Pulse thermal imaging detection temperature response prediction method

Abstract

The invention belongs to the technical field of nondestructive testing, and particularly relates to a pulse thermal imaging detection temperature response prediction method. In order to solve the problem of temperature response prediction distortion of a classical pulse thermal imaging detection theoretical model, the invention provides a pulse thermal imaging detection temperature response prediction method, which comprises the following steps of: 1, calculating the thermal diffusion coefficient, the thermal conductivity, the light absorption rate, the density and the specific heat capacity of a test piece 5 to be detected according to the thermal diffusion coefficient, the thermal conductivity, the light absorption rate, the density and the specific heat capacity; the power density, the pulse duration and the environment temperature of the light pulse emitted by the flash lamp 3 and the depth of the defect 6 expected to be detected are brought into an improved pulse thermal imaging detection theoretical model; and 2, by means of an improved pulse thermal imaging detection theoretical model, calculating temperature response of the surface of the test piece 5 after pulse heating is finished, and if the calculated temperature difference between a defect area and a non-defect area is clear and distinguishable, maintaining current experimental parameters to carry out an experiment.

Description

TECHNICAL FIELD

[0001] The present application belongs to the technical field of nondestructive testing, and particularly relates to a pulse thermography detection temperature response prediction method. BACKGROUND

[0002] Pulse thermography is one of the most mature and widely used infrared nondestructive testing methods. This method uses a high-intensity flash lamp with a single pulse energy generally greater than 5000J to emit a large-size light pulse with a duration of 1-50ms to the material surface to achieve instantaneous thermal excitation, and simultaneously uses an infrared thermal imager to record the temperature change of the material surface in real time (the entire measurement period usually only takes a few seconds). When there is a defect in the material, the local heat conduction is blocked, which in turn causes an abnormally high temperature area on the material surface. The operator can effectively judge and locate the defect in the material by analyzing the abnormally high temperature area in the infrared image. This method has the advantages of convenient operation, good visualization effect, and high detection efficiency, and has broad application prospects in the rapid screening of defects in the fields of aerospace and vehicle manufacturing.

[0003] In the early 1990s, foreign researchers first built a theoretical model for pulse thermography detection. This team simplified the heat conduction process in the material after pulse heating as a one-dimensional Fourier heat conduction problem, and used the Laplace transform method for analytical solution, and derived the thermal response of the material surface after pulse heating. In fact, the team has realized that there is a large deviation between the predicted value of the theoretical model and the actual measured value, and speculated that this phenomenon is due to the boundary effect or the defect size feature not being fully considered. However, due to its simplicity and universality, this model has been used as a classic theoretical framework in the field of pulse infrared thermography detection to date, and has been widely used in key links such as defect depth quantitative evaluation, experimental parameter optimization, and data processing algorithm development. SUMMARY

[0004] In order to solve the problem of temperature response prediction distortion of the classical pulse thermography detection theoretical model, the present application proposes a pulse thermography detection temperature response prediction method, which comprises the following steps:

[0005] Step 1: The thermal diffusivity, thermal conductivity, light absorption rate, density, specific heat capacity of the test piece 5, the power density and pulse duration of the light pulse emitted by the flash lamp 3, the environmental temperature, and the expected defect 6 depth to be detected are brought into the improved pulse thermography detection theoretical model;

[0006] Step 2: Using the improved pulse thermal imaging detection theoretical model, calculate the temperature response of the surface of the test piece 5 after the pulse heating is completed. If the calculated temperature difference between the defective area and the non-defective area is clearly discernible, maintain the current experimental parameters and continue the experiment. If the temperature difference between the two is not discernible, optimize the power density and pulse duration of the light pulse emitted by the flash lamp 3 until the calculated result meets the expectations, that is, the temperature difference between the defective area and the non-defective area is clearly discernible, and continue the experiment with the current parameters as guidance, that is, step 3. If the power density of the light pulse emitted by the flash lamp 3 is adjusted to the maximum value and a clear temperature difference is still not obtained, it means that the defect is too deep and cannot be detected by the pulse thermal imaging system.

