PSI - Issue 17

Michal Vyhlídal et al. / Procedia Structural Integrity 17 (2019) 690–697 Vyhlídal et al./ Structural Integrity Procedia 00 (2019) 000 – 000

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2.1. Linear elastic fracture mechanics

Linear elastic fracture mechanics (LEFM) deals with the study of stress and displacement field in the vicinity of cracks for the case of homogenous, isotropic and linearly elastic material, i.e. Hook’s law is valid. Another assumption is that the size of the plastic zone is small compared to the body dimensions and crack length – small scale yielding. Stress and strain fields can be obtained by solving the biharmonic partial differential equation (1) including the conditions of stress-free edges, see e.g. Anderson (2005). ( 2 2 + 1 ⋅ + 1 2 ⋅ 2 2 ) ( 2 2 + 1 ⋅ + 1 2 ⋅ 2 2 ) = 0 (1) where Φ is Airy stress function and r, θ are the polar coordinates. Best known solutions are by the complex variable technique Westergaard (1939) and by infinite power series Williams (1957). Stress field in the close vicinity of the crack tip according to Williams (1957) is described by followi ng equation (Williams’s expansion): ij = ∑ ( n ⋅ 2 ) ∞ =1 ⋅ n 2 −1 ⋅ ij ( , ) , (2) where σ ij is the stress tensor component, A n are terms of Williams ’s expansion and f ij (n, θ) is shape function. In the vicinity of crack tip ( → 0 ) higher order terms of the infinite series can be neglected and the stress and strain fields are represented only by the first (singular) term and the second (constant) term. Components of the stress tensor are given by the superposition of three basic failure modes – opening (mode I), in-plane shear (mode II) and out-of-plane shear (mode III) – Irwin (1957): →0 ij = I √2 ij I ( ) + II √2 ij II ( ) + III √2 ij III ( ), (3) where i for i = I, II, III are stress intensity factor in loading modes I, II, III. The values K i are ascertained from a numerical solution of the studied geometry, materials and boundary conditions. For cracks in homogeneous media, procedure for K i determination is usually included in the FEM software (e.g. Ansys, Inc. Software), while for general singular stress concentrators various direct or integration methods are used, see Ping et al. (2008), Klusák et al. (2008) or Profant et al. (2008). The most important and predominant failure mode in engineering experience is the opening mode (mode I). This dominance is due to several reasons, but the most important is that, unlike other two failure modes, the crack loaded by mode I propagates in its own plane and there are also no frictional forces between crack surfaces – see Karihaloo (1995). According to the above reasons, we will pay attention to pure mode I. 2.2. Criterion of stability based on average stress ahead of the crack tip One of the well-known LEFM conditions of stability says that a crack initiation occurs if the stress intensity factor K I reaches its critical value K I,c . Critical value K I,c is also called fracture toughness and is the material constant, see e.g. Anderson (2005). Another theory says that a crack will propagate in the direction where the tangential stress σ θθ is maximal. This maximum tangential stress (MTS) criterion was established by Erdogan and Sih (1963) and seems to be suitable especially in the case of brittle fracture failure.