A method for predicting the conductivity of composite materials based on layer-by-layer differential integration

Abstract

The present application belongs to the field of composite material conductivity prediction, and particularly relates to a composite material conductivity prediction method based on layer-by-layer differential integration, which comprises the following steps: 1) judging the geometric structure of the reinforced phase of the composite material through a scanning electron microscope or X-ray diffraction, simultaneously obtaining the volume fraction of the corresponding geometric structure, simplifying the volume fraction into a corresponding composite structure volume unit, and sending the volume unit to an upper computer; 2) the upper computer uses a layer-by-layer differential-integral method to establish a differential equation about the conductivity for the corresponding composite structure volume unit; 3) the upper computer obtains the material parameters and geometric parameters corresponding to each component in the composite structure volume unit, and obtains the final conductivity of the composite material in combination with the differential equation about the conductivity. The present application establishes a quantitative relationship between the composite material structure and the conductivity, realizes the rapid prediction of the conductivity of the composite material, and thus significantly reduces the experimental amount required in the material process optimization process and the component performance verification process.

Description

Technical Field

[0001] This invention belongs to the field of composite material conductivity prediction, specifically a composite material conductivity prediction method based on layer-by-layer differential-integral. Background Technology

[0002] Material conductivity, commonly including elastic modulus, thermal conductivity, electrical conductivity, magnetic permeability, liquid permeability coefficient, and gas diffusion coefficient, is a physical quantity that evaluates the conductive properties of materials. It is of great significance to both scientific and engineering fields. For example, elastic modulus largely determines the safety of engineering components, while conductivity determines the energy conversion efficiency of electronic components and power transmission metal lines.

[0003] In fact, real-world materials are all multiphase composite structures. Composite materials consist of two or more phases, and the conductivity of a single component phase is usually known. However, due to the complexity of the bonding methods and geometry between the components in composite materials, accurately predicting the conductivity of composite materials remains a challenging problem. Although some simplified theoretical models exist that treat the components in composite materials as being in series or parallel relationships, namely the Voigt parallel approximation and the Reuss series approximation, these models can only provide upper and lower limits for conductivity, not precise values. While some methods can narrow down the conductivity prediction range, they still cannot obtain accurate values. Therefore, current theoretical models still struggle to accurately predict the conductivity of composite materials. Summary of the Invention

[0004] The purpose of this invention is to provide a novel layer-by-layer differential-integral method for rapidly predicting the conductivity of composite materials. This method, based on rigorous theoretical derivation and combined with systematic finite element simulation and experimental verification, establishes a quantitative relationship between the composite structure and conductivity. Using this method, only the conductivity and geometric parameters of each component phase need to be input to quickly and accurately predict the conductivity of the composite material. Compared with existing theoretical models, this method offers higher accuracy and convenience in predicting conductivity.

[0005] The technical solution adopted by this invention to achieve the above objectives is: a method for predicting the conductivity of composite materials based on layer-by-layer differential integration, comprising the following steps:

[0006] 1) Determine the geometric structure of the reinforcing phase of the composite material by scanning electron microscopy or X-ray diffraction, obtain the volume fraction of the corresponding geometric structure, simplify it into the corresponding volume unit of the composite structure, and send it to the host computer.

[0007] 2) For the corresponding composite structural volume element, the host computer establishes a differential equation about the conductivity using the layer-by-layer differential-integral method;

[0008] 3) The host computer obtains the material parameters and geometric parameters corresponding to each component in the volume unit of the composite structure, and obtains the final conductivity of the composite material by combining the differential equation of conductivity.

[0009] Determine the geometric structure of the reinforcing phase in the composite material and simplify it into the corresponding composite structural volume element, specifically:

[0010] The determination of the geometric structure of the reinforcing phase:

[0011] 1-1) When the reinforcing phase consists of randomly and uniformly distributed particles, it is simplified to a discrete composite structure volume element.

[0012] 1-2) When the reinforcing phase is an interwoven strip, it is simplified to an interpenetrating composite structural volume unit;

[0013] 1-3) Obtain the geometric parameters of the composite structure volume element based on the volume fraction of the reinforcing phase.

[0014] In step 1-1), when the reinforcing phase consists of randomly and uniformly distributed particles, it is simplified to a discrete composite structural volume unit, specifically:

[0015] Take any composite structure with a similar geometry in a representative volume unit of composite material, and set its conductivity to P. Add an infinitesimally thin layer dm to it, and its conductivity will also increase by an increment dP. Therefore, the conductivity of the composite structure after increasing the thickness is P+dP.

