Determination of stress intensity factors for elements with sharp corner located on the interface of a bi-material struc
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O R I G I NA L PA P E R
Grzegorz Mieczkowski
Determination of stress intensity factors for elements with sharp corner located on the interface of a bi-material structure or homogeneous material
Received: 14 July 2020 / Revised: 30 September 2020 / Accepted: 2 October 2020 © The Author(s) 2020
Abstract This paper presents a new universal analytical and numerical method allowing for determination of stress intensity factors (SIFs) for notches and cracks located in both a homogeneous and a heterogeneous material. An advantage of the proposed method is that it does not require knowledge of an analytical description of singular stress/displacement fields or connection of SIFs to energy parameters such as energy release factor. In this method, a universal analytical function has been used, which in combination with data obtained using finite element method (FEM) allows for a direct determination of stress intensity factors. One parameter that is necessary to know when using the proposed method is the eigenvalue λ. The characteristic equation allowing for determination of the eigenvalues for any corner has been given herein. What is more, also a criterion clearly defining the nodes is determined, from which FEM numerical results are implemented to the developed analytical function. In order to verify the proposed method, for a selected group of geometrical and material structures with sharp corner, values of stress intensity factors were determined and compared to data available in the literature. Satisfactory compliance of obtained results with literature ones was found. 1 Introduction With the determination of durability and strength of structural elements with sharp notches and cracks, some difficulty is involved. According to theoretical solutions based on elasticity theory, around the concentrator tip, values of both stresses and strains go ad infinitum. Use of the regular method involving comparison of these values to permissible stresses or strains is pointless, and thus a completely different approach is required. Therefore, for forecasting the durability and strength of materials and structural elements with material defects, methods based on fracture mechanics are used. Determination of critical service loads most frequently requires knowledge of stress fields around the concentrator tip area where a local stress concentration is formed. In the literature, analytical solutions can be found which are based on elasticity theory, where particular components of stresses (in a polar coordinate system located in the concentrator tip, Fig. 1) are described by N N the following equation—σi j (r, ϕ) = n=1 K In r λn −1 f iIj (ϕ) + n=1 K IIn r λn −1 f iIIj (ϕ) where f iIj (ϕ), f iIIj (ϕ) are known eigenfunctions including boundary conditions of the issue, λn —eigenvalues, n—number of terms of the asymptotic solution, K In , K IIn —stress intensity factors (SIFs). The description of mechanical fields most frequently requires the use of the first power series term (n = 1, λn = λ1 = λ, K In = K I1 = K I , K IIn = K II1
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