Lecture note_09_Micromechanics-Schemes

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(I)
U:σ
0
ε
(I)
R :ε 0
0 0
ε (σ )
Matrix
I h Inhomogeneity i An inhomogeneity is embedded in an infinite matrix for calculating effective bulk and shear moduli of composite
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Analogy between elasticity and conductivity problems Conceptual p relation between elasticity y and conduction
Elasticity u: displacement di l t ε: strain σ: stress T :traction C: elastic moduli S: compliances g phase p rigid empty phase
J(x) K(x) F(x)
J K F
F : field applied on a material -- cause J : response of the material to F -- effect K : coefficient
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Analogy between elasticity and conductivity problems Parameter analogy between the steady state conduction Physical Subject F K J φ thermal temperature gradient thermal heat flux y conduction conductivity electrical potential intensity electrical current conduction conductivity density electrostatics l i potential i l intensity i i permittivity i i i electric l i induction magnetostatics t t ti potential t ti l intensity i t it magnetic ti magnetic ti permeability induction elasticity displacement strain elastic stress moduli
(I ) (m) S* S ( m ) c S S :U
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Micromechanical framework Linear displ. boundary condition:
u ε x
0
Define a strain concentration tensor R:
11/1rk
(Duan H L et al., ( , JMPS, 2005) )
Uniform traction boundary condition:
Σ σ0 n
0 ( 0 )
Define a stress concentration tensor J:
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Th k you for Thank f your attention i !
Matrix (Mori-Tanaka method) Inhomogeneity Remote strain (stress) which is equal to the volume average strain (stress) ( ) in the matrix is applied pp to an infinite matrix containing ga single inhomogeneity for calculating effective bulk and shear moduli of composite
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Nanomechanics and Micromechanics ---Effective Effective elastic moduli H. L. Duan
School of Engineering Engineering, Peking University
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Analogy between elasticity and conductivity problems Framework of p physical y and mechanical p properties p Heterogeneous medium Equivalent homogeneous medium
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Mi Micromechanical h i l framework f k
Micromechanical scheme 2: Self-Consistent 0 ( 0 )

(I)
U:σ
0
ε
(I)
R :ε 0
Composite
ε 0 (σ 0 )
Inhomogeneity
An inhomogeneity g y is embedded in an infinite equivalent q medium ( (ie the composite) for calculating effective bulk and shear moduli of composite
(Duan H L et al., JMPS, 2005)
Uniform traction boundary condition: Define a stress concentration tensor U:
Σ σ0 n
0 ( 0 )
(I) U:σ 0
Effective compliance tensor:
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Mi Micromechanical h i l framework f k
Micromechanical scheme 4: GSCM
(Generalized self-consistent method)
0 ( 0 )
Matrix
ε (σ )
0
0
Composite p
Inhomogeneity A composite sphere composed of an inhomogeneity and matrix shell is embedded in an infinite equivalent q medium ( (ie the composite) for calculating effective bulk and shear moduli of composite
ε R: Rε
C C
* (m)
(I)
0
Effective stiffness tensor:
c[C C ]:R
(I) (m)
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Mi Micromechanical h i l framework f k
Micromechanical scheme 1: Dilute 0 ( 0 )
u ε x
0
Define a strain concentration tensor M:
ε M: Mε
(I) ( )
(m)
Effective stiffness tensor:
C* C(m) c [C(I) C(m) ]:M :[I (4 s ) f (M I (4 s ) )]1
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Micromechanical framework
Micromechanical scheme 3: CSA
(Hashin’s composite assemblage model)
0 ( 0 )
Matrix
Inhomogeneity Hydrostatic loading is applied to a composite sphere consisting of an inhomogeneity and a matrix shell for calculating effective bulk modulus of composite
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Micromechanical framework for nano-inhomogeneities
Micromechanical c o ec a ca scheme sc e e 5: MTM
0 (m) 0 (m) ε = ε ( σ = ) 0 0 matrix matrix ( )


(I)
J:
(m)
Effective compliance tensor:
(I ) (m) (4 s ) (4 s ) 1 S* S ( m ) c S S : J :[ I c ( J I )]

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Micromechanical framework Linear displ. boundary condition:
Conduction φ: temperature t t -φ,i : gradient q: flux qn: normal flux component μ: conductivities ρ: resistivities superconductor p insulator
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Micromechanical framework