On self-similarly expanding Eshelby inclusions: Spherical inclusion with dilatational eigenstrain

• For self-similarly expanding spherical inclusions the pde’s of elastodynamics become elliptic.
• The Eshelby property of constant stress is valid also for self-similarly expanding inclusions.
• The driving force (Eshelby force) on the boundary of the expanding inclusion is calculated.

For a subsonically self-similarly expanding spherical inclusion with dilatational transformation strain in a linear elastic solid, the governing system of partial differential equations is shown to be elliptic under scaling of uniform stretching of the variables, and the resulting elliptic equation is solved by satisfying the Hadamard jump conditions on the moving boundary. The solution has the Eshelby constant stress property for the interior domain, and can thus be used for the expanding inhomogeneity with transformation strain according to Eshelby (1957). The driving force on the moving boundary is also obtained.