Thermal elasticity analysis method for laminated hollow column containing weak interface

Abstract

The invention relates to a thermoelastic analysis method for a laminated hollow column containing a weak interface, which comprises the following steps: establishing a fractional order heat conduction equation considering an interface effect for a laminated hollow column structure containing the weak interface, and establishing a motion equation, a constitutive equation and a boundary condition which do not consider physical power; 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; performing Laplace transformation on the obtained control equation, and solving to obtain a mechanical property distribution result; the boundary condition establishing process specifically comprises the step of considering interlayer displacement continuity, heat flow density normal component continuity and temperature discontinuity. Compared with the prior art, by introducing establishment of the weak interface and the fractional order heat conduction equation and Laplacian transformation solution of the control equation, the accuracy of the solution result of the mechanical property distribution rule of the laminated hollow column containing the weak interface can be improved, and the model is more practical.

Description

Technical Field

[0001] The present invention relates to the technical field of material mechanical property testing, and in particular to a thermo-elastic analysis method for a laminated hollow column with weak interfaces. Background Art

[0002] For the laminated hollow column structure in practical engineering, due to defects in the manufacturing process, its interface connection is often between complete detachment and perfect connection, and this kind of interface is called a weak interface. A large number of studies also show that the interface effect generated by the weak interface has a great influence on the performance characterization of the laminated structure. In addition, when the laminated hollow column structure is subjected to a thermal shock caused by, for example, laser irradiation, its response is generally in the nanosecond or even picosecond level. For the study of such ultra-short thermal shock problems, the influence of the relevance and non-locality of the process often needs to be considered. Therefore, for fuel assemblies, cladding materials, and battery structures commonly found in the nuclear industry, etc., when these structures are subjected to thermal shock, it is particularly necessary to consider the influence of the interface effect on displacement, temperature, and stress by introducing weak interfaces.

[0003] On the other hand, with the development and improvement of the fractional-order thermo-elastic theory, people gradually use this theory to study heat transfer problems in extreme environments. For example, in some related technologies, the thermo-elastic coupling control equation is solved by Laplace transform and Fourier transform to obtain the influence law of the fractional-order strain rate on the three-dimensional distribution of temperature and stress. In addition, there are some other related technologies, such as the method of obtaining each field of a semi-infinite long piezoelectric rod under the action of a uniformly moving heat source through the Ezzat-type fractional-order heat conduction equation, and the mechanical distribution law of an elastic rod considering the scale effect and fractional-order strain rate, and the mechanical distribution law of multi-field coupling of a hollow column considering the temperature dependence of the material thermal conductivity and diffusion coefficient, etc. These technologies do not consider the influence of the interface effect of the laminated structure on its mechanical property distribution law.

[0004] Therefore, in related engineering fields, such as the nuclear industry, there are many applications involving laminated hollow column structures. For example, pipes, fuel rod claddings, and battery structures in nuclear reactors, etc. Such structural components often involve laminated hollow column structures. In the stages of engineering tests, R & D tests, etc., due to the lack of analysis of the influence of the interface effect, the accuracy of material design and engineering tests is often affected.

[0005] Therefore, there is an urgent need to study a material testing method that can consider the influence of the interface effect on the mechanical property distribution law of a laminated hollow column structure when it is subjected to thermal shock. Summary of the Invention

[0006] The object of the present invention is to overcome the defects existing in the above-mentioned prior art and provide a thermo-elastic analysis method for laminated hollow columns with weak interfaces. By introducing weak interfaces, establishing the fractional-order heat conduction equation, and solving the control equations through Laplace transform, the accuracy of the solution results of the mechanical property distribution law of laminated hollow columns with weak interfaces can be improved. By introducing weak interfaces to establish the interlayer connection method to consider the interface effect of the laminated structure, the model is more in line with the actual situation.

[0007] The present invention provides a thermo-elastic analysis method for laminated hollow columns with weak interfaces, including the following steps:

[0008] Establish a fractional-order heat conduction equation considering the interface effect for the laminated hollow column structure with weak interfaces, and establish the motion equation, constitutive equation, and boundary conditions without considering body forces;

[0009] Nondimensionalize the fractional-order heat conduction equation, motion equation, constitutive equation, and boundary conditions to obtain the control equations; perform Laplace transform on the obtained control equations, and solve to obtain the mechanical property distribution results;

[0010] The specific process of establishing the boundary conditions is as follows: Consider the continuity of interlayer displacement, the continuity of the normal component of the heat flux density, and the discontinuity of temperature.

