1represents linear stiffness;
Figure BDA0002185532210000123
K2the nonlinear rigidity of the contact surface is expressed, namely the second-order rigidity; h is the gap distance of the contact interface.
And d, substituting the formula (15) and the formula (16) into the formula (12) to obtain a first-order nonlinear ordinary differential equation related to Y, wherein the expression form is as follows:
Figure BDA0002185532210000124
e, solving an approximate solution by using a perturbation method as follows:
consider an approximate solution of the equation Y ═ Y1+Y2. Wherein, Y1Is a first order trace, Y2Is a second order trace and is an approximate solution to the equation. At the same time, the user can select the desired position,Y1,Y2the following equation is satisfied:
Figure BDA0002185532210000125
Figure BDA0002185532210000126
obtaining by solution:
Figure BDA0002185532210000127
wherein the content of the first and second substances,
Figure BDA0002185532210000128
δ1＝arctan(ω1/a)，δ2＝arctan(ω2/a)
Figure BDA0002185532210000131
wherein, take Δ ω ═ ω2-ω1，Σω＝ω1+ω2，
Figure BDA0002185532210000132
Figure BDA0002185532210000133
ψ1＝θ1+δ1-δ2，ψ2＝θ2-δ1-δ2，θ1＝arctan[a/(Δω)]，θ2＝arctan(a/Σω)。
Step f, finally obtaining an expression of the reflected wave and the transmitted wave:
Figure BDA0002185532210000134
Figure BDA0002185532210000141
and step three, analyzing and obtaining the nonlinear effect of the aliasing wave beam incident on the solid-solid bonding interface according to the expression of the reflected wave and the transmitted wave obtained by deduction in the step two, and defining four nonlinear parameters to evaluate the contact characteristic of the solid-solid bonding interface by combining the power law relation of the linear rigidity and the contact stress of the interface.
The expression of the power law relation of the interface linear rigidity and the contact stress is as follows:
K1＝Cσ0 m (24)
wherein C and m are normal numbers, will
Figure BDA0002185532210000142
And
Figure BDA0002185532210000143
and substituting to obtain a relation function of second-order rigidity and contact pressure:
Figure BDA0002185532210000144
defining a non-linearity parameter beta1,β2The nonlinear parameter gamma is the ratio of the amplitude of the difference frequency component and sum frequency component to the amplitude of the fundamental frequency in the transmitted wave1,γ2The ratio of the amplitude of the difference frequency component and the sum frequency component in the reflected wave to the amplitude of the fundamental frequency is:
Figure BDA0002185532210000151
Figure BDA0002185532210000152
Figure BDA0002185532210000153
Figure BDA0002185532210000154
the four non-linear parameters (beta) for evaluating the contact characteristics of the solid-solid bonding interface are obtained by the steps of theoretical calculation1,β2,γ1,γ2)。
And step four, drawing a change curve of the four nonlinear parameters along with the contact stress under the condition of aliasing of beams with different frequencies according to the numerical calculation result, thereby analyzing the nonlinear characteristics of the solid-solid interface. Meanwhile, the dependence of the harmonic on the contact stress of the solid-solid bonding interface was evaluated.
And fifthly, selecting the excitation signal aliasing with proper frequency by using the ultrasonic signal generator. The ultrasonic probe is used for exciting the test piece, and then the signal collector is used for collecting signals on the other side of the test piece.
And sixthly, the acquired signals are subjected to fast Fourier transform to obtain corresponding frequency spectrum images. And measuring to obtain the nonlinear parameters of the test piece under different pressures based on the theoretical basis of the third step.
And seventhly, analyzing to obtain that the nonlinear parameter has a monotonic relation with the bonding strength of the test piece by comparing the theoretical value with the actual value, so that the contact characteristic of the solid-solid bonding interface can be evaluated by using the nonlinear parameter, and the correctness and the feasibility of the method can be verified.
According to the theoretical calculation, under the condition of analyzing aliasing of beams with different frequencies, the change curves of four nonlinear parameters along with the contact stress are specifically as follows: as shown in fig. 2, the nonlinear coefficient β is obtained at different difference frequencies1The change curve of stress; as shown in fig. 3, the nonlinear coefficient β is different at different sum frequencies2The change curve of stress; as shown in fig. 4, the nonlinear coefficient γ is obtained at different difference frequencies1The change curve of stress; as shown in fig. 5, the nonlinear coefficient γ is different for different sum frequencies2And the change curve of stress. The following conclusions were reached: non-linear parameter increasing with pressureLarge and small; the larger the difference of the incident frequencies is, the smaller the nonlinear parameter is; the non-linearity parameter is smaller as the sum of the incident frequencies is larger. Therefore, theoretical analysis shows that the nonlinear parameter has a monotonous relation with the bonding strength of the test piece, and the method can be used for evaluating the contact characteristics of the solid-solid bonding interface.
Finally, the feasibility of the method of the invention was verified by experimental practical measurements. FIG. 6 is a schematic diagram of the experimental apparatus. In the experiment, two rows of sine waves with frequencies of 2MHz and 2.5MHz respectively are selected, mixed by an ultrasonic signal generator and used as an excitation source, the two rows of sine waves are excited on a tested piece by an ultrasonic probe, and then signals are collected by a signal collector on the other side. The acquired signals are subjected to fast fourier transform to obtain spectral images as shown in fig. 7 and 8. The measurement results in the nonlinear parameters of the test piece under different pressures. As shown in fig. 9, by comparing the theoretical value and the actual value, the nonlinear parameter obtained by analysis has a monotonic relationship with the bonding strength of the test piece, and the fitting degree of the actual measurement curve and the theoretical curve is high. Therefore, the feasibility and the reliability of evaluating the contact characteristics of the solid-solid bonding interface by utilizing the nonlinear parameters are verified.
The above description is only of the preferred embodiments of the present invention, and it should be noted that: it will be apparent to those skilled in the art that various modifications and adaptations can be made without departing from the principles of the invention, and such modifications and adaptations are intended to be within the scope of the invention.

