CN102819647B - A kind of heterogeneous material random microscopic structure finite element modeling method - Google Patents

Abstract

The invention proposes a random microstructure finite element modeling method for heterogeneous materials, specifically based on microstructure probability distribution information and a random algorithm to establish a random microstructure finite element grid model for heterogeneous materials. In this method, the probability distribution function of the microstructure is first determined by using the real or fictitious physical distribution of the components and phases of the heterogeneous material, and the probability distribution function is transformed into the discrete space of the model based on the establishment of the finite element grid topology model of the material. , and then use the random real numbers uniformly distributed in the interval [0,1] generated by the pseudo-random number generator to determine the material properties of each unit in the finite element model, thereby establishing a random microstructure model of heterogeneous materials. This method is applicable to different forms of heterogeneous materials, and the established finite element mesh model can be directly used to analyze the microscopic properties of heterogeneous materials, the relationship between microstructure and macroscopic properties, etc., and provide new materials for the development and preparation of new materials. in accordance with.

Description

A Finite Element Modeling Method for Stochastic Microstructure of Heterogeneous Materials

technical field

The invention belongs to the technical field of finite element modeling, and relates to a finite element modeling method, in particular to a finite element modeling method of random microstructure of heterogeneous materials.

Background technique

The macroscopic properties (such as stiffness, strength, and toughness, etc.) of heterogeneous materials composed of multiple components (such as composite materials, porous materials, etc.) are mainly determined by their microstructure. The influence of microstructure on its macroscopic properties is of great significance for the design and development of new high-performance heterogeneous materials. The finite element method is one of the most effective methods to study the relationship between the microstructure and macroscopic properties of materials, and this method needs to establish a finite element model of the microstructure of heterogeneous materials first. For the materials that have been developed and prepared, we can reconstruct the finite element mesh model reflecting its real microstructure on the basis of collecting X-ray tomography images, and use it to accurately predict the macroscopic mechanical properties of the material. However, in order to establish the relationship between the microstructure of heterogeneous materials and their macroscopic properties, microstructure information should appear as a variable in the research process. A variety of materials is not realistic. The development of finite element modeling methods that can reflect the random changes in the microstructure of heterogeneous materials can help overcome this shortcoming and provide a basis for the design of new heterogeneous materials.

Contents of the invention

The purpose of the present invention is to overcome the shortcoming of above-mentioned prior art, provide a kind of heterogeneous material stochastic microstructure finite element modeling method, this method is based on real or fictitious heterogeneous material component phase physical distribution form, According to the probability distribution function of the microstructure of heterogeneous materials, the stochastic microstructure model is established, which is helpful to establish the cross-scale correlation between the microstructure and macroscopic properties of heterogeneous materials and to explore the microstructure with the best macroscopic properties.

The purpose of the present invention is solved by the following technical solutions:

This finite element modeling method for random microstructure of heterogeneous materials includes the following steps:

1) Determine the probability distribution function of the component phase according to the microstructural characteristics of each component phase of the heterogeneous material;

2) Establish a finite element mesh topology model for heterogeneous materials;

3) Transform the probability distribution function into a discrete space, and determine the material properties of each unit in the finite element mesh model by a stochastic algorithm.

Further, the above step 1) is specifically carried out according to the following method:

The shape and distribution of each component phase cluster in heterogeneous materials has a specific form, that is, the probability of each component phase appearing in the global space of the material can be expressed by a specific mathematical distribution function; for each component phase, random and uniform distribution of M-phase heterogeneous materials, the volume fraction of each phase material is v _n (n=1,2,…,M), then its probability distribution function is v _n (X)=v _n (n=1, 2,…,M), and the corresponding cumulative distribution function is For a two-phase heterogeneous material with a gradient microstructure distribution and a gradient distribution of the component phases along the x _k direction, the probability distribution functions are v ₁ (X)=2v ₁ /(1+exp(g-2gx _k / X _k )) and v ₂ (X)=1-v ₁ (X), where g is the gradient index, the larger the value is, the larger the gradient between the two component phase materials is, and X _k is the overall material The total size of the model along the x _k direction.

