CN107451345A - A kind of method for obtaining material macroscopic three dimensional equivalent properties - Google Patents

A kind of method for obtaining material macroscopic three dimensional equivalent properties Download PDF

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CN107451345A
CN107451345A CN201710597759.6A CN201710597759A CN107451345A CN 107451345 A CN107451345 A CN 107451345A CN 201710597759 A CN201710597759 A CN 201710597759A CN 107451345 A CN107451345 A CN 107451345A
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高亮
高杰
肖蜜
李好
张严
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Huazhong University of Science and Technology
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Abstract

The invention belongs to mechanics of materials technical field, and a kind of method for obtaining material macroscopic three dimensional equivalent properties is disclosed, comprised the following steps：(1) double yardstick coordinate systems for describing material and microstructure are established, it is theoretical based on gradual extension, build material macro equivalent attribute model；(2) the periodic boundary model of three-dimensional microstructures is built；(3) the periodicity restriction relation of three-dimensional microstructures is established by finite element analysis；(4) three-dimensional microstructures interior lines elastic equilibrium equation is established based on periodicity restriction relation, and the displacement field of microstructure is obtained according to linear elasticity equilibrium equation；(5) displacement field based on finite element analysis and microstructure, unit strain energy is solved, obtains material macroscopic three dimensional equivalent properties；Method provided by the invention introduces unit interactivity energy in finite element analysis is carried out, and establishes total strain energy of the microstructure under unit testing strain and establishes equivalence relation, effective acquisition material properties with material properties.

Description

A kind of method for obtaining material macroscopic three dimensional equivalent properties
Technical field
The invention belongs to mechanics of materials technical field, and material macroscopic three dimensional equivalent properties are obtained more particularly, to one kind Method.
Background technology
Research for material equivalent properties is long-standing, initially relates only to the performance and its body of material each component Divide the voiget-Reuss bounds of ratio, such as most basic material equivalent performance, and more accurate two-dimentional isotropism material The Hill-Hashin bounds of material and the Hashin-shtrikman bounds of Three-Dimensional Isotropic material.Develop into more later Bound for the anisotropic material equivalent performance of complexity is predicted, such as Torquato introduction points correction function so that respectively to different The bound of property material equivalent properties achieves further development.And the accurate Forecasting Methodology of material Effective Elastic Properties The material that the emphasis, Eshelbyl etc. that even more everybody studies is made up of isotropic material using Equivalent Inclusion theoretical calculation Effective Elastic Properties.Hershey and Kroner successively proposes the Effective Elastic Properties of self-consistancy theory research polycrystalline material, Budiansky and Hill have further developed Self -consistent method, are generalized to the equivalent elastic modulus of heterogeneous material.
Different from these methods, the homogenization method that nineteen seventies grow up is based on continuum theory, Think that macroscopic material is made up of the periodically continued repeated arrangement of microstructure, therefore based on the periodicity of microstructure and company Continuous property characteristic uses perturbation theory, tests and strains for microstructure applying unit, secondly presses physical field and characterizes microstructure Yardstick correlation small parameter carries out asymptotic series expansion, realizes the calculating of material macro equivalent attribute.For grinding for homogenization technology Study carefully, disturbed from initial asymptote Expansion Theory with applying, establish material properties asymptote expanded form, realize the attribute of attribute Model inference.The mathematical theory of the class model is very rigorous, but derivation is extremely complex, is secondly directed to the numerical value meter of such method Calculation method is more time-consuming, is highly detrimental to its application in engineering.
The content of the invention
For the disadvantages described above or Improvement requirement of prior art, it is equivalent to obtain material macroscopic three dimensional the invention provides one kind The method of attribute, its object is to improve the efficiency of acquisition material macroscopic three dimensional equivalent properties.
To achieve the above object, according to one aspect of the present invention, there is provided one kind obtains the equivalent category of material macroscopic three dimensional The method of property, it is characterised in that comprise the following steps：
(1) double yardstick coordinate systems for describing material and microstructure are established, and are based on gradual extension the Theory Construction Material macro equivalent attribute model；
(2) the periodicity side of three-dimensional microstructures is built with continuity based on the periodicity of localized micro structure in material Boundary's model；
(3) tested by applying unit and strain the finite element analysis for carrying out microstructure to establish the week of three-dimensional microstructures Phase property restriction relation；
(4) three-dimensional microstructures interior lines elastic equilibrium equation is established based on periodicity restriction relation, and put down according to linear elasticity The equation that weighs obtains the displacement field of microstructure；
(5) the displacement field acquiring unit strain energy based on finite element analysis and microstructure, is asked according to unit strain energy With acquisition material macroscopic three dimensional equivalent properties.
