‘the mechanics and mechanisms of fracture of … · ‘the mechanics and mechanisms of fracture...
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3rd Progress Meeting
Imperial College London
‘The Mechanics and Mechanisms of Fracture of Nano-Modified Epoxy Polymers'
Tony Kinloch
Department of Mechanical Engineering
Adhesion, Adhesives and Composites Group Imperial College London, UK
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1. Introduction This talk is concerned with thermosetting epoxy polymers
for adhesives and matrices for fibre-composites. These are formed from, for example:
OO
OO
S
O
O
H2N NH2+
Epoxy (e.g. DGEBF) monomer: Hardener:
• These react to give a three-dimensional, crosslinked, amorphous epoxy polymer.
• This microstructure leads to inherently very brittle polymers, but with good stiffness, creep and thermal properties.
• And they can be used as adhesives and matrices for fibre-composites, since they may start life as low viscosity ‘monomers + hardener’.
and
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Crack Propagation Event
Bulk Fracture Mechanics Tests << Sharp Pre-cracks and Measure Gc >>
Measure the fracture energy, Gc, via Standard ISO Methods.
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‘Added’ Nano-Particles << J. Materials Sci., 2002, 37, 433 >>
Cyanate ester resin + 10 wt.% ‘added’ nano-particles.
20 nm or 100 nm diameter.
No surface treatment.
0
20
40
60
80
100
120
140
160
Unmodified Al2O3(20nm)
Al2O3(100nm)
SiO2(20nm)
TiO2(20nm)
Y2O3(20nm)
Frac
ture
ene
rgy,
J/m
2
Decrease in fracture energy.
Due to agglomeration (light areas in micrograph below).
Thus, need excellent dispersion!
20 µm _____
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Nano-Silica Composites << ‘In-situ’ Nano-SiO2 Particles >>
We have therefore been working on sol-gel materials, since here the nano-silica particles are formed in-situ - thus, overcoming agglomeration (and health and safety) problems.
The matrix viscosity is virtually unchanged even at high loadings.
Any transparency is maintained.
The Tg of the cured polymer matrix is unaffected.
Opposite is shown a TEM of a cured nano-silica/epoxy.
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Single-Component (‘1K’) Epoxy/Anhydride << Effect of Nano-SiO2 (wt%) >>
Effect of type of epoxy and why do the nano-silica particles increase the toughness ?
DGEBA epoxy/methylhexahydrophthalic acid anhydride. (Tg = 139oC)
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0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.140
1
2
3
4
5
6 Polyether-amine cured DGEBA/F Polyether-amine cured DGEBA Anhydride-cured DGEBA Amine-cured TGMDA Linear fit (Polyether-amine cured epoxies) Linear fit (Anhydride- and Amine-cured epoxies)
Norm
alise
d fra
ctur
e en
ergy
Volume fraction, vf, of silica nanoparticles
Normalised Fracture Energy versus the Volume Fraction of Nano-silica Particles
High Tg epoxies
Low Tg epoxies
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Will show that the toughening micromechanisms arise from: 1. Localised shear-yield bands initiated by the particles; and: 2. Particle debonding which enables plastic void growth of the epoxy matrix.
2. Toughening Micromechanisms
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2.1 Localised Shear-Yield Bands Initiated by the Particles
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Transmission Optical Micrograph (Polarised Light) << Sectioned region: normal to fracture plane, of nano-SiO2 epoxy >>
fractured half of specimen
birefringent plastic shear-yield bands zone
tip of crack
20µm
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The Plane-Strain Compression Test
Thin polymer sheet
Hardened steel anvils
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0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.10
50
100
150
200
250
300
True
stre
ss (M
Pa)
True strain
Amine-cured TGMDA Anhydride-cured DGEBA Polyether-amine cured DGEBA Polyether-amine cured DGEBA/F
True Stress versus True Strain Curves from Plane-Strain Compression Tests
Low Tg epoxies High Tg epoxies
Note more strain-softening and higher failure strains in the low Tg epoxies.
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2 mm
2 mm
2 mm
2 mm Anhydride-cured DGEBA
Polyether-amine cured DGEBA/F
Polyether-amine cured DGEBA
Amine-cured TGMDA
Transmission Optical Micrograph (Polarised Light) of Epoxy Polymers << Sectioned regions: normal to applied compressive stress >>
Taken from the plane-strain compression test specimens:
Shows formation of plastically-deformed shear bands.
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2.2 Particle Debonding which Enables Plastic Void Growth of the Epoxy Matrix
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Field Emission Gun SEM << Epoxy-Anhydride, 14.8% wt% nano-SiO2 >>
Voids around debonded nano-particles.
