Chapter 4, Problem 4.40P

Interpretation Introduction

(a)

Interpretation:

The cation to anion ratio of zinc HCP metal needs to be calculated. The slip processes needs to be determined and the critical resolved shear stress needs to be estimated.

Concept Introduction:

Shear stress is defined as the component of stress in plane with the cross section of the material. It arises due to force vector component being parallel to the cross section of the material. The cation to anion ratio is simply ratio of radius of cation and anion present in the crystal. The value is different for different metal crystals. The component of the shear stress that resolved in the slip direction, resulting initiation of slip in a grain is known as crystal resolved shear stress.

Answer to Problem 4.40P

The c/a ratio is $1.856$.

The critical resolved shear stress is low in nature.

Explanation of Solution

From appendix A, for zinc metal the lattice parameter c is $4.9470\stackrel{\text{o}}{\text{A}}$ and $a\text{ is 2}\text{.6648 }\stackrel{\text{o}}{\text{A}}$.

The ratio can be calculated as follows:

  $\begin{array}{c}\frac{c}{a}=\frac{8.9470\stackrel{\text{o}}{\text{A}}}{2.6648\stackrel{\text{o}}{\text{A}}}\\ =1.856\end{array}$

The critical resolved shear stress is low in nature.

Interpretation Introduction

(b)

Interpretation:

The cation to anion ratio of magnesium HCP metal needs to be calculated. The slip processes needs to be determined and the critical resolved shear stress needs to be estimated.

Concept Introduction:

Shear stress is defined as the component of stress in plane with the cross section of the material. It arises due to force vector component being parallel to the cross section of the material. The cation to anion ratio is simply ratio of radius of cation and anion present in the crystal. The value is different for different metal crystals. The component of the shear stress that resolved in the slip direction, resulting initiation of slip in a grain is known as crystal resolved shear stress.

Answer to Problem 4.40P

The $c/a$ ratio is $1.62$.

The critically resolved shear stress is medium in nature.

Explanation of Solution

From appendix $A$, for magnesium the lattice parameter $c$ is $5.209\stackrel{\text{o}}{\text{A}}$ and $a\text{ is }3.2087\stackrel{\text{o}}{\text{A}}$.

The ratio can be calculated as follows:

  $\begin{array}{c}\frac{c}{a}=\frac{5.209\stackrel{\text{o}}{\text{A}}}{3.2087\stackrel{\text{o}}{\text{A}}}\\ =1.62\end{array}$

The critical resolved shear stress is medium in nature.

Interpretation Introduction

(c)

Interpretation:

The cation to anion ratio of titanium HCP metal needs to be calculated. The slip processes needs to be determined and the critical resolved shear stress needs to be estimated.

Concept Introduction:

Shear stress is defined as the component of stress in plane with the cross section of the material. It arises due to force vector component being parallel to the cross section of the material. The cation to anion ratio is simply ratio of radius of cation and anion present in the crystal. The value is different for different metal crystals. The component of the shear stress that resolved in the slip direction, resulting initiation of slip in a grain is known as crystal resolved shear stress.

Answer to Problem 4.40P

The $c/a$ ratio is $1.587$.

The critically resolved shear stress is high in nature.

Explanation of Solution

From appendix $A$, for titanium, the lattice parameter $c$ is $4.6831\stackrel{\text{o}}{\text{A}}$ and $a\text{ is }2.9503\stackrel{\text{o}}{\text{A}}$.

The ratio is

  $\begin{array}{c}\frac{c}{a}=\frac{4.6831\dot{A}}{2.9503\dot{A}}\\ =1.587\end{array}$

The critical resolved shear stress is high in nature.

Interpretation Introduction

(d)

Interpretation:

The cation to anion ratio of zirconium HCP metal needs to be calculated. The slip processes needs to be determined and the critical resolved shear stress needs to be estimated.

Concept Introduction:

Shear stress is defined as the component of stress in plane with the cross section of the material. It arises due to force vector component being parallel to the cross section of the material. The cation to anion ratio is simply ratio of radius of cation and anion present in the crystal. The value is different for different metal crystals. The component of the shear stress that resolved in the slip direction, resulting initiation of slip in a grain is known as crystal resolved shear stress.

Answer to Problem 4.40P

The $c/a$ ratio is $1.593$.

The critically resolved shear stress is high in nature.

Explanation of Solution

From appendix $A$, for zirconium, the lattice parameter $c$ is $5.1477\stackrel{\text{o}}{\text{A}}$ and $a\text{ is }3.2312\stackrel{\text{o}}{\text{A}}$.

The ratio can be calculated as follows:

  $\begin{array}{c}\frac{c}{a}=\frac{5.209\stackrel{\text{o}}{\text{A}}}{3.2087\stackrel{\text{o}}{\text{A}}}\\ =1.593\end{array}$

The critical resolved shear stress is high in nature.

Interpretation Introduction

(e)

Interpretation:

The cation to anion ratio of rhenium HCP metal needs to be calculated. The slip processes needs to be determined and the critical resolved shear stress needs to be estimated.

Concept Introduction:

Shear stress is defined as the component of stress in plane with the cross section of the material. It arises due to force vector component being parallel to the cross section of the material. The cation to anion ratio is simply ratio of radius of cation and anion present in the crystal. The value is different for different metal crystals. The component of the shear stress that resolved in the slip direction, resulting initiation of slip in a grain is known as crystal resolved shear stress.

Answer to Problem 4.40P

The $c/a$ ratio is $1.615$.

The critically resolved shear stress is medium in nature.

Explanation of Solution

From appendix $A$, for rhenium, the lattice parameter $c$ is $4.458\stackrel{\text{o}}{\text{A}}$ and $a\text{ is }2.760\stackrel{\text{o}}{\text{A}}$.

The ratio is calculated as follows:

  $\begin{array}{c}\frac{c}{a}=\frac{4.458\stackrel{\text{o}}{\text{A}}}{2.760\stackrel{\text{o}}{\text{A}}}\\ =1.615\end{array}$

The critical resolved shear stress is high in nature.

Interpretation Introduction

(f)

Interpretation:

The cation to anion ratio of beryllium HCP metal needs to be calculated. The slip processes needs to be determined and the critical resolved shear stress needs to be estimated.

Concept Introduction:

Shear stress is defined as the component of stress in plane with the cross section of the material. It arises due to force vector component being parallel to the cross section of the material. The cation to anion ratio is simply ratio of radius of cation and anion present in the crystal. The value is different for different metal crystals. The component of the shear stress that resolved in the slip direction, resulting initiation of slip in a grain is known as crystal resolved shear stress.

Answer to Problem 4.40P

The $c/a$ ratio is $1.568$.

The critically resolved shear stress is high in nature.

Explanation of Solution

From appendix $A$, for beryllium the lattice parameter $c$ is $3.5842\stackrel{\text{o}}{\text{A}}$ and $a\text{ is }2.2858\stackrel{\text{o}}{\text{A}}$.

The ratio is calculated as follows:

  $\begin{array}{c}\frac{c}{a}=\frac{3.5842\stackrel{\text{o}}{\text{A}}}{2.2858\stackrel{\text{o}}{\text{A}}}\\ =1.568\end{array}$

The critical resolved shear stress is high in nature.