Preferably, step (1) includes following sub-step：
(1.1) establishing includes double yardstick coordinate systems of global coordinate system and local coordinate system；Wherein, global coordinate system is used for Macroscopic material is described, local coordinate system is used to describe microstructure；
(1.2) to microstructure interior nodes displacement field uε(x) gradual expansion is carried out, it is specific as follows：
uε(x)=u0(x, y)+ε u1(x, y)+ε2u2(x, y)+...+εmum(x, y) ...+εnun(x, y)；
Wherein, ε refers to the scale factor between global coordinate system x and local coordinate system y, ε=x/y；uε(x) refer to microcosmic Structure intrinsic displacement field；um(x, y) refers to the m rank displacement fields of expansion, and m takes the positive integer for being less than n more than 3, and n refers to the of expansion N ranks, take infinity；
(1.3) displacement field after expansion is simplified, obtains the homogenization elasticity tensor attribute of material：
Wherein,Refer to the homogenization elasticity tensor attribute of material, i, j, k, l each mean applying unit test strain Direction, corresponds to abscissa direction and longitudinal direction and ordinate direction, and H is used to represent to homogenize；Ω refers to that microstructure is set Domain is counted, Y refers to the area or volume in microstructure design domain,WithThe unitary elasticity test for referring to apply strains first Initial value；WithRefer to unknown strain field in microstructure design domain；DpqrsRefer to any point in microstructure design domain Elasticity modulus of materials value, d refers to the dimension in microstructure design domain, 2 taken when two-dimentional, 3 are taken when three-dimensional；P, q, r, s refer to sit The different directions of parameter.
Preferably, the step (2) is specially：
The periodic boundary model for building three-dimensional microstructures is as follows：
Wherein, w corresponds to the displacement relation between the counterpart node of microscopic units,Corresponding unit test displacement field field, Represent Unknown Displacement field in microstructure, p refers to the position of current point, comprising coordinate system direction, during two-dimensional coordinate system it is corresponding it is horizontal, Vertical both direction, correspond to horizontal, vertical, perpendicular three directions during three-dimensional system of coordinate；K refers to the normal direction on microscopic units border, k+Refer to boundary normal positive direction, k- refers to boundary normal negative direction.
Preferably, step (3) includes following sub-step：
(3.1) the modal displacement field in three-dimensional microcosmic unit is classified according to node position；Microscopic units are whole Displacement body field U point is 8 parts, respectively U1,U2,U3,U4,U5,U6,U7,U8；
(3.2) based on described periodic boundary model, establish three-dimensional microstructures different piece modal displacement field it Between periodicity restriction relation it is as follows：
Wherein, interior nodes, symmetrical line node and face node keep balancing, will according to the classification of modal displacement field F points of global load in microscopic units is F1~F88 parts, meet F2=0, F3+F4+F5+F6=0, F7+F8=0；Node position Shifting meets periodicity restriction relation between field；
w1Refer to U3With U4Between periodicity constraints；w2Refer to U3With U5Between periodicity constraints；w3It is Refer to U3With U6Between periodicity constraints；w4Refer to U7With U8Between periodicity constraints.
