Certain problems of an arbitrarily oriented stringer in a composite an isotropic plane

Abstract

The fundamental solution for a composite anisotropic plane is constructed taking defects into account on the line separating the materials. On this basis, a mathematical formulation is given of the contact problem for an arbitrarily oriented stringer located in one of the half-planes, in the form of a singular integral equation with a fixed singularity. The solvability of the equation obtained is proved. The power nature of the behaviour of the solution is clarified (by using the asymptotic properties of Cauchy-type integrals), and the conditions are established for the satisfaction of which the mentioned asymptotic form is strengthened by a logarithmic polynomial. An exact solution is constructed for the problem of a semi-infinite stringer leaving the line of material separation at an arbitrary angle, and an analogous problem for a finite inextensible stringer. It is hown that the asymptotic form of the contact shear stresses at the point of departure does not depend on the elastic properties of the stringer. Solutions of analogous problems for anisotropic and orthotropic half-planes can be obtained by a passage to the limit. Such problems were examined earlier for a finite stringer in an anisotropic half-plane /1/ and for a semi-infinite stringer perpendicular to the boundary of an orthotropic half-plane /2, 3/. An incorrect assumption was made here about the power-logarithmic nature of the behaviour of the solution at the point of stringer departure at the boundary (the authors incorrectly utilized the asymptotic properties of Cauchy-type integrals). The error of the result /2/ is mentioned in /3/ where an exact solution of the problem is constructed. However, the true reason for the error is not given here, in which connection a false deduction is made about the fact that the asymptotic form of Cauchy-type integrals does not permit a unique solution of the question of the nature of the singularity if it is assumed to be power-logarithmic. The error of this deduction is shown below.