Crack shape

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... 

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). 

As for approximate methods of finding crack shapes we refer the reader to (Banichuk, 1970). Qualitative properties of solutions to boundary value problems in nonsmooth domains are in (Oleinik et al., 1981 Nazarov, 1989 Nazarov, Plamenevslii, 1991 Nicaise, 1992 Maz ya, Nazarov, 1987 Gris-vard, 1985,1991 Kondrat ev et al., 1982 Kondrat ev, Oleinik, 1983 Dauge, 1988 Costabel, Dauge, 1994 Sandig et al., 1989 Movchan A.B., Movchan N.V., 1995). 

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. 

Given W G we have q = —aijn n > 0, and hence, the density q is defined by the normal component of the surface forces at At the end, in Section 2.8.3, we establish the stability of solutions with respect to perturbations in the crack shape. 

We now aim to study the stability of the solution with respect to the crack shape. Let y = 5 x) be the crack shape, and 5 be a parameter which will subsequently tend to zero. 

The nonpenetration condition considered in this section leads to new effects such as the appearance of interaction forces between crack faces. It is of interest to establish the highest regularity of the solution up to the crack faces and thus to analyse the smoothness of the interaction forces. The regularity of the solution stated in this section entails the components of the strain and stress tensors to belong to in the vicinity of the crack and the interaction forces to belong to T. If the crack shape is not regular, i.e. 0 1), the interaction forces can be characterized by the nonnegative... 

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... 

This obviously means that the limiting function w is a solution of the equilibrium problem for the shell having the crack shape y = x) =0,xG [0,1]. Thus the following statement has been proved. 

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. 

Banichuk N.V. (1970) The small parameter method in finding a curvilinear crack shape. Izvestiya USSR Acad. Sci., Mechanics of Solid (2), 130-137 (in Russian). 

Double torsion test specimens take the form of rectangular plates with a sharp groove cut down the centre to eliminate crack shape corrections. An initiating notch is cut into one end of each specimen (Hill Wilson, 1988) and the specimen is then tested on two parallel rollers. A load is applied at a constant rate across the slot by two small balls. In essence the test piece is subjected to a four-point bend test and the crack is propagated along the groove. The crack front is found to be curved. 

A 62-year-old man with diabetes was given metformin 750 mg bd and his blood glucose concentration fell from 22 to 15 mmol/1 within 4 days. The dose of metformin was increased to 850 mg bd and the blood glucose concentration fell to 8.7 mmol/1 over the next week. Within 2 days of starting therapy his vision became blurred. Slit lamp examination 2 weeks later showed cracked shaped lines on the lens. The cracks resolved spontaneously by 3 months. 

For Keff> 0, depending on the loading and crack/specimen geometry conditions, one can draw on known32,33 solutions for the crack shape and Keff. If u(x) is the crack opening at some location, x (see Fig. 10.6), then, in the presence of a remote applied stress, [Pg.347]

If the crack is considered to be a wake zone and p = p(u, t) in Eqns. (19) and (21) then, for a stationary crack, both the crack shape and Keff are time dependent, given respectively by u(x,t) and KejAt) from Eqns. (19) and (21) for the case of partial shielding. Cox and Rose35 recently considered an elastic time-dependent bridging law of the form... 

Clearly, the function p(u,t) for a stationary crack depends on the crack shape as well as on crack/specimen geometry. 

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