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<h2 class="hd hd-2 unit-title">Problem 2.1</h2>
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Problem 2.1A
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>A steel bar \(100 \;mm\) long and having a square cross section \(20 \; mm\) on an edge is pulled in tension with a load of \(89000 \;N\), and experiences an elogation of \(0.1 \; mm\). Assuming that the deformation is entirely elastic, calculate the elastic modulus (\(E\)) of the steel.</p>
<p>\(E\) (in \(GPa\)):</p>
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Problem 2.1B
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<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><p>The response of dislocations in uniaxial loading of single-crystal specimens is very sensitive to the crystallographic orientation of the specimen relative to the applied stress. The resolved shear stress is the component of the force acting in the direction of shear divided by the area of the slip plane. Under uniaxial tensile loading it can be expressed as: </p>
<p>\(\tau_{rss} = \sigma cos (\lambda) cos (\phi)\),</p>
<p>where \(\sigma\) is the applied uniaxial stress, \(\lambda\) is the angle between slip direction and specimen axis, and \(\phi\) is the angle between specimen axis and slip plane normal. </p>
<p>Consider a single crystal of silver oriented such that a tensile stress is applied along a \([001]\) direction. If slip occurs on a \((111)\) plane and in a \([\overline 1 0 1 ]\) direction, and is initiated at an applied tensile stress of \(1.1 \; MPa\) \((160 \; psi)\), compute the critical resolved shear stress.</p>
<p>\(\tau_{crss}\) (in \(MPa\)):</p>
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<h2 class="hd hd-2 unit-title">Problem 2.2</h2>
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<h3>Problem 2.2</h3>
<p>The dislocations described below are imperfections in a monatomic, primitive cubic crystal with one atom per unit cell. Answer the following questions for the two dislocations described below: What is the relationship between the Burgers vector and the dislocation line? Is the dislocation edge or screw in character? What is the glide plane of the dislocation? (Or , if the glide plane is not unique, what are the constraints on the glide plane?)</p>
<p>Note: This problem is adapted from Allen and Thomas, <em>The Structure of Materials</em></p>
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Problem 2.2A
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<p><b>Dislocation 1:</b> \(\mathbf{b_1} = a[010]\), \(\hat{t}_2 = [0 \overline 1 0]\) </p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 1" role="group"><div class="multi-inputs-group" role="group"><p style="display:inline">Dislocation 1 has the tangent vector </p>
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<option value="parallel"> parallel</option>
<option value="antiparallel"> antiparallel</option>
<option value="perpendicular"> perpendicular</option>
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</div><p style="display:inline">, thus it is a </p>
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<option value="edge"> edge</option>
<option value="right-handed screw"> right-handed screw</option>
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</div><p style="display:inline">dislocation. </p>
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<p> What is the slip plane of the dislocation? Please choose the best answer below: </p>
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<input type="radio" name="input_9a1652bbc5e04b47a5c34bf3e16d12d3_3_1" id="input_9a1652bbc5e04b47a5c34bf3e16d12d3_3_1_choice_1" class="field-input input-radio" value="choice_1"/><label id="9a1652bbc5e04b47a5c34bf3e16d12d3_3_1-choice_1-label" for="input_9a1652bbc5e04b47a5c34bf3e16d12d3_3_1_choice_1" class="response-label field-label label-inline" aria-describedby="status_9a1652bbc5e04b47a5c34bf3e16d12d3_3_1"> (101)
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<input type="radio" name="input_9a1652bbc5e04b47a5c34bf3e16d12d3_3_1" id="input_9a1652bbc5e04b47a5c34bf3e16d12d3_3_1_choice_2" class="field-input input-radio" value="choice_2"/><label id="9a1652bbc5e04b47a5c34bf3e16d12d3_3_1-choice_2-label" for="input_9a1652bbc5e04b47a5c34bf3e16d12d3_3_1_choice_2" class="response-label field-label label-inline" aria-describedby="status_9a1652bbc5e04b47a5c34bf3e16d12d3_3_1"> (010)
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<input type="radio" name="input_9a1652bbc5e04b47a5c34bf3e16d12d3_3_1" id="input_9a1652bbc5e04b47a5c34bf3e16d12d3_3_1_choice_3" class="field-input input-radio" value="choice_3"/><label id="9a1652bbc5e04b47a5c34bf3e16d12d3_3_1-choice_3-label" for="input_9a1652bbc5e04b47a5c34bf3e16d12d3_3_1_choice_3" class="response-label field-label label-inline" aria-describedby="status_9a1652bbc5e04b47a5c34bf3e16d12d3_3_1"> (001)
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<input type="radio" name="input_9a1652bbc5e04b47a5c34bf3e16d12d3_3_1" id="input_9a1652bbc5e04b47a5c34bf3e16d12d3_3_1_choice_4" class="field-input input-radio" value="choice_4"/><label id="9a1652bbc5e04b47a5c34bf3e16d12d3_3_1-choice_4-label" for="input_9a1652bbc5e04b47a5c34bf3e16d12d3_3_1_choice_4" class="response-label field-label label-inline" aria-describedby="status_9a1652bbc5e04b47a5c34bf3e16d12d3_3_1"> The dislocation can slip on any plane of the {010}
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Problem 2.2B
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<p><b>Dislocation 2:</b> \(\mathbf{b_2} = a[010]\), \(\hat{t}_2 = [0 0 \overline 1 ]\) </p>
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<option value="perpendicular"> perpendicular</option>
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</div><p style="display:inline">, thus it is a </p>
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<p> What is the glide plane of the dislocation? Please choose the best answer below: </p>
<div class="wrapper-problem-response" tabindex="-1" aria-label="Question 2" role="group"><div class="choicegroup capa_inputtype" id="inputtype_62cba59bfbdf42ef8c365c9ad5b2e33f_3_1">
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Problem 2.2C
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A slip system for the bcc crystal structure is \(\{110\}&lt;111&gt;\). A \(\{110\}\) type plane for the bcc structure is shown below. Drag and drop the arrows below to indicate two different \(&lt;111&gt;\) slip directions within this plane. (Your arrows should originate from the atom indicated in blue.)
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<h2 class="hd hd-2 unit-title">Problem 2.3</h2>
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Problem 2.3A
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<p>The Burgers vectors typically from in the closest packed directions.
BCC metals typically have dislocations of the type
\[
\vec{b} = \frac{a}{2} (1,1,1)
\]
Calculate the magnitude of the Burgers vector for iron in Angstroms The lattice constant for iron is \(2.87 &#197;\).
</p>
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Problem 2.3B
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Problem 2.3C
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