A crack originating from a semicircular notch in a non homogeneous sheared half-plane

An interfacial crack originates from a semi circular notch in a non-homogeneous semi infinite elastic material subjected to constant shear loads of magnitudes Tj, j = 1, 2. Asymptotic deformation fields near the crack tip are derived in a closed form. The stress intensity factor is shown to depend on material constants except when equal and opposite tractions are applied on load sites of equal length. Our results agree with known ones.


INTRODUCTION
The notched homogeneous elastic half plane under remote loading has been studied in Mitchel (1965), Rice (1967) and Ejike (1973).The non-homogeneous case was studied in Nnadi, (2004a) under more practicable finite surface loading.In this paper we study the elastic half plane made from two materials.Each material is a quarter plane from which a quarter circle of radius c is removed.The resultant figures are then perfectly bonded along their interface to form the half plane with a semicircular notch of radius c.A crack of length b-c emanates from the edge of the notch along the interface.The free surface of each quarter plane is subjected to anti-plane shear loads of magnitude T j , on interval [a j , b j ], j = 1, 2. Every other surface is stress free (fig 1).The subscript j = 1 refers to the upper quarter plane while the subscript j = 2 refers to the other material.

Fig. 1: Geometry Of The Problem
The polar stresses are:

TRANSFORMATION OF THE PROBLEM
The notched half plane is transformed on to a plane with a slit on the left half line 0 Re < ℑ with the aid of the conformal mapping function: Due to the transformation the boundary value problem becomes

SOLUTION OF THE TRANSFORMED PROBLEM
The behaviours are . The Mellin integral transform applied to ( 8) -( 10) gives where the Mellin transform is defined by In view of the expression Andrews (1992 p.21), Let the solution of (12) be of the form Application of ( 13) and ( 14) to ( 19) yields where Hwang et al (1994) The inversion formula yields the displacements as Residue method and Jordan's lemma, Walker (1974) are applied in seeking ( ) , by evaluating the following integrals: The results for j=1,2, are: , (10a) gives simultaneously.The integrals to be evaluated for Res < 0 are:

FIELDS NEAR THE CRACK TIP
As 0 → ρ in (27) the asymptotic displacement fields near the rack tip are: , which applied to (29), leads to Hence the crack tip displacement fields as The conventional form of ( ) Tada et al (1985) ( ) ( ) where b-c is the crack length and is the mode III stress intensity factor.From ( 17) we derive ( ) ( ) and (32) it is not difficult to apply (11a,b) to get

CONCENTRATED SHEAR FORCE
From the relation ( ) The intensity factor corresponding to such concentrated loading is obtained from (31) as the configuration is that of a non-inclined interfacial edge crack of depth b in two bonded quarter planes Hwang et al (1994).If in addition to c=0, the condition holds, the configuration is that of an edge crack in a homogeneous half plane Nnadi (2004c).In all the cases our results agree with those for the corresponding configuration.The stress intensity factor for a load mixture of traction T j distributed on [a j , b j ], j=1 or 2 and shear force, Q i concentrated at a i , i=2 when j=1 or i=1 when j=2, can be written as The result for equal and opposite concentrated loads is These mean that dependence on material constants are suppressed when equal and opposite loads are applied at equal intervals or concentrated at points that are symmetric with respect to the crack line.
Fig. 2: Geometry of The Transformed Problem