 Problems for Chapter 5

Analytical Techniques and Solutions for Linear Elastic Solids

5.9.  Energetics of Dislocations

5.9.1.      Calculate the stress induced by a straight screw dislocation in an infinite solid using the formula in Section 5.8.4.  Compare the solution with the result of the calculation in Problem 5.3.4. 5.9.2.      The figure shows two nearby straight screw dislocations in an infinite solid, with line direction perpendicular to the plane of the figure.  The screw dislocations can be introduced into the solid by cutting the plane between the dislocations and displacing the upper of the surfaces created by the cut ( ${S}^{-}$ ) by $b{e}_{3}/2$, and the lower ( ${S}^{+}$ ) by $-b{e}_{3}/2$, and re-connecting the surfaces.   The solid deforms in anti-plane shear, with a displacement field of the form $u=u\left({x}_{1},{x}_{2}\right){e}_{3}$

5.9.2.1.            Write down nonzero components of stress and strain in the solid

5.9.2.2.            Show that the total strain energy of the solid (per unit out of plane distance) can be expressed as

$U=\frac{1}{2}\underset{A}{\int }{\sigma }_{3\alpha }\frac{\partial u}{\partial {x}_{\alpha }}dA$

5.9.2.3.            Show that the potential energy can be re-written as

$U=-\frac{1}{2}\underset{-d/2}{\overset{d/2}{\int }}{\sigma }_{32}\left({x}_{1}\right)bd{x}_{1}$

where the integral is taken along the line ${x}_{2}=0$.

5.9.2.4.            Use the solution for a screw dislocation given in Problem 5.3.4 (or 5.9.1) to show that the energy can be calculated as

$U=\frac{\mu {b}^{2}}{8\pi }\underset{-d/2}{\overset{d/2}{\int }}\left(\frac{1}{{x}_{1}+d/2}+\frac{1}{d/2-{x}_{1}}\right)d{x}_{1}$

Note that the integral is unbounded, as expected.  Calculate a bounded expression by truncating the integral at $d/2-\rho$ and $-d/2+\rho$

5.9.2.5.            Calculate the force exerted on one dislocation by the other by differentiating the expression for the energy.  Is the force attractive or repulsive?

5.9.2.6.            Check your answer using the Peach-Koehler formula.

5.9.3.      Calculate the stress induced by an edge dislocation in an infinite solid using the formula in Section 5.8.4.  Compare the solution with the result given in 5.3.4

5.9.4.      Calculate the nonsingular stress ${\sigma }_{ij}^{\left(\rho \right)}$ induced by a screw dislocation in an infinite solid using the formula in Section 5.9.2.  Compare the solution with the result of the calculation in Problem 5.3.4.

5.9.5.      Calculate the nonsingular self- energy per unit length of a straight dislocation, using the approach discussed in Section 5.9.2.  (To do this, you have to calculate the energy of a dislocation segment with finite length, then take the limit of the energy per unit length as the dislocation length goes to infinity). 5.9.6.      Calculate the self-energy of a square prismatic dislocation loop with side length L.  Use nonsingular dislocation theory, and give your answer to zeroth order in the parameter $\rho$

5.9.7.      Suppose that the dislocation loop described in the preceding problem is subjected to a uniaxial tensile stress ${\sigma }_{ij}={\sigma }_{0}{\delta }_{i3}{\delta }_{j3}$.  Calculate the total potential energy of the system.   Display your result as a graph of normalized potential energy ${V}^{D}\left(1-{\nu }^{2}\right)/EL{b}^{2}$ as a function of L/b, for various values of ${\sigma }_{0}\left(1-{\nu }^{2}\right)/E$.  Take $\rho =b/4$ as a representative value.  Hence, estimate (i) an expression for the activation energy required for homogeneous nucleation of a prismatic dislocation loop, as a function of ${\sigma }_{0}\left(1-{\nu }^{2}\right)/E$; and (ii) the critical size required for a pre-existing dislocation loop to grow, as a function of ${\sigma }_{0}\left(1-{\nu }^{2}\right)/E$. 5.9.8.      Calculate the self-energy of a rectangular glide dislocation loop with burgers vector $b=b{e}_{1}$ and side lengths a,d.  Use nonsingular dislocation theory, and give your answer to zeroth order in the parameter $\rho$. 5.9.9.      A composite material is made by sandwiching thin layers of a ductile metal between layers of a hard ceramic. Both the metal and the ceramic have identical Young’s modulus $E$ and Poisson’s ratio $\nu$. The figure shows one of the metal layers, which contains a glide dislocation loop on an inclined slip-plane.  The solid is subjected to a uniaxial tensile stress ${\sigma }_{0}$ perpendicular to the layers.

5.9.9.1.            Calculate the total energy of the dislocation loop, in terms of the applied stress and relevant geometric and material parameters.  Use non-singular dislocation theory to calculate the self-energy of the loop.

5.9.9.2.            Suppose that the layer contains a large number of dislocation loops with initial width ${d}_{0}$.  The layer starts to deform plastically if the stress is large enough to cause the loops to expand in the plane of the film (by increasing the loop dimension d).   Calculate the yield stress of the composite.   How does the yield stress scale with film thickness?