A method for determining the volume average Eshelby tensor of arbitrary shaped inclusions in particle reinforced composites

Abstract

The present invention relates to a method for determining a volume average Eshelby tensor of an inclusion of arbitrary shape in a particle reinforced composite material. The method comprises the following steps: step 1, establishing a two-phase microscopic equivalent geometric model of a particle reinforced composite material: the microscopic equivalent geometric model comprises a cubic matrix phase and particles of arbitrary shape as two phases of an inclusion phase, wherein the inclusion phase is located at the exact center of the cubic phase; step 2, establishing a quantitative correlation mechanism between a volume average Eshelby tensor and a volume average stress and strain of each phase: in an existing microscopic equivalent geometric model, the volume average Eshelby tensor is derived according to the Eshelby equivalent inclusion theory and the Mori-Tanaka inclusion random distribution hypothesis; step 3, obtaining a volume average Eshelby tensor by reverse analysis: using a finite element numerical method to calculate the volume average stress and strain in the inclusion phase and the matrix phase in the two-phase microscopic equivalent geometric model of the particle reinforced composite material, and obtaining a volume average Eshelby tensor of an inclusion of arbitrary shape by reverse analysis.

Description

Technical Field

[0001] The invention discloses a method for measuring the volume average Eshelby tensor of inclusions of arbitrary shapes in a particle-reinforced composite material, and belongs to the technical field of composite material mechanical behavior research. Background Art

[0002] Particle-reinforced composites are widely used in aerospace, shipbuilding, automobile and other engineering applications because they can accurately improve the overall effective performance characteristics of metals, polymers, cementitious materials, ceramics and other materials. Accurately predicting their effective mechanical properties is of great significance for actual production. Starting from the most basic effective elastic properties, the homogenization method based on the Eshelby ellipsoid inclusion theory is often used for evaluation and prediction. However, the inclusions in real situations, such as aggregates, pores, cracks, etc., are complex and changeable in shape and cannot be simply approximated as ellipsoid shapes for calculation. At the same time, when the particle inclusion shape in the particle-reinforced composite is non-ellipsoidal, the stress and strain in the inclusion are no longer uniform, and the key Eshelby tensor is no longer uniform and there are many singular points in the inclusion. So far, its universal theoretical solution has not been obtained, which greatly limits the observation of the effective performance of particle-reinforced composites.

[0003] The volume average Eshelby tensor eliminates the inhomogeneity within the inclusion on the basis of the Eshelby tensor, and can be perfectly applied to classical homogenization models (such as the Hashin-Strikman limit model, self-consistent model, Mori-Tanaka model, double / multi-inclusion model and other classical mesomechanics). Therefore, the volume average Eshelby tensor is regarded as one of the simplest and most effective means to study the effective performance of complex-shaped inclusions. At present, the solution methods for this complex integral can be divided into two categories. One is that for certain specific inclusions, such as hyperspheres and hyperellipsoids, the volume average Eshelby tensor is simplified and numerically solved due to factors such as symmetry in the integral region. This method is limited to specific inclusion shapes and cannot be extended to other arbitrary shapes. The other is to directly numerically solve the volume average Eshelby tensor in the entire inclusion region, convert the three-dimensional volume integral into the boundary surface integral, and obtain it by interpolation after removing the neighborhood of the singular point. The number of integrals on the boundary surface increases in direct proportion to the total number of Gaussian points. However, in order to ensure accuracy, a sufficient number of integral points is required, resulting in a large amount of calculation in the entire integral process, complex operation, too cumbersome, and too low applicability. Therefore, it is urgent to invent a simple and efficient method to obtain the volume average Eshelby tensor containing three-dimensional arbitrary shape inclusions in particle reinforced composite materials. Summary of the invention

[0004] In order to solve the deficiencies of the prior art, the present invention proposes a simple and efficient method for obtaining the volume average Eshelby tensor of complex non-ellipsoidal inclusions in particle reinforced composite materials. The method can effectively solve the volume average Eshelby tensor of inclusions of arbitrary complex shapes, with accurate results, simple operation, and strong usability. Moreover, it can be directly applied to the study of effective elasticity and transmission performance of particle reinforced composite materials containing inclusions of arbitrary complex shapes through various mean field models, and further implemented in engineering applications.

