MMComposites_NTKien

Faculty of Civil Engineering and Applied Mechanics

Department of Structures

Mechanics of Composite Materials

� PhD. Nguyễn Trung KiênEmail: [email protected] of Civil Engineering and Applied Mechanics1 Vo Van Ngan Street, Thu Duc DistrictHo Chi Minh City, Viet Nam

� Keywords: - Mechanics of Composite Materials- Laminated materials and Structures- Homogenization- Theory of plates and beams

Contents

�C1 : Introduction to composite materials�C2 : Mechanical behaviors of composite materials�C3 : Elastic behavior of unidirectional composite

2

�C3 : Elastic behavior of unidirectional composite materials

�C4 : Elastic behavior of orthotropic composite�C5 : Off-axis behavior of composite materials�C6 : Fracture and damage of composite materials�C7 : Modeling of mechanical behaviors of laminated �C7 : Modeling of mechanical behaviors of laminated

plates�C8 : Homogenization of composite materials

References

� Autar K. Kaw, Mechanics of Composite Materials, Taylor & Francis, NewYork, 2006

� Jean-Marie Berthelot, Composite Materials –

3

� Jean-Marie Berthelot, Composite Materials –Mechanical behavior and Structural analysis, Springer, 1999

� J. N. Reddy, Mechanics of laminated composite plates and shells – Theory and Analysis, CRC Press, 2004. Press, 2004.

Contents


4




INTRODUCTION TO COMPOSITE MATERIALS

� Introduction

� Composite materialsMatrix materials

5

o Matrix materialso Fiberso Architecture of composite materials

o Study the mechanical behavior of composite materials

� Composite materials for civil engineering applications� Composite materials for civil engineering applications

Introduction

� Composite materials used more and more for primary structures in aerospace, marine, energy,…

6

Introduction

� Composite materials used more and more for primary structures in civil engineering, etc

7

Composite materials

� Definition:

o “Composite” means "made of two or more different parts

� Classification:

8

� Classification:o Form of constituents

� Fiber composite� Particle composite

o Nature of Constituents� Organic matrix composites� Metallic matrix composites� Metallic matrix composites� Mineral matrix composites

Composite materials

� Classification by class of constituents

9

Fiber Reinforcement Matrix Composite

Particle Matrix CompositeParticle Matrix Composite

� Mechanical properties of composites� the nature of the constituents� the proportions of the constituents� the orientation of the fibers

Composite materials

� Matrix comprises a resin (polyester, epoxide, etc.) and fillers which is to improve the characteristics of the resin:o Thermosetting Resins:

10

o Thermosetting Resins: � Polyester Resins

� Condensation Resins � Epoxide Resins

o Thermoplastic Resins: � polyvinyl chloride (PVC), polyethylene, polypropylene, polystirene,

polyamide, and polycarbonate

o Thermostable Resins: o Thermostable Resins: o Bismaleimide Resins, Polyimide Resins

Composite materials

� Epoxide Resins:

11

Advantages of epoxide resins are the following:� good mechanical properties (tension, bending, compression, shock, etc.) superior to those of polyesters� good behavior at high temperatures: up to 150-190°C in continuous use� excellent chemical resistance� low shrinkage in molding process and during cure (from 0.5-1 %)� very good wettability of reinforcements� excellent adhesion to metallic materials

Disadvantages:� High cost, manufacture, sensibility to cracking

Composite materials

� Polypropylene, polyamide:

12

Advantages of epoxide resins are the following:� low cost, fabrication

Disadvantages:� mechanical and thermomechanical properties : low� mechanical and thermomechanical properties : low

�� Limited development

� Thermostable Resins : Bismaleimide Resins, Polyimide Resins

� Thermal performance developed especially in the aviation and space

Composite materials

� Fillers and additives : function of improving the mechanical and physical characteristics of the finished product or making their manufacture easier

� Fillers : Reinforcing Fillers, Nonreinforcing Fillers

13

� Fillers : Reinforcing Fillers, Nonreinforcing Fillerso Reinforcing Fillers : improve the mechanical properties of a resin

