The fracture strength of glass may be increased by etching away a thin surface layer. It is believed that the etching may alter surface crack geometry (i.e., reduce the crack length and increase the tip radius). Compute the ratio of the etched and origi- nal crack-tip radii for a fourfold increase in frac- ture strength if half of the crack length is removed.

The stress concentration factor K is given by the following formula: \[ K = 1 + 2 \left(\frac{a}{\rho}\right)^\frac{1}{2} \] where \( K \) = stress concentration factor \( a \) = crack length \( \rho \) = crack-tip radius

Let's define the given conditions: 1. For the original crack length and crack-tip radius: original fracture strength, \(a_o\) and \(\rho_o\) 2. For the etched crack (after removing half of the crack length): fourfold increase in fracture strength, \(\frac{1}{2}a_o\) and \(\rho_e\)

For original crack condition: \[ K_o = 1 + 2 \left(\frac{a_o}{\rho_o}\right)^\frac{1}{2} \] For etched crack condition: \[ K_e = 1 + 2 \left(\frac{\frac{1}{2}a_o}{\rho_e}\right)^\frac{1}{2} \]

The problem states that the fracture strength of etched glass is 4 times the original. Fracture strength is inversely proportional to the stress concentration factor. Therefore, \[ K_o = 4 K_e \]

We now substitute the expressions of \(K_o\) and \(K_e\) found in steps 3: \[ 1 + 2 \left(\frac{a_o}{\rho_o}\right)^\frac{1}{2} = 4 \left[1 + 2 \left(\frac{\frac{1}{2}a_o}{\rho_e}\right)^\frac{1}{2}\right] \]

Now, we solve the equation for the desired ratio \(\frac{\rho_e}{\rho_o}\): \[ 1 + 2 \left(\frac{a_o}{\rho_o}\right)^\frac{1}{2} = 4 + 8 \left(\frac{a_o}{2 \rho_e}\right)^\frac{1}{2} \] \[ -3 = -2 \left(\frac{a_o}{\rho_o}\right)^\frac{1}{2} + 8 \left(\frac{a_o}{2 \rho_e}\right)^\frac{1}{2} \] To isolate the ratio \(\frac{\rho_e}{\rho_o}\), we divide the equation by \(-2 \left(\frac{a_o}{\rho_o}\right)^\frac{1}{2}\): \[ \frac{3}{2} = 1 - 4 \left(\frac{\rho_o}{\rho_e}\right)^\frac{1}{2} \] Solving the equation for \(\frac{\rho_e}{\rho_o}\): \[ \left(\frac{\rho_o}{\rho_e}\right)^\frac{1}{2} = \frac{1}{2} \] \[ \frac{\rho_o}{\rho_e} = \left(\frac{1}{2}\right)^2 \] \[ \frac{\rho_e}{\rho_o} = \left(\frac{2}{1}\right)^2 \] \[ \frac{\rho_e}{\rho_o} = 4 \] So, the ratio of the etched and original crack-tip radii is 4.