# Advanced scientific computing (B673) by Meglicki Z. PDF By Meglicki Z.

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The plane contains a linear crack located on x1 axis (x2 = 0) described by equation |x1 | < a, the tips of which have no outer loads (Fig. 22a). In this case the edges of the cut are displaced with respect to each other along the line perpendicular to the front of thickness (called the thickness of a transversal shear). The solution of the plane problem for cracks of transversal shear, in the framework of elasticity theory, may also be represented in the form of a sum of solutions of two problems: in the ﬁrst problem (Fig.

138) we can get the factor of the density intensity of charges, which in the general case is determined by Kσ = im σ ∗ (ρ) ρ→R 2π (R − ρ) . 155) Now consider in a more detail the electric ﬁeld created by a charged conductive circular disk of radius R. 4 Singular Physical Fields 43 Here Q=8 0 RV, (hence, the capacitance of a circular disk in vacuum is given by C = Q/V = 8 0 R). Let us now work in the cylindrical system of coordinates. 157) describe the distribution of the potential of a circular disk in the whole space.

Transition from the cavity to the mathematic cut in an unbounded body: a cavity in the form of an elliptic cylinder (a), a mathematical tunnel of a rectangular cut (b) At the limiting transition b → 0, the ellipse becomes a linear tunnel cut in section (−a, a) of Ox1 - axis. The upper half-ellipse x2 = b 1 − x21 /a2 moves to the upper edge of the cut |x4 | < a, x2 = +0, and the lower half-ellipse x2 = −b 1 − x21 /a2 to the lower edge of the cut |x| < a, x2 = −0 (Fig. 17b). Here, we should mention that the linear tunnel cut in an unbounded body or linear through cut in a thin plate is a basic idealized form of a real crack: in an arbitrary small vicinity of the point, O, of the front (Fig.