  This article is the second of a three-part series that derives a
self-consistent theoretical framework of the electromechanics of arbitrarily
curved lipid membranes. Existing continuum theories commonly treat lipid
membranes as strictly two-dimensional surfaces. While this approach is
successful in many purely mechanical applications, strict surface theories fail
to capture the electric potential drop across lipid membranes, the effects of
surface charges, and electric fields within the membrane. Consequently, they do
not accurately resolve Maxwell stresses in the interior of the membrane and its
proximity. Furthermore, surface theories are generally unable to capture the
effects of distinct velocities and tractions at the interfaces between lipid
membranes and their surrounding bulk fluids. To address these shortcomings, we
apply a recently proposed dimension reduction method to the three-dimensional,
electromechanical balance laws. This approach allows us to derive an effective
surface theory without taking the limit of vanishing thickness, thus
incorporating effects arising from the finite thickness of lipid membranes. We
refer to this effective surface theory as $(2 + \delta)$-dimensional, where
$\delta$ indicates the thickness. The resulting $(2 + \delta)$-dimensional
equations of motion satisfy velocity and traction continuity conditions at the
membrane-bulk interfaces, capture the effects of Maxwell stresses, and can
directly incorporate three-dimensional constitutive models.
