Tóm tắt Luận án Equivalent - Inclusion approximation for effective properties of compound-inclusion composites

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 equivalent- inclusion 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 The RVE consists of n components occupying regions VV  and having isotropic conductivities c , elastic modulus k ,  (α = 1,, n). 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 IcII I ccDcc tdt dc 1 , 1 1    (1.34) and coupled differential equations:                          n IIII I n IIkII I kkD tdt d kkDkk tdt dk 1 1 ,,, 1 1 ,,, 1 1             n II,t 1 10   (1.36) with conditions       MMM kkcc   0,0,0  D,D,D kc - the dilute suspension expressions for every inclusion type α. The effective values are solutions of correspon- ding equations when t = 1. 1.5.1.2. Self-consistent approximations     0 1   n IcII c,cDcc   (1.38) and coupled differential equations:                      0 0 1 1 n IIII n IIkII ,k,,kD ,k,,kDkk       (1.39) 1.5.1.3. Mori – Tanaka approximation 4 For two-component isotropic composites, Mori-Tanaka approximation can be given as:     1 1 2 31 , 3 MTA M I I M M I M I M p p p c c c c c c c                        1 1 1 ,I MMTA M I I M M k I M k k k k k k P d k d                    .   1 1 1 . 2 2 I M MTA M I I M M I M P                           (1.40) 1.5.1.4. Some other approximation - Three-point correlation approximation: related to three- point correlation information about components’ microgeome- tries. - Polarization approximations: was started from the mini- mum energy principle and polarization field Hashin-Strikman. 1.5.2. Bounds on the effective propreties 1.5.2.1. Hill – Paul bounds 1.5.2.2. 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.     max min 2 1 2 1eff k k d d P k P d d                  (1.45)    *max *min effP P     (1.46)      max min1 1 eff c cP d c c P d c    (1.47) 1.5.2.3. Pham Duc Chinh bounds 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 appro- ximation and equivalent-inclusion model to find the conduc- tivity 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 physico- mechanical properties , author have an overview and a know- ledge base. With the selection an equivalent-inclusion appro- ximation, 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 effc is solution of equation: 1/2 1 2 1 2 1 eff eff c c c c cc        (2.23) 2.1.2.3. Three-point correlation approximation (TT3Đ) 1 1 2 2 0 0 0 1 0 2 ( )eff cc P c c c c c c               (2.24) c0 is the positive solution of equation:  0 2 0cc P c ,  - three point correlation parameters. 2.1.3. Equivalent-inclusion approximation for coated circles (a) (b) 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):  1 1 1 cos ; B T A r r          2 2 2 cos ; B T A r r          3 3 cosT A r  Constants A, B are determined by the following conditions: - Continuous temperature between phases:        1 22 2, , ;T r T r         2 33 3, ,T r T r  (2.38) - Continuous flux between phases:     1 2   q n q n ;     2 3   q n q n (2.39) - Boundary temperature r1:    1 1 1, . cos .T r r   (2.40) β – gradient temperature, β<<1.                 