Theoretical aspects of fracture mechanics

Abstract

In this review we try to cover various topics in fracture mechanics in which mathematical analysis can be used both to aid numerical methods and cast light on key features of the stress field. The dominant singular near crack tip stress field can often be parametrised in terms of three parameters K 1, K II and K III designating three fracture modes each having an angular variation entirely specified for the stress tensor and displacement vector. This is true for cracks in homogeneous elastic media, and for cracks against bimaterial interfaces although the stress singularity is different in this latter case. For cracks lying on bimaterial interfaces the classical elastic solution produces complex stress singularities and associated unphysical interpenetration of the crack faces. These results and contact zone models for removing the interpenetration anomaly are described. Generalisations of the above results to viscoelastic media are described. For homogeneous media with constant Poisson's ratio the angular variation of singular crack tip stresses and displacements are shown to be the same for all time and the same inverse square root singularity as occurs in the elastic medium case is found (this being true for a time varying Poisson ratio too). Only the stress intensity factor varies through time dependence of loads and relaxation properties of the medium. For cracks against bimaterial interfaces both the stress singularity and angular form evolve with time as a function of the time dependent properties of the bimaterial. Similar behaviour is identified for sharp notches in viscoelastic plates. The near crack tip behaviour in material with non-linear stress strain laws is also identified and stress singularities classified in terms of the hardening exponent for power law hardening materials. Again for interface cracks the near crack tip behaviour requires careful analysis and it is shown that more than one singular term may be present in the near crack tip stress field. Relations between stress intensity factors and elastic energy release rates can be connected by means of the configurational force on a defect and the elastic energy momentum tensor. Ideas based on a ‘pseudo’ energy momentum tensor and associated invariants can be used to get useful checks on results, as an ancillary aid to numerical calculations and simple solutions to special problems. A variety of such theory and applications is presented for inhomogeneous elastic media, coupled thermoelasticity etc. Methods based on reciprocal theorems and dual functions which can also aid in getting awkward singular stress behaviour from numerical solutions are also reviewed. Finally theoretical calculations of fibre reinforced and particulate composite toughening mechanisms are briefly reviewed.