A method for obtaining thermo-elastic-diffusion behavior of a hollow column structure, a detection system and a storage medium

Abstract

The application provides a kind of hollow column structure thermal-elastic-diffusion behavior acquisition method, including corresponding fractional order generalized thermal-elastic-diffusion model, motion model, constitutive model are established by model analysis, boundary conditions and initial conditions are determined;For fractional order generalized thermal elastic model, fractional order generalized diffusion model, motion model and constitutive model are dimensionless, to obtain corresponding control equation;By Laplace transformation to the dimensionless control equation, and combining the eigenvalue method to solve the above control equation, and Laplace transformation to the control equation and combining the eigenvalue method, displacement field, temperature field, chemical potential field, stress field and concentration field are obtained;The distribution law of displacement field, temperature field, chemical potential field, stress field and concentration field is obtained by solving Laplace numerical inverse transformation. The application can obtain the displacement field, temperature field, stress field distribution law of real field, which is helpful for material design and experimental comparison.

Description

Technical Field

[0001] This invention relates to the field of material mechanical property testing technology, and in particular to a method, testing system and storage medium for obtaining the thermo-elastic-diffusion behavior of a hollow column structure. Background Technology

[0002] With the continuous development of science and technology, more and more hollow column structures are being applied in practical engineering, such as tunnels, pipelines, and heat pipes. However, due to the complexity of the working environment, the properties of these structural materials are easily altered. For example, prolonged exposure to high temperatures or unequal ion concentrations often leads to changes in thermal conductivity and ion diffusion problems. Fractional calculus theory, due to its good time memory, is increasingly being applied in materials research. Therefore, the monitoring and characterization of the thermo-elastic-diffusion behavior of these hollow column structures is particularly important.

[0003] For example, Chinese Patent Publication No. CN111625952A discloses a method for detecting the three-dimensional distribution of temperature and stress, including: establishing a three-dimensional generalized thermo-elastic coupling model based on fractional strain rate; solving the thermo-elastic coupling control equations using Laplace transform and Fourier transform; and obtaining the influence of fractional strain rate on the three-dimensional distribution of temperature and stress. Although this scheme involves a three-dimensional generalized thermo-elastic coupling model based on fractional strain rate and performs Laplace transform and Fourier transform, it primarily fails to consider the change in thermal conductivity of the material with temperature during the analysis and solution process, assuming that the material parameters are constant. Furthermore, it only performs related thermo-elastic analysis for elastic materials, neglecting diffusion issues. However, for materials in actual engineering, their thermal conductivity is not constant and is often accompanied by particle diffusion problems. This fractional-order generalized thermo-elastic model cannot accurately describe practical engineering problems. Summary of the Invention

[0004] To address the shortcomings of existing technologies, the present invention aims to provide a method and system for obtaining the thermo-elastic-diffusion behavior of hollow column structures. By combining the fractional-order generalized heat conduction equation and the fractional-order generalized diffusion equation, the distribution laws of displacement, temperature, chemical potential, stress, and concentration of an infinitely long hollow column with thermal conductivity varying with temperature are obtained. The results are then numerically solved using Laplace transform, eigenvalue method, and inverse Laplace numerical transform, resulting in higher accuracy.

[0005] It should be understood that the general description above and the detailed description below are exemplary and illustrative, and are intended to provide further explanation of this disclosure.

[0006] To address the aforementioned technical problems, this invention provides a method for obtaining the thermo-elastic-diffusion behavior of a hollow column structure, characterized in that the method includes:

[0007] Step 1: Establish the corresponding fractional-order generalized thermo-elastic-diffusion model, kinematic model, and constitutive model through model analysis, and determine the boundary conditions and initial conditions;

[0008] Step 2: Dimensionless processing is performed on the physical quantities involved, including the fractional-order generalized thermoelastic model, the fractional-order generalized diffusion model, the kinematic model, and the constitutive model, to obtain the corresponding governing equations.

[0009] Step 3: By performing a Laplace transform on the dimensionless control equations and solving the control equations using the eigenvalue method, and by performing a Laplace transform on the control equations and using the eigenvalue method, the displacement field, temperature field, chemical potential field, stress field, and concentration field are obtained.

[0010] Step 4: The distribution patterns of displacement field, temperature field, chemical potential field, stress field and concentration field are obtained by solving the Laplace numerical inverse transform.

[0011] Preferably, this invention further discloses a method for obtaining the thermo-elastic-diffusion behavior of a hollow cylindrical structure, characterized in that the coefficients in the thermo-elastic-diffusion coupling model of the hollow cylindrical structure satisfy:

[0012]

[0013]

[0014]

[0015]

[0016]

[0017]

[0018] By combining equations (1) and (6), the coefficient A can be obtained. i (i = 1, 2, ..., 6).

[0019] Preferably, the present invention further discloses a method for obtaining the thermo-elastic-diffusion behavior of a hollow column structure, characterized in that step 2 further includes:

[0020] The process of establishing the model of the hollow column structure is as follows:

[0021] (1) By introducing a thermal relaxation coefficient into the traditional Fourier heat conduction equation and performing a Taylor series expansion of the heat flow components with fractional-order time parameters, the corresponding fractional-order generalized heat conduction equation can be obtained:

[0022]

[0023] Where κ is the thermal conductivity coefficient, ν is the temperature field, and θ = T - T0, where T is the absolute temperature, κ1 is a small variable parameter, T0 is the reference temperature, τ1 is the thermal relaxation coefficient, α is a fractional-order parameter, ρ is the material density, and c E Let λ be the specific heat coefficient, u be the displacement tensor, t be time, Γ(1+α) be the Gamma function with independent variable α, λ and μ be the Lamé coefficients, and r be the coordinate variable. β1=(3λ+2μ)α t β2=(3λ+2μ)α c α t α is the coefficient of linear expansion. c The linear diffusion coefficient is given by ψ, and φ represents the thermal diffusion effect and diffusion effect coefficient, respectively.

