CN105069203B - The thermoelasticity homogenization process of three-dimensional random heterogeneous material under a kind of finite deformation - Google Patents
The thermoelasticity homogenization process of three-dimensional random heterogeneous material under a kind of finite deformation Download PDFInfo
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Abstract
The invention discloses a kind of thermoelasticity homogenization process of three-dimensional random heterogeneous material under Large Elastic plastic Deformation, including：1) composite based on random sequence addition RSA method structure particle random distributions represents elementary volume, volume element three-dimensional RVE numerical models, and eliminates the particle overlapping phenomenon when grain volume fraction is larger；2) finite element analysis computation is carried out to composite three dimensional RVE numerical models under thermoelasticity environment, tries to achieve the numerical solution of RVE numerical model effective properties；3) macroscopical effective properties of random homogenizing model solution composite are established, and it is used as its authentic and valid property using macroscopical effective properties of the composite, more structurally sound homogenization result is provided for macroscopical effective properties of composite, the optimization design of use and structure for new advanced material provides sufficient foundation.
Description
Technical Field
The invention relates to composite material computational mechanics, in particular to a thermoelastic homogenization method of a three-dimensional random heterogeneous material under limited deformation.
Background
The research and analysis of heterogeneous materials is a classical problem, and in the continuous medium mechanical range, two research methods of macroscopic mechanics and microscopic mechanics exist. Macro-mechanical methods focus on studying the properties of structures (e.g., laminates) made of composite materials, such as the calculation of stiffness, strength, thermal stress, etc.; the mesomechanics method mainly researches the interaction of each phase material on the mesostructure scale and establishes the relationship between the macroscopic property of the material and the mesostructure parameter. Macroscopic properties, also referred to herein as effective properties, refer to material properties that can be experimentally measured on a macroscopic scale, including effective elastic constants, effective conductivities, and the like.
Since the macro-length dimension of the heterogeneous material is significantly larger than the length dimensions of the different components at the micro-scale, if all the micro-scale components are explicitly solved, both the analytic method and the numerical solution are very challenging and costly works. One way to significantly reduce the cost of analyzing such problems is to replace the original heterogeneous material with an effective homogeneous material and obtain the macro-effective properties of the material by constructing an effective constitutive equation that reflects the macro-properties, i.e., homogenization (see fig. 1). The method was proposed and applied by french scientists in the 70 s of the 20 th century to the analysis of materials with periodic structures, by using potential energy to introduce and solve the effective characteristic quantities of the materials. This method uses a Representative Volume Element (RVE) taken from the material. The method can greatly reduce the number of composite design parameters, and is continuously developed along with the change of engineering requirements, particularly after a finite element method is combined with the finite element method, one of the most effective methods for internationally analyzing the properties of the heterogeneous material is formed at present, and meanwhile, the famous Chinese scholars such as Liu Shuan, jun Zhi and Zhang hongwu also make important contributions, and the elastic constant, the thermal conductivity coefficient, the thermal expansion coefficient, the elastoplasticity problem, the viscoelasticity problem and the like of the material are predicted by using the method.
The problem of homogenization of heterogeneous materials in multiple physical fields has been explored by the prior art. Boundary values for equivalent specific heat and coefficient of thermal expansion were earlier derived and used by Rosen, hashin, and Torquto et al. However, determining such boundary values is difficult under limited deformation conditions, even in the case of purely mechanical analysis. Currently, the focus of research is in characterizing the macro-scale thermoelastic response of heterogeneous materials. For this purpose, francfort proposes a method of homogenizing a linear thermoelastic periodic medium. This method was subsequently introduced and developed by multiple people, L' Hostis and Devries, yu and Tang, terada, etc. In addition, ghosh and Liu propose homogenization methods based on the micropolar theory.
In summary, the problem of thermoelastic homogenization of heterogeneous materials in the linear case has been achieved. However, the limited deformation with large temperature excursion is not a good conditionThere have been developments. Lasdet and Alzina et al derive a nonlinear heat transfer equation that ignores the deformation phase. In addition, if the homogenization of the energy balance is neglected, the thermally induced effects in the nonlinear mechanical analysis can be analyzed. The study in this respect starts with Aboudi and Arnold, which both establish an approximate solution for homogenization. Furthermore, khisaeva and Osroja-Starzewski and Miehe et al also contribute in this respect, but only Aboudi andthe framework of limited thermoelastic analysis of heterogeneous materials in a coupling field is more fully presented.
Disclosure of Invention
Based on the defects of the traditional homogenization method, the invention analyzes the thermodynamic response of the microstructure of the heterogeneous material in a thermoelastic physical field under the condition of limited deformation, and provides a thermoelastic random homogenization framework of the heterogeneous material. Based on thermoelastomechanics, a fundamental scale conversion method for homogenization is proposed. The macrostructure temperature is applied to obtain microstructure parameters based on the temperature field, and the randomness and the correlation of the parameters of the material microstructure are fully considered. The micromechanics analysis step can be decomposed into two steps of pure mechanical analysis and pure thermal analysis, so as to obtain the homogenization response of the macro scale. Finally, a refinement numerical solution under the condition of any particle volume fraction is obtained, so that more accurate macroscopic effective parameters are obtained, and objective, sufficient and real basis is provided for the optimization design of novel advanced materials and structures.
