Method for calculating equivalent thermal expansion coefficient of porous composite material based on energy homogenization

Abstract

The invention relates to a porous composite material equivalent thermal expansion coefficient calculation method based on energy homogenization, and belongs to the technical field of calculation materials, and the method comprises the following steps: S1, establishing an energy homogenization theoretical basis for calculating an equivalent thermal expansion coefficient; s2, deducing a finite element equation of the equivalent thermal stress coefficient based on energy homogenization based on the theoretical basis of energy homogenization for calculating the equivalent thermal stress coefficient established in the step S1; and S3, solving the equivalent thermal expansion coefficient of the porous composite material based on the finite element equation of the equivalent thermal stress coefficient based on energy homogenization deduced in the step S2. According to the new method provided by the invention, the obtained equivalent thermal expansion coefficient is under the double-precision condition of 15-bit effective digits, and the same effective digits of the calculation result reach 12-13 digits; and under the grid density of 100 * 100-1000 * 1000, the calculation efficiency is improved by about 60%-70%.

Description

Technical Field

[0001] The invention relates to the technical field of computational materials, and in particular to a method for calculating equivalent thermal expansion coefficient of porous composite materials based on energy homogenization. Background Art

[0002] Porous composite materials exhibit excellent mechanical properties due to their ultra-light weight, high specific strength, high specific stiffness and excellent energy absorption capacity. In addition, the unique properties of porous composite materials in terms of sound absorption, shock absorption, heat dissipation, electromagnetic shielding and permeability make them a high-performance multifunctional engineering structural material with both structural and functional functions. These excellent properties make porous composite materials show broad application prospects in aerospace, national defense, military, transportation and energy.

[0003] In practical applications, porous composite materials are often exposed to extreme thermal environments, such as thermal protection systems of hypersonic vehicles, heat exchangers in the chemical industry, and tail nozzles of aircraft engines. Under high temperature conditions, composite materials will produce thermal strain, which poses a threat to the service safety of the structure. Therefore, in order to accurately evaluate the safety of service structures, it is of great significance to determine the thermal expansion coefficient of porous composite materials. In order to solve this problem, a number of invention patents have provided corresponding measurement and calculation methods. For example, the invention patent with publication number CN113155891A proposes a method for determining the thermal expansion coefficient of fine aggregate, the invention patent with publication number CN101482526A discloses a method for determining the thermal expansion coefficient of early-age concrete, the invention patent with publication number CN112629491A provides a method for calculating the thermal expansion coefficient of a composite material plate based on an optical fiber strain conversion matrix, and the invention patent with publication number CN111488698A proposes a multi-scale prediction method for the thermal expansion coefficient of hardened concrete. Although these methods can achieve the calculation of thermal expansion coefficient through complex experimental equipment and monitoring data, they require a lot of equipment, manpower and time costs during implementation, which makes it difficult to meet the requirements of high efficiency in engineering practice.

[0004] In order to further reduce costs and improve efficiency, the use of numerical methods that do not require complex experimental preparation for simulation calculations is often a realistic and feasible first choice. In view of this, Andresen, E., and CS Andresen (Reference 1) disclosed a calculation procedure for the equivalent thermal expansion coefficient of composite materials based on the homogenization theory; Shang Shipeng and Wu Jinwei respectively proposed a homogenization calculation method for the equivalent thermal expansion coefficient using a finite element platform; the invention patent with publication number CN107451309A discloses a method for multi-scale calculation of the equivalent thermal expansion coefficient of complex composite structures. However, the asymptotic homogenization theory based on which the above-mentioned method for calculating the equivalent thermal expansion coefficient of composite materials is based has the disadvantage of low computational efficiency compared to energy homogenization. Due to the imperfection of the energy homogenization theory, there is currently no method for calculating the thermal expansion coefficient based on energy homogenization. Therefore, it is urgent to develop an energy homogenization method for calculating the equivalent thermal expansion coefficient with higher computational efficiency to meet the efficiency requirements of the industry for the numerical calculation of the thermal expansion coefficient of porous composite materials. Summary of the invention

[0005] In view of the deficiencies of the prior art, an object of the present invention is to provide a method for calculating the equivalent thermal expansion coefficient of a porous composite material based on energy homogenization, which method has higher calculation efficiency without losing calculation accuracy.