[0005] Technical solution: The technical solution adopted by the present invention to solve the above technical problems is: a method for obtaining the average Eshelby tensor of the inclusion volume of any shape, the method comprising the following steps:

[0006] Step 1) establishing a microscopic equivalent geometric model of a two-phase particle reinforced composite material including a cubic matrix phase and an inclusion phase of arbitrary shape;

[0007] Step 2) Based on Eshelby classical solution, Eshelby effective inclusion theory and Mori-Tanaka inclusion random distribution theory, a quantitative correlation mechanism expression between the volume average Eshelby tensor and the volume average stress and strain of each phase is established;

[0008] Step 3) The volume average stress and strain in the inclusion phase and the matrix phase in the microscopic equivalent geometric model of the two-phase particle reinforced composite material are calculated using the finite element numerical method, and the volume average Eshelby tensor of inclusions of arbitrary shapes is obtained through inverse analysis.

[0009] Furthermore, the specific method of step 1) is as follows: the microscopic equivalent geometric model of the two-phase particle reinforced composite material is composed of a composite of two-phase materials, a matrix phase and an inclusion phase. The matrix phase is considered to be a cubic unit D′ with a finite volume, in which a single particle of arbitrary shape is enclosed as an inclusion phase Ω′. The inclusion phase is located at the center of the cubic matrix. Both phases are isotropic homogeneous materials, which together constitute a two-phase microscopic equivalent geometric model.

[0010] Further, the specific method of step 2) is as follows:

[0011] Step 2.1) Consider a cubic unit D′, denote a subdomain of arbitrary shape inside as Ω and apply a uniform plastic strain ε* to the subdomain. Then, considering the uniform stress boundary condition σ 0 Under the condition of , the composition of the volume average stress and strain of the subdomain Ω and its outer region D′-Ω is calculated:

[0012]

[0013]

[0014] Among them, V Ω and V D′-Ω are the volumes of the subdomain Ω and its outer region D′-Ω respectively; σ′ ij (x) and ε′ ij (x) is the component expression of stress and strain in the cubic unit D′ at this time. The left side of the equation in formula (1) is the integration of the stress and strain about the position point x in the integration area Ω, and then divided by the volume of the corresponding integration domain. The right side of the equation is: σ 0 and ε 0 is due to the uniform stress boundary condition σ 0 Directly causes the volume average stress and strain in the region Ω, <σ ∞ > Ω and <ε ∞ > Ω is the stress and strain part of the Eshelby classical solution after volume averaging in the region Ω, <σ I > Ω and <ε I > Ω It is the stress and strain part of the difference different from the classical solution after volume averaging in the region Ω; the left side of the equation in formula (2) is the integration of the stress and strain about the position point x in the integration region D′-Ω, and then divided by the volume of the corresponding integration domain. The right side of the equation consists of three parts: σ 0 and ε 0 is due to the uniform stress boundary condition σ 0 Directly causes the volume average stress and strain in the region D′-Ω, <σ ∞ > D′-Ω and <ε ∞ > D′-Ω is the stress and strain part of the Eshelby classical solution after volume averaging in the region D′-Ω, <σ I > D′-Ω and <ε I > D′-Ω It is the stress and strain part of the difference from the classical solution after volume averaging in the region D′-Ω;

[0015] Step 2.2) Replace the material in the subdomain Ω in step 2.1) with particles of another material, recorded as inclusion phase Ω′. According to the above derivation and Eshelby equivalent inclusion theory, calculate the volume average stress <σ in the inclusion phase (1) > and strain <ε (1) >:

