Problem 1
Compute the binding energy of \(\mathrm{CsCl}\). Use \(\beta_{0}=\) \(5.95 \times 10^{-11} \mathrm{~Pa}^{-1}, \rho_{0}=3988 \mathrm{~kg} \mathrm{~m}^{-3},\) and \(A=\) 1.7627. The molecular weight of \(\mathrm{CsCl}\) is 168.36 , and thermodynamic data give \(-U_{0}=660 \mathrm{~kJ} \mathrm{~mol}^{-1}\)
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 \(10^{3} \mathrm{~kJ} \mathrm{~mol}^{-1}\) and its mean atomic volume is \(6.26 \times 10^{-6} \mathrm{~m}^{3} \mathrm{~mol}^{-1}\). The presence of grain boundaries and dislocations weakens a crystalline solid considerably below its theoretical strength.
Problem 4
According to the law of Dulong and Petit the specific heats of solids should differ only because of differences in \(M_{a}\). Calculate \(M_{a}\) and \(c\) for \(\mathrm{MgSiO}_{3}\) and \(\mathrm{MgO} .\) The measured values of \(c\) at standard conditions of temperature and pressure are 815 \(\mathrm{J} \mathrm{kg}^{-1} \mathrm{~K}^{-1}\) for \(\mathrm{MgSiO}_{3}\) and \(924 \mathrm{~J} \mathrm{~kg}^{-1} \mathrm{~K}^{-1}\) for \(\mathrm{MgO} .\) A MATLAB solution to this problem is provided in Appendix \(D\).
Problem 5
Obtain an order of magnitude estimate for the spring constant \(\bar{k}\) associated with the interatomic forces in a silicate crystal such as forsterite by assuming \(\bar{k} \sim E b\), where \(E\) is Young's modulus and \(b\) is the average interatomic spacing. Young's modulus for forsterite is \(1.5 \times 10^{11}\) Pa. Obtain a value for \(b\) by assuming \(b^{3}\) is the mean atomic volume. The density of forsterite is \(3200 \mathrm{~kg} \mathrm{~m}^{-3}\). Estimate the maximum amplitude of vibration of an atom in a forsterite crystal at a temperature of \(300 \mathrm{~K}\). How does it compare with the mean interatomic spacing? What is the Einstein frequency at this temperature? The spring constant may also be estimated from the compressibility of forsterite using \(\bar{k} \sim 3 b / \beta,\) where the factor of 3 arises from the relation between fractional volume changes and fractional changes in length. How does this estimate of \(\bar{k}\) compare with the previous one? The compressibility of forsterite is \(0.8 \times 10^{-11} \mathrm{~Pa}^{-1}\).
Problem 10
Show that the effective viscosity \(\mu_{\text {eff }}\) for the channel flow of a power-law fluid is given by $$ \mu_{\mathrm{eff}} \equiv \frac{\tau}{d u / d y}=\left(\frac{p_{1}-p_{0}}{L}\right) \frac{h^{2}}{4(n+2) \bar{u}}\left(\frac{2 y}{h}\right)^{1-n} $$ or $$ \frac{\mu_{\text {eff }}}{\mu_{\text {eff, wall }}}=\left(\frac{2 y}{h}\right)^{1-n} $$ where \(\mu_{\text {eff,wall }}\) is the value of \(\mu_{\text {eff }}\) at \(y=\pm h / 2\). Plot \(\mu_{\text {eff }} / \mu_{\text {eff, wall as a function of }} y / h\) for \(n=1,3\), and 5 .
Problem 19
Another model of viscoelastic behavior is the Kelvin model, in which the stress \(\sigma\) in the medium for a given strain \(\varepsilon\) and strain rate \(\dot{\varepsilon}\) is the superposition of linear elastic and linear viscous stresses, \(\sigma_{e}\) and \(\sigma_{f}\). Show that the rheological law for the Kelvin viscoelastic material is $$ \sigma=\varepsilon E+2 \mu \frac{d \varepsilon}{d t} $$ Show also that the response of the Kelvin viscoelastic material to the sudden application of a stress \(\sigma_{0}\) at time \(t=0\) is $$ \varepsilon=\frac{\sigma_{0}}{E}\left(1-e^{-t / \tau_{w}}\right) $$ Assume that \(\sigma=\sigma_{0}\) for \(t>0 .\) While stresses decay exponentially with time in a Maxwell material subjected to constant strain, strain relaxes in the same way in a Kelvin material subjected to constant stress.
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?
Problem 24
Consider a long circular cylinder of elastic-perfectly plastic material that is subjected to a torque \(T\) at its outer surface \(r=a\). The state of stress in the cylinder can be characterized by an azimuthal shear stress \(\tau\). Determine the torque for which an elastic core of radius \(c\) remains. Assume that the yield stress in shear is \(\sigma_{0}\). In the elastic region the shear stress is proportional to the distance from the axis of the cylinder \(r\). What is the torque for the onset of plastic yielding? What is the maximum torque that can be sustained by the cylinder?