[0007] Step 3: Computer 1 issues a control instruction to make signal generator 2 generate two high-energy synchronous voltages, driving flash lamp 3 to emit high-energy light pulses to illuminate the surface of specimen 5 (the irradiation angles of the two flash lamps 3 are as follows: Figure 1 As shown in the figure, as long as it can cover the field of view of the infrared thermal imager), the defect 6 hinders the heat flow to the inside of the test piece 5, and an abnormal temperature distribution will appear on the surface of the test piece 5. During this process, the infrared thermal imager 4 continuously collects (the angle between the infrared thermal imager 4 and the flash lamp 3 remains unchanged during the entire experiment) the temperature distribution data on the surface of the test piece 5 to be inspected, and transmits it back to the computer 1. The operator detects the defect based on the temperature difference distribution in the infrared image.

[0008] As a preferred embodiment, in step 2, the clearly discernible temperature difference means that the temperature difference is ≥ 1°C.

[0009] As another preferred solution, in step 1, the derivation process of the improved pulse thermal imaging detection theoretical model is as follows:

[0010] S1: Assume that the test piece 5 is an isotropic, uniform plate with no internal heat source. The upper surface of the test piece 5 is heated by a uniform short-term heat source. The interior of the plate is considered to undergo one-dimensional unsteady-state heat conduction. The mathematical description of this problem is:

[0011]

[0012] Where a = λ / ρc is the thermal diffusivity of the specimen 5, λ is the thermal conductivity of the specimen 5, ρ is the density of the specimen 5, c is the specific heat capacity of the specimen 5, T is the temperature, t is the time, and x represents the depth;

[0013] S2: At the initial moment, the temperature inside the test piece 5 is T0:

[0014] T (x,0) =T0;

[0015] S3: The boundary condition of the lower surface of the test piece 5 is expressed as:

[0016]

[0017] S4: The boundary condition of the upper surface of the test piece 5 is expressed as:

[0018]

[0019] where ε represents the light absorption rate of the test piece 5, q0is the power density of the heat source, g (t) is the time distribution function of the power density.

[0020] S5: A rectangular pulse function is selected to describe the time distribution characteristics of the power density of the heat source, which is in the form of:

[0021]

[0022] where u (t) is a unit step function, and t0is the pulse width.

[0023] S6: The Laplace transform of T (x,t) with respect to t is expressed as:

[0024]

[0025] S7: The Laplace transform of S1 with respect to t is obtained as:

[0026]

[0027] S8: S2 is brought into S7 to obtain:

[0028]

[0029] S9: The general solution of the ordinary differential equation S8 is:

[0030]

[0031] S10: The original function T (x,t) is a finite value, and the image function (indicating the corresponding function) T (x,t) obtained by the Laplace transform is also a finite value, based on which, C1=0 is obtained, and S9 is simplified as:

[0032]

[0033] S11: The partial derivative of S10 with respect to x is obtained as:

[0034]

[0035] S12: g (t)Substitute S4 into S3 and take Laplace transform on both sides with respect to t, according to the linearity and delay properties of Laplace transform, we have:

[0036]

[0037] S13: Substitute S12 into S11, we have:

[0038]

[0039] S14: Substitute S13 into S10, we have:

[0040]

[0041] S15: Take inverse Laplace transform on S14 and simplify, we have:

[0042]

[0043] where Δt = t - t0;

[0044] S16: erfc(x) is the error function, which is expressed as:

[0045] erfc(x) = 1 - erf(x);

[0046] S17: Substitute erf into S15, we have the thermal response of each point inside the specimen 5 after pulse excitation:

[0047]

[0048] S18: where erf is the error function, which is solved by power series expansion:

[0049]

[0050] S19: Substitute x = 0 into S17, we have the thermal response of the surface of the object after pulse excitation:

[0051]

