Problems for Chapter 5
Analytical Techniques and Solutions for
Linear Elastic Solids
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 ( ) by ,
and the lower ( ) by ,
and re-connecting the surfaces. The
solid deforms in anti-plane shear, with a displacement field of the form
nonzero components of stress and strain in the solid
Show that the
total strain energy of the solid (per unit out of plane distance) can be
Show that the
potential energy can be re-written as
the integral is taken along the line .
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
that the integral is unbounded, as expected.
Calculate a bounded expression by truncating the integral at and
force exerted on one dislocation by the other by differentiating the expression
for the energy. Is the force attractive
Check your answer
using the Peach-Koehler formula.
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 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
5.9.5. 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
5.9.7. Suppose that the dislocation loop described in the
preceding problem is subjected to a uniaxial tensile stress . Calculate the total potential energy of the
system. Display your result as a graph
of normalized potential energy as a function of L/b, for various values of . Take 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 ;
and (ii) the critical size required for a pre-existing dislocation loop to
grow, as a function of .
5.9.8. Calculate the self-energy of a rectangular glide
dislocation loop with burgers vector and side lengths a,d. Use nonsingular
dislocation theory, and give your answer to zeroth order in the parameter .
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 and Poisson’s ratio .
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 perpendicular to the layers.
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
Suppose that the
layer contains a large number of dislocation loops with initial width . 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?