Fracture surface: crack growth direction
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Atomic Force Microscopy << Epoxy-Anhydride, 11.1wt % nano-SiO2 >>
AFM height image
Surface profile of a line drawn across the particle encircled
AFM of fracture surface. Showing void around a debonded nano-particle.
Particle ~ 30 nm
(This sample was not coated with Pt or Au as for FEG-SEM.)
200 nm
Vertical distance (nm) H
eigh
t (nm
)
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We need to model the effects of the two micromechanisms: 1. Localised shear-yield bands initiated by the particles; and: 2. Particle debonding and plastic void growth of the epoxy matrix.
3. Modelling the Toughening Micromechanisms
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3.1 The Model
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Modelling of the Debonding and Plastic Deformation of the Epoxy
From calculation, then the debonding of the particle absorbs little energy - i.e. less than about 1 J/m2.
And we therefore have two types of plastic deformation to model:
Plastic shear-yield bands which form around the particles, due to the local stress concentrations.
Plastic void expansion of the epoxy matrix, which develops after particle debonding.
Ψ+= cuc GG
Overall toughening contribution
Fracture energy of the unmodified epoxy
Measured fracture energy
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Modelling of the Plastic Deformation of the Epoxy Polymer
Toughening contribution from the plastic shear-yield band mechanism.
vs GΔG ∆+=Ψ
Toughening increment.
Toughening contribution from the plastic void expansion mechanism.
ΔGs = 2� Us(r)drry
rp
ΔGv = 2� Uv (r)drry
0
U(r): strain-energy density ry: plastic zone radius at the crack tip rp: particle radius
We can solve the above equations - and with no ‘fitting’ factors !’ !
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Modelling of the Debonding and Plastic Deformation of the Epoxy
Vf: volume fraction of nano-SiO2 particles. σyc: compressive yield stress. γf : compressive fracture strain. F/(ry): a geometric function. µm: yield-criteria pressure-dependency constant. Vfv: volume fraction of voids caused by debonding/plastic deformation of epoxy. Vfp: volume fraction of nano-SiO2 particles which debond and void. ryu: plastic zone radius of unmodified epoxy polymer. Kvm: von Mises stress concentration.
Toughening contribution from the plastic shear-yield band mechanism.
vs GΔG ∆+=Ψ
Total toughening increment.
Toughening contribution from the plastic void expansion mechanism.
ΔGs = 0.5VfσycγfF′ �ry�
∆Gv = (1 − μm2 /3)�Vfv − Vfp�σyc ryu Kvm
2
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Calculation of the Predicted Toughness
Ψ+= cuc GG
Overall toughening contribution
Fracture energy of the unmodified epoxy
Measured fracture energy
Ψ = 0.5VfσycγfF′ �ry�+ (1−μm2 /3)�Vfv−Vfp�σyc ryu Kvm
2
Therefore, we have:
And, so can calculate the predicted toughness from:
All the terms above are known except for (i) the volume fraction of particles which do debond and so enable plastic void growth of the epoxy polymer, and
(ii) the volume fraction of voids.
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3.2 Parameters for the Plastic Void Growth Term
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What volume fraction of the particles will debond?
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Field Emission Gun SEM << Epoxy-Anhydride, 9.6% vol. fraction of nano-SiO2 >>
Voids around debonded nano-particles.
Fracture surface: crack growth direction
But note that not all of the silica nanoparticles debond ! WHY ? AND WHAT IS THE VALUE OF Vfp?
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A. Void
2
1
2
4
3
4
6
7
7
Deformed FEA Mesh containing a Void at Point A, showing the Numbered Particles
Are the nearest neighbours (Particles 1 and 2) ‘shielded’ from debonding? If so, can we then predict the percentage of particles which will debond
(and so enable void growth to occur) - and hence obtain Vfp ?
Volume fraction of particles = 13.7% subjected to a hydrostatic tensile stress with a void at point A.
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A. Void
2
1
2
4
3
4
6
7
7
Comparison of values of the strain-energy required to debond a particle at various positions around a void normalised with respect to the strain-energy
required for an isolated particle.
Basically, the deformed void ‘shields’ its nearest neighbours from the applied stress field - and hence they are most unlikely to debond. Can we then apply a
statistical analysis to calculate the percentage of such ‘nearest neighbours’.
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representation to the ‘Voronoi Tessellation’, termed the ‘Delaunay Triangulation’, may be calculated. The number of nearest neighbours is equivalent to the number
of the Delaunay triangles that contain that particle as a vertex. (The ‘Voronoi Tessellation’ of the material is shown in the background and breaks the material
into polygons with exactly one particle within each cell.)
These studies predict that only 14.3% of the silica nanoparticles will see a sufficiently high applied strain-energy to debond.
FEG-SEM experiments give a value of 15±5%.
