Thermoelasticity analysis method for hollow drum structure

Abstract

The invention relates to a thermoelastic analysis method for a hollow drum structure, which comprises the following steps: S1, establishing a fractional order heat conduction equation considering a viscoelastic effect for the hollow drum structure, and establishing a motion equation and a constitutive equation not considering physical strength; s2, determining boundary conditions of the hollow drum structure under the thermal shock action; s3, performing dimensionless processing on the fractional order heat conduction equation, the motion equation, the constitutive equation and the boundary condition to obtain a control equation; and S4, carrying out Laplace inverse transformation on the obtained control equation, solving through a characteristic value method to obtain a mechanical property distribution result, and giving a corresponding distribution rule diagram. Compared with the prior art, the method is more suitable for actual working conditions. The calculation precision is improved due to the influence of structural rotation on a mechanical property distribution rule; an efficient solving method is provided, and numerical stability is ensured; the method has the advantages of wide applicability and the like.

Description

Technical Field

[0001] This invention relates to the field of material mechanical property testing technology, and in particular to a thermoelastic analysis method for a hollow rotating cylinder structure. Background Technology

[0002] For hollow rotating cylinder structures in practical engineering, characterizing their mechanical properties under rotational and thermal shock is particularly important. Furthermore, due to temperature variations, material deformation is not necessarily purely elastic and may lead to viscous deformation. In addition, studies of thermal shock or femtosecond-level heat conduction often require consideration of nonlocality. Therefore, for rotating cylinder structures commonly found in the nuclear industry, it is more appropriate to characterize their mechanical properties under thermal shock by introducing fractional-order viscoelastic constitutive relations.

[0003] Among the existing related technologies, such as solving the thermo-elastic coupling control equations through inverse Laplace transform and Fourier transform to obtain the influence of fractional strain rate on the three-dimensional distribution of temperature and stress, there are also some other related technologies, such as considering the mechanical distribution law of elastic rods by taking into account the scale effect and fractional strain rate, and considering the mechanical distribution law of multi-field coupling of hollow columns with temperature related to the thermal conductivity and diffusion coefficient of the material. None of these technologies have considered the influence of structural rotation on the distribution law of its mechanical properties.

[0004] Therefore, hollow rotary structures are commonly used in related engineering fields, such as the nuclear industry. These structures operate in complex environments, making direct experimental measurement for performance characterization difficult, which has always been a challenge.

[0005] Therefore, there is an urgent need to study a performance characterization method that can take into account the effect of rotation on the distribution of mechanical properties of hollow rotary cylinder structures. Summary of the Invention

[0006] The purpose of this invention is to overcome the shortcomings of the existing technology, such as not considering the influence of structural rotation on the distribution of mechanical properties, and being unable to accurately describe the nonlocal heat conduction effect and viscoelastic deformation under thermal shock, by providing a thermoelastic analysis method for hollow rotating cylinder structures. This method is more in line with actual working conditions (the influence of structural rotation on the distribution of mechanical properties), improves calculation accuracy, provides an efficient solution method, ensures numerical stability, and has wide applicability.

[0007] This invention provides a thermoelastic analysis method for a hollow rotating cylinder structure, comprising the following steps:

[0008] S1: Establish a fractional-order heat conduction equation considering viscoelastic effects for the hollow rotating cylinder structure, and establish the motion equation and constitutive equation neglecting body forces;

[0009] S2: Determine the boundary conditions for the hollow rotating cylinder structure subjected to thermal shock;

[0010] S3: The fractional heat conduction equation, motion equation, constitutive equation and boundary conditions are dimensionlessly processed to obtain the governing equations;

[0011] S4: Perform the inverse Laplace transform on the obtained governing equations, and solve them by the eigenvalue method to obtain the mechanical property distribution results, and give the corresponding distribution law diagram;

[0012] The formula for calculating the fractional-order heat conduction equation is as follows:

[0013]

[0014] Where k represents the thermal conductivity of the rotating cylinder, Γ represents the Gamma function, and C E denoted by , where α represents the fractional order parameter, γ represents the coefficient of thermal expansion, e represents the strain diffusivity, τ1 represents the strain relaxation coefficient, τ2 represents the thermal relaxation coefficient, θ represents the temperature, θ = T - T0, where T is the absolute temperature, T0 is the reference temperature, and r is the radius coordinate.