Further, the above step 2) is specifically carried out as follows:

First determine the number of elements W, H, and T of the finite element model along the x, y, and z directions, and the element sizes w, h, and t, and then establish a finite element mesh topology model composed of eight-node cuboid elements or four-node rectangular elements, The coordinates of the nth node in the 3D and 2D models are determined by the following two sets of expressions, respectively:

In the formula, "%" is the remainder of integer division, Take the largest integer smaller than the operand; the nodes of the nth unit in the model are:

Further, the above step 3) is specifically carried out as follows:

MersenneTwister and Mitchell-Moore pseudo-random number generators are used to generate random numbers required in microstructure modeling. In order to improve the initial randomness, the following random seed generation algorithm is used:

In the formula, seed[n](n=1,2,…) is the seed of the random number generator, the number of seeds is determined by the specific pseudo-random number generator, t _S is the current time of the computer system, t _B and t _E are respectively is the system time at the beginning and end of the previous program, p is the binary number factor, determined by the unit size of the microstructure modeling, & and < are bitwise AND and left shift operators, respectively;

Transform the continuous probability distribution function of heterogeneous material components into the discrete space of the finite element grid model, and the random real number R uniformly distributed in the interval [0,1] generated by the pseudo-random number generator determines the finite element The material properties of each unit in the model; for multi-phase heterogeneous materials with randomly distributed microstructures, the material properties of the nth unit in the finite element model are determined by the following formula:

${\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; \&Element;</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; ]</span>\\ \mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; \&Element;</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ]</span>\end{array}\right.<span\; class="notranslate">\; ;</span>$

In the formula, s _n is the value at unit n after the probability distribution function v ₁ (X) is discretized, that is:

${\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; g</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; gi</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; ;</span>$

In the formula, I _k is the total number of units of the model along the x _k direction, and i _k is the discrete coordinate of unit n along the x _k direction. For 3D and 2D models, i _m (m=1,2,3) are respectively:

The present invention has the following beneficial effects:

The invention provides an economical and effective method for establishing a model of random changes in microstructural characteristics such as the phase distribution form and aggregation state of each component of the heterogeneous material. The method is applicable to different forms of heterogeneous materials, including multi-phase Materials, functionally graded materials, porous media, etc.; the finite element mesh model established by this method can be directly used to analyze the relationship between microstructure and macroscopic properties such as the morphology, aggregation characteristics, and distribution of each component phase in heterogeneous materials ; The physical distribution of each component phase of heterogeneous materials can be determined by real materials, which is helpful to study the relationship between the microstructure and macroscopic properties of existing materials, and provide a basis for the improvement of material performance; the physical distribution of each component phase The distribution form can also be freely designed fictitiously, so as to explore the microstructure with the best macroscopic performance, and provide reference for the development and preparation of new materials.

Description of drawings

Fig. 1 flow chart of the present invention;

Figure 2 Establishment of two-dimensional finite element mesh topology model;

Figure 3 Establishment of three-dimensional finite element mesh topology model;

Figure 4 The two-dimensional finite element mesh model of the random distribution of the microstructure of the two-phase heterogeneous material;

Figure 5. The three-dimensional finite element mesh model of the random distribution of the microstructure of the three-phase heterogeneous material;

Fig. 6 The two-dimensional finite element mesh model of the gradient distribution of the microstructure of the two-phase heterogeneous material;

Fig. 7 The three-dimensional finite element mesh model of the gradient distribution of the microstructure of the two-phase heterogeneous material.

detailed description

The present invention is described in further detail below in conjunction with accompanying drawing:

The present invention establishes the finite element grid topology model of the heterogeneous material on the basis of using the physical distribution form of each component phase of the heterogeneous material to determine the probability distribution function, and determines each The material properties of the elements, thereby modeling the stochastic microstructure of heterogeneous materials. The specific implementation process of the method is shown in Figure 1, and the specific technical problems will be described in detail below according to the process.

1. Determine the probability distribution function of the component phase according to the microstructural characteristics of each component phase of the heterogeneous material:

The shape and distribution of clusters of component phases in heterogeneous materials have a specific form, that is, the probability of each component phase appearing in the global space of the material can be expressed by a specific mathematical distribution function. For the materials that have been developed and prepared, the probability distribution function of each component phase can be determined according to a series of measures such as tomographic images or metallographic methods, and some microstructure distribution forms can also be fabricated to construct the probability of each component phase. Distribution function, so as to explore the microstructure form that can optimize the macroscopic properties of the material.

