Exam Prep - Formulas Flashcards
Force Balance Formula
F = Ff + Fm σA = σf Af + σm Am
When εmu > εfu, state which fractures first and the corresponding formulas to use.
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.
When εfu > εmu, state which fractures first and the corresponding formulas to use.
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.
What is the average number molar mass formula?
Mn = W/n
Where n=W/M
How do you find the volume of crystallinity?
Rules of mixtures
ρ = ρc Xv + ρa (1-Xv)
Xv = ρ - ρa / ρc - ρa
Where:
ρc = 1/Vc
ρa = 1/Va
How do you find the continuous fibre stress?
σfmax = (σc/Ec) * Ef
How do you find the length of the fibre to reach the continuous fibre stress?
Li = σfmax* d / 2*τ
What is the formula for critical fibre length?
Lc = σfu* d / 2*τ
How would you calculate the interfacial shear strength improvement?
Improvement = (τnew - τ/τ) *100
What is the critical aspect ratio formula?
Lc/d = σfu/ 2*τ
How do you find the survival probability?
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.
How do you find the longitudinal tensile failure strain?
εcu = σcu/Ec
where σcu is calculated using the Kelly Tyson model.
How do you calculate the weight fraction?
Wf = (ρf/ρc)* Vf
Or
Wf = Weight of fibres/ Weight of composite
State the formula you need for the boltzmann superposition principle.
ε(t) = (1/E + t-u/η)σ + ∑(1/E + t-u/η)Δσ
What is the formula for thickness of composite to wood ratio?
Lc/Lw = [(1/Vc) - 1]^-1
Where Vc = 2tc/2tc-tw
Or
Vc = Epanel - Ewood/ Ecomposite - Ewood
In the composite and wood question how do you manipulate the rules of mixtures?
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
What is the degree of polymerisation formula?
Dp = Mn / M of the monomer
M = 104 unless given
Mass fraction crystallinity formula?
Xm = mc / mc + ma Xm = ρc*Vc/ρV Xm = (ρc/ρ) * Xv
What is the bending moment formula? state the flexibility formula too.
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
What is the maximum crack length formula?
a = 2Eγ / Pi*σfu^2
What is the surface to volume ratio in a fibre?
Sf/Vf = 4/df
State the atoms numbers for carbon, hydrogen and oxygen?
Carbon = 12 Hydrogen = 1 Oxygen = 16
How do you calculate one month strain using the tensile creep compliance in two directions?
ε1 (1month) = D(1month) * σ1
ε2 (1month) = εr (1month) = ν * D(1month) * σ2
How do you calculate one month change in diameter and wall thickness using the tensile creep strain?
Δd = ε1 (1month) * d
Δt = εr (1month) * t
What is the void fraction formula?
Vv = 1 - Vf - Vm
What is the ratio of fibre strain energy to matrix strain energy relation?
Uf/Um = Ef * εf^2 * Vf / Em * εm^2 * Vm
What is the inplane shear modulus relations?
G12 = Gm*Gf / VmGf + VfGm
G12 = Vf/Gf + (1-Vf)/Gm
Gf = Ef / 2(1+vf) Gm = Em / 2(1+vm)