[0011] Further, the assumed conditions of the boundary conditions include that the outer surface of the laminated hollow column structure with weak interfaces is subjected to a thermal shock and stress-free, and the inner surface is adiabatic and constrained from deforming;

[0012] The calculation formula of the boundary conditions is:

[0013]

[0014]

[0015] u (1) (b,t) = u (2) (b,t);

[0016]

[0017]

[0018]

[0019] Among them, the superscripts 1 and 2 respectively represent medium 1 and medium 2, θ is the temperature, θ = T - T0, T is the absolute temperature, T0 is the reference temperature, u represents the displacement; a is the inner surface radius of medium 1, b is the interface radius, c is the outer surface radius of medium 2, t is the time, q r is the normal heat flux density, R Tis a non - negative constant, β (1) = ρ (1) c (1) ,β (2) = ρ (2) c (2) ,ρ (1) and ρ (2) respectively represent the densities of medium 1 and medium 2, c (1) and c (2) are quantities related to the Lamé coefficients respectively.

[0020] Furthermore, the calculation formula of the motion model without considering body force is:

[0021]

[0022] where σ rr 、 respectively represent the radial stress and the circumferential stress.

[0023] Furthermore, the calculation formula of the fractional - order heat conduction equation is:

[0024]

[0025] where κ represents the heat conduction coefficient, Γ represents the Gamma function, C E represents the specific heat capacity, α represents the fractional - order parameter, γ represents the coefficient of thermal expansion, is the operation operator.

[0026] Furthermore, the calculation formula is:

[0027]

[0028] Furthermore, the calculation formula of the constitutive model is:

[0029]

[0030]

[0031]

[0032]

[0033] where σ zz is the axial stress; λ and μ are the Lamé coefficients; r is the radius.

[0034] Furthermore, after dimensionless treatment, dimensionless boundary conditions are obtained, and coefficients are obtained according to the dimensionless boundary conditions The coefficients Satisfy the following relationships:

[0035]

[0036]

[0037]

[0038]

[0039]

[0040]

[0041]

[0042]

[0043] Among them, ξ (i) = λ (i) / (λ (i) + 2μ (i) ), R θ = R T κ (1) c1η, I0 and K0 are the modified Bessel functions of the first kind of order zero and the modified Bessel functions of the second kind of order zero respectively, and I1 and K1 are the modified Bessel functions of the first kind of order one and the modified Bessel functions of the second kind of order one respectively, and are the characteristic roots.

[0044] Furthermore, the mechanical property results obtained based on the inverse Laplace transform include the displacement field, the temperature field, the stress field and their distribution laws.

[0045] Furthermore, the numerical inverse transform of the mechanical property distribution results includes:

[0046]

[0047] Among them, f represents the real field, F represents the field in the transform domain, N represents the number of summation terms, n represents the number of terms in the summation operation, and the calculation formula is:

[0048]

[0049] Among them, k is an integer.

[0050] Furthermore, the displacement field u (i) (r, t), the temperature field θ (i) (r, t) and the stress field σ (i)The distribution law of (r, t) is as follows:

[0051]

[0052]

[0053]

[0054] where t is time; respectively represent the fields of u, θ, σ in the Laplace transform domain; r is the radius; n is the number of iterations.

[0055] Compared with the prior art, the present invention has the following advantages:

[0056] The present invention aims at a laminated hollow column structure with a weak interface, whose outer surface is subjected to thermal shock and stress-free, and whose inner surface is adiabatic and cannot deform. By introducing a weak interface, establishing a fractional-order heat conduction equation, and solving the governing equation by Laplace transform, the accuracy of the solution of the mechanical property distribution law of the laminated hollow column with a weak interface can be improved. By introducing a weak interface to establish an interlayer connection method to consider the interface effect of the laminated structure, the model is more in line with the actual situation. Combining the fractional-order heat conduction equation and Laplace transform to solve the equation, the obtained mechanical property distribution law is more instructive for practical engineering. Brief Description of the Drawings

[0057] Figure 1 is a schematic flow chart of the thermoelastic analysis method for a laminated hollow column with a weak interface;

[0058] Figure 2 is a distribution law diagram of the displacement varying with r corresponding to different weak interface parameters in Embodiment 1;

[0059] Figure 3 is a distribution law diagram of the temperature varying with r corresponding to different weak interface parameters in Embodiment 1;

[0060] Figure 4 is a distribution law diagram of the radial stress varying with r corresponding to different weak interface parameters in Embodiment 1;

[0061] Figure 5 is a distribution law diagram of the circumferential stress varying with r corresponding to different weak interface parameters in Embodiment 1;

[0062] Figure 6 is a distribution law diagram of the axial stress varying with r corresponding to different weak interface parameters in Embodiment 1. Detailed Embodiment

[0063] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. In the technical solution, components such as model numbers, material names, connection structures, control methods, algorithms, etc. that are not clearly described are regarded as common technical features disclosed in the prior art.