[0006] To achieve the above object, the present invention adopts the following technical solution: A method for calculating the equivalent thermal expansion coefficient of a porous composite material based on energy homogenization comprises the following steps: S1. Establish the energy homogenization theoretical basis for calculating equivalent thermal stress coefficient; S2, based on the energy homogenization theoretical basis for calculating the equivalent thermal stress coefficient established in step S1, deriving a finite element equation of the equivalent thermal stress coefficient based on energy homogenization; S3. Solving the equivalent thermal expansion coefficient of the porous composite material based on the finite element equation of the equivalent thermal stress coefficient based on energy homogenization derived in step S2.

[0007] Preferably, in step S1, a specific method for establishing a theoretical basis for energy homogenization for calculating the equivalent thermal stress coefficient includes: S101. Establish the constitutive equation of equivalent thermal stress coefficient based on energy homogenization as follows: ; In the formula, is the equivalent thermal stress coefficient based on energy homogenization, Y is the domain of the representative volume element of the porous composite material, E pars is the local variation elasticity tensor, is a set of strain field components resulting from the unit test strain applied directly to the periodic boundary conditions, is the thermal strain under free thermal expansion conditions; S102, rewrite the constitutive equation of equivalent thermal stress coefficient based on energy homogenization; The thermal strain under free thermal expansion conditions is further expressed as: ; In the formula, α rs is the thermal expansion coefficient of the material, Δ T is the unit temperature field; Based on the thermal strain formula under free thermal expansion conditions, the constitutive equation of the equivalent thermal stress coefficient based on energy homogenization is rewritten to obtain the following rewritten constitutive equation of the equivalent thermal stress coefficient based on energy homogenization: ; Complete the establishment of the energy homogenization theoretical basis for calculating the equivalent thermal stress coefficient.

[0008] Preferably, in step S2, based on the energy homogenization theoretical basis for calculating the equivalent thermal stress coefficient established in step S1, a specific method for deriving a finite element equation of the equivalent thermal stress coefficient based on energy homogenization includes: S201. In finite element analysis, the coefficient of thermal expansion of the representative material α rs The discretization is performed as follows: ; In the formula, α e is the thermal expansion coefficient of the discrete unit; Ф is the formal matrix, which is expressed in a 2D structure as , which is expressed in 3D as ; N T is the heat transfer shape function; T e is a constant temperature field and is constant in the 2D structure under linear unit conditions. , which is always ; S202, the thermal coupling matrix of the structural unit is as follows: ; In the formula, H e is the unit thermal coupling matrix, Y e is the unit domain, B is the strain matrix, B T is the transposed matrix of the strain matrix, and D is the elasticity matrix; S203, constructing a finite element equation of an equivalent thermal stress coefficient based on energy homogenization; Combining the discretization formula of the thermal expansion coefficient representing the material and the unit thermal-mechanical coupling matrix formula, the rewritten constitutive equation of the equivalent thermal stress coefficient based on energy homogenization is discretized using finite element analysis, and the finite element equation of the equivalent thermal stress coefficient based on energy homogenization is obtained as follows: ; In the formula, β H is the equivalent thermal stress coefficient, yes The corresponding displacement solution is, e is a unit pointer variable, N is the total number of cells.

