Crack boundaries

Fig. 8.35 Reduction in area versus applied potential for 316 stainless steel, in 5 ppm chloride content water at 250°C. Horizontal arrows indicate RA% for tests in argon gas open-annealed, filled-sensitised. Vertical downward pointing arrows on the annealed curve indicate cracking-non-cracking boundaries. Full downward pointing arrow on the sensitised curve indicates commencement of cracking. Open upward pointing arrow on the sensitised curve indicates transition from transgranular to intergranular cracking...

Fracture surface in 3D space and its piecewise planar representation is shown in Fig. 6. Through the boundary 5 fracturing fluid is pumped from the wellbore to the crack. Boundary S p is the fluid s front. 

The electron microscopy data confirm the made conclusion. In Fig. 8.2, the microphotograph of stable crack boundary in PASF sample (solvent -chloroform) is adduced. As one can see, the fracture surface has microroughnesses at any rate of two levels ( 1 mcm and 20 mn) that allows to apply fractal models for PASF samples fracture process [4, 5]. 

FIGURE 8.2 The electron microphotograph of stable crack boundary for PASF sample (solvent - chloroform). Enlargement 15,000 [10],... 

This paper compares experimental data for aluminium and steel specimens with two methods of solving the forward problem in the thin-skin regime. The first approach is a 3D Finite Element / Boundary Integral Element method (TRIFOU) developed by EDF/RD Division (France). The second approach is specialised for the treatment of surface cracks in the thin-skin regime developed by the University of Surrey (England). In the thin-skin regime, the electromagnetic skin-depth is small compared with the depth of the crack. Such conditions are common in tests on steels and sometimes on aluminium. 

At sufficiently high frequency, the electromagnetic skin depth is several times smaller than a typical defect and induced currents flow in a thin skin at the conductor surface and the crack faces. It is profitable to develop a theoretical model dedicated to this regime. Making certain assumptions, a boundary value problem can be defined and solved relatively simply leading to rapid numerical calculation of eddy-current probe impedance changes due to a variety of surface cracks. 

Installation for Ultrasonic Testing AKV-S is designed for testing of diesel motors pistons. Particularly, this device identifies the areas with cracks and lowered adhesion on interfacial boundary between niresist ring and base material. 

A problem obviously exists in trying to characterise anomalies in concrete due to the limitations of the individual techniques. Even a simple problem such as measurement of concrete thickness can result in misleading data if complementary measurements are not made In Fig. 7 and 8 the results of Impact Echo and SASW on concrete slabs are shown. The lE-result indicates a reflecting boundary at a depth corresponding to a frequency of transient stress wave reflection of 5.2 KHz. This is equivalent to a depth of 530 mm for a compression wave speed (Cp) of 3000 m/s, or 706 mm if Cp = 4000 m/s. Does the reflection come from a crack, void or back-side of a wall, and what is the true Cp ... 

Submitting the main topic, we deal with models of solids with cracks. These models of mechanics and geophysics describe the stationary and quasi-stationary deformation of elastic and inelastic solid bodies having cracks and cuts. The corresponding mathematical models are reduced to boundary value problems for domains with singular boundaries. We shall use, if it is possible, a variational formulation of the problems to apply methods of convex analysis. It is of importance to note the significance of restrictions stated a priori at the crack surfaces. We assume that nonpenetration conditions of inequality type at the crack surfaces are fulfilled, which improves the accuracy of these models for contact problems. We also include the modelling of problems with friction between the crack surfaces. 

The model describing interaction between two bodies, one of which is a deformed solid and the other is a rigid one, we call a contact problem. After the deformation, the rigid body (called also punch or obstacle) remains invariable, and the solid must not penetrate into the punch. Meanwhile, it is assumed that the contact area (i.e. the set where the boundary of the deformed solid coincides with the obstacle surface) is unknown a priori. This condition is physically acceptable and is called a nonpenetration condition. We intend to give a mathematical description of nonpenetration conditions to diversified models of solids for contact and crack problems. Indeed, as one will see, the nonpenetration of crack surfaces is similar to contact problems. In this subsection, the contact problems for two-dimensional problems characterizing constraints imposed inside a domain are considered. 

The stress free boundary condition (1.45) for crack surfaces implies... 