[0016] Establishing the relationship between the thickness dm and the increment dP, and setting P1 = P2, the differential equation of the discrete composite structure volume element is expressed as:

[0017]

[0018] Where P is the conductivity of the composite material, P1 and P2 are the conductivity of the reinforcing phase and the matrix phase respectively, a is the size of the reinforcing phase, m is the size of the matrix in the solution process, and A is the size of the matrix; P is a function of m, and the conductivity of the composite material can be obtained by setting m = A, where A is the size of the matrix.

[0019] In step 1-1), when the reinforcing phase is an interwoven strip, it is simplified to an interpenetrating composite structural volume unit, specifically:

[0020] Take any composite structure with a similar geometry in a representative volume unit of composite material, and set its conductivity to P. Add an infinitesimally thin layer dm to it, and its conductivity will also increase by an increment dP. Therefore, the conductivity of the composite structure after increasing the thickness is P+dP.

[0021] Establishing the relationship between the thickness layer dm and the incremental dP, the volume element of the interpenetrating composite structure is:

[0022]

[0023] Where P is the conductivity of the composite material, P1 and P2 are the conductivity of the reinforcing phase and the matrix phase respectively, a is the size of the reinforcing phase, m is the size of the matrix in the solution process, and A is the size of the matrix; P is a function of m, and the conductivity of the composite material can be obtained by setting m = A, where A is the size of the matrix.

[0024] In steps 1-3), obtaining the geometric parameters of the representative volume element of the composite structure based on the volume fraction of the reinforcing phase specifically involves:

[0025] (1) Obtain the volume fractions of the matrix and the reinforcing phase using scanning electron microscopy or X-ray diffraction, and set them as f1 and f2, respectively;

[0026] (2) Based on the geometric relationships, the geometric parameters A and a in the representative volume element are deduced, that is:

[0027] f1 = 1 - a 3 / A 3

[0028] f2=a 3 / A 3

[0029] Where a is the size of the reinforcing phase and A is the size of the matrix.

[0030] In step 3), the material parameters corresponding to each component in the composite structural volume element are obtained, specifically as follows:

[0031] The material parameters for the conductivity of the corresponding single-phase material can be found in relevant physical property manuals or literature.

[0032] In step 3), the conductivity includes any one of the following: elastic modulus, thermal conductivity, electrical conductivity, magnetic permeability, permeability coefficient, and expansion coefficient.

[0033] This also includes: verifying the predicted results of the theoretical conductivity, specifically:

[0034] Conduct conductivity tests to obtain the experimental conductivity, and compare the experimentally measured conductivity results with the conductivity results predicted by theoretical methods.

[0035] The present invention has the following beneficial effects and advantages:

[0036] 1. This invention can effectively solve the problem of frequently needing to adjust the volume fraction and geometric structure of the component phases. It only requires collecting the conductivity data of each component phase of the composite material and analyzing the geometric characteristics of the reinforcing phase to establish a quantitative relationship between the composite structure and conductivity. No additional experimental steps are required; only the governing equations need to be solved.

[0037] 2. This invention establishes a quantitative relationship between the structure and conductivity of composite materials, enabling rapid prediction of the conductivity of composite materials, thereby significantly reducing the amount of experiments required in the material process optimization and component performance verification process.

[0038] 3. By using the principles of calculus to establish a differential equation for conductivity, the conductivity of different composite structures can be predicted quickly and accurately through numerical solution of this equation. Attached Figure Description

[0039] Figure 1 The flowchart of the composite material conductivity prediction method based on layer-by-layer differential-integral is shown in the present invention.

[0040] Figure 2a The interpenetrating composite structural volume unit of the present invention;

[0041] Figure 2b The discrete composite structural volume unit of the present invention;

[0042] Figure 3a This invention compares the theoretical equation prediction and finite element simulation of discrete composite structures.

[0043] Figure 3b Comparison of theoretical equation prediction and finite element simulation of the interpenetrating composite structure of this invention;

[0044] Figure 4 Scanning electron microscope (SEM) characterization images of the composite material morphology provided in this embodiment of the invention;

[0045] Figure 5 A comparison chart of theoretical prediction results and experimental test results from embodiments of the present invention. Detailed Implementation

[0046] The present invention will now be described in further detail with reference to the accompanying drawings and embodiments.