[0024] (2) By introducing a diffusion relaxation coefficient into the traditional Fick's law and performing a Taylor series expansion of the mass diffusion flux with fractional-order time parameters, the corresponding fractional-order generalized diffusion equation can be obtained:

[0025]

[0026] Where D is the diffusion coefficient, and τ2 is the diffusion relaxation coefficient.

[0027] (3) The motion model that does not consider physical strength is:

[0028]

[0029] Where, σ rr Radial stress, It is circumferential stress;

[0030] (4) The constitutive relation is:

[0031]

[0032]

[0033]

[0034]

[0035] Where, σ zzThis is axial stress;

[0036] (5) The boundary conditions and initial conditions are as follows:

[0037]

[0038] σ rr (a,t)=0,u(b,t)=0 (15)

[0039]

[0040]

[0041] Where H(t) is the Heaviside function;

[0042] (6) Introduce the following dimensionless quantities:

[0043]

[0044] For ease of description, the asterisks (*) above each physical quantity are omitted, resulting in the following governing equations and boundary conditions:

[0045]

[0046]

[0047]

[0048]

[0049]

[0050]

[0051]

[0052] σ rr (a,t)=0,u(b,t)=0 (26)

[0053]

[0054]

[0055] in,

[0056]

[0057]

[0058] (7) Applying the Laplace transform to equations (19) to (25) yields:

[0059]

[0060]

[0061]

[0062]

[0063]

[0064]

[0065]

[0066] in, Represent u, ν, p, σ respectively rr , σ zz C is the field in the Laplace transform domain, and s is the field variable corresponding to time t;

[0067] Taking the first derivatives of equations (29) and (30) with respect to r and combining them with equation (35), we can express them in the form of a vector matrix, as follows:

[0068] LV = BV (36)

[0069] in, And, b 11 =εs 2 ,

[0070] The characteristic equation corresponding to matrix B is

[0071]

[0072] Therefore, by using the eigenvalue method and combining equations (31) to (34), the forms of the solutions for the displacement field, temperature field, chemical potential field, stress field, and concentration field can be obtained as follows:

[0073]

[0074]

[0075]

[0076]

[0077]

[0078]

[0079]

[0080] Where I0 and K0 are the zeroth-order modified Bessel function of the first kind and the zeroth-order modified Bessel function of the second kind, respectively, and I2 and K2 are the second-order modified Bessel function of the first kind and the second-order modified Bessel function of the second kind, respectively. k i (i = 1, 2, 3) represent the three roots corresponding to the characteristic equation (37), respectively. This represents the i-th component of the eigenvector corresponding to the j-th eigenvalue;

[0081] Based on the boundary conditions (26) to (28), the coefficient A is obtained. i (i = 1, 2, ..., 6);

[0082] (8) Perform the Laplace numerical inverse transform on equations (38) to (44) according to the following formula to obtain the distribution law of the displacement field, temperature field, chemical potential field, stress field and concentration field in the real field:

[0083]

[0084] Where f represents the real field, F represents the field in the transform domain, N represents the number of terms in the summation, and n and k represent the number of terms in the summation operation, respectively. ! indicates factorial operation;

[0085] Thus, the displacement field, temperature field, chemical potential field, stress field, and concentration field of the real field can be obtained as follows:

[0086]

[0087]

[0088]

[0089]

[0090]

[0091]

[0092]

[0093] The present invention also discloses a computer device, characterized in that the computer device includes a memory and a processor, the memory storing a computer program, and when the computer program is executed by the processor, the processor performs the following steps:

[0094] Step 1: Establish the corresponding fractional-order generalized thermo-elastic-diffusion model, kinematic model, and constitutive model through model analysis, and determine the boundary conditions and initial conditions;

[0095] Step 2: Dimensionless processing is performed on the physical quantities involved, specifically including dimensionless processing of the fractional-order generalized thermoelastic model, fractional-order generalized diffusion model, kinematic model and constitutive model to obtain the corresponding governing equations.

[0096] Step 3: By performing a Laplace transform on the dimensionless governing equations and solving the governing equations using the eigenvalue method, and by performing a Laplace transform on the governing equations and using the eigenvalue method, the displacement field, temperature field, chemical potential field, stress field, and concentration field are obtained.

[0097] Step 4: The distribution patterns of displacement field, temperature field, chemical potential field, stress field and concentration field are obtained by solving the Laplace numerical inverse transform.

[0098] This invention discloses a computer-readable storage medium storing a computer program, which, when executed by a processor, causes the processor to perform the following steps:

[0099] Step 1: Establish the corresponding fractional-order generalized thermo-elastic-diffusion model, kinematic model, and constitutive model through model analysis, and determine the boundary conditions and initial conditions;

[0100] Step 2: Dimensionless processing is performed on the physical quantities involved, specifically including dimensionless processing of the fractional-order generalized thermoelastic model, fractional-order generalized diffusion model, kinematic model and constitutive model to obtain the corresponding governing equations.

[0101] Step 3: By performing a Laplace transform on the dimensionless governing equations and solving the governing equations using the eigenvalue method, and by performing a Laplace transform on the governing equations and using the eigenvalue method, the displacement field, temperature field, chemical potential field, stress field, and concentration field are obtained.

[0102] Step 4: The distribution patterns of displacement field, temperature field, chemical potential field, stress field and concentration field are obtained by solving the Laplace numerical inverse transform.

[0103] This invention discloses a displacement, temperature, chemical potential, stress, and concentration distribution detection system that operates any one of the aforementioned displacement, temperature, chemical potential, stress, and concentration distribution detection methods, characterized in that the displacement, temperature, chemical potential, stress, and concentration distribution detection system comprises:

[0104] The model building module is used to build a thermo-elastic-diffusion coupling model of a hollow cylindrical structure with an analysis time fraction;

[0105] The dimensionless processing module is used to solve the thermo-elastic-diffusion coupling control equations using Laplace transform and eigenvalue method;

[0106] The distribution law determination module is used to obtain the distribution laws of displacement field, temperature field, chemical potential field, stress field and concentration field through Laplace numerical inverse transform.