[0015] Furthermore,

[0016] The process of establishing the boundary conditions is as follows: considering the continuity of interlayer displacement, the continuity of the normal component of heat flux density, and the discontinuity of temperature.

[0017] The assumptions of the boundary conditions include that the outer surface of the hollow rotary cylinder structure is subjected to thermal shock and stress is free, while the inner surface is insulated and constrained so as not to deform.

[0018] Furthermore, the boundary conditions are:

[0019] σ rr (b,t)=0;

[0020] θ(b,t)=θ0H(t);

[0021] u(a,t) = 0;

[0022]

[0023] Where θ is temperature, θ = T - T0, T is absolute temperature, T0 is reference temperature, u represents displacement; a is inner diameter of the rotating cylinder, b is outer diameter of the rotating cylinder, k is thermal conductivity of the rotating cylinder, t is time, r is radius coordinate, θ0 is the magnitude of thermal shock temperature, and H(t) is the Heaviside function.

[0024] Furthermore, the calculation formula for the motion model that disregards physical exertion is as follows:

[0025]

[0026] Where, σ rr , Let represent radial stress and circumferential stress respectively, ρ represent density, ω represent drum rotation speed, u represent displacement, t represent time, and r represent radius coordinates.

[0027] Furthermore, the formula for calculating the constitutive equation is as follows:

[0028]

[0029] Where σ rr , σ zz λ and μ represent radial stress, circumferential stress, and axial stress, respectively; λ and μ are Lamé coefficients, Γ represents the Gamma function, α represents the fractional-order parameter, γ represents the coefficient of thermal expansion, e represents the strain diffusivity, τ1 represents the strain relaxation coefficient, θ is the temperature, θ = T - T0, T is the absolute temperature, T0 is the reference temperature, r is the radial coordinate, u represents the displacement, and t is the time.

[0030] Furthermore, the formula for calculating the strain diffusivity e is as follows:

[0031]

[0032] r represents the radius coordinate, and u represents the displacement.

[0033] Furthermore, in S3, dimensionless processing yields dimensionless boundary conditions, and coefficient A is obtained based on these dimensionless boundary conditions. i (i=1,2) and A′ i (i = 1, 2) satisfy the following relationship:

[0034]

[0035]

[0036] in, η1 and η2 are two eigenvalues. Let Γ represent the i-th component of the j-th eigenvector, I0 and K0 represent the zero-order modified Bessel function of the first kind and the zero-order modified Bessel function of the second kind, respectively, I1 and K1 represent the first-order modified Bessel function of the first kind and the first-order modified Bessel function of the second kind, respectively, Γ represents the Gamma function, α represents the fractional-order parameter, γ represents the coefficient of thermal expansion, τ1 represents the strain relaxation coefficient, θ0 represents the magnitude of the thermal shock temperature, and r represents the radius coordinate.

[0037] Furthermore, the numerical inverse Laplace transform of the mechanical property distribution results includes:

[0038]

[0039] Where f represents the true field, F represents the field in the transformed domain, N represents the number of terms in the summation, n represents the number of terms in the summation operation, r is the radius coordinate, and t is time; the calculation formula is:

[0040]

[0041] Where k is an integer.

[0042] Furthermore, the mechanical properties obtained based on the inverse Laplace transform include the displacement field, temperature field, and stress field, as well as their distribution patterns.

[0043] Furthermore, the displacement field u(r,t), temperature field θ(r,t), and stress field σ rr (r,t), σ zz The distribution law of (r,t) is as follows:

[0044]

[0045] Where t is time; Let u, θ, and σ represent the fields in the Laplace transform domain, respectively; r is the radius; and n is the iteration number.