[0047] like Figure 1 The diagram shown is a flowchart of a method for predicting the conductivity of composite materials based on layer-by-layer differential integration according to the present invention. The present invention includes the following steps:

[0048] 1) Determine the geometric structure of the reinforcing phase of the composite material by scanning electron microscopy or X-ray diffraction, obtain the volume fraction of the corresponding geometric structure, simplify it into the corresponding volume unit of the composite structure, and send it to the host computer.

[0049] 2) For the corresponding composite structural volume element, the host computer establishes a differential equation about the conductivity using the layer-by-layer differential-integral method;

[0050] 3) The host computer obtains the material parameters and geometric parameters corresponding to each component in the volume unit of the composite structure, and obtains the final conductivity of the composite material by combining the differential equation of conductivity.

[0051] Determine the geometric structure of the reinforcing phase in the composite material and simplify it into the corresponding composite structural volume element, specifically:

[0052] The determination of the geometric structure of the reinforcing phase:

[0053] 1-1) When the reinforcing phase consists of randomly and uniformly distributed particles, it is simplified to a discrete composite structure volume element.

[0054] 1-2) When the reinforcing phase is an interwoven strip, it is simplified to an interpenetrating composite structural volume unit;

[0055] 1-3) Obtain the geometric parameters of the composite structure volume element based on the volume fraction of the reinforcing phase.

[0056] like Figures 2a-2b As shown, this invention employs the principles of calculus, differentiating layer by layer from the inside out to establish the relationship between the thickness increment dm and the conductivity increment dP, thus obtaining a differential equation regarding conductivity. Starting with the innermost yellow structure, which contains only a single phase, its conductivity is set as P0. Figure 2b Then, using the series-parallel model of conductivity, the conductivity can be calculated layer by layer outwards until the outermost layer is reached. To simplify this process and improve computational accuracy and efficiency, the concept of calculus can be utilized. During the layer-by-layer calculation, an arbitrary position is selected, for example... Figure 2b The red box in the image represents the conductivity, which is set to P. Based on this, an infinitesimally thin layer dm is added, i.e. Figure 1 The blue box in the diagram represents the conductivity, which will also have an increment dP. Therefore, the conductivity of the blue box is P+dP. This establishes the relationship between dP and dm. Ignoring higher-order infinitesimals, we can obtain the differential equation for conductivity, i.e., Equation 1. The above describes the conductivity calculation process for an interpenetrating structure. For discrete structures, Equation 1 can be directly derived. Substituting P1 = P2 into the differential equation and keeping the initial conditions of Equation 1 unchanged, i.e., P(a) = P1, Equation 1 can be transformed into Equation 2, which is the governing equation for the discrete composite structure.

[0057] This method connects complex composite structures using a series-parallel model, which is based on the assumptions of equal load gradient and equal mass flow density. However, for interpenetrating and discrete composite structures, the load gradient and mass flow density distribution near the reinforcing phase are not uniform, thus the series-parallel model cannot be directly applied. This method connects one infinitesimal layer at a time, thus essentially guaranteeing the assumptions of equal load gradient and mass flow density. Calculation results show that this approach converges to a single value, namely the conductivity of the composite structure.

[0058] like Figure 2b As shown, this is the discrete composite structure volume unit of the present invention. In step 1-1), when the reinforcing phase is randomly and uniformly distributed particles, it is simplified to a discrete composite structure volume unit, specifically:

[0059] The differential equation for a discrete composite structural volume element is as follows:

[0060]

[0061] Where P is the conductivity of the composite material, P1 and P2 are the conductivity of the reinforcing phase and the matrix phase respectively, a is the size of the reinforcing phase, m is the size of the matrix in the solution process, and A is the size of the matrix; P is a function of m, and the conductivity of the composite material can be obtained by setting m = A, where A is the size of the matrix.

[0062] For the aforementioned first-order ordinary differential equations, there are already many mature algorithms and software programs that can easily solve them numerically. This method only requires consulting the conductivity of each component phase of the composite material and the geometric characteristics of the composite structure. A differential equation solving program can then be written using software such as Mathematica, MATLAB, or Python to quickly predict the conductivity of the composite material.

[0063] In step 1-1), when the reinforcing phase is an interwoven strip, it is simplified to an interpenetrating composite structural volume unit, specifically:

[0064] First, using an interpenetrating composite structure as a typical case, the main steps and basic principles of this method for predicting conductivity are explained. This method uses the principles of calculus to establish a differential equation concerning conductivity. This equation consists of a first-order ordinary differential equation and initial conditions, as follows: Figure 2a As shown, the interpenetrating composite structural volume element of the present invention is expressed by the differential equation as follows:

[0065]

[0066] Where P is the conductivity of the composite material, P1 and P2 are the conductivity of the reinforcing phase and the matrix phase respectively, a is the size of the reinforcing phase, m is the size of the matrix in the solution process, and A is the size of the matrix; P is a function of m, and the conductivity of the composite material can be obtained by setting m = A, where A is the size of the matrix.