[0016]

[0017] Among them, C (1) and C (0)are the stiffness tensors of the inclusion phase material and the matrix phase material, respectively;

[0018] Step 2.3) Introduce the Mori-Tanaka random distribution inclusion theory and obtain the volume average stress <σ when a new inclusion Ω′ is introduced at any position in the matrix phase newp > Ω′ and strain <ε newp > Ω′ The stress of the original inclusion phase <σ (1) > and strain <ε (1) >same, expressed as:

[0019]

[0020] In the above formula, <σ (0) > is the volume average stress of the matrix phase, <ε (0) > is the volume average strain of the matrix phase, which is obtained by formula (2), that is:

[0021]

[0022] In formula (4), the second part on the right side of the equation <σ ∞ >′ Ω′ and <ε ∞ >′ Ω′ is the volume averaged perturbation stress and strain caused by the new inclusion Ω′, which is expressed by the volume averaged Eshelby tensor <s>The relationship is obtained as:

[0023] <σ ∞ >' Ω' =C (0) :(<ε ∞ >' Ω' -ε ** )=C (0) :( <s>:ε ** -ε ** ) (6)

[0024] Among them, the equivalent eigenstrain ε** is different from the specific value of ε* in formulas (1) and (3), and is distinguished by **;

[0025] Step 2.4) Combining the above three steps, the quantitative correlation mechanism expression of the volume average Eshelby tensor and the volume average stress and strain of each phase is obtained as follows:

[0026] <s>:[(<ε (1) >-<ε (0) >)-F (0) :(<σ (1) >-<σ (0) >)]=<ε (1) >-<ε (0) > (7)

[0027] Among them, F (0) Represents the flexibility tensor of the matrix phase.

[0028] Further, the specific method of step 3) is as follows:

[0029] Step 3.1) Establish the required two-phase particle reinforced composite microstructure model, and numerically calculate the stress and strain under the action of six directions, three axial pressures of x, y, and z and shear stresses of x, y, and z planes, and their orthogonal loads;

[0030] Step 3.2) For each set of stress-strain results under the action of each load, take the product of the stress-strain at the centroid of each unit in the same phase and the unit volume, add them together, and then divide them by the total volume of the phase to obtain the volume average stress of the inclusion phase <σ (1) > strain < ε (1) >Results and matrix phase volume average stress <σ (0) > strain < ε (0) >Results:

[0031]

[0032] Among them, V j represents the volume of the jth unit, the inclusion phase has N units in total, and the matrix phase has M units in total, and represents the stress-strain value at the centroid of the jth unit in the matrix phase, and represents the stress and strain values ​​at the centroid of the jth unit in the matrix phase;

[0033] Step 3.3) Each set of volume averaged stress or strain results in the previous step is converted into a 6-row column vector by the Voigt method. The 6 sets of stress or strain results are then combined into a 6×6 matrix and brought into formula (7). After inverse analysis, the volume averaged Eshelby tensor is obtained: <s>The matrix expression of is as follows:

[0034]

[0035] in, 1111 >、 1122 >、 1133 >、 1123 >、 1113 >、 1112 >、 2211 >、 2222 >、 2233 >、 2223 >、 2213 >、 2212 >、 3311 >、 3322 >、 3333 >、 3323 >、 3313 >、 3312 >、 2311 >、 2322 >、 2333 >、 2323 >、 2313 >、 1312 >、 1311 >、 1322 >、 1333 >、 1323 >、 1313 >、 1312 >、 1211 >、 1222 >、 1233 >、 1223 >、 1213 >、 1212 > are the non-zero components of the fourth-order volume average Eshelby tensor, denoted by four subscripts.

[0036] Beneficial effects: Compared with the prior art, the technical solution of the present invention has the following beneficial effects:

[0037] 1. The present invention obtains the volume average Eshelby tensor through inverse analysis through the quantitative correlation mechanism of the volume average stress strain of each phase and the volume average Eshelby tensor in the microstructure model. Compared with the numerical method directly solving from the integral definition formula, it avoids complex integral operations and singular integral points, is easy to operate, and has high calculation efficiency.