[0107] This invention discloses a material processing system, characterized in that the system is equipped with the aforementioned thermo-elastic-diffusion distribution detection system. This invention addresses the problem of difficulty in experimentally characterizing the mechanical behavior of such hollow cylinders in practical engineering by using Laplace transform, eigenvalue method, and inverse Laplace numerical transform for numerical solutions of infinitely long hollow cylinders with varying thermal conductivity with temperature. The results are more accurate, solving the problem of difficulty in experimentally characterizing the mechanical behavior of such hollow cylinders in practical engineering. By combining a fractional-order generalized thermo-elastic-diffusion model for mechanical behavior analysis, and considering the influence of fractional-order parameters and thermal conductivity on various fields, the distribution laws of the displacement field, temperature field, and stress field in the real field are obtained, which is helpful for material design and experimental comparison.

[0108] Based on the novel fractional-order generalized thermoelastic diffusion theory proposed by Sherief, this invention combines Laplace transform, eigenvalue method and Laplace numerical inverse transform method, focusing on the distribution law of the mechanical properties of an infinitely long hollow column under thermal and chemical potential excitation as a function of fractional-order parameters and thermal conductivity. Attached Figure Description

[0109] Embodiments of the present disclosure will now be described in detail with reference to the accompanying drawings. Preferred embodiments of the present disclosure will now be described in detail, examples of which are illustrated in the drawings. Wherever possible, the same reference numerals will be used in all the drawings to denote the same or similar parts. Furthermore, although the terminology used in this disclosure is selected from commonly known and used terminology, some terms referenced in this disclosure may have been chosen by the applicant at his or her judgment, and their detailed meanings are explained in the relevant sections of the description herein. Moreover, this disclosure should be understood not only by the actual terms used, but also by the meaning implied by each term.

[0110] The above and other objects, features and advantages of the present invention will become apparent to those skilled in the art from the detailed description thereof, with reference to the accompanying drawings.

[0111] Figure 1 This is a flowchart of the present invention according to one or more embodiments;

[0112] Figure 2 This is a distribution diagram of the displacement u along r corresponding to different fractional-order parameters when t = 0.2s and κ1 = 0, according to one or more embodiments of the present invention.

[0113] Figure 3 This is a graph showing the distribution of temperature ν along r for different fractional-order parameters when t = 0.2s and κ1 = 0, according to one or more embodiments of the present invention.

[0114] Figure 4 This is a graph showing the distribution of chemical potential p along r for different fractional-order parameters when t = 0.2s and κ1 = 0, according to one or more embodiments of the present invention.

[0115] Figure 5 This invention relates to the stress σ corresponding to different fractional-order parameters when t = 0.2s and κ1 = 0, according to one or more embodiments. rr Distribution pattern along r;

[0116] Figure 6 This invention relates to the stress corresponding to different fractional-order parameters when t = 0.2s and κ1 = 0, according to one or more embodiments. Distribution pattern along r;

[0117] Figure 7 This invention relates to the stress σ corresponding to different fractional-order parameters when t = 0.2s and κ1 = 0, according to one or more embodiments. zz Distribution pattern along r;

[0118] Figure 8 This is a graph showing the distribution of concentration C along r for different fractional-order parameters when t = 0.2s and κ1 = 0, according to one or more embodiments of the present invention.

[0119] Figure 9 This is a graph showing the distribution of displacement u along r corresponding to different thermal conductivity coefficients when t = 0.2s and α = 0.5, according to one or more embodiments of the present invention.

[0120] Figure 10 This is a graph showing the distribution of temperature ν along r for different thermal conductivity coefficients when t = 0.2s and α = 0.5, according to one or more embodiments of the present invention.

[0121] Figure 11 This invention relates to the chemical potentials corresponding to different thermal conductivityes at t = 0.2s and α = 0.5, based on one or more embodiments. p Distribution pattern along r;

[0122] Figure 12 This invention relates to the stress σ corresponding to different thermal conductivity coefficients when t = 0.2s and α = 0.5, based on one or more embodiments. rrDistribution pattern along r;

[0123] Figure 13 This invention relates to the stress corresponding to different thermal conductivity coefficients when t = 0.2s and α = 0.5, based on one or more embodiments. Distribution pattern along r;

[0124] Figure 14 This invention relates to the stress σ corresponding to different thermal conductivity coefficients when t = 0.2s and α = 0.5, based on one or more embodiments. zz Distribution pattern along r;

[0125] Figure 15 This is a graph showing the distribution of concentration C along r for different thermal conductivity coefficients when t = 0.2s and α = 0.5, according to one or more embodiments of the present invention.

[0126] Figure 16 The diagram illustrates the composition of a system for obtaining a method for characterizing the thermo-elastic-diffusion behavior of a hollow column structure according to the present invention. Detailed Implementation

[0127] To more clearly illustrate the technical solutions of the embodiments of this application, the accompanying drawings used in the description of the embodiments will be briefly introduced below. Obviously, the drawings described below are merely some examples or embodiments of this application. For those skilled in the art, these drawings can be applied to other similar scenarios without creative effort. Unless obvious from the context or otherwise specified, the same reference numerals in the drawings represent the same structures or operations.

[0128] As indicated in this application and claims, unless the context clearly indicates otherwise, the words "a," "an," "an," and / or "the" are not specifically singular and may include plural forms. Generally speaking, the terms "comprising" and "including" only indicate the inclusion of explicitly identified steps and elements, which do not constitute an exclusive list, and the method or apparatus may also include other steps or elements.