[0009] Preferably, in step S3, based on the finite element equation of the equivalent thermal stress coefficient based on energy homogenization derived in step S2, a specific method for solving the equivalent thermal expansion coefficient of the porous composite material includes: S301. Establish the equivalent elastic property tensor constitutive equation based on energy homogenization as follows: ; In the formula, is the equivalent elastic property tensor, is another set of strain field components generated by the unit test strain directly applied on the periodic boundary conditions; S302. Discretize the equivalent elastic property tensor constitutive equation based on energy homogenization by finite element analysis, and obtain the equivalent elastic property equation as follows: ; In the formula, E H is the equivalent elastic property, yes The corresponding displacement solution, K e is the element stiffness matrix; S303, solving the equivalent thermal expansion coefficient; Using the finite element equation of the equivalent thermal stress coefficient based on energy homogenization and the equivalent elastic property equation, the equivalent thermal expansion coefficient solved by energy homogenization is obtained as follows: ; In the formula, α H is the equivalent thermal expansion coefficient solved using energy homogenization.

[0010] Compared with the prior art, the present invention has the following beneficial effects: The equivalent thermal expansion coefficient obtained by the new method proposed in the present invention is the same as the result of the method of document 1 mentioned in the background technology; under the double precision condition of 15 significant digits, the same significant digits of the calculated result reach 12-13 digits; Through the new method proposed in the present invention, the calculation efficiency of the equivalent thermal expansion coefficient is greatly improved; at a grid density of 100×100-1000×1000, the calculation efficiency is improved by about 60%-70% compared with the method of document 1 mentioned in the background technology. BRIEF DESCRIPTION OF THE DRAWINGS

[0011] In order to more clearly illustrate the technical solutions in the present invention or the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, for ordinary technicians in this field, other drawings can be obtained based on the drawings in the following description without paying any creative work.

[0012] Figure 1 The figure is a schematic diagram of a method flow of an embodiment of the present invention.

[0013] Figure 2 This is the representative volume unit model of Experimental Example 1 of the present invention.

[0014] Figure 3 This is the representative unit model of Experimental Example 2 of the present invention.

[0015] Figure 4 This is the representative unit model of Experimental Example 3 of the present invention.

[0016] Markings in the figure: 1, material one; 2, material two. DETAILED DESCRIPTION

[0017] In order to make the purpose, technical solutions and advantages of the present invention clearer, the technical solutions in the present invention will be clearly and completely described below in conjunction with the drawings in the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, rather than all of the embodiments. Based on the embodiments in the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention. In order to make the above-mentioned features and advantages of the present invention more obvious and easy to understand, the following embodiments are specifically cited and described in detail with the drawings.

[0018] like Figure 1 As shown, an embodiment of the present invention provides a method for calculating the equivalent thermal expansion coefficient of a porous composite material based on energy homogenization, comprising the following steps: S1. Establish the energy homogenization theoretical basis for calculating equivalent thermal stress coefficient; S2, based on the energy homogenization theoretical basis for calculating the equivalent thermal stress coefficient established in step S1, deriving a finite element equation of the equivalent thermal stress coefficient based on energy homogenization; S3. Solving the equivalent thermal expansion coefficient of the porous composite material based on the finite element equation of the equivalent thermal stress coefficient based on energy homogenization derived in step S2.

[0019] In this embodiment, in step S1, a specific method for establishing a theoretical basis for energy homogenization for calculating equivalent thermal stress coefficients includes: S101. Establish the constitutive equation of equivalent thermal stress coefficient based on energy homogenization as follows: ; In the formula, is the equivalent thermal stress coefficient based on energy homogenization, Y is the domain of the representative volume element of the porous composite material, E pars is the local variation elasticity tensor, is a set of strain field components resulting from the unit test strain applied directly to the periodic boundary conditions, is the thermal strain under free thermal expansion conditions; S102, rewrite the constitutive equation of equivalent thermal stress coefficient based on energy homogenization; The thermal strain under free thermal expansion conditions is further expressed as: ; In the formula, α rs is the thermal expansion coefficient of the material, Δ T is the unit temperature field; Based on the thermal strain formula under free thermal expansion conditions, the constitutive equation of the equivalent thermal stress coefficient based on energy homogenization is rewritten to obtain the following rewritten constitutive equation of the equivalent thermal stress coefficient based on energy homogenization: ; Complete the establishment of the energy homogenization theoretical basis for calculating the equivalent thermal stress coefficient.

