Exam Prep - Formulas

Question 1

Force Balance Formula

Answer 1

F = Ff + Fm
σA = σf Af + σm Am

Question 2

When εmu > εfu, state which fractures first and the corresponding formulas to use.

Answer 2

Fibres fracture first, ductile matrix.

At low vf: σcu = σmu (1 - Vf)
At high vf: σcu = σfuVf + σmu’ * (1 - Vf)
where σmu’ = εfuEm

Combine the two equations to solve for Vf, the volume fraction at which a change in failure mode occurs.

Vf = σmu - σmu’ / (σfu - σmu’ + σmu)

To obtain the critical volume fraction:

Vfc = σmu - σmu’ / (σfu - σmu’)

Draw the graph.

Question 3

When εfu > εmu, state which fractures first and the corresponding formulas to use.

Answer 3

Matrix fails first, brittle matrix.

At low vf: σcu = σfu’ * Vf + σmu * (1 - Vf)
where σfu’ = εmu * Ef

At high vf: σcu = σfu*Vf

Combine the two equations to solve for Vf, the volume fraction at which a change in failure mode occurs.

Vf = σmu / (σfu - σfu’ + σmu)

Draw the graph.

Question 4

What is the average number molar mass formula?

Answer 4

Mn = W/n

Where n=W/M

Question 5

How do you find the volume of crystallinity?

Answer 5

Rules of mixtures
ρ = ρc Xv + ρa (1-Xv)

Xv = ρ - ρa / ρc - ρa

Where:
ρc = 1/Vc
ρa = 1/Va

Question 6

How do you find the continuous fibre stress?

Answer 6

σfmax = (σc/Ec) * Ef

Question 7

How do you find the length of the fibre to reach the continuous fibre stress?

Answer 7

Li = σfmax* d / 2*τ

Question 8

What is the formula for critical fibre length?

Answer 8

Lc = σfu* d / 2*τ

Question 9

How would you calculate the interfacial shear strength improvement?

Answer 9

Improvement = (τnew - τ/τ) *100

Question 10

What is the critical aspect ratio formula?

Answer 10

Lc/d = σfu/ 2*τ

Question 11

How do you find the survival probability?

Answer 11

s = exp [ -L/Lo * (σ/σo)^m]
ln (s) = -L/Lo * (σ/σo)^m
Rules of logs:
ln(-ln (s))= - ln(L) - mln(σ) + mln(σo)

Obtain two sets of equations using the numbers in the question and then do eq1 - eq2. Find m, sub back in to find σo. Find the s. NOTE the -L/Lo = -V/Vo = -d/do and the o term is a unit length = 1 and cancels.

Question 12

How do you find the longitudinal tensile failure strain?

Answer 12

εcu = σcu/Ec

where σcu is calculated using the Kelly Tyson model.

Question 13

How do you calculate the weight fraction?

Answer 13

Wf = (ρf/ρc)* Vf

Or

Wf = Weight of fibres/ Weight of composite

Question 14

State the formula you need for the boltzmann superposition principle.

Answer 14

ε(t) = (1/E + t-u/η)σ + ∑(1/E + t-u/η)Δσ

Question 15

What is the formula for thickness of composite to wood ratio?

Answer 15

Lc/Lw = [(1/Vc) - 1]^-1

Where Vc = 2tc/2tc-tw

Or

Vc = Epanel - Ewood/ Ecomposite - Ewood

Question 16

In the composite and wood question how do you manipulate the rules of mixtures?

Answer 16

ρpanel = ρcomposite*Vc + ρwood (1-Vc)

ρcomposite = ρfibre*Vf + ρmatrix (1-Vf)

or

Epanel = Ecomposite*Vc + Ewood (1-Vc)

Ecomposite = Efibre*Vf + Ematrix (1-Vf)

or

σpanel = σcomposite*Vc + σwood' (1-Vc)
σwood' = εw * Ew
σcomposite = σfibre*Vf + σmatrix' (1-Vf)
σmatrix' = εfu * Em,     if εmu > εfu

Question 17

What is the degree of polymerisation formula?

Answer 17

Dp = Mn / M of the monomer

M = 104 unless given

Question 18

Mass fraction crystallinity formula?

Answer 18

Xm = mc / mc + ma
Xm = ρc*Vc/ρV
Xm = (ρc/ρ) * Xv

Question 19

What is the bending moment formula? state the flexibility formula too.

Answer 19

M/I = σ/y = E/R,

I = bd^3/12 for a rectangular block
R = 1/k

Thus the flexibility formula:
k/m = 1/IE
k/m = 64/Epid^4 for a circular diameter

Question 20

What is the maximum crack length formula?

Answer 20

a = 2Eγ / Pi*σfu^2

Question 21

What is the surface to volume ratio in a fibre?

Answer 21

Sf/Vf = 4/df

Question 22

State the atoms numbers for carbon, hydrogen and oxygen?

Answer 22

Carbon = 12
Hydrogen = 1
Oxygen = 16

Question 23

How do you calculate one month strain using the tensile creep compliance in two directions?

Answer 23

ε1 (1month) = D(1month) * σ1

ε2 (1month) = εr (1month) = ν * D(1month) * σ2

Question 24

How do you calculate one month change in diameter and wall thickness using the tensile creep strain?

Answer 24

Δd = ε1 (1month) * d

Δt = εr (1month) * t

Question 25

What is the void fraction formula?

Answer 25

Vv = 1 - Vf - Vm

Question 26

What is the ratio of fibre strain energy to matrix strain energy relation?

Answer 26

Uf/Um = Ef * εf^2 * Vf / Em * εm^2 * Vm

Question 27

What is the inplane shear modulus relations?

Answer 27

G12 = Gm*Gf / VmGf + VfGm

G12 = Vf/Gf + (1-Vf)/Gm

Gf = Ef / 2(1+vf)
Gm = Em / 2(1+vm)