Preferably, step (4) includes following sub-step：
(4.1) it is as follows to establish global lines elastic equilibrium equation in microscopic units：
KU=F
Wherein, K refers to microstructure overall situation stiffness matrix, and U refers to microstructure global displacement field, and F refers to microstructure Overall situation load；
(4.2) above-mentioned global lines elastic equilibrium equation is converted to following form by the classification based on three kinds of displacement fields：
(4.3) according to the periodicity restriction relation of three-dimensional microstructures and the global lines elastic equilibrium equation after conversion, obtain Microstructure overall situation stiffness matrix K after must reducingr, microstructure global displacement field U after reductionrWith the microcosmic knot after reduction The global load F of structurerIt is specific as follows：
Wherein, F points of microstructure global load is F1~F88 parts, field U points of microstructure global displacement is U1~U88 Individual part, K points of microstructure overall situation stiffness matrix is K11, K12..., Kmn…K8864 parts, m=1~8, n=1~8；
Preferably, step (5) includes following sub-step：
(5.1) the interactivity energy of microscopic units is obtained according to the displacement field of microstructure；Wherein, finite elements e friendship Interaction performance amount
Wherein,Represent the interactivity energy that unit e is tested under should changing direction in i, j, k, l, keRefer to finite elements e Stiffness matrix,WithRefer to finite elements e initial cell test displacement field,WithRefer to finite elements E Unknown Displacement field, T refer to the transposition of matrix；
Wherein, finite elements e stiffness matrix keThis structure attribute and test strain based on material obtain,
Wherein, BeRefer to unit e strain displacement matrix, CeRefer to unit e constitutive matrix, ε0Represent that initial cell is surveyed Examination strain, ΩeRefer to finite elements e domain；T refers to matrix BeTransposition；
(5.2) the interactivity energy of all finite elements is subjected to summation and obtains the total interactivity energy of microscopic unitsWherein, QijklRepresent applying i, j, k, total interaction of microstructure when the unit testing in l directions strains Performance amount, NE refer to the quantity of the finite elements of the discrete acquisition of microscopic units；
(5.3) macro equivalent that material is obtained according to the total interactivity energy of microscopic units homogenizes elasticity tensor
In general, by the contemplated above technical scheme of the present invention compared with prior art, it can obtain down and show Beneficial effect：
(1) method provided by the invention for obtaining material macroscopic three dimensional equivalent properties, is asked for existing homogenization technology The problem of difficult is solved, when microstructure carries out finite element analysis, unit interactivity energy is introduced, establishes microstructure in list Total strain energy under member test strain establishes equivalence relation, effective acquisition material properties with material properties；
(2) method provided by the invention for obtaining material macroscopic three dimensional equivalent properties, by introducing unit interactivity energy, Material macro equivalent attribute is calculated based on ess-strain principle, traditional progressive extended method is instead of, simplifies homogenization Technology；
(3) method provided by the invention for obtaining material macroscopic three dimensional equivalent properties, is introduced periodically in microscopic units Boundary model, periodicity constraint equation is established for different nodes, based on constraint equation, realized to global lines elastic equilibrium The reduction of equation；Global stiffness matrix, global displacement matrix and the reduction of global load matrix yardstick are brought, and then reduces meter The yardstick of calculation, calculating cost is saved, calculate and analyze suitable for the material properties of large scale macrostructure.
Brief description of the drawings
Fig. 1 is the flow chart of the method for acquisition material macroscopic three dimensional equivalent properties provided by the invention；
Fig. 2 macroscopic materials and microstructure schematic diagram；
Fig. 3 is 3D microscopic units node schematic diagrames；
Fig. 4 a~Fig. 4 c are three kinds of different microstructure schematic diagrames for intending obtaining homogenization attribute；
Fig. 5 is to be obtained using existing homogenization method from two kinds of different methods of uniformity of energy method of the invention The efficiency variance schematic diagram of material properties；
Fig. 6 is the relation schematic diagram calculated between time and the gridding of finite elements for obtaining material properties.
Embodiment
In order to make the purpose , technical scheme and advantage of the present invention be clearer, it is right below in conjunction with drawings and Examples The present invention is further elaborated.It should be appreciated that the specific embodiments described herein are merely illustrative of the present invention, and It is not used in the restriction present invention.As long as in addition, technical characteristic involved in each embodiment of invention described below Conflict can is not formed each other to be mutually combined.
The method provided by the invention for obtaining material macroscopic three dimensional equivalent properties, its flow is as shown in figure 1, including following step Suddenly：
(1) double yardstick coordinate systems are established；Macroscopic material is described using global coordinate system (x), retouched using local coordinate system (y) Microstructure is stated, the Theory Construction material macro equivalent attribute model is extended based on asymptote；
Macroscopic material as shown in Figure 2 includes two kinds of microstructures, and each microstructure has its specific material properties And material distribution form, it is distributed in the different zones of macroscopic material；
It is as follows that the displacement field of microstructure is subjected to expansion acquisition material macro equivalent attribute solution formula：
Wherein,Refer to the homogenization elasticity tensor attribute of material, H is used to represent to homogenize；Y refers to microstructure design The area or volume in domain, Ω refer to microstructure design domain,WithThe unitary elasticity test for referring to apply strains first Initial value；WithRefer to unknown strain field in microstructure design domain；I, j, k, l refer to the side of applying unit test strain To corresponding to abscissa direction (value 1) and longitudinal direction (value 2) and ordinate direction (value 3)；DpqrsRefer to microcosmic knot The elasticity modulus of materials value at any point in structure design domain,Refer to that unknown strain field, d refer to micro- in microstructure design domain The dimension for seeing structure design domain (takes 2, taken when three-dimensional 3), p, q, r, s also refers to the direction of reference axis when two-dimentional.