1 2 2 3 2 1 2 3 3 1 1 1 2 2 3 2 2 1 2 3 32 2 1 2 3 2 1 2 3 1 1 2 2 3 2 2 1 2 3 32 2 2 4 2 4 eff c c c c c c c c c c c c c c c c r rq c c c c c cT c c c c c c c c r r                            (2.60) If the coated inclusions are substituted by equivalent homogeneous inclusions with a volume ratio 23 2 3    , conductivity 23c (fig. 2.4b): 1 1 1 23 23 23 1 1 1 23 23 1 2 2 eff c c c c c c c c c          (2.62) Identify (2.60) – (2.62), conductivity of equivalent inclusion is defined: 8 ' 3 23 2 ' 2 3 2 2 , 1 2 c c c c c       (2.64) with ' ' 322 3 2 3 2 3 ;            . (2.65) ceff in (2.62) have expression: 1 23 23 1 1 1 1 23 1 1 23 1 1 1 2 2 effc c c c c c c c c                 (2.66) Applying the differential approximation scheme of (2.23) to the equivalent medium, one can get the differential equivalent- inclusion approximation for the suspension of coated circles as the solution of : 1/2 231 1 23 1 eff eff c cc c cc        (2.67) If additional correlation information about microgeome- try 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 2 31 0 1 0 0 23 effc c c c c c             (2.68) where c0 is solution of: 1 1 2 0 0 1 0 0 23 c c c c c c             (2.69) 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). 9 1 1 1 1 1 231 23 2 c ccc c n eff                    (2.70) For example 1 11 2 3 2 31, 5, 20 ( . . );c c c W m K        and 1 11 2 3 2 320, 5, 1( . . );c c c W m K        . Effective conduc- tivity depend on volume fraction, is shown on figures 2.5, 2.6. Fig. 2.5. 1 2 31, 5, 20c c c   Fig. 2.6 1 2 320, 5, 1c c c   where HSU, HSL – upper and lower bounds Hashin- Strikman, VP-CTĐ : equation results (2.67), ĐTLN: equation results (2.66) , TT-CTĐ: euqation results (2.68) with random distribution of inclusion. 2.1.4. Experiment result Fig 2.8. Transverse conductivity of abaca fiber composites Liu et al. measured the transverse conductivity of abaca fiber composite by flash diffusion technique (TN) 10 2.2. Unidirectional coated-fiber composite with anisotropic coating 2.2.1. Model (a) (b) Fig 2.9. Unidirectional coated-fiber composite with anisotropic coating. Suppose coated-fiber with thin coating of small volume proportion Δ 1  , which itself is composed of 2m ultra-thin coatings of volume proportion Δ / 2m , and conductivities 12c and 22c alternately. Conductivity coated-fiber 23c :         1 1 2 2 2 3 3 2 2 3 3 2 2 23 3 1 2 2 2 2 2 2 c c c c c c c cΔ c c m m Ο Δ m c c                      22 223 3 3 Δ 2. I N c c c c Ο Δ c         (2.74) with:   1 2 1 2 1 22 2 2 2 2 21 2 2 2 21 , , . 2 T N I T N c c c c c c c c c c c c c       (2.75) When volume ratio of coating is significant, similar to the differential approximation, from (2.74) we construct:     2 2 1 1 2 I N c cdc d c     (2.77) with boundary condition:    3 23 20 ;c c c c   (2.78) In the case cN = const, cT =const, equation (2.77) can be integrated explicitly: c1 (a) (b) c 3 cT cn c1 (a) ( ) c 3 cT cn 11           ' 3 3 3 ' 3 23 3 3 2' 3 3 3 , I N I N c c I I I I c c I I c c c c c c c c c c c                  (2.80) Effective conductivity effc 1 23 1 1 23 1 12 effc c c c c            (2.81) 2.3. Conclusion Chapter 2 has developed approximate formulas for determining the effective conductivity coefficient of coated-fiber composite with isotropic or anisotropic coating by equivalent- inclusion approach. Research results in this chapter have been published in scientific works [2, 4, 7, 8]. CHAPTER 3. EQUIVALENT-INCLUSION APPROXIMATION FOR ELASTIC MODULUS OF COATED-INCLUSION COMPOSITES 3.1. Elastic modulus of coated spheres Fig. 3.1. (a) a three-component suspension of coated spheres ; (b) - Equivalent homogeneous-inclusion suspension 3.1.2. Bulk modulus We assume that the loading applied to domain is defined by the tensor E representing