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Chapter 8: Problem 4

A polystyrene component must not fail when a tensile stress of \(1.25 \mathrm{MPa}(180 \mathrm{psi})\) is applied. Determine the maximum allowable surface crack length if the surface energy of polystyrene is \(0.50 \mathrm{~J} / \mathrm{m}^{2}\left(2.86 \times 10^{-3}\right.\) in.-lb \(\left._{\mathrm{t}} / \mathrm{in} .^{2}\right)\). Assume a modulus of elasticity of \(3.0 \mathrm{GPa}\) \(\left(0.435 \times 10^{6} \mathrm{psi}\right)\)

Short Answer

Expert verified
Answer: The maximum allowable surface crack length in the polystyrene component under the given tensile stress is approximately 5.73 mm.

Step by step solution

01

Write down the given information

We are given the following information: - Tensile stress: \(\sigma = 1.25 \mathrm{MPa}\) - Surface energy: \(\gamma = 0.50 \mathrm{J/m^2}\) - Modulus of elasticity (Young's modulus): \(E = 3.0 \mathrm{GPa}\)
02

Apply the Griffith's theory equation for critical stress

The Griffith's theory for brittle materials with cracks relates the critical stress, crack length, modulus of elasticity, and surface energy as follows: \(\sigma_{c} = \sqrt{\frac{2 E \gamma}{\pi a}}\) where \(\sigma_{c}\) is the critical stress at which the material fails, \(E\) is the modulus of elasticity, \(\gamma\) is the surface energy, and \(a\) is the length of the crack.
03

Rearrange the equation to solve for the crack length 'a'

To determine the maximum allowable surface crack length, 'a', we will rearrange the Griffith's equation as follows: \(a = \frac{2 E \gamma}{\pi \sigma_{c}^2}\)
04

Substitute the given values into the equation

Now, we will substitute the given values for tensile stress, surface energy, and modulus of elasticity into the equation for 'a': \(a = \frac{2 \times 3.0 \times 10^9 \mathrm{\,Pa} \times 0.50 \mathrm{\,J/m^2}}{\pi \times (1.25 \times 10^6 \mathrm{\,Pa})^2}\)
05

Solve for the maximum allowable surface crack length 'a'

Performing the calculation, we find the maximum allowable surface crack length 'a': \(a \approx 5.73 \times 10^{-3} \mathrm{m}\) So, the maximum allowable surface crack length in the polystyrene component under the given tensile stress is approximately \(5.73 \mathrm{mm}\).

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Most popular questions from this chapter

Following is tabulated data that were gathered from a series of Charpy impact tests on a tempered 4140 steel alloy. $$ \begin{array}{rc} \hline \text { Temperature }\left({ }^{\circ} \boldsymbol{C}\right) & \text { Impact Energy }(\boldsymbol{J}) \\ \hline 100 & 89.3 \\ 75 & 88.6 \\ 50 & 87.6 \\ 25 & 85.4 \\ 0 & 82.9 \\ -25 & 78.9 \\ -50 & 73.1 \\ -65 & 66.0 \\ -75 & 59.3 \\ -85 & 47.9 \\ -100 & 34.3 \\ -125 & 29.3 \\ -150 & 27.1 \\ -175 & 25.0 \\ \hline \end{array} $$ (a) Plot the data as impact energy versus temperature. (b) Determine a ductile-to-brittle transition temperature as that temperature corresponding to the average of the maximum and minimum impact energies. (c) Determine a ductile-to-brittle transition temperature as that temperature at which the impact energy is \(70 \mathrm{~J}\).

List four measures that may be taken to increase the resistance to fatigue of a metal alloy.

Cite three metallurgical/processing techniques that are employed to enhance the creep resistance of metal alloys.

A specimen of a 4340 steel alloy having a plane strain fracture toughness of \(45 \mathrm{MPa} \sqrt{\mathrm{m}}\) (41 ksi \(\sqrt{\text { in. }})\) is exposed to a stress of 1000 MPa (145,000 psi). Will this specimen experience fracture if it is known that the largest surface crack is \(0.75 \mathrm{~mm}(0.03\) in.) long? Why or why not? Assume that the parameter \(Y\) has a value of \(1.0\).

Suppose that the fatigue data for the cast iron in Problem \(8.20\) were taken for bendingrotating tests, and that a rod of this alloy is to be used for an automobile axle that rotates at an average rotational velocity of 750 revolutions per minute. Give maximum lifetimes of continuous driving that are allowable for the following stress levels: (a) \(250 \mathrm{MPa}(36,250\) psi), (b) \(215 \mathrm{MPa}(31,000 \mathrm{psi})\), (c) \(200 \mathrm{MPa}\) \((29,000\) psi). and (d) \(150 \mathrm{MPa}(21,750 \mathrm{psi})\)
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