Science Applied 278
December Exam - 2006
THE UNIVERSITY OF BRITISH COLUMBIA
Department of Materials Engineering
APPLIED SCIENCE 278
Engineering Materials
FINAL EXAMINATION, December, 2006
This is a Closed Book Examination.
The use of calculators containing stored information of relevance to this course is forbidden.
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Time: 2.5 hours
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Answer 4 of 5 questions. Only 4 questions will be marked. Each question is worth 25 marks.
The complete exam is 10 pages in length.
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Some useful formulae, material data and phase diagrams are given on the last 5 pages of the
exam. The use of sketches or schematic diagrams is encouraged wherever these will aid in
solving or discussing a problem.
Marks
a) An aluminum alloy which is presently being used in a commercial airliner has a yield
stress of 450MPa, a tensile stress of 550 MPa, an elastic modulus of 70 GPa and a Poisson
ratio of 0.3. Given a member which has a circular cross-section with a diameter = 8.75 mm :
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Question #1
i) what is the maximum tensile load that may be applied without causing plastic
deformation ?
(2)
ii) what is the maximum tensile load the member can support ?
(3)
iii) if the original length of the component is 350 mm, what would the length of the
component be just prior to the onset of plastic deformation ?
(3)
iv) what would be the change in diameter of the component just prior to the onset of
plastic deformation ?
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(2)
b)
calculate the atomic packing factor for the FCC unit cell assuming the atoms to be hard
spheres.
(4)
c) Prove that the true stress is related to the engineering stress in a tensile test is given by:
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E
(3)
T = (1 + )
E
(3)
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Under what conditions can this equation not be used.
d) A brass alloy to be used for a spring application must have a modulus of resilience of at
least 1.4 MPa. What must be its minimum yield stress ? Given: E = 100 GPa, = 0.3.
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Applied Science 278
December Exam - 2006
(5)
e) the combination of slip plane and slip direction defines what is know as the slip system. For
face-centred cubic materials, the slip plane is the (111) which has 3 possible slip directions, i.e.
[1 10], [1 01] and [0 1 1]
i. in the x-y-z reference frame, sketch this slip plane and the associated slip
directions
ii. using a 2-dimensional sketch, show the hard sphere representation of this plane
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Question #2
(5)
a) A cylindrical aluminum alloy 2024-T6 bar with a very gentle radius of curvature (for the
transition from the grip to the gauge section) is subjected to repeated compression-tension
stress cycling along its axis. If the load amplitude is 18 000 N, compute the minimum
allowable diameter to ensure that fatigue failure will not occur. Assume a factor of safety
of 2.0
b) If the sample described in part a) had been machined with the geometry shown below,
predict the number of cycles to failure (in this case, assume no factor of safety) , Given:
applied load = 7.5 kN, thickness = 5 mm, w = 20 mm, h = 10, r = 1 mm
Figure 1 stress concentration factor for geometry shown.
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(3)
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Applied Science 278
December Exam - 2006
(4)
c) The aluminum alloy, AA5754, is used for structural applications in automobiles. This alloy
has been processed in different ways to give the following properties:
Treatment
a
b
Grain size
20 m
50 m
Yield stress
130 MPa
90 MPa
Estimate the yield stress if the alloy could be processed to have a grain size of 10 m.
d) A single piece of annealed brass strip is cold rolled in 3 stages from 1.0 mm to 0.8 mm,
from 0.8 mm to 0.6 mm and finally from 0.6 mm to 0.4 mm.
i. Sketch the appearance of the grains at the end of each of the three rolling stages.
ii. Use the graph on below to determine the yield strength after each of the rolling
stages.
iii. Briefly discuss why the strength increases with increased cold working.
iv. If the sample was then annealed at 600 oC for 1 hour, discuss the change that you
would expect to observe in the microstructure (the grains and the dislocation
density) and in the mechanical properties (yield and tensile strength and ductility).
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(10)
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Figure 2 mechanical properties for 70/30 brass as a function of cold work
e) Determine the safe operating stress for a component fabricated from a low carbon nickel
steel if it is to survive for 10 years at 649 oC without failure. If the temperature was decreased
to 538 oC, what would be the new level of allowable stress.
