Assuming that a charge density ρ*c* exists everywhere in the domain, the net electrical potential energy is given by ρ*c* φ*e* , where φ*e* is the electric potential. The total electrical energy of the system at steady state is given by
Display Formula

$\Psi e=\u222b\Omega \rho c\phi ed\Omega $

(4)

The rate of change of this quantity, assuming equilibrium is reached for electrical potential much faster than the rest of the system, is obtained by taking the material time derivative [

11] of Eq.

4Display Formula$\Psi \xb7e=\u222b\Omega \u2202(\rho c\phi e)\u2202td\Omega +\u222b\Gamma ext(\rho c\phi e)vnd\Gamma +\u222b\Gamma int[[(\rho c\phi e)vn]]d\Gamma $

(5)

where [[]] is the jump in the quantities across the interface defined as [[

*a*]] =

*a*+ +

*a*− , referring to the quantity on either side of the interface. Assuming that the material velocity is significantly slower than the time taken to reach steady state, we can assume that the system is always at steady electrical state. Hence, the spontaneous change term,

$\u2202(\rho c\phi e)\u2202t\u21920$. Also, if the external boundaries of the system are fixed,

$\u222b\Gamma ext(\rho c\phi e)vnd\Omega =0$. Further, for a material–void system, one side of the jump term can be ignored. Hence, the free energy rate due to electrical work can be approximated in electromigration-induced void evolution problems as

Display Formula$\Psi \xb7e=\u222b\Gamma int(\rho c\phi e)vnd\Gamma $

(6)

Similarly, the contribution to free energy from mechanical loads is

Display Formula$\Psi m=\u222b\Omega \phi md\Omega $

(7)

where φ

*m* is the strain energy density. To evaluate the mechanical energy contributions, from the balance of mechanical energy on the system, we get

Display Formula$\u222b\Gamma t\u2192\xb7vd\Gamma =\u222b\Omega \sigma :Dd\Omega $

(8)

where

*D* is the rate of deformation tensor. Assuming linear elasticity (σ =

*E*: ɛ), and therefore the strain energy density

$\phi m=(12)\u025b:E:\u025b$, we have, after similar considerations for equilibrium of the stresses

Display Formula$\u222b\Gamma t\u2192\xb7vd\Gamma =\u222b\Omega \u2202\phi m\u2202t\u2003d\Omega +\u222b\Gamma ext\phi mvnd\Gamma +\u222b\Gamma int[[\phi mvn]]d\Gamma $

(9)

Arguing, as before, that the system reaches equilibrium instantly relative to the times involved in mass transport, we have

$\u2202\phi m\u2202t\u21920$. Assuming fixed external boundaries, and no loads in the void, we get

Display Formula$\Psi \xb7m=\u222b\Gamma t\u2192\xb7vd\Gamma =\u222b\Gamma int\phi mvnd\Gamma $

(10)

Combining Eqs.

3,

6,

10, we get

Display Formula$\Psi \xb7=\u222b\Gamma int[\gamma \kappa +\rho c\phi e+\phi m]vnd\Gamma $

(11)

Following the arguments from Ref. [

12], and introducing surface and bulk mobilities

*Ms * and

*Mb *, we get the following equations for the velocity of the front.