� Spherical fillers: diameter usually lying between 10 and 150µm. They can be glass, carbon, or organic (epoxide, phenolic, polystirene, etc.),

� Nonspherical fillers: mica used most (dimension: 100-500µm, thickness: 1-20 µm)

o Nonreinforcing Fillers: reducing the cost of resins, preserving their performance � carbonates, silicates

� Additives : pigments and colorants, antishrinkage agents, antiultraviolet agents

Composite materials

� Fibers:� Improve mechanical characteristics: stiffness,strength, hardness, etc� Improve certain of the physical properties: thermal properties, fire

14

� Improve certain of the physical properties: thermal properties, fire resistance, resistance to abrasion, electrical properties

� Reinforcements origins: vegetable, mineral, artificial, synthetic � fibers� linear forms (strands, yarns, rovings, etc.)�surfacing tissues (woven fabrics, mats, etc.)�multidirectional forms (preforms, complex cloths, etc.)

Composite materials

� Specific mechanical characteristics of materials, made in the form of fibers

15

Composite materials

� Architecture of composite materials� Laminates

16

Composite materials

� Architecture of composite materialso Sandwich

17

Composite materials

� Study the mechanical behavior of composite materials

18

Composite materials

� Study the mechanical behavior of composite materials

19

Composite materials for civil engineering applications

20

Contents


21




MECHANICAL BEHAVIORS OF COMPOSITE MATERIALS

� Linear elastic scheme

� Elastic behavior of a unidirectional composite material� Elastic behavior of an orthotropic composite material

22

� Elastic behavior of an orthotropic composite material� Elastic behavior of composite materials outside of main

axes� Strength failure theories

Mechanical behaviors of composite materials

� Linear elastic scheme

o Stiffness and compliance matrix

σ ε

23

11 12 13 14 15 161 1

12 22 23 24 25 262 2

13 23 33 34 35 363 3

14 24 34 44 45 4623 23

15 25 35 45 55 5613 13

16 26 36 46 56 6612 12

C C C C C C

C C C C C C

C C C C C C

C C C C C C

C C C C C C

C C C C C C

σ εσ εσ εσ γσ γσ γ

=

ε σ11 12 13 14 15 161 1

12 22 23 24 25 262 2

13 23 33 34 35 363 3

14 24 34 44 45 4623 23

15 25 35 45 55 5613 13

16 26 36 46 56 6612 12

S S S S S S

S S S S S S

S S S S S S

S S S S S S

S S S S S S

S S S S S S

ε σε σε σγ σγ σγ σ

=

� C, S 6x6-matrix is called the stiffness matrix and compliance matrix having 21 independent constants: S = C-1


� Change of coordinate system

, = =σ Cε ε Sσ

� Vector:

24

where

� Tensor:

Rotation of a ɵ angle of coordinate system around 3-axis



, = =σ Cε ε Sσ

� Stress:

25

� In coordinate system e: σ = C ε� In coordinate system e': σ’ = C’ ε’

1' −=C T CTwhere :

Rotation of a ɵ angle of coordinate system around 3-axis

1' −=σ ε

C T CTwhere :

( )1

' , '

−

= =

=

σ ε

ε σ

σ T σ ε T ε

T TT

� Stresses and strains in e’:



, = =σ Cε ε Sσ

26

2 2

2 2

2 2

cos sin 0 0 0 2sin cos

sin cos 0 0 0 2sin cos

0 0 1 0 0 0

0 0 0 cos sin 0

0 0 0 sin cos 0

sin cos sin cos 0 0 0 cos sin

θ θ θ θθ θ θ θ

θ θθ θ

θ θ θ θ θ θ

−

= −

− −

σT

2 2

2 2

2 2

cos sin 0 0 0 sin cos

sin cos 0 0 0 sin cos

0 0 1 0 0 0

0 0 0 cos sin 0

0 0 0 sin cos 0

2sin cos 2sin cos 0 0 0 cos sin

θ θ θ θθ θ θ θ

θ θθ θ

θ θ θ θ θ θ

−

= −

− −

εT

( )1

' , '

−

= =

=

σ ε

ε σ

σ T σ ε T ε

T TT

� Stresses and strains in e’:


