Method for acquiring mechanical property distribution rule of tubular structure

Abstract

The invention relates to a tubular structure mechanical property distribution rule obtaining method. Establishing a fractional order heat conduction model considering the scale effect for the tubular structure, and establishing a motion model and a constitutive model which do not consider physical strength; performing dimensionless processing on the fractional order heat conduction model, the motion model and the constitutive model to obtain a control equation; and performing Laplace transformation on the obtained control equation, and solving to obtain a mechanical property distribution result. Compared with the prior art, the method aims at a tubular structure of which the outer surface is subjected to constant-temperature thermal shock, the stress is free, the inner surface is adiabatic, and the tubular structure is restrained and cannot deform through establishment of a fractional order heat conduction model and Laplacian transformation solution of a control equation; and the accuracy of the solution result of the tubular structure mechanical property distribution rule can be improved.

Description

Technical Field

[0001] The invention relates to the technical field of material mechanical property testing, and in particular to a method for acquiring a distribution law of mechanical properties of a tubular structure. Background Art

[0002] For most heat conduction problems, Fourier's law can be accurately described. However, for micro-nano structures and short-term heat release problems, Fourier's law often needs to be corrected by considering the thermal relaxation coefficient. Typical examples include fuel assemblies, cladding materials, pipeline structures in the nuclear industry, semiconductors irradiated by lasers in the chip field, and battery structures. When these structural parts are subjected to thermal shock, changes in their temperature differences will cause deformation and stress changes in the structure. Therefore, it is particularly necessary to master the distribution law of the mechanical properties of structural parts under such extreme conditions.

[0003] On the other hand, with the continuous development of fractional calculus theory, it is increasingly being applied to the research of engineering thermal physics, and a fractional thermoelastic theory has been formed. For example, in some related technologies, the thermoelastic coupling control equation is solved by Laplace transform and Fourier transform to obtain the influence of fractional strain rate on the three-dimensional distribution of temperature and stress, but the solution process is mainly for the strain rate fractional heat conduction equation, and does not involve the scale effect of the structure. In addition, there are some other related technologies that consider the method of obtaining various fields of a semi-infinite piezoelectric rod under the action of a uniformly moving heat source through the Ezzat-type fractional heat conduction equation, but the scale effect is also not involved. The scale effect of structural parts often affects the accuracy of detection or testing. Therefore, it is particularly important to study the heat conduction mechanism by considering both the scale effect and the time fractional order.

[0004] In related engineering fields, such as the nuclear industry, there are many applications involving tubular structures. For example, pipes and fuel rod cladding in nuclear reactors can be considered as tubular structures. In the stages of engineering experiments, R&D testing, etc., there is still a lack of methods to obtain the distribution law of mechanical properties that consider both scale effects and time fractional orders, which affects the accuracy of material design and engineering experiments.

[0005] Therefore, there is an urgent need to study a material testing method that can simultaneously consider the scale effect and the time fractional order distribution law of mechanical properties. Summary of the invention

[0006] The purpose of the present invention is to provide a method for obtaining the distribution law of mechanical properties of tubular structures in order to overcome the defects of the above-mentioned prior art. By establishing a fractional-order heat conduction model and solving the Laplace transform of the control equation, the accuracy of the solution of the distribution law of mechanical properties of tubular structures can be improved.

[0007] The purpose of the present invention can be achieved by the following technical solutions:

[0008] The present invention provides a method for obtaining a distribution law of mechanical properties of a tubular structure, the method comprising:

[0009] A fractional-order heat conduction model considering scale effect is established for tubular structures, and a motion model and constitutive model ignoring body force are established;

[0010] The fractional-order heat conduction model, the motion model and the constitutive model are dimensionlessly processed to obtain control equations; the obtained control equations are Laplace transformed and mechanical property distribution results are obtained by solving them;

[0011] The process of establishing the fractional-order heat conduction model is specifically as follows: introducing the scale effect coefficient and the thermal relaxation coefficient into the Fourier heat conduction equation, performing Taylor series expansion on the heat flux component according to the fractional-order time parameter, so as to obtain the fractional-order heat conduction model.

