Implications of mechanical equilibrium

Arclength s along S is measured from some arbitrary origin. Local surface orientation is represented by the tangent line, with orientation described by a unit vector in the direction of increasing s, and by a unit vector the two unit vectors satisfy mini = 0. It is also useful to represent local surface orientation at any time t by means of the angle 6 s, t) = sin [ea (ei x m)]. 

In terms of this angle, the surface curvature in the reference configuration is 

The strain at a point of the surface S is understood to be consistent with the limiting value of bulk strain Cjj as the observation point approaches the surface from within R. The surface strain expressed as a three-dimensional tensor field over the surface with outward unit normal n, is 

The expression (8.124) faithfully reproduces strain components for all material lines or directions within the surface, but includes no contribution from lines or directions normal to the surface. Thus, it is convenient for expressing energy changes of the surface associated with deformation. In particular, for the two-dimensional state being considered, the extensional 

The total time rate of change of extensional strain of the surface includes contributions due to the time dependence of strain at a material point, the time dependence of the orientation of the surface, and the motion of the surface with respect to the material particles instantaneously on it. If these effects are taken into account, then 

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