**EN224: Linear Elasticity **

**Division
of Engineering**

3.5 Eigenstrains

One can also use the Kelvin solution to derive fields due to eigenstrains within an infinite region.

Consider an unbounded, homogeneous linear elastic solid, which is free of stress. Suppose we introduce an inelastic strain distribution into the solid, by heating it, deforming it plastically, or inducing a phase transformation. What are the resulting displacement and stress fields?

This turns out to be a remarkably simple problem. Assume that the total strain field is made up of an elastic part and the prescribed, inelastic part. (Neither the elastic or plastic strains need be compatible, but the total strain is). Then

Stresses are induced by the elastic deformation only:

In the absence of body forces, the equilibrium equations are

We could solve this using Fourier transforms. Alternatively, we may write down an integral representation for the solution. Define a fictitious body force distribution

then

We know the general solution to this equation . It is generated by Papkovich-Neuber potentials