JPH09180002A - Finite element model preparing method for composite materials - Google Patents

Abstract

PROBLEM TO BE SOLVED: To improve micro static stress in composite materials by longitudinally and laterally dividing the rectangle of micro image of composite materials, finding the ratio of area occupied by a 2nd phase material in that divided area and generating a finite element model by quantifying the dispersibility of 2nd phase material corresponding to the relation of average value, standard deviation and number of divided areas. SOLUTION: The rectangle of micro observation image of composite materials of matrix materals 1 and 2nd phase materials 2 is divided into longitudinal (n) × lateral (n) pieces of areas by several kinds of integers (n). In each divided area, the ratio of area occupied by the 2nd phase materials 2 such as fibers or fillers is found the dispersibility of the 2nd phase materials 2 is quantified corresponding to the relation of average value, standard deviation and number (n) of divided areas, and the finite element model depending on this dispersibility is generated. Thus, analytic precision for the miro static stress, thermal stress and transient temperature distribution, etc., can be improved.

Description

Detailed Description of the Invention

[0001]

BACKGROUND OF THE INVENTION 1. Field of the Invention The present invention relates to a method for creating a finite element model of a microscopic structure of a composite material, which includes analysis of microscopic thermal stress inside the composite material, prediction of macroscopic elastic coefficient and Poisson's ratio. Or analyze the microscopic thermal stress inside the composite material, predict the linear expansion coefficient, or analyze the transient temperature distribution during transient thermal conduction inside the composite material, thermal shock stress analysis, thermal conductivity It is used for the purpose of predicting.

[0002]

2. Description of the Related Art As a conventional method of creating a finite element model of a microscopic structure of a composite material, for example, "Advances
in Electronic Packaging 1
992 “Technical Pro
party Estimation of Encap
sulants by Microstructura
l Analysis using F.I. E. FIG. M. As shown in “,” it is general that particles of the second phase material are uniformly dispersed in the matrix material, and element division is performed on the homogenized unit structure. is there.

[0003]

However, the dispersion form of the second phase material in an actual material cannot be an ideal homogeneous dispersion as represented by the above-mentioned conventional finite element model. Also, various physical properties and phenomena such as material destruction are greatly affected by the quality of dispersibility. That is, in order to predict various physical properties and phenomena of the composite material, an analytical model in consideration of the dispersion heterogeneity of the second phase material is required.

On the other hand, faithfully modeling a microscopic image of the inside of a composite material observed with an electron microscope or the like is a very laborious operation. Further, since a completely different image is observed depending on the observation position, faithful modeling of the microscopically observed image is meaningless. In other words, the homogenized unit structure cannot capture the actual phenomenon, and the faithful modeling of the partial observation image having the dispersion heterogeneity of the actual material is meaningless. I have a problem.

The present invention has been made in view of the above problems, and an object thereof is to quantify the dispersibility of the second phase material such as fibers and fillers in the composite material, and to consider this. It is to provide a simple method for creating the finite element model.

[0006]

According to the invention of claim 1, in order to solve the above-mentioned problems, the finite element model creating method of the present invention uses a rectangular shape of a microscopically observed image of a composite material. It is divided into vertical n × horizontal n regions by some kind of integer n, and the area ratio occupied by the second phase material such as fiber or filler in each divided region is calculated. From the relationship, the dispersibility of the second phase material is quantified, and a finite element model depending on this is generated.

The method of the present invention enables simple modeling and analysis in consideration of dispersibility of the second phase material such as fibers and fillers in the composite material. Further, from the analysis result, it is possible to obtain knowledge such as grasping the microscopic stress concentration degree in the material as a relationship between the volume ratio of the second phase material and the dispersibility.

[0008]

DETAILED DESCRIPTION OF THE INVENTION The present invention will be described below with reference to FIGS. FIG. 1 is a conceptual diagram of a microscopic image inside the composite material observed by an electron microscope or the like. In the figure, 1
Denotes a matrix material, and 2 denotes a second phase material such as a fiber or a filler. FIG. 2 shows, for example, "Advances in
Electronic Packaging 199
2 ”,“ Mechanical Prope ”
rty Estimation of Encapsul
Ants by MicrostructuralAn
alysis using F. E. FIG. M. FIG. 1 is a conceptual diagram for modeling a microscopic structure of a conventional composite material as seen in FIG. 1, wherein particles of a second phase material and the like are uniformly dispersed in one matrix material. Assuming that
Element division is performed on the uniformed unit structure.