[0067] In steps 1-3), obtaining the geometric parameters of the representative volume element of the composite structure based on the volume fraction of the reinforcing phase specifically involves:

[0068] (1) Obtain the volume fractions of the matrix and the reinforcing phase using scanning electron microscopy or X-ray diffraction, and set them as f1 and f2, respectively;

[0069] (2) Based on the geometric relationships, the geometric parameters A and a in the representative volume element are deduced, that is:

[0070] f1 = 1 - a 3 / A 3

[0071] f2=a 3 / A 3

[0072] Where a is the size of the reinforcing phase and A is the size of the matrix.

[0073] In step 3), the material parameters corresponding to each component in the composite structural volume element are obtained, specifically as follows:

[0074] The material parameters for the conductivity of the corresponding single-phase material can be found in relevant physical property manuals or literature.

[0075] In step 3), the conductivity includes any one of the following: elastic modulus, thermal conductivity, electrical conductivity, magnetic permeability, permeability coefficient, and expansion coefficient.

[0076] like Figures 3a-3b As shown, the present invention also includes: verifying the predicted results of the theoretical conductivity, specifically:

[0077] Conduct conductivity tests to obtain the experimental conductivity, and compare the experimentally measured conductivity results with the conductivity results predicted by theoretical methods.

[0078] In this invention, the predicted results of the governing equations for the conductivity of the composite structure are in high agreement with the finite element simulation results. To verify the effectiveness of the layer-by-layer differential-integral method, we conducted a systematic finite element simulation, considering two different composite structures: discrete and interpenetrating composite structures. The comparative results are presented in [the table / document / etc.]. Figure 3a and Figure 3bThe results show that the conductivity (including elastic modulus, thermal conductivity, electrical conductivity, magnetic permeability, permeability coefficient, and expansion coefficient) predicted by the layer-by-layer differential-integral method is consistent with the finite element simulation results, demonstrating the effectiveness of this theoretical method. Furthermore, predicting conductivity using the governing equations is quick and simple; for different composite structures, only material parameters need to be changed. In contrast, finite element simulation requires rebuilding the model each time, a cumbersome process. Therefore, this theoretical method significantly accelerates the prediction speed and reduces computational costs.

[0079] Example:

[0080] This embodiment uses the prediction and experimental verification of the conductivity of iron-based titanium carbide particle-reinforced composite materials;

[0081] (1) Materials

[0082] Iron-based titanium carbide particle-reinforced composite materials were rolled into three composite materials with titanium carbide volume fractions of 20%, 40%, and 60%, respectively.

[0083] (2) Process

[0084] Step 1: Analyze the structure of the material's reinforcing phase using scanning electron microscopy, such as... Figure 4 The image shown is a scanning electron microscope (SEM) characterization image of the composite material morphology given in an embodiment of the present invention. It can be seen that the reinforcing phase is randomly and uniformly dispersed in the matrix in a discrete form.

[0085] Step 2: Find relevant data to obtain the elastic modulus, thermal conductivity, electrical conductivity, and magnetic permeability of iron and titanium carbide.

[0086] Step 3: Substitute the conductivity of iron and titanium carbide into the governing equation for the conductivity of the discrete composite structure, as follows:

[0087]

[0088] Where P is the conductivity of the composite material, P1 and P2 are the conductivity of the reinforcing phase and the matrix phase, respectively, and a is the size of the reinforcing phase. P is a function of m, and the conductivity of the composite material can be obtained by setting m = A, where A is the size of the matrix.

[0089] Step 4: Solve the differential equations using Mathematica to obtain the conductivity of the iron-based titanium carbide composite material.

[0090] Step 5: Conduct conductivity tests to obtain the elastic modulus, thermal conductivity, electrical conductivity, and magnetic permeability.

[0091] Step 6: Compare the experimentally measured conductivity results with the theoretically predicted results. The comparison results are as follows: Figure 5As shown in the figure, (a) represents the elastic modulus; (b) represents the thermal conductivity; (c) represents the electrical conductivity; and (d) represents the magnetic permeability.

[0092] The above description is merely an embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, extensions, etc., made within the spirit and principles of the present invention are included within the scope of protection of the present invention.

