CHEM 120-PART II
STRUCTURE AND PROPERTIES OF MATERIALS
SAMPLE PROBLEMS
M1.
A cylindrical pure titanium wire 3.0 mm diameter and 2.5 x 104 mm long is
subjected to a load of 500 N. Calculate the resulting elongation. Assume elastic
deformation.
M2.
In normal motion, the load exerted on the hip joint is 2.5 times body weight.
a) Calculate the stress on an artificial hip implant with a cross-sectional area of
5.64 cm2 in a patient weighing 68 kg.
b) Calculate the corresponding elastic strain if the implant is made of Ti-6Al-4V
alloy.
M3.
The dimensions of a bicycle cross tube are 25.6 mm diameter, 0.5 mm wall
thickness and 900 mm long.
a) Calculate the mass of a tube of 6061 Al alloy.
b) Using the buckling equation, calculate the maximum compressive load which
the tube will support.
M4.
A cross tube is made of pure Ti, having the same diameter, length and maximum
allowable load as in Ques. M3.
a) Calculate the minimum wall thickness of the Ti tube.
b) Calculate the mass of this tube.
M5.
The interatomic potential energy (U) as a function of spacing (r) for a pair of
atoms is given by
U = -A/r +B/rn
a) Calculate the equilibrium bond length (ro) in terms of the constants A,B and n.
b) Calculate the equilibrium bond energy (Uo).
M6.
Using the interatomic bond energy function given in Ques. M5,
with A = 0.92 x 10-27 [J-m], B = 0.40 x 10-105 [J-m9] and n = 9,
a) Calculate the interatomic spacing at which the interatomic force is maximum.
b) Calculate the maximum value of the interatomic force (FMAX).
M7.
a) Make a drawing of the unit cell of a face-centered cubic (F.C.C) crystal
showing the positions of all the atoms.
b) Draw the atomic arrangement on the plane perpendicular to the close-packed
direction in a F.C.C. crystal.
M8.
Copper has the F.C.C. crystal structure with atom radius (R) = 0.128 nm.
a) Calculate the specific volume (i.e. m3/atom) of Cu.
b) Calculate the density of Cu.
M9.
Using the interatomic potential energy function given in Ques. M5,
a) Show that
A = (nUoro)/(n-1) and B = (Uoron)/(n-1)
b) If the atoms form a simple cubic structure, show that the elastic modulus (E)
for stretching in a direction along the edge of the unit cell is given by
E = (nUo)/(ro3)
M10. In the absence of stress, the centre-to-centre atomic separation distance of 2 Fe
atoms along the body-diagonal direction of the unit cell is 0.2480 nm. Under a
tensile stress of 1000 MPa along this direction, the atomic separation distance
increases to 0.2489 nm. Calculate the modulus of elasticity in this direction.
M11. The equilibrium structure of Fe is F.C.C. above 912 C and B.C.C. below 912 C.
a) Calculate the volume per atom for the F.C.C. and B.C.C. unit cells.
b) Calculate the percent volume change which accompanies the
F.C.C.  B.C.C. phase change at 912 C.
(Atomic radii @ 912 C: RFCC = 0.1292 nm, RBCC = 0.1258 nm
Hint: carry 4 significant digits throughout your calculation.)
M12. Magnesium has the H.C.P. crystal structure and a density of 1.74 Mg-m-3.
a) Make a drawing of the unit cell of Mg showing the positions of all the atoms.
b) Calculate the atom radius of Mg. (Recall: c/a = 1.633 for H.C.P)
M13. A silicon wafer is doped with 5 x 1021 aluminum atoms per m3 in solid solution.
a) Calculate the atom percent of Al in the doped Si wafer.
b) Calculate the weight percent of Al.
M14. Using the drawing in the notes showing the interstitial sites for C atoms in
B.C.C. Fe,
a) Calculate the maximum number of interstitial C atoms per unit cell.
b) Calculate the maximum concentration (wt. pct.) of C in B.C.C. Fe.
M15. a) For the binary alloys i) Cu –Ni, ii) Al – Cu, iii) Al – Mg, calculate the percent
difference in atomic radii.
b)Cu – Ni exhibits complete solid solubility, whereas Al – Cu and Al- Mg
have limited mutual solubility. Explain this behaviour in terms of the HumeRothery Rules.
M16. An alloy of iron – 12 wt. pct. chromium (a stainless steel) is a substitutional solid
solution having the B.C.C. crystal structure. Calculate the following: i) the Cr
concentration in the alloy in atom pct., ii) the lattice parameter of the alloy, iii) the
density of the alloy.
M17. Pernod is a transparent yellow fluid consisting of water, alcohol and esters. The
esters dissolve in concentrated water-alcohol solutions, but precipitate out as
whitish droplets if the solution is diluted with more water.
An experiment performed in a sidewalk café in Paris produced the following
values of the state variables (T, wt.% H2O) at which Pernod turned cloudy:
T (C):
0
20
40
% H20 (wt.%):
60
70
85
a) Sketch an appropriate phase diagram for the Pernod – H20 system.
b) Label the single-phase and 2-phase regions on your phase diagram.
M18. Below 100C, Al – 4.5 wt.% Cu alloy is a 2 – phase mixture of pure Al and
CuAl2 ().
a) What is the wt.% Cu in ?
b) What is the wt.%  in Al – 4.5 wt.% Cu at room temperature (R.T.)?