The invention is solved by the following technical scheme:
a method for thermoelastic homogenization of a three-dimensional random heterogeneous material under limited deformation, the method comprising the steps of:
1) Constructing a three-dimensional numerical model of composite material characterization volume units (RVEs) with randomly distributed particles based on a random sequence addition method (RSA) by using a FORTRAN language;
for the composite material containing the irregular particle inclusions, the particle shape is described by setting different boundary curves; the coordinates of the center point of the particles are generated by random numbers;
2) Carrying out finite element analysis calculation on a three-dimensional numerical model of a composite material characterization volume unit (RVE) with randomly distributed particles to obtain a numerical solution of effective properties of the numerical model of the characterization volume unit (RVE);
(1) meshing the characterized volume elements (RVEs);
(2) applying thermodynamic boundary conditions to an interface of the composite material consisting of the matrix and the particles and distributed randomly to obtain a numerical solution representing the effective thermodynamic properties of volume units (RVEs) under a finite element method;
3) Establishing a random homogenization model to solve the macroscopic effective properties of the composite material;
selecting a sample space n according to unknown parameters in a three-dimensional numerical model of composite material characterization volume units (RVEs) with randomly distributed particles, calculating the n composite material characterization volume units (RVEs) with randomly distributed particles to obtain a series of random numerical solutions, performing statistical processing on the numerical solutions by using a mathematical statistics method, and taking the average value of the mathematical statistics as a predicted value of the macroscopic effective property of the composite material with randomly distributed particles.
Further, in the step 1), a FORTRAN language is used to construct a three-dimensional numerical model of the composite material characterization volume units (RVEs) with randomly distributed particles based on a random sequence addition method (RSA) method, wherein spherical or ellipsoidal particle inclusions are generated in a three-dimensional matrix material one by one, and the center-to-center distance between any two particles is larger than or equal to the diameter of the particles, namely the particles cannot overlap.
Further, in the step 1), a composite material characterization volume unit (RVE) three-dimensional numerical model with randomly distributed particles is constructed by adopting a random sequence addition method, and the method comprises the following steps:
1a) V is found from the particle volume fraction 2 And number of particles N p Calculating the radius R of the particles:
1b) Generation of primary particles P by means of random numbers 1 Coordinate of center point (x) 1 ,y 1 ,z 1 ) Generating second particles P 2 Coordinate of center point (x) 2 ,y 2 ,z 2 ) Calculating (P) 1 ，P 2 ) Center distance L of 12 ：
1c) If L is 12 Not less than 2R, the second particle meets the requirement, and then the third particle P is generated 3 And judge (P) 1 ，P 3 ) Center distance L of 13 、(P 2 ，P 3 ) Center distance L of 23 Whether each is greater than 2R; if L is 12 &2R, then the second particle P is regenerated 2 Until no particle overlap occurs.
Further, in the step 2), thermodynamic boundary conditions are applied to obtain a numerical solution of effective thermodynamic properties of the volume units (RVEs) under a finite element method, and the method is implemented by the following steps:
2a) Carrying out mesh division on the characterization volume units RVE, and selecting eight-node hexahedron units;
2b) For application to heterogeneous materialsThe general thermodynamic boundary value problem ofThe linear momentum balance equation above is:
coupling its solution to the energy balance equation yields:
wherein, div [ ·]Expressing divergence, P being the first Petalo-kirchhoff stress, P 0 For reference to mass density, x is the position vector, e isInternal energy per unit volume, F is deformation gradient, q 0 Representing the heat flow vector, f (r) is the bulk stress per unit mass independent of deformation, r represents the heat supply per unit mass of the material;
for application to heterogeneous materialsThe constitutive equation of the above general thermodynamic boundary value problem is as follows:
the constitutive constraints derived from the second law of thermodynamics are as follows:
wherein η isEntropy per unit volume, theta denotes temperature, g 0 Indicating a temperature gradient,. Psi.Helmholtz free energy per unit volume, heat dissipationJ＝det[F]And is provided withAnd is provided withIs the only dissipation source in the thermodynamic boundary value problem.
In the first-order homogenization framework, by using Legendre transformObtaining thermodynamic compatibility conditions that ensure the response of the macro-scale components:
wherein c isSpecific heat per unit volume at constant deformation;
2c) In the finite element solution method, the numerical solution microstructure characterizing the effective thermodynamic properties of the volume elements RVE is obtained by a two-step solution process: a mechanical solving phase and a thermal solving phase.
In the mechanical solution phase, by applying macroscopic deformations F to the original microstructureAnd maintaining the temperature atThe microstructure after deformation can be obtainedAnd the numerical solution of the mechanical macroscopic parameter is expressed as
In the thermal solution phase, the macroscopic temperature gradient isApplied to a fixed microstructureThereby obtaining a numerical solution of the thermal macroscopic parameter, the equation of which is expressed as
Further, in the step 3), a random homogenization model is established to solve the macroscopic effective properties of the composite material, and the method is realized by the following steps:
3a) Establishing a random homogenization model
Using the property parameters (k) of the matrix and the particles 1 ,μ 1 ,k 2 ,μ 2 ,α 0 ,k c ,γ 1 ,γ 2 ,γ 3 ,β 1 ,β 2 ,β 3 ) And volume fraction v of particles 2 To indicate the macroscopically effective nature of the composite material, i.e. the effective amount, i.e.