[0129] Unless otherwise specifically stated, the relative arrangement, numerical expressions, and values ​​of the components and steps described in these embodiments do not limit the scope of this application. It should also be understood that, for ease of description, the dimensions of the various parts shown in the drawings are not drawn to actual scale. Techniques, methods, and devices known to those skilled in the art may not be discussed in detail, but where appropriate, such techniques, methods, and devices should be considered part of the specification. In all examples shown and discussed herein, any specific values ​​should be interpreted as merely exemplary and not as limitations. Therefore, other examples of exemplary embodiments may have different values. It should be noted that similar reference numerals and letters in the following drawings denote similar items; therefore, once an item is defined in one drawing, it need not be further discussed in subsequent drawings.

[0130] In the description of this application, it should be understood that the orientation or positional relationship indicated by directional terms such as "front, back, up, down, left, right", "horizontal, vertical, horizontal" and "top, bottom" is usually based on the orientation or positional relationship shown in the accompanying drawings, and is only for the convenience of describing this application and simplifying the description. Unless otherwise stated, these directional terms do not indicate or imply that the device or element referred to must have a specific orientation or be constructed and operated in a specific orientation, and therefore should not be construed as a limitation on the scope of protection of this application; the directional terms "inner" and "outer" refer to the inner and outer contours relative to the outline of each component itself.

[0131] For ease of description, spatial relative terms such as "above," "on top of," "on the upper surface of," "above," etc., are used herein to describe the spatial positional relationship of a device or feature as shown in the figures to other devices or features. It should be understood that spatial relative terms are intended to encompass different orientations in use or operation beyond the orientation of the device as described in the figures. For example, if the device in the figures were inverted, a device described as "above" or "on top of" other devices or structures would subsequently be positioned as "below" or "under" other devices or structures. Thus, the exemplary term "above" can include both "above" and "below." The device may also be positioned in other different ways (rotated 90 degrees or in other orientations), and the spatial relative descriptions used herein will be interpreted accordingly.

[0132] Furthermore, it should be noted that the use of terms such as "first" and "second" to define components is merely for the purpose of distinguishing the corresponding components. Unless otherwise stated, these terms have no special meaning and therefore should not be construed as limiting the scope of protection of this application. In addition, although the terminology used in this application is selected from commonly known and used terms, some terms mentioned in this application's specification may have been chosen by the applicant according to his or her judgment, and their detailed meanings are explained in the relevant sections of this description. Moreover, this application should be understood not only through the actual terms used, but also through the meaning implied by each term.

[0133] Flowcharts are used in this application to illustrate the operations performed by the system according to embodiments of this application. It should be understood that the preceding or following operations are not necessarily performed in exact order. Instead, various steps can be processed in reverse order or simultaneously. Furthermore, other operations may be added to these processes, or one or more steps may be removed from these processes.

[0134] Example 1:

[0135] In a typical embodiment of the present invention, a method for obtaining the thermo-elastic-diffusion behavior of a hollow column structure is provided, comprising:

[0136] A generalized heat conduction model is obtained by introducing a thermal relaxation coefficient into the Fourier heat conduction equation and performing a Taylor series expansion of the heat flux component with fractional-order time parameters; a fractional-order generalized diffusion model is obtained by introducing a thermal relaxation coefficient into the Fick diffusion equation and performing a Taylor series expansion of the mass diffusion flux with fractional-order time parameters; at the same time, a motion model and a constitutive model that neglect body forces are established.

[0137] The fractional-order generalized heat conduction model, fractional-order generalized diffusion model, motion model, and constitutive model are subjected to dimensionless processing to obtain the corresponding governing equations;

[0138] The Laplace transform of the governing equations is performed, and the displacement field, temperature field, chemical potential field, stress field, and concentration field are obtained by combining the eigenvalue method. The distribution laws of the displacement field, temperature field, chemical potential field, stress field, and concentration field are obtained based on the inverse numerical Laplace transform.

[0139] Specifically, including, for example Figure 1 The following steps are shown:

[0140] Step S1: Establish the corresponding fractional-order generalized thermo-elastic-diffusion model, kinematic model, and constitutive model through model analysis, and determine the boundary conditions and initial conditions;

[0141] Step S2: Dimensionless processing is performed on the physical quantities involved, specifically including dimensionless processing of the fractional-order generalized thermoelastic model, fractional-order generalized diffusion model, kinematic model and constitutive model to obtain the corresponding governing equations;

[0142] Step S3: By performing a Laplace transform on the dimensionless governing equations and solving the governing equations using the eigenvalue method, and by performing a Laplace transform on the governing equations and using the eigenvalue method, the displacement field, temperature field, chemical potential field, stress field, and concentration field are obtained.

[0143] Step S4: Solve the distribution patterns of displacement field, temperature field, chemical potential field, stress field and concentration field by Laplace numerical inverse transformation.

[0144] The process of establishing the model for an infinitely long hollow cylinder is as follows:

[0145] (1) By introducing a thermal relaxation coefficient into the traditional Fourier heat conduction equation and performing a Taylor series expansion of the heat flow components with fractional-order time parameters, the corresponding fractional-order generalized heat conduction equation can be obtained:

[0146]

[0147] Where κ is the thermal conductivity coefficient, ν is the temperature field, and θ = T - T0, where T is the absolute temperature, T0 is the reference temperature, κ1 is a small variable parameter, τ1 is the thermal relaxation coefficient, α is a fractional-order parameter, ρ is the material density, and c E Let λ be the specific heat coefficient, u be the displacement tensor, t be time, Γ(1+α) be the Gamma function with independent variable α, λ and μ be the Lamé coefficients, and r be the coordinate variable. β1=(3λ+2μ)α t β2=(3λ+2μ)α c α t α is the coefficient of linear expansion. c denoted as the linear diffusion coefficient, and ψ and φ represent the thermal diffusion effect and diffusion effect coefficients, respectively.

