Extreme crack shapes

In the following we analyse the behaviour of the solution as 5 — 0. It will enable us in the sequel to prove the existence of extreme crack shapes. The formulation of this problem is given below. So, for every fixed 5 there exists a solution = iyV of the problem 

It is clear that the Jacobian qg = 1 — of this transformation converges uniformly to the unit on as J — 0. Introduce the notations 

A substitution of a fixed test function % in (2.116) results in the relation 

Omitting the sign 5 in the functions it is easy to rewrite this inequality in the new variables 

Herein f x,y) = f x,y), a 2, (5 1. A dependence of the function g on its arguments is fully determined by the transformation (2.117). It is of importance that this function has quadratic growth in the higher order derivatives. In view of the inequality q 1/2 holding for small 6, from (2.118) we conclude that 

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This result enables us to investigate the extreme crack shape problem. The formulation of the last one is as follows. Let C Hq 0, 1) be a convex, closed and bounded set. Assume that for every -0 G the graph y = %j) x) describes the crack shape. Consequently, for a given -0 G there exists a unique solution of the problem... 

This precisely means that the limiting function -0 is a solution of the extreme crack shape problem (2.125). 

The crack shape is defined by the function -ip. This function is assumed to be fixed. It is noteworthy that the problems of choice of the so-called extreme crack shapes were considered in (Khludnev, 1994 Khludnev, Sokolowski, 1997). We also address this problem in Sections 2.4 and 4.9. The solution regularity for biharmonic variational inequalities was analysed in (Frehse, 1973 Caffarelli et ah, 1979 Schild, 1984). The last paper also contains the results on the solution smoothness in the case of thin obstacles. As for general solution properties for the equilibrium problem of the plates having cracks, one may refer to (Morozov, 1984). Referring to this book, the boundary conditions imposed on crack faces have the equality type. In this case there is no interaction between the crack faces. 

This section is concerned with an extreme crack shape problem for a shallow shell (see Khludnev, 1997a). The shell is assumed to have a vertical crack the shape of which may change. From all admissible crack shapes with fixed tips we have to find an extreme one. This means that the shell displacements should be as close to the given functions as possible. To be more precise, we consider a functional defined on the set describing crack shapes, which, in particular, depends on the solution of the equilibrium problem for the shell. The purpose is to minimize this functional. We assume that the... 

Here [ ] is a jump of a function at the crack faces, v is the unit normal vector to the crack shape, and 2h is the thickness of the shell. A similar extreme crack shape problem for a plate was considered in Section 2.4. 

At the beginning we study the (5-dependence of the solution and next we consider the problem of finding extreme crack shapes. First, let us note that the problem (4.168) has a solution owing to the coercivity and the weak lower semicontinuity of II5 on the space The solution is unique for... 

Consider the problem of finding the extreme crack shapes. The setting of this problem is as follows. Let C be a convex, closed and... 

The problem of finding an extreme crack shape is formulated as follows ... 

In what follows we prove the existence of the extreme crack shape. 

We have studied the the fracture properties of such elastic networks, under large stresses, with initial random voids or cracks of different shapes and sizes given by the percolation statistics. In particular, we have studied the cumulative failure distribution F a) of such a solid and found that it is given by the Gumbel or the Weibull form (3.18), similar to the electrical breakdown cases discussed in the previous chapter. Extensive numerical and experimental studies, as discussed in Section 3.4.2, support the theoretical expectations. Again, similar to the case of electrical breakdown, the nature of the competition between the percolation and extreme statistics (competition between the Lifshitz length scale and the percolation correlation length) is not very clear yet near the percolation threshold of disorder. 

Rg. 18. Localized finger-shaped epsilon CTPZ development in PEC. The DCO crack arrest bands are not visible in this optical micrograph. The irregular lines at the top of the Cmger-Ukc regions arc due to moisture in the cracks. A faint com x craze profile can be seen at the extreme tips ... 

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