In the formula (I), the compound is shown in the specification,represents the macroscopically effective property, k, of the composite material 1 And mu 1 The bulk and shear moduli, k, of the matrix, respectively 2 And mu 2 The bulk modulus and shear modulus of the particles, respectively; alpha is alpha 0 Is the coefficient of thermal expansion, k, of the substrate c Is the thermal conductivity of the substrate; { gamma. } was prepared from a mixture of two or more of the above-mentioned compounds 1 ,γ 2 ,γ 3 ,β 1 ,β 2 ,β 3 Is a parameter of the Ogden material, satisfyingf denotes that each effective amount is related to a material parameter (k) 1 ,μ 1 ,k 2 ,μ 2 ,α 0 ,k c ,γ 1 ,γ 2 ,γ 3 ,β 1 ,β 2 ,β 3 ,v 2 ) Subscript FEM indicates solving using finite element method;
when considering the randomness of the distribution of the positions of the particles in the representative volume units (RVEs), firstly selecting the sample capacity, namely selecting n representative volume units (RVEs), wherein the positions of the particles in each RVE are different, and the difference of the positions of the particles directly determines the difference of effective properties;
3b) Efficient characterization of individual volume units (RVEs)
For the ith RVE, the effective properties are determined
Wherein, the first and the second end of the pipe are connected with each other,represents the effective properties of the i-th characterized volume unit (RVE), i = (1, 2, … n);
3c) Mathematical statistics of the effective properties of n characterized volume units (RVEs)
The effective properties of the n RVEs are obtained through the steps, and mathematical statistics is carried out on the effective properties.
Further, in the step 3 a), the macroscopic effective properties of the composite material comprise stress tensorTensor of effective heat fluxThe deformation gradient tensor F.
Further, in the step 3 c), the mathematical statistics of the effective properties of the n volume units (RVE) are performed, including the following steps:
the 2-norm of the tensor matrix is solved, and then the mean value and the mean square error of the tensor matrix are solved, wherein the expression is as follows:
in the formula (I), the compound is shown in the specification,a 2-norm representing each effective tensor,is the average of n volume units of interest (RVE) tensor,is the mean square error of the n volume units of interest (RVE) effective tensors; calculated by the above formulaValue sumI.e. the effective constant under the homogenization model when the particles are randomly distributed.
Further, in the step 3 c), the mathematical statistics of the effective properties of the n volume units (RVE) are performed, including the following steps:
directly solving the mean value and mean square error of each element in the tensor matrix, wherein the result is also in a matrix form and is expressed as follows:
wherein a represents each element in each tensor matrix, and E (a) and D (a) represent the mean and mean square error thereof, respectively;
the mean and mean square error matrix obtained in this way are in the same form as the original tensor matrix.
Compared with the prior art, the invention has the following advantages:
the research content of the invention has obvious prospective and challenging properties, the research work belongs to the leading-edge subject in the fields of continuous medium mechanics, thermodynamics and material science, and the invention has practical application prospect, academic value and theoretical significance. In terms of the present invention, it has the following features:
(1) The randomness of the component parameters of the heterogeneous material and the correlation among the parameters are comprehensively considered;
(2) A three-dimensional model is established, and the numerical modeling problem of the complex RVE is solved;
(3) The powerful calculation function of the Fortran language is fully utilized, a more accurate calculation result than the traditional mesomechanics boundary value analysis is obtained, and a foundation is laid for the optimization design of the heterogeneous material structure.
Drawings
FIG. 1 is a flow diagram of an embodiment of the present invention.
Fig. 2 is a schematic representation of the replacement of macrostructural heterogeneous materials with homogeneous equivalent materials.
Figure 3 is a graph of the stress, deformation gradient, heat flow profile of the RVE considering only the randomness of the elements.
FIG. 3 (a) is a μ -3 σ distribution diagram of deformation, stress, and energy.
FIG. 3 (b) is a graph of the mean μ distribution of deformation, stress, and energy.
Fig. 3 (c) is a μ +3 σ distribution diagram of deformation, stress, and energy.
Figure 4 is a thermal flow profile of an RVE taking into account randomness and correlation of elements.
FIG. 4 (a) is a heat flow μ -3 σ profile of RVE.
FIG. 4 (b) is a heat flow μ profile of RVE.
FIG. 4 (c) is a heat flow μ +3 σ distribution diagram of RVE.
Detailed Description
The following further describes embodiments of the present invention with reference to the drawings.
As shown in FIG. 1, there are three main processes of the present invention. Firstly, a three-dimensional RVE (volume characterized Unit) model of random distribution of particles is constructed by using FORTRAN language based on a random sequence addition method (RSA). Meanwhile, on the basis of a finite element method, a numerical solution of the macroscopic effective quantity of the material under the condition of a linear boundary is obtained through a Monte Carlo simulation method, a random homogenization model is established to solve the macroscopic effective property of the composite material, and a three-dimensional distribution diagram of an RVE internal deformation gradient field, a stress field and a thermal flow field is drawn. The specific technical scheme is as follows:
1. construction of a stochastic RVE numerical model
A method for constructing a composite material volume characterized (RVE) three-dimensional numerical model of random distribution of particles by using FORTRAN language based on a random sequence addition method, namely an RSA method, is to generate spherical or ellipsoidal particle inclusions in a three-dimensional matrix material one by one, and the center distance between any two particles is larger than or equal to the diameter of the particles, namely the particles cannot overlap, and comprises the following steps:
according to the volume fraction v of the particles 2 And number of particles N p Calculating the radius R of the particles:
generating random numbers to obtain primary particles P 1 Coordinate of center point (x) 1 ,y 1 ,z 1 ) And a second particle P 2 Coordinate of center point (x) 2 ,y 2 ,z 2 ) Calculating (P) 1， P 2 ) Center distance L of 12 ：
If L is 12 Not less than 2R, the second particle meets the requirement, and then the third particle P is generated 3 And judge (P) 1 ，P 3 ) Center distance L of 13 、(P 2 ，P 3 ) Center distance L of 23 Whether each is greater than 2R; if L is 12 &2R, then the second particle P is regenerated 2 Until no particle overlap occurs.