[0038] 2. Through comparative verification, it is confirmed that this method has good accuracy; it is suitable for solving arbitrary two-dimensional and three-dimensional shapes and has wide applicability.​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​

[0039] 3. The present invention can be directly applied to the homogenization model to further solve the effective performance of composite materials reinforced with particles of arbitrary shapes.

[0040] 4. This method obtains the volume average Eshelby tensor of inclusions of arbitrary shapes through inverse analysis based on the quantitative correlation mechanism between the volume average Eshelby tensor and the volume average stress and strain of each phase material, avoiding the solution of complex integrals in arbitrary shape regions and the resulting singularities. This method is simple to operate and has accurate results, and can be extended to the precise prediction of the effective physical and mechanical properties of composite materials containing inclusions of arbitrary shapes. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] Figure 1 This is a flow chart of the present invention patent;

[0042] Figure 2 Schematic diagram of the mesoscopic structural model of a finite volume matrix phase containing a single inclusion phase of arbitrary shape;

[0043] Figure 3 Schematic diagram of the evolution of the mesostructure model, including two sub-figures (a) and (b). Figure (a) is a schematic diagram of the first stage, including a cubic matrix D′ and an arbitrary-shaped internal subdomain Ω, and Figure (b) is a hypothetical two-phase equivalent mesostructure model, including a cubic matrix D′ and an arbitrary-shaped inclusion phase Ω′.

[0044] Figure 4 It is a schematic diagram of multiple inclusions to a single inclusion;

[0045] Figure 5 It is a comparison diagram between the results of the present invention and the finite element results, including two sub-diagrams (a) and (b), wherein Figure (a) is a comparison diagram of finite elastic parameters, and Figure (b) is a comparison diagram of the results of effective transmission coefficients. DETAILED DESCRIPTION

[0046] The technical solution of the present invention is now further described in conjunction with the accompanying drawings and embodiments (the following embodiments are merely illustrative and non-restrictive, and the protection scope of the present invention cannot be limited thereto).

[0047] This example is dedicated to demonstrating a method for determining the volume average Eshelby tensor of inclusions of arbitrary shapes in a particle reinforced composite material. Figure 1 First, a two-phase particle reinforced composite microstructure model containing a single inclusion of arbitrary shape in a finite volume matrix is ​​set, as shown in Figure 2 As shown; then determine the volume average stress and strain composition of each phase of the matrix phase and the particle inclusion phase, establish the quantitative correlation mechanism between the volume average Eshelby tensor and the volume average stress and strain of each phase material in the microstructure of this particle reinforced composite material; finally, combine the numerical simulation method or software to directly calculate the volume average stress and strain of each phase material in the microstructure model of the two-phase particle reinforced composite material, and obtain the volume average Eshelby tensor of inclusions of arbitrary shapes through inverse analysis. Specifically, it includes the following steps:

[0048] 1. Establish a mesostructure model of a two-phase particle-reinforced composite material containing a cubic matrix phase and an inclusion phase of arbitrary shape:

[0049] The microstructure model of two-phase particle reinforced composite materials is composed of two-phase materials: matrix phase and particle inclusion phase. Figure 2 As shown, the matrix phase is considered to be a cubic unit D′ with a finite volume, in which a single particle of arbitrary shape is enclosed as the inclusion phase Ω′. The inclusion phase is located in the center of the cubic matrix. Both phase materials are isotropic homogeneous materials, which together constitute a two-phase microscopic equivalent geometric model.