Now we intend to derive nonpenetration conditions for plates and shells with cracks. Let a domain Q, d B with the smooth boundary T coincide with a mid-surface of a shallow shell. Let L, be an unclosed curve in fl perhaps intersecting L (see Fig.1.2). We assume that F, is described by a smooth function X2 = i ixi). Denoting = fl T we obtain the description of the shell (or the plate) with the crack. This means that the crack surface is a cylindrical surface in R, i.e. it can be described as X2 = i ixi), —h < z < h, where xi,X2,z) is the orthogonal coordinate system, and 2h is the thickness of the shell. Let us choose the unit normal vector V = 1, 2) at F,, ... 

Substituting here the corresponding geometrical and constitutive relations of Sections 1.1.3 and 1.1.4, we obtain H = H(17, w). The set of admissible displacements K is defined by the boundary conditions at F and nonpenetration conditions at the crack F, stated in Section 1.1.7. The variational form of the equilibrium problem is the following ... 

In this section we define trace spaces at boundaries and consider Green s formulae. The statements formulated are applied to boundary value problems for solids with cracks provided that inequality type boundary conditions hold at the crack faces. 

Let a solid occupy the domain C B with the crack Sc such that its boundary dflc belongs to the class in accord with Section 1.4.1. Introduce the space... 

At this point we have to mention different approaches to the crack problem with equality type boundary conditions (Osadchuk, 1985 Panasyuk et ah, 1977 Duduchava, Wendland, 1995). 

In this chapter we analyse a wide class of equilibrium problems with cracks. It is well known that the classical approach to the crack problem is characterized by the equality type boundary conditions considered at the crack faces, in particular, the crack faces are considered to be stress-free (Cherepanov, 1979, 1983 Kachanov, 1974 Morozov, 1984). This means that displacements found as solutions of these boundary value problems do not satisfy nonpenetration conditions. There are practical examples showing that interpenetration of crack faces may occur in these cases. An essential feature of our consideration is that restrictions of Signorini type are considered at the crack faces which do not allow the opposite crack faces to penetrate each other. The restrictions can be written as inequalities for the displacement vector. As a result a complete set of boundary conditions at crack faces is written as a system of equations and inequalities. The presence of inequality type boundary conditions implies the boundary problems to be nonlinear, which requires the investigation of corresponding boundary value problems. In the chapter, plates and shells with cracks are considered. Properties of solutions are established existence of solutions, regularity up to the crack faces, convergence of solutions as parameters of a system are varying and so on. We analyse different constitutive laws elastic, viscoelastic. 

The results on contact problems for plates without cracks can be found in (Caffarelli, Friedman, 1979 Caffarelli et al., 1982). Properties of solutions to elliptic problems with thin obstacles were analysed in (Frehse, 1975 Schild, 1984 Necas, 1975 Kovtunenko, 1994a). Problems with boundary conditions of equality type at the crack faces are investigated in (Friedman, Lin, 1996). 

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. 

In this case the crack is said to have a zeroth opening. The cracks of a zeroth opening prove to possess a remarkable property which is the main result of the present section. Namely, the solution % is infinitely differentiable in a vicinity of T, dT provided that / is infinitely differentiable. This statement is interpreted as a removable singularity property. In what follows this assertion is proved. Let x G T dT and w > (f in O(x ), where O(x ) is a neighbourhood of x. For convenience, the boundary of the domain O(x ) ia assumed to be smooth. 

Let a plate occupy a bounded domain fl c with smooth boundary F. Inside fl there is a graph Fc of a sufficiently smooth function. The graph Fc corresponds to the crack in the plate (see Section 1.1.7). A unit vector n = being normal to Fc defines the surfaces of the crack. 

In so doing, the boundary value of on F is assumed to provide nonemptiness of the set K. The equilibrium problem for the plate contacting with the punch and having the crack can be formulated as a variational one ... 

A thin isotropic homogeneous plate is assumed to occupy a bounded domain C with the smooth boundary T. The crack Tc inside 0 is described by a sufficiently smooth function. The chosen direction of the normal n = to Tc defines positive T+ and negative T crack faces. 

We consider a boundary value problem for equations describing an equilibrium of a plate being under the creep law (1.31)-(1.32). The plate is assumed to have a vertical crack. As before, the main peculiarity of the problem is determined by the presence of an inequality imposed on a solution which represents a mutual nonpenetration condition of the crack faces... 

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