A Continuum Perspective on Point Defects

We begin by examining what continuum mechanics might tell us about the structure and energetics of point defects. In this context, the point defect is seen as an elastic disturbance in the otherwise unperturbed elastic continuum. The properties of this disturbance can be rather easily evaluated by treating the medium within the setting of isotropic linear elasticity. Once we have determined the fields of the point defect we may in turn evaluate its energy and thereby the thermodynamic likelihood of its existence. 

We begin with the special case in which it is assumed that the displacements resulting from the presence of the point defect are spherically symmetric. We know that the fields must satisfy the Navier equations derived in section 2.4.2, namely 

Because of the presumed spherical symmetry of the solution, we expect the displacement field to have only a nonvanishing radial component, Ur which implies the need to consider only the equation in the radial degree of freedom which is given by 

In addition to satisfying the differential equation itself, there are boundary conditions that must be respected as well. Indeed, this is when the modeling begins for it is through the application of these boundary conditions that we are forced to posit the elastic implications of such a defect. 

In light of this solution, we may now compute the continuum estimate for the elastic energy stored in the displacement fields as a result of the presence of the 

---