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MatS 2001
Introduction to Materials Science
Exam 1 – Solution
Summer 2010
1. TRUE-FALSE circle Correct Answer (-1 pt for incorrect; +1 pt for correct; 0 pt for no answer; 10 pts total) 2.
(a) T F Metals have elastic moduli in the range of 40-400 GPa.
(b) T F Typical yield stresses for metals are in the range of 20-1400 GPa. MPa
(c) T F In tensile tests, necking occurs at the yield point Tensile Strength.
(d) T F The most important factor in determining a material’s resistance to fracture is the number SIZE of cracks in it.
(e) T F Fatigue is a type of materials failure that happens when the material is exposed to cyclic tensile and compressive stresses.
(f) T F Tensile loads can cause crack propagation.
(g) T F Ceramics have higher LOWER K (^) IC values than metals.
(h) T F K (^) IC is a material parameter indicating a material’s resistance to fracture
(i) T F K (^) IC is small LARGE for metals because they have some ductility.
(j) T F A hole in a material can serve as a stress concentrator.
2. Fracture/Fatigue (35 points)
A metal part is used in an application that is susceptible to fatigue and for which failure would be unacceptable. A preventive maintenance program includes periodic inspections to detect cracks. The smallest edge crack that can be reliably detected is 0.25 mm.
Material properties are: σy = 505 MPa, TS = 680 MPa, K (^) IC = 24.8 MPa-m 1/2, %elongation = 11%
Application: σmax = 350 MPa (tension), σmin = -50 MPa (compression), Geometry factor Y = 1.27 (independent of crack length), crack growth parameters; A = 1.2 x 10 -12^ and m = 4.0 such that crack growth is in m/cycle.
(a) What crack length will cause failure by fracture? (5 pts)
2 2 1 1 24.8 - (^) 9.9 10 4 0. (1.27)(350 )
IC c
K MPa m a x m or mm π Y σ π MPa
= ^ = ^ = −
(b) What is the maximum stress that a new part (with no detectable cracks) can withstand without failure? Careful…check your answer. (7 pts)
For fracture:
-
24.8 MPa- m 696.8 MPa 1.27 0.25x
IC c
K
Y a (^) m
= = = but, TS = 680 MPa
so failure will occur at 680 MPa, or 505 MPa for plastic yield.
(c) What is the crack growth rate at the detection limit? (8 pts)
12 -3^4
8
1.2 10 1.27 350 MPa 0.25x10 m
m 2.89 10 cycle
da m A K x dN da x dN
−
= ∆ = ^
(d) What is the maximum number of cycles that can be allowed between inspections to insure no fatigue failure IF the critical crack length is 5 mm (NOT the answer to part (a)). Set up equations including values of constants, etc., then compute. (15 pts)
( 2 ) / 2 ( 2 ) / 2 / 2
2[( ) ( ) ]
m m c i m m m
a a N
m AY σ π
( 1) ( 1) 12 4 4 2
2[(0.005 m) (0.00025 m) ] 8219 Cycles ( 2)(1.2 10 )(1.27 )(350 MPa)
N
x π
− − −
b. ) Estimate the diameter of the rod at the onset of necking. (10 pts)
At onset of necking ( plastic deformation) ∆ V = 0 2 2 0 0 0 0 Neck 2 1/ 2^2 1/ 2 0 0
or Estimate necking at 1.23 m
(2 mm) 1 m 1.80 mm 1.23 m
A A d d
d d
×
or
0 0
0
ln 0.207 ln 2 ln 1
exp 2 mm exp 1.80 mm 2 2
i i T T
T i
A d A d
d d
ε ε
ε
^ ^ ^
−^ −
c. ) What is the strain hardening exponent (n) for this material? (5 pts)
At necking, n = ε T , TS
0
ln ln 0. TTS TS 1.
ε n
^
d. ) What is the strength coefficient K? (5 pts)
3 2
, at T.S. - 1.25 and 1.80 (from above)
491, 218 Pa or 0.491 MPa 1.80 10 2 0.491 MPa 0.681 MPa
n T T
T
T n T
K F N d F N A (^) m
K
−
×
e. ) What is the maximum mass that can be suspended from the rod without causing irreversible deformation? (5 pts)
Point where NO irreversible deformation occurs is below yield stress.
2
0.88 N at =1.007 m
; or 0.
F
F N
F mg m kg g m s
5. Creep (15 points)
The hook shown to the right will support a container of molten steel and must carry a load of 40,000 lb. The hook operates continuously at a temperature of 500 C. The relationship between stress and the Larson – Miller parameter is shown on the graph below:
where, for this material:
3
L-M T 38.316 1.796 log tday x 10
−
Temperature is in deg. Kelvin and rupture time is in days.
(a) If the useful life of the hook, with a safety factor of 2, is to be 10 years of continuous operation, what must be the diameter (in inches) of the indicated portion of the hook? (10 pts)