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For the particles that do debond, how large will the
voids grow ?
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Prediction of Toughness: Extent of Plastic Void Growth
• Assume that any void that develops after the particles
debonds will grow until the hoop strain around the void equals the strain to failure of the epoxy.
• Therefore:
Radius of void = (1+γf) x radius of particle.
• Where γf is the measured strain to failure of the epoxy polymer.
• Hence, can predict the volume fraction of voids, Vfv.
Page 30
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3.3 The Results of the Model
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Comparison of Predicted and Measured Toughness Increases for the Anhydride-Cured Epoxy
Model assumes that toughening mechanisms are via: • Plastic shear-yield bands initiating and growing
around the particles. • And the debonding of the nano-silica particles
followed by plastic void growth of the epoxy.
Nano-silica (vol %)
Predicted toughness (J/m2) Measured Toughness, Gc (J/m2) ∆Gs ∆Gv Gc
2.5 34 12 123 123
4.9 48 23 148 179
7.1 58 34 169 183
9.6 68 46 191 191
13.4 78 64 219 212
[Gcu = 77 (J/m2)]
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Agreement with Other Literature Results
Also, very good agreement between the model proposed and experimental results in the literature: Ma et al. Polymer 49 (2008)
3510), see below, and Pearson et al. Polymer 53 (2012) 1890.
Page 33 0 5 10 15 200
200
400
600
800
1000
1200
Analytical model Present Study Ma et al.
Frac
ture
ene
rgy,
G c (J/m
2 )
Nanosilica content (wt. %)
0 10 200
100
200
300
400
∆Gs
0.15∆Gv
∆ G
s and
∆G v (
J/m2 )
Nanosilica content (wt. %)
Cancan can W can
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4. Cyclic Fatigue Properties
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Fatigue Data for Nano-SiO2 Epoxy << log da/dN versus Gmax >>
• Gmax is the maximum G value applied in the fatigue cycle.
• da/dN is the rate of crack growth per cycle.
• Frequency: 5 Hz. • Note the presence of
a fatigue threshold, Gth, value.
4.0wt.%
14.8wt.%
0 wt%.
20.2 wt.%
7.8 wt.%
-8
-7
-6
-5
-4
-3
-2
0.5 1 1.5 2 2.5 3
log G max (J.m-2)
log
da/d
N (m
m/c
ycle
)
Control7.8N0R4N0R14.8N0R20.2N0R
Note: Micron-sized particles often do not increase the fatigue threshold, Gth, value - being larger than the plastic zone radius, ry, at the threshold value (ry(threshold) ≅ 0.5 to 1 µm.)
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5. Does the Size of the Silica Phase Matter ?
(Or: ‘Why bother with nano ?’)
Recall that size did matter for the optimum fatigue properties - but for the initial toughness, Gc ?
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Effect of Particle Size on Toughness of Silica-Epoxy Polymers
Average diameter of
silica particles
(μm)
vf of silica
particles (%)
Gc (silica-particle epoxy)/Gc (epoxy)
(Measured)
Gc (silica-particle epoxy)/Gc (epoxy)
(Predicted)
0.02 10 2.5 2.4
16 10 2.0 1.7
32 10 1.6 1.6
47 10 1.1 1.4
Note: Values of measured toughness ratios in ‘green’ from Spandoukis and Young, J. Materials Sci., 1984, 19, 473.
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6. Concluding Remarks
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The highest values of toughness, Gc, which can be modelled, occur when: The silica nanoparticles are present as a very well-dispersed phase in the epoxy polymer. This is an essential requirement ! The epoxy polymer exhibits strain-softening followed by strain-hardening, which allows the ready formation, and then stabilisation, of plastic deformation associated with the silica nanoparticles. The epoxy polymer possesses a relatively low Tg, and high Mc, which lead to a relatively high plastic failure strain to be achieved. There is relatively low adhesion at the nanoparticle/polymer interface, which allows the silica nanoparticles to debond in the triaxial stress-field ahead of the crack tip and so enables plastic void-growth in the epoxy polymer to develop. The cyclic fatigue properties can also be significantly improved.
The ‘Best’ Microstructure of Particles and Matrix for High Toughness?
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7. Acknowledgments Academic Faculty Colleague: Ambrose Taylor, Felicity Guild Our Post-Docs/PhD Students: Bernt Johnsen, Reza Mohammed, Joon Lee, Kunal Masania, James Hsieh, David Bray, Mana Techapaitoon. Industrial Collaborators: Stephan Sprenger (Evonik, Germany) and Dave Egan (Emerald Materials, USA) Indian National Aerospace Laboratory (NAL): Manjunatha Chikkamadal Manchester University: Ian Kinloch SIMTech: Wern Sze Teo, Mana Techapaitoon