Fracture Mechanics

If we consider an unfaulted material with a crystalline structure, such as the simple cubic cell, which is characterised on the dimensional scale by the lattice parameter ao of the order of 0.5 mn, the possibilities of ordered and bulk deformation (Fig. 1.1), at temperatures that are low compared to the melting point, correspond to the answer to the two components of a force acting on a reticular or crystallographic plane: a stress normal to this plane and a shear stress parallel to this plane. The critical values required to ensure these motions and break the interatomic bond are called the theoretical cohesive stress (or cleavage stress) and the theoretical gliding stress (or shear stress).

We consider in this chapter an isotropic homogeneous continuum in which there is a geometric discontinuity at rest. The discontinuity is also said to be static or stationary and is subjected to an increasing load. The case of the discontinuity in motion will be treated afterwards.

We consider in this chapter a homogeneous isotropic continuum body in which there is a notch or a crack at rest and which is subjected to a monotone increasing load. The case of a crack in motion will be treated hereafter.

With the aim of evaluating the risk of brittle fracture of structural components with the possible existence of laws, a first approach proposed by Pellini was to consider the mechanical energy required to break specimens and representative components. This approach is not developed here, though it is very interesting for illustrating real behaviour. The second approach, which is the mechanics applied to fractures, also called fracture mechanics, allows the quantification of the fracture risk. This takes into account three aspects:

In Chapter 1, the theoretical cohesive stress of the material on the scale of the crystalline lattice, i.e., on the scale of nm, has been estimated from the theoretical surface energy. This energy for some metals is given in Table 5.1. The value is of the order of 10^-5daJ • cm ^-2.In Chapters 2 and 3, it was emphasised that the fracture theories based only on this theoretical stress and on purely elastic behaviour were insufficient and in fault, because for microcracks and cracks, a plastic deformation that appears around the crack normally for physical reasons is to be taken into account. This plastic behaviour will have to particularly explain why energies of the order of G _IC = 0.5daJ • cm ^-2 (with K _IC = 100M Pa \( \sqrt m \) for a steel) are associated with fracture.

When we consider as a whole the behaviour of a metallic material during a tensile test, two points of view are to be considered: On the one hand the nature of the material and on the other hand the deformation range (Fig. 6.1). On the one hand, as already presented in Chapter 4, the materials belong either to the category of low or medium strength, with a high hardening rate, or to the category of high strength materials and with low strain hardening rate. On the other hand, we are interested either on the whole deformation range, but with a natural correction for the necking phenomenon, and then the elastic range is considered as negligible and the material can be considered as fully plastic with or without strainhardening, or we are mainly interested in the beginning of deformation, i.e., by low strains, and the materials considered as an elasto-plastic material.

In Chapter 6, the considered material possessed a fully plastic behaviour under monotonically increasing loading or equivalently a non-linear elastic behaviour. In this chapter, to this plastic component is added the linear elastic component. The material is said to present an elastic-plastic behaviour. And for this new material, like for the preceding materials, linear elastic and non linear elastic, we try

The first part of Chapter 4 presented toughness for an overall elastic behaviour of the cracked component. This toughness is given in terms of a critical value K _lC or under the representation of a crack growth resistance curve K — R, these quantities being amenable of an experimental determination. The effect of thickness was presented: for great thickness, toughness appears as a characteristic of only material and for small thickness, toughness is a function of material and thickness. Here toughness is described in the general case by utilising the microscopic models of fracture that were treated in Chapter 5 and the stress and strain fields that were described in Chapters 6 and 7. Moreover some assumption has already been introduced in Chapter 3 in the case of brittle fracture with the a-priori assumption of a critical tangential stress. This postulate is here more thoughtfully evaluated with a complete analysis of mixed mode fracture, which allows a discrimination between the postulated basic fracture criteria.