[0052] S20: The above calculation results are applicable to the specimen 5 thickness L > (2a / ω), where ω is the modulation frequency of the heat source, and the thermal disturbance caused by the lower boundary of the specimen 5 is less than the thermal sensitivity of the infrared thermal imager. For a finite thickness L or defect depth h, the temperature at the boundary between the defect 6 and the specimen 5 should satisfy the temperature continuity:

[0053] T (h,t) = T (s,t) = T (d,t) ;

[0054] S21: Energy conservation:

[0055]

[0056] wherein the subscripts s and d represent the interface of the normal region and the defect 6 region and the interface of the defect 6 region and the normal region, respectively;

[0057] S22: When the heat is conducted to the defect interface, it is considered that part of the heat continues to conduct in the opposite direction, and then goes back and forth between the upper surface of the test piece 5 and the defect 6 multiple times until it is completely attenuated. The temperature change of the surface of the test piece 5 is considered to be the temperature drop caused by the heat conduction in the semi-infinite region and the temperature accumulation caused by the multiple reflections of the bottom surface of the test piece 5 or the defect 6 and the surface of the test piece 5. The reflection coefficient is r, and the thickness of the test piece 5 or the depth of the defect 6 is h. The temperature change of the surface layer of the test piece 5 caused by the reflection of the defect 6 is represented as:

[0058]

[0059] S23: Assuming that the reflection coefficient r = 1 (total reflection), S22 is simplified as:

[0060]

[0061] S24: The temperature of the upper surface of the test piece 5 in the defect 6 region:

[0062]

[0063] S25: Similarly, the temperature of the surface of the test piece 5 in the intact structure is:

[0064]

[0065] S26: The temperature difference between the defect-free region and the defect 6 region on the surface of the test piece 5 is:

[0066] ΔT = T i(0，t) -T d(0，t) = T br(0，t) -T dr(0，t) .

[0067] Advantages of the present application

[0068] Step 1 of the present application is to obtain calculation parameters according to experimental conditions, step 2 is to calculate according to the above parameters (determine the power density and pulse duration of the light pulse emitted by the flash lamp 3), and step 3 is to carry out experiments according to the power density and pulse duration determined in step 2.

[0069] The step 2 is a predicted value, and the step 3 is a real detected value. The experimental parameters are optimized through the predicted value, if the experimental parameters are calculated without the aid of the model of the application, the experimental parameters can only be set according to experience, if a group of experimental results are not good, the next group of experiments does not need to wait for the test piece to cool down, and repeated adjustment is carried out until the experimental results meet the expectation, so it is blind and time-wasting.

[0070] The classical pulse thermal imaging theoretical model adopts the Dirac function to describe the time distribution characteristics of the heat source power density, the action time of the function is far lower than the relaxation time of the material, which not only cannot reflect the actual working condition, but also violates the applicable condition of the Fourier heat conduction law, so that there is a serious deviation between the calculation result and the actual value. The application improves the classical pulse thermal imaging detection theoretical model, and uses the rectangular pulse function to depict the time distribution characteristics of the heat source power density, and the calculation result is very close to the result measured by the experiment. The model is expected to be widely applied in the key links of pulse thermal imaging experimental parameter optimization, defect depth quantitative evaluation and data processing algorithm development. BRIEF DESCRIPTION OF DRAWINGS

[0071] Figure 1 The schematic diagram of the pulse thermal imaging detection experimental system used in the application.

[0072] Figure 2 (a) is a schematic diagram of the Dirac function used when the classical pulse thermal imaging detection theoretical model depicts the time distribution characteristics of the heat source power density;

[0073] Figure 2 (b) is a schematic diagram of the rectangular pulse function used when the improved pulse thermal imaging detection theoretical model of the application depicts the time distribution characteristics of the heat source power density.

