By definition the derivative in , etc. are ``material'' in that it takes the differences of quantities evaluated at the same point on the reference crack. This means that one may have a nonzero etc. which does not correspond to an actual change of the crack geometry. Indeed, an which is tangential everywhere on S and vanishes at both tips gives rise to no geometry change of S. The corresponding is obviously equal to
where . This consideration suggests another definition of the derivative given by
with . Then a nonzero certainly corresponds to a change of crack geometry, and an which vanishes at both tips is irrelevant to the change of S.
One would then expect that is written in terms of and . That this is true is easily seen from (8) since one has
Also the variational equation (9) for is rewritten as
The hypersingularity in the LHS comes only from the singularity of , as we shall see. At the first sight it might appear that vanishes for a purely tangential move of the crack () since the RHS of (13) vanishes then. As a matter of fact, this is incorrect. Indeed, one has near the tips of S if does not vanish there, since and then. A homogeneous solution of (13) satisfying is identically zero, but there exist nontrivial solutions of (13) which are proportional to at the tips (see Bueckner for related topics). Hence (12) and (13) tell that the variation of J vanishes unless the crack moves into the normal direction or one of the tips moves.