M19. Suppose that you have a crucible containing 1 kg of an alloy of composition
90 wt.% Sn – 10 wt.% Pb at a temperature of 184C. How much Sn would you
have to add to the crucible to completely solidify the alloy without changing the
system temperature? (Use the Sn – Pb phase diagram in your notes.)
M20. A Cu – 60 wt.% Ni alloy is slowly cooled from 1500C. Using the Cu – Ni phase
diagram in your notes, determine
a) the temperature at which the 1st solid phase forms and the composition of this
solid,
b) the temperature at which the last solid forms, the composition of this solid and
the composition of the last drop of liquid at this point.
M21. A Pb – Sn alloy containing 80 at.% Pb is cooled slowly from 325 C to 125 C.
Using the Pb – Sn phase diagram in your notes, describe the sequence of phases
which are present as the alloy is cooled.
M22. Consider Canada as an alloy with 2 phases, English Canada ( phase) and Quebec
( phase). The  phase contains 19 M A atoms (Anglophones) and 1 M B atoms
(Francophones). The  phase contains 1 M A atoms and 4 M B atoms. The A
atom is 10 % heavier than the B atom.
a) What is the overall composition (wt.%) of the Canada alloy?
b) What is the composition of the  phase ?
c) What is the weight fraction of the  phase in the alloy?
M23. A batch of solder is made by melting together 64 g of a Pb – 60 wt.% Sn alloy
with 53 g of a Pb – 40 wt.% Sn alloy.
a) Calculate the composition (wt.% Sn) of the resulting solder alloy.
b) Calculate the weights of  and  phases in the final solder alloy, assuming it
is slowly cooled to R.T. and there is no further change in the phase mixture
below 100C. (Use the Pb- Sn phase diagram in your notes.)
M24. Automotive engine blocks are made with an Al – 5 wt.% Si alloy by melting and
casting. Using the attached Al – Si phase diagram,
a) What is the melting temperature of the Al – 5 wt.% Si alloy? (i.e., the
temperature at which the first liquid forms upon heating the solid alloy.)
b) What is the solidification temperature? (i.e., the temperature at which the first
solid forms upon cooling from the liquid phase.)
c) Determine the phases present at R.T., the Si concentration of each phase and
the amount of each phase. (Assume that the equilibrium phase mixtures
established at 500 C will not change during subsequent cooling to R.T.)
d) For a 1.00 kg ingot of this alloy, calculate the total mass of Si in each phase
present at R.T.
M25. The Fe – 12 Cr alloy in Ques. M16 is heated to 1500 C, then rapidly quenched to
room temperature.
a) Calculate the vacancy supersaturation immediately following the quench.
(supersaturation = actual concentration/equilibrium concentration.)
b) Eventually, the vacancy concentration at R.T. will return to equilibrium.
Explain with the aid of sketches how this occurs.
(vacancy formation energy for B.C.C. Fe = 1.6 eV.)
M26. A polcrystalline metal has a mean grain diameter of 50 m.
a) Calculate the no. of grains per m3.
b) A planar section of metal is cut, polished and viewed in an optical microscope
at a magnification of 500X. How many grains will be seen in a 10 cm x 10 cm
field of view?
c) If each grain boundary is 3 atom spacings wide and the atom spacing is
0.1 nm, calculate the total volume of grain boundary in 1 m3 of metal.
(Hint: For these calculations, approximate the tetrakaidecahedral grains by cubes.)
M27
A brittle ceramic material has a simple cubic crystal structure with nearest –
neighbour atom spacing = 0.3 nm and a bond-breaking force (FMAX) = 10-9 N.
Calculate the fracture stress of this material.
M28. A 10 mm diam. bar of steel is loaded in tension to 50,000N, at which point the
total strain is 0.1. The yield stress for this material is 600 MPa, and the elastic
modulus E = 200 GPa.
a) If the load is removed at this point, what is the permanent plastic strain?
b) If the sample is then reloaded to a load of 60,000N, at what stress will plastic
deformation recommence?
(Hint: Draw a  -  diagram for the complete test.)
M29. From the mechanical property data for the 5 materials, given in the table below,
a) Sketch the  -  curve for each material. (Assume materials which plastically
yield before fracture deform to a plastic strain of at least 0.1)
b) Rank each material according to i) stiffness, ii) strength, iii) toughness.
Material
E (GPa)
YIELD (MPa)
FRAC.(MPa)
SiO2 window glass
SiC grinding wheel
Al2O3 fibres
pipeline steel
Al bicycle tube
94
450
390
200
70
7200
10,000
5000
600
300
60
3000
1000
800
400
Data For Sample Problems
Material
Al
Cu
Ni
Pb
Fe
Cr
Ti
Mg
Sn
Si
Cr-Mo steel
6061 Al alloy
Ti-6Al-4V alloy
Crystal Structure
FCC
FCC
FCC
FCC
BCC
BCC
HCP
HCP
Tetrag. (>13.2°C)
Cubic
Atom Radius (nm)
0.143
0.128
0.125
0.175
0.124
0.125
0.147
0.160
0.158
0.117
E (GPa)
69
124
214
14
196
289
116
41
43
107
210
69
124
 (kg·m-3)
2.7 ×103
9.0 ×103
8.9 ×103
11.3 ×103
7.9 ×103
7.1 ×103
4.5 ×103
1.7 ×103
7.3 ×103
2.4 ×103
7.9 ×103
2.7 ×103
4.5 ×103