When determining the center point coordinates of the nth (n is more than or equal to 2) particles, calculating the center distance between the nth particle and all the first n-1 particles, if L is 1n ,L 2n ,...,L n-1,n If the n-1 center distances are all larger than 2R, the generation of the nth particle is successful, otherwise, the coordinates of the center point of the nth particle are determined again until the nth particle does not intersect with all generated particles. Fig. 2 is a schematic representation of the substitution of a macrostructural heterogeneous material with a homogeneous equivalent material.
2. Thermodynamic analysis using finite element method
2.1 Meshing of characterized volume elements RVE
And an eight-node hexahedron unit is selected to perform gridding division on the RVE, and the density of the gridding can be properly increased according to the increase of the volume fraction of the particles.
2.2 Homogenization framework based on continuous media mechanics
2.2.1 Equation of equilibrium
Heterogeneous materialIs of the reference (now) configurationThe position vector is denoted X (X), the boundaryThe unit of upper external normal vector is N (N), infinitesimal volume is dV (dV), infinitesimal surface area dA (dA), absolute temperature value θ.The gradient operator above is expressed asThe divergence operator is represented as Div [ ·](Div[·]). Functional of position changeThe passing time t willAndin connection with this:the deformation gradient of the existing configuration is expressed asIf J = det [ F ]]Then dv = JdV. From the deformation gradient, the right (left) cauchy-green strain C = F can be obtained T F(b＝FF T ) And projecting a first Pierce kirchhoff stress P to kirchhoff stress τ = PF T And Coxis stress T = J -1 τ. The temperature gradient and the heat flow vector obtained in the reference (present) configuration are respectively represented as And q is 0 (q＝J -1 Fq 0 )。
For application toThe general thermodynamic Boundary Value Problem (BVP) ofThe linear momentum balance equation of
Coupling its solution to an energy balance equation
Where ρ is 0 (ρ＝J -1 ρ 0 ) For reference to the (now) mass density, f (r) is the unit mass stress independent of deformation, and e is the internal energy. For the sake of simplicity, the following thermodynamic quantities are all expressed asIn the case of a unit volume.
2.2.2 Limitation of the structure
The constitutive equation for the thermodynamic BVP problem is as follows:
due to the particularity of the thermoelastic response, there are no internal variables in the constitutive equation, and since η = η (F, θ), there areThe internal energy derives the Helmholtz free energy, the Legendre transformation form of which is psi = psi (F, theta) = e-theta eta, and can be obtained therefromThus, act on P and q 0 The constitutive limits of (c) can be derived from the second law of thermodynamics as follows:
wherein the heat is dissipatedAnd is provided withIs the only dissipation source in the existing problem. While other constitutive constraints derived from the observation invariance requirements for the functional form are such that ψ and q 0 Independent of the curl of F.
The energy balance equation (4) can be expressed using ψ as follows:
whereinTo representThe specific heat per unit volume at constant deformation,indicating the koff-joule effect. This effect is generally considered to be very weak and ignored.
However, this amount is reserved here to show the general applicability of the invention. In addition, specific heat is considered to be constant in many thermodynamic analyses.
2.2.3 Thermodynamic potential and entropy in thermoelastic analysis
In the temperature interval theta 0 And theta is in pairIntegration is performed to obtain the following expression
Here, the notation [ ·] 0 (F) Denotes e is at θ 0 Explicit valuation of (1). Also, in the same temperature intervalBy integration, it is possible to obtain
Finally, substituting expressions (8) and (9) into Legendre transformation ψ = e- θ η to obtain
Wherein the content of the first and second substances,therefore, it is necessary to know the isothermal characteristic and the c value at the reference temperature to solve the thermal force potential under the thermoelasticity. The invention neglects the influence of F on the heat conduction performance of rigid media when the rigid media is used for researching the heat conduction performance of the rigid media.
2.3 Problem of homogenization of boundary value
For the macro-structured BVP problem, the solution would be very cumbersome due to the lack of a series of necessary microscopic parameters. The solution is to introduce a suitably small volumetric unit RVE for characterizationBy usingThe effective macroscopic physical properties of the heterogeneous material are determined by the scale-switching relationship between the microscale features and the macroscale response. Characterizing the parameters of the volume units RVE toIn a form described inRepresenting the ordinary parameter tensor of the microstructure. The parameters characterizing the volume units RVE are substituted into the homogenization framework described above to give the so-called primary homogenization framework. Each equation is the same as the above framework, and it is worth mentioning that by using the macroscopic Legendre transformationThe following so-called thermodynamic compatibility conditions may be obtained which ensure the response of the macro-scale components:
wherein c isSpecific heat per unit volume at constant deformation.
