Composite Materials: Mechanical and Fracture Characterization

Dr Rajesh Kitey

School on Mechanics of Reinforced Polymer Composites Knowledge Incubation for TEQIP

Indian Institute of Technology Kanpur January 22-25, 2017

Classification

2

3

10% spherical particle 10% Milled fibers

45% continuous fibers

Distribution uniformity

Ref – Yesgat and Kitey, EFM 2016

4

Volume fraction characterization

pf

total

LV

L= f

intpp

1 VlN−

=

( )fThintp

f

2 13

D Vl

V−

=

Grey level histogram

Pixe

l Fre

quen

cy

Pixel Intensity

Ref – Kitey and Tippur, Acta Mater 2005

cev

ct

1V ρρ

= −

ct f f m mV Vρ ρ ρ= +

• Density increases with filler Vf

• Density is lower for short fiber due to the entrapped air

5

Void volume fraction

Ref – Yesgat and Kitey, EFM 2016

6

Mechanical Characteristics

7

Uniaxial tension test

ASTM D638

Ref – Gayathri, M Tech Tesis, IITK 2014

Strain softening

Strain hardening

Compression test

0

PA

σ =( )

0

SS

LL

∆ε =

Total Machine( ) ( )SL L PC∆ = ∆ −

ASTM D695-10

L/D < 2

Ref – Sarthak, Kitey and Venkitnarayanan, IITK 2017

9

Test specimens - Laminated composites

ASTM D3039 ASTM D695

Ref – Paiva, Mayer and Rezende, Mat Res 2005, 2006

0 2 4 6 80

50

100

150

200

δ(mm)

Load

(N)

250C450C550C

Spherical

m

ASTM D790M 3

f 34S mEWD

=

2

f 2

3 1 6 42

PS DWD S S S

δ δσ = + −

10

2

6f

DSδε =

Flexural test

Ref – Yesgat, PhD Dissertation, IITK 2016

11

Shear test

Arcan test ASTM D 5379 (Ioscipescu test)

Three rail shear test Uniaxially loading [ ]2

45S

±

12

12

2xx

xx yy

σσ =

γ = ε − ε

Ref – Mohite, nptel

Ref – Hasan PhD 2015

Split Hopkinson’s Pressure Bar (SHPB)

Dynamic compression test

( )( ) 2• ε

ε = − RS b

S

tt Cl 0

( ) 2 ( )ε = − ε∫t

bS R

S

Ct t dtl

( ) ( )σ = εb bS T

S

E At tA

=ρ

bb

b

ECRef – Sarthak, Kitey and Venkitnarayanan, IITK 2017

Stress histories

( )( ) 2• ε

ε = − RS b

S

tt Cl

0( ) 2 ( )ε = − ε∫

tb

S RS

Ct t dtl

( ) ( )σ = εb bS T

S

E At tA

Ref – Sarthak, Kitey and Venkitnarayanan, IITK 2017

14

Failure or Fracture

15

Maximum normal stress (Rankine) theory

Maximum shear stress (Tresca) theory

1 3max

12 2 y

σ στ σ−= ≥

( )1 1 2 3uσ σ σ σ σ≥ > >… for brittle material

Maximum distortion energy (Von-Mises) theory

( ) ( ) ( )2 2 2 21 3 1 3 1 3 2 yσ σ σ σ σ σ σ− + − + − ≥

… distortion energy

Failure theories

16

( )1ij f

rσ θ∝

2a 2b

σ0

2a

σ0

r θ

max 0 1 2 ab

σ σ = +

…Linear theory of elasticity

Flaws

17

Uniaxial tensile strength to characterize fracture is specimen dependent

- Fracture strength decreases with increasing fiber diameter

- Fracture strength of bulk glass ~ 100 MPa whereas theoretical stress required to break atomic bonds is ~ 10,000 MPa

Microscopic flaws

20

0

Eγσδ

= 0 0.1Eσ

….. theoretical predictions

δ

σ

σ0

Leonard-Jones potential

δ0 σ – Applied stress

δ – Atomic distance

γ – Surface energy per unit area

Griffith’s Observations (1925)