Above formula is solved based on Micro Instructional Design domain interior lines elastic equilibrium equation as follows：
Wherein DijpqReferring to this structure property value of arbitrfary point in microscopic units, Ω refers to microstructure design domain,Refer to Unknown strain field in microstructure design domain,Refer to initial cell strain field, v in microstructure design domainiRefer to microcosmic list Virtual displacement field in member, it can use arbitrary value；yjRefer to the coordinate on j directions of microscopic units, in j corresponding three-dimensional coordinate systems It is horizontal, vertical, erect three directions, respectively value 1,2,3；R3Refer to three dimensional design space, HperRefer to allow in Micro Instructional Design domain Dynamics displacement field.
(2) the cyclic fluctuation displacement field in microscopic units is eliminated, establishes the periodic boundary mould of three-dimensional microstructures Type is as follows：
Wherein, periodic boundary condition is formed by known variable；W correspond between the counterpart node of microscopic units Displacement relation,Corresponding unit tests strain field；Represent Unknown Displacement field in microstructure；P represents the position of current point, bag Direction containing coordinate system when three-dimensional system of coordinate (corresponding horizontal during two-dimensional coordinate system, vertical both direction, correspond to horizontal, vertical, perpendicular three directions)； K refers to the normal direction on microscopic units border, k+Refer to boundary normal positive direction, k-Refer to boundary normal negative direction.
(3) above-mentioned periodic boundary model is based on, node-classification is carried out to microscopic units using cell node as research object, The periodicity constraints for establishing all kinds of nodes is obtained, the periodicity of three-dimensional microstructures is obtained according to the periodicity constraints Restriction relation；
In embodiment, 3D microscopic units nodes are as shown in Figure 3；Modal displacement field in all microscopic units is divided into as follows Several classes listed by table 1：
The modal displacement field classification of the 3D microscopic units of table 1
The relation that above-mentioned 8 kinds of displacement places meet is as follows：
Displacement field U1, it is that each node is defined in different displacements based on the position of periodic boundary condition and stationary nodes Shift value off field.
Displacement field U2, it is the displacement field in 3D microscopic units.
Displacement field U3 U4 U5With U6, correspond to boundary line node, and U4 U5With U6Relative to U3With periodicity w1 w2 w3；And such node spatially meets dynamic balance relation, meets F3+F4+F5+F6=0.
Displacement field U7With U8, equal corresponding sides interface node, and U7Relative to U8With periodicity w4, such node is in space On meet dynamic balance relation, meet F7+F8=0；
The periodicity restriction relation obtained according to above-mentioned relation in microscopic units between 8 class displacement fields is as follows：
(4) the linear elasticity equilibrium equation of microscopic units is established based on above-mentioned periodicity restriction relation, is obtained according to equilibrium equation Take the displacement field U of microstructure its node in endurance；
Wherein, F points of microstructure global load is F1~F88 parts, field U points of microstructure global displacement is U1~U8 8 parts, K points of microstructure overall situation stiffness matrix is K11, K12..., Kmn…K8864 parts, m=1~8, n=1~8；
Finite elements balance side is established according to the periodicity restriction relation between 8 class displacement fields and linear elasticity equilibrium equation Journey KrUr=Fr；
Wherein：
Wherein, by can be seen that compared with initial linear elasticity equilibrium equation：Global stiffness matrix K, the overall situation load to The yardstick of amount F and global displacement field U vectors reduces respectively, corresponds to K after reduction respectivelyr,FrWith Ur。
(5) the displacement field U in microscopic units obtained according to above-mentioned steps (4)r, obtain the finite elements of microscopic units Interactivity energyThe total strain energy of microstructure is obtained by summing
Wherein,Represent the interactivity energy that unit e is tested under should changing direction in i, j, k, l, keRefer to finite elements e Stiffness matrix,WithRefer to finite elements e initial cell test displacement field,WithRefer to finite elements e Unknown Displacement field, T refers to the transposition of matrix；
This structure attribute and test strain based on material obtain microscopic units overall situation stiffness matrix K and load matrix F：
Wherein, BeRefer to unit e strain displacement matrix, CeRefer to unit e constitutive matrix, ε0Represent that initial cell is surveyed Examination strain, ΩeRefer to finite elements e domain；T refers to matrix transposition；
The macro equivalent homogenization elasticity tensor of material is equivalent to the total strain energy of microstructure, i.e.,：Thus Obtain the macro equivalent homogenization elasticity tensor attribute of material
The method of the above-mentioned acquisition material macroscopic three dimensional equivalent properties provided using embodiment, to three kinds shown in Fig. 4 not Same microscopic units obtain the even macroscopic resilient property of material；Fig. 4 includes three kinds of different microstructures, wherein, Fig. 4 (a) it is microstructure 1, the structure has maximum volume modulus；Fig. 4 (b) is microstructure 2, and the structure has maximum shear mould Amount；Fig. 4 (c) is microstructure 3, is full of all material in the microstructure.