2.3.1 Scale conversion
Macro-scale to micro-scale transformation: is at the same timeApplying boundary conditions to the basic variables of the macro structure at specific pointsUsed in microstructure analysis and obtained therefromSolution { x, θ } in (1).
Micro-scale to macro-scale transformation: substituting the microscale solution { x, θ } into the component constitutive equation can yield the overall tensor field variable a, which will be used in the following volume averaging process over region Ω to solve the macroscopic BVP problem:
the macro-scale basic thermodynamic variables can be expressed as:
2.3.2 applying macroscopic temperatures
Assuming a macrostructureReference configuration ofNo stress or heat flow restriction, and the reference temperature of the heterogeneous material is always theta 0 . The temperature of the homogenized material can be conveniently selected asHere, the formulas (8) and (9) are substituted intoAndby the definition of (12), can be obtained
Herein, willThe key to associating with its corresponding microscopic portion isIf a material subject to the conditions of the primary homogenization frame is strictly purely thermal, then it is inC = c (θ) in all points of (1) and in the thermodynamic equations (14), (15) can be simplified to
WhereinAndare all constant.
In the current homogenization method, a necessary condition for both equation (16) to be satisfied and for a constant first-order homogenization framework to be obtained is to ensure that all temperature-containing functional forms are at macroscopic temperature in the micromechanics solving stepThe evaluation is performed. These forms contain variables
2.3.3 Specific heat at macro scale
To obtainThe temperature under the macro scale is substituted into the expression (14) to obtain
Wherein the micro field quantity F and the macro state thereofAnd (4) associating. To make it possible toSatisfying the compatibility conditions described in formula (11), willAll derived forms of (A) are substituted to obtain
Wherein, the first and the second end of the pipe are connected with each other,expressed in a macroscopically deformed state ofThe gradient of the microscopic deformation field to the macroscopic temperature. The variable and inhomogeneous material units are in constant deformation stateTime temperatureThe energy change caused by the change. When in useIs a constant valueThe deformation of the microstructure can vary, and the corresponding change in internal energy can also result inA change in (c). The following are some special cases:
in the microstructure, when c is independent of deformation, there areAs can be seen from this, it is,still depend onAnd
only in the case of purely nonelastic thermal materials is thereThis is true.
2.4 Micro-mechanics solving procedure
The solution of the deformation field only depends onRegardless of the temperature field caused by the macroscopic temperature gradient. Therefore, the microstructure solving process can be divided into two steps: a mechanical solving phase and a thermal solving phase. In the mechanical solving stage, by deforming macroscopicallyIs applied toAnd maintaining the temperature atThe microstructure after deformation can be obtainedAnd macroscopic parameters independent of the temperature gradient are obtained in this way. In the thermal solution phase, the macroscopic temperature gradient isApplied to a fixed microstructureThereby obtaining a macroscopic parameter based on the temperature gradient.
2.4.1 Mechanics solution phase
Compatibility conditions
For a macroscopic solution applied to the microstructure volume samples:
due to the fact thatAre still balanced, the above equation can be decomposed into the following two equations
And due to the fact thatEstablishedAnd is andindependently of
The compatibility condition of formula (11) can be obtained. However, althoughIndependent process variables at the microscopic scale, F, however, remain dependentOn the other hand, if the following formula is used
Is established by applyingIt is ensured that the compatibility condition (22) is satisfied becauseAndall tend to zero. And, because ofIndependently of F, obtainable from formula (21)
Andimplicit relations
Problem of boundary value
The content of the foregoing compatibility condition section is based on the assumption of the formula (23). The following formula
Can be restated as
The expression is generally called micro-macro work standard and represents the equivalence of mechanical work under macro scale and work under micro scale. Here, it serves as a condition for ensuring thermodynamic compatibilityOne requirement of (a). The expression can be regarded as the constraint of a mechanical solving stage on conditions such as boundaries and the like. The volume average of kirchhoff stress can also be used for the calculation of the fundamental macroscopic stress:
to satisfy equation (26), the use of bulk stress and dynamic conditions is avoided here, since they increase the macroscopic stress, and here only the intrinsic response needs to be found. Thus can ensureP in (1) does not diverge. On the other hand, the discontinuity of the displacement can cause the non-uniformity of the material to be hardly strictly suitable for thermoelastic homogenization frames. Thus, x is inThere is no jump. The mechanics solution phase can be expressed as
And selecting the linear displacement boundary conditions (epsilon-LN-BCs), i.e. applyingAnd obtainFrom the current settings, the appropriate F can be obtained and the ideal applied<P&And can also be used for<P>, and<F&and gt, quantifying the sensitivity of the sample.
2.4.2 Thermal solution phase
From the definition in formula (13)And the corresponding compatibility conditions in formula (11) areApplied to the micro-scale dissipation functionCan obtain
This micro-to-macro heat dissipation criterion ensures that the micro-scale heat dissipation is equivalent to the macro-scale heat dissipation.
The problem of BVP is as followsThe equations of the thermal solution phase can be expressed as
Where F is obtained from the mechanics solving phase. This equation can be appliedUnder the condition of linear displacement boundary, solving. And is of the formula
In this case by<q 0 >, toValue of (2)It is ensured that the formula (28) is satisfied. It is emphasized thatIs only to derive g 0 Its size is independent of dissipation and does not affect its value.
