Chapter 4 Mechanical Properties

Chapter 4 Mechanical Properties

Glasses are brittle.

Fracture behaviour is mainly determined by flaws in the glass or by the environment.

σ = E ε

Elastic modulus

Glasses exhibit elastic deformation (reversible and temporary).

The relationship between stress and strain is given by Hooke's law.

Shear modulus

τ = shear stress (MPa) γ = shear strain

G = shear modulus

τ = Gγ

1

Dashed line: before tensile stress Solid line: after tensile stress

When tensile stress imposed in z direction:

Elastic elongation and strain ε

in z direction;

Elastic compression and strains ε

and ε

in x and y directions.

For isotropic material, ε

= ε

. Poisson’s ratio:

z y z

x

ε

− ε ε =

− ε

= ν

Negative sign included so that ν is positive.

(Compressive strains are negative).

( ^{+ ν} )

= 2G 1 E

G = shear modulus

E = modulus of elasticity

Poisson's ratio

3 r0

dr

E dF 



 



∝ 

• E is a measure of the material's stiffness, or resistance to elastic deformation.

• The magnitude of E is a measure of the strength of the interatomic bonding forces.

r₀= equilibrium interatomic spacing The magnitude of E is proportional to the

slope of the F – r curve at F = 0 i.e.

m N n

r b r

F = − a + a, b, n, m = constants Condon-Morse curve

This model applies to ionic, close-packed structures.

For glasses, E is also affected by dimensionality and connectivity.

Chain structure → layered structure → 3D network E increases

• Non-bridging oxygen ions cause E to decrease.

• Glasses with high T_ghave high E.

• Low thermal expansion glasses have high E.

Fracture Strength of Glasses

Theoretical strength of glasses

• The force need to overcome the maximum total force F_N in the F vs. r curve.

• Once r exceeds the distance corresponding to the maximum value of F_N, continued application of a force will cause interatomic bonds to break and a crack will propagate.

• Orowan: the stress needed to break a bond depends on the energy needed to create two new fracture surfaces.

σ

σ

0

t r

= Eγ

σ σ_t= Orowan stress

γ = surface energy of fracture surface r₀ = equilibrium interatomic spacing γ has values between 2 - 4 Jm^-2

If we use typical values for a silicate glass of E = 70 GPa, γ = 3 Jm^-2 and r₀= 0.2 nm, σ_t = 32 GPa.

Practical strength of glasses

In practice, the strength of glasses is much lower than the theoretical strength.

In any glass, there are surface cracks.

Stress at the crack tip is magnified.

2 1

2 



 



 σ ρ

≅

σ c

m

σ

= stress at crack tip σ = applied stress

c = crack length

ρ = radius of crack tip

ρ can be as small as an interatomic spacing.

∴ σ

can be large. ∴ glasses weak in tension.

N.B. compressive stress closes cracks, so they do not affect compressive strength.

For a brittle material, the critical stress σ

required for crack propagation is:

where

– E = modulus of elasticity – γ = specific surface energy

– c* = critical crack length for crack growth

When σ

> σ

in a brittle material, crack will propagate and sample will fracture.

2 1

f

c

E

2

*





 



 π

= γ σ

• E and γ do not change much with glass composition.

• Flaws are caused by external factors and can vary in size by several orders of magnitude.

• The theoretical strength of a glass has little effect on its practical strength.

• An increase in hardness can increase the resistance to flaw formation i.e. scratch

resistance.

7

Sample problem.

A glass plate contains an atomic-scale surface crack. (Take the crack-tip radius ≈ diameter of an O

ion.) If the crack length is 1 µm and the theoretical strength of defect-free glass is 7.0 GPa, calculate the breaking strength of the glass.

2 1

2  

 



 σ ρ

≅

σ c

m

Rearranging gives:

1

2

1 



 

 σ  ρ

=

σ

c

( ^nm ) ^{= 0.264 nm.}

=

=

ρ 2 r

2 0 . 132

O

( ) ^{57 MPa.}

m

Pa m   =

 





×

× ×

= σ

∴

10 1

10 264 10 0

0 2 7

1 .

