Issue 33

G. Laboviciute et alii, Frattura ed Integrità Strutturale, 33 (2015) 167-173; DOI: 10.3221/IGF-ESIS.33.21 168 I NTRODUCTION he CJP model of crack tip stresses is a modified version of the Williams crack tip stress field which takes account of simplified stress distributions that arise from the presence of a zone of plastic deformation associated with the crack flanks and crack tip, and that act on the elastic field that drives crack growth [1]. The elastic stress field responsible for crack growth is therefore controlled by the applied loading and by the induced boundary stresses at the interface with the plastic zone. The CJP model is essentially an extension of the crack tip stress model which underpins the use of the classic stress intensity factor K and that uses a ‘plastic inclusion’ approach for dealing with the stresses induced by local plasticity which is concomitant with a growing fatigue crack [1]. A localised zone of plasticity arises from crack growth mechanisms and essentially blunts the crack and creates a reversed cyclic plastic zone. In addition the compatibility requirement for displacements at the elastic-plastic boundary induces interfacial shear stresses along the crack flanks, along with the possible generation of wake contact stresses. These induced stresses act on the applied elastic stress field at the boundary of the elastic-plastic enclave surrounding the crack. This meso-scale model of crack tip stresses leads to a modified set of crack tip stress intensity factors that include the resultant influence of plastic wake- induced crack tip shielding, and which therefore have the potential to help resolve some long-standing controversies associated with plasticity-induced closure. This full-field approach has been developed for stress using photoelasticity [1] and also for displacement using digital image correlation [2]. The CJP model was initially derived for a uniaxial stress field and was then extended to deal with biaxial loading; the appropriate equations being presented at the Malaga Crack Tip fields conference in 2013 [2]. It is worth noting that Tada and Paris [3] used a similar approach in considering the influence that additional terms in the Westergaard stress function might have on fatigue crack growth when the crack surfaces are not stress-free. Their examples were focussed on specific cases of applied forces, e.g. an internal crack subject to concentrated Mode 1 splitting forces. They did not consider plastic constraint effects and were not focussing on crack tip shielding, but did give some attention to the possible presence of terms which cannot be included in the series expansion of the form given below:   2 0 1 2 2 n n K K z K z K z Z z z       (1) In the period between the Malaga and Urbino Crack Tip Fields conferences, an internet-mediated research group has been established between research groups at Plymouth (James, Christopher) and Liverpool (Patterson) in the UK and Jaen (Díaz Garrido) in Spain. The intention has been to characterise the plastic zone size and shape using sophisticated experimental techniques (e.g. thermoelastic stress analysis, electronic speckle pattern interferometry, and digital image correlation) and materials with several strain hardening exponents, and to compare this data with the predictions of the Williams crack tip stress model and the CJP model. As the CJP model contains five terms, the Williams crack tip stress expansion is considered with two terms and with five terms. Preliminary results from this work have indicated that the CJP model gives a closer correlation to the size and shape of the experimentally obtained plastic zones than the Williams model. The part of the work that will be presented in this paper has been aimed at exploring the characterisation of crack growth rate data with the biaxial CJP model. Crack growth rates have been measured with compact tension specimens that contain inclined cracks at the notch tip with initial angles of 30°, 45° and 60° to the horizontal axis. Significant experimental difficulties are experienced in growing cracks in a biaxial field under uniaxial tensile loading, as the natural tendency of the crack is to turn so that it becomes perpendicular to the maximum principal stress direction. Coupled with this tendency, the tension, bending and buckling stress field in thin compact tension (CT) specimens also lead to changes in crack direction as the crack extends. The CJP model resolves the displacement field around the crack tip to obtain stresses and hence stress intensity factors ( K F , K R and K S ) parallel with, and perpendicular to, the crack plane. K F represents the total driving force on the crack, K R the retarding influences arising from the plastic enclave and K S reflects the compatibility-induced shear stresses at the elastic-plastic interface along the complete plastic boundary. In the CJP model crack angle is not an issue, and the stress components can be rotated into directions comparable with the usual K I and K II directions and used to calculate stress intensity parameters that should be directly comparable with the standard stress intensity formulations. Another difficulty arises, however, in finding published expressions for K I and K II for CT specimens with curved or kinked cracks. This paper will present experimental crack growth rate data obtained from testing inclined notch specimens at R=0.1 and characterised using the three stress intensity factors derived from the biaxial CJP model, which is given in Eq. (2) [2]: T