ABSTRACT
Regarding boundary data, we recall that yˆ(u, ς) is assigned on ∂κC .This implies that its midsurface value, r, and those of its through-thickness derivatives, d and g, are assigned on ∂. However, the latter two fields cannot be assigned arbitrarily if the foregoing model is to apply on the closure of . The assigned values must agree with the continuous extensions to ∂ of the functions d¯ and g¯ delivered by equations (28)1,2. The first requirement effectively means that r and its normal derivative r,ν are assigned. For, the midsurface deformation gradient may be decomposed in the form
where τ and ν are the unit tangent and normal to the edge and the tangential derivative r,s is obtained by differentiating r with respect to arclength on ∂. The continuous extension to ∂ of the field d¯ derived from (28)1 is thus controlled by the boundary values of r and r,ν. In the same way we stipulate that the values of g on ∂ agree with the continuous extension of the solution g¯ to (28)2. We then consider d and g to be assigned accordingly and thus require their variations to vanish on ∂. In general these edge values are fully specified only after the problem has been solved, and so some parts of the data on ∂κC must effectively adjust a posteriori to suit the problem at hand, rather than being imposed a priori. The alternative to this state of affairs is to use three-dimensional theory in a region adjoining the boundary and then match its predictions to those of the foregoing interior equations. Here, however, we intend that the two-dimensional model should apply on the closure of , and this requires a compromise with respect to boundary data.