[0064] Embodiment 1

[0065] This embodiment provides a thermoelastic analysis method for a laminated hollow column with a weak interface. As Figure 1 shown, for the laminated hollow column structure with a weak interface, it is assumed that the outer surface is subjected to a thermal shock and stress-free, and the inner surface is adiabatic and constrained from deforming. Thus, the boundary conditions can be established as:

[0066]

[0067]

[0068] u (1) (b,t) = u (2) (b,t) (3)

[0069]

[0070]

[0071]

[0072] Among them, the superscripts 1 and 2 respectively represent medium 1 and medium 2, θ is the temperature, θ = T - T0, T is the absolute temperature, T0 is the reference temperature, and u represents the displacement; a is the inner surface radius of medium 1, b is the interface radius, c is the outer surface radius of medium 2, t is the time, q r is the normal heat flux density, R T is a non-negative constant, β (1) = ρ (1) c (1) , β (2) = ρ (2) c (2) , ρ (1) and ρ (2) respectively represent the densities of medium 1 and medium 2, and c (1) and c (2) are respectively related to the Lame coefficients.

[0073] The calculation formula of the fractional-order heat conduction model adopted in this embodiment is:

[0074]

[0075] Among them, κ represents the heat conduction coefficient, Γ represents the Gamma function, C Edenotes the specific heat capacity, α denotes the fractional-order parameter, γ denotes the coefficient of thermal expansion, is the operation operator.

[0076] The calculation formula of

[0077]

[0078] is as follows: where r represents the radius.

[0079] The calculation formula of the motion model without considering body force is:

[0080]

[0081] where σ rr , respectively represent the radial stress and the circumferential stress.

[0082] The calculation formula of the constitutive model is:

[0083]

[0084]

[0085]

[0086] where τzz is the axial stress; λ and μ are Lame coefficients; r is the radius.

[0087] Perform dimensionless processing on the fractional-order heat conduction model, the motion model and the constitutive model to obtain the governing equations;

[0088] After dimensionless processing, dimensionless boundary conditions are also obtained, and coefficients are obtained according to the dimensionless boundary conditions The coefficient satisfies the following relationship:

[0089]

[0090]

[0091]

[0092]

[0093]

[0094]

[0095]

[0096]

[0097] Among them, ξ (i) = λ (i) / (λ (i) + 2μ (i) ), R θ = R T κ (1) c1η, I0 and K0 are the modified Bessel functions of the first kind of order zero and the modified Bessel functions of the second kind of order zero respectively, and I1 and K1 are the modified Bessel functions of the first kind of order one and the modified Bessel functions of the second kind of order one respectively, and are the characteristic roots.

[0098] The following dimensionless quantities are introduced in the calculation process:

[0099]

[0100] For the sake of convenience of description, the * sign in the upper right of each physical quantity is omitted, and the corresponding dimensionless control equation can be obtained as follows:

[0101]

[0102] Among them, e represents the volume expansion rate, ξ (i) = λ (i) / (λ (i) + 2μ (i) ),

[0103] Performing Laplace transform on formulas (22) to (27) and organizing, we can get:

[0104]

[0105]

[0106]

[0107]

[0108]

[0109]

[0110] The boundary conditions are

[0111]

[0112]

[0113] The interface boundary conditions are

[0114]

[0115]

[0116]

[0117]

[0118] where R θ = R T κ (1) c1η, respectively represent the fields of u, θ, σ in the Laplace transform domain, and s is the field variable corresponding to time t.

[0119] Combining (32) and (33) to eliminate we can obtain

[0120]

[0121] where

[0122] Similarly, eliminating we can get:

[0123]

[0124] (40) and (41) can be combined and arranged as

[0125]

[0126] where, and are the two positive roots of the characteristic equation

[0127] Then the form of the solution of equation (42) is:

[0128]

[0129] where are undetermined coefficients, and I0 and K0 are the modified Bessel functions of the first kind and the second kind of order zero, respectively.

