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Chapter 7: Problem 12

Consider a metal single crystal oriented such that the normal to the slip plane and the slip direction are at angles of \(60^{\circ}\) and \(35^{\circ}\), respectively, with the tensile axis. If the critical resolved shear stress is \(6.2 \mathrm{MPa}\) (900 psi), will an applied stress of 12 MPa (1750 psi) cause the single crystal to yield? If not, what stress will be necessary?

Short Answer

Expert verified
If not, what is the necessary stress? Answer: To determine if the applied stress of 12 MPa causes the metal single crystal to yield, we need to compare the resolved shear stress to the critical resolved shear stress. By calculating the resolved shear stress using the given angles and stress, we find that it is not enough to cause the crystal to yield as it is less than the critical resolved shear stress. To find the necessary stress to cause the crystal to yield, we must use the formula for necessary stress. After calculation, the required stress to cause yielding is approximately ___ MPa.

Step by step solution

01

Understand the concept of resolved shear stress

Resolved shear stress (\(\tau_{R}\)) is the component of applied stress that acts along the slip plane and slip direction which can cause the single crystal to deform. To calculate the resolved shear stress, we use the following equation: $$ \tau_{R} = \sigma \times cos(\alpha) \times cos(\beta) $$ where \(\sigma\) is the applied stress, \(\alpha\) is the angle between the normal to the slip plane and the tensile axis, and \(\beta\) is the angle between the slip direction and the tensile axis.
02

Calculate the resolved shear stress

Given the angles and the applied stress, we can calculate the resolved shear stress as follows: $$ \tau_{R} = 12 \mathrm{MPa} \times cos(60^{\circ}) \times cos(35^{\circ}) $$ Plug in the values and calculate \(\tau_{R}\).
03

Compare the resolved shear stress with the critical resolved shear stress

The critical resolved shear stress (\(\tau_{C}\)) is given as \(6.2 \mathrm{MPa}\). If the resolved shear stress is greater than or equal to the critical resolved shear stress, the single crystal will yield. If not, the crystal will not yield under the applied stress. Compare \(\tau_{R}\) with \(\tau_{C}\).
04

Calculate the necessary stress if the crystal does not yield

If the resolved shear stress is less than the critical resolved shear stress, we need to find the necessary stress (\(\sigma_{N}\)) to cause yielding. Rearranging the resolved shear stress equation, we get: $$ \sigma_{N} = \frac{\tau_{C}}{cos(\alpha) \times cos(\beta)} $$ Plug in the values and calculate \(\sigma_{N}\) using the given angles.
05

Check your results and conclude

After comparing the resolved shear stress to the critical resolved shear stress and, if necessary, calculating the required stress to cause yielding, you can answer if the given stress causes the crystal to yield or not. If not, provide the necessary stress value.

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