[0148] (2) By introducing a diffusion relaxation coefficient into the traditional Fick's law and performing a Taylor series expansion of the mass diffusion flux with fractional-order time parameters, the corresponding fractional-order generalized diffusion equation can be obtained:

[0149]

[0150] Where D is the diffusion coefficient, and τ2 is the diffusion relaxation coefficient.

[0151] (3) The motion model that does not consider physical strength is:

[0152]

[0153] Where, σ rr Radial stress, This is circumferential stress.

[0154] (4) The constitutive relation is:

[0155]

[0156]

[0157]

[0158]

[0159] Where, σ zz This is axial stress.

[0160] It should be noted that the thermal conductivity changes linearly through a small empirical coefficient throughout the entire process.

[0161] (5) The boundary conditions and initial conditions are as follows:

[0162]

[0163] σ rr (a,t)=0,u(b,t)=0 (61)

[0164]

[0165]

[0166] Here, H(t) is the Heaviside function.

[0167] (6) Introduce the following dimensionless quantities:

[0168]

[0169] For ease of description, the asterisks (*) above each physical quantity are omitted, resulting in the following governing equations and boundary conditions:

[0170]

[0171]

[0172]

[0173]

[0174]

[0175]

[0176]

[0177] σ rr (a,t)=0,u(b,t)=0 (72)

[0178]

[0179]

[0180] in,

[0181]

[0182]

[0183] (7) Applying the Laplace transform to equations (65) to (71) yields:

[0184]

[0185]

[0186]

[0187]

[0188]

[0189]

[0190]

[0191] in, Represent u, ν, p, σ respectively rr , σ zz C is the field in the Laplace transform domain, and s is the field variable corresponding to time t.

[0192] Taking the first derivatives of equations (75) and (76) with respect to r and combining them with equation (81), we can express them in the form of a vector matrix, as follows:

[0193] LV = BV (82)

[0194] in, And, b 11 =εs 2 ,

[0195] The characteristic equation corresponding to matrix B is

[0196]

[0197] Therefore, by using the eigenvalue method and combining equations (77) to (80), the forms of the solutions for the displacement field, temperature field, chemical potential field, stress field, and concentration field can be obtained as follows:

[0198]

[0199]

[0200]

[0201]

[0202]

[0203]

[0204]

[0205] Where I0 and K0 are the zeroth-order modified Bessel function of the first kind and the zeroth-order modified Bessel function of the second kind, respectively, and I2 and K2 are the second-order modified Bessel function of the first kind and the second-order modified Bessel function of the second kind, respectively. k i (i = 1, 2, 3) represent the three roots corresponding to the characteristic equation (83), respectively. It represents the i-th component of the eigenvector corresponding to the j-th eigenvalue.

[0206] Combining the boundary conditions, we can obtain:

[0207]

[0208]

[0209]

[0210]

[0211]

[0212]

[0213] By combining equations (91) and (96), the coefficient A can be obtained. i (i = 1, 2, ..., 6).

[0214] (8) By performing the Laplace numerical inverse transformation on equations (84) to (90) according to the following formula, the distribution law of displacement field, temperature field, chemical potential field, stress field and concentration field in the real field can be obtained.

[0215]

[0216] Where f represents the real field, F represents the field in the transform domain, N represents the number of terms in the summation, and n and k represent the number of terms in the summation operation, respectively. ! indicates factorial operation.

[0217] Thus, the displacement field, temperature field, chemical potential field, stress field, and concentration field of the real field can be obtained as follows:

[0218]

[0219]

[0220]

[0221]

[0222]

[0223]

[0224]

[0225] This embodiment focuses on infinitely long hollow cylinders, employing Laplace transform, eigenvalue method, and inverse Laplace numerical transform for numerical solution, resulting in higher accuracy. This addresses the challenge of characterizing the mechanical behavior of such infinitely long hollow cylinders experimentally in practical engineering. By combining a fractional-order generalized thermo-elastic diffusion model for mechanical behavior analysis, and considering the influence of fractional-order parameters and thermal conductivity on various fields, the distribution patterns of the displacement field, temperature field, chemical potential field, stress field, and concentration field in the real field are obtained, which is helpful for material design and experimental comparison.

[0226] Example 2:

[0227] This embodiment verifies the method described in Embodiment 1 using actual data, wherein the material parameters are taken as follows:

[0228] λ = 7.76 × 10 10 Kg / ms 2 μ=3.86×10 10 Kg / ms 2 ρ=8.954×10 3 Kg / m 3 κ=386W / mK

[0229] D = 8.5 × 10 -9 Kg·s / m 3 ,α c =1.98×10 -4 m 3 / Kg,α t =1.78×10 -5 K -1 ,c E = 383.1 J / (kg·K),

[0230] ψ = 1.2 × 10 4 m 2 / (s 2 vK),φ=9×10 5 m 5 / (Kg·s 2 ), T0=293K, τ1=0.02, τ2=0.2, a=1m, b=2.5m

[0231] And assume that N = 10 in the inverse transform.