the strain state at the macroscopic scale, displacement fields of phases: 2 1 2 2 1 12 2 2 , ,I I MrI I rI I M M B B B u A r u A r u A r r r r       (3.4) Stress fields of phases: 12 2 1 2 2 2 2 1 1 1 13 3 3 3 4 , 3 4 , 3 4 . I I rrI I I I rrI I I I M rrM M M M B B k A k A r r B k A r             (3.5) Boundary condition: When 0r  then 20 0r Iu B   ; 𝑟 → ∞ then 0 0r ME A E    . A constant E0 satisfies E = E01. The remaining A and B constants are determined by continuous conditions of displacement and stress between phases: 2 1 2 1,rI rI rI rIu u    ; 1 1, .rI rIM rI rMu u    Similar, for model on fig. 3. 1b, displacement fields of phases: 2 2 , .EI MrEI EI M M B B u A r u A r r r     (3.13) with 00,EI MB A E  , a different constants are determined by continuous conditions of displacement and stress between phases 1 ,EI I rEI rM rEI rMr R R u u       . (3.14) Displacement on boundary RI1 in both models are the same     11 1 1 1 1 12 1 I rI I rEI I I I EI I I B u R u R A R A R R     (3.17) 3 2 1 1 1 2 1 1 1 2 3 2 1 2 1 1 4 4 3 4 , 3 3 3 4 I I I I I I I I I EI I I I I I ak ak ak k ak R k a ak ak k R              (3.19)   * * * 4 , 3 eff M M M EI EI M M M EI M k k k k k k k k k         (3.20) In addition, based on Hashin's two- component coated- sphere assemblage model, we proposed the formula for calculating the bulk modulus of equivalent-inclusion, as follows: 1 ' ' 2 1 * 1 * 1 1 2 * 1 1 * 1 4 , 3 I I EI I I I I I I I k k k k k k k               (3.21) The values of bulk modulus of equivalent-inclusion established in two methods presented by Eqs. (3.19) and (3.21) are identical. 13 3.1.2. Shear modulus Based on Eshelby dilute suspension results, shear modulus of equivalent-inclusion configuration on fig.3.1b has form: 1 1 eff d EI M EI EI M A                  (3.27) The shear modulus of the coated inclusion configuration (fig. 3.1a) take also the similar form: 1 2 1 1 2 21 1 1 eff d dI I M I I I I M M A A                             (3.30) Combining Eqs. (3.27) and (3.30), the equivalent-inclu- sion expression is obtained:  * * * d M M M M EI d M M M C C              (3.32) with * 9 8 6 12 M M M M M M K K        1 21 1 2 2 1 1 1d d dI II I I I EI M M C A A                           (3.33) To determine the shear modulus of equivalent-inclusion in (3.32), it is necessary to determine the deviatoric components d I d I A,A 21 . Consider material in pure shear state. The displacement components in spherical coordinate system have the form:    2sin cos2 , sin cos cos2 ,r ru U r u U r        sin sin 2u U r    (3.34) with 3 4 2 6 3 5 4 . , 1 2 1 2 r c d U ar br r r            (3.41) 3 4 2 7 4 2 2 , . 1 2 c d U ar br U U r r              (3.42-3.43) Boundary conditions: 14 - When r = 0 : 2 20 0r I IU c d    . - When r → ∞ : 0 0 , 0r M MU E r a E b    . When 1 2,I Ir R r R  : 8 remaining constants will be determined from continuous conditions of displacement  , ,ru u u  and stress  , ,rr r r    between phases. Displacement field in phase:   r ru z u e u e u e     The average strain in phase I2 is determined by the expression:     .dSznzu IRr I    2 2ε  2 2 0 1 1 2 2ε A:E , d I IA E e e e e           2 22 2 2 2 2 3 2 02 2 4 4 521 1 . 