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(3)
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Applied Science 278
December Exam - 2006
Question 3.
(8)
a) A lead-tin alloy of composition 30 wt%Sn-70 wt%Pb is slowly heated from a temperature
of 150C (300F).
(a) At what temperature does the first liquid phase form?
(b) What is the composition of this liquid phase?
(c) At what temperature does complete melting of the alloy occur?
(d) What is the composition of the last solid remaining prior to complete melting?
(e) At 150C, determine, which phases are present, their composition and the
weight fraction of the phases?
b)
compare the heat treatment of a precipitation hardening alloy and a quenched and
tempered steel. For each step in the process, i) qualitatively discuss the strength and
ductility of the material and ii) sketch the microstructure of the materials.
(6)
c) Consider the slow cooling of Fe-C alloy of carbon concentration of 1.5 wt% C from a
temperature of 1100 oC
i) sketch the expected microstructures at 700, 800 and 1100 oC
(3)
d)
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(8)
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list 3 methods for making dislocation motion more difficult and thereby increasing the
yield stress of a metal.
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i) a rapid hardening cement
ii) a low heat evolution cement.
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Question 4
(6)
a) Given the composition of Portland cement as follows, suggest and justify compositional
changes in the cement to produce:
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Table 1 characteristics of Portland cement
K
Constituent
heat evolution
(kJ/kg)
225
490
1200
-
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C2S
C3S
C3A
C4AF
weight fraction
(%)
50
20
12
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December Science Exam - 2006
Figure 3 - Kinetics for hardening of constituent phases in cement
b) what do the symbols C, H, S and A represent when discussing various cements and cement
reactions
(5)
c) describe the steps involved in the tempering of glass. Name two advantages of tempered glass
over regular glass
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(4)
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d) the average strength of Al2O3 tested in three-point bending is 400 MPa while testing in simple
tension gives a strength of 200 MPa
i) explain the difference in these strength values (two sentences, sketch may be useful).
(3)
ii) upon inspection after the test, it was determined that the critical flaw size in the tensile
sample was 0.13 mm. Calculate KIc (assume Y = 1 and the crack is located on the surface)
(3)
iii) based on your result in ii), predict the failure stress in simple tension if the critical flaw
size is increased to 0.5 mm.
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(4)
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December Exam - 2006
Question 4
(5)
a) sketch a typical stress-strain curve for an amorphous polymer such as polycarbonate
(Tradename, LEXAN). For each characteristic region of the stress-strain curve, sketch the
appropriate changes in the polymer structure which are occurring in the tensile sample.
(6)
b) Given the following molecular weight data, determine the weight average degree of
polymerization given that the mer unit is polymethyl methacrylate.
xi
wi
8 000 20 000
0.04
0.01
20 000 32 000
0.1
0.05
32 000- 44 000
0.16
0.12
44 000- 56 000
0.26
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Molecular Weight Range
56 000- 68 000
0.23
0.27
68 000- 80 000
0.15
80 000-92 000
0.06
M
(gmol-1)
0.21
0.09
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0.25
c) an alternating copolymer has a number average molecular weight of 250 000 g/mol and a
number average degree of polymerization of 3420. If one of the mers is styrene, identify
which of the following polymers is the other mer: ethylene, propylene, tetrafluorethylene and
vinyl chlorine. Given: mstyrene = 104.14 g/mol, methylene = 28.05 g/mol, mpolypropylene =
42.08g/mol, mtetrafluoroethylene = 100.02 g/mol, mvinyl chlorine = 62.49 g/mol.
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(3)
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Hint: the mer molecular weight in alternating co-polymer of ababab=0.5 x ma+0.5 x mb
d) describe the molecular differences between a thermoplastic, a thermoset and an elastomer.
(5)
e) what is meant by crystallinity in polymer materials (sketch may be useful). Identify 3 factors
which favour the formation of crystalline regions in polymers.
