Elastic fields of a dislocation loop in a two-phase material

Abstract

The purpose of this study is to set forth the mathematical framework for an efficient treatment of a plane dislocation loop in a two-phase material, and to carry the elastostatic part of the analysis far enough, so that the resolts are immediately tractable for applications to solid state. The two-phase material is idealized as two isotropic elastic half-space~ with perfect adhesion, while the loop is placed in a plane parallel to the interface. It is shown that the elastic fields can be obtained by differentiating two types of integrals and, thus, are readily evaluated for any shape of the loop for which the integrals are available from potential theory. Explicit results are given for circular prismatic and glide loops. ZUSAMMENFASSUNGMit der vorliegenden Arbeit wird die Absicht verfolgt, den mathematischen Rahmen zu einer leistungsf/ihigen Behandlung von ebenen Versetzungsschleifen in einem Zweiphasenmaterial zu erweitern und den elastostatischen Teil der Berechnung so weit fortzufiihren ,dab die Resultate fiir die Anwendung in der Festk6rperphysik unmittelbar zur Verfiigung stehen. Das Zweiphasen-Material wird idealisiert durch zwei isotrope elastische Halbr~ume mit perfekter Adh~ision, w~hrend die Versetzungsschleife in einer Ebene parallel zur Trennfl/iche angenommen wird. Es wird gezeigt, dab die elastischen Felder durch Differentiation zweier Integraltypen und damit fiir jede beliebige Form der Versetzungsschleife erhalten werden k6nnen, fi]r die die Integrale sich aus der Potentialtheorie ergeben. Explizite Resultate werden flit kreisf6rmige prismatische und gleitende Schleifen angegeben.

The effect of path cut on Somigliana ring dislocation elastic fields

“…Lipschitz-Hankel integrals are heavily drawn upon in many of the papers to date relating to the solutions for circular dislocations (Korsunsky, 1996a;Salamon and Dundurs, 1971;Kolesnikova and Romanov, 2004). They are defined, with three parameters l, m, k and two (coordinate) variables q, f, as…”

“…Two types of Volterra type edge dislocation can be identified: the climb or prismatic loop, when the Burgers vector lies in the z-direction, and the glide loop when it lies in the x À y plane. The literature includes solutions for climb loops in an infinite space (Korsunsky, 1996a;Kroupa, 1960;Salamon and Comninou, 1979) and a half-space (Korsunsky, 1996b;Salamon and Dundurs, 1971). The glide loop is less comprehensively covered, but solutions for an infinite and half spaces are given for example in Salamon and Dundurs (1977).…”

“…BothKorsunsky (1996a) andSalamon and Dundurs (1971) have fewer terms. The second expression for r rr is found using Laplace's equation, and results in less complex, differential operations with respect to r.…”

The effect of path cut on Somigliana ring dislocations in a half-space

“…The discontinuity may be expressed as u zjz (q < 1,f = d)=b z . This basic solutions for the full space (Kroupa, 1960), half-space (Basteck, 1964) and isotropic bimaterial (Salamon and Dundurs, 1971;Dundurs and Salamon, 1972) have been known for some time but these papers do not discuss the effect of path cut. It is included here to provide a complete set in the same terms, and as a guide because it does not have path cut dependency on the stresses.…”

Triangular Dislocation Loop Model for Surface Displacement in Indented Thin Films

“…In the literature [2,3], the indentation residual stress has been investigated by the elastic field of prismatic dislocation loops. The applied pressure needed for Vickers indentation was investigated by Tanaka [3], where the elastic recovery associated with the circular prismatic loop arrays were numerically solved by summing an analytical solution for a single prismatic loop field in two phase material [4]. The square dislocation loop model for Vickers indentation was presented by Mura [3].…”