[0012] Furthermore, the assumptions of the fractional-order heat conduction model include that the tubular structure is subjected to a constant-temperature thermal shock on the outer surface and is stress-free, and the inner surface is insulated and constrained to be unable to deform.

[0013] Furthermore, the calculation formula of the fractional-order heat conduction model is:

[0014]

[0015] Where k is the thermal conductivity; θ is the temperature, θ = T-T0, T is the absolute temperature, T0 is the reference temperature; τ0 is the thermal relaxation coefficient; α is the fractional parameter; ρ is the material density; C E is the specific heat coefficient; e is the volume expansion rate, and e=div u, u represents displacement; γ is the thermal conductivity, t is the time, Γ(1+α) is the Gamma function with independent variable α, is the operation operator.

[0016] Further, The calculation formula is:

[0017]

[0018] Here, r represents the radius.

[0019] Furthermore, the calculation formula of the exercise model without considering physical strength is:

[0020]

[0021] Among them, σ rr , represent radial stress and hoop stress respectively.

[0022] Furthermore, the calculation formula of the constitutive model is:

[0023]

[0024] Wherein, ξ is the scale effect coefficient of the tubular structure; σ zz is the axial stress; λ and μ are the Lame coefficients; r is the radius; u is the displacement; θ is the temperature; γ is the thermal conductivity; and e is the volume expansion rate.

[0025] Furthermore, dimensionless boundary conditions are obtained after dimensionless processing, and coefficients A1, A2, B1, and B2 are obtained according to the dimensionless boundary conditions. The coefficients A1, A2, B1, and B2 satisfy the following relationship:

[0026]

[0027] Where I1 is the first-order modified Bessel function of the first kind;

[0028]

[0029] Among them, k 1,2 is the eigenvalue; λ1, λ2, A and ε represent the formula on the right side of the equation with symbols λ1, λ2, A and ε respectively.

[0030] Furthermore, the mechanical properties results obtained based on the inverse Laplace transform include displacement field, temperature field and stress field, as well as the distribution laws of the displacement field, temperature field and stress field.

[0031] Furthermore, the numerical inverse transformation of the mechanical property distribution result includes:

[0032]

[0033] Where 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 summation operations, and c n is the symbolic notation, and the calculation formula is:

[0034]

[0035] Wherein, k is an integer.

[0036] Furthermore, the distribution law of the displacement field u(r, t), temperature field θ(r, t) and stress field σ(r, t) is:

[0037]

[0038] Where, t is time; They represent the fields of u, θ, σ in the Laplace transform domain respectively; r is the radius; n is the number of iterations.

[0039] Compared with the prior art, the present invention has the following advantages and beneficial effects:

[0040] The present invention aims at a tubular structure whose outer surface is subjected to constant temperature thermal shock and stress is free, and whose inner surface is insulated and constrained to be unable to deform. By establishing a fractional-order heat conduction model and solving the Laplace transform of the control equation, the accuracy of the solution results of the distribution law of the mechanical properties of the tubular structure can be improved. The thermoelastic behavior of micro-nano-scale micro-nano pipes is calculated and analyzed through non-local effects and fractional-order thermoelasticity theory; the mechanical behavior analysis is carried out in combination with the fractional-order thermoelastic model, and the analysis of the influence of fractional-order parameters on each field is considered to obtain the distribution law of the displacement field, temperature field, and stress field of the real field, which is helpful for material design and experimental comparison. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 A schematic diagram of a flow chart of a method for obtaining the distribution law of mechanical properties of a tubular structure;

[0042] Figure 2 is a distribution diagram of displacement changes along r corresponding to different fractional order parameters in Example 1;

[0043] Figure 3 is a distribution diagram of temperature change along r corresponding to different fractional order parameters in Example 1;

[0044] Figure 4 is a distribution diagram of radial stress along r corresponding to different fractional order parameters in Example 1;

[0045] Figure 5 is a distribution diagram of the hoop stress along r corresponding to different fractional order parameters in Example 1;

[0046] Figure 6 This is a distribution diagram of the axial stress along r corresponding to different fractional order parameters in Example 1. DETAILED DESCRIPTION

[0047] The specific implementation methods of the present invention are described in detail below through examples. These examples are implemented on the premise of the scheme described in the present invention, and provide detailed implementation methods and specific operating processes, but the protection scope of the present invention is not limited to the following examples.

