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MINISTRY OF EDUCATION VIETNAM ACADEMY OF AND TRAINING SCIENCE AND TECHNOLOGY GRADUATE UNIVERSITY SCIENCE AND TECHNOLOGY ----------------------------------- NGUYEN THI HUONG GIANG EQUIVALENT-INCLUSION APPROXIMATION FOR EFFECTIVE PROPERTIES OF COMPOUND-INCLUSION COMPOSITES Major: Mechanics of Solid Code: 944 01 07 SUMMARY OF MECHANICS DOCTORAL THESIS HA NOI 2020 The thesis has been completed at: Graduate University Science and Technology-Vietnam Academy of Science and Technology. Supervision: 1. Prof. Dr. of Sci. Pham Duc Chinh 2. Assoc. Prof. Dr. Tran Bao Viet Riviewer 1: Prof. Dr. Hoang Xuan Luong Riviewer 2: Assoc. Prof. Dr. Nguyen Trung Kien Riviewer 3: Assoc. Prof. La Duc Viet Thesis is defended at Graduate University Science and Technology Vietnam Academy of Science and Technology at................on................. Hardcopy of the thesis be found at: - Library of Graduate University Science and Technology - Vietnam national library 1 PREFACE 1. Urgency of thesis In the producing composite materials, due to the chemical reaction between the matrix phase and inclusion phase or by the coating technique of fibers to form an intermediate phase (the shell surrounding the inclusion - interface), which in the thesis is called materials with compound inclusion. Compound inclusion is understood as the inclusion and the shell surrounding its. This shell has mechanical - physical properties unlike the matrix or inclusion and affects the macroscopic properties of the material. Then, if the upper and lower bounds method is used to determine the effective values, the bounds are often far apart, with little actual value. The effective medium approximation methods analyzed using a cylindrical or multilayered spherical model are relatively complex when used for calculation engineers. Therefore, the research direction in the thesis is determination of macro mechanical - physical properties of compound - inclusion composite materials, using equivalent inclusion method to give simple approximation formulas. They are suitable for engineers to initially assess the mechanical properties of the materials used. Numerical simulations by finite element method are also applied to test the correctness of the approximation formula. 2. Objectives of the thesis Building approximation formulas using equivalentinclusion method to determine the effective values of the conductivity, the elastic modulus of compound-inclusion composite materials and applying numerical methods using finite elements (FE) calculated for some specific material models. 3. Contents of the thesis Determine the effective conductivity of unidirectional coated-fiber composite materials, the coating shell isotropic or anisotropic (chapter 2). Determine the effective elastic modulus of compound – spherical inclusion composite and unidirectional coated-fiber composite materials (chapter 3). 2 Applying numerical method using finite element to calculate for some periodic composite materials (chapter 4). CHAPTER 1. OVERVIEW 1.1. Classification composite material 1.1.1. Based on inclusions Composite has 2 types: particle reinforced and fiber reinforced. 1.1.2. Based on matrices Composite has 3 types: polymer matrix, metal matrix, ceramic matrix. 1.2. Conductivity Thermal conductivity c, electrical conductivity c, diffusion D, fluid permeability k, electric permittivity ϵ, magnetic permeability μ. They have the same mathematical formula and satisfies the same equilibrium equation. Therefore, in the thesis uses thermal conductivity in calculations and illustrative examples to characterize the conductivity. 1.3. Elastic modulus Young’s modulus E, shear modulus µ, bulk modulus k, Poisson’s ratio ν. 1.4. Representative Volume Element The representative volume element (RVE) must be large enough for the microstructures to represent the properties of the composite material and at the same time small enough for the size of the object to determine the macroscopic properties. Fig 1.3 . Representative volume element RVE 3 V and having isotropic conductivities c , elastic modulus k ,  (α = 1,…, n). The RVE consists of n components occupying regions V 1.5. Approximations and bounds on the effective properties of composite materials 1.5.1. Effective medium approximations 1.5.1.1. Differential approximation n dc 1   I cI  c Dc cI , c  dt 1   I t  1 (1.34) and coupled differential equations: n 1  dk  dt  1   t  I k I  k Dk k I ,  I , k ,    I  1  n  d  1     D k ,  , k ,   I I  I I   dt 1   I t  1 0  t  1, n  I   I (1.36)  1 with conditions c0  cM , k 0  k M ,  0   M Dc , Dk , D - the dilute suspension expressions for every inclusion type α. The effective values are solutions of corresponding equations when t = 1. 1.5.1.2. Self-consistent approximations n  I c I  c Dc c I ,c   0 (1.38)  1 and coupled differential equations:  n   I k I  k Dk k I ,  I , k ,    0  1  n      D k ,  , k ,    0  I I  I I   1 1.5.1.3. Mori – Tanaka approximation (1.39) 4 For two-component isotropic composites, Mori-Tanaka approximation can be given as:   cMTA  cM   I  cI  cM  M   1   p1  p2  p3   cI  cM   I  , 1  3cM     kMTA   kM   I  k I  kM  M  MTA    M   I   I  M  M   1 k I  kM   Pk d kM  1  1    I  , .  