[0050] 2. Establish the quantitative correlation mechanism expression between the volume average Eshelby tensor and the volume average stress and strain of each phase:

[0051] Step 2.1) Consider a cubic unit D′, denote a subdomain of arbitrary shape inside as Ω and apply a uniform plastic strain (or eigenstrain) ε* to the subdomain, such as Figure 3 As shown in (a) in . Then, considering the uniform stress boundary condition σ 0 Under the condition of , the composition of the volume average stress and strain of the subdomain Ω and its outer region D′-Ω is calculated:

[0052]

[0053]

[0054] Among them, V Ω and V D′-Ω are the volumes of the subdomain Ω and its outer region D′-Ω respectively; σ′ ij (x) and ε′ ij (x) is the stress and strain in the cubic unit D′ at this time. The left side of the equation in formula (1) is the integration of the stress and strain about the position point x in the integration area Ω, and then divided by the volume of the corresponding integration domain. The right side of the equation is: σ 0 and ε 0 is due to the uniform stress boundary condition σ 0 Directly causes the volume average stress and strain in the region Ω, <σ ∞ > Ω and <ε ∞ > Ω is the stress and strain part of the Eshelby classical solution after volume averaging in the region Ω, <σ I > Ω and <ε I > Ω It is the stress and strain part of the difference different from the classical solution after volume averaging in the region Ω; the left side of the equation in formula (2) is the integration of the stress and strain about the position point x in the integration region D′-Ω, and then divided by the volume of the corresponding integration domain. The right side of the equation consists of three parts: σ 0 and ε 0 is due to the uniform stress boundary condition σ 0 Directly causes the volume average stress and strain in the region D′-Ω, <σ ∞ > D′-Ω and <ε ∞ > D′-Ω is the stress and strain part of the Eshelby classical solution after volume averaging in the region D′-Ω, <σ I > D′-Ω and <ε I > D′-Ω It is the stress and strain part of the difference from the classical solution after volume averaging in the region D′-Ω;

[0055] Step 2.2) Replace the material in the subdomain Ω in the previous step with another particle material, recorded as the inclusion phase Ω′, such as Figure 3 According to the above derivation and Eshelby equivalent inclusion theory, the volume average stress in the inclusion phase is calculated as <σ (1) > and strain <ε (1) >:

[0056]

[0057] Among them, C (1) and C (0) are the stiffness tensors of the inclusion phase material and the matrix phase material, respectively.

[0058] Step 2.3) Introduce the Mori-Tanaka random distribution inclusion theory (consider that the inclusions Ω′ with sufficient number and small size are randomly and uniformly distributed in the matrix phase D′, such as Figure 4 As shown in Figure 2), the volume average stress <σ is obtained when a new inclusion Ω′ is introduced at any position in the matrix phase. newp > Ω′ and strain <ε newp > Ω′ The stress of the original inclusion phase <σ (1) > and strain <ε (1) > is the same and can be expressed as:

[0059]

[0060] In the above formula, <σ (0) > is the volume average stress of the matrix phase, <ε (0) > is the volume average strain of the matrix phase, which can be obtained by formula (11), that is:

[0061]

[0062] The second part of the right side of the equation in formula (13) <σ ∞ >′ Ω′ and <ε ∞ >′ Ω′ is the volume average perturbation stress and strain caused by the new inclusion Ω′, which can be expressed by the volume average Eshelby tensor <s>The relationship is obtained as:

[0063] <σ ∞ >' Ω' =C (0) :(<ε ∞ >' Ω' -ε ** )=C (0) :( <s>:ε ** -ε ** ) (15)

[0064] The equivalent eigenstrain ε** is different from the specific value of ε* in formulas (10) and (12) and is distinguished by **.

[0065] Step 2.4) Combining the above three steps, the quantitative correlation mechanism expression of the volume average Eshelby tensor and the volume average stress and strain of each phase is calculated as follows:

[0066] <s>:[(<ε (1) >-<ε (0) >)-F (0) :(<σ (1) >-<σ (0) >)]=<ε (1) >-<ε (0) > (16)

[0067] Among them, F (0) Represents the flexibility tensor of the matrix phase.