It is worth mentioning that if q is obtained by the heat conduction coefficient tensor K = K (F, θ) at the microscopic scale 0 If it is not variable, then q is 0 ＝-Kg 0 Can obtainDue to K and g 0 Neither the modulus nor the direction of (a) is relevant, so that the thermal solution after a given mechanical solution phase has the same architecture as the classical linear purely thermal problem solution. So as to haveCan obtainThat is, the heat transfer coefficient tensor at the macroscopic scale is not dependent on the temperature gradient
3. Construction of a random homogenization model
3.1 Randomness and cross-correlation of variables
Consider the randomness of the following variables: bulk modulus k of the matrix 1 And shear modulus mu 1 Bulk modulus k of the particles 2 And shear modulus mu 2 The coefficient of thermal expansion of the substrate alpha 0 And coefficient of thermal conductivity k c And the parameters of the Ogden material used { gamma } 1 ,γ 2 ,γ 3 ,β 1 ,β 2 ,β 3 }. Wherein { gamma 1 ,γ 2 ,γ 3 ,β 1 ,β 2 ,β 3 The equation must be satisfiedAssuming that the above variables are all normally distributed, i.e.WhereinRepresenting the above parameters, μ and msd represent the mean and mean square error of the variables, respectively. For each set of determined parameters, a Monte Carlo method can be used to generate n sets of samples that follow a normal distribution. These n sets of samples were substituted into the homogenization step to obtain n different sets of results. And finally, carrying out mathematical statistics on the n groups of effective performance data to obtain the digital characteristic values such as the mean value, the mean square error and the like of the results.
Consider the linear correlation between the following variables: bulk modulus k of the matrix 1 And shear modulus mu 1 Bulk modulus k of meso, granular 2 And shear modulus mu 2 Coefficient of thermal expansion alpha of matrix 0 And coefficient of thermal conductivity k c And Ogden material parameter { gamma } 1 ,γ 2 ,γ 3 ,β 1 ,β 2 ,β 3 }. Wherein, { gamma., (gamma.) 1 ,γ 2 ,γ 3 ,β 1 ,β 2 ,β 3 The correlation between the parameters should be as follows: ρ is a unit of a gradient γ1γ2 ＝ρ γ1γ3 ＝ρ γ2γ3, ρ β1β2 ＝ρ β1β3 ＝ρ β2β3 。
3.2 Performing statistical processing
Using the property parameters (k) of the matrix and the particles 1 ,μ 1 ,k 2 ,μ 2 ,α 0 ,k c ,γ 1 ,γ 2 ,γ 3 ,β 1 ,β 2 ,β 3 ) And volume fraction v of particles 2 To express the macroscopic effective properties of the composite material, i.e. effective amount, such as stress tensor P, effective heat flow tensorDeformation gradient tensor F, etc., i.e
In the formula (I), the compound is shown in the specification,representing macroscopic effective properties of composite materials, e.g. stress tensorTensor of effective heat flowDeformation gradient tensor F, k 1 And mu 1 The bulk and shear moduli, k, of the matrix, respectively 2 And mu 2 The bulk modulus and shear modulus of the particles, respectively; alpha is alpha 0 Is the coefficient of thermal expansion, k, of the substrate c Is the thermal conductivity of the substrate; { gamma. } was prepared from a mixture of two or more of the above-mentioned compounds 1 ,γ 2 ,γ 3 ,β 1 ,β 2 ,β 3 Is a parameter of the Ogden material, satisfyingf denotes that each effective amount is related to a material parameter (k) 1 ,μ 1 ,k 2 ,μ 2 ,α 0 ,k c ,γ 1 ,γ 2 ,γ 3 ,β 1 ,β 2 ,β 3 ,v 2 ) The subscript FEM indicates solving using finite element methods.
When considering the randomness of the distribution of the location of the particles in the RVEs, a sample volume is first selected, i.e. n RVEs are selected, the location of the particles in each RVE being different, and the difference in location of the particles directly determines the difference in effective properties.
3.2.1 Efficient characterization of a Single RVE
For the ith RVE, the effective properties are determined
Wherein, the first and the second end of the pipe are connected with each other,represents the effective properties of the ith RVE, i = (1, 2, \8230n);
3.2.2 Mathematical statistics of n RVE effective properties
The effective properties of n RVEs are obtained through the steps, and the numerical statistics is carried out on the effective properties, and the effective properties have two forms:
the 2-norm of the tensor matrix is found, and then its mean and mean square error are found, expressed as:
in the formula (I), the compound is shown in the specification,a 2-norm representing each effective tensor,is the average of the n RVE effective tensors,is the mean square error of the n RVE effective tensors. Calculated by the above formulaValue sumI.e. the effective constant under the homogenization model when the particles are randomly distributed.
Directly solving the mean value and mean square error of each element in the tensor matrix, wherein the result is also in a matrix form and is expressed as follows:
in the formula, a represents each element in each tensor matrix, and E (a) and D (a) represent the mean value and the mean square error thereof, respectively.
The mean and mean square error matrix obtained in this way are in the same form as the original tensor matrix.
