Effective Plane Strain Bulk Modulus           

We need to find the effective plane strain bulk modulus such that the axial strain is zero and there is same strain in two transverse directions. This state of strain can be applied as follows:

(7.219)

For this state of strain, the displacement field in cylindrical coordinates system on the outer boundary of the cylinders becomes

(7.220)

The displacement field at the outer boundary of the cylinders reflects a pure radial stress tractions, that is,   will give us the same boundary conditions of a homogeneous cylinder. If these tractions are transferred to the Cartesian coordinate system, then we get

(7.221)

(7.222)

and the strains can be given in terms of stresses as

(7.223)

However, one can give the strains from the displacement field in Equation (7.220).