[0046] Compared with the prior art, the present invention has the following advantages:

[0047] (1) Better reflects actual working conditions (the influence of structural rotation on the distribution of mechanical properties) and improves calculation accuracy. This invention introduces fractional derivatives (α-order) to characterize thermal relaxation (τ2) and strain relaxation (τ1) effects. Unlike traditional thermoelastic analysis, which usually uses classical heat conduction theory and pure elastic constitutive relations, this invention is more consistent with the mechanical response of transient thermal shock and viscoelastic materials, making the calculation results closer to real physical phenomena. Through dimensionless processing, the interaction of thermal expansion, inertial force, and viscoelastic effects is uniformly characterized, improving the prediction accuracy under complex working conditions.

[0048] (2) Provides efficient solution methods to ensure numerical stability. The inverse Laplace transform and eigenvalue method are used to analytically solve the governing equations, avoiding the time step limitations of the traditional finite element method in transient problems, resulting in higher computational efficiency. An analytical solution is constructed by modifying the Bessel function, and combined with the inverse Laplace numerical transform, ensuring the stability of the time-domain solution, making it suitable for long-term dynamic response analysis. Simultaneously, the spatiotemporal distributions of displacement, temperature, and stress fields are output, intuitively reflecting the structural deformation, temperature gradient, and stress concentration regions under thermal shock.

[0049] (3) Wide applicability. Taking into account the coupling effect of rotation and thermal shock, this invention explicitly introduces the rotational speed ω into the equation of motion, making it suitable for thermo-mechanical coupling analysis of high-speed rotating machinery (such as turbines and nuclear reactor rotors). The influence of material properties is quantified through dimensionless parameters, facilitating engineering optimization design. Attached Figure Description

[0050] Figure 1 This is a flowchart illustrating a thermoelastic analysis method for a hollow rotating cylinder structure.

[0051] Figure 2 This is a diagram showing the distribution of displacement along r corresponding to different rotational speeds in Example 1;

[0052] Figure 3 This is a graph showing the temperature distribution along r for different rotational speeds in Example 1.

[0053] Figure 4 This is a diagram showing the distribution of radial stress along r at different rotational speeds in Example 1.

[0054] Figure 5 This is a diagram showing the distribution of circumferential stress along r at different rotational speeds in Example 1.

[0055] Figure 6 This is a diagram showing the distribution of axial stress along r at different rotational speeds in Example 1. Detailed Implementation

[0056] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments. Component models, material names, connection structures, control methods, algorithms, and other features not explicitly described in this technical solution are considered common technical features disclosed in the prior art.

[0057] Example 1

[0058] This embodiment provides a thermoelastic analysis method for a hollow rotating cylinder structure, such as... Figure 1 As shown,

[0059] Includes the following steps:

[0060] S1: Establish a fractional-order heat conduction equation considering viscoelastic effects for the hollow rotating cylinder structure, and establish the motion equation and constitutive equation neglecting body forces;

[0061] S2: Determine the boundary conditions for the hollow rotating cylinder structure subjected to thermal shock;

[0062] S3: The fractional heat conduction equation, motion equation, constitutive equation and boundary conditions are dimensionlessly processed to obtain the governing equations;

[0063] S4: Perform the inverse Laplace transform on the obtained governing equations, and solve them by the eigenvalue method to obtain the mechanical property distribution results, and give the corresponding distribution law diagram.

[0064] In a specific implementation, for the hollow rotating cylinder structure, it is assumed that the outer surface is subjected to thermal shock and stress is free, while the inner surface is insulated and constrained, preventing deformation. Therefore, the boundary conditions can be established as follows:

[0065] σ rr (b,t)=0 (1)

[0066] θ(b,t)=θ0H(t) (2)

[0067] u(a,t)=0 (3)

[0068]

[0069] Where θ is temperature, θ = T - T0, T is absolute temperature, T0 is reference temperature, u represents displacement; a is inner diameter of the rotating cylinder, b is outer diameter of the rotating cylinder, k is thermal conductivity of the rotating cylinder, t is time, r is radius coordinate, θ0 is the magnitude of thermal shock temperature, and H(t) is the Heaviside function.

[0070] The calculation formula for the fractional-order heat conduction model used in this embodiment is as follows:

[0071]

[0072] Where k represents the thermal conductivity, Γ represents the Gamma function, and C E Let represent specific heat capacity, α represent fractional-order parameters, γ represent the coefficient of thermal expansion, e represent strain diffusivity, τ1 represent the strain relaxation coefficient, and τ2 represent the thermal relaxation coefficient. The formula for calculating e is:

[0073]

[0074] Where r represents the radius.