d  1  I  M   P M 2 1  1     I  . 2    (1.40) 1.5.1.4. Some other approximation - Three-point correlation approximation: related to threepoint correlation information about components’ microgeometries. - Polarization approximations: was started from the minimum energy principle and polarization field Hashin-Strikman. 1.5.2. Bounds on the effective propreties 1.5.2.1. 1.5.2.2. Hill – Paul bounds Hashin – Strikman bounds Hashin and Strikman have developed their own variational principle and introduced a polarization field with different medium values on different phases and made a new estimation that is better than Hill – Paul’s.  2  d  1   2  d  1  Pk  max   k eff  Pk  min  d d     P  *max    eff  P  *min  Pc   d  1 cmax   ceff  Pc   d  1 cmin  1.5.2.3. Pham Duc Chinh bounds (1.45) (1.46) (1.47) 5 Derived from minimum energy principle and constructed a polarization field similar to the Hashin field, Pham found tighter estimates than Hashin bounds thanks to the three-point correlation information about the micro-structure of a composite 1.5.3. Equivalent-inclusion approximation In some studies, when calculating the effective values of compound-inclusion composite material, coated-inclusion was substituted by the equivalent homogeneous one of the same size and corresponding mechanical properties. Some works such as Hashin calculate thermo-elastic properties of fiber-reinforced composites with thin interface. Qui and Weng searched the bulk modulus of particular composite material with a thin layer of inclusion. D. C. Pham and B. V. Tran used Maxwell's approximation and equivalent-inclusion model to find the conductivity of compound-inclusion composites. 1.6. Numerical method One of researche methods in the homogenization of materials is the numerical method, in which classical digital has constructed approximately dynamically possible fields. Common is the numerical method using finite elements (PTHH). 1.7. Conclusion With understanding the development history of materials science for the determination of macroscopic physicomechanical properties , author have an overview and a knowledge base. With the selection an equivalent-inclusion approximation, the author desires to have an simple analytical approach, suitable for wide application to calculation engineers and will be presented in later chapters of the thesis. CHAPTER 2. EQUIVALENT-INCLUSION APPROXIMATION FOR CONDUCTIVITY OF COMPOUND-INCLUSION COMPOSITES 2.1. Unidirectional coated-fiber composite with isotropic coating 2. 1.1. Model 6 Fig 2.1. Unidirectinal fiber composite 2.1.2. Formulas for conductivity of circle inclusion 2.1.2.1. Hashin-Shtrikman bounds (HS) 2.1.2.2. Differential approximation (VP) Effective conductivity c eff is solution of equation: 1/2  c1   eff  c  c2  ceff  1 c2  c1 (2.23) 2.1.2.3. Three-point correlation approximation (TT3Đ) 1   2  c  Pc 2 (c0 )   1  (2.24)   c0  c0  c1 c0  c2  c0 is the positive solution of equation: c0  Pc 2  c0  ,  - three eff point correlation parameters. 2.1.3. Equivalent-inclusion approximation for coated circles (b) (a) Fig 2.4. (a) - Coated circles in an infinite matrix (b) - Equivalent inclusion in an infinite matrix 7 1 2 3 T   ;T   ;T   are temperature fields on phases (1), (2), (3): B  B    1 2 T     A1r  1  cos ; T     A2 r  2  cos ; T 3  A3r cos r  r    Constants A, B are determined by the following conditions: - Continuous temperature between phases: T    r2 ,   T    r2 ,  ; T    r3 ,   T    r3 ,  (2.38) - Continuous flux between phases: 2 3 1 2 (2.39) q   n  q    n ; q   n  q    n - Boundary temperature r1: 1 T    r1 ,    .r1 cos . (2.40) β – gradient temperature, β<<1. 1 2 2 3 2c1c2  c2  c3 2  4c1c2 c33 c  c c  c   c2  c1   c2  c3   1 2 2 2 3 r32 r2 q   2c1  c2  c3 2  4c1c23  T  1  c  c c  c   c2  c1   c2  c3   1 2 2 2 3 r32 r2 1c1  c eff (2.60) If the coated inclusions are substituted by equivalent homogeneous inclusions with a volume ratio 23  2  3 , conductivity c23 (fig. 2.4b): 2c1 1c1  23c23 c23  c1 (2.62) c eff  2c1 1  23 c23  c1 Identify (2.60) – (2.62), conductivity of equivalent inclusion is defined: 8 c23  c2  with 2'  2 3' ' 1  2 c3  c2 2c2 2  3 ; 3'  , 3 2  3 (2.64) . (2.65) ceff in (2.62) have expression: c eff 1  23    c1    1   c1  1  1  c23  c1 2c1  c23  c1 2c1 23 (2.66) Applying the differential approximation scheme of (2.23) to the equivalent medium, one can get the differential equivalentinclusion approximation for the suspension of coated circles as the solution of : 1/2 eff  c1  c23  c (2.67)  eff  c  c  1 c  23 1 If additional correlation information about microgeometry of a particular equivalent medium suspension is available, then one can apply equation (2.24) to get the correlation equivalent-inclusion approximation for our suspension of coated circles 1     3  ceff   1  2   c0  c1  c0 c0  c23  where c0 is solution of: 1 (2.68)   2  c0   1  (2.69)   c0  c1  c0 c0  c23  Generally, different coated circular inclusions may be made from different materials, or even be made from the same materials, but with different volume proportions of the inner circles and coated (fig. 2.1 c).