� Engineering Matrix Notation

, = =σ Cε ε Sσ

� Stress:

27

� Strains:


� Anisotropic materials

11 12 13 14 15 161 1C C C C C C

C C C C C C

σ εσ ε

28

12 22 23 24 25 262 2

13 23 33 34 35 363 3

14 24 34 44 45 4623 23

15 25 35 45 55 5613 13

16 26 36 46 56 6612 12

C C C C C C

C C C C C C

C C C C C C

C C C C C C

C C C C C C

σ εσ εσ γσ γσ γ

=

� Matrix C: 21 independent constants at a pointat a point� Note that these constants can vary from point to point if the material is nonhomogeneous


� Monoclinic material

29

1–2 symmetry plane of a monoclinic material

Note:� A monoclinic material is a material that has a symmetry plane� 13 independent elastic constants


� Orthotropic material

30

Note:� Three mutually perpendicular planes of material symmetry� 9 independent elastic constants


� Transverse isotropic material (Unidirectional material)

31

Note:� Orthotropic material having one axis of revolution � 5 independent elastic constants


� Isotropic material

32

Note:� Properties are independent of the choice of its reference axis� 2 independent elastic constants


� Isotropic material

33

Relations between elasticity coefficients of an isotropic material

� Elasticity modulus: E, νo Uniaxial Tension or Compression


� Exercise 1: In the case of a monoclinic material with the symmetry plane (1,2) show that the stiffness matrix has the form (a).

34

(a) (b)

� Exercise 2: The symmetry plane (1,3) is added to a monoclinic material in order to obtain an orthotropic material. Show that the stiffness matrix has the form (b)

(a) (b)

C2 – Mechanical behaviors of composite materials

� Exercises 3: Consider a rotation through an angle e about the I-axis of an orthotropic material. Write the stiffness matrix in the new axes and deduce the form (a) of the stiffness matrix of a transverse isotropic material.

35

isotropic material.

� Exercise 4: In case of isotropic material, show that the stiffness matrix has the form (b)

(a) (b)

Contents


36




ELASTIC BEHAVIOR OF UNIDIRECTIONAL COMPOSITE MATERIALS

� Effective moduli:

37

� Microscopic scale : scale of constituents.

Microscopic ↔ Macroscopic

� Macroscopic scale of size δ : properties of the material can be averaged to agood approximation. The properties measured in a sample of size δ areindependent of the place (of the point)� Homogenized problems of designing structures can be solved by considering theaverage properties measured on the scale δ.� Macroscopic homogeneity or statistical homogeneity � Homogenization.

Effective properties of composites

�How to determine the homogenized properties� Phenomenological approach� Homogenization method

38

� Homogenization method

Homogenization method Phenomenological method

Effective properties of composites

� An element of volume V and size δ:

39

Basic equations of the average strain and stress field:

� C: Effective stiffness matrix� S: Effective compliance matrix

Homogenization of composite materials

�Homogenization method: 3 main stages• Representative Volume Element (RVE)• Localization problem

40

• Localization problem • Homogenization

Periodic materials Random materials

Periodic material media

� Localization problem of periodic composites

41

� Strain energy:

� Macroscopic constitutive equation:

� Resolution method of the elastic problem on a unit cell: FEM, Fourier(Gusev, Kanit et al., Suquet et al., Mishneavsky, etc)

� Strain energy:

Elastic behavior of unidirectional composite

�Unidirectional composite material

42

L: Longitudinal direction T, T’: Transverse direction

� Hook’s law: � 5 independent coefficients: � 5 independent coefficients: C11, C12, C22, C23, C66

How ?