.

If σ

= 7.0 GPa, the crack will grow and the glass will break.

Flaw sources and removal

• Oxide glasses have Vickers' hardness between 2 - 8 GPa. (Compare 100 GPa for diamond)

• Contact with a harder material can cause a flaw.

• Contact with another piece of glass or with metal tools used to handle glass can create flaws.

• Chemical attack also creates flaws: touching a glass surface with a fingertip will deposit NaCl and cause chemical attack.

• Thermal stresses caused during rapid cooling produce flaws through thermal shock.

• Heating glasses for a long time causes some crystallisation on the surface or causes dust particles to bond to the surface. This causes thermal expansion mismatches during cooling, creating flaws.

• Flaw generation can be reduced by coating the fresh glass surface with lubricant.

• Flaws can be removed by chemical etching, mechanical polishing or flame polishing.

• Etching blunts the flaw tip and reduces flaw length.

• Mechanical polishing reduces flaw length.

• Flame polishing removes flaws through viscous flow in the surface region.

Strengthening of Glass

• For a crack to grow, there must be a tensile stress at the crack tip.

• If a residual compressive stress is created at the surface of the glass, it will impede crack growth.

• No growth will happen unless the applied stress is large enough to overcome the residual compressive stress.

Glass piece is heated to T_g < Temp < softening point.

Rapidly cooled to room temp with air jets.

Surface cools more rapidly than interior and becomes rigid when its temp drops below the strain point.

Interior is at higher temp and still plastic.

At the glass piece cools, the interior attempts to contract to a greater degree than the rigid surface will allow.

After the glass piece has cooled to room temp, it has a compressive stress at the surface and a tensile stress in the interior.

To cause failure, the applied tensile stress must overcome the residual compressive surface stress before it can cause crack growth.

Room temp residual stress distribution over cross- section of a tempered glass plate

Tempered Glass

9

Another way of describing this:

Volume

• The interior of the glass cools more slowly than the surface.

• The interior has a lower fictive temperature and a higher density than the surface region.

• The surface region and interior are bonded together, so elastic strains form to counter the difference in densities.

• The surface is placed in compression, the interior in tension.

• The difference in fictive temperature between the surface and interior depends on the difference in cooling rate between the surface and interior.

• The compressive surface stress increases with cooling rate and glass thickness.

• Compressive stress also increases with glass thermal expansion coefficient (TEC) and the difference in TEC between the glass and supercooled liquid.

• Thermal tempering is not effective for: thin pieces of glass or fibres SiO₂ glass (low TEC)

borosilicate glasses (small volume change with fictive

Other Strengthening Methods

Ion exchange strengthening (chemical tempering)

• Glass containing alkali ions e.g. Na is exposed to a source of different alkali ions e.g K at T << T_gto avoid stress relaxation).

• This can be done by placing the glass in a molten salt bath of KNO₃ at 450°C.

• Na⁺ ions diffuse out of the surface of the glass and K⁺ ions diffuse in.

• The larger K⁺ ions form a compressive stress in the glass surface (up to 400 MPa).

• Special glasses have been developed for chemical strengthening - equimolar concentrations of Na and Al - high Na diffusivity and T_g.

• Alternate method: a smaller ion e.g. Li⁺is introduced at T > T_g.

• Surface region contains Li instead of Na and has lower TEC.

• On cooling, the surface is forced into compression and the bulk into tension.

Na⁺ K⁺

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Surface crystallisation

• Used in alkali aluminosilicate glasses.

• Replace Na⁺ with Li⁺in surface region.

• Crystallisation of phase with very low TEC e.g. spodumene LiAlSi₂O₆) may happen.

• On cooling, crystallised surface region is placed in compression. bulk in tension.