Claims (4)

1. A method for evaluating the contact characteristic of a solid-solid interface by utilizing a nonlinear effect is characterized by comprising the following steps: the method comprises the following steps:
s01, establishing an isotropic solid-solid bonding interface model, giving a longitudinal wave motion equation and a stress-displacement relation, transmitting a one-dimensional elastic longitudinal wave along the X-axis direction, and setting the reference plane position of a contact surface to be X ═ X respectively-,x＝X+In the region X < X-Incident one-dimensional elastic longitudinal wave fincTo obtain a reflected wave frefIn the region X > X+Obtaining a transmitted wave ftra；
The one-dimensional elastic longitudinal wave is positively propagated along the x axis, and the relation between the motion equation and the stress displacement is as follows:
Figure FDA0003487163840000011
Figure FDA0003487163840000012
wherein u represents the displacement of the one-dimensional elastic longitudinal wave propagating along the x direction; σ represents the stress variation; sigma0Two solid contact static stresses; t represents time; ρ represents the density of the medium;
s02, deriving expressions of reflected waves and transmitted waves by using a perturbation method according to a nonlinear wave equation of aliasing incident beams and the boundary conditions of a solid-solid interface, and specifically deriving the expressions of the reflected waves and the transmitted waves by using the perturbation method comprises the following steps:
the solutions of step a, formula (1) and formula (2) are as follows:
u(x,t)＝finc(x-ct)+fref(x+ct) x＜X- (7)
u(x,t)＝ftra(x-ct) x＞X+ (8)
in the formula, function finc(x-ct),fref(x + ct) and ftra(x-ct) represents an expression of an incident wave propagating right along the x-axis on the left side of the interface, a reflected wave propagating left along the x-axis, and a transmitted wave propagating right along the x-axis on the right side of the interface, respectively, and the wave velocity c of the acoustic wave is (λ/ρ)1/2λ is the elastic constant;
step b, in order to solve the equations of the formula (1) and the formula (2), introducing the following variables:
Z(t)＝[u(X+,t)+u(X-,t)]/2 (9)
Y(t)＝u(X+,t)-u(X-,t)＝h(t)-h0 (10)
one-dimensional elastic longitudinal wave propagates along the direction of the X axis, and the reference plane positions of the contact surfaces are respectively set as X ═ X-,x＝X+(ii) a Wherein u is(X+T) is one-dimensional elastic longitudinal wave at the interface X ═ X+A displacement of (a); u (X)-T) is one-dimensional elastic longitudinal wave at the interface X ═ X-A displacement of (a); h (t) is the gap distance between the contact surfaces when the sound waves are transmitted; h is0The gap distance between the contact surfaces is constant when the contact surface is silent, and is related to the roughness of the solid medium and the contact surface; z represents the half of the displacement sum of the two ends of the contact interface, and Y represents the displacement difference of the two ends of the contact interface;
the derivation of equation (9) and equation (10) yields:
Figure FDA0003487163840000021
Figure FDA0003487163840000022
wherein, σ (h)0+ Y) is the stress at the contact interface when the acoustic wave propagates;
and solving to obtain an expression of the reflected wave and the transmitted wave:
Figure FDA0003487163840000023
Figure FDA0003487163840000024
wherein Y represents the displacement difference between the two ends of the interface;
c, exciting to obtain a frequency f1，f2Two ultrasonic waves will generate nf due to the nonlinear interaction between the two ultrasonic waves at the interface1，nf2，f1+f2Sum frequency wave sum f1-f2The difference frequency wave of (a) is,
regardless of the acoustic wave attenuation and the initial phase difference, the introduced excitation signal expression is as follows:
Figure FDA0003487163840000025
in the formula, A1And A2，ω1And omega2Respectively the amplitude and angular frequency of the two excited sound waves,
solving a reflected wave equation and a transmitted wave equation by using a perturbation method according to a wave equation and an incident wave expression; considering that the gap distance changes very little when the displacement of the incident wave is small, the function σ (h) may be derived from this in this case when h is h0The nearby taylor expansion is replaced, taken to a second order term, as follows:
σ(h)＝σ(h0+Y)＝σ0-K1Y+K2Y2 (16)
wherein, the first and the second end of the pipe are connected with each other,
Figure FDA0003487163840000026
K1represents linear stiffness;
Figure FDA0003487163840000027
K2representing the nonlinear rigidity of the contact surface, namely the second-order rigidity; h is the gap distance of the contact interface;