[0074] Figure 3 (a) is the temperature-time resolution curve of the defect area and the intact area obtained by the pulse thermal imaging detection experiment, and the temperature difference (corresponding to the right coordinate axis) between the two;

[0075] Figure 3 (b) is the temperature-time resolution curve of the defect area and the intact area obtained by the improved pulse thermal imaging detection theoretical model, and the temperature difference (corresponding to the right coordinate axis) between the two;

[0076] Figure 3 (c) is the temperature-time resolution curve of the defect area and the intact area obtained by the classical pulse thermal imaging detection theoretical model, and the temperature difference (corresponding to the right coordinate axis) between the two;

[0077] Figure 4 It is a schematic diagram of the pulse thermal imaging detection result.

[0078] Figure 5 Schematic diagram of heat diffusion inside the test piece after it is subjected to surface thermal excitation. DETAILED DESCRIPTION

[0079] The present invention will be described in further detail below with reference to the accompanying drawings.

[0080] The pulse thermal imaging detection experimental system used includes a computer, a signal generator, a flash lamp and an infrared thermal imager.

[0081] Step 1: If Figure 1 As shown, the thermal diffusivity, thermal conductivity, light absorptivity, density, specific heat capacity of the test piece 5, the power density and pulse duration of the light pulse emitted by the flash lamp 3, the ambient temperature and the depth of the expected defect 6 are introduced into the improved pulse thermal imaging detection theoretical model;

[0082] Parameters such as thermal diffusion coefficient vary depending on the object being detected, detection environment, detection distance, etc., and can be set according to actual working conditions.

[0083] Step 2: With the help of the improved pulse thermal imaging detection theoretical model proposed in this invention, the Figure 1 The temperature response of the surface of specimen 5 is calculated as follows: Figure 3 If the temperature difference between the defect 6 area and the non-defect area in (b) is clearly discernible (≥1°C), the experiment can be carried out with the current experimental parameters. If the temperature difference between the two is not discernible, the calculation parameters need to be optimized (e.g., increasing the power density of the light pulse) according to the actual working conditions (the actual working conditions during each set of experiments, including the thermal diffusivity, thermal conductivity, light absorptivity, density, and specific heat capacity mentioned above) until the calculation results meet expectations, and the experiment can be carried out with the current parameters as a guide. If the power density of the light pulse emitted by the flash lamp 3 is adjusted to the maximum value and no clear temperature difference is obtained, it means that the defect is too deep and cannot be detected by the pulse thermal imaging system.

[0084] Step 3: If Figure 1 As shown in the figure, the computer 1 issues a control instruction to make the signal generator 2 generate two high-energy synchronous voltages, driving the flash lamp 3 to emit high-energy light pulses to illuminate the surface of the test piece 5. The defect 6 hinders the heat flow to the inside of the test piece 5, and the surface of the test piece 5 will have an abnormal temperature distribution. During this process, the infrared thermal imager 4 continuously collects the temperature distribution data of the surface of the test piece 5 to be tested and sends it back to the computer 1. The operator can Figure 4 The temperature difference distribution in the infrared image shown enables the detection of defects.

[0085] In step 1 of the present invention, the derivation process of the improved pulse thermal imaging detection theoretical model is as follows:

[0086] S1: Assume that the specimen 5 to be tested is an isotropic, uniform, semi-infinite plate with no internal heat source. The upper surface of the specimen 5 is heated by a uniform short-term heat source, and convection and radiation heat dissipation are neglected (in reality, convection and radiation heat dissipation are included, but because this process is short, the interference caused by them is not significant and can be ignored. In addition, it is very difficult to obtain an analytical solution, so these factors are generally ignored). The interior of the plate can be regarded as a one-dimensional unsteady-state heat conduction. The mathematical description of this problem is:

[0087]

[0088] Where a = λ / ρc (λ is the thermal conductivity of the specimen 5, ρ is the density of the specimen 5, and c is the specific heat capacity of the specimen 5) is the thermal diffusivity of the specimen 5, T is the temperature, t is the time, and x represents the depth;

[0089] S2: At the initial moment, the temperature inside the test piece 5 is T0:

[0090] T (x,0) =T0;

[0091] S3: The boundary condition of the lower surface of the test piece 5 can be expressed as:

[0092]

[0093] S4: The boundary conditions on the upper surface of the test piece 5 can be expressed as:

[0094]

[0095] Where ε represents the light absorption rate of the test piece 5, q0 is the power density of the heat source, g (t) is the time distribution function of power density.

