Composite Materials

(1)

Composite Materials

(2)

Metal Matrix Composites

Long fiber composites SiC–Ti (fiber diameter : 500 µm)

(3)

Polycrystalline Morphology

zinc coating shape memory alloy Cu–Zn–Al

(4)

Two–phase Materials

Titanium alloy Ti6242 nickel–base single crystal superalloy

(5)

Characterization of thin layers and coatings

coating of a galvanized steel sheet

EBSD analysis

RD = (1) TD = (2)

RD TD ND

1 2 3 4

5 6

7 8

9 10

11 12

14 13 15

16 17

18 19

20

21 22

23 24 25 26

27 28

29 30

31

32 33 34

ND = (3)

TD = (2) RD = (1)

Symmetrical

2D geometry but 3D deformation modes!

(6)

Multi–crystalline specimens

0 5 10 16 21 26 31 37 42 48 53 58 101

eveq (x1.0E-3)

(011) (001)

(-111) 1

4 2 3 0

5 10 16 21 26 31 37 42 48 53 58

x1.0E-3 etot 33

3 1 2 4 sample A

(7)

Metallic foams

Aluminum foam nickel foam

(8)

X-ray tomography

(9)

(10)

Quasi–transparent materials

−→ confocal microscopy

(11)

Influence of phase morphology and distribution on material behaviour

100 µm

(12)

Influence of phase morphology on Young’s modulus

0 200 400 600 800 1000 1200 1400 1600 1800

0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8

E (MPa)

f₁ fine microstructure coarse microstructure E (H+) E (H−)

(13)

(14)

Models for random materials : Polycrystals

3D Vorono¨ı cells

(15)

3D Vorono¨ı cells

(16)

How to prescribe a mean deformation to a material volume element?

possible boundary conditions :

• Kinematic uniform boundary conditions

• Traction uniform boundary conditions

• Periodicity conditions

• mixed boundary conditions

(17)

Kinematic uniform boundary conditions

u = E_∼.x, ∀x ∈ ∂V, u_i = E_{i j}x_j

< ε_∼ > =1 V

Z

V

∼εdV = E_∼ Proof (divergence theorem) :

< εi j > = 1 V

Z

V

u_(i,_j)dV = 1 V

Z

∂V

u_(in_j)dS

= E_(ik1 V

Z

∂V

x_kn_j)dS = E_(ik1 V

Z

V

x_k,_j)dV

= E_(ikδk j) = E_{i j} Mean work of internal forces :

σ_∼^? statically admissible stress field (divergence free) ε∼

0 compatible strain field fulfilling the kinematic uniform boundary conditions

< σ_∼^? : _∼ε⁰ >= Σ_∼ : E_∼ where the macroscopic stress tensor is defined by

Σ_∼ =< σ_∼^? >

return

(18)

Divergence, Stokes, Gauss ... theorem

Z

V

u

_,i

dV =

Z

∂V

u n

_i

dV where u

_,i

=

_∂^∂_x^u

i

1D :

Z

_b

a

u

⁰

(x) dx = [ u ]

^b_a

return

(19)

Traction uniform boundary conditions

T = σ_∼.n = Σ_∼.n, ∀x ∈ ∂V, T_i = Σi jn_j

< σ_∼ > =1 V

Z

V

σ_∼ dV = Σ_∼ Proof (divergence theorem) :

< σi j > = 1 V

Z

V

σi jdV = 1 V

Z

V

σikδk jdV = 1 V

Z

V

(σik x_j)_,kdV

= 1 V

Z

∂V

σikx_jn_kdS = 1 V

Z

∂V

Σikx_jn_kdS

= Σi j

Mean work of internal forces :

σ_∼^? statically admissible stress field (divergence free) and fulfilling the traction uniform boundary conditions

ε∼0 compatible strain field

< σ_∼^? : _∼ε⁰ >= Σ_∼ : E_∼ where the macroscopic strain tensor is defined by

E_∼ =< ε_∼⁰ >

return

(20)

Periodicity conditions

u = E_∼.x+v where v is a periodic fluctuation

-

x y

z

+

v(x⁺) = v(x⁻), u(x⁺)−u(x⁻) = E_∼.(x⁺−x⁻)

=⇒ u is not periodic!!!

σ_∼(x⁻).n⁻ = −σ_∼(x⁺).n⁺, (n⁻ = −n⁺)

< ε_∼ >= E_∼ Proof : < v_i,_j >= 1

V

Z

V

v_i,_j dV = 1 V

Z

∂V

v_in_j dS = 0 Mean work of internal forces :

< σ_∼ : ε_∼ >= Σ_∼ : E_∼ where the macroscopic stress tensor is defined by

Σ_∼ =< σ_∼ >

return

(21)

Example

compute the effective elastic properties of a periodic composite...

local properties





 σ11

σ22

σ33

σ12







=







c₁₁ c₁₂ c₁₃ 0 c₁₂ c₂₂ c₂₃ 0 c₁₃ c₂₃ c₃₃ 0 0 0 0 c₄₄











 ε11

ε22

ε33

2ε12







effective properties





 Σ11

Σ22

Σ33

Σ12







=







C₁₁ C₁₂ C₁₃ 0 C₁₂ C₂₂ C₂₃ 0 C₁₃ C₂₃ C₃₃ 0

0 0 0 C₄₄













E₁₁ E₂₂ E₃₃ 2E₁₂







(22)

Extension test

impose < ε_∼ >, compute < σ_∼ >...

E_∼ =





1 0 0 0 0 0 0 0 0





Σ11 = C₁₁ Σ22 = C₁₂ Σ33 = C₁₃ Σ12 = 0 homogeneous boundary conditions

x y

z

periodicity conditions

x y

z

(23)

Shear test

E_∼ =





0 1/2 0 1/2 0 0

0 0 0



 Σ11 = Σ22 = Σ33 = 0 Σ12 = C₄₄

homogeneous boundary conditions

x y

z

periodicity conditions

x y

z

(24)

Application to two–phase Vorono¨ı mosaics

x y

z

two elastic phases : E=2500 MPa, ν = 0.3 E = 25 MPa, ν = 0.49

(25)

Apparent shear modulus

0 200 400 600 800 1000

10 100 1000 10000

µapp (MPa)

V

KUBC

Bounds of Voigt−Reuss

(26)

Apparent shear modulus

0 200 400 600 800 1000

10 100 1000 10000

µapp (MPa)

V

KUBC SUBC Bounds of Voigt−Reuss

(27)

Apparent shear modulus

0 200 400 600 800 1000

10 100 1000 10000

µapp (MPa)

V

KUBC SUBC PERIODIC Bounds of Voigt−Reuss

(28)

Parallel computing

turbine blade in 8 subdomains

FETI method

local equilibrium on each subdomain

compatibility and equilibrium at the interface : iterative

algorithm

cluster of PCs under Linux

(29)

Sub–domain decomposition

x

y

z

x y

z

(30)