[0232] The distribution law of dimensionless displacement u with distance r under different fractional-order parameters is as follows: Figure 2 As shown, the displacement at r = 2.5 is 0, which is consistent with the boundary conditions, and the maximum displacement is at... The distribution of dimensionless temperature ν with distance r under different fractional-order parameters is as follows: Figure 3 As shown, the temperature value is 1 at r = 1, which is consistent with the boundary conditions. The distribution of the dimensionless chemical potential p with distance r under different fractional-order parameters is as follows: Figure 4 As shown, the chemical potential is 1 at r = 1, which is consistent with the boundary conditions. The distribution of dimensionless stress with distance r under different fractional-order parameters is as follows: Figures 5-7 As shown. By Figure 5 It can be seen that the radial stress σ rr The value is 0 at r = 1, which is consistent with the boundary conditions. The distribution of dimensionless concentration C with distance r under different fractional-order parameters is as follows: Figure 8 As shown, the concentration at r = 1 is 0.2058, which is consistent with the boundary conditions. The distribution of dimensionless displacement u with distance r under different thermal conductivity is as follows: Figure 9 As shown, the displacement at r = 2.5 is 0, which is consistent with the boundary conditions, and the maximum displacement is at... The dimensionless temperature ν with distance r under different thermal conductivity is obtained from [location]. Figure 10As shown, when κ1 takes values ​​of 0.2, 0.4, and 0.6, the temperature values ​​at r = 1 are 1.1, 1.2, and 1.3, respectively, which is consistent with the boundary conditions. The distribution of the dimensionless chemical potential p with distance r under different thermal conductivity is as follows: Figure 11 As shown, the chemical potential is 1 at r = 1, which is consistent with the boundary conditions. The distribution of dimensionless stress with distance r under different thermal conductivity is as follows: Figures 12-14 As shown. Figure 12 It can be seen that the radial stress σ rr The value is 0 at r = 1, which is consistent with the boundary conditions. The distribution of dimensionless concentration C with distance r under different thermal conductivity is as follows: Figure 15 As shown, the concentrations at r=1 are 0.2236, 0.2414, and 0.2592, which is consistent with the boundary conditions. From Figures 2-9 It can be seen that considering the influence of fractional-order parameters on each physical quantity provides a more comprehensive understanding. Figures 10-15 It is evident that the material's temperature coefficient is a crucial influencing factor when performing design calculations.

[0233] Example 3:

[0234] Please see Figure 16 As shown, this embodiment provides a system for characterizing the thermo-elastic-diffusion behavior of hollow cylindrical structures, including:

[0235] Model building module 11 is configured to: introduce a thermal relaxation coefficient into the Fourier heat conduction equation and perform a Taylor series expansion of the heat flux component with fractional-order time parameters to obtain a generalized heat conduction model; by introducing a diffusion relaxation coefficient into the traditional Fick's law and performing a Taylor series expansion of the mass diffusion flux with fractional-order time parameters, the corresponding fractional-order generalized diffusion equation can be obtained; at the same time, a motion model and constitutive model that neglect body forces are established.

[0236] The dimensionless processing module 12 is configured to perform dimensionless processing on the fractional-order generalized heat conduction model, fractional-order generalized diffusion model, motion model and constitutive model to obtain the corresponding governing equations.

[0237] The distribution law determination module 13 is configured to: perform a Laplace transform on the governing equations and solve them using the eigenvalue method to obtain the displacement field, temperature field, chemical potential field, stress field, and concentration field; and obtain the distribution laws of the displacement field, temperature field, chemical potential field, stress field, and concentration field based on the inverse Laplace numerical transform.

[0238] Example 4:

[0239] This embodiment provides an electronic device, including a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the program, it implements the method for obtaining the thermo-elastic-diffusion behavior of a hollow column structure as described in Embodiment 1.

[0240] Example 5:

[0241] This embodiment provides a computer-readable storage medium storing a computer program that, when executed by a processor, implements the method for obtaining the thermo-elastic-diffusion behavior of a hollow column structure as described in Embodiment 1.

[0242] The above description is merely a preferred embodiment of this application and is not intended to limit this application. Various modifications and variations can be made to this application by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of this application should be included within the protection scope of this application.

[0243] The basic concepts have been described above. Obviously, for those skilled in the art, the above disclosure is merely illustrative and does not constitute a limitation of this application. Although not explicitly stated herein, those skilled in the art may make various modifications, improvements, and corrections to this application. Such modifications, improvements, and corrections are suggested in this application, and therefore remain within the spirit and scope of the exemplary embodiments of this application.

[0244] Furthermore, this application uses specific terms to describe embodiments of the application. For example, "an embodiment," "one embodiment," and / or "some embodiments" refer to a particular feature, structure, or characteristic related to at least one embodiment of the application. Therefore, it should be emphasized and noted that "an embodiment," "one embodiment," or "an alternative embodiment" mentioned twice or more in different locations in this specification do not necessarily refer to the same embodiment. In addition, certain features, structures, or characteristics in one or more embodiments of the application can be appropriately combined.

[0245] Some aspects of this application can be executed entirely by hardware, entirely by software (including firmware, resident software, microcode, etc.), or by a combination of hardware and software. The aforementioned hardware or software may be referred to as a "data block," "module," "engine," "unit," "component," or "system." The processor may be one or more application-specific integrated circuits (ASICs), digital signal processors (DSPs), digital signal processing devices (DAPDs), programmable logic devices (PLDs), field-programmable gate arrays (FPGAs), processors, controllers, microcontrollers, microprocessors, or combinations thereof. Furthermore, aspects of this application may manifest as computer products residing in one or more computer-readable media, including computer-readable program code. For example, computer-readable media may include, but are not limited to, magnetic storage devices (e.g., hard disks, floppy disks, magnetic tapes, etc.), optical discs (e.g., compressed CDs, digital multifunction DVDs), smart cards, and flash memory devices (e.g., cards, sticks, key drives, etc.).

[0246] A computer-readable medium may contain a propagated data signal containing computer program code, for example, on baseband or as part of a carrier wave. This propagated signal may take various forms, including electromagnetic, optical, and so on, or suitable combinations thereof. A computer-readable medium can be any computer-readable medium other than a computer-readable storage medium, which can be connected to an instruction execution system, apparatus, or device to enable communication, propagation, or transmission of a program for use. The program code located on the computer-readable medium can be propagated through any suitable medium, including radio, cable, fiber optic cable, radio frequency signals, or similar media, or any combination of the above media.

[0247] The basic concepts have been described above. Obviously, for those skilled in the art, the above disclosure is merely illustrative and does not constitute a limitation of this application. Although not explicitly stated herein, those skilled in the art may make various modifications, improvements, and corrections to this application. Such modifications, improvements, and corrections are suggested in this application, and therefore remain within the spirit and scope of the exemplary embodiments of this application.

