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Materials IA > Equations (NOT IN DATA BOOK) > Flashcards

Equations (NOT IN DATA BOOK) Flashcards

1
Q

equation for a dipole moment

A

μ = qr
(vector)

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2
Q

give the two equations for polarisation (dipole moments and charge)

A

P = n μ
P = Q/A
(vector)

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3
Q

give the 2 equations for resultant charge density (displacement fields)

A

D = εo E + P
(E = electric field, P = polarisation)

D = ε E = k εo P

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4
Q

give the equation for the change in voltage due to stress on a piezoelectric material

A

ΔV = dσL /ε = dσL / k*εo

d = piezoelectric constant

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5
Q

give the equation for the change in voltage due to a change in temperature for a pyroelectric material

A

ΔV = ΔQL / kεoA = pΔT L / k εo

p = pyroelectric constant

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6
Q

give the Goldschmidt tolerance factor, what should it equal

A

t = (rA + ro) / (sqrt(2)*(rB + ro))

it should equal 1

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7
Q

give the equation for the number of diffusing ions in an electric field (Boltzmann stats)

A

n = no exp(-qv/kt)

k = Boltzmann constant
q = charge on ion
V = voltage
T = temp

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8
Q

give the equation derived from the number of diffusing ions in an electric fields for the conc. gradient

A

dn/dx = -qn/KT dv/dx = qn/KT E

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9
Q

give the thin lens equation for optical microscopes

A

1/f = 1/u + 1/v

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10
Q

give the equation for maxima spacing from a double slit

A

x = lλ/d

l = distance to slits from screen
λ = wavelength
d = slit spacing

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11
Q

give the general equation for the phase difference from scattering from the nth atom in a structure (in terms of its coords)

A

ϕn = 2π (h xn + k yn + l zn)

(xn,yn,zn) are the coords of the nth atom

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12
Q

give the equation for the multiplicity of a plane (hkl)

A

M(hkl) = (3!)(2^A) / (n!)

A = number of non-zero indices
n = number of repeated indices

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13
Q

give the equation for the intensity of a peak intensity for a plane (hkl)

A

I(hkl) α M(hkl) |F(hkl)|^2 g(θ)

where g(θ) increases with θ

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14
Q

give the equation for the distance between two points representing planes in reciprocal space in terms of the real spacing of the planes

A

|d(hkl)*| = 1/|d(hkl)|

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15
Q

give the equation for determining the planar spacing from a diffraction pattern from a TEM

A

d(hkl) = λL/R

λ = wavelength of electrons
L = distance from specimen to detector
R = distance on detector from origin to spot

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16
Q

give the equation for internal energy (differential in three forms, one at constant volume)

A

dU = δq + δw
dU = CdT - pdV

at const, vol
dU = C(v)dT

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17
Q

give the equation for absolute internal energy at a given temperature

A

U = Uo + ∫CvdT (0 to T1) + Σli (on i)

where li is the ith latent heat of transition

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18
Q

give the non differential AND differential forms of enthalpy, H

give the equation for enthalpy at a constant pressure

A

H = U +PV
dH = δq + Vdp

dH = δq = C(p) dT
at a const. pressure

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19
Q

give the equation for absolute enthalpy at a given temperature

A

H = Ho + ∫CpdT (0 to T1) + Σli (on i)

where li is the ith latent heat of transition

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20
Q

give the non-differential form for G

A

G = H - TS

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21
Q

give the starting equation for ΔHmix

A

ΔHmix = Hs - Hmm

Hs = enthalpy of the solution
Hmm = enthalpy of the mechanical mixture

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22
Q

give the equation for the enthalpy of a solution of one mole of atoms

A

Hs = (NA*Z / 2) ( XA^2 EAA + XB^2 EBB + 2 XA XB EAB)

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23
Q

give the equation for the enthalpy of a mechanical mixture

A

Hmm = (NA*Z / 2) (XA EAA + XB EBB)

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24
Q

using the equations for Hmm and Hs give the final equation for ΔHmix

A

ΔHmix = Hs - Hmm = (NA*Z /2) XA XB (2EAB - EAA - EBB) = XA XB ψ

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25
Q

give the defining equation for entropy

A

S = k ln(w)

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26
Q

what are the equations for the number of distinguishable ways to arrange NA particles in a mechanical mixture and a solution

A

Ωmm = 1
(A atoms on A sites, B atoms on B)

Ωs = NA! / ((XANA)! ((1-XA)NA)!)