The present invention illustrates the determination of component phase probability distribution functions with heterogeneous materials with random distribution and gradient distribution of microstructure. For a heterogeneous material in which the clusters of each component phase are randomly and uniformly distributed, the volume fraction of each phase material is v _n (n=1,2,…,M), where M is the number of component phases of the material, Then the probability of the nth component phase appearing at point X is v _n (X) = v _n , that is, the probability distribution function of each component phase of the material in the global space is v _n (X) = v _n (n =1,2,…,M), the corresponding cumulative distribution function is $c_{\mathrm{no}}\left(x\right)=\underset{m\&le;\mathrm{no}}{\mathrm{\&Sigma;}}{v}_m\left(x\right)(\mathrm{no}=\mathrm{1,2},...,m).$

For two-phase heterogeneous materials with a gradient distribution of microstructure, the clusters of its component phases are distributed in a gradient along one direction, while they are still randomly and uniformly distributed along other directions. Assuming that the volume fraction of each phase material is v _n (n=1,2), and the material is distributed along the x _k direction, the probability distribution function of each phase material is v ₁ (X)=2v ₁ /(1+ exp(g-2gx _k /X _k )) and v ₂ (X)=1-v ₁ (X), where g is the gradient index, the larger the value, the greater the gradient between the two phase materials The bigger is; X _k is the total size of the overall material model along the x _k direction.

2. Establish a finite element mesh topology model for heterogeneous materials:

(1) Two-dimensional model

Firstly, according to the goal of random microstructure modeling of heterogeneous materials, the number of elements W and H and the element sizes w and h of the two-dimensional finite element model along the x and y directions are determined, and then the four-node rectangular element is established as shown in Figure 2 The topological model of the finite element mesh. According to the numbering of nodes and elements in the figure, the coordinates of the nth node in the model can be simply determined by the following mathematical relationship

In the formula, x _n and y _n are the coordinates of node n along the x and y directions respectively, "%" is the remainder of integer division, Takes the largest integer less than the operand. The four nodes arranged in the counterclockwise direction of the nth unit in the model are determined by the following formula

(2) 3D model

Also according to the modeling objective, the number of elements T and element size t of the three-dimensional finite element model along the z direction are obtained, and the finite element mesh topology model shown in Figure 3 is established, which is composed of eight-node cuboid elements. According to the node and unit numbering method of the model in the figure, the coordinates of the nth node in the model can be obtained as

The eight nodes arranged in the counterclockwise direction of the nth unit in the model are determined by the following formula

3. Microstructural Modeling

(1) Generation of random numbers

The present invention adopts the higher random degree, MersenneTwister and Mitchell-Moore pseudo-random number generator with longer period to generate the random numbers required in the microstructure modeling, in order to improve the initial random degree, adopt the following random seed generation algorithm

In the formula, seed[n](n=1,2,…) is the seed of the random number generator, and the number of seeds is determined by the specific pseudo-random number generator; t _S is the current time of the computer system; t _B and t _E are the system time at the start and end of the previous program, respectively; p is the binary number factor, determined by the cell size of the microstructure modeling; & and < are the bitwise AND and left shift operators, respectively.

(2) Determination of model unit attributes

Converting the continuous probability distribution function of heterogeneous material components into the discrete space of the finite element grid topology model, the random real numbers uniformly distributed in the interval [0,1] generated by the pseudo-random number generator can determine the finite Material properties of individual elements in the metamodel. For multiphase heterogeneous materials with randomly distributed microstructures, the material properties of the nth element in the finite element model are determined by the following formula:

${\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; \&le;</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}\\ \mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; \&Element;</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; -</span>\mathrm{<span\; class="notranslate">\; 1</span>}}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; c</span>}}_{\mathrm{<span\; class="notranslate">\; r</span>}}<span\; class="notranslate">\; ]</span><span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; <</span>\mathrm{<span\; class="notranslate">\; r</span>}<span\; class="notranslate">\; \&le;</span>\mathrm{<span\; class="notranslate">\; m</span>}\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 6</span>}<span\; class="notranslate">\; )</span>$

In the formula, R is a random real number uniformly distributed in the interval [0,1] generated by a pseudo-random number generator.

