Experimentally, it has been observed for \(\sin -\) gle crystals of a number of metals that the critical resolved shear stress \(\tau_{\mathrm{crss}}\) is a function of the dislocation density \(\rho_{D}\) as \\[\tau_{\mathrm{crss}}=\tau_{0}+A \sqrt{\rho_{D}}\\] where \(\tau_{0}\) and \(A\) are constants. For copper, the critical resolved shear stress is \(0.69 \mathrm{MPa}\) \((100 \mathrm{psi})\) at a dislocation density of \(10^{4} \mathrm{mm}^{-2}\) If it is known that the value of \(\tau_{0}\) for copper is 0.069 MPa \((10 \mathrm{psi}),\) compute the \(\tau_{\mathrm{crss}}\) at a dislocation density of \(10^{6} \mathrm{mm}^{-2}\)

Answer: To find the new critical resolved shear stress (\(\tau_{\mathrm{crss_{new}}}\)), follow these steps: 1. Calculate the constant \(A\) using the given values of \(\tau_{\mathrm{crss}}\), \(\rho_{D}\), and \(\tau_{0}\): \\[0.69 = 0.069 + A \sqrt{10^{4}}\\] 2. Use the equation with the calculated value of \(A\) and the new dislocation density (\(10^{6} \mathrm{mm}^{-2}\)) to compute the new \(\tau_{\mathrm{crss_{new}}}\) value: \\[\tau_{\mathrm{crss_{new}}} = \tau_{0} + A \sqrt{10^6}\\] Remember to substitute the given values of \(\tau_{0}\) and the calculated \(A\) into this equation to find the new \(\tau_{\mathrm{crss}}\) value at a dislocation density of \(10^{6} \mathrm{mm}^{-2}\).