[0048] The present invention is further described below in conjunction with the accompanying drawings and specific embodiments. Any features such as component models, material names, connection structures, preparation methods, materials, structures or composition ratios not clearly described in this technical solution are deemed to be common technical features disclosed in the prior art.

[0049] Example 1

[0050] This embodiment provides a method for obtaining the distribution law of mechanical properties of a tubular structure. Figure 1 As shown, a fractional-order heat conduction model is established for the tubular structure, and a motion model and a constitutive model that ignore body force are established;

[0051] The process of establishing the fractional-order heat conduction model is as follows: introducing the scale effect coefficient and the thermal relaxation coefficient into the Fourier heat conduction equation, performing Taylor series expansion on the heat flux component according to the fractional-order time parameter, so as to obtain the fractional-order heat conduction model. The calculation formula of the fractional-order heat conduction model used in this embodiment is:

[0052]

[0053] Where k is the thermal conductivity; θ is the temperature, θ = T-T0, T is the absolute temperature, T0 is the reference temperature; τ0 is the thermal relaxation coefficient; α is the fractional parameter; ρ is the material density; C E is the specific heat coefficient; e is the volume expansion rate, and e=div u, u represents displacement; γ is the thermal conductivity, t is the time, Γ(1+α) is the Gamma function with independent variable α, is the operation operator;

[0054] The calculation formula is:

[0055]

[0056] Here, r represents the radius.

[0057] The calculation formula of the exercise model without considering physical strength is:

[0058]

[0059] Among them, σ rr , represent radial stress and hoop stress respectively.

[0060] The calculation formula of the constitutive model is:

[0061]

[0062]

[0063] Wherein, ξ is the scale effect coefficient of the tubular structure; σ zz is the axial stress; λ and μ are the Lame coefficients; r is the radius; u is the displacement; θ is the temperature; γ is the thermal conductivity; and e is the volume expansion rate.

[0064] Performing dimensionless processing on the fractional-order heat conduction model, the motion model and the constitutive model to obtain a control equation;

[0065] After the dimensionless processing, dimensionless boundary conditions are obtained, and coefficients A1, A2, B1, B2 are obtained according to the dimensionless boundary conditions. The coefficients A1, A2, B1, B2 satisfy the following relationship:

[0066]

[0067] Where I1 is the first-order modified Bessel function of the first kind.

[0068]

[0069] Among them, k 1,2 is the eigenvalue; λ1, λ2, A and ε represent the formula on the right side of the equation with symbols λ1, λ2, A and ε respectively.

[0070] The specific calculation and derivation process of formula (1) to formula (5) is as follows:

[0071] The boundary conditions (6) and initial conditions (7) are:

[0072]

[0073] The following dimensionless quantities are introduced in the calculation process:

[0074]

[0075] For the convenience of description, the * sign on the upper right of each physical quantity is omitted, and the corresponding dimensionless control equation can be obtained as follows:

[0076]

[0077] Among them, β=λ / (λ+2μ), ε=γ 2 T0 / [ρC E (λ+2μ)].

[0078] By performing Laplace transformation on formula (9) to formula (13) and rearranging them, we can obtain:

[0079]

[0080] in, They represent the fields of u, θ, σ in the Laplace transform domain respectively, and s is the field variable corresponding to time t.