However, as shown in FIG. 1, in a general composite material, the dispersed state of the second phase material in the matrix material cannot be the regular uniform dispersion as shown in FIG. On the other hand, faithfully modeling a microscopic image inside a composite material observed with an electron microscope or the like is a very laborious operation. Further, since a completely different image is observed depending on the observation position, faithful modeling of the microscopically observed image is meaningless. Therefore, the dispersion state of the second phase material is quantified, and a substitute finite element model is simply formed using this quantitative index.

1 and 3, the document "Application of Fractal Dimension to Evaluation of Filler Dispersion State in Composite Material (Material Vol. 42, No. 478, pp. 836-8).
42) shows an example of a method for quantifying the dispersion state of the second phase material 2 with respect to the matrix material 1 in the composite material.

1) The microscopic image of the inside of the composite material observed with an electron microscope or the like in FIG. 1 is divided into n × n areas in which each side is divided into n, as shown in FIG. 2) In each of the divided regions, the ratio of the area occupied by the second phase material to the entire area of the region is determined, and this is calculated as A
(I, j); i = 1 to n and j = 1 to n. 3) A
Let Am be the average value of (i, j) and σ (A) be the standard deviation,
Using the value of n dividing each side as a parameter, σ (A) is calculated for each n division.

The area ratio and dispersibility of the second phase material in the cross section of the composite material are represented by the arrangement of the unit structure of the second phase material 2 with respect to the matrix material 1. Arrangement of the second phase material in the finite element method analysis model is performed by dividing the number n of one side of the image and the area ratio A (i, j) of the second phase material to the total area of each divided area; i
= 1 to n, j = 1 to n standard deviation value σ (A).

The average value Am of A (i, j) is expressed by the following equation. Am = ΣA (i, j) / n ² = Vf Equation 1 Am can be obtained by Am = Vf (Vf: area ratio of the second phase material in the entire observation area), without depending on n. it can.

On the other hand, the standard deviation σ (A) is expressed by the following equation. σ (A) = √ (A ² m−Am ² ) ... Equation 2 (A ² m is the average value of A (i, j) ² ). The variance V (A) is expressed by the following equation. It V (A) = σ (A) ² = A ² m-Am ² = (ΣA (i, j) ² / n ² ) -Am ² ... Equation 3

As shown in FIG. 4, when n = 2, the area ratio occupied by the second phase material in each divided region is A (i, j); i = 1 to 2, j = 1 to 2 To do. In order to easily obtain A (i, j) in each region, when i = j, A (i, j) = Am, that is, A ₁₁ = A ₂₂ = Am (Equation 4) and A ₁₂ = (1 + α ) Am, A ₂₁ = (1−α) Am ... Note that α ≧ 0.

Let V (A ₂ ) and σ (A ₂ ) be the variance and standard deviation of A (i, j) when n = 2. When these are substituted into Equation _{3, V (A 2) =} {2Am 2 + (1 + α) 2A 2 + (1-α) 2A 2} / 4-Am 2 = α 2 Am 2/2 Therefore, σ (A ₂ ) = Α · Am / √2 α = (√2) · σ (A ₂ ) / Am (Equation 6) From the observed microscopic image inside the composite material, Am and σ
Since (A ₂ ) is known, α is obtained, and equations 4 and 5 are used.
Thus, the area ratio A (i, j); i = 1 to 2, j = 1 to 2 occupied by the second phase material in each of the divided regions can be obtained.

