مرکز منطقه ای اطلاع رساني علوم و فناوري - A STUDY OF THE ELASTIC FIELDS OF INTERFACIAL EDGE DISLOCATIONS (STRAIGHT AND SINUSOIDAL) USING GALERKIN VECTORS WITH THREE-DIMENSIONAL BIHARMONIC FUNCTIONS IN FOURIER FORMS

Abstract :

In this study, we consider two elastic solids (S1) and (S2), of infinite sizes, welded along a non-planar surface S in the form of a corrugated sheet; more specifically, with respect to a Cartesian coordinate system x_i , the interface has the same sinusoidal shape ξ = ξ_n k_n x_3 sin in the x_2 x_3 planes and is rectilinear in the x_1 x_2 planes. We investigate the elastic fields (displacement and stress) due to a dislocation lying on that interface at the origin and running indefinitely along the x_3 direction. The approach used is to treat the elastic fields as the difference of two quantities : 1) the first corresponds to the elastic fields of a sinusoidal dislocation at the origin in an infinitely extended homogeneous medium and 2) the second satisfies the equilibrium equations with a discontinuity, when crossing the interface, identical to that given by the elastic fields of the sinusoidal dislocation from the change in the elastic constants on the passage from (S2) to (S1). This second quantity is set using Galerkin vectors whose components are expressed in the form of Fourier series and integrals. Then equations are written that reflect the continuity of the elastic fields at the crossing of the interface. These interface boundary conditions split into two distinct groups: those corresponding to a planar interface with a straight edge dislocation at the origin and those (in the linear approximation with respect to ξ , assuming ξ to be small) proportional to the sinusoid or its spatial derivative with respect to x_3 . We then restrict our treatment by satisfying only to the boundary conditions associated with a planar interface with a straight edge dislocation. The displacement and stress fields of an interface straight edge dislocation, thus obtained, reflect the presence of the Dirac delta function in the shear stresses on the interface. Finally, a comparison is made of our findings with those previously published on the same subject.