and d, substituting the formula (15) and the formula (16) into the formula (12) to obtain a first-order nonlinear ordinary differential equation related to Y, wherein the expression form is as follows:
Figure FDA0003487163840000031
e, solving an approximate solution by using a perturbation method as follows:
consider an approximate solution of the equation Y ═ Y1+Y2Wherein Y is1Is a first order trace, Y2Is a second order trace, is an approximate solution of the equation, and, at the same time, Y1,Y2The following equation is satisfied:
Figure FDA0003487163840000032
Figure FDA0003487163840000033
obtaining by solution:
Figure FDA0003487163840000034
wherein the content of the first and second substances,
Figure FDA0003487163840000035
δ1＝arctan(ω1/a)，δ2＝arctan(ω2/a)
Figure FDA0003487163840000036
wherein, take Δ ω ═ ω2-ω1，Σω＝ω1+ω2，
Figure FDA0003487163840000037
ψ1＝θ1+δ1-δ2，ψ2＝θ2-δ1-δ2，θ1＝arctan[a/(Δω)]，θ2＝arctan(a/Σω)；
Step f, finally obtaining an expression of the reflected wave and the transmitted wave:
Figure FDA0003487163840000041
Figure FDA0003487163840000042
s03, analyzing to obtain a nonlinear effect of an aliasing beam incident on a solid-solid bonding interface, and defining four nonlinear parameters to evaluate the contact characteristics of the solid-solid bonding interface by combining the power law relation of the interface linear rigidity and the contact stress;
s04, drawing a curve of four nonlinear parameters changing along with contact stress under the condition of aliasing of beams with different frequencies according to the numerical calculation result;
s05, selecting excitation signal aliasing with proper frequency by using an ultrasonic signal generator, exciting on the tested piece through an ultrasonic probe, receiving an acquisition signal by using another ultrasonic probe on the other side of the tested piece, and visually acquiring the signal through a signal acquisition device;
s06, carrying out fast Fourier transform on the acquired signals to obtain corresponding frequency spectrum images, and measuring to obtain nonlinear parameters of the test piece under different pressures based on the theoretical basis of S03;
and S07, comparing the theoretical value with the actual value, and analyzing to obtain the monotonic relation between the nonlinear parameter and the bonding strength of the test piece.
2. The method for evaluating the contact characteristic of the solid-solid interface by utilizing the nonlinear effect as claimed in claim 1, wherein: in S02, the aliased beam nonlinear wave equation expression is as follows:
Figure FDA0003487163840000051
where phi is the velocity potential and the velocity expression of the acoustic wave is
Figure FDA0003487163840000052
Wherein, c0The propagation velocity of small-amplitude sound waves in ideal gas is constant and depends on a medium; gamma is the ratio of specific heat at constant pressure to specific heat at constant volume; p is pressure, P0Is the initial pressure; rho0Is the initial density.
3. The method for evaluating the contact characteristic of the solid-solid interface by utilizing the nonlinear effect as claimed in claim 1, wherein: in S02, the solid-solid interface boundary condition satisfies:
when the elastic wave has interaction across the boundary, the displacement and stress conditions are as follows:
Figure FDA0003487163840000053
when the elastic waves do not interact across the boundary, i.e. there is no contact between the two solids at this time, the displacement and stress conditions are as follows:
Figure FDA0003487163840000061
wherein σ (+0, t) is that sound waves are in the region X > X+The stress of (a); sigma (-0, t) is that the sound wave is in the region X < X-Of the stress of (c).
4. The method for evaluating the contact characteristics of the solid-solid interface by utilizing the nonlinear effect as claimed in claim 3, wherein: in S03, the expression of the power law relationship between the interface linear stiffness and the contact stress is as follows:
K1＝Cσ0 m (24)
wherein C and m are normal numbers, will
Figure FDA0003487163840000062
And
Figure FDA0003487163840000063
and substituting to obtain a relation function of second-order rigidity and contact pressure:
Figure FDA0003487163840000064
defining a non-linearity parameter beta1,β2The nonlinear parameter gamma is the ratio of the amplitude of the difference frequency component and sum frequency component to the amplitude of the fundamental frequency in the transmitted wave1,γ2The ratio of the amplitude of the difference frequency component and the sum frequency component in the reflected wave to the amplitude of the fundamental frequency is:
Figure FDA0003487163840000065
Figure FDA0003487163840000066
Figure FDA0003487163840000067
Figure FDA0003487163840000068
CN201910812907.0A 2019-08-30 2019-08-30 Method for evaluating solid-solid interface contact characteristics by utilizing nonlinear effect Active CN110455919B (en)

