Grain Boundaries, Mechanical Properties of Materials
homework and all required material is attached
ENG45: Properties of Materials – Winter 2017 Homework 3 – 58 pts Due Date: Wednesday, February 15th, 2017 at 5:00 pm SHARP to Homework Box in Kemper 2131 Note: You
must show all the steps you used to obtain your answer in order to receive full credit. Be sure to include your name and section number on your answer sheet. 1.
Grain Boundaries A material composed of only one crystal is called a single crystal, a material which consists of two crystals is called a bi-crystal, while a
material composed of many grains is referred to as being polycrystalline. In class we described a low angle, tilt grain boundary as being one which was composed of a
series of edge dislocations spaced along the plane of intersection between the two grains (crystals). In this exercise, you will investigate the relationship between
the magnitude of the tilt angle between the two grains, and the spacing between the edge dislocations along the grain boundary. This exercise will be carried out using
the PowerPoint file entitled “ENG45_Homework3_2017” saved in the Homework folder of the Files section of the Canvas site. In the PowerPoint file, you will find a slide
with pairs of grains which have been tilted by 2?, 4?, or 6? from one another. Each square represents a cubic unit cell and the axis of rotation is vertical, out of
the page. (a) For each pair of grains, move the grains so that they just overlap at the top of the drawing. You do NOT need to perform any additional rotation of the
grains. Print a copy of your slide for this step. (b) A darker grey region should form in your image where the two crystal overlap. For each pair of grains, remove
unit cells from the left grain to eliminate the regions of overlap between the two crystals. It is Ok to have some white regions between the two grains as the atoms
adjacent to the white area will shift their atomic positions to better accommodate the void. In general there will be much more open volume along the grain boundary
than in the unaffected region of the grain. Print a copy of your slide for this step. (c) For each grain boundary, indicate the location of the edge dislocations with
a ? symbol. (d) For each grain boundary, determine the spacing between edge dislocations, D, in terms of the lattice parameter, a. Print a copy of your slide for steps
(c) and (d). (e) Circle the correct answer that completes the sentence below. The spacing between edge dislocations, D, _______________ the tilt angle of the grain
boundary. (i) is independent of (ii) is inversely proportional to (iii) has a linear relationship with (iv) has a quadratic relationship with (f) Circle the correct
answer that completes the sentence below. A tilt grain boundary is one where the axis of rotation of the grains is _____________ the plane of the grain boundary. (i)
parallel to (ii) perpendicular to (iii) is canted out of the plane by an angle ? 90? from (g) On your drawing of the 2? tilt boundary, draw a Burger’s circuit around a
perfect region of the crystal AND around an edge dislocation. Clearly mark the start and end positions of your Burger’s circuit AND identify the Burger’s vector, B as
the vector pointing from the end to the start of the Burger’s circuit. Print a copy of your slide for this step. Note: Twist grain boundaries are related to screw
dislocations in a similar manner, but the pictures are much more difficult to draw.
ENG 45 © Prof. Y. Takamura, 2017
2. Mechanical Properties of Materials The mechanical properties of materials used in musical instruments have a direct impact on their acoustical properties. In this
problem, we will investigate the mechanical properties of piano wires. Piano wires experience continuous high tensile loads, but also must survive additional
stretching and slackening during tuning, and repeated blows while during playing. Piano wires consist of cylindrical wires with the wire diameter and length increasing
as the pitch decreases. The pitch is also affected by the density of the wire, and the applied tension load. The lower pitch wires are formed of an inner core wire,
with a Cu wire coiled around it to increase its mass without causing the wire to be too stiff. Table I: Mechanical properties for candidate metal alloys.
Material Alloy composition
Young’s Modulus (GPa)
Yield Strength (MPa)
Tensile Strength (MPa)
Poisson’s Ratio
Density (g/cm3)
High carbon steel
Fe with C: 0.70-1.00 wt% and Mn: 0.20-0.60 wt% 207 700 800 0.30 7.85 Monel 400 Ni: 67 wt%, Cu: 31.5 wt%; balance: C, Mn, Fe, S, Si 173 283 579 0.32 8.80
Ductile iron
Fe with C: 3.5-3.9 wt%; Si: 2.25-3.00 wt%, Mn: 0.15- 0.35 wt%, trace S and P
169 329 461 0.35 7.10
Ti-6Al-4V Ti with Al: 5 wt%, V: 4 wt%, trace Fe and O 110 825 862 0.33 4.48
(a) Calculate the stress on a piano wire assuming a tensile load of 640 N (i.e. the load of being strung in a piano), and an initial diameter, d0 = 1.2 mm. (b) For
each of the four candidate materials listed in Table I, state whether the load on the piano wire corresponds to the elastic regime, the plastic regime before necking,
or the plastic regime after necking. Explain your answer in each case in one sentence. (c) For the material(s) which remain in the elastic regime, calculate the
strain on the piano wire under the load of 640 N. Hint: You should have at least one material in the elastic regime if your calculations are correct. (d) Assume that
the length of the piano wire was 51.35 cm before being strung (l0), calculate its final length (lf) AND final diameter (df) under the load of 640 N for the material(s)
which remain in the elastic regime. Show your answer in a sufficient number of significant digits to see the small difference in dimensions. (e) Calculate the mass
of the piano wire before being strung for the material(s) which remain in the elastic regime. (f) The dimensions given correspond to the case that the piano wire is
composed of high carbon steel. In the early development of the piano, the materials used for the piano wires (e.g. ductile iron) and the piano frame were not so
sophisticated, requiring much lower loads on the wire. Assuming the same dimensions of wires, what is the maximum load that a ductile iron could sustain to just remain
in the elastic regime? Would you recommend operating the piano under these conditions (i.e. tension and wire diameter)? Why or why not? (g) The frequency of the piano
wire (Hz) can be expressed as: 1 10 where l = length of wire in cm, d = diameter of wire in cm, T = tensile load on wire in Newtons, and ? = density of wire in g/cm3.
Calculate the frequency of the high carbon steel wire and ductile iron wire from (f) assuming they have the same dimensions noted in (a) and (d).
Source: Coltharp Piano Wire
