Issue 33

J.M. Vasco-Olmo et alii, Frattura ed Integrità Strutturale, 33 (2015) 191-198; DOI: 10.3221/IGF-ESIS.33.24 192 F UNDAMENTALS OF THE MODEL FOR CHARACTERISING CRACK TIP DISPLACEMENTS FIELDS JP model is a novel mathematical methodology developed by Christopher, James and Patterson [6] based on Muskhelishvili’s complex potentials [7] which considers that the plastic enclave that exists around a fatigue crack tip and along the crack flanks will shield the crack from the full influence of the applied elastic stress fields. Crack tip shielding includes the effect of crack flank contact forces and a compatibility-induced interfacial shear stress at the elastic-plastic boundary. Fig. 1 illustrates schematically the forces acting at the interface of the plastic zone and the surrounding elastic field material. F Ax and F Ay are the reaction forces to the applied remote load. This load generates the T-stress and the corresponding force F T . The above forces cause a plastic zone at the crack tip, deforming permanently the material and inducing the forces F Px and F Py during unloading. F S is the induced force by the interfacial shear at the elastic-plastic boundary of the crack wake. F C is the force generated when the plastic wakes contact during unloading, so the crack closes under the action of the elastic field. Figure 1 : Schematic idealisation of the forces acting at the elastic-plastic boundary [6]. According to this model, the crack tip displacements field can be described as follows:       1 1 1 2 2 2 1 1 2 2 1 1 1 2 2 2 2 2 2 4 2 ln 4 2 ln 4 ` ln 2 4 C F G u iv B E z z Ez z z C F z B E z Ez z C F Az Dz z Dz z                                         (1) Where   2 1 Y G    is the shear modulus, Y and ν are the Young’s modulus and the Poisson’s ratio of the material respectively, 3 1       for plane stress or 3 4     for plane strain, z is the complex coordinate and A , B , C , D , E and F are unknown coefficients. Thus, this new model employs four parameters to characterise the stress and displacement fields around the crack tip generated by the forces as indicated in Fig. 1. Where K F is an opening mode stress intensity factor, K R a retardation stress intensity factor, K S shear stress intensity factor and T the T-stress. K F is characterised by the driving crack growth force F A generated by the remote load: C