5 1 2 5 1 2 I Id I I I I I I I d A a b R ER              (3.67) Similar, the average strain in phase I1+ I2:        1 1212 1 1 1 1 1 2 23 1 1 1 4 4 521 ε , 5 1 2 5 1 2 I I I I I I I I I d a b R e e e e R                      1 12 12 1 1 1 3 1 01 1 4 4 521 1 5 1 2 5 1 2 I Id I I I I I I I d A a b R ER             (3.69) Using  21 2 12 2 1 d d d dI I I I I I A A A A      (3.71) From eqs. (3.67), (3.71) the deviatoric components d I d I A,A 21 are determined. The simple Maxwell approximation-type expression of the equivalent bulk modulus KEI invites us to suggest a similar Maxwell approximation-type simplified (approximate) equi- valent shear modulus: 1 ' ' 1 2 1 1 * 1 * 1 1 1 * 1 2 * 1 1 1 9 8 ; 6 12 I I I I SEI I I I I I I I I I K K                          (3.72) 15 The values KEI, μSEI given in Eqs. (3.21) and (3.72) shall be used as simplified (approximate) equivalentinclusion moduli in our approach to determine the effective characteristics of media containing coated inclusions. 3.1.4. A general approximation 1 * * 1 * * 4 ; 3 n eff EI M M M M EI M M M k k k k k k k                    (3.73) 1 * * 1 * * 9 8 ; . 6 12 n eff EI M M M M M M EI M M M M M k k                               (3.74) 3.1.5. Comparison 1 2 1 21 , 5 , 25 , 0.3,M I I I I MGPa GPa GPa           1 22I I  . Fig 3.2. Effective bulk modulus Fig 3.3. Effective shear modulus 3.2. Effective modulus of unidirectional coated-fiber composite 3.2.1. Model Fig 3.6. Unidirectional coated-fiber composite 3.2.2. Effective transverse area modulus 16 ' ' 1 2 1 1 1 2 2 ,I IEI I I I I I K K K               (3.79) where KI1, KI2 are transverse area moduli of phases I1, I2. Effective transverse area modulus : 1 23 . eff EI M M EI M M M K K K                (3.83) 3.2.3. Effective longitudinal shear modulus 1 12 12 , 2 eff EI M M MEI M                (3.95) with ' 12 2 1 ' 1 2 1 1 . 1 2 I EI I I I I I            (3.96) 3.2.4. Longitudinal Young’s modulus and Poisson’s ratio        1 2' ' 2 1 2 1' ' 2 2 1 ' 2 ' 2 1 2 2 2 1 1 1 2 1 1 4 , 2 1 2 2 1 2 2 1 I I I I I EI I I I I I I I I I I I I I E E E E E E                                     ' ' 2 2 2 1 2 1 2 1 1 1 2 2' ' 2 2 1 1 ' 2 ' 2 1 1 2 2 2 2 1 1 2 1 1 2 1 2 . 1 2 1 2 1 I I I I I I I I I I EI I I I I I I I I I I I I I I E E E E E                                               2 1 2 2 4 , 2 1 2 2 1 2 2 1 I M EI Meff I EI M M M EI EI I M M M EI M M E E E E E E                                     2 2 12 2 2 1 2 1 2 . 1 2 1 2 1 EI M EI M EI M M M EI EIeff EI EI M M M M EI EI EI EI M M EI M E E E E E                                       3.2.5. Effective transverse shear modulus 17 1 23 * * * * * ; , 2 eff EI M M M M M EI M M M M M K K                        1 ' ' 1 2 1 1 * 1 * 1 2 * 1 * 1 1 1 * ; . 2 I I I I EI I I I I M I I I K K                        3.3. Conclusion Chapter 3 has developed approximate formulas for determining the effective elastic moduli of coated-sphere composite and unidirectional coated-fiber composite by equivalent-inclusion approach. Research results in this chapter have been published in scientific works [1, 3, 5, 6]. CHAPTER 4. NUMERICAL SIMULATION USING FINITE ELEMENT FOR COATED-INCLUSION COMPOSITE 4.1. Periodic material (a) (b) (c) (c) (d) Fig 4.2. Some periodic cells, (a) simple cubic, (b) body cen- tered cubic, (c) face-centered cubic, (d) square, (e) hexagonal 4.2. The original formulas 4.2.1. Elastic modulus For this elementary cell, we have a periodic field of stress σ(x) and displacement u(x): 18    1 1 2 2 3 3 1 2 3; , ,n a n a n a n n n N    1 2 3σ x σ x e e e . (4.3)    'u x = E.x + u x (4.4) where u’(x) - periodic perturbation displacement field    1 1 2 2 3 3 .n a n a n a    ' ' 1 2 3u x u x e e e (4.5) The equations