Yardstick of three kinds of microstructures on the direction of three, space in Fig. 4 is respectively defined as 1, in finite element analysis When, finite element mesh is 15 × 15 × 15；The Young's modulus of material is defined as 1, and Poisson's ratio is defined as 0.3；Using the present invention The method based on uniformity of energy and the homogenization technology of prior art that there is provided obtain the elasticity tensor of above-mentioned 3 kinds of structures As listed by table 2 below：
The elasticity tensor list of table 2
From table 2 it can be seen that it is completely the same based on the structural elasticity tensor value that above two method obtains, demonstrate this hair The correctness of bright provided method.13 kinds of different griddings are defined, are respectively adopted provided by the invention uniform based on energy The method of change and the homogenization technology of prior art obtain the structural elasticity tensor value of material, provided by the present invention to verify Method can improve obtain material macroscopic three dimensional equivalent properties efficiency.As shown in figure 5, wherein dark circles dot pattern is formed Curve be using conventional uniform method obtain material structural elasticity tensor value needed for time；Black bars shape pattern institute The curve of composition is the time needed for the structural elasticity tensor value of the homogenization method acquisition material improved using the present invention；Pass through Compare as can be seen that being less than existing homogenization technology using the time required for being invented improved uniformity of energy method The required time, and as the increase of gridding yardstick, gap are more and more obvious.The time difference curve that Fig. 6 is provided illustrates Obtain the relation between the required time of material properties and the gridding of finite elements, it can be seen that energy provided by the present invention Amount homogenization method is more suitable for being directed to engineering structure.
As it will be easily appreciated by one skilled in the art that the foregoing is merely illustrative of the preferred embodiments of the present invention, not to The limitation present invention, all any modification, equivalent and improvement made within the spirit and principles of the invention etc., all should be included Within protection scope of the present invention.

Claims (6)

  1. A kind of 1. method for obtaining material macroscopic three dimensional equivalent properties, it is characterised in that comprise the following steps：
    (1) double yardstick coordinate systems for describing material and microstructure are established, and based on gradual extension the Theory Construction material Macro equivalent attribute model；
    (2) the periodic boundary mould of three-dimensional microstructures is built with continuity based on the periodicity of localized micro structure in material Type；
    (3) tested by applying unit and strain the finite element analysis for carrying out microstructure to establish the periodicity of three-dimensional microstructures Restriction relation；
    (4) three-dimensional microstructures interior lines elastic equilibrium equation is established based on periodicity restriction relation, and according to linear elasticity balance side Journey obtains the displacement field of microstructure；
    (5) the displacement field acquiring unit strain energy based on finite element analysis and microstructure, obtained according to the summation of unit strain energy Obtain material macroscopic three dimensional equivalent properties.
  2. 2. the method as described in claim 1, it is characterised in that step (1) includes following sub-step：
    (1.1) establishing includes double yardstick coordinate systems of global coordinate system and local coordinate system；Wherein, global coordinate system is used to describe Macroscopic material, local coordinate system are used to describe microstructure；
    (1.2) to microstructure interior nodes displacement field uε(x) gradual expansion is carried out, it is specific as follows：
    uε(x)=u0(x,y)+εu1(x,y)+ε2u2(x,y)+…+εmum(x,y)…+εnun(x,y)；
    Wherein, ε refers to the scale factor between global coordinate system x and local coordinate system y, ε=x/y；uε(x) microstructure is referred to Intrinsic displacement field；um(x, y) refers to the m rank displacement fields of expansion, and m takes the positive integer for being less than n more than 3, and n refers to the n-th order of expansion, Take infinity；
    (1.3) displacement field after expansion is simplified, obtains the homogenization elasticity tensor attribute of material：
    Wherein,Referring to the homogenization elasticity tensor attribute of material, i, j, k, l each mean the direction of applying unit test strain, Abscissa direction and longitudinal direction and ordinate direction are corresponded to, H is used to represent to homogenize；Ω refers to microstructure design domain, Y Refer to the area or volume in microstructure design domain,WithRefer to the initial value of unitary elasticity test strain applied；WithRefer to unknown strain field in microstructure design domain；DpqrsRefer to the material at any point in microstructure design domain Expect elastic mould value, d refers to the dimension in microstructure design domain, 2 are taken when two-dimentional, 3 are taken when three-dimensional；P, q, r, s refer to reference axis Different directions.