[0020] In this embodiment, in step S2, based on the energy homogenization theoretical basis for calculating the equivalent thermal stress coefficient established in step S1, a specific method for deriving a finite element equation of the equivalent thermal stress coefficient based on energy homogenization includes: S201. In finite element analysis, the coefficient of thermal expansion of the representative material α r The discretization is performed as follows: ; In the formula, α e is the thermal expansion coefficient of the discrete unit; Ф is the formal matrix, which is expressed in a 2D structure as , which is expressed in 3D as ; N T is the heat transfer shape function; T e is a constant temperature field and is constant in the 2D structure under linear unit conditions. , which is always ; S202, the thermal coupling matrix of the structural unit is as follows: ; In the formula, H e is the unit thermal coupling matrix, Y e is the unit domain, B is the strain matrix, B T is the transposed matrix of the strain matrix, and D is the elasticity matrix; S203, constructing a finite element equation of an equivalent thermal stress coefficient based on energy homogenization; Combining the discretization formula of the thermal expansion coefficient representing the material and the unit thermal-mechanical coupling matrix formula, the rewritten constitutive equation of the equivalent thermal stress coefficient based on energy homogenization is discretized using finite element analysis, and the finite element equation of the equivalent thermal stress coefficient based on energy homogenization is obtained as follows: ; In the formula, β H is the equivalent thermal stress coefficient, yes The corresponding displacement solution is, e is a unit pointer variable, N is the total number of cells.

[0021] In this embodiment, in step S3, based on the finite element equation of the equivalent thermal stress coefficient based on energy homogenization derived in step S2, a specific method for solving the equivalent thermal expansion coefficient of the porous composite material includes: S301. Establish the equivalent elastic property tensor constitutive equation based on energy homogenization as follows: ; In the formula, is the equivalent elastic property tensor, is another set of strain field components generated by the unit test strain directly applied on the periodic boundary conditions; S302. Discretize the equivalent elastic property tensor constitutive equation based on energy homogenization by finite element analysis, and obtain the equivalent elastic property equation as follows: ; In the formula, E H is the equivalent elastic property, yes The corresponding displacement solution, K e is the element stiffness matrix; S303, solving the equivalent thermal expansion coefficient; Using the finite element equation of the equivalent thermal stress coefficient based on energy homogenization and the equivalent elastic property equation, the equivalent thermal expansion coefficient solved by energy homogenization is obtained as follows: ; In the formula, α H is the equivalent thermal expansion coefficient solved using energy homogenization.

[0022] Experimental Example 1: Figure 2 As shown in the figure, the geometric model of the representative unit of the porous composite material is a square with a side length of 1 mm. The distributed material 2 in the model is a square concentric with the entire representative unit with a side length of 0.6 mm, and the rest is distributed with material 1. The Young's modulus of material 1 is 70 GPa, the Poisson's ratio is 0.3, and the thermal expansion coefficient is 235×10 -7 / ℃, the Young's modulus of material 2 is 200Gpa, the Poisson's ratio is 0.3, and the thermal expansion coefficient is 125×10 -7 / ℃. The entire representative volume unit is divided into 100×100 square first-order linear grids in the finite element calculation.

[0023] Based on steps S1-S3 of the method of the present invention, the equivalent thermal expansion coefficient under the conditions of Experimental Example 1 is calculated under the framework of energy homogenization, and the obtained results are compared with the results of Document 1 as shown in Table 1.

