Compatibility and the genesis of residual stress by volumetric growth

The equations of compatibility which are pertinant for growth strain fields are collected and examples are given in simply-connected and multiply-connected regions. Compatibility conditions for infinitesimal strains are well known and the possibilities of Volterra dislocations in multiply-connected regions are enumerated. For finite growth strains in a multiply-connected regions, each case must be examined individually and no generalizations in terms of Volterra dislocations are available. Any incompatible growth strains give rise to residual stresses which are known to occur in many tissues such as the heart, arterial wall, and solid tumors.