EN224: Linear Elasticity


Homework 5: Complex variable methods for plane problems
Due Friday April 22, 2005



Division of Engineering

Brown University




1. Let  be two complex potentials that generate stresses and displacements according to the usual formulation (no continuation)

Show that the displacement and stress components in the  basis shown in the figure can be calculated as













2. Complex variable solution to a pressurized cylinder.


2.1 Using the results of the preceding problem, show that the complex potentials that generate stress and displacement fields in a pressurized cylinder (see above) must satisfy



2.2 By expanding  and  as Laurent series, find the potentials that solve this problem.  To simplify the algebra, note that , and assume that the solution can be generated from terms in the series that after substitution in the boundary conditions, are independent of z.



















3. Dislocation near a rigid interface




3.1 Begin by finding an analytic continuation that automatically satisfies D=0 on . To do this, start with the standard complex variable formulation

Express the boundary condition in terms of  and  defined in .  Next, express the boundary condition in terms of potentials  which are analytic in .   Use the result to show that

where L denotes the real axis. Hence, conclude that this implies that

is analytic in .  Use this to calculate an expression for , and hence show that a solution with D=0 on  can be generated by finding a single potential  that is analytic in , and calculating displacements and stresses from



3.2  Let  generate the solution for a dislocation at position  in an infinite solid.  Deduce that, to satisfy D=0 on L, we must superpose a second potential  satisfying

and calculating stresses and displacements from this potential using the formulation in 3.1.


3.3 Using (3.1) as a guide, write down the potential  in terms of  and




4. Stress induced by indentation with a rigid wedge


Suppose that an elastic half-space is indented by a rigid frictionless wedge, with profile .  Calculate the potential  that generates the stress field in the solid in terms of  and . Hence, determine the contact pressure distribution and the slope of the surface for .  Use the conditions that the contact pressure cannot be tensile, and the two solids cannot overlap to deduce the relationship between contact width a and the force applied to the punch.















5. An alternative solution for the pressurized crack



There is often more than one choice of analytic continuation for a particular boundary value problem.  To illustrate this, in this problem we will devise an alternative procedure to solve the pressurized crack problem that was discussed in class.


Consider a crack that is subjected to equal and opposite tractions  on its faces. Symmetry conditions imply that  on  outside the crack.  Moreover, it is evidently sufficient to find a solution in the upper half-plane, since the solution in the lower half-plane follows by symmetry.


5.1 Starting with the standard complex variable formulation

show that setting  will automatically satisfy  on .


5.2 With this choice of , show that the condition that  on  implies that

showing that

is continuous outside the crack, and analytic in the whole plane.  Deduce that .


5.3 Hence show that traction boundary condition on the crack faces leads to a Hilbert problem for



5.4 Write down the general solution to the Hilbert problem.



5.5 Hence, find an expression for the stress intensity factors induced at the right hand crack tip by a pair of equal and opposite point forces acting on the crack faces.