[0024] Table 1 Equivalent thermal expansion coefficient α of Experimental Example 1 and Reference 1 H Comparative analysis of calculations:

[0025] Experimental Example 2: Figure 3 As shown in the figure, the geometric model of the representative unit of the porous composite material is a square with a side length of 1 mm. The center of the model is distributed with a "cross-shaped" material 1, the single wing width is 0.2 mm, and the rest is distributed with material 2. The Young's modulus of material 1 is 40 GPa, the Poisson's ratio is 0.35, and the thermal expansion coefficient is 187×10 -7 / ℃, the Young's modulus of material 2 is 115Gpa, the Poisson's ratio is 0.33, and the thermal expansion coefficient is 172×10 -7 / ℃. The entire representative volume unit is divided into 500×500 square first-order linear grids in the finite element calculation.

[0026] Based on steps S1-S3 of the present invention, the equivalent thermal expansion coefficient under the conditions of Experimental Example 2 is calculated under the framework of energy homogenization. The obtained results are compared with the results of Document 1 as shown in Table 2.

[0027] Table 2 Equivalent thermal expansion coefficient α of Experimental Example 2 and Reference 1 H Comparative analysis of calculations:

[0028] Experimental Example 3: Figure 4 As shown in the figure, the geometric model of the representative unit of the porous composite material is a square with a side length of 1 mm. The center of the model is distributed with a vertical strip of material 1 with a width of 0.4 mm, and the rest is distributed with material 2. The Young's modulus of material 1 is 207 GPa, the Poisson's ratio is 0.29, and the thermal expansion coefficient is 133×10 -7 / ℃, the Young's modulus of material 2 is 102Gpa, the Poisson's ratio is 0.3, and the thermal expansion coefficient is 102×10 -7 / ℃. The entire representative volume unit is divided into 1000×1000 square first-order linear grids in the finite element calculation.

[0029] Based on steps S1-S3 of the present invention, the equivalent thermal expansion coefficient under the conditions of Experimental Example 3 is calculated under the framework of energy homogenization. The obtained results are compared with the results of Document 1 as shown in Table 3: Table 3 Equivalent thermal expansion coefficient α of Experimental Example 3 and Reference 1 H Comparative analysis of calculations

[0030] It can be seen from Experimental Examples 1-3 that the equivalent thermal expansion coefficient obtained by the new method proposed in the present invention is the same as the result of the method in the existing document 1. Under the double precision condition of 15 significant digits, the same number of significant digits of the calculated result reaches 12-13 digits; the calculation efficiency of the equivalent thermal expansion coefficient is greatly improved by the new method proposed in the present invention. Under the grid density of 100×100-1000×1000, the calculation efficiency is improved by about 60%-70% compared with the method in the existing document 1.

[0031] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, rather than to limit it. Although the present invention has been described in detail with reference to the aforementioned embodiments, those skilled in the art should understand that they can still modify the technical solutions described in the aforementioned embodiments, or make equivalent replacements for some of the technical features therein. However, these modifications or replacements do not deviate the essence of the corresponding technical solutions from the spirit and scope of the technical solutions of the embodiments of the present invention.

Claims

1. A method for calculating the equivalent thermal expansion coefficient of porous composite materials based on energy homogenization, characterized in that: The following steps are involved: S1. Establish the energy homogenization theoretical basis for calculating equivalent thermal stress coefficient; S2. Based on the energy homogenization theoretical basis for calculating the equivalent thermal stress coefficient established in step S1, a finite element equation of the equivalent thermal stress coefficient based on energy homogenization is derived, and the finite element equation of the equivalent thermal stress coefficient based on energy homogenization is obtained as follows: ； In the formula, β H is the equivalent thermal stress coefficient, Y is the domain of the representative volume element of the porous composite material, yes The corresponding displacement solution is, is a set of strain field components resulting from the unit test strain applied directly to the periodic boundary conditions, T is the transpose symbol, e is a unit pointer variable, N is the total number of units, H e is the unit thermal coupling matrix, T e is a constant temperature field; S3, based on the finite element equation of the equivalent thermal stress coefficient based on energy homogenization derived in step S2, solving the equivalent thermal expansion coefficient of the porous composite material, and obtaining the equivalent thermal expansion coefficient solved by energy homogenization as follows: ； In the formula, α H is the equivalent thermal expansion coefficient solved using energy homogenization, β H is the equivalent thermal stress coefficient, E H is the equivalent elastic property.