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27
Q

give the expression for ΔSmix and give the final equation for ΔSmix, roughly explain how to get from one to the other

A

ΔSmix = kln(Ωs) - kln(Ωmm) = kln(Ωs)

once the Stirling approximation has been used this gives

ΔSmix = -R(XA ln(XA) + XB ln(XB))

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28
Q

by combining the two expressions for ΔSmix and ΔHmix, give the overall equation for ΔGmix

A

ΔGmix = ΔHmix - TΔSmix

= XA XB ψ + RT(XA ln(XA) + XB ln(XB))

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29
Q

give Ficks second law, when is it used

A

∂C/∂t = D ∂^2C/ ∂x^2

where D = diffusivity
C = conc.

30
Q

give the equation for the driving force/change in free energy
for the nucleation of a new phase

A

ΔG = ΔS(Te - T)
where Te is the equil. temp

31
Q

give the expression for the work of nucleation (assuming the nucleus that forms is a sphere) and the work of nucleation if we include a strain term

A

Wn = (4/3)πr^3 ΔGv + 4π r^2 γ
excluding strain

Wn = (4/3)πr^3 (ΔGv + U) + 4π r^2 γ
where U is the strain energy per unit vol.

32
Q

from the equations for work done in nucleation what are the equations for critical nucleus radius and critical work done

A

r* = -2γ / ΔGv

Wn* = (16π/3) (γ^3 / (ΔGv)^2)

33
Q

when considering homogeneous nucleation, what are the two terms that we must consider, give the equations that each of these are given by and hence the equation for nucleation frequency, I

A

1) population of critical nuclei
prop. to exp(-(Wn*/ kT))

2) rate at which particles can add to make nuclei post-critical
prop. to exp(-(Q/kT))

overall
I = Cn exp( - ((Wn*+Q)/kT) )

34
Q

give the relationship between critical work done for heterogenous nucleation and critical work done for homogenous nucleation

A

Wnhetero = Wnhomo((2+cosθ)(1-cosθ)^2) /4)

35
Q

give the equation for the growth rate of a new phase

A
  • the growth rate of a new phase is given by

ν = Cg exp(-Q/KT)(1 - exp(VaΔGv / KT))

Cg = const.
Q = activation energy
K = Boltz. const.
T = temp
Va = vol of atom
ΔGv = driving force per unit vol.

36
Q

give the equation (relating to TTT diagrams) for the critical rate of cooling to NOT form the phase in question for the TTT diagram

A

critical rate of T = ΔT(nose)/t(nose)

t(nose) = time at nose of 0% curve

37
Q

give the equation for engineering stress

A

σ(eng) = F/Ao

Ao = init. cross-sectional area

38
Q

give the equation for engineering strain

A

ε(eng) = li/lo - 1
li = final length
lo = init. length

39
Q

give the equation for shear stress

A

τ = F/Ao

40
Q

give the equation for shear strain

A

γ = ΔYo/xo = tanφ

i.e. distance the side length has shifted

41
Q

give the equation/definition for poisson’s ratio

A

for normal stress on z
εx = εy = -v εz

v = poisson’s ratio

42
Q

give the equation linking E and G using poisson’s ratio

A

E = 2G(1+v)

43
Q

give the equation for how to approximate E using U (potential energy curve) and interatomic spacing, ro