satisfy on the periodic cell:   0 σ x ;   T 1 2     ε x u u ;       σ x C x : E+ε' x σ(x) n : anti-periodic U 4.2.2. Conductivity For this elementary cell, we have a periodic field of flux q(x):    1 1 2 2n a n a  1 2q x q x e e (4.13) The equations satisfy on the periodic cell: 0 q(x) ;    q(x) = -c x T x ;    0 *.T T T T  x x x Where T0 – an initial temperature of composite,  *T x - a periodic perturbation field induced by the presence of the coated circles:    * * 1 1 2 2T T n a n a  1 2x x e e (4.18) q(x) n : anti-periodic U (4.19) 4.3. Cast3M software 4.4. Calculate for some models and comparation 4.4.1. Unidirectional coated-fiber Composite 4.4.1.1. Effective transverse conductivity for isotropic coating Fig. 4.4. (a) – hexagonal cell, (b) – square cell 19 (a) 1 2 31, 5, 20c c c   (b) 1 2 320, 5, 1c c c   Fig 4.6. Effective transverse conductivity for square array (a) 1 2 31, 5, 20c c c   (b) 1 2 320, 5, 1c c c   Fig 4.7. Effective transverse conductivity for hexagonal array 4.4.1.2. Effective transverse conductivity for anisotropic coating (a) (b) Fig 4.8. (a) – periodic cell with anisotropic coating, (b) – coated element x 2 x 1 x 1 x 2 ' ' 25°  20 (a) (b) Fig 4.9. Effective transverse conductivity for anisotropic coating, (a) - 1 3 2 31, 100, 30, 50, 0.1n Tc c c c       ; (b) - 1 3 2 3100, 1, 50, 70, 0.1n Tc c c c       4.4.1.3. Effective elastic modulus (a) (b) Fig 4.10. (a) Hexagonal periodic cell for unidirectional coated- fiber composite; (b) discrete model 3 2 1 I M 2 I1 21 Fig. 4.11. Effective elastic modulus 1 , 0.2,M ME GPa   1 1 2 2 1 25 , 0.3, 10 , 0.4,I I I I I IE GPa E GPa        4.4.2. Composite with coated-sphere inclusions (a) (b) Fig. 4.13. (a) – Cubic cell; (b) - discrete model. 22 Fig. 4.14. Effective elastic moduli for cubic array when KM =1, μM =0.4, KI1 =4, μI1 =2, KI2 =20, μI2 =12 (GPa). Fig 4.15. Effective elastic moduli for body-centered cubic array when KM =1, μM =0.4, KI1 =4, μI1 =2, KI2 =20, μI2 =12 (GPa). Fig. 4.16. Effective elastic moduli for face-centered cubic array when KM =1, μM =0.4, KI1 =4, μI1 =2, KI2 =20, μI2 =12 (GPa). with PTHH: direct finite element results for the coated inclusion composite, PTHH – CTĐ : finite element results for the equivalent homogeneous-inclusion composite, PTHH – CTĐĐG: finite element results for the simple equivalent homogeneous-inclusion composite. 23 4.5. Conclusion In this chapter 4, with using Cast3M software, the thesis has determined the effective conductivity and effective elastic modulus of unidirectional coated-fiber composite and coated- sphere composite. Research results in this chapter have been published in scientific works [1-8]. CONCLUSIONS AND RECOMMENDATIONS New contributions of the thesis: - Formulating approximation formulas to calculate the effective conductivity of unidirectional coated-fiber composite with isotropic or anisotropic coating. - Formulating approximation formulas to calculate the effective elastic modulus of coated-sphere composite. - Formulating approximation formulas to calculate the effective elastic modulus of unidirectional coated-fiber composite. - Numerical methods using finite elements (CAST3M) calculated for some periodic material. The thesis opens some issues that can be further studie - Formulating approximation formulas to calculate the effective properties of coated-ellipsoid composite by equivalent-inclusion approach. - Numerical simulation for calculating conductivity of