‘the mechanics and mechanisms of fracture of … · ‘the mechanics and mechanisms of fracture...

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



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



Crack Propagation Event

Bulk Fracture Mechanics Tests << Sharp Pre-cracks and Measure Gc >>

Measure the fracture energy, Gc, via Standard ISO Methods.



‘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 _____



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.



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)



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



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



2.1 Localised Shear-Yield Bands Initiated by the Particles



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



The Plane-Strain Compression Test

Thin polymer sheet

Hardened steel anvils



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.



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.



2.2 Particle Debonding which Enables Plastic Void Growth of the Epoxy Matrix



Field Emission Gun SEM << Epoxy-Anhydride, 14.8% wt% nano-SiO2 >>

Voids around debonded nano-particles.

Fracture surface: crack growth direction



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

)



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



3.1 The Model



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



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 !’ !



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



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.



3.2 Parameters for the Plastic Void Growth Term



What volume fraction of the particles will debond?



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?



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.



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’.


Imperial College London To calculate the total number of nearest neighbours around a particle, a dual

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%.



For the particles that do debond, how large will the

voids grow ?



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.



3.3 The Results of the Model



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)]



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.

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



4. Cyclic Fatigue Properties



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.)



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 ?



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.



6. Concluding Remarks



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?



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

‘the mechanics and mechanisms of fracture of … · ‘the mechanics and mechanisms of fracture...

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