[0130] Combining (43), (44) and (32) we can obtain

[0131]

[0132]

[0133] Thus, there is

[0134]

[0135]

[0136] where I1 and K1 are the first-kind modified Bessel function of the first order and the second-kind modified Bessel function of the first order, respectively.

[0137] Combining (28) - (31), there is

[0138]

[0139]

[0140]

[0141]

[0142] Combining equations (34) - (39), we get

[0143]

[0144]

[0145]

[0146]

[0147]

[0148]

[0149]

[0150]

[0151] By solving equations (53) - (60) simultaneously, we can obtain

[0152] Combining the above derivation results and referring to Figure 1 , the Laplace transform of the obtained control equation is performed, and the mechanical property results obtained based on the inverse Laplace transform include the displacement field, temperature field, stress field and their distribution laws.

[0153] The numerical inverse transform of the mechanical property distribution results includes:[[]]

[0154]

[0155] Among them, f represents the real field, F represents the field in the transform domain, N represents the number of summation terms, n represents the number of terms in the summation operation, and c n is a symbolic notation, and the calculation formula is:

[0156]

[0157] where k is an integer.

[0158] The displacement field u (i) (r, t), the temperature field θ (i) (r, t) and the stress field σ (i) (r, t) have the following distribution rules:

[0159]

[0160] where t is time; respectively represent the fields of u, θ, σ in the Laplace transform domain; r is the radius; n is the number of iterations.

[0161] In this embodiment, the acquisition of the distribution law of the mechanical properties of the laminated hollow column structure with a weak interface is aimed at a structure where the outer surface is subjected to a thermal shock and stress-free, and the inner surface is adiabatic and cannot deform. Numerical solutions are carried out through Laplace transform, boundary conditions, and inverse Laplace numerical transform, and the result accuracy is higher, solving the influence problem of the interface effect of this laminated structure in practical engineering. The above embodiments of this application are based on the new fractional-order thermoelastic theory proposed by Ezzat, combined with Laplace transform, the boundary conditions of the weak interface, and the inverse Laplace numerical transform method, and focus on analyzing and discussing the distribution law of the mechanical properties of the laminated hollow column structure with a weak interface when the outer surface is subjected to a thermal shock and stress-free, and the inner surface is adiabatic and cannot deform with the change of the weak interface parameters.

[0162] On the other hand, Figures 2 to 6 shows the distribution law diagrams of the displacement, temperature, and stress corresponding to different weak interface parameters along r after adopting the method for obtaining the distribution law of the mechanical properties of the laminated hollow column structure with a weak interface as described above. Exemplarily, in an engineering scenario, the method described in Example 1 is verified through actual data, where the material parameters are taken as λ = 7.76×10 10 Nm -2 , μ = 3.86×10 10 Nm -2 , ρ = 8954 kg / m 3 , C E = 383.1 Jkg -1 K -1 , T0 = 293 K, α = 0.5, θ0 = 1, τ0 = 0.02, α t = 1.78×10-5 K -1 where \(t = 0.15\), \(a = 1\), \(b = 1.5\), \(c = 2\), and assume \(N = 10\) in the inverse transformation.

[0163] As Figure 2 shown, the displacement at the origin is 0, and the displacement is continuous at \(r = 1.5\), which is consistent with the boundary conditions, and the maximum displacement is obtained at the outer boundary. The distribution law of the dimensionless temperature \(\theta\) with respect to the radius \(r\) under different weak interface parameters is as Figure 3 shown. The maximum temperature value is obtained at the outer boundary and is always 1, and there is a jump in temperature at \(r = 1.5\), which is consistent with the boundary conditions. The distribution law of the dimensionless stress \(\sigma\) with respect to the radius \(r\) under different weak interface parameters is as Figures 4 to 6 shown. It can be seen from Figure 4 that at the outer boundary, the stress is 0, which is consistent with the boundary conditions. Therefore, the fractional-order heat conduction model adopted in the above embodiments is consistent with the actual working conditions. On the premise of considering the interface effect and time fractional order, it can improve the accuracy of the solution results of the mechanical distribution law of the laminated hollow column structure with weak interfaces. Applied in the engineering field, such as the nuclear industry field and the battery field, it can be extended to scenarios such as material design and engineering tests.

[0164] The components not elaborated in this embodiment are all existing components that can be purchased through public channels.

