PSI - Issue 45

Koji Fujimoto et al. / Procedia Structural Integrity 45 (2023) 74–81 Koji Fujimoto / Structural Integrity Procedia 00 (2019) 000 – 000 The lowest order terms of are related to the stress intensity factors. Stress intensity factors have been playing a very important role in the research of fractures of solid materials. On the other hand, the term of the zero power of (the constant term) is known as T -stress and appears only in the stress component , not in the other stress components. It is said that T -stress may be related to the stability in the crack path (Cotterell (1966)) or the size of the plastic zone at the crack tip. However, the engineering effectiveness of T -stress is controversial and Gupta et al. (2015) reviewed on a great many researches about T -stress in detail. Moreover, it is difficult to obtain T -stresses accurately in general. The aim of this study is to develop a method for determining T -stress of cracks in two-dimensional elasticity by the method of continuously distributed dislocations model in high precision. For obtaining the dislocation density on the crack surface by solving a singular integral equation, the method of Theocaris and Ioakimidis (1977) was used and extended in order to obtain T -stress. In this paper, T -stress is determined for three examples using the proposed method in order to demonstrate its effectiveness and accuracy of calculated values. The first problem is a mode I crack, the tip of which is in the vicinity of the interface of dissimilar materials and the second is an infinite row of mode I cracks which are parallel to each other. Furthermore, as the third problem, the method will be applied for determining T stresses of two parallel cracks which is a mixed mode crack problem (modes I and II). Nomenclature I , II stress intensity factors T -stress ( ) dislocation density function on crack surface (plane strain condition), (3 − )/(1 + ) (plane stress condition) 2. Formulation Consider a simple crack problem in which the boundary condition on the crack surface is described as the following singular integral equation. ∫ ( , ) ( ) = ( ) ( < < ) ∙∙∙∙∙ (2) where ( ) ( < < ) is the dislocation density (unknown function), and ( , ) and ( ) are the known functions. The integral kernel ( , ) ( < , < ) has a singularity proportional to 1/( − ) . In order to ensure the uniqueness of the solution of the singular integral equation, the following additional condition is used. ∫ ( ) =0 ∙∙∙∙∙ (3) This equation means that the crack closes at its both ends. The transformations of variables = + 2 + − 2 , = + 2 + − 2 (−1< , <1) ∙∙∙∙∙(4) make the singular integral equation (2) into ∫ ̅( , ) ̅( ) 1 −1 = ̅( ) (−1< <1) ∙∙∙∙∙(5) where ̅( )≡ ( + 2 + − 2 ), ̅( ) ≡ ( + 2 + − 2 ), 75 2 shear modulus Poisson’s ratio =3−4