composite materials: mechanical and fracture characterization on mechanics/ppt... · composite...
TRANSCRIPT
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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
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Classification
2
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3
10% spherical particle 10% Milled fibers
45% continuous fibers
Distribution uniformity
Ref – Yesgat and Kitey, EFM 2016
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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
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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
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6
Mechanical Characteristics
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Uniaxial tension test
ASTM D638
Ref – Gayathri, M Tech Tesis, IITK 2014
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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
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Test specimens - Laminated composites
ASTM D3039 ASTM D695
Ref – Paiva, Mayer and Rezende, Mat Res 2005, 2006
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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
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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
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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
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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
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Failure or Fracture
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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
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( )1ij f
rσ θ∝
2a 2b
σ0
2a
σ0
r θ
max 0 1 2 ab
σ σ = +
…Linear theory of elasticity
Flaws
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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)
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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
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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
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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)
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Mode II In-plane shear Mode I
Opening
Mode III Out-of-plane shear
Fracture modes
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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
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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
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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
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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
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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
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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
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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
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( )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
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In-plane displacement contours
2Npu =
Moiré Interfeometry
Ref – Savalia and Tippur, Exp Mech 2007
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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
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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
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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
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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
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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
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