[0075] In a specific implementation, the calculation formula for the motion model that disregards physical exertion is as follows:

[0076]

[0077] Where, σ rr , Let represent radial stress and circumferential stress, respectively; ρ represent density; and ω represent the rotational speed of the drum.

[0078] The formula for calculating the constitutive model is as follows:

[0079]

[0080] Where, σ zz λ represents the axial stress; λ and μ are the Lamé coefficients; r is the radius.

[0081] In a specific implementation, in S3, after dimensionless processing, dimensionless boundary conditions are obtained, and the coefficient A is obtained based on the dimensionless boundary conditions. i (i=1,2) and A′ i (i = 1, 2) satisfy the following relationship:

[0082]

[0083] in, η1 and η2 are two eigenvalues. Let Ii represent the i-th component of the j-th eigenvector, and let I0 and K0 be the zero-order modified Bessel function of the first kind and the zero-order modified Bessel function of the second kind, respectively. Let I1 and K1 be the first-order modified Bessel function of the first kind and the first-order modified Bessel function of the second kind, respectively.

[0084] The following dimensionless quantities are introduced during the calculation process:

[0085]

[0086] For ease of description, the asterisks (*) above each physical quantity are omitted, resulting in the dimensionless governing equations:

[0087]

[0088] Among them, β=λ / (λ+2μ), ε=γ 2 T0 / [ρC E (λ+2μ)。

[0089] By performing Laplace transformation on equations (16) to (20) and rearranging, we can obtain:

[0090]

[0091] The boundary conditions are:

[0092]

[0093] in, Represent u, θ, σ respectively rr , σ zz In the Laplace transform domain, s is the field variable corresponding to time t.

[0094] Combining (25) and (24), we can obtain

[0095]

[0096] Combining formulas (30) and (24), we have:

[0097] LV = BV (31)

[0098] in, And there are

[0099]

[0100] At this point, the characteristic equation of matrix B is:

[0101] η 2 -(b 11 +b 22 )η+b 11 b 22 -b 12 b 21 =0 (36)

[0102] Let the eigenvectors of matrix B be Then there is

[0103] X1 = b 12 (37)

[0104] X2=η-b 11 (38)

[0105] Let η1 and η2 be the two roots of the characteristic equation, and denote them as follows:

[0106]

[0107] The solution to equation (31) is in the form of:

[0108]

[0109] Among them, I i (·) represent the i-th order first-kind modified Bessel function, and K represents the i-th order, first-kind modified Bessel function, respectively. j (·) denotes the j-th order modified Bessel function of the second kind. A1, A2, A1', A'2 are undetermined coefficients related to the boundary conditions.

[0110] Therefore,

[0111]

[0112] Where I1 and K1 are the first-order modified Bessel function of the first kind and the first-order modified Bessel function of the second kind, respectively.

[0113] Combining (21) to (23) we have

[0114]

[0115] Combining equations (26) to (29), we get

[0116]

[0117]

[0118] By combining (47) to (50), we can find A1, A2, A1', and A'2.

[0119] Based on the above derivation and calculation results, and referring to Figure 1 The obtained governing equations are subjected to inverse Laplace transform. The mechanical properties obtained based on the inverse Laplace transform include displacement field, temperature field and stress field and their distribution law.

[0120] In a specific implementation, the numerical Laplace inverse transform of the mechanical property distribution results includes:

[0121]

[0122] Where f represents the real field, F represents the field in the transform domain, N represents the number of terms in the summation, n represents the number of terms in the summation operation, and c n The formula is for symbolic notation:

[0123]

[0124] Where k is an integer.

[0125] The displacement field u(r,t), temperature field θ(r,t), and stress field σ rr (r,t), σ zz The distribution law of (r,t) is as follows:

[0126]

[0127] Where t is time; Let u, θ, and σ represent the fields in the Laplace transform domain, respectively; r is the radius; and n is the iteration number.

