Elasticity_ Barber

134 9 Calculation of displacements
8. A rigid circular inclusion of radius a in a large elastic plate is subjected to
a force F in the x-direction.
Find the stress field in the plate if the inclusion is perfectly bonded to the
plate at r =a and the stresses tend to zero as r →∞.
9. A rubber bushing comprises a hollow cylinder of rubber a < r < b, bonded
to concentric thin-walled steel cylinders at r =a and r =b, as shown in Figure
9.3. The steel cylinders may be considered as rigid compared with the rubber.
Figure 9.3
The outer cylinder is held fixed, whilst a force F per unit axial length
is applied to the inner cylinder in the x-direction. Find the stiffness of the
bushing under this loading — i.e. the relationship between F and the resulting
displacement δ of the inner cylinder in the x-direction.
In particular, find the stiffness for a long cylinder (plane strain conditions)
and a short cylinder (plane stress), assuming that rubber has a shear modulus
µ and is incompressible (ν =0.5).
Plot a graph of the dimensionless stiffness F/µδ as a function of the ratio
a/b in the range F/µδ < 100. Can you provide a simple physical explanation
of the difference between the plane stress and plane strain results?
Hint: The hollow cylinder is a multiply connected body.