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3.4.4 Linearization of External Virtual Work
The linearization of in (3.4-3) depends on whether natural boundary conditions are prescribed as area densities or total net values over an area. Thus, in the case when (net force), (net volumetric flow rate), or (net effective molar flow rate) are prescribed over the elemental area , there is no variation in and it follows that . Alternatively, in the case when , or are prescribed, the linearization may be performed by evaluating the integral in the parametric space of the boundary surface , with parametric coordinates . Accordingly, for a point on , surface tangents (covariant basis vectors) are given by and the outward unit normal is The elemental area on is . Consequently, the external virtual work integral may be rewritten as where The directional derivative of may then be applied directly to its integrand, since the parametric space is invariant .
If we restrict traction boundary conditions to the special case of normal tractions, then where is the prescribed normal traction component. Then it can be shown that the linearization of along produces The linearizations along and reduce to zero, and .