[0096] S5: Optional Figure 2 The rectangular pulse function shown in (b) is used to describe the time distribution characteristics of the heat source power density, which is as follows:

[0097]

[0098] where u (t) is the unit step function, and t0 is the pulse width.

[0099] S6: T (x,t) The Laplace transform of t is expressed as:

[0100]

[0101] S7: Perform Laplace transform on both ends of S1 with respect to t, and we get:

[0102]

[0103] S8: Bring S2 into S7 to get:

[0104]

[0105] S9: The general solution of ordinary differential equation S8 is:

[0106]

[0107] S10: Since the original function T (x,t) is a finite value, the image function obtained by Laplace transform, must also be a finite value, based on which C1=0 can be obtained, and S9 can be simplified as:

[0108]

[0109] S11: Take the partial derivative of S10 with respect to x to get:

[0110]

[0111] S12: Substitute g (t) in S5 into S4, and take the Laplace transform of both ends with respect to t, according to the linearity and delay properties of Laplace transform, we can get:

[0112]

[0113] S13: Bring S12 into S11 to get:

[0114]

[0115] S14: Substitute S13 into S10 to get:

[0116]

[0117] S15: Take the inverse Laplace transform of S14 and simplify to get:

[0118]

[0119] where Δt=t-t0;

[0120] S16: erfc(x) is the error function, which can be expressed as:

[0121] erfc(x)=1-erf(x);

[0122] S17: Substitute erf into S15 to get the thermal response of each point inside the test piece 5 after pulse excitation:

[0123]

[0124] S18: where erf is the error function, which can be solved by power series expansion:

[0125]

[0126] S19: Substitute x=0 into S17 to obtain the thermal response of the object surface after pulse excitation:

[0127]

[0128] S20: The above calculation results are applicable to the case where the thickness of the specimen 5 is thick enough and the lower boundary of the specimen 5 does not affect the thermal diffusion; Figure 5 As shown, for a finite thickness L (or defect depth h), without considering the internal heat loss of specimen 5 (this is only theoretical, but various formulas are basically acceptable approximations under certain circumstances. Experimental results show that this approximation is reasonable because the prediction is very accurate), the boundary between defect 6 and specimen 5 should satisfy temperature continuity:

[0129] T (h,t) =T (s,t) =T (d,t) ;

[0130] S21: Conservation of Energy:

[0131]

[0132] like Figure 5 As shown, where the subscripts s and d represent the interface between the normal area and the defect 6 area and the interface between the defect 6 area and the normal area, respectively;

[0133] S22: When heat is conducted to the defect interface, it can be considered that part of the heat continues to conduct in the opposite direction, and then travels back and forth between the upper surface of the specimen 5 and the defect 6 multiple times until it is completely attenuated. The temperature change on the surface of the specimen 5 can be regarded as the temperature drop caused by heat conduction in the semi-infinite area and the temperature accumulation caused by multiple reflections between the bottom surface of the specimen 5 (or the defect 6) and the surface of the specimen 5. Assuming that the reflection coefficient is r and the thickness of the specimen 5 or the depth of the defect 6 is h, the temperature change on the surface of the specimen 5 due to the reflection from the defect 6 can be expressed as:

[0134]

[0135] S23: Assume that the reflection coefficient r = 1 (total reflection) and ignore higher-order attenuation (which cannot be ignored in practice, but the theoretical calculation must be approximated. This approximate calculation result is far superior to the traditional method. In addition, there is no solution that can take all interference factors into account). S22 can be simplified to:

[0136]

[0137] S24: Temperature of the upper surface of the specimen 5 in the defect 6 region:

[0138]

[0139] S25: Similarly, temperature of the upper surface of the specimen 5 in the intact region:

[0140]

[0141] S26: Temperature difference between the intact region and the defect 6 region on the upper surface of the specimen 5:

[0142] ΔT = T i(0,t) - T d(0,t) = T br(0,t) - T dr(0,t) .