Engineering constants - Tests

� Engineering constants: Young's moduli (E), Poisson ratios (ν), shear moduli (G).� Longitudinal Tensile Test

43

� Longitudinal Tensile Test

� Stress and strain:

� Elastic moduli:


� Transverse tensile test

44


� Elastic moduli:

(ν21)

(ν23)

Nota:


� Longitudinal shear test � Transverse shear test

45


� Elastic moduli:


� Elastic moduli:

Nota:

(G23)


�Lateral hydrostatic compression

46


� Lateral compression modulus:


� Moduli as functions of the stiffness

47

� Only 5 independent moduli, practically: E , E , νννν , G , G� Only 5 independent moduli, practically: EL, ET, ννννLT, GLT, GTT'


� Moduli as functions of the stiffness

48

� Stiffness as functions of Moduli� Stiffness as functions of Moduli

Engineering constants – Theoretical approach

� Different approach to the problem� Find 5 independent constants as functions of the mechanical and

geometric properties of the constituents (engineering constants of

49

the matrix and fibers, volume fraction of the fibers),…

� Periodic fiber arrangements:

� Random fiber arrangements:� Random fiber arrangements:

How to estimate elastic constants ?


� Estimation of elastic constants:� Bounds (upper and lower bounds) using energy variational

theorems (total potential energy theorem, Hashin-Shtrikman,…) :

50

not accurate for high contrast of materials

� Exact solutions: simple geometry

� Numerical methods (FEM, Fourier)


� Bounds on engineering constants:� Total potential energy theorem (displacement approach): upper

bounds

51

� Complementary potential energy theorem (stress approach): lower approach

Random fibers Periodic fibers


� Bounds on engineering constants:

52


� Simplified approach:� Longitudinal Young’s modulus:

53


� Transverse Young’s modulus:

54


� Longitudinal Poisson ratio:

55


� Longitudinal shear modulus:

56

Exercises

57

Contents


58




Elastic behavior of orthotropic composite

�Orthotropic composite material� 9 independent coefficients: C11, C12, C13, C22, C23, C33, C44 ,C55, C66

59

� Hook’s law:

C11, C12, C13, C22, C23, C33, C44 ,C55, C66

How ?

S11, S12, S13, S22, S23, S33, S44 , S55, S66

Elastic behavior of orthotropic composite

�Orthotropic composite material

60

C11, C12, C13, C22, C23, C44 ,C55, C66

S , S , S , S , S , S , S , SS11, S12, S13, S22, S23, S44 , S55, S66


� Engineering constants: Young's moduli (E), Poisson ratios (ν), shear moduli (G).� Tensile test in direction 1

61

� Tensile test in direction 1




� Shear test

62

Similarly,

Conclusion:


� Compliance constants:

63

� Stiffness constants:

Engineering constants – orthotropic composite

Exercise 1: Calculate the stiffness and compliance constants of an orthotropic composite with the following characteristics:

64

Exercise 2: Calculate the stiffness and compliance constants of an orthotropic composite with the following characteristics:

Contents


65




Off-axis behavior of composite materials

� Constitutive equations of off-axis layers

66

1, 2, 3: principal directions1’, 2’, 3’: reference system

Stiffness and compliance constants:


� Elastic constants

67


� Exercise: o Determine elastic constants of an orthotropic material whose principal direction 1 makes an angle ɵ with the x-axis directiono Plot the variation of C’11 and C’12 in terms of ɵ (0 - 90o)

68

11 12

Two dimensional stress state

� Plane stress state:

69


� Plane stress state:

70

wherewhere

Q: reduced stiffness matrix


� Plane stress state in principal directions:

71


� Plane stress state in off-axis:

72


� Exercise:

� Consider a plane stress state, determine:- Stresses: σxx, σyy, σxy

73

- Stresses: σxx, σyy, σxy- Stresses in principal axes (L, T)

Contents


74




Fracture and Damage of Composite Materials

� Fracture Processes Induced in Composite Materials :

75


� Failure Criteria:� Maximum stress criterion

76

+ Xt, Xc: the tensile and compressive strengths in the longitudinal direction+ Yt , Yc: the tensile and compressive strengths in the transverse direction+ S: the in-plane shear strength of the layer


� Failure Criteria:� Maximum strain criterion

77

+ Xεt, Xεc: the tensile and compressive strains in the longitudinal direction+ Yεt , Yεc: the tensile and compressive strains in the transverse direction+ S: the in-plane shear strain of the layer


� Failure Criteria:� Interactive criteria

Hill’s criterion:

78

Tsai - Hill’s criterion: plane stress state

Hoffman’s criterion:

Tsai - Wsu criterion: (book)

