Cracks of zero opening

The crack is said to have a zero opening in this case. As it turned out there is no singularity of the solution provided the crack has a zero opening. What this means is the solution of (3.144), (3.147), (3.148) coincides with the solution of (3.140)-(3.142) found in the domain Q with the initial and boundary conditions (3.144), (3.145) (and without (3.143)). In the last case the equations (3.141), (3.142) hold in Q. This removable singularity property is of local character. Namely, if O(x ) is a neighbourhood of the point and 

we recall two Green formulae. Let D C be a bounded domain with a smooth boundary 7 having the external normal n = (ni,ri2). Consider the following two operators defined on 7  

Here 7 can be equal to 7+ as well as to 7 . We should remark at this point that, in fact, the integration is fulfilled over O(x ) in the right-hand side of (3.155). In other words, we integrate over and use the condition [d9 t)/di ] = 0 on holding true due to the regularity of 9. The existence of two angular points on 7= = presents no problems since the function (p has a compact support. It follows from (3.156) for almost all t G (0,T) that 

By taking into account (3.157), (3.159) we are in a position to prove the result related to the cracks of minimal opening. 

The right-hand side of (3.164) is equal to zero because of (3.153), (3.157), which completes the proof of (3.163). 

---

The arguments given below are concerned with a justification of C °°-regularity of the solution for the crack of zero opening. We shall prove the solution regularity in the neighbourhood of the line x (0,t°), where = (0,0), > 0, i.e. in the vicinity of the crack tip. The solution... 

The main statement related to the cracks of zero opening, i.e. to the cracks with the property [x] = 0, is as follows. 

---