Introduction to Elasticity/Equilibrium example 2

From Wikiversity
Jump to navigation Jump to search

Example 2[edit | edit source]

Given: The displacement equation of equilibrium for an isotropic inhomogeneous linear elastic material can be written as

where

and and are the Lamé moduli.

Show:

Show that the displacement equation of equilibrium can be expressed as

Solution[edit | edit source]

The skew part of the tensor does not affect the stress because it leads to a rigid displacement field. Therefore, the displacement equation of equilibrium may be written as

where

In index notataion,

and

Therefore,

Now,

Hence,

Taking the divergence,

Recall that

Therefore,

Hence,

Therefore, the displacement equation of equilibrium can be expressed as required, i.e,