PSI - Issue 23

Hector A. Tinoco et al. / Procedia Structural Integrity 23 (2019) 529–534 H. A. Tinoco/ Structural Integrity Procedia 00 (2019) 000 – 000

530

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1. Introduction In fracture mechanics, different mechanical parameters have been proposed to correlate the crack growth behaviour either by static or cyclic loading, e.g. stress intensity factor (SIF), crack-tip opening displacement (CTOD), J -integral among others (Anderson, 1995; Caputo et al., 2013). The SIF has revealed to be more effective to evaluate the fracture in brittle materials; on the contrary case in ductile materials, the use of CTOD, J -integral and the plastic zone size (at the crack tip) are criteria more practical according to Hoh et al. (2010). Though, different researchers have pointed out that the determination of plastic zone size is the most suitable criterion for assessment of fatigue crack propagation (Yi et al., 2010; Caputo et al., 2013; Jingjie et al., 2014; Paul, 2016a). Exact analytical solutions (not approximations) are not available in the literature taking into account that the size refers to shape and dimensions. To overcome this difficulty, finite element method has been applied to estimate the plastic zones (shape and size) at the crack tip as performed by different authors (Caputo et al., 2013; Jingjie et al., 2014; Paul, 2016a; Paul 2016b; Chen et al., 2017). It indicates that the application of numerical solutions is an important tool to understand the failure mechanisms. In the present work, a finite element analysis is applied to asses CTOD and the plastic zones size of a stationary crack subjected to one and a half load cycle under plane stress conditions. Kinematic hardening is included in the material model used for a railway axle steel EA4T and a geometric strategy to construct a multiscale meshing with dimensional high resolution is presented.

Nomenclature 

Crack-tip opening displacement (CTOD) Crack opening displacement field ( Δ CTOD) Stress intensity factor for fracture mode I

y u

I K

P r

Maximum distance of plastic zone for loading conditions Maximum distance of plastic zone for unloading conditions

CP r

2. Materials and methods

2.1. Crack-tip opening displacement (CTOD) Linear-elastic fracture mechanics considers that the SIF range Δ K eff = ( K max – K min ) helps to describe fatigue crack growth in different materials ( K min = 0 for this study) . Classical fracture mechanics relationships are usually used for calculation of these quantities which are based on simplified theories that do not give accurate values. The crack tip opening displacement CTOD = δ is determined by the following relation under plane stress conditions: 2 1 , I y c K E    (1) where 1 1 c  in its classical presentation and E is the Young’s modulus. For plane stress, Eq. (1) remains as   2 2 (1 ) ( 3 ) I y v K E     . The associated monotonic plastic zone size around the crack tip is defined by the SIF and the yield stress σ y which are related by Eq. (2) for plane stress 2

1       

P 2 I y r c K  

,

(2)

where 2 1 c  in the classical theory. For plane strain, Eq. (2) can be rewritten as     2 P 1 3 . I y r K    Under cycling loading, the cyclic (also called reversed) plastic zone size can be computed as CP P 4 r r  for the stress ratio R = 0. This relation assumes a material simplification for an elastic-perfectly plastic material.