[0068] 3. Inverse analysis to obtain the volume average Eshelby tensor of inclusions of arbitrary shapes:

[0069] Step 3.1) Establish the required two-phase microstructure model, and use numerical methods to calculate the stress and strain under the action of orthogonal loads in six directions (three axial pressures of x, y, and z and shear stresses of x, y, and z planes).

[0070] Step 3.2) For each set of stress-strain results under the action of each load, take the product of the stress-strain at the centroid of each unit in the same phase and the unit volume, add them together, and then divide them by the total volume of the phase to obtain the volume average stress of the inclusion phase <σ (1) > strain < ε (1) >Results and matrix phase volume average stress <σ (0) > strain < ε (0) >Results:

[0071]

[0072] Among them, V j represents the volume of the jth unit, the inclusion phase has N units in total, and the matrix phase has M units in total, and represents the stress-strain value at the centroid of the jth unit in the matrix phase, and Represents the stress and strain value at the centroid of the jth unit in the matrix phase.

[0073] Step 3.3) Each set of volume averaged stress (or strain) results in the previous step is converted into a 6-row column vector by the Voigt method. The 6 sets of stress (or strain) results will be combined into a 6×6 matrix and brought into formula (16). After inverse analysis, the matrix expression of the volume averaged Eshelby tensor can be obtained as follows:

[0074]

[0075] in, 1111 >、 1122 >、 1133 ​​​>、 1123 >、 1113 >、 1112 >、 2211 >、 2222 >、 2233 >、 2223 >、 2213 >、 2212 >、 3311 >、 3322 >、 3333 >、 3323 >、 3313 >、 3312 >、 2311 >、 2322 >、 2333 >、 2323 >、 2313 >、 1312 >、 1311 >、 1322 >、 1333 >、 1323 >、 1313 >、 1312 >、 1211 >、 1222 >、 1233 >、 1223 >、 1213 >、 1212 > are all non-zero components of the fourth-order volume-averaged Eshelby tensor, denoted collectively by four subscripts.

[0076] 4. Example verification:

[0077] In order to verify the accuracy of this method, we adopted two approaches to verify. One is to compare the volume average Eshelby tensor of individual complex shape inclusions calculated by direct numerical integration, and the other is to use the results of this method into the homogenization model to calculate the effective performance of the composite material and compare it with the direct finite element calculation results. Table 1 shows the comparison results of the volume average Eshelby tensor components of 2 hyperspheres, 2 donuts and 2 helical rods when the matrix Poisson's ratio is 0, 0.3 and 0.4 respectively, as shown below:

[0078] Table 1. Comparison of volume averaged Eshelby tensor results

[0079]

[0080]

[0081] Next, we will show the verification results of another approach.​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​ Figure 5 (a) Figure 5 (b) shows the comparison between the effective elastic properties and effective transmission properties of the two-phase particle reinforced composite material obtained by directly bringing the volume average Eshelby tensor obtained by this method into the homogenization model and the results of direct finite element calculation. The inclusion shape is a 20-sided non-convex body evolved from a regular icosahedron. Among them, for the elastic behavior part ( Figure 5 In (a), the Young's modulus of the inclusion phase E is selected (1) and Poisson's ratio ν (1) , Young's modulus E of the matrix phase (0) and Poisson's ratio ν (0) They are: E (1) =80Gpa, E (0) =2Gpa and ν (1) =0.2, ν (0) = 0.3, and the normalized effective bulk modulus K is obtained eff / K (0) , shear modulus μ eff / μ (0) The result diagram of the change of the volume fraction of the inclusion phase f; for the transport behavior part ( Figure 5 In (b)), the transmission coefficients of the inclusion phase and the matrix phase are selected as 0 and 1 respectively, and the normalized effective transmission coefficient D is obtained. eff / D (0) Result diagram of the change with the volume fraction f of the inclusion phase.