18

f a Cσ ≈

Linear elasticity theory can not explain the experimental observation

Crack growth increases the surface energy, γ

Thermodynamic approach

f2E

aγσ

π=

E – Elastic modulus

σf – Critical stress

γ– Surface energy per unit area

a – Crack length

….. works good for brittle material

Reduced strength due to flaws

19

Strain energy is converted into surface energy. The crack growth occurs when the energy available is sufficient to overcome the material’s resistance.

kdEG Rda

= +

where andu dEdEG Rda da

γ= − =

G – Energy release rate

R – Resistance to crack growth

Ek – Kinetic energy

Eu – Potential energy

Eγ – Surface energy

G R=

For quasi-static crack growth

….. for linear elastic material

G

a a0

R σ1

σ2

σ3

Gc

Griffith’s energy criteria

20

2a Eσ π γ=

s pγ γ γ= +

….. material constant

Plastic wake Energy dissipation - Grain rotation

- Heat dissipation

- Dislocation motion

γp – Energy due to plastic dissipation

G

a a0

R

σ1

σ2

σ3

Gc

2aGE

πσ=

Irwin’s model (1950)

21

Mode II In-plane shear Mode I

Opening

Mode III Out-of-plane shear

Fracture modes

22

σyy

σxx

r θ

σxy

x

y

3cos 1 sin sin2 2 22

3cos 1 sin sin2 2 22

3cos sin sin2 2 22

Ixx

Iyy

Ixy

Kr

Kr

Kr

θ θ θσπ

θ θ θσπ

θ θ θσπ

= − = +

=

√r singularity

KI – Stress intensity factor

….. Amplitude of crack tip singularity

….. Valid only near the crack tip

….. Enough to define stress field

….. from Westergaard’s analysis

For linear elastic material

( )

2

22

...plane stress

1 ...planestrain

II

II

KGE

KGE

ν

=

= −

Stress field around a crack-tip

23

σyy

σxx

r θ

σxy

x

y

( )

( )

1 12 12 2

1 2 3 4

1 122 2

1 2 3

3cos 1 sin sin 2 cos 1 sin 2 cos ....2 2 2 2 2

3cos 1 sin sin 0 cos 1 sin2 2 2 2 2

xx

yy

A r A A r A r

A r A A r

θ θ θ θ θσ θ

θ θ θ θ θσ

−

−

= − + + + + + = + + + −

( )

( ) ( )

4

1 122 2

1 2 3 4

0 ....

3cos sin cos 0 sin cos sin ....2 2 2 2 2xy

A

A r A A r A rθ θ θ θ θσ θ−

+ +

= + − − +

K dominant terms

William’s asymptotic expansion

24

SIF measuring techniques

• Load cell

• Strain gages

• Photoelasticity

• Interferometry

• Digital image correlation

KI = KIc at crack initiation

KIc – critical stress intensity factor (fracture toughness)

Fracture toughness

25

a

w

1.25 w

P

a w

P

P/2 P/2 S

L 2a

2w

P

a w P

L

a 2w P

L a

a w

M

L

Ref – Fracture Mechanics, T. L. Anderson

CT SENB MT

SENT

DENT

IP aK f

WB W =

Test samples

Fracture test specimen (SENB)

ASTM D5045

S = 60 mm

B = 5 mm

W = 15 mm a

( )32I

P SK fBW

ξ=

( )( )( ){ }( )( )