A

E = (1/ro) (d^2U/dr^2)|r=ro

44
Q

using the block slab model, give the expression for axial modulus of a composite, Ec, in terms of the moduli of the matrix, Em and fibres, Ef

A

Ec = EfVf + Em(1-Vf)

45
Q

using the block slab model, give the expression for transverse modulus of a composite, Ec, in terms of the moduli of the matrix, Em and fibres, Ef

A

Ec = (EfEm) / (EmVf + Ef(1-Vf))

46
Q

give the equation for expansion due to temperature using the coefficient of thermal expansion, α

A

εt = α ΔT

47
Q

give the equation for how we define radius when considering bending of beams

A

R = l / θ

48
Q

give the equation for the deflection at any point on a rectangular beam and hence the max deflection

A

y = (Fx^2/6EI)(3L-x)

δ = FL^3 / 3EI

49
Q

give the expression for the Peierls-Nabarro stress

A

τ = 3G exp(-2πw/ b)

w = dislocation width
G = shear modulus
b = |b|

50
Q

give the expression for the glide ‘force’ (per unit length) of a dislocation

A

F = τb
b = burgers !vector!

51
Q

give two expressions for the work done in moving a dislocation

A

W = FLd
d = dislocation distance
F = glide force (per unit length)
L = length of line of dislocation

W = τbLd

52
Q

give the equation for the critical resolved shear stress

A

τ(crit) = σy cos(φ) cos(λ)

φ = angle of slip plane normal to tensile axis
λ = angle of slip direction to tensile axis

53
Q

what is the Schmid factor

A

cos(φ) cos(λ)

54
Q

what is the equation we should consider when thinking about how the tensile axis rotated towards the slip direction on maintained slip

A

TA = TA(orig.) + n [slip direction]

55
Q

when two dislocations combine, what is the new burgers vector

A

b3 = b1+b2

56
Q

what is franks rule, when is dislocation combination favourable

A

(b3)^2 < (b1)^2 + (b2)^2

57
Q

give the equation for dislocation density

A

ρ = 1/L^2

L = avr. spacing of dislocations on an area

58
Q

Give the Hall-Petch relationship regarding grain size and yield stress

A

σy = σo + k/sqrt(d)

σy = yield stress
σo = Peierls-Nabarro stress
k = const.
d = avr. grain diameter

59
Q

give the relationship for the increase in shear stress due to precipitate cutting

A

Δτ prop. to sqrt(r)

60
Q

Give the griffith criterion

A

G = π(σo^2)(c) / E

G >= Gc = 2γ

61
Q

Give the griffith criterion for ductile materials

A

G >= Gc = 2(γ + γp)

62
Q

give the equation for the stress intensity paramater

A

K = σo sqrt(pi*c)

63
Q

give the equation for viscous liquid flow

A

τ = -η dγ/dt

64
Q

Give the equation for the critical undercooling for ice formation on an Ice nucleating agents

A

ΔT(crit) = -2γ/ ΔSv R

65
Q

give the Clausius-Clapeyron eq.

A

dp/dt = ΔS/ΔV

66
Q

give the inequality which explains why surface melting generally occurs

A

γsv > γls + γlv

67
Q

give the equation for strain rate due to creep (generally)

A

dε/dt = Aσ^n

68
Q

give the equation for strain rate due to diffusion (coble) creep

A

dε/dt = (B’σ / d^2) exp(-Q/RT)

69
Q

give the equation for strain rate due to dislocation creep

A

dε/dt = (A’ σ^n) exp(-Q/RT)

70
Q

give the general equations for the efficiency of the carnot and brayton cycles for jet engines

A

Carnot
Efficiency = 1 - Tc/Th

Brayton
Efficiency = 1- T2/T1

71
Q

give the eq.’s for the max and average transferable energy in radiation damage and define the coeffecient

A

Max. transferable energy = ζ En
Av. transferable energy = ζ En /2

ζ = 4A / (1+A)^2
A = atomic mass number

Materials IA (9 decks)

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