For a two-phase heterogeneous material with a gradient microstructure distribution, the material properties of the nth element in the finite element model are determined by the following formula:

${\mathrm{<span\; class="notranslate">\; p</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>\left\{\begin{array}{cc}\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; \&Element;</span><span\; class="notranslate">\; [</span>\mathrm{<span\; class="notranslate">\; 0</span>}<span\; class="notranslate">\; ,</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; ]</span>\\ \mathrm{<span\; class="notranslate">\; 2</span>}<span\; class="notranslate">\; ,</span>& \mathrm{<span\; class="notranslate">\; R</span>}<span\; class="notranslate">\; \&Element;</span><span\; class="notranslate">\; (</span>{\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; ,</span>\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; ]</span>\end{array}\right.<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 7</span>}<span\; class="notranslate">\; )</span>$

In the formula, s _n is the value at unit n after the probability distribution function v ₁ (X) is discretized, namely

${\mathrm{<span\; class="notranslate">\; the\; s</span>}}_{\mathrm{<span\; class="notranslate">\; no</span>}}<span\; class="notranslate">\; =</span>\frac{{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; v</span>}}_{\mathrm{<span\; class="notranslate">\; 1</span>}}}{\mathrm{<span\; class="notranslate">\; 1</span>}<span\; class="notranslate">\; +</span>\mathrm{<span\; class="notranslate">\; exp</span>}<span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; g</span>}<span\; class="notranslate">\; -</span>{\mathrm{<span\; class="notranslate">\; 2</span>}\mathrm{<span\; class="notranslate">\; gi</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; /</span>{\mathrm{<span\; class="notranslate">\; I</span>}}_{\mathrm{<span\; class="notranslate">\; k</span>}}<span\; class="notranslate">\; )</span>}<span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; -</span><span\; class="notranslate">\; (</span>\mathrm{<span\; class="notranslate">\; 8</span>}<span\; class="notranslate">\; )</span>$

In the formula, I _k is the total number of elements in the finite element model along the x _k direction, that is, I ₁ =W, I ₂ =H, I ₃ =T; i _k is the discrete coordinate of element n in the finite element model along the x _k direction , for a two-dimensional model, i _m (m=1,2) are:

For the 3D model, i _m (m=1,2,3) are respectively:

4. Example

(1) Microstructure random distribution model

By taking parameters W=H=50, w=h=0.1, v ₁ =0.6, v ₂ =0.4, a two-dimensional finite element grid with randomly distributed microstructure of two-phase heterogeneous materials can be established as shown in Figure 4 Model, in the model, the black area is the component phase with volume fraction v ₁ =0.6, and the white area is the component phase with volume fraction v ₁ =0.4; take parameters W=H=50, T=20, w=h=t =0.1, v ₁ =0.5, v ₂ =0.3, v ₃ =0.2 can establish a three-dimensional finite element mesh model in which the microstructure of the three-phase heterogeneous material is randomly distributed as shown in Figure 5. In the figure, the gray Larger regions represent component phases with smaller volume fractions.

(2) Microstructure gradient distribution model

Taking the parameters W=H=50, w=h=0.1, v ₁ =0.6, v ₂ =0.4, g=5, x _k =x, the microstructure of the two-phase heterogeneous material as shown in Figure 6 can be established The two-dimensional finite element mesh model with gradient distribution. In the model, the black area is the component phase with volume fraction v ₁ =0.6, and the white area is the component phase with volume fraction v ₁ =0.4. The transition between the two phase materials It is relatively smooth; take parameters W=H=50, T=20, w=h=t=0.1, v ₁ =0.7, v ₂ =0.3, g=20, x _k =y, can establish as shown in Figure 7 The three-dimensional finite element mesh model of the gradient distribution of the microstructure of the two-phase heterogeneous material. In the figure, the black area is the component phase with volume fraction v ₁ =0.7, and the white area is the component phase with volume fraction v ₁ =0.3. The gradient between the two phase materials changes sharply.