Priority Applications (1)

Application Number Priority Date Filing Date Title
CN201910812907.0A CN110455919B (en) 2019-08-30 2019-08-30 Method for evaluating solid-solid interface contact characteristics by utilizing nonlinear effect

Applications Claiming Priority (1)

Application Number Priority Date Filing Date Title
CN201910812907.0A CN110455919B (en) 2019-08-30 2019-08-30 Method for evaluating solid-solid interface contact characteristics by utilizing nonlinear effect

Publications (2)

Publication Number Publication Date
CN110455919A CN110455919A (en) 2019-11-15
CN110455919B true CN110455919B (en) 2022-06-17

Family

ID=68490035

Family Applications (1)

Application Number Title Priority Date Filing Date
CN201910812907.0A Active CN110455919B (en) 2019-08-30 2019-08-30 Method for evaluating solid-solid interface contact characteristics by utilizing nonlinear effect

Country Status (1)

Country Link
CN (1) CN110455919B (en)

Families Citing this family (4)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN111257419A (en) * 2020-02-03 2020-06-09 天津大学 Extra-high voltage insulation pull rod interface defect detection device
CN111257417A (en) * 2020-02-03 2020-06-09 天津大学 Method for detecting interface defects of extra-high voltage insulating pull rod
CN111505116B (en) * 2020-04-25 2021-05-14 西安交通大学 Material near-surface macro-micro defect integrated ultrasonic detection method based on spatial modulation laser ultrasonic spectrum
CN115950955A (en) * 2023-03-13 2023-04-11 江苏交科能源科技发展有限公司 Gantry crane bolt vibration monitoring method and system