[0128] This embodiment focuses on obtaining the mechanical property distribution law of a hollow rotating cylinder at different rotational speeds. For structures where the outer surface is subjected to thermal shock and stress is free, while the inner surface is thermally insulated and cannot deform, numerical solutions are obtained using Laplace transform, boundary conditions, eigenvalue method, and inverse Laplace numerical transform. This yields more accurate results and solves the problem of obtaining the mechanical property distribution law of such structures in practical engineering. Based on fractional-order thermoviscoelasticity theory, and combining Laplace transform, eigenvalue method, and inverse Laplace numerical transform, the above embodiments of this application focus on analyzing and discussing the distribution law of mechanical properties of a hollow rotating cylinder structure under different rotational speeds when the outer surface is subjected to thermal shock and stress is free, while the inner surface is thermally insulated and cannot deform.

[0129] on the other hand, Figures 2-6 The diagram illustrates the distribution patterns of displacement, temperature, and stress along rotational speed at different rotational speeds after adopting the method for obtaining the mechanical property distribution of a hollow rotary cylinder structure. For example, in an engineering scenario, the method described in Example 1 is verified using actual data, where the material parameter is λ = 7.76 × 10⁻⁶. 10 Nm -2 μ = 3.86 × 10 10 Nm -2 ρ=8954kg / m 3 C E =383.1 Jkg -1 K -1 , T0=293K, α=0.5, θ0=1, τ1=0.02, τ2=0.02, α t =1.78×10 -5 K -1 t = 0.15, a = 1, b = 2, and assume N = 20 in the inverse transform.

[0130] The distribution law of dimensionless displacement u with radius r at different rotational speeds is as follows: Figure 2 As shown, the displacement at the origin is 0, which is consistent with the boundary conditions, and the maximum displacement is obtained at the outer boundary. The distribution law of dimensionless temperature θ with radius r at different rotational speeds is as follows: Figure 3 As shown, the temperature value reaches its maximum at the outer boundary and remains constant at 1, which is consistent with the boundary conditions. The distribution law of dimensionless stress σ with radius r at different rotational speeds is as follows: Figures 4-6 As shown. From Figure 4 It can be seen that the stress at the outer boundary is 0, which is consistent with the boundary conditions. From Figures 2-6It is also evident that rotational speed has a negligible impact on the temperature field distribution, but a significant impact on other quantities; the amplitude of each quantity increases with increasing rotational speed. In summary, the fractional-order heat conduction model used in the above embodiments is consistent with actual working conditions. Considering thermal viscoelasticity and the fractional time order, it can improve the accuracy of the solution results for the mechanical distribution law of the hollow rotating cylinder structure. Its application in engineering fields, such as the nuclear industry and battery field, can be extended to material design and engineering testing scenarios.

[0131] Components not described in detail in this embodiment are all existing components that can be purchased through public channels.

[0132] The above description of the embodiments is provided to enable those skilled in the art to understand and use the invention. It will be apparent to those skilled in the art that various modifications can be made to these embodiments, and the general principles described herein can be applied to other embodiments without inventive effort. Therefore, the present invention is not limited to the above embodiments, and any improvements and modifications made by those skilled in the art based on the disclosure of the present invention without departing from the scope of the invention should be within the protection scope of the present invention.

Claims

1. A thermoelastic analysis method for a hollow rotating cylinder structure, characterized in that, Includes the following steps: S1: Establish a fractional-order heat conduction equation considering viscoelastic effects for the hollow rotating cylinder structure, and establish the motion equation and constitutive equation neglecting body forces; S2: Determine the boundary conditions for the hollow rotating cylinder structure subjected to thermal shock; S3: The fractional heat conduction equation, motion equation, constitutive equation and boundary conditions are dimensionlessly processed to obtain the governing equations; S4: Perform the inverse Laplace transform on the obtained governing equations, and solve them by the eigenvalue method to obtain the mechanical property distribution results, and give the corresponding distribution law diagram; The formula for calculating the fractional-order heat conduction equation is as follows: Where k represents the thermal conductivity of the rotating cylinder, Γ represents the Gamma function, and C E denoted by , where α represents the fractional order parameter, γ represents the coefficient of thermal expansion, e represents the strain diffusivity, τ1 represents the strain relaxation coefficient, τ2 represents the thermal relaxation coefficient, θ represents the temperature, θ = T - T0, where T is the absolute temperature, T0 is the reference temperature, and r is the radius coordinate.