Contents


79


�C4 : Elastic behavior of orthotropic composite�C5 : Off-axis behavior of composite materials�C6 : Fracture and damage of composite materials�C7 : Modeling of mechanical behaviors of �C7 : Modeling of mechanical behaviors of

laminated plates�C8 : Homogenization of composite materials

Modeling the Mechanical Behavior of Laminated Plates

� Basics of Laminate Theory :

80

Plate element Laminated element

� Equivalent single-layer theories (2D) � Classical laminate theory � Shear deformation laminate theories

� Three-dimensional elasticity theory � 3D elasticity formulations � Layerwise theories

Modeling the Mechanical Behavior of Laminated Plates

� Laminate Theories :

81

Laminated element

Functions to be determined

Various plate theories

Plate models

� Plate model of Love-Kirchhoff:

82

Plate models

� Plate model of Reissner-Mindlin: First-order shear deformation theory

83

Plate models

� High-Order Shear Deformation Plate Model� In-plane displacements varied in the thickness:

84

TSDT :

SSDT : SSDT :

� Transverse displacement varied in the thickness:

Plate models

� Strains and stress

85

� Stress at point of kth-layer:

Plate models

� 2D plate theories� Assumption: σzz=0

86

Plate models

� Resultants and Moments:

87

Plate models

� Resultants and Moments:

88

Classical laminated plate theory

� Hypothesis� Straight lines perpendicular to

the midsurface before

89

the midsurface beforedeformation remain straightafter deformation

� Transverse normals are inextensible

� Transverse normals rotatesuch that they remainperpendicular to theperpendicular to themidsurface after deformation

0, 0zz xz yzε γ γ⇒ = = =


� Kinematics

90

� Strains:


� Constitutive relations

91


� Constitutive relations� A : extensional stiffness� D: bending stiffness

92

� D: bending stiffness� B: bending-extensional coupling stiffness


� Constitutive relations� A, B, D : extensional stiffness, bending stiffness, bending-extensional coupling stiffness

93

coupling stiffness


� Equations of motion� Governing equations derived using the principle of virtual

displacements

94

displacements

where δU, δV, δK are virtual strain energy, virtual work done and virtual kinetic energy


� Equations of motion

95

where



(1)ε κ

� Note:

96

(1)

(1)

(1)

xx xx

yy yy

xy xy

ε κε κγ κ

=



97

� Equilibrium equation:



98



99



100

First-order shear deformation laminated plate theory

� Hypothesis� Straight lines perpendicular to

the midsurface before

101

the midsurface beforedeformation remain straightafter deformation

� Transverse normals are inextensible

� Kinematics

� Exercise: Equations of motion ?� Shear correction factor : k=5/6

Laminated plate theory

�Laminate stiffnesses� Angle-ply laminates, cross-ply laminates

102

General angle-ply laminate Cross-ply laminate

� General angle-ply laminate:ɵ and - ɵ orientation in [0,90]

� Cross-ply laminate:ɵ orientation of 0o or 90o


�Laminate stiffnesses� Symmetric and asymmetric laminates

103

� Notation : (-25/35/0/90)s

�Laminate stiffnesses� Material stiffnesses, layer thicknesses, lamination scheme


�Laminate stiffnesses� Isotropic layers

104

Examples: � Single isotropic layer� Single orthotropic layer

Analytical solution

�Navier solution

105

Analytical solution

�Characteristic equations

106

Analytical solution

� Bending analysis

107

� Buckling analysis

� Free vibration

Contents


108





� Introduction:

109

� Material can be considered to be homogeneous or heterogeneous according tothe scale at which it is observed.

Microscopic ↔ Macroscopic

Homogenization

the scale at which it is observed.

� Material can be described in the framework of continuum mechanics by twomodels: one at microscopic scale where the behavior is heterogeneous, the otherat macroscopic scale where the behavior is homogeneous.

� Objective of homogenization: study the relation of these two models, especiallydetermination of behavior at macroscopic scale in terms of one at microscopicscale.


� Introduction:

110

� Determination of behavior at macroscopic scale in terms of one at microscopicscale:

� Representative volume element (RVE) whose boundary subjected to

HomogenizationPhenomenological method

� Representative volume element (RVE) whose boundary subjected tohomogeneous boundary conditions in strain and stress.