[0143] The difference between the rectangular pulse function depicting the time distribution characteristics of the heat source power density in S5 and the classical pulse thermal imaging theoretical model lies in:

[0144] It is well known that the Fourier heat conduction law is actually based on the assumption that the transfer speed of thermal disturbance is infinite. In some cases, the Fourier heat conduction differential equation is not applicable, for example:

[0145] 1) The temperature of the object is close to absolute zero;

[0146] 2) The heat transfer process is extremely short, and the relaxation time of the material itself is close (copper: 2.7 x 10 -14 s);

[0147] 3) The spatial scale of the heat transfer process is extremely small, close to the average free path of microscopic particles;

[0148] 4) The heat flux is extremely high (such as 10 10 W / m 2 ).

[0149] The heat conduction problem that the above Fourier heat conduction law is not applicable is collectively referred to as non-Fourier heat conduction, which belongs to the research scope of modern micron and nanometer heat transfer. The excitation time (1-50 ms) of the pulse thermal imaging detection is far beyond the relaxation time of the material, and the heat flux applied is far below the critical threshold of the non-Fourier heat conduction problem. It is obvious that the pulse thermal imaging still follows the classical Fourier heat conduction law. The classical pulse thermal imaging theoretical model adopts the Dirac function δ Figure 2 (a) as shown: (t)The time distribution characteristic of the heat source power density is described, which only acts on the time t=0s (the action time is much lower than the relaxation time of the material), which not only cannot reflect the actual working condition, but also violates the applicable condition of Fourier heat conduction law, resulting in a serious deviation between the calculation result and the actual value. The calculation result of the classical pulse thermal imaging detection theoretical model is as follows Figure 3 (c) shown, which seriously deviates from the experimental result Figure 3 (a).

[0150] It can be understood that the above specific description of the present application is only used to illustrate the present application and is not limited to the technical solutions described in the embodiments of the present application. Those skilled in the art should understand that the present application can still be modified or replaced equivalently to achieve the same technical effect; as long as the use needs are met, it is within the protection scope of the present application.

Claims

1. A method of predicting the temperature response of a pulsed thermographic inspection, characterized by The method comprises the following steps: Step 1: the thermal diffusivity, thermal conductivity, light absorption rate, density, specific heat capacity of the test piece (5), the power density and pulse duration of the light pulse emitted by the flash lamp (3), the ambient temperature and the expected defect (6) depth to be detected are brought into the improved pulsed thermographic testing theoretical model; Step 2: the temperature response of the surface of the test piece (5) after the pulse heating is calculated by means of the improved pulsed thermographic testing theoretical model, if the temperature difference between the defect area and the non-defect area calculated is clear and distinguishable, the current experimental parameters are kept to carry out the experiment, if the temperature difference between the two is indistinguishable, the power density and pulse duration of the light pulse emitted by the flash lamp (3) need to be optimized until the calculation result meets the expectation, that is, the temperature difference between the defect area and the non-defect area is clear and distinguishable, and the current parameters are used to guide the experiment, that is, step (3); if the power density of the light pulse emitted by the flash lamp (3) has been adjusted to the maximum value and still cannot obtain a clear and distinguishable temperature difference, it indicates that the defect depth is too deep and the pulsed thermographic system cannot detect it; Step 3: the computer (1) issues a control instruction to make the signal generator (2) generate two high-energy synchronous voltages to drive the flash lamp (3) to emit a high-energy light pulse to irradiate the surface of the test piece (5), the defect (6) hinders the heat from propagating to the inside of the test piece (5), and the surface of the test piece (5) will appear abnormal temperature distribution, in the process, the infrared thermal imager (4) continuously collects the temperature distribution data of the surface of the test piece (5) and returns them to the computer (1), and the operator realizes the detection of the defect according to the temperature difference distribution in the infrared image.

2. The method of claim 1, wherein the temperature response is predicted by the method of claim 1. In step 2, the clear and distinguishable temperature difference refers to a temperature difference ≥1℃.

