Problem 3
A theoretical estimate of the strength of a crystalline solid is its binding energy per unit volume. Evaluate the strength of forsterite if its binding energy is \(10^{3} \mathrm{~kJ} \mathrm{~mole}^{-1}\) and its mean atomic volume is \(6.26 \times 10^{-6} \mathrm{~m}^{3} \mathrm{~mole}^{-1}\). The presence of grain boundaries and dislocations weakens a crystalline solid considerably below its theoretical strength.
Problem 8
The average kinetic energy \(\bar{\phi}\) of an atom in a crystalline solid is given by $$\bar{\phi}=\frac{1}{n} \int_{0}^{\infty} \phi d n_{\phi}$$ Verify that the Maxwell-Boltzmann distribution gives \(\bar{\phi}=\frac{3}{2} k T\) by carrying out the integration.
Problem 9
Consider the one-dimensional diffusion of radioactive tracer atoms initially absent from a crystalline solid but deposited uniformly at time \(t=0\) on the surface \(x=0\) of the semi-infinite solid. The number of radioactive atoms deposited at \(t=0\) is \(C\) per unit surface area. Show that the concentration of radioactive atoms \(n\) (number per unit volume) in the solid must satisfy the diffusion equation $$\frac{\partial n}{\partial t}=D \frac{\partial^{2} n}{\partial x^{2}} $$Equation \((7-79)\) can be obtained by first deriving the equation of conservation of tracer atoms $$\frac{\partial n}{\partial t}=-\frac{\partial J}{\partial x}$$ where we assume that tracer atoms diffuse in the \(x\) direction only. The actual decay of the tracer atoms has been ignored in formulating the mass balance. Solve Equation \((7-79)\) subject to the initial and boundary conditions $$\begin{array}{ll}n(x, t=0)=0 & (7-81) \\\\\int_{0}^{\infty} n(x, t) d x=C . & (7-82)\end{array}$$ Show that \(n(x, t)\) is given by $$n(x, t)=\frac{C}{(\pi D t)^{1 / 2}} \exp \left(\frac{-x^{2}}{4 D t}\right)$$
Problem 23
Consider the state of stress \(\sigma_{x x}=\sigma_{y y}=\) \(\sigma_{z z}=\sigma\) and \(\sigma_{x y}=\sigma_{y x}=\tau, \sigma_{x z}=\sigma_{z x}=\sigma_{y z}=\sigma_{z y}=0\) Determine the yield conditions on the basis of the Tresca and von Mises criteria. How does hydrostatic loading affect plastic yielding?