[0081] Combining (14) and (15) eliminates Available

[0082]

[0083] in,

[0084] Similarly, combining (14) and (15) eliminates We can get:

[0085]

[0086] (20) and (21) can be combined and organized as

[0087]

[0088] Where k1 and k2 are given by the characteristic equation k 4 -λ1k 2 +λ2=0 is determined and satisfies

[0089]

[0090] Then the solution of formula (22) is:

[0091]

[0092] Among them, A i ,B i ,A' i ,B' i (i=1,2) are unknown coefficients related to s, I0 and K0 are first-kind zero-order modified Bessel functions and second-kind zero-order modified Bessel functions, respectively.

[0093] Combining (24), (25) and (15), we can obtain

[0094]

[0095] Substituting (26) into (25), we get

[0096]

[0097] From formula (24), we can get:

[0098]

[0099] make Combining (16) to (18), we can get

[0100]

[0101] Combining equations (19) and (27) to (30), we get

[0102]

[0103] Combining (33) to (36) we can find A i ,B i (i=1,2).

[0104] Combined with the above derivation and calculation results, refer to Figure 1 , the obtained control equation is subjected to Laplace transform, and the mechanical performance results obtained based on inverse Laplace transform include displacement field, temperature field and stress field, as well as the distribution law of displacement field, temperature field and stress field.

[0105] The numerical inverse transformation of the mechanical property distribution result includes:

[0106]

[0107] Where 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 summation operations, and c n is the symbolic notation, and the calculation formula is:

[0108]

[0109] Wherein, k is an integer.

[0110] The distribution law of the displacement field u(r, t), temperature field θ(r, t) and stress field σ(r, t) is:

[0111]

[0112]

[0113] Where, t is time; They represent the fields of u, θ, σ in the Laplace transform domain respectively; r is the radius; n is the number of iterations.

[0114] The distribution law of the mechanical properties of the tubular structure in this embodiment is obtained for the tubular structure whose outer boundary is subjected to constant temperature thermal shock and stress is free, and the inner surface is insulated and constrained to be unable to deform. The result is numerically solved through Laplace transformation, boundary conditions and Laplace numerical inverse transformation, and the result is more accurate, which solves the problem that the mechanical behavior of the tubular structure in actual engineering is difficult to characterize by experiment. The above embodiment of the present application is based on the new fractional-order generalized thermoelasticity theory proposed by Ezzat, combined with scale effect and Laplace transformation, boundary conditions and Laplace numerical inverse transformation method, and focuses on analyzing and discussing the distribution law of mechanical properties of the tubular structure with fractional-order parameters when the outer boundary is subjected to constant temperature thermal shock and stress is free, and the inner surface is insulated and constrained to be unable to deform.

[0115] on the other hand, Figure 2 to Figure 6 The figure shows the distribution law of displacement, temperature and stress corresponding to different fractional order parameters after adopting the above method for obtaining the distribution law of mechanical properties of tubular structures. For example, in an engineering scenario, the method described in Example 1 is verified by actual data, where the material parameter is λ = 7.76 × 10 10 Nm -2 , μ=3.86×10 10 Nm -2 ,ρ=8954kg / m 3 , C E =383.1Jkg -1 K -1 , T0=293K, ξ=0.1, τ0=0.02, α t =1.78×10 -5 K -1 , t=0.15, a=2, b=1, and assume N=20 in the inverse transform.

[0116] like Figure 2 As shown in , 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 absolute temperature θ with radius r under different fractional order parameters is as follows Figure 3 As shown in , the temperature value reaches the maximum value at the outer boundary and is always 1, which is consistent with the boundary conditions. The distribution law of dimensionless stress σ with radius r under different fractional order parameters is as follows Figures 4 to 6 As shown. Figure 4It can be seen that at the outer boundary, the stress is 0, which is consistent with the boundary condition. Therefore, the fractional-order heat conduction model used in the above embodiment is consistent with the actual working conditions. Under the premise of considering the scale effect and the time fractional order, it can improve the accuracy of the solution results of the mechanical distribution law of the tubular structure. It can be applied in engineering fields, such as the nuclear industry, the battery field, and the micro-nanostructure semiconductor field, and can be extended to scenarios such as material design and engineering experiments.

