## EN224: Linear Elasticity

##### Homework 4: Complex variable methods for anti-plane shear problems

##### Due Friday April 8, 2005

**Division of Engineering**

**Brown University**

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**1. **Consider the
analytic function ,
with a branch cut along the negative real axis. Find expressions for the anti-plane shear stress and displacement
fields generated by this potential, and hence identify the anti-plane shear
boundary value problem that this potential solves.

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**2. **Anti-plane
shear solution to rigid circular inclusion in an infinite solid**.**

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An infinite solid contains a rigid cylindrical inclusion
of radius *a* at the origin. The solid is subjected to remote anti-plane
shear loading:

By expanding the potential in a
Laurent series, find the complex potential *f(z)
* such that

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Hence, find the displacement and stress fields.

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**3. **Hypotrochoidal hole in an infinite solid**.** The mapping

maps the unit circle onto a
shape that resembles (vaguely) a square. Clearly, *a* just makes the
square larger or smaller. Plot the
image of |z|=1 under this mapping for a few values of and to figure out what they do (keep small or you will get something weird).

Suppose that an infinite solid
contains a hole whose boundary is the image of the unit circle under this
mapping. Assume that the solid is
subjected to remote anti-plane shear loading as in Problem 3. Find the state of
stress in the solid. (you can leave your answer in terms of ).

Find the greatest stress
concentration factor (you can use your intuition to figure out what orientation
of the hole will lead to the greatest stress concentration, you don’t have to
prove it).

Plot some stress and
displacement contours using your favorite symbolic manipulation program (Maple
has a useful `transform’ function in the plottools package that makes this very
simple)

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