  3. 3. method as claimed in claim 1 or 2, it is characterised in that the step (2) is specially：
    The periodic boundary model for building three-dimensional microstructures is as follows：
    Wherein, w corresponds to the displacement relation between the counterpart node of microscopic units,Corresponding unit test displacement field field,Represent micro- Unknown Displacement field in structure is seen, p refers to the position of current point, corresponding horizontal, vertical two during two-dimensional coordinate system comprising coordinate system direction Direction, correspond to during three-dimensional system of coordinate it is horizontal, vertical, erect three directions；K refers to the normal direction on microscopic units border, k+Refer to Boundary normal positive direction, k-Refer to boundary normal negative direction.
  4. 4. method as claimed in claim 1 or 2, it is characterised in that step (3) includes following sub-step：
    (3.1) the modal displacement field in three-dimensional microcosmic unit is classified according to node position；Microscopic units entirety position It point is 8 parts to move field U, respectively U1,U2,U3,U4,U5,U6,U7,U8；
    (3.2) based on described periodic boundary model, establish between the modal displacement field of three-dimensional microstructures different piece Periodicity restriction relation is as follows：
    Wherein, interior nodes, symmetrical line node and face node keep balancing, will be microcosmic according to the classification of modal displacement field F points of global load in unit is F1~F88 parts, meet F2=0, F3+F4+F5+F6=0, F7+F8=0；Modal displacement field Between meet periodicity restriction relation；
    w1Refer to U3With U4Between periodicity constraints；w2Refer to U3With U5Between periodicity constraints；w3Refer to U3With U6Between periodicity constraints；w4Refer to U7With U8Between periodicity constraints.
  5. 5. method as claimed in claim 4, it is characterised in that step (4) includes following sub-step：
    (4.1) it is as follows to establish global lines elastic equilibrium equation in microscopic units：
    KU=F
    Wherein, K refers to microstructure overall situation stiffness matrix, and U refers to microstructure global displacement field, and F refers to the microstructure overall situation Load；
    (4.2) above-mentioned global lines elastic equilibrium equation is converted to following form by the classification based on three kinds of displacement fields：
    (4.3) according to the periodicity restriction relation of three-dimensional microstructures and the global lines elastic equilibrium equation after conversion, contracted Microstructure overall situation stiffness matrix K after subtractingr, microstructure global displacement field U after reductionrIt is complete with the microstructure after reduction Office load FrIt is specific as follows：
    Wherein, F points of microstructure global load is F1~F88 parts, field U points of microstructure global displacement is U1~U88 Part, K points of microstructure overall situation stiffness matrix is K11, K12..., Kmn…K8864 parts, m=1~8, n=1~8.
  6. 6. method as claimed in claim 4, it is characterised in that step (5) includes following sub-step：
    (5.1) the interactivity energy of microscopic units is obtained according to the displacement field of microstructure；Wherein, finite elements e interactivity Energy
    Wherein,Represent the interactivity energy that unit e is tested under should changing direction in i, j, k, l, keRefer to that finite elements e's is firm Spend matrix,WithRefer to finite elements e initial cell test displacement field,WithRefer to finite elements e not Know displacement field, T refers to the transposition of matrix；
    Wherein, finite elements e stiffness matrix keThis structure attribute and test strain based on material obtain,
    Wherein, BeRefer to unit e strain displacement matrix, CeRefer to unit e constitutive matrix, ε0Represent that initial cell test should Become, ΩeRefer to finite elements e domain；T refers to matrix BeTransposition；
    (5.2) the interactivity energy of all finite elements is subjected to summation and obtains the total interactivity energy of microscopic unitsWherein, QijklRepresent applying i, j, k, total interaction of microstructure when the unit testing in l directions strains Performance amount, NE refer to the quantity of the finite elements of the discrete acquisition of microscopic units；
    (5.3) macro equivalent that material is obtained according to the total interactivity energy of microscopic units homogenizes elasticity tensor
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陈永强 等: ""应用格形模型和统计方法分析两相材料宏观等效力学性质"", 《固体力学学报》 *

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