� Macroscopic behavior is the relation between volume average of stress andstrain in RVE

� Size of RVE should be large enough at micro scale to well describeheterogeneity, and small enough at macro scale in which calculatedmechanical fields are very few variables in RVE


� Introduction:

111

� Scale of structure: macroscopic scale

Heterogeneity scale RVE scale

� Scale of structure: macroscopic scale

� Scale of heterogeneity : microscopic scale

� Scale of RVE: mesoscopic scale

� Concept of RVE: linear elasticity, nonlinear elasticity, elastoplasticity, limitanalysis


� Objective:

� Definition of homogenized linear elasticproperties of heterogeneous materials

112

properties of heterogeneous materials

� Presentation of bounds that enable toestimate the properties in terms ofconstituent properties

� Hypothesis:� Small deformation

� Heterogeneous linear elastic materials

� No cavities, cracks

� Constituents are perfectly adherent


� Average value on RVE:

( )1D

f f dVV ∈

= ∫x V

xDefinition:

113

V ∈∫

x V

( ) ( ){ }0

0

0 in , . .

V

div= = =

⇒ =

σ x σ x V σ n σ n

σ σ

� Average of stress :


� Average of strain on RVE:

Consider : ( ) ( ) ( ) ( ) 00

1, . on V

2t= ∇ + ∇ = = ∂ε x u u u x u x ε x

114

Average strain field of RVE:


� Average value on RVE:

( )1 1 . . .dV dS= =∫ ∫σ ε σε u σ n

� Homogeneous boundary condition in stress:

115

( ) . . .

, . . . .

VV

V

V V V

dV dSV V

V

∈ ∈∂

= =

=∀ ∈∂ = ⇒ =

∫ ∫x D x

σ ε σε u σ n

σ Σ

x σ n Σnσ ε σ ε

� Homogeneous boundary condition in strain:

( ), . VV =∀ ∈∂ = ⇒ ε E

x u x E x( ), . . .V

V V V

V =∀ ∈∂ = ⇒

=

ε Ex u x E x

σ ε σ ε

� Hill – Mandel’s principle :

. .V V V

=σ ε σ ε


� Homogenization with strain approach:( ) . V= ∈∂u x E x x AV∈ V and BCs �Direct method:

116

Heterogeneous : a(x) Homogeneous : A

( )� Elastic problem on RVE: solutions (σE, εE, uE) � Nota :

A ? for a (x)

E VV∀ ⇒ = =E Σ σ A E

( )( ) ( ) ( )

( ) ( )( )

0

1

2.

t

div

V

=

=

= ∇ + ∇

= ∈∂

σ x

σ x a x ε x

ε x u u

u x E x x

( )( )( )( )

, if ,

,

.

E

E

E

V

V

∀ ∈ =

=

⇒∀ ∈ = =

x a x A

σ x AE

x ε x E

u x E x

� E : uniform strain in RVE

. .E E E EV V V= =σ ε σ ε ΣE


� Potential energy method

( ) ( ) ( ) ( )( ): .

E E KAW Min W

KA V

∈− Φ = − Φ

∈ = ∈∂uu u u u

u u x E x x

117

( )

( ) ( ) ( ) ( ) ( )

: .

1. . , 0

2

KA V

W dV∈

∈ = ∈∂

= Φ =∫x D

u u x E x x

u ε x a x ε x u

Minimum principle of potential energy:

( ) ( ) ( )( ) ( )

. 0 V= + ⇒ = ∀ ∈ ∂

= +

u x u x E x u x x

ε x ε u E

ɶ ɶ

ɶ

( )( ) ( ) ( )( ) ( )( ) ( ) ( )( )( ) ( ) ( ) ( )( )

( )( ) ( ) ( )( )

( )( ) ( ) ( )( )

. . . .

,

. .

, . .