4. Numerical simulation results
On the basis of a finite element method, a random homogenization model for solving macroscopically effective response of the heterogeneous material under the thermoelasticity condition is constructed according to a Monte Carlo simulation method. Under the condition of linear displacement boundary, the volume modulus and the shear modulus of the matrix are respectively set as k 1 ＝4，μ 1 =1; the bulk modulus and shear modulus of the particles are each k 1 ＝40，μ 1 =10. The thermal expansion coefficient and the thermal conductivity coefficient of the base are respectively alpha 0 ＝0.001，k c =0.1. The Ogden material has the following parameters: gamma ray 1 ＝0.660,γ 2 ＝-0.231,γ 3 ＝0.050；β 1 ＝1.8,β 2 ＝-2.0,β 3 =7.0. The correlation coefficient between the parameters was taken to be 0.8.
4.1 Calculated value of effective amount
When volume fraction of particles V 2 Step-wise increment from 0.05 to 0.25 for each V 2 N =1000 times of simulation are carried out according to the homogenization model to obtain the macroscopic effective stress tensor of the materialTensor of effective heat flowAnd the like. When the particle volume fraction is 0.25 and all the parameters are considered to be random, the correlation between the parameters is considered and the correlation is not consideredAndthe mean and mean square error of (d) are listed in the table below.
TABLE 0.25 volume fraction, correlation vs. non-correlationAndmean and variance of
4.2 Physical field
Further, boundaries between the mean values μ and 3 σ of physical fields such as RVE internal stress field, deformation gradient field, and thermal flow field were determined, and the distributions were as shown in fig. 3 (a), (b), (c), and fig. 4 (a), (b), and (c).
The invention has the beneficial effects that:
aiming at random heterogeneous materials under limited deformation, the invention adopts a new random analysis method to construct a random homogenization model, develops an effective thermoelasticization calculation method, and solves each random macroscopic response quantity and statistical characteristic value thereof so as to achieve the purposes of effectively supplementing and predicting experimental observation and providing objective, full and true basis for the optimization design of novel advanced materials and structures.
Claims (9)
1. A method for thermoelastic homogenization of three-dimensional random heterogeneous materials under limited deformation, characterized in that it comprises the following steps:
1) Constructing a three-dimensional numerical model of a composite material characterization volume unit RVE with randomly distributed particles by using a FORTRAN language based on a random sequence addition method RSA;
for the composite material containing the irregular particle inclusions, the particle shape is described by setting different boundary curves; the coordinates of the center point of the particles are generated by random numbers;
2) Carrying out finite element analysis calculation on the RVE three-dimensional numerical model of the composite material characterization volume units with randomly distributed particles to obtain a numerical solution of the effective properties of the RVE numerical model of the characterization volume units;
2a) Carrying out mesh division on the characterization volume units RVE, and selecting eight-node hexahedron units;
2b) Applying thermodynamic boundary conditions to the interface of the composite material which is composed of the matrix and the particles and distributed randomly, and solving a numerical solution for representing the effective thermodynamic properties of the volume units RVE under a finite element method;
for application to heterogeneous materialsThe general thermodynamic boundary value problem ofThe linear momentum balance equation above is:
coupling its solution to the energy balance equation yields:
wherein, div [ ·]Denotes the divergence, P is the first Piorel-Kirchoff stress, P 0 For reference to mass density, x is the position vector, e isInternal energy per unit volume, F is deformation gradient, q 0 Representing the heat flow vector, f (r) is the bulk stress per unit mass independent of deformation, r represents the heat supply per unit mass of the material;
for application to heterogeneous materialsThe above general thermodynamic boundary value problem, its constitutive equation is as follows:
the constitutive constraints derived from the second law of thermodynamics are as follows:
wherein eta isEntropy per unit volume, theta denotes temperature, g 0 Indicating a temperature gradient,. Psi.Helmholtz free energy per unit volume, heat dissipationJ＝det[F]And is provided withAnd isIs the only dissipation source in the thermodynamic boundary value problem;
in the first-order homogenization framework, by using Legendre transformationObtaining thermodynamic compatibility conditions that ensure the response of the macro-scale components:
wherein c isSpecific heat per unit volume at constant deformation;
2c) In the finite element solution method, the numerical solution microstructure characterizing the effective thermodynamic properties of the volume elements RVE is obtained by a two-step solution process: a mechanical solving stage and a thermal solving stage;
3) Establishing a random homogenization model to solve the macroscopic effective property of the composite material;
according to unknown parameters in a three-dimensional numerical model of a composite material characterization volume unit RVE with randomly distributed particles, selecting a sample space n, performing calculation on the n composite material characterization volume units RVE with randomly distributed particles to obtain a series of random numerical solutions, performing statistical processing on the numerical solutions by using a mathematical statistics method, and taking the average value of the mathematical statistics as a predicted value of the macroscopic effective property of the composite material with randomly distributed particles.
2. The method for thermoelastic homogenization of three-dimensional random heterogeneous materials under limited deformation according to claim 1, characterized in that in the step 1), the random sequence addition method RSA is used to construct a RVE three-dimensional numerical model of the composite material characterization volume unit with randomly distributed particles based on FORTRAN language, and spherical or ellipsoidal particle inclusions are generated in a three-dimensional matrix material one by one, and the center-to-center distance between any two particles must be greater than or equal to the particle diameter, i.e. the particles cannot overlap.