Next, each region divided with n = 4 is further divided into four regions, and the area ratio occupied by the second phase material in each region is determined by the same procedure. Second of each subdivided area
The area ratio occupied by the phase material is as shown in FIG. Β ≧ 0 in the figure. The variance and standard deviation of A (i, j) when n = 4 are V (A ₄ ) and σ (A ₄ ). n = 2
Substituting these into Equation 3 with n = 4 as in the case of, V (A ₄ ) = {2 (1 + α) ² Am ² +2 (1-α) ² Am ² +2 (1 + β) ² Am ² +2 (1-β) ² Am ² + (1 + β) ² (1-α) ² Am ² + (1-β) ² (1-α) ² Am ² + (1 + β) ² (1 + α) ² Am ² + (1 −β) ² (1 + α) ² Am ² + 4Am ² } / 16-Am ² = [2Am ² {(1 + α) ² + (1-α) ² } + 2Am ² {(1 + β) ² + (1-β) ² } + Am ² {(1 + β) ² + (1-β) ² } (1-α) ² + Am ² {(1 + β) ² + (1-β) ² } (1 + α) ² + 4Am ² ] / 16-Am ² = { ^{4Am 2 (1 + α 2)} + 4Am 2 (1 + β 2) + 2Am 2 (1 + β 2) (1-α) 2 + 2Am 2 (1 + β 2) (1 + α) 2 + 4Am 2} / 16-Am 2 = {4Am 2 (2 + α 2 + ^{2) + 4Am 2 (1 +} β 2) (1 + α 2) + 4Am 2} / 16-Am 2 = {4Am 2 (2 + α 2 + β 2) + 4Am 2 (1 + β 2 + α 2 + β 2 α 2) + 4Am 2} / 16-Am 2 = {4Am ² (α ² + β ² ) + 4Am ² (β ² + α ² + β ² α ² )} / 16 = Am ² {2α ² + β ² (α ² +2)} / 4 Therefore, β ² = (4V (A ₄ ) / Am ² -2α ² ) / (α
² +2) Substituting the equation 6, β ² = {4 (σ (A ₄ ) / Am) ² -4 (σ (A ₂ ) / Am) ² } / {2 (σ (A ₂ ) / Am) ² + 2} = 2 {(σ (A ₄ ) / Am) ² − (σ (A ₂ ) / Am) ² } / {(σ (A ₂ ) / Am) ² +1} ・ ・ ・ Equation 7 Observed composite From the microscopic image inside the material, σ (A ₄ ),
Since σ (A ₂ ) and Am are known, β is obtained by Equation 7, and the area ratio A (i, j) of the second phase material in the divided regions shown in FIG. 5 is i = 1 to 4 , J = 1 to 4
Can be requested. If necessary, the same procedure is repeated for the subdivided area.

After the second phase material is arranged in correspondence with A (i, j) in each divided area determined as described above and element division is performed, material physical properties are defined for each corresponding element. . That is, the rectangle of the microscopic image inside the observed composite material is divided into vertical n × horizontal n regions by some integer n, and the standard deviation of the area ratio of the second phase material in each divided region is calculated. By quantifying the dispersibility of the second phase material in the composite by determining it and determining its alternative placement through the above procedure, a finite element model having a second phase material dispersibility equivalent to this is obtained. Simple creation is possible.

An example of boundary conditions of various analyzes for a finite element model considering the heterogeneous dispersion of the second phase material inside the composite material will be described with reference to FIG. When performing microscopic static stress analysis inside the composite material in the elastic range, the physical properties (elastic modulus, Poisson's ratio) of each material forming the composite material are defined at the corresponding locations in each element in the model. A node on the X axis has a displacement of 0 in the Y direction, and a node on the Y axis has a displacement of 0 in the X direction. The same Y-direction displacement corresponding to the desired Y-direction strain εY is applied to a node whose initial Y coordinate is on the straight line of Y = b. For nodes having initial X coordinates on a straight line of X = a, the degree of freedom of displacement in the X direction is set so that all nodes have the same displacement in the X direction. The above boundary conditions enable microscopic stress analysis inside the composite material against a uniaxial tensile load.

Further, the macroscopic elastic coefficient is obtained from the relationship between the macroscopic Y-direction strain and the nodal reaction force at this time, and the macroscopic Y
The macroscopic Poisson's ratio can be obtained from the relationship between the directional strain and the macroscopic X-direction strain.

When performing microscopic thermal stress analysis inside the composite material and predicting the linear expansion coefficient, each element in the model is
Physical properties (elasticity coefficient, Poisson's ratio, linear expansion coefficient) of each material constituting the composite material are defined at corresponding positions.

As a boundary condition, a node on the X axis has a displacement of 0 in the Y direction, and a node on the Y axis has a displacement of 0 in the X direction. For nodes with an initial Y coordinate on the straight line of Y = b,
The condition is set to the degree of freedom of the Y-direction displacement so that all nodes generate the same Y-direction displacement dY. With respect to the nodes whose initial X coordinate is on the straight line of X = a, all nodes are the same in the X direction displacement d.
A condition is set for the degree of freedom of displacement in the X direction so as to generate X.
Under this boundary condition, the initial temperature T ₀ is set for the entire model.
To give the temperature change amount dT.