2

32

3 1.99 1 2.15 3.93 2.7

2 1 2 1f

ξ ξ ξ ξ ξξ

ξ ξ

− − − +=

+ −

Fracture toughness – Mode I

Notch

Crack tip

Ref – Yesgat and Kitey, EFM 2016

27 Ref – Dally and Sanford, Experimental Mechanics

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

1 122 2

1 2 3

1 122 2

1 2 3

3cos 1 1 sin sin 2 cos 1 1 sin2 2 2 2 2

3cos 1 1 sin sin 2 cos 1 1 sin2 2 2 2 2

xx

yy

E A r A A r

E A r A A r

θ θ θ θ θε ν ν ν ν

θ θ θ θ θε ν ν ν ν ν

−

−

= − − + + + − + + = − + + − + − − +

( ) ( )1 12 2

1 332 sin cos sin cos2 2xy A r A rθ θµγ θ θ

−

= −

( ) ( ) 1for tan cot 2 , where cos 2 ,2 1θ να α

ν− = − = − +

( ) ( ) ( ) ( )' '1 1 3 1 32 cos sin sin cos 2 sin cos sin 21 2 2 2 2 22

Ix x

Kr

ν θ θ θµε θ α θ ανπ

− = − + +

r θ x

y α

x’

y’ Quasi-static and dynamic K

SIF gage

28

Polarizer

Analyzer

¼ wave plate

¼ wave plate Specimen

Light source

Camera

Maximum in-plane shear stress contours

( )max 1 212 2

Nfhστ σ σ= − =

Photoelasticity

29

( )1 2 | sin |2

INf Kh r

σσ σ θπ

− = =

31 sin sin2 2

3cos 1 sin sin2 2 22

3sin sin2 2

xxI

yy

xy

Kr

θ θ

σθ θ θσ

πσθ θ

− = +

θ x

y

r

Solve by over deterministic least square analysis

K measurement

30

In-plane displacement contours

2Npu =

Moiré Interfeometry

Ref – Savalia and Tippur, Exp Mech 2007

31

( )

( )0

2along ,8

lim

I app

I I appr

ENpKr

K K

πθ π

→

= ± =

=

( )( )

122 2sin 1 sin cos

2 2 2 22I app

KNpu r θ θ θνπ

= = − +

K measurement

Ref – Savalia and Tippur, Exp Mech 2007

32

y

x

G1

G2

L

D

Specimen

Gratings Filtering lens Filter

plane

Collimated laser beam

Mirror Load

Argon ion Laser

( )2 2x y

w B Npx x E

ν σ σ =

∂ ∂= − +∂ ∂ ∆

Coherent Gradient Sensing Interferometer

Ref – Kitey and Tippur, Acta Mater 2005

33

( )2 2x y

w B Npx x E

ν σ σ =

∂ ∂= − +∂ ∂ ∆

Using asymptotic expansion for σx and σy

Least-squares analysis to get An where

p grating pitch ∆ grating distance B sample thickness N fringe order

22

11 cos 2

2 2 2 2

n

nn

B n n NpA rE

ν θ ∞ −

=

− − − = ∆ ∑

1 2IK A π=

0 10

10

30

60

90

120

150

180

210

240

270

300

330

0o

180o

120o

N = 2 N = 1

N = -1

N = 3

N = -2

(r,θ) … CGS

CGS – KI measurement

Ref – Kitey and Tippur, Acta Mater 2005

Double Cantilever Beam Test

34

Interfacial fracture energy – Mode I

0 20 40 60Extension (mm)

0

20

40

60

Load

(N)

IdGbda

∏= −

ASTM D5528-01

Ref – Deepak, M Tech Thesis, IITK 2012

35

Data reduction – Modified beam theory

0 20 40 60Extension (mm)

0

20

40

60

Load

(N)

0 0.0002 0.0004 0.0006 0.0008

0.001 0.0012 0.0014 0.0016 0.0018

0.002

0 20 40 60 80 100 120 140

Com

plia

mce

(m/N

)

Delamiantion Length (mm)

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

-20 30 80 130

C1/

3 ((m

/N)1/

3 )

Delamination Length (mm) 0

200

400

600

800

1000

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Stra

in e

nerg

y re

leas

e rat

e (J/

m2 )

Delamination Length (mm)

A

B

C GIc

Fiber Bridging

Steady state crack propagation

P

δ

a

C

CPδ

=

∆

( )

2

23

2

IP dCGb daP

b aδ

=

=+ ∆

Ref – Deepak, M Tech Thesis, IITK 2012

36