American Journal of Computational Mathematics, 2012, 2, 156-162 doi:10.4236/ajcm.2012.22021 Published Online June 2012 (http://www.SciRP.org/journal/ajcm) Solution of Singular Integrals in Mathematical Model of Mode I Crack Near Strength Mismatched Interface Sunil Bhat 1 , Vijay G. Ukadgaonker 2 1 School of Mechanical and Building Sciences, VIT University, Vellore, India 2 Department of Mechanical Engineering, Indian Institute of Technology, Mumbai, India Email: sbhat_789@rediffmail.com Received April 5, 2012; revised May 2, 2012; accepted May 10, 2012 ABSTRACT Characteristics of Mode I crack near the interface of elasticity matched but plasticity and strength mismatched materials differ from those of the crack in a homogenous body. Interface body of different strength influences the plastic or cohe- sive zone at the crack tip in parent body. The mathematical model for load line opening of the crack near the interface in linear elastic regime involves singular integrals. The paper presents explicit solution of these integrals with the help of Cauchy’s principal value theorem. Cases of thin and thick welds between the materials are investigated. Solutions of the integrals are well substantiated. Final results are provided in a consolidated form. Keywords: Crack Opening Displacement; Singular Integrals; Strength Mismatch; Weld Interface; Cauchy’s Principal Value Theorem 1. Introduction The material behaviour at the tip of the Mode I crack in a homogenous body is in general very complex and diffi- cult to describe by continuum mechanical models. The crack tip region where the material undergoes degrada- tion or damage is known as the process region. Refer Figure 1(a). Micro-mechanical processes, viz. micro- cracking in brittle materials and void initiation and coa- lescence in ductile materials create new traction free sur- faces or cracks in process region. Yielding occurs outside the process region. This zone is called as the plastic or cohesive zone. Cohesive zone is considered as the crack extension under the action of closing cohesive stress generated by elastic constraint exerted by surrounding non-yielded material over the cohesive zone. The cohe- sive stress is assumed equal to material yield strength in plane stress and 3 times the yield strength in plane strain conditions. Qualitative characteristics of the cohe- sive zone were experimentally verified by Hahn et al. [1]. They conducted experiments on cracked steel specimens and found the cohesive zone, as shown in Figure 1(b), by etching the polished surface in front of the crack tip. In a bimaterial comprising elasticity identical but plas- ticity and strength mismatched constituents (like steels), the Mode I crack near the interface has the characteristics similar to the one in homogeneous parent body as long as the cohesive zone is in the parent body alone. The effect of approaching interface body of different strength is not felt by the crack in such a stage because of similar elastic properties across the interface. But as the crack grows and reaches nearer to the interface, the increasing mag- nitude of crack tip stress field causes the cohesive zone to develop in the interface body. Consequently, the part of cohesive zone in the interface body is subjected to cohesive stress different from that acting over its portion in the parent body that triggers the effect of strength mis- match across the interface over the crack tip. The effect continues with increasing intensity as the cohesive zone spreads deeper into the interface body with crack growth and reaches the maximum when the crack tip touches the interface body with the cohesive zone fully in the inter- face body Cases of thin and thick welds between the steels are examined. Thin weld, obtained by non-fusion, solid state like friction welding between dissimilar steels leads to a single thin interface whereas a thick weld by fusion bonding from electron or laser beam welding results in two interfaces, one between the parent body and the weld and the other between the weld and the interface body. The parent body, the weld and the interface body have similar elastic properties but variable strengths of com- parable magnitudes. 2. Problem Definition Solution for load line opening of the crack is obtained by modeling its cohesive zone. Complex potentials are used Copyright © 2012 SciRes. AJCM