Family Cites Families (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101666780B (en) * 2009-10-13 2011-11-09 北京工业大学 Imperfect interface contact state nonlinear ultrasonic evaluation method based on micro texture
CN101949894B (en) * 2010-08-16 2012-06-27 南京大学 Method for detecting interface contact strength by double frequency ultrasound
CN102721747B (en) * 2012-06-08 2014-05-21 北京航空航天大学 Non-colinear non-linear ultrasonic nondestructive testing method
CN108593775A (en) * 2018-06-01 2018-09-28 北京工业大学 A kind of non-linear ultrasonic guided wave detecting method for contacting state evaluation between conductor casing
CN108872393B (en) * 2018-06-21 2021-01-05 北京工业大学 Nonlinear ultrasonic frequency mixing method for detecting structural fatigue crack direction

Also Published As

Publication number Publication date
CN110455919A (en) 2019-11-15

Similar Documents

Publication Publication Date Title
CN110455919B (en) Method for evaluating solid-solid interface contact characteristics by utilizing nonlinear effect
Belanger et al. Guided wave diffraction tomography within the born approximation
Huthwaite et al. Mode selection for corrosion detection in pipes and vessels via guided wave tomography
Clorennec et al. Laser ultrasonic inspection of plates using zero-group velocity lamb modes
US11092573B2 (en) Apparatus, systems, and methods for determining nonlinear properties of a material to detect early fatigue or damage
CN101949894B (en) Method for detecting interface contact strength by double frequency ultrasound
Liu et al. Measuring bulk material nonlinearity using harmonic generation
Li et al. Modeling and simulation of backward combined harmonic generation induced by one-way mixing of longitudinal ultrasonic guided waves in a circular pipe
US20140102200A1 (en) Methods for non-destructively evaluating a joined component
Fierro et al. Nonlinear imaging (NIM) of flaws in a complex composite stiffened panel using a constructive nonlinear array (CNA) technique
Zhang et al. Damage detection of fatigue cracks under nonlinear boundary condition using subharmonic resonance
Dunn et al. Critical aspects of experimental damage detection methodologies using nonlinear vibro-ultrasonics
Druet et al. Passive guided wave tomography for structural health monitoring
Gresil et al. Predictive model of fatigue crack detection in thick bridge steel structures with piezoelectric wafer active sensors
Li et al. Micro-defect imaging with an improved resolution using nonlinear ultrasonic Lamb waves
de Castro et al. Baseline-free damage imaging algorithm using spatial frequency domain virtual time reversal
Barzegar et al. Polar coordinate for damage imaging of adhesively bonded plates using ultrasonic guided waves and laser Doppler vibrometer measurements
Drewry et al. One-dimensional time-domain finite-element modelling of nonlinear wave propagation for non-destructive evaluation
Revel et al. The development of a shock-tube based characterization technique for air-coupled ultrasonic probes
JP4673686B2 (en) Surface inspection method and surface inspection apparatus
Barth et al. Experimental determination of Lamb wave dispersion diagrams using 2d fourier transform and laser vibrometry
Zagrai et al. N‐Scan®: New Vibro‐Modulation System for Crack Detection, Monitoring and Characterization
Cheng et al. The pitch-catch nonlinear ultrasonic imaging techniques for structural health monitoring
CN114061804B (en) Air coupling ultrasonic stress detection system based on collinear frequency mixing technology and detection method thereof
KR101452442B1 (en) Elasticity Test method

Legal Events

Date Code Title Description
PB01 Publication
PB01 Publication
SE01 Entry into force of request for substantive examination
SE01 Entry into force of request for substantive examination
GR01 Patent grant
GR01 Patent grant
TR01 Transfer of patent right

Effective date of registration: 20221212

Address after: Floor 1, Building 08, High tech Development Zone, Nanjing, Jiangsu 210031

Patentee after: NANJING YOUQIAO ELECTRONIC TECHNOLOGY Co.,Ltd.

Address before: 213000, No. 200 Jinling North Road, Jiangsu, Changzhou

Patentee before: CHANGZHOU CAMPUS OF HOHAI University

TR01 Transfer of patent right