[0117] The above description of the embodiments is to facilitate the understanding and use of the invention by those skilled in the art. It is obvious that those skilled in the art can easily make various modifications to these embodiments and apply the general principles described herein to other embodiments without creative work. Therefore, the present invention is not limited to the above embodiments, and 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 present invention should be within the scope of protection of the present invention.

Claims

1. A method for obtaining the distribution law of mechanical properties of a tubular structure, characterized in that: The method comprises: A fractional-order heat conduction model considering scale effect is established for tubular structures, and a motion model and constitutive model ignoring body force are established; The fractional-order heat conduction model, the motion model and the constitutive model are dimensionlessly processed to obtain control equations; the obtained control equations are Laplace transformed and mechanical property distribution results are obtained by solving them; The process of establishing the fractional-order heat conduction model is specifically as follows: introducing the scale effect coefficient and the thermal relaxation coefficient into the Fourier heat conduction equation, performing Taylor series expansion on the heat flux component according to the fractional-order time parameter, so as to obtain the fractional-order heat conduction model.

2. A method for obtaining the distribution law of mechanical properties of a tubular structure according to claim 1, characterized in that: The assumptions of the fractional-order heat conduction model include that the tubular structure is subjected to a constant-temperature thermal shock on the outer surface and is stress-free, and the inner surface is insulated and constrained from deformation.

3. The method for obtaining the distribution law of mechanical properties of a tubular structure according to claim 1, characterized in that: The calculation formula of the fractional-order heat conduction model is: Where k is the thermal conductivity; θ is the temperature, θ = T-T0, T is the absolute temperature, T0 is the reference temperature; τ0 is the thermal relaxation coefficient; α is the fractional parameter; ρ is the material density; C E is the specific heat coefficient; e is the volume expansion rate, and e=div u, u represents displacement; γ is the thermal conductivity, t is the time, Γ(1+α) is the Gamma function with independent variable α, is the operation operator.

4. A method for obtaining the distribution law of mechanical properties of a tubular structure according to claim 3, characterized in that: The calculation formula is: Here, r represents the radius.

5. The method for obtaining the distribution law of mechanical properties of a tubular structure according to claim 1, characterized in that: The calculation formula of the exercise model without considering physical strength is: Among them, σ rr , represent radial stress and hoop stress respectively.

6. A method for obtaining the distribution law of mechanical properties of a tubular structure according to claim 1, characterized in that: The calculation formula of the constitutive model is: Wherein, ξ is the scale effect coefficient of the tubular structure; σ zz is the axial stress; λ and μ are the Lame coefficients; r is the radius; u is the displacement; θ is the temperature; γ is the thermal conductivity; and e is the volume expansion rate.

7. The method for obtaining the distribution law of mechanical properties of a tubular structure according to claim 1, characterized in that: After the dimensionless processing, dimensionless boundary conditions are obtained, and coefficients A1, A2, B1, B2 are obtained according to the dimensionless boundary conditions. The coefficients A1, A2, B1, B2 satisfy the following relationship: Where I1 is the first-order modified Bessel function of the first kind; Among them, k 1,2 is the eigenvalue; λ1, λ2, A and ε represent the formula on the right side of the equation with symbols λ1, λ2, A and ε respectively.

8. The method for obtaining the distribution law of mechanical properties of a tubular structure according to claim 1, characterized in that: The mechanical properties results obtained based on the inverse Laplace transform include displacement field, temperature field and stress field, as well as the distribution laws of displacement field, temperature field and stress field.

9. The method for obtaining the distribution law of mechanical properties of a tubular structure according to claim 1, characterized in that: The numerical inverse transformation of the mechanical property distribution result includes: Where 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 summation operations, and c n Indicated by the symbol c n The formula on the right side of the equation is: Wherein, k is an integer.

10. The method for obtaining the distribution law of mechanical properties of a tubular structure according to claim 8, characterized in that: The distribution law of the displacement field u(r, t), temperature field θ(r, t) and stress field σ(r, t) is: Where, t is time; They represent the fields of u, θ, σ in the Laplace transform domain respectively; r is the radius; n is the number of iterations.

Images