2. The method for calculating equivalent thermal expansion coefficient of porous composite materials based on energy homogenization according to claim 1, characterized in that: In step S1, a specific method for establishing a theoretical basis for energy homogenization for calculating equivalent thermal stress coefficients includes: S101. Establish the constitutive equation of equivalent thermal stress coefficient based on energy homogenization as follows: ； In the formula, is the equivalent thermal stress coefficient based on energy homogenization, Y is the domain of the representative volume element of the porous composite material, E pars is the local variation elasticity tensor, is a set of strain field components resulting from the unit test strain applied directly to the periodic boundary conditions, is the thermal strain under free thermal expansion conditions; S102, rewrite the constitutive equation of equivalent thermal stress coefficient based on energy homogenization; The thermal strain under free thermal expansion conditions is further expressed as: ； In the formula, α rs is the thermal expansion coefficient of the material, Δ T is the unit temperature field; Based on the thermal strain formula under free thermal expansion conditions, the constitutive equation of the equivalent thermal stress coefficient based on energy homogenization is rewritten to obtain the rewritten constitutive equation of the equivalent thermal stress coefficient based on energy homogenization: ； Complete the establishment of the energy homogenization theoretical basis for calculating the equivalent thermal stress coefficient.

3. The method for calculating the equivalent thermal expansion coefficient of porous composite materials based on energy homogenization according to claim 2, characterized in that: In step S2, based on the energy homogenization theoretical basis for calculating the equivalent thermal stress coefficient established in step S1, a specific method for deriving a finite element equation of the equivalent thermal stress coefficient based on energy homogenization includes: S201. In finite element analysis, the coefficient of thermal expansion of the representative material α rs The discretization is performed as follows: ； In the formula, α e is the thermal expansion coefficient of the discrete unit; Ф is the formal matrix, which is expressed in a 2D structure as , which is expressed in 3D structure as ; N T is the heat transfer shape function; T e is a constant temperature field and is constant in the 2D structure under linear unit conditions. , which is always ; S202, the thermal coupling matrix of the structural unit is as follows: ； In the formula, H e is the unit thermal coupling matrix, Y e is the unit domain, B is the strain matrix, B T is the transposed matrix of the strain matrix, and D is the elasticity matrix; S203, constructing a finite element equation of an equivalent thermal stress coefficient based on energy homogenization; Combining the discretization formula representing the thermal expansion coefficient of the material and the unit thermal-mechanical coupling matrix formula, the rewritten constitutive equation of the equivalent thermal stress coefficient based on energy homogenization is discretized using finite element analysis to obtain the finite element equation of the equivalent thermal stress coefficient based on energy homogenization.

4. The method for calculating equivalent thermal expansion coefficient of porous composite materials based on energy homogenization according to claim 3, characterized in that: In step S3, based on the finite element equation of the equivalent thermal stress coefficient based on energy homogenization derived in step S2, a specific method for solving the equivalent thermal expansion coefficient of the porous composite material includes: S301. Establish the equivalent elastic property tensor constitutive equation based on energy homogenization as follows: ； In the formula, is the equivalent elastic property tensor, is another set of strain field components generated by the unit test strain directly applied on the periodic boundary conditions; S302. Discretize the equivalent elastic property tensor constitutive equation based on energy homogenization by finite element analysis, and obtain the equivalent elastic property equation as follows: ； In the formula, E H is the equivalent elastic property, yes The corresponding displacement solution, K e is the element stiffness matrix; S303, solving the equivalent thermal expansion coefficient; By using the finite element equation of the equivalent thermal stress coefficient based on energy homogenization and the equivalent elastic property equation, the equivalent thermal expansion coefficient solved by energy homogenization is obtained.

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