[0082] 5. Method application guide:

[0083] Once the inclusion shape is determined, first follow Figure 2 A two-phase model is established in the form of a cube (the cube size is greater than 4 times the maximum semi-axis size of the inclusion or the volume fraction of a single inclusion is guaranteed to be less than 1%), the matrix material is guaranteed to be consistent with the requirements when set, and the stress and strain under 6 loads are calculated; secondly, the volume average stress and strain of each phase are obtained by post-processing; finally, the volume average Eshelby tensor of the shape under the Poisson's ratio is obtained by substituting it into formula (16). In summary, the method proposed in this patent can ensure sufficient accuracy, is simple to operate, and has good promotion value.< / s> < / s> < / s> < / s> < / s> < / s> < / s>

Claims

1. A method for determining the volume average Eshelby tensor of inclusions of arbitrary shapes in a particle reinforced composite material, characterized in that: The method comprises the following steps: Step 1) establishing a two-phase mesoscopic equivalent geometric model of a particle reinforced composite material including a cubic matrix phase and an inclusion phase of arbitrary shape; Step 2) Based on Eshelby classical solution, Eshelby effective inclusion theory and Mori-Tanaka inclusion random distribution theory, a quantitative correlation mechanism expression between the volume average Eshelby tensor and the volume average stress and strain of each phase is established; Step 3) using the finite element numerical method to calculate the volume average stress and strain in the inclusion phase and the matrix phase in the micro-equivalent geometric model, and obtaining the volume average Eshelby tensor of inclusions of arbitrary shapes through inverse analysis; The specific method of step 2) is as follows: Step 2.1) Consider a cubic unit D′, denote a subdomain of arbitrary shape inside as Ω, and apply a uniform plastic strain ε* to the subdomain. Then, considering the uniform stress boundary condition σ 0 Under the condition of , the composition of the volume average stress and strain of the subdomain Ω and its outer region D′-Ω is calculated: Among them, V Ω and V D′-Ω are the volumes of the subdomain Ω and its outer region D′-Ω respectively; σ′ ij (x) and ε′ ij (x) is the component expression of stress and strain in the cubic unit D′ at this time. The left side of the equation in formula (1) is the integration of the stress and strain about the position point x in the integration area Ω, and then divided by the volume of the corresponding integration domain. The right side of the equation is: σ 0 and ε 0 is due to the uniform stress boundary condition σ 0 Directly causes the volume average stress and strain in the region Ω, <σ ∞ > Ω and <ε ∞ > Ω is the stress and strain part of the Eshelby classical solution after volume averaging in the region Ω, <σ I > Ω and <ε I > Ω It is the stress and strain part of the difference different from the classical solution after volume averaging in the region Ω; the left side of the equation in formula (2) is the integration of the stress and strain about the position point x in the integration region D′-Ω, and then divided by the volume of the corresponding integration domain. The right side of the equation consists of three parts: σ 0 and ε 0 is due to the uniform stress boundary condition σ 0 Directly causes the volume average stress and strain in the region D′-Ω, <σ ∞ > D′-Ω and <ε ∞ > D′-Ω is the stress and strain part of the Eshelby classical solution after volume averaging in the region D′-Ω, <σ I > D′-Ω and <ε I > D′-Ω It is the stress and strain part of the difference from the classical solution after volume averaging in the region D′-Ω; Step 2.2) Replace the material in the subdomain Ω in step 2.1) with another granular material, recorded as the inclusion phase Ω′. According to the above derivation and Eshelby equivalent inclusion theory, calculate the volume average stress <σ in the inclusion phase (1) > and strain <ε (1) >: Among them, C (1) and C (0) are the stiffness tensors of the inclusion phase material and the matrix phase material, respectively; Step 2.3) Introduce the Mori-Tanaka random distribution inclusion theory and obtain the volume average stress <σ when a new inclusion Ω′ is introduced at any position in the matrix phase newp > Ω′ and strain <ε newp > Ω′ The stress of the original inclusion phase <σ (1) > and strain <ε (1) >same, expressed as: In the above formula, <σ (0) > is the volume average stress of the matrix phase, <ε (0) > is the volume average strain of the matrix phase, which is obtained by formula (2), that is: In formula (4), the second part on the right side of the equation <σ ∞ >′ Ω′ and <ε ∞ >′ Ω′ is the volume-averaged perturbation stress-strain due to the new inclusion Ω′, which is expressed by the volume-averaged Eshelby tensor <s> The relationship is obtained as:< / s> <s> <s ∞ >' Ω' =C (0) :(<e ∞ >' Ω' -e**)=C (0) :( <s> :ε**-ε**)(6)< / s> <s> Among them, the equivalent eigenstrain ε** is different from the specific value of ε* in formulas (1) and (3), and is distinguished by **; Step 2.4) Combining the above three steps, the quantitative correlation mechanism expression of the volume average Eshelby tensor and the volume average stress and strain of each phase is obtained as follows: <s>:[(<ε (1) >-<ε (0) >)-F( 0) :(<σ (1) >-<σ (0) >)]=<ε (1) >-<ε (0) > (7)< / s> <s> Among them, F (0) Represents the flexibility tensor of the matrix phase.