3. The method of claim 1, wherein the method further comprises: In step 1, the derivation process of the improved pulsed thermographic testing theoretical model is as follows: S1: assuming that the test piece (5) is an isotropic, uniform and non-thermal source flat plate, a uniform short-time heat source is used to heat the upper surface of the test piece (5), the inside of the flat plate is regarded as one-dimensional non-steady-state heat conduction, the mathematical description of this problem is: Wherein a = λ / ρc is the thermal diffusivity of the test piece (5), λ is the thermal conductivity of the test piece (5), ρ is the density of the test piece (5), c is the specific heat capacity of the test piece (5), T is the temperature, t is the time, and x represents the depth; S2: the temperature of each part inside the test piece (5) at the initial moment is T0: T (x,0) = T0; S3: the boundary condition of the lower surface of the test piece (5) is expressed as: S4: the boundary condition of the upper surface of the test piece (5) is expressed as: where ε represents the light absorption rate of the test piece (5), q0is the power density of the heat source, g (t) is the time distribution function of the power density; S5: a rectangular pulse function is selected to describe the time distribution characteristics of the heat source power density, which is as follows: where u (t) is the unit step function, t0is the pulse width; S6: T (x,t) The Laplace transform representation with respect to t is given by S7: Laplace transform is performed on both ends of S1 with respect to t to obtain: S8: S2 is brought into S7 to obtain: S9: the general solution of the ordinary differential equation S8 is: S10: the original function T (x,t) is a finite value, the image function T (x,t) is also a finite value, based on which, C1=0, and S9 is simplified as: S11: the partial derivative of S10 with respect to x is obtained: S12: Change the g in S5 (t) Substitute into formula S4 and perform Laplace transform on both ends with respect to t. According to the linearity and delay properties of Laplace transform, we can get: S13: S12 is brought into S11 to obtain: S14: S13 is substituted into S10 to obtain: S15: inverse Laplace transform is performed on S14 and simplified to obtain: Wherein Δt = t-t0; S16: the error function erfc(x) is expressed as: erfc(x) = 1-erf(x) S17: erf is brought into S15 to get the thermal response of each point inside the test piece (5) after pulse excitation: S18: erf is the error function, which is solved by power series expansion: S19: x = 0 is brought into S17 to get the thermal response of the surface of the object after pulse excitation: S20: The above calculation results are applicable to the thickness of the test piece (5) >> (2a / ω), where ω is the modulation frequency of the heat source, and the thermal disturbance caused by the lower boundary of the test piece (5) is less than the thermal sensitivity of the infrared thermal imager. For a finite thickness L or defect depth h, the defect (6) and the boundary of the test piece (5) should satisfy the temperature continuity: T (h,t) = T (s,t) = T (d,t) ; S21: Energy conservation: Where subscript s and d represent the interface between the normal region and the defect (6) region and the interface between the defect (6) region and the normal region, respectively; S22: When heat is conducted to the defect interface, it is considered that part of the heat continues to conduct in the opposite direction, and then repeatedly goes back and forth between the upper surface of the test piece (5) and the defect (6), until it is completely attenuated. The temperature change on the surface of the test piece (5) is considered to be the temperature drop caused by heat conduction in the semi-infinite region, and the sum of the temperature accumulation caused by multiple reflections of the bottom surface of the test piece (5) or the defect (6) and the surface of the test piece (5). The reflection coefficient is r, the thickness of the test piece (5) or the depth of the defect (6) is h, and the temperature change of the surface layer of the test piece (5) caused by the reflection of the defect (6) is represented as: S23: Set the reflection coefficient r = 1, and S22 is simplified as: S24: The temperature of the upper surface of the test piece (5) in the defect (6) region: S25: Similarly, the temperature of the surface of the test piece (5) in the intact structure: S26: The temperature difference between the defect-free region and the defect (6) region on the surface of the test piece (5): ΔT = T i(0，t) - T d(0，t) = Tbr (0，t) - T dr(0，t) .

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