An evolving free surface

The elastic strain energy density U(eij) is determined locally by the elastic strain field Cij, and the energy density function incorporates consti- 

With these constitutive assumptions, the free energy of the system is the sum of elastic energy and surface energy, 

The change of shape of S can be expressed in terms of the normal velocity of the surface Vn, viewed as a function of position over S for the time range of interest. This is the normal speed of the surface with respect to the material particles instantaneously on the surface, as determined in the reference configuration of the material. Then 

The quantity dU/dcij is the stress dij which is necessary to enforce the elastic strain eij. The stress field is an equilibrium field which, in the 

This expression leads naturally to a definition of chemical potential as a field over the evolving surface S. Note that VndS represents a local rate of material volume addition (u 0) or removal vn 0). If it is understood that all quantities are represented in the reference configuration of the material, then there is an unambiguous equivalence between volume of material added and mass of material added. The free energy per unit volume of material being added locally is then 

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