As described above, a microscopic thermal stress analysis due to a physical property mismatch inside the composite material with respect to temperature change becomes possible. In addition, the following equation using the analysis results enables the prediction of the macroscopic linear expansion coefficient of the composite material. αx = dx / a / dT αy = dy / b / dT αx: Macroscopic linear expansion coefficient of composite material in X direction (/ ° C.) αy: Macroscopic linear expansion coefficient of composite material in Y direction (/ ° C.) dx: Amount of displacement in the X direction of the analytical model dy: Amount of displacement in the Y direction of the analytical model a: Initial X direction dimension of the analytical model b: Initial Y direction dimension of the analytical model dT: Temperature change (° C) During thermal shock inside the composite material When performing microscopic transient temperature distribution, thermal shock stress analysis, and thermal conductivity prediction, each element in the model includes physical properties (thermal conductivity, thermal conductivity, Density, specific heat). An initial temperature T ₀ is given to the entire model, and a temperature change amount dT is given to a node on the Y axis, and unsteady heat conduction analysis is performed. This makes it possible to analyze the microscopic transient temperature distribution inside the composite material during thermal shock.

Further, each element has physical properties of each material (heat conductivity, density, specific heat, elastic coefficient, Poisson's ratio, linear expansion coefficient).
Under the same boundary conditions as the thermal stress analysis,
The thermal shock stress can be analyzed by a method such as the same analysis using the temperature-displacement coupled element. In these analyzes, the time from when the temperature change amount dT is given to the node on the Y axis until the temperature of each node on the straight line Y = b reaches T ₀ + dT is used as a macroscopic view of the composite material in the Y direction. It is possible to predict the thermal conductivity. The heat conduction behavior in the X direction is similarly obtained by giving a temperature change amount dT to a node on the X axis and performing unsteady heat conduction analysis.

[0025]

As described above, according to the present invention, a rectangle of a microscopically observed image of a composite material is represented by several integers n, vertical n × horizontal n.
The area ratio occupied by the second phase material such as fiber or filler in each of the divided areas is obtained, and the dispersibility of the second phase material is determined from the relationship between the average value and standard deviation and the number of divisions n. Since it quantifies and generates a finite element model that depends on this, the dispersibility of the second phase material such as fibers and fillers in the composite material is quantified, and a finite element model having equivalent dispersibility is created. It can be created easily. Therefore, the accuracy of analysis of microscopic static stress, thermal stress, transient temperature distribution, etc. in the actual composite material can be improved.

[Brief description of the drawings]

FIG. 1 is a conceptual diagram of a microscopic image inside a composite material observed with an electron microscope or the like.

FIG. 2 is a conceptual diagram of a microstructure of a conventional composite material assuming uniform dispersion of a second phase material.

FIG. 3 is a diagram in which each side of FIG. 1 is divided into n for calculating a standard deviation of an area ratio of a second phase material.

FIG. 4 is an explanatory diagram showing an area ratio occupied by a second phase material in each divided region when n = 2.

FIG. 5 is an explanatory diagram showing an area ratio occupied by a second phase material in each divided region when n = 4.

FIG. 6 is an explanatory diagram showing an example of boundary conditions of various microscopic analysis methods inside the composite material.

[Explanation of symbols]

 1 Matrix material 2 Second phase material

Claims (4)

[Claims] 1. A rectangular of a microscopically observed image of a composite material is divided into vertical n × horizontal n regions by some integer n, and an area ratio occupied by a second phase material in each divided region is obtained. , The average value and standard deviation and the number of divisions n,
The dispersibility of the second phase material such as fibers and fillers in the material is quantified, and the second deviation in each region is calculated from this standard deviation calculation formula.
A method for creating a finite element model of a microscopic structure of a composite material that considers the heterogeneous dispersion of the second phase material by determining the placement of the phase material. 2. The analysis model of claim 1 is used to analyze the microscopic stress inside the composite material, predict the macroscopic elastic modulus, and the Poisson's ratio. Finite element model creation method. 3. A finite element model of a microscopic structure of a composite material, characterized by analyzing microscopic thermal stress inside a composite material and predicting a linear expansion coefficient using the analytical model according to claim 1. How to make. 4. A composite material characterized by using the analytical model of claim 1 to analyze transient temperature distribution during transient heat conduction in the composite material, analyze thermal shock stress, and predict thermal conductivity. Method for Creating Finite Element Model of Microscopic Structures of.