E E KAV V

E E E E

E E E E E E VV V VV

V KA V

Min

Min

∈

∈

+ + = + +

= + = +

+ + = = =

⇒∀ = + +

u

u

ε u E a x ε u E ε u E a x ε u E

ε u ε u E σ a x ε u E

ε u E a x ε u E σ ε σ ε EA E

E EA E ε u E a x ε u E

ɶ ɶ ɶ ɶ

ɶ ɶ

ɶ ɶ

ɶ ɶ


� Complementary energy method

( ) ( ) ( ) ( ),

: 0

E E SAW Min W

SA div

∈∀ − Φ = − Φ

∈ =σ

E σ σ σ σ

σ σ

118

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

: 0

1. . , . . .

2 V V

SA div

W dV dS dV∈ ∈∂ ∈

∈ =

= Φ = =∫ ∫ ∫x D x x

σ σ

σ σ x s x σ x u E x σ n E σ x

( ) ( )( )

. . 2 . . 2

. .

E E E SAV VV V

E E E E E E VV V VV

Min ∈ − = −

= = =

σσ s x σ E σ σ s x σ E σ

σ s x σ σ ε σ ε EA E

Minimum principle of complementary energy:

( ), 2 . .V SA V VMax ∈

⇒∀ = − σE EA E E σ σ s x σ

Conclusions:

( ) ( )( ) ( ) ( )( ), 2 . . . .SA V KAV V VMax Min∈ ∈

∀ − = = + + σ uE E σ σ s x σ EA E ε u E a x ε u Eɶ ɶ


� Homogenization with stress approach:

( ) V= ∈∂σ x n Σn x�Direct method:

119

( ) 0 div =σ x

� Elastic problem on RVE: solutions (σ∑, ε ∑, u ∑):

Heterogeneous : s(x) Homogeneous : S

( ): 0, SA div V∈ = = ∈∂σ σ σ x n Σn x( )

( ) ( ) ( )

( ) ( )( )

0

1

2

t

div

V

=

=

= ∇ + ∇

= ∈∂

σ x

σ x a x ε x

ε x u u

σ x n Σn x

VVΣ =ε S Σ

( ) ( ) ( ) ( ) ( ) ( ) ( ), 2 . . . .V SAV V VMax Min ∈

∀ − = = + + u σΣ Σ ε u ε u a x ε u ΣS Σ σ Σ s x σ Σɶ ɶ


� Homogenized properties of a heterogeneous medium:

120


� Bounds on Ahom:

� Voigt and Reuss

121

( ), V Vf aα α

α

∀ ≤ = ∑E EA E E a x E E E

( ) ( )( ) ( ) ( )( ), 2 . . . .SA V KAV V VMax Min∈ ∈

∀ − = = + + σ uE E σ σ s x σ EA E ε u E a x ε u Eɶ ɶ

(Voigt’s bound)

1f α −

( )( )

0 0 0 0

1

1

hom

, 2

,

,

VV

VV

f sα α

α

−

−

∀ − ≤

⇒∀ ≤

⇒∀ ≤

∑

E σ Eσ σ s x σ EA E

E E s x E EA E

E E E EA E

(Reuss’s bound)

1

hom

1

hom

9 3 2,

3 6 2

fK f K

K

ff

K KE

K K

αα α

αα α

αα α

αα α

µ µµ

µ µνµ µ

−

−

≤ ≤

≤ ≤

−= =+ +

∑ ∑

∑ ∑


� Bounds on Ahom:

� Hashin-Shtrikman

K K V Vµ µ− −− −

122

( ) ( )

( ) ( )

( ) ( )

1 2 1 2

2 1 2 12 1 2 12 2

1 1 1 1

1 2 1 2

2 1 2 12 1 2 12 2

1 2 1 2

;1 1 1 1

4 / 3

;1 1 1 1

4 / 3

9 8 / 6 2

9 3 2

K K V VK KK K V V

K f

K K V VK KK K V V

K f

f K K

K Kα α α α α α

µ µµ µµ µ

µ µµ µ

µ µµ µµ µ

µ µ µµ µ

− −

+ +

− −= =− −− −+ − + −+ +

− −= =− −− −+ − + −+ +

= + +−

2 1 2 1

9 3 2,

3 6 2

: ,

K KE

K K

Nota K K

µ µνµ µ

µ µ

−= =+ +

≥ ≥

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