Chapter 6, Problem 6.82P

(a)

To determine

The expression for average pressure in plain-strain compression with Tresca friction.

Answer to Problem 6.82P

The expression for average pressure in plain-strain compression with Tresca friction is ${\sigma }_y={S^{\prime }}_y\left(1+\frac{ma}{2h}\right)$ .

Explanation of Solution

The figure (1) shows the force analysis in forging method,

  

Figure (1)

Using force balance in x-direction,

  $\left(\sigma +d{\sigma }_x\right)h+2mkdx-{\sigma }_{x}h=0$

Here, $m$ is the coefficient of Tresca friction, $h$ is the height of the sample, $k$ is the frictional stresses and ${\sigma }_x$ is the stress in x-direction.

On solving the above equation,

  $\begin{array}{l}d{\sigma }_x=-\frac{2mk}{h}dx\\ {\displaystyle \int d{\sigma }_x}=-{\displaystyle \int \frac{2mk}{h}dx}\\ {\sigma }_x=-\frac{2mk}{h}x+c\end{array}$ …… (1)

Applying the boundary conditions,

  $\begin{array}{l}x=a\\ {\sigma }_x=0\end{array}$

The value of $c$ can be calculated as,

  $\begin{array}{l}0=\frac{2mka}{h}+c\\ c=-\frac{2mka}{h}\end{array}$

The normal stress can be calculated by equation (1),

  $\begin{array}{l}{\sigma }_x=-\frac{2mk}{h}x+\frac{2mka}{h}\\ {\sigma }_x=\frac{2mk}{h}\left(a-x\right)\end{array}$

The condition for the die pressure is given as,

  ${\sigma }_y-{\sigma }_x={S^{\prime }}_y$

Here, ${\sigma }_y$ is the stress in y-direction and ${S^{\prime }}_y$ is the yield strength in plain strength.

Substitute the values of ${\sigma }_x$ in above equation,

  $\begin{array}{l}{\sigma }_y-\frac{2mk}{h}\left(a-x\right)={S^{\prime }}_y\\ {\sigma }_y-\left(\frac{{S^{\prime }}_y}{2}\right)\frac{2m}{h}\left(a-x\right)={S^{\prime }}_y\\ {\sigma }_y={S^{\prime }}_y\left(1+\frac{m}{h}\left(a-x\right)\right)\end{array}$

For $x=0$ ,

  ${\sigma }_y={S^{\prime }}_y\left(1+\frac{ma}{h}\right)$ …… (2)

For $x=a$ ,

  ${\sigma }_y={S^{\prime }}_y$ …… (3)

Adding equation (2) and (3),

  ${\sigma }_y={S^{\prime }}_y\left(1+\frac{ma}{2h}\right)$

Conclusion:

Therefore, the expression for average pressure in plain-strain compression with Tresca friction is ${\sigma }_y={S^{\prime }}_y\left(1+\frac{ma}{2h}\right)$ .

(b)

To determine

The expression for average pressure in plain-strain compression with sticking friction with $m=1$ .

Answer to Problem 6.82P

The expression for average pressure in plain-strain compression with Sticking friction is ${\sigma }_y={S^{\prime }}_y\left(1+\frac{ma}{2h}\right)$ .

Explanation of Solution

The figure (2) shows the force analysis in forging method,

  

Figure (2)

Using force balance in x-direction,

  $\left(\sigma +d{\sigma }_x\right)h+2mkdx-{\sigma }_{x}h=0$

Here, $m$ is the coefficient of Tresca friction, $h$ is the height of the sample, $k$ is the frictional stresses and ${\sigma }_x$ is the stress in x-direction.

On solving the above equation,

  $\begin{array}{l}d{\sigma }_x=-\frac{2mk}{h}dx\\ {\displaystyle \int d{\sigma }_x}=-{\displaystyle \int \frac{2mk}{h}dx}\\ {\sigma }_x=-\frac{2mk}{h}x+c\end{array}$ …… (1)

Applying the boundary conditions,

  $\begin{array}{l}x=a\\ {\sigma }_x=0\end{array}$

The value of $c$ can be calculated as,

  $\begin{array}{l}0=\frac{2mka}{h}+c\\ c=-\frac{2mka}{h}\end{array}$

The normal stress can be calculated by equation (1),

  $\begin{array}{l}{\sigma }_x=-\frac{2mk}{h}x+\frac{2mka}{h}\\ {\sigma }_x=\frac{2mk}{h}\left(a-x\right)\end{array}$

The condition for the die pressure is given as,

  ${\sigma }_y-{\sigma }_x={S^{\prime }}_y$

Here, ${\sigma }_y$ is the stress in y-direction and ${S^{\prime }}_y$ is the yield strength in plain strength.

Substitute the values of ${\sigma }_x$ in above equation,

  $\begin{array}{l}{\sigma }_y-\frac{2mk}{h}\left(a-x\right)={S^{\prime }}_y\\ {\sigma }_y-\left(\frac{{S^{\prime }}_y}{2}\right)\frac{2m}{h}\left(a-x\right)={S^{\prime }}_y\\ {\sigma }_y={S^{\prime }}_y\left(1+\frac{m}{h}\left(a-x\right)\right)\end{array}$

For $x=0$ ,

  ${\sigma }_y={S^{\prime }}_y\left(1+\frac{ma}{h}\right)$ …… (2)

For $x=a$ ,

  ${\sigma }_y={S^{\prime }}_y$ …… (3)

Adding equation (2) and (3),

  ${\sigma }_y={S^{\prime }}_y\left(1+\frac{ma}{2h}\right)$

For sticking friction $m=1$ .

  ${\sigma }_y={S^{\prime }}_y\left(1+\frac{a}{2h}\right)$

Conclusion:

Therefore, the expression for average pressure in plain-strain compression with Sticking friction is ${\sigma }_y={S^{\prime }}_y\left(1+\frac{a}{2h}\right)$ .