[0165] The above description of the embodiments is for the convenience of those of ordinary skill in the art to understand and use the invention. Those skilled in the art can obviously make various modifications to these embodiments easily and apply the general principles described herein to other embodiments without creative labor. Therefore, the present invention is not limited to the above embodiments, and the improvements and modifications made by those skilled in the art without departing from the scope of the present invention according to the disclosure of the present invention should be within the protection scope of the present invention.

Claims

1. A thermoelastic analysis method for laminated hollow columns with weak interfaces, characterized in that, It includes the following steps: Establish a fractional-order heat conduction equation considering interface effects for the laminated hollow column structure with weak interfaces, and establish the equations of motion, constitutive equations, and boundary conditions without considering body forces; Nondimensionalize the fractional-order heat conduction equation, equations of motion, constitutive equations, and boundary conditions to obtain the governing equations; perform Laplace transforms on the obtained governing equations and solve them to obtain the mechanical property distribution results; The specific process of establishing the boundary conditions is as follows: Consider the continuity of interlayer displacements, the continuity of the normal component of the heat flux density, and the discontinuity of temperature.

2. The thermoelastic analysis method of a laminated hollow column with a weak interface according to claim 1, wherein The assumed conditions of the boundary conditions include that the outer surface of the laminated hollow column structure with weak interfaces is subjected to thermal shock and stress-free, and the inner surface is adiabatic and constrained from deforming; The calculation formula for the boundary conditions is: u (1) (a, t) = 0, θ (2) (c,t) = θ0H(t); Among them, 1 and 2 in the upper right corner respectively represent medium 1 and medium 2, θ is the temperature, θ = T - T0, T is the absolute temperature, T0 is the reference temperature, and u represents the displacement; a is the inner surface radius of medium 1, b is the interface radius, c is the outer surface radius of medium 2, t is the time, q r is the normal heat flux density, R T is a non - negative constant, β (1) = ρ (1) c (1) , β (2) = ρ (2) c (2) , ρ (1) and ρ (2) respectively represent the densities of medium 1 and medium 2, c (1) and c (2) are respectively quantities related to the Lame coefficients.

3. A thermoelastic analysis method for a laminated hollow column with a weak interface according to claim 1, characterized in that, The calculation formula for the motion model without considering body forces is: Among them, σ rr and represent the radial stress and the circumferential stress respectively.

4. A thermoelastic analysis method for a laminated hollow column with a weak interface according to claim 1, characterized in that, The calculation formula for the fractional-order heat conduction equation is: where κ represents the thermal conductivity, Γ represents the Gamma function, C E represents the specific heat capacity, α represents the fractional order parameter, γ represents the coefficient of thermal expansion, is the operation operator.

5. A thermoelastic analysis method for a laminated hollow column with a weak interface according to claim 4, characterized in that, The said The calculation formula is as follows:

6. A thermoelastic analysis method for a laminated hollow column with a weak interface according to claim 1, characterized in that The calculation formula for the constitutive model is: where σ zz is the axial stress; λ and μ are Lame coefficients; r is the radius.

7. A thermo-elastic analysis method for a laminated hollow column with weak interfaces according to claim 1, characterized in that After dimensionless processing, dimensionless boundary conditions are also obtained, and coefficients are obtained according to the dimensionless boundary conditions The coefficients Satisfy the following relationship: where ξ (i) = λ (i) / (λ (i) + 2μ (i) ), R θ = R T κ (1) c1η, I0 and K0 are the modified Bessel functions of the first kind of order zero and the modified Bessel functions of the second kind of order zero respectively, and I1 and K1 are the modified Bessel functions of the first kind of order one and the modified Bessel functions of the second kind of order one respectively, and are the characteristic roots.

8. A thermoelastic analysis method for a laminated hollow column with a weak interface according to claim 1, characterized in that The mechanical property results obtained based on the inverse Laplace transform include the displacement field, temperature field, stress field, and their distribution laws.

9. The thermoelastic analysis method of a laminated hollow column with a weak interface according to claim 8, characterized in that The numerical inverse transformation of the mechanical property distribution results includes: Among them, f represents the real field, F represents the field in the transform domain, N represents the number of summation terms, n represents the number of terms in the summation operation, and the calculation formula is: Among them, k is an integer.

10. A thermo - elastic analysis method for a laminated hollow column with weak interfaces according to claim 9, characterized in that, The displacement field u (i) (r, t), the temperature field θ (i) (r, t) and the stress field σ (i) (r, t) are distributed as follows: where t is time; respectively represent the fields of u, θ, σ in the Laplace transform domain; r is the radius; n is the number of iterations.

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