The application of the present invention is not limited to the above modeling examples. It can be known from the above-mentioned technical principles that it is applicable to the establishment of two-dimensional and three-dimensional finite elements with different volume fractions of component phases for heterogeneous materials of any multiphase The mesh model, on the other hand, is suitable for establishing two-dimensional and three-dimensional finite element mesh models with different component phase volume fractions and gradients for any two-phase heterogeneous material whose microstructure is distributed along a certain direction with a gradient.

Claims (2)

1. a heterogeneous material random microscopic structure finite element modeling method, is characterized in that, comprise the following steps:

1) according to the probability distribution function of the microstructure features determination component phase of each component phase of heterogeneous material;

2) the finite element grid topological model of heterogeneous material is set up; Specifically carry out in accordance with the following methods:

First determine that finite element model is along number of unit W, H and T in x, y and z direction and unit size w, h and t, then set up the finite element grid topological model be made up of eight node rectangular parallelepiped unit or the four nodes element of rectangle, in three peacekeeping two dimensional models, the coordinate of the n-th node is determined by following two groups of expression formulas respectively:

In formula, " % " is division of integer remainder, get the maximum integer of less-than operation object; In model, the node of Unit n-th is respectively:

3) probability distribution function is transformed in discrete space, and by the material properties of each unit in random algorithm determination finite element grid model; Specifically carry out in accordance with the following methods:

Adopting MersenneTwister and Mitchell-Moore pseudorandom number generator to generate random number required in micromechanism modeling, in order to improve initial random degree, adopting following random seed generating algorithm:

Seed [n] wherein n=0 in formula, 1,2 ... for the seed of randomizer, number seeds is determined by concrete pseudorandom number generator, t _sfor the current time of computer system, t _band t _ebe respectively the last period program start and at the end of system time, p is the number of bits factor, is determined by the unit scale of micromechanism modeling, & and << be respectively step-by-step with and accord with to left shift operation;

Be transformed in the discrete space of finite element grid model by the probability distribution function of heterogeneous material component phase, the random real number R of the Uniformly distributed produced by pseudorandom number generator in interval [0,1] determines the material properties of unit in finite element model; For the heterogeneous heterogeneous material of micromechanism stochastic distribution, in finite element model, the material properties of Unit n-th is determined by following formula:

$p_n=\left\{\begin{array}{cc}1,& R\&Element;\&lsqb;0,s_n\&rsqb;\\ 2,& R\&Element;(s_n,1)\end{array}\right.;$

S in formula _nfor probability distribution function ν ₁(X) the discrete value afterwards at unit n place, that is:

$s_n=\frac{2v_1}{1+\exp (g-2{\mathrm{gi}}_k/I_k)};$

I in formula _kfor model is along x _kthe unit total number in direction, i _kfor unit n is along x _kthe discrete coordinates in direction, x _kdirection is x, y, z direction, and g is graded index, ν ₁for the volume fraction of material, for three peacekeeping two dimensional models, i _mwherein m=1,2,3 are respectively:

2. heterogeneous material random microscopic structure finite element modeling method according to claim 1, is characterized in that, described step 1) specifically carry out in accordance with the following methods:

The shape that each component of heterogeneous material is assembled bunch mutually and distribution thereof have specific form, and namely the probability that occurs in the universe space of material of each component can with specific mathematical distribution function representation; For the mutually random equally distributed M phase heterogeneous material of each component, the volume fraction of each phase material is respectively v _n, wherein n=1,2 ..., M, then its probability distribution function is v _n(X)=v _nwherein, n=1,2 ..., M, corresponding cumulative distribution function is n=1,2 ..., M; For the two-phase heterogeneous material of micromechanism distribution gradient, component is handed down x _kdirection distribution gradient, then its probability distribution function is respectively ν ₁(X)=2 ν ₁/ (1+exp (g-2gx _k/ X _k)) and ν ₂(X)=1-ν ₁(X), in formula, g is graded index, and the larger variable gradient then between two component phase materials of graded index is larger, X _kfor material monolithic model is along x _kthe overall dimensions in direction.