Micromechanics- Shear Lag Theory 2

Question 1

Load balance do a segment of an aligned short fibre composite under longitudinal stress

Answer 1

σ1=ffσbarf+(1-ff)σbarm
Where σ1 is the applied longitudinal stress
σbarf is volume averaged stress in the fibre
σbarm is volume averages stress in the matrix
Can multiply both sides by CSA of composite (A) for force balance

Question 2

Assumption for strain in the matrix of short fibre composites

Answer 2

Assume the strain in the strip is uniform and equal to that on the composite (ε1)
Means σbarm is about Emε1
Less uniform for long fibre composites

Question 3

Formula for composite modulus in 1 direction for aligned short fibre composites in databook

Answer 3

E1=ffEf(1-(tanh(ns)/ns)+(1-ff)Em
Where n is dimensionless constant with formula in databook
Ef and Em are stiffness of fibre and matrix
s is fibre aspect ratio L/r
In databook there is η at start but this is 1 in this case (aligned)

Question 4

Modified shear lag version of formula for composite modulus in 1 direction for aligned short fibre composites not in databook

Answer 4

E1’=ffEf(1-(Ef-E’m(tanh(ns))/Efns)+(1-ff)Em
The difference is the Ef-E’m over Ef stuff
E’m has formula in databook
Takes into account stress transfer across fibre ends

Question 5

How does composite modulus vary with ff for different aspect ratios (unmodified shear lag theory)?

Answer 5

For carbon fibre in epoxy: start nearly 0, curve up to high modulus when ff=1. Lower aspect ratio means E1 stays lower for longer and more curvy. Higher aspect ratio more linear like long fibre composites.
For SiC in glass: start higher E1 then same trend to higher E1 at ff=1. Same effect of aspect ratio but not as noticeable and each aspect ratio more linear than for C fibre in epoxy (less mismatch in properties between fibres and matrix)

Question 6

How does composite modulus vary with ff for different aspect ratios (modified shear lag theory)?

Answer 6

Even lower effect of aspect ratio and all lines more linear than for unmodified theory. Still same start and end points

Question 7

What happens to E1 formula for increasing aspect ratio?

Answer 7

At high aspect ratios fibres are effectively continuous. Equation reverts to simple longitudinal rule of mixtures because
tanh(ns)/ns tends to 0

Question 8

Formula for minimum aspect ratio for effective stress transfer

Answer 8

This is the aspect ratio which maximises stiffness (where rule of mixtures is observed).
Estimated by s sub RM = 10/n
sRM is about 100 for polymer composites
sRM is about 25 for metal matrix composites

Question 9

How does sRM vary with fibre volume fraction (still aligned fibres)?

Answer 9

Starts high, curves down with decreasing stiffness, then nearly straight line until curves downwards near end. Highest curve for C fibre in epoxy, lowest for SiC in glass, middle for glass fibre in unsaturated polyester.

Question 10

How does formula for composite modulus change when short fibres are not aligned?

Answer 10

This is the formula in the databook with η in front. This is the Krenchel factor which is the efficiency of reinforcement.
For randomly oriented short fibres in a 2D plane it is 3/8 (e.g for chopped strand mat).
For randomly oriented short fibres in 3D it is 1/5

Question 11

How does being randomly aligned in 2D or 3D affect composite modulus vs ff graph?

Answer 11

For every aspect ratio the curve starts at same point but remains lower and finishes at lower value when ff=1. Lowest for random in 3D. Even when fibres effectively continuous the near linear line is just less steep

Question 12

Where is interfacial shear stress τi the highest?

Answer 12

At the fibre ends

Question 13

What happens when interfacial shear stress exceeds a critical value τic?

Answer 13

Either get local plasticity in the matrix
Or interfacial sliding between the matrix and fibre

Question 14

Formula for interfacial shear stress

Answer 14

τi=(n/2)Efε1sinh(nx/r)sech(ns)
In databook
x is distance from centre of fibre so is equal to L at fibre ends
When looking at fibre ends get nL/r which is ns

Question 15

Formula for critical interfacial shear stress

Answer 15

τic=(n/2)Efε1cTanh(ns)
In databook
ε1c is critical strain
Tanh(ns) comes from simplification of sinh(nL/r)sech(ns)

Question 16

Critical stress meaning and formula

Answer 16

σ1c is the stress at which the composite stress strain curve will start to deviate from linearity, but this yielding or interfacial slinging effect is localised around fibre ends.
σ1c=(2τic/nEf)((ffEf+(1-ff)Em)coth(ns)-ffEf/ns)
In databook

Question 17

How does critical fibre stress σ1c vary with short fibre volume fraction.

Answer 17

Very similar shaped curves and effect of aspect ratio to graphs of composite modulus vs ff. Each for a fixed ε1c

Question 18

How does critical interfacial shear stress τic vary with ff?

Answer 18

Starts very low and curves up very slowly until fast curve almost vertical neat ff=1. Aspect ratio makes very little difference. Higher values for SiC in glass compared to C fibre in epoxy

Question 19

Where is the peak fibre stress at the onset of yielding or interfacial sliding?

Answer 19

At the centre of the fibre (x=0)

Question 20

What happens before fibres start tot fracture?

Answer 20

Yielding or sliding occurs first. Further strain leads to sliding or yielding spreading along the fibre increasing the tensile stress in the fibre

Question 21

Formula and meaning of critical fibre half length

Answer 21

The half length below which no further fibre fracture can occur.
Lc=σ(fu)r/2τic
Or sc=σfu/2τic
Where σ sub fu is the ultimate failure stress of the fibres (assuming it is well defined)

Question 22

What is expected in terms of length for a suitably high applied strain?

Answer 22

Lc

Question 23

What is the critical aspect ratio?

Answer 23

s sub c. The aspect ratio below which no further fibre fracture is possible. About 100 for polymer matrix composites. About 2-3 for metal matrix composites

Question 24

What is the stress transfer aspect ratio?

Answer 24

s sub t. The aspect ratio at which peak stress is attained in a short fibre.

Question 25

What is the rule of mixtures aspect ratio?

Answer 25

s sub RM. the aspect ratio required for rule of mixtures like behaviour in a short fibre