[0248] Furthermore, this application uses specific terms to describe embodiments of the application. For example, "an embodiment," "one embodiment," and / or "some embodiments" refer to a particular feature, structure, or characteristic related to at least one embodiment of the application. Therefore, it should be emphasized and noted that "an embodiment," "one embodiment," or "an alternative embodiment" mentioned twice or more in different locations in this specification do not necessarily refer to the same embodiment. In addition, certain features, structures, or characteristics in one or more embodiments of the application can be appropriately combined.

[0249] Similarly, it should be noted that, in order to simplify the description of the present application and thus aid in the understanding of one or more embodiments of the invention, the foregoing description of the embodiments of the present application sometimes combines multiple features into a single embodiment, drawing, or description thereof. However, this disclosure method does not imply that the subject matter of the application requires more features than those mentioned in the claims. In fact, the embodiments contain fewer features than all the features of the single embodiments disclosed above.

[0250] In some embodiments, numbers describing the quantity of components and attributes are used. It should be understood that such numbers used in the description of embodiments are modified in some examples with the terms "approximately," "approximately," or "generally." Unless otherwise stated, "approximately," "approximately," or "generally" indicates that the numbers are allowed to vary by ±20%. Accordingly, in some embodiments, the numerical parameters used in the specification and claims are approximate values, which may be changed depending on the characteristics required by individual embodiments. In some embodiments, numerical parameters should take into account specified significant digits and employ a general method of digit reservation. Although the numerical ranges and parameters used to confirm their breadth of scope in some embodiments of this application are approximate values, in specific embodiments, such values ​​are set as precisely as feasible.

[0251] Although this application has been described with reference to specific embodiments, those skilled in the art should recognize that the above embodiments are only used to illustrate this application, and various equivalent changes or substitutions can be made without departing from the spirit of this application. Therefore, any changes or modifications to the above embodiments within the essential spirit of this application will fall within the scope of the claims of this application.

Claims

1. A method for obtaining the thermo-elastic-diffusion behavior of a hollow column structure, characterized in that, The method includes: Step 1: Establish the corresponding fractional-order generalized thermo-elastic-diffusion model, kinematic model, and constitutive model through model analysis, and determine the boundary conditions and initial conditions; Step 2: Dimensionless processing is performed on the physical quantities involved, including the fractional-order generalized thermoelastic model, the fractional-order generalized diffusion model, the kinematic model, and the constitutive model, to obtain the corresponding governing equations. Step 3: By performing a Laplace transform on the dimensionless control equations and solving the control equations using the eigenvalue method, and by performing a Laplace transform on the control equations and using the eigenvalue method, the displacement field, temperature field, chemical potential field, stress field, and concentration field are obtained. Step 4: The distribution patterns of the displacement field, temperature field, chemical potential field, stress field, and concentration field are obtained by solving the Laplace numerical inverse transform. The process of establishing the model for the hollow column structure includes: (1) By introducing a thermal relaxation coefficient into the traditional Fourier heat conduction equation and performing a Taylor series expansion of the heat flow components with fractional-order time parameters, the corresponding fractional-order generalized heat conduction equation can be obtained: (7) in, The thermal conductivity coefficient, It is a temperature field, and , , Absolute temperature For small variable parameters, For reference temperature, The thermal relaxation coefficient is... For fractional-order parameters, For material density, Specific heat coefficient, For the displacement tensor, For time, The independent variable is The Gamma function, and Represents the Lamé coefficient. Represents coordinate variables, , , , , , The coefficient of linear expansion is 1 / 3. The linear diffusion coefficient is... and These represent the thermal diffusion effect and the diffusion effect coefficient, respectively. (2) By introducing a diffusion relaxation coefficient into the traditional Fick's law and performing a Taylor series expansion of the mass diffusion flux with fractional-order time parameters, the corresponding fractional-order generalized diffusion equation can be obtained: (8) in, Where is the diffusion coefficient. The diffusion relaxation coefficient is... , , .

2. The method for obtaining the thermo-elastic-diffusion behavior of a hollow column structure according to claim 1, characterized in that, The coefficients in the thermo-elastic-diffusion coupling model of the hollow cylindrical structure satisfy the following: (1) (2) (3) (4) (5) (6) By combining equations (1) and (6), the coefficients can be obtained. , , , , and .