2. The thermoelastic analysis method for a hollow rotating cylinder structure according to claim 1, characterized in that, The process of establishing the boundary conditions is as follows: considering the continuity of interlayer displacement, the continuity of the normal component of heat flux density, and the discontinuity of temperature. The assumptions of the boundary conditions include that the outer surface of the hollow rotary cylinder structure is subjected to thermal shock and stress is free, while the inner surface is insulated and constrained so as not to deform.

3. The thermoelastic analysis method for a hollow rotating cylinder structure according to claim 1, characterized in that, The boundary conditions are as follows: s rr (b,t)=0; θ(b,t)=θ0H(t); u(a,t) = 0; Where θ is temperature, θ = T - T0, T is absolute temperature, T0 is reference temperature, u represents displacement; a is inner diameter of the rotating cylinder, b is outer diameter of the rotating cylinder, k is thermal conductivity of the rotating cylinder, t is time, r is radius coordinate, θ0 is the magnitude of thermal shock temperature, and H(t) is the Heaviside function.

4. The thermoelastic analysis method for a hollow rotating cylinder structure according to claim 1, characterized in that, The calculation formula for the motion model that disregards physical exertion is as follows: Where, σ rr , Let represent radial stress and circumferential stress respectively, ρ represent density, ω represent drum rotation speed, u represent displacement, t represent time, and r represent radius coordinates.

5. The thermoelastic analysis method for a hollow rotating cylinder structure according to claim 1, characterized in that, The formula for calculating the constitutive equation is as follows: Where σ rr , σ zz λ and μ represent radial stress, circumferential stress, and axial stress, respectively; λ and μ are Lamé coefficients, Γ represents the Gamma function, α represents the fractional-order parameter, γ represents the coefficient of thermal expansion, e represents the strain diffusivity, τ1 represents the strain relaxation coefficient, θ is the temperature, θ = T - T0, T is the absolute temperature, T0 is the reference temperature, r is the radial coordinate, u represents the displacement, and t is the time.

6. The thermoelastic analysis method for a hollow rotating cylinder structure according to claim 5, characterized in that, The formula for calculating the strain diffusivity e is: r represents the radius coordinate, and u represents the displacement.

7. The thermoelastic analysis method for a hollow rotating cylinder structure according to claim 1, characterized in that, In S3, dimensionless boundary conditions are obtained after dimensionless processing, and coefficient A is obtained based on the dimensionless boundary conditions. i (i=1,2) and A' i (i = 1, 2) satisfy the following relationship: in, η1 and η2 are two eigenvalues. Let Γ represent the i-th component of the j-th eigenvector, I0 and K0 represent the zero-order modified Bessel function of the first kind and the zero-order modified Bessel function of the second kind, respectively, I1 and K1 represent the first-order modified Bessel function of the first kind and the first-order modified Bessel function of the second kind, respectively, Γ represents the Gamma function, α represents the fractional-order parameter, γ represents the coefficient of thermal expansion, τ1 represents the strain relaxation coefficient, θ0 represents the magnitude of the thermal shock temperature, and r represents the radius coordinate.

8. The thermoelastic analysis method for a hollow rotating cylinder structure according to claim 1, characterized in that, The numerical inverse Laplace transform of the mechanical property distribution results includes: Where f represents the true field, F represents the field in the transformed domain, N represents the number of terms in the summation, n represents the number of terms in the summation operation, r is the radius coordinate, and t is time; the calculation formula is: Where k is an integer.

9. The thermoelastic analysis method for a hollow rotating cylinder structure according to claim 1, characterized in that, The mechanical properties obtained based on the inverse Laplace transform include the displacement field, temperature field, and stress field, as well as their distribution patterns.

10. The thermoelastic analysis method for a hollow rotating cylinder structure according to claim 9, characterized in that, The displacement field u(r,t), temperature field θ(r,t), and stress field σ rr (r,t), σ zz The distribution law of (r,t) is as follows: Where t is time; Let u, θ, and σ represent the fields in the Laplace transform domain, respectively; r is the radius; and n is the iteration number.

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