2. A method for determining the volume average Eshelby tensor of inclusions of arbitrary shapes in a particle reinforced composite material according to claim 1, characterized in that: The specific method of step 1) is as follows: The two-phase microscopic equivalent geometric model of the particle-reinforced composite material is composed of a composite of two-phase materials, a matrix phase and an inclusion phase. The matrix phase is considered to be a cubic unit D′ with a finite volume, in which a single particle of arbitrary shape is enclosed as an inclusion phase Ω′. The inclusion phase is located at the center of the cubic matrix. Both phases are isotropic uniform materials, which together constitute a two-phase microscopic equivalent geometric model of the particle-reinforced composite material.

3. A method for determining the volume average Eshelby tensor of inclusions of arbitrary shapes in a particle reinforced composite material according to claim 1, characterized in that: The specific method of step 3) is as follows: Step 3.1) Establish the required two-phase microstructure model, and numerically calculate the stress and strain under the three axial pressures in six directions, x, y, and z, and the shear stresses on the three planes x, y, and z, respectively, orthogonal loads; Step 3.2) For each set of stress-strain results under the action of each load, take the product of the stress-strain at the centroid of each unit in the same phase and the unit volume, add them together, and then divide them by the total volume of the phase to obtain the volume average stress of the inclusion phase <σ (1) > strain < ε (1) >Results and matrix phase volume average stress <σ (0) > strain < ε (0) >Results: Among them, V j represents the volume of the jth unit, the inclusion phase has N units in total, and the matrix phase has M units in total, and represents the stress-strain value at the centroid of the jth unit in the matrix phase, and represents the stress and strain values ​​at the centroid of the jth unit in the matrix phase; Step 3.3) Each set of volume averaged stress or strain results in the previous step is converted into a 6-row column vector by the Voigt method. The 6 sets of stress or strain results are then combined into a 6×6 matrix and brought into formula (7). After inverse analysis, the volume averaged Eshelby tensor is obtained: <s> The matrix expression of is as follows:< / s> <s> in, 1111 >、 1122 >、 1133 >、 1123 >、 1113 >、 1112 >、 2211 >、 2222 >、 2233 >、 2223 >、 2213 >、 2212 >、 3311 >、 3322 >、 3333 >、 3323 >、 3313 >、 3312 >、 2311 >、 2322 >、 2333 >、 2323 >、 2313 >、 1312 >、 1311 >、 1322 >、 1333 >、 1323 >、 1313 >、 1312 >、 1211 >、 1222 >、 1233 >、 1223 >、 1213 >、 1212 > are the non-zero components of the fourth-order volume average Eshelby tensor, denoted by four subscripts.​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​ < / s> < / s> < / s> < / s>

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