3. The method for obtaining the thermo-elastic-diffusion behavior of a hollow column structure according to claim 2, characterized in that, Step 2 further includes: The modeling process for the hollow column structure also includes: (3) The motion model that does not consider physical strength is: (9) in, Radial stress, It is circumferential stress; (4) The constitutive relation is: (10) (11) (12) (13) in, For axial stress; (5) The boundary conditions and initial conditions are as follows: (14) (15) (16) (17) in, For the Heaviside function; (6) Introduce the following dimensionless quantities: (18) For ease of description, the upper right corner of each physical quantity is omitted. Then, the corresponding governing equations and boundary conditions can be obtained as follows: (19) (20) (21) (22) (23) (24) (25) (26) (27) (28) in, 。 (7) Applying the Laplace transform to equations (19) to (25) yields: (29) (30) (31) (32) (33) (34) (35) in, They represent In the field of the Laplace transform domain, In order to keep pace with time Corresponding field variables; Apply equations (29) and (30) respectively to Taking the first derivative and combining it with equation (35), it can be expressed in the form of a vector matrix, as follows: (36) in, , , ,and, , , , , , , , , ; matrix The corresponding characteristic equation is (37) Therefore, by using the eigenvalue method and combining equations (31) to (34), the forms of the solutions for the displacement field, temperature field, chemical potential field, stress field, and concentration field can be obtained as follows: (38) (39) (40) (41) (42) (43) (44) in, and These are the zeroth-order modified Bessel function of the first kind and the zeroth-order modified Bessel function of the second kind, respectively. and These are the second-order modified Bessel function of the first kind and the second-order modified Bessel function of the second kind, respectively. , These represent the three roots corresponding to the characteristic equation (37). Indicates the first The eigenvector corresponding to the eigenvalue of the eigenvalue is the th eigenvector. One component; By combining the boundary conditions (26)~(28), the coefficients are obtained. ; (8) Perform the Laplace numerical inverse transform on equations (38) to (44) according to the following formula to obtain the distribution law of the displacement field, temperature field, chemical potential field, stress field and concentration field in the real field: (45) in, Represents the real field, Represents the field in the transform domain. Indicates the number of terms to be summed. n and k These represent the number of terms in the summation operation. , This represents the factorial operation; Thus, the displacement field, temperature field, chemical potential field, stress field, and concentration field of the real field can be obtained as follows: (46) (47) (48) (49) (50) (51) (52)。 4. A computer device, characterized in that, The computer device includes a memory and a processor. The memory stores a computer program, which, when executed by the processor, causes the processor to perform the following steps: Step 1: Establish the corresponding fractional-order generalized thermo-elastic-diffusion model, kinematic model, and constitutive model through model analysis, and determine the boundary conditions and initial conditions; Step 2: Dimensionless processing is performed on the physical quantities involved, specifically including dimensionless processing of the fractional-order generalized thermoelastic model, fractional-order generalized diffusion model, kinematic model and constitutive model to obtain the corresponding governing equations. Step 3: By performing a Laplace transform on the dimensionless governing equations and solving the governing equations using the eigenvalue method, and by performing a Laplace transform on the governing equations and using the eigenvalue method, the displacement field, temperature field, chemical potential field, stress field, and concentration field are obtained. Step 4: The distribution patterns of the displacement field, temperature field, chemical potential field, stress field, and concentration field are obtained by solving the Laplace numerical inverse transform. The modeling process for the hollow column structure includes: (1) By introducing a thermal relaxation coefficient into the traditional Fourier heat conduction equation and performing a Taylor series expansion of the heat flow components with fractional-order time parameters, the corresponding fractional-order generalized heat conduction equation can be obtained: (7) in, The thermal conductivity coefficient, It is a temperature field, and , , Absolute temperature For small variable parameters, For reference temperature, The thermal relaxation coefficient is... For fractional-order parameters, For material density, Specific heat coefficient, For the displacement tensor, For time, The independent variable is The Gamma function, and Represents the Lamé coefficient. Represents coordinate variables, , , , , , The coefficient of linear expansion is 1 / 3. The linear diffusion coefficient is... and These represent the thermal diffusion effect and the diffusion effect coefficient, respectively. (2) By introducing a diffusion relaxation coefficient into the traditional Fick's law and performing a Taylor series expansion of the mass diffusion flux with fractional-order time parameters, the corresponding fractional-order generalized diffusion equation can be obtained: (8) in, Where is the diffusion coefficient. The diffusion relaxation coefficient is... , , .

5. A computer-readable storage medium storing a computer program, which, when executed by a processor, causes the processor to perform the following steps: Step 1: Establish the corresponding fractional-order generalized thermo-elastic-diffusion model, kinematic model, and constitutive model through model analysis, and determine the boundary conditions and initial conditions; Step 2: Dimensionless processing is performed on the physical quantities involved, specifically including dimensionless processing of the fractional-order generalized thermoelastic model, fractional-order generalized diffusion model, kinematic model and constitutive model to obtain the corresponding governing equations. Step 3: By performing a Laplace transform on the dimensionless governing equations and solving the governing equations using the eigenvalue method, and by performing a Laplace transform on the governing equations and using the eigenvalue method, the displacement field, temperature field, chemical potential field, stress field, and concentration field are obtained. Step 4: The distribution patterns of the displacement field, temperature field, chemical potential field, stress field, and concentration field are obtained by solving the Laplace numerical inverse transform. in, The process of establishing a model for a hollow column structure includes: (1) By introducing a thermal relaxation coefficient into the traditional Fourier heat conduction equation and performing a Taylor series expansion of the heat flow components with fractional-order time parameters, the corresponding fractional-order generalized heat conduction equation can be obtained: (7) in, The thermal conductivity coefficient, It is a temperature field, and , , Absolute temperature For small variable parameters, For reference temperature, The thermal relaxation coefficient is... For fractional-order parameters, For material density, Specific heat coefficient, For the displacement tensor, For time, The independent variable is The Gamma function, and Represents the Lamé coefficient. Represents coordinate variables, , , , , , The coefficient of linear expansion is 1 / 3. The linear diffusion coefficient is... and These represent the thermal diffusion effect and the diffusion effect coefficient, respectively. (2) By introducing a diffusion relaxation coefficient into the traditional Fick's law and performing a Taylor series expansion of the mass diffusion flux with fractional-order time parameters, the corresponding fractional-order generalized diffusion equation can be obtained: (8) in, Where is the diffusion coefficient. The diffusion relaxation coefficient is... , , .

6. A thermal-elastic-diffusion distribution detection system for a method of obtaining the thermal-elastic-diffusion behavior of a hollow column structure according to any one of claims 1 to 3, characterized in that, The thermo-elastic-diffusion distribution detection system includes: The model building module is used to establish a thermo-elastic-diffusion coupling model of a hollow cylindrical structure with an analysis time fractional order. The dimensionless processing module is used to solve the thermo-elastic-diffusion coupling control equations using Laplace transform and eigenvalue method; The distribution law determination module is used to obtain the distribution laws of displacement field, temperature field, chemical potential field, stress field and concentration field through Laplace numerical inverse transform.

7. A material processing system, characterized in that, The material processing system is equipped with the thermo-elastic-diffusion distribution detection system as described in claim 6.

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