1. 1. The following data for zinc single crystal were obtained by Jillson:
A = 122 mm2
Establish that Schmid’s law is obeyed.
Demonstrate the orientation dependence of the yield stress by plotting σ0 versus cosΦcosλ.
90-Φ λ
P, kg
6.5 18
20.730
19.5 29
7.870 30
30.5
5.280 40
4.600 61
62.5
5.600 85
85.5
28.500

2. 2. Copper alloy is subjected to the stress state σx = 100, σy = -200, τxy = 100 (all in MPa). Determine whether yield occur according to the (a) Tresca and (b) von Mises criterion.

3. 3. A region on the surface of a 6061-T4 aluminum alloy component has strain gage attached, which indicate the following stresses:
11 = 70 MPa
22 = 120 MPa
12 = 60 MPa
Determine the yielding for both Tresca’s and von Mises’ criteria, given that σy = 150 MPa (the yield stress).

4. 4. If the yield strength of a steel is 950MPa, determine whether yielding will have occurred on the basis of both von Mises and Tresca criteria. The stress state is given by
Unit : MPa

5. 5. The following data were obtained for a carbon steel and an aluminum alloy.
(a) Show that the yield strengths of the steel and aluminum alloy obey the Hall-Petch relationship. Determine σ0 and ky for each material.
(b) Certain microalloyed steels contain small additions of vanadium or niobium that permit the grain size to be reduced to about 2 μm if the processing of the steel is carefully controlled. Likewise, advanced aluminum alloys containing special types of particles can be produced to yield a grain size of about 2 μm. Suppose we reduce the grain size of steel and aluminum from 150 μm to 2 μm by such processing. Would a substantial increase in the strengths of these materials result?
Carbon steel
d (μm)
σy (MN/m2)
406 93
106
129 75
145 43
158 30
189 16
233
Aluminum alloys
d (μm)
σy (MN/m2) 42
223 16
225 11
8.5
226
5.0
231
3.1
238

6. 6. A tensile specimen with a 12 mm initial diameter and 50 mm gage length reaches maximum load at 90 kN and fractures at 70 kN. The minimum diameter at fracture is 10 mm.
(a) Determine the engineering stress at maximum load (the ultimate tensile strength) and the true fracture stress.
(b) Determine the true strain at fracture.
(c) What is the engineering strain at fracture?