Alkali removal

• Removal of alkali from surface leave SiO₂- rich region with lower TEC than bulk glass.

• On cooling, the surface is forced into compression and the bulk into tension.

• Exposing the glass to SO₂vapour during annealing can remove alkali.

• Na⁺ions exchange with H⁺/H₃O⁺ ions.

• This method is not used now for safety reasons.

• Instead, an aqueous solution of ammonium sulfate is sprayed on the glass before annealing.

Statistical Nature of Fracture of Glass

Fracture strength is controlled by number and size of flaws on glass.

A set of identical glass samples can have a wide variation in fracture strength.

Experimental method also affects fracture strength e.g. 3- point vs. 4-point bend test.

Compressive stress

Tensile stress

Experimental results can be represented by a Gaussian distribution

( )











− −













= π ₂

2 m

2 2d

S S d

2

P 1 exp

P = probability of finding a sample with failure stress S S_m= strength of greatest probability

d = standard deviation of distribution

13

A Weibull distribution can also be used.















 



−

−

=

m

S0

1 S

F exp

F = the fraction of samples which fail at stresses below S.

S₀ = scaling factor

m = width of the distribution (Weibull modulus)

Small m means a broad distribution.

The equation is often changed to:

log[-ln(1 - F)] = m(log S - log S₀)

Data are plotted as log[-ln(1 - F)] vs. log S (called a Weibull plot).

m and S₀ are determined from a least-squares fit of the data.

Fatigue of Glasses

•No cyclic loading! (i.e. const. σ).

•Water-containing environment.

•Temperature ∼ room temp.

Fracture can occur by slow crack growth under a constant static stress.

• H

O reacts with SiO

network to cause breaks.

• Reaction at crack tip lengthens crack.

• Static fatigue important for -100 °C < T < 150°C.

• Fatigue rate increases with humidity and temperature.

Static Fatigue

15

Thermal Shock

Tension develops at surface

( ^{α − ∆ ν} )

= σ 1

T E

σ

rapid quench

resists contraction

tries to contract during cooling

T

T

Consider a glass rapidly cooled from T_g. The surface cools first and tries to contract more than the interior, which is still hot. The surface tries to “pull” the interior into compression. The interior

constrains the surface and “pulls” it into tension. If the surface is instantaneously cooled to room temp while the bulk is still at T_g, the surface stress is given by:

• Thermal shock is a problem when glasses are rapidly cooled over large temperature ranges.

• When a body is heated or cooled, the outside changes temperature more rapidly than the interior.

• Temporary stresses form which can cause fracture.

∆T = T₂ - T₁

α = thermal expansion coefficient ν = Poisson's ratio

σ_f= failure stress

( )

α ν

−

= σ

∆ E

T ^f 1 or

For a glass plate cooled at a finite cooling rate:

( )







φ ρ ν

−

= α

σ 3k

C L 1

E p

2

φ = cooling rate, K.sec^-1 k = thermal conductivity ρ = density

C_p= heat capacity at constant pressure L = one half of the plate thickness

Annealing of Thermal Stresses

Temporary stresses during cooling

Permanent stresses by thermal tempering.

Uncontrolled permanent stresses can also be formed during processing.

Uncontrolled permanent stresses can cause: static fatigue

dimensional changes

changes in refractive index stress birefringence

changes in fictive temperature

These permanent stresses are removed by annealing.

17

• The glass object is placed in an annealing lehr.

• The object is heated up to a temperature 5 °C above the annealing point or T_g.

• The glass is held at that temp fpr 30 - 60 minutes.

• The glass is cooled slowly (1 °C.min^-1) to a temperature below the strain point.

• Once well below the strain point, the glass can be more rapidly cooled (e.g. 10 °C.min^-1)

without causing permanent stress. Excessive temporary stresses must be avoided.

During annealing:

1. existing permanent stresses must be removed

2. new residual stresses must be minimized 3. temporary stresses large enough to cause

failure must be avoided