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MAT2210-MAT376

Spring 2020-2021 Exam

DEFORMATION AND FAILURE OF MATERIALS

SECTION A

1. Write down the formula for the shear modulus G of an ideal rubber, and explain briefly all the symbols used in the formula. [3 marks]

2. The shear modulus G of a polymeric elastomer is measured to be 2.0 x 106 Pa at 2 0°C. Using the formula for the shear modulus G of an ideal rubber, estimate the number of submolecules in the elastomer with a volume of 0.5 m3 .

[4 marks]

Note: Boltzmann’s constant k = 1.38 x 10-23 J K-1 .

3. Describe the free volume theory of glass transition. On the basis of the theory, explain the fact that, even though poly(vinyl chloride) (PVC) and polypropylene (PP) have similar chemical structures and the size of a chlorine atom is about the same as that of a CH3 methyl group, PVC has a much higher glass transition temperature than PP. [4 marks]

4. The strain-time curve of a viscoelastic polymer, with a constant stress applied at time zero and removed later, is shown schematically below. Identify the three different mechanisms that contribute to the induced strain, explain their origins, and identify which of them are fully recoverable. [4 marks]

5. The shift factor of a viscoelastic polymer can be calculated using the WLF equation

The relaxation modulus of a viscoelastic polymer can be measured with a lab experimental setup in the range between 10-2 to 102 hours. In order to predict the relaxation modulus of the polymer at glass transition temperature Tg , but at a longer time t = 106 hours, how much do we need to raise the experimental

temperature above Tg?

[5 marks]

SECTION B

Answer ALL Questions

6. A single-crystal sample of Cu-45 wt.% Zn is subjected to a tensile test at 500 ˚C. The sample is loaded parallel to the [1 4 5] tensile axis and its yield strength is found to be 300 MPa.

(a) Define the slip system family applicable to the Cu-45 wt. % alloy. [1 mark]

(b) Determine the slip system that will operate first during the tensile test at 500˚C. [2 marks]

(c) Define Schmid’s Law and calculate the critical resolved shear stress in MPa for the alloy at 500˚C. Show your working. [6 marks]

(d) What is the new tensile axis?

[1 mark]

(e) Outline the four key strengthening strategies that can be utilised to improve

the strength of polycrystalline metals. [2 marks]

Four alloys from the Cu-Zn system have been manufactured with the compositions shown and processed such that the grain size across all samples was consistent and equal to about ~ 20 µm, and the dislocation density in all alloys was equivalent. Tensile testing was performed on each alloy at 500˚C to obtain the alloy yield strengths. The data is presented in Table 1.

Table 1: Cu-Zn alloys and their associated yield strengths at 500˚C

Alloy Composition	Yield Strength at 500˚C (MPa)
Cu – 15 wt.% Zn	386
Cu – 30 wt.% Zn	427
Cu – 40 wt.% Zn	483
Cu – 45 wt.% Zn	300

(f) With reference to the phase diagram and the alloy compositions, indicate the

strengthening mechanisms in operation in each alloy and explain the marked

changes in strength observed. [6 marks]

(g) The alloys with compositions of Cu-15wt.% Zn and Cu-45 wt.% Zn were also

subjected to tensile testing at room temperature, the results of which are shown in Table 2. With reference to the crystal structures adopted by each alloy, explain the temperature dependence of the yield strengths. [2 marks]

Table 2: Tensile properties at 500˚C and room temperature of two Cu-Zn alloys.