Claims

1. A method for predicting the conductivity of composite materials based on layer-by-layer differential-integral methods, characterized in that, Includes the following steps: 1) Determine the geometric structure of the reinforcing phase of the composite material by scanning electron microscopy or X-ray diffraction, obtain the volume fraction of the corresponding geometric structure, simplify it into the corresponding volume unit of the composite structure, and send it to the host computer; When the reinforcing phase consists of randomly and uniformly distributed particles, it simplifies to a discrete composite volumetric unit structure, specifically: Take any composite structure with a similar geometry in a representative volume unit of composite material, and set its conductivity to P. Add an infinitesimally thin layer dm to it, and its conductivity will also increase by an increment dP. Therefore, the conductivity of the composite structure after increasing the thickness is P+dP. Establish the relationship between the thickness layer dm and the incremental dP, and... The differential equation for a discrete composite structural volume element is expressed as: ； Where P is the conductivity of the composite material, P1 and P2 are the conductivity of the reinforcing phase and the matrix phase respectively, a is the size of the reinforcing phase, m is the size of the matrix in the solution process, and A is the size of the matrix; P is a function of m, and the conductivity of the composite material can be obtained by setting m=A, where A is the size of the matrix. When the reinforcing phase consists of interwoven strips, it simplifies to an interpenetrating composite structural volumetric unit, specifically: Take any composite structure with a similar geometry in a representative volume unit of composite material, and set its conductivity to P. Add an infinitesimally thin layer dm to it, and its conductivity will also increase by an increment dP. Therefore, the conductivity of the composite structure after increasing the thickness is P+dP. Establishing the relationship between the thickness layer dm and the incremental dP, the volume element of the interpenetrating composite structure is: ； Where P is the conductivity of the composite material, P1 and P2 are the conductivity of the reinforcing phase and the matrix phase respectively, a is the size of the reinforcing phase, m is the size of the matrix in the solution process, and A is the size of the matrix; P is a function of m, and the conductivity of the composite material can be obtained by setting m=A, where A is the size of the matrix. 2) For the corresponding composite structural volume element, the host computer establishes a differential equation about the conductivity using the layer-by-layer differential-integral method; 3) The host computer obtains the material parameters and geometric parameters corresponding to each component in the volume unit of the composite structure, and obtains the final conductivity of the composite material by combining the differential equation of conductivity.

2. The method for predicting the conductivity of composite materials based on layer-by-layer differential-integral transformation according to claim 1, characterized in that, Determine the geometric structure of the reinforcing phase in the composite material and simplify it into the corresponding composite structural volume element, specifically: The determination of the geometric structure of the reinforcing phase: 1-1) When the reinforcing phase consists of randomly and uniformly distributed particles, it simplifies to a discrete composite structure volume element. 1-2) When the reinforcing phase is an interwoven strip, it is simplified to an interpenetrating composite structural volume unit; 1-3) Obtain the geometric parameters of the composite structure volume element based on the volume fraction of the reinforcing phase.

3. The method for predicting the conductivity of composite materials based on layer-by-layer differential-integral transformation according to claim 1, characterized in that, In steps 1-3), obtaining the geometric parameters of the representative volume element of the composite structure based on the volume fraction of the reinforcing phase specifically involves: (1) The volume fractions of the matrix and the reinforcing phase are obtained by scanning electron microscopy or X-ray diffraction and are set as f1 and f2, respectively; (2) Based on the geometric relationship, the geometric parameters A and a in the representative volume element are deduced, that is: f1=1-a 3 / THE 3 ; f2=a 3 / A 3 ; Where a is the size of the reinforcing phase and A is the size of the matrix.

4. The method for predicting the conductivity of composite materials based on layer-by-layer differential-integral transformation according to claim 1, characterized in that, In step 3), the material parameters corresponding to each component in the composite structural volume element are obtained, specifically as follows: The material parameters for the conductivity of the corresponding single-phase material can be found in relevant physical property manuals or literature.

5. The method for predicting the conductivity of composite materials based on layer-by-layer differential-integral transformation according to claim 1, characterized in that, In step 3), the conductivity includes any one of the following: elastic modulus, thermal conductivity, electrical conductivity, magnetic permeability, permeability coefficient, and expansion coefficient.

6. The method for predicting the conductivity of composite materials based on layer-by-layer differential-integral transformation according to claim 1, characterized in that, This also includes: verifying the predicted results of the theoretical conductivity, specifically: Conduct conductivity tests to obtain the experimental conductivity, and compare the experimentally measured conductivity results with the conductivity results predicted by theoretical methods.

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