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(6)
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December Exam - 2006
Question 5
(5)
a) prove that the modulus perpendicular to the fibre axis for an aligned continuously reinforced
polymer is given by:
EmE f
Ec =
Em Vf + E f Vm
(10)
b) Consider a carbon fibre/polyester continuous fibre composite. For a volume fraction of
carbon fibres = 40 % with the fibres oriented along the loading axis:
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Quantify:
i) the modulus of elasticity
ii) the yield strain of the matrix,
iii) the fracture strain of the fibres
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Sketch the expected tensile curve for the composite material showing and identifying the 4
stages of deformation
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Useful data:
Carbon fibres:
modulus of elasticity = 400 GPa, fracture stress = 1000 MPa
Polyester matrix: modulus of elasticity = 7 GPa, yield stress = 20 MPa, fracture strain = 0.3
c) For a continuous and oriented fibre-reinforced composite, the moduli of elasticity in the
longitudinal and transverse direction are 24.5 and 5.2 GPa, respectively. Given the
volume fraction of fibres is 32 %, determine the moduli of elasticity for the fibres and the
matrix. (note: Ef >> Em)
(5)
d) you have been asked to design a hollow tube for a tension/compression application. The tube
must have the minimum weight possible. A carbon fibre composite is being considered. Using
the design of cell wall in wood as your inspiration, suggest a design for the composite tube.
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(5)
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Applied Science 278
December Exam - 2006
R = 8.314 Jmol-1K-1
= x = y
z
z
&
s = K 2 n exp
Q
RT
E = 2G(1 + )
1
Ur = y =
2
2E
a
t
m = 2 o
n
K T
nA / N A
Vc
rate = Ae Q
y =
E
2
f y ( )
K
2r
K = Y a
K
B 2.5 Ic
y
A
E
3F f L
2bd 2
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t
t = o exp
Er ( t ) =
( t)
o
Ewll = Es
s
1
Ew = Es
s
2
2
Ec = E f V f + E mVm
Ec =
Em E f
EmV f + E f Vm
2
&
s = K 1 n
fs =
Ef Vf
Ff
=
Em Vm
Fm
G
1
N
E
2
2
i.e. d
2
E
)
1
RT
t recrx = Ae Q RT
= o + kyd
2
K
+k +l
2
1
2
2 E ( s + p )
c =
a
R
(h
a
2
1
2 E s
c =
a
n = 2d sin
d hkl =
1
.C
a
m = o 1 + 2
t
T = ln (1 + )
=
r=d n
or r = n
R = cos cos
f
Mw
m
R
2
E
10
Mn
m
DPw = nw =
E
y +u
theoretical
T = (1 + )
T =
DPn = nn =
Qv
N v = N exp
RT
2
y
toughness
M w = wi M i
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Useful Formulae
R = min
max
M n = xi M i
Ff
E f / Em
=
Fc
E f / Em + Vm / Vf
lc =
*d
f
2 c
Ec=KEfVf + EmVm
c l = m(1 - Vf) +* Vf
*
'
f
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Applied Science 278
December Exam - 2006
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Figure A.1 Fe-C Phase diagram
Figure A.2 Lead Tin phase diagram
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Applied Science 278
December Exam - 2006
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Figure A.3 - S-N curve for several different engineering materials (mean stress = 0).
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Figure A.4 - Rupture life plot for a low carbon-nickel steel.
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December Exam - 2006
Table A1 Characteristics of selected elements
Density
(gcm-3)
2.71
2.25
8.94
19.32
8.90
4.51
19.3
7.13
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O
13
35
6
17
29
9
79
1
28
7
8
22
74
30
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Al
Br
C
Cl
Cu
F
Au
H
Ni
N
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Ti
W
Zn
Atomic
Weight
(gmol-1)
26.98
79.90
12.011
35.45
63.55
19.00
196.97
1.008
58.69
14.007
16.00
47.88
183.85
65.39
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Aluminum
Bromine
Carbon
Chlorine
Copper
Fluorine
Gold
Hydrogen
Nickel
Nitrogen
Oxygen
Titanium
Tungsten
Zinc
Atomic
Number
.C
Symbol
R
Element
11
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December Exam - 2006
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