O Pewnym Zagadnieniu Niejednorodnej Półprzestrzeni Sprężystej

This paper presents solutions of two problems of axially-symmetric state of stress in a non homogeneous semi-infinite elastic body, which are: a) the problem of a punch, b) the problem of a crack.
Both problems are solved by means of Hankel's integral transformation and by reducing the boundary-value problem to that of dual integral equations. The point of departure for solving both boundary value problems is discussed in Sec. 2 of the present work, which is of general nature and in which are given general (integral) formulae for stresses and displacements.
Sec. 3 is devoted to the solution of the problem of a punch of any form and gives, in the case: of a small nonhomogeneity (small parameter n), an effective equation for the contact stresses (3.38). and (3.41). Also a numerical example is discussed giving a diagram of stresses under the punch. (Fig. 3).
Sec. 4 brings an effective solution of the second problem. The case of p (r) = const is considered, equations for stresses and displacements being obtained ((4.20) and (4.21)). Figure 5 represents a diagram of these quantities. In conclusion it is found that, with the type of nonhomogeneity assumed E = E (z), the solution of the problem of the type under consideration depends in an essential manner on the parameter I (the radius of the punch or the crack).