Boundary conditions at the crack faces

Let us elucidate the boundary conditions on F, for the solution (IF, w) of (2.100) assuming that w in some neighbourhood IV of the graph F,. To this end, we first note that the equation 

Let us put functions of the form (IF, w) as test ones in (2.100), where w is the third component of the solution (IF, w). This yields 

In so doing, the test functions W should satisfy the inequality 

One can represent the vector aijPj on F as a sum of the normal and tangential components 

A similar formula can be written on F. Choosing the functions IF having the property W]p 0 on F,, the test elements IF = IF + IF can be substituted in (2.106). Since the boundary dfl of domain is a combination of the sets F, F, F, the integration by parts is easily carried out. This implies 

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We prove an existence theorem for elastoplastic plates having cracks. The presence of the cracks entails the domain to have a nonsmooth boundary. The proof of the theorem combines an elliptic regularization and the penalty method. We show that the solution satisfies all boundary conditions imposed at the external boundary and at the crack faces. The results of this section follow the paper (Khludnev, 1998). 

In this subsection we prove the solvability of the elastoplastic problem for a plate having a nonsmooth boundary. A solution of the problem will satisfy all boundary conditions both at the exterior boundary and at the crack faces. 

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. 

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). 

Note that conditions (4.23), (4.24) hold in the weak sense. We see that boundary conditions considered at the crack faces have the equality type in this section. 

Figure 5. Description of (a) boundary conditions for the elastoplastic problem and (b) initial and boundary conditions for the hydrogen diffusion problem at the blunting crack tip in the MBL formulation. The parameter bCl denotes the crack tip opening displacement in the absence of hydrogen. The parameter C, (P) denotes NILS hydrogen concentration on the crack face in equilibrium with hydrogen gas pressure P. and / is hydrogen flux.

Figure 9. MBL formulation results plotted against normalized domain size L lb. under zero concentration boundary condition on the remote boundary while the crack faces are in equilibrium with 15 MPa hydrogen gas (a) non-dimensionalized effective time to steady state = >t lb (b) peak values of the normalized hydrogen concentration in NILS at / =/ (effective time to steady state).

If the crack faces are taken as stress free, then we have the boundary conditions Oifl = oe = 0 at 0 = 7T so that ... 

We recall that, in Sect. 3.1, (//(z, t) was eliminated by requiring essentially that the discontinuity in 0(z, /) over the real axis be zero where the stresses are zero. A somewhat different tactic is employed in the present case. First, let us state our boundary conditions explicitly. All stresses are zero at infinity. Off the crack face, the displacements are continuous everywhere, in particular along the x-axis. Let the region of the x-axis, on the crack face, be F(/), made up of 0(/) and C(0, two disjoint sets, 0(/) being the region on which the crack face is open and C t) being the region on which it is closed. On 0(/), we have... 

To a degree greater than their elastic counterparts, stress analysis problems for cracked viscoelastic bodies are subject to difficulties arising from the facts that cracks may extend with time and that boundary conditions may not be definitely known until the problem is solved. Valid solutions should involve no material overlap over the crack face. Furthermore, at points where the crack is closed, the normal traction should be a pressure while at points where it is open, this quantity is usually prescribed. 

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