3. The thermoelastic homogenization method of three-dimensional random heterogeneous materials under limited deformation according to claim 1, wherein in the step 1), a random sequence addition method RSA is adopted to construct a composite material characterization volume unit RVE three-dimensional numerical model with randomly distributed particles, and the method comprises the following steps:
1a) According to the volume fraction v of the particles 2 And number of particles N p Calculating the radius R of the particles:
1b) Generation of primary particles P by means of random numbers 1 Coordinate of center point (x) 1 ,y 1 ,z 1 ) Generating second particles P 2 Coordinate of center point (x) 2 ,y 2 ,z 2 ) Calculate (P) 1 ,P 2 ) Center distance L of 12 ：
1c) If L is 12 Not less than 2R, the second particle meets the requirement, and then the third particle P is generated 3 And judge (P) 1 ，P 3 ) Center distance L of 13 、(P 2 ，P 3 ) Center distance L of 23 Whether each is greater than 2R; if L is 12 If < 2R, a second particle P is formed 2 Until no particle overlap occurs.
4. Method for the thermoelastic homogenization of three-dimensional random heterogeneous materials under limited deformation according to claim 1, characterized in that said mechanical solving phase is carried out by means of macroscopic deformationApplied to the original microstructureAnd maintaining the temperature atObtaining a deformed microstructureAnd obtaining a numerical solution of the mechanical macroscopic parameter, wherein the equation is expressed as
5. The method for thermoelastic homogenization of three-dimensional random heterogeneous materials under limited deformation according to claim 1, wherein the thermal solution phase is a macroscopic temperature gradientApplied to a fixed microstructureThereby obtaining a numerical solution of the thermal macroscopic parameter, the equation of which is expressed as
6. The thermoelastic homogenization method of three-dimensional random heterogeneous materials under limited deformation according to claim 1, wherein in the step 3), a random homogenization model is established to solve the macroscopic effective properties of the composite material, which is realized by the following way:
3a) Establishing a random homogenization model
Using the property parameters (k) of the matrix and the particles 1 ,μ 1 ,k 2 ,μ 2 ,α 0 ,k c ,γ 1 ,γ 2 ,γ 3 ,β 1 ,β 2 ,β 3 ) And volume fraction v of particles 2 To indicate the macroscopically effective, i.e. effective, amount of the composite material, i.e.
In the formula (I), the compound is shown in the specification,represents the macroscopically effective property, k, of the composite material 1 And mu 1 The bulk and shear moduli, k, of the matrix, respectively 2 And mu 2 The bulk modulus and shear modulus of the particles, respectively; alpha is alpha 0 Is the coefficient of thermal expansion, k, of the substrate c Is the thermal conductivity of the substrate; { Gamma 1 ,γ 2 ,γ 3 ,β 1 ,β 2 ,β 3 Is a parameter of the Ogden material, fullFoot (A)f denotes that each effective amount is related to a material parameter (k) 1 ,μ 1 ,k 2 ,μ 2 ,α 0 ,k c ,γ 1 ,γ 2 ,γ 3 ,β 1 ,β 2 ,β 3 ,v 2 ) Subscript FEM indicates solving using a finite element method;
when considering the randomness of the distribution of the positions of the particles in the RVEs, firstly selecting a sample volume, namely selecting n RVEs, wherein the positions of the particles in each RVE are different, and the difference of the positions of the particles directly determines the difference of effective properties;
3b) Efficient characterization of individual volume characterizing units RVE
For the ith characteristic volume element RVE, the effective properties are determined
Wherein, the first and the second end of the pipe are connected with each other,represents the effective properties of the i-th characterized volume unit RVE, i = (1, 2.. N);
3c) Performing mathematical statistics on effective properties of n characterization volume units RVE
The effective properties of n characterization volume units RVE are obtained through the steps, and the mathematical statistics is carried out on the effective properties.
7. The method for thermoelastic homogenization of three-dimensional random heterogeneous materials under limited deformation according to claim 6, characterized in that in step 3 a) the macroscopically effective properties of the composite material comprise the stress tensorTensor of effective heat flowThe deformation gradient tensor F.
8. The method for thermoelastic homogenization of three-dimensional random heterogeneous materials under limited deformation according to claim 6, wherein the step 3 c) of performing mathematical statistics on the RVE effective properties of n volume units characterized comprises the following steps:
the 2-norm of the tensor matrix is solved, and then the mean value and the mean square error of the tensor matrix are solved, wherein the expression is as follows:
in the formula (I), the compound is shown in the specification,a 2-norm representing each effective tensor,for the average of the n RVE effective tensors characterizing the volume element,is the mean square error of the RVE effective tensors of the n characterizing volume units; calculated by the above formulaValue sumI.e. the effective constant under the homogenization model when the particles are randomly distributed.
9. The method for thermoelastic homogenization of three-dimensional random heterogeneous materials under limited deformation according to claim 6, wherein the step 3 c) of performing mathematical statistics on the RVE effective properties of n volume units characterized comprises the following steps:
directly solving the mean value and mean square error of each element in the tensor matrix, wherein the result is also in a matrix form and is expressed as follows:
wherein a represents each element in each tensor matrix, and E (a) and D (a) represent the mean and mean square error thereof, respectively; the mean and mean square error matrix obtained in this way are in the same form as the original tensor matrix.
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