Author/Authors :
anongba, p. n. b. université félix-houphouët-boigny de cocody - ufr sciences des structures de la matière et de technologie, Côte d’Ivoire
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 sin k_n x_3 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 indefinitelyalong 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_n , assuming ξ_n to be small) proportional to the sinusoid or its spatial derivative with respect to x_3 . The displacement and stress fields, provided by our analysis, for an interface straight edge dislocation, reflect thepresence of the Dirac delta function in the shear stresses on the interface; a comparison is made of these findings with those previously published on the same subject. Among the additional oscillating stresses, σ 23 is the only one that possesses a singularity of the 1 / x_1 type on the interface. Consequently, this stress contributes a non-zero value to the crack extension force when a non-planar interface crack, with an oscillating front perpendicular to the x_1 direction, is loaded in tension.
NaturalLanguageKeyword :
linear elasticity , interface dislocations , Galerkin vector , threedimensional biharmonic functions , Fourier forms , linear systems of equations
