Introduction

The fracture mechanics based on stress intensity factor (SIF) helped to characterize fracture in terms of the critical SIF, or fracture toughness. The application of linear elastic fracture mechanics (LEFM) became very limited for metals, in which plastic deformation preceded any crack extension. Wells (1961) experimented with variety of metals. He observed that before the onset of extension, the crack-tip blunts and there is a definite opening at the original crack-tip location. The extent of the opening is dependent on the fracture resistance of the material. The opening increases as the resistance of the material to fracture increases. He estimated the crack opening displacement (COD) at the original crack-tip location and presented the condition of fracture in terms of this parameter. This forms the basis of COD criterion of fracture mechanics. For small-scale plastic deformation at the tip, this condition is equivalent to the fracture condition in terms of Griffith potential energy release rate GI for Mode I, that is, GI = GIC, where GIC is the fracture resistance of the material.

In the presence of linear or nonlinear elastic deformation at the crack-tip, the deformation field is conservative. Stress—strain relation for a material showing plastic deformation at the crack-tip, under monotonically increasing loading, is very similar to that of a nonlinear elastic material. Provided there is no unloading or crack extension, the material obeys the deformation theory of plasticity, that is, the total strain at any stage is related to the total stress, and the relationship is path independent. Rice (1968) showed that under such an elastic (linear or non-linear) deformation of a component with a crack, there exists an integral, called J integral, which is path independent when calculated joining any two points on the opposite crack flanks. Further, this integral indicates the potential energy release rate associated with the crack extension. It can characterize the onset of crack growth in the same fashion as the SIF, but it is valid even beyond the linear elastic limit. The path independence of this parameter along with its energy release rate character is shown in this chapter. Further its graphical interpretation is also given.

It has been shown in the Chapter 2 that the fracture resistance of a material is constant for a purely elastic material.

Introduction

The foundation for the understanding of brittle fracture originating from a crack in a component was laid by Griffith (1921), who considered the phenomenon to occur within the framework of its global energy balance. He gives the condition for unstable crack extension in terms of a critical strain energy release rate (SERR) per unit crack extension. The next phase of development, which is due to Irwin (1957a and b), is based on the crack-tip local stress–strain field and its characterization in terms of stress intensity factor (SIF). The condition of fracture is given in terms of the SIF reaching a critical value, and the parameter is shown to be related to the critical energy release rate given by Griffith. Later, the scope of the SIF approach was amended to take care of small-scale plastic deformation ahead of the crack-tip. Most of the present applications of the principles of linear elastic fracture mechanics (LEFM) for design or safety analysis have been based on this SIF.

This chapter presents the gradual developments that have taken place to advance the understanding of fracture of brittle materials and other materials that give rise to small-scale plastic deformation before the onset of crack extension. Examples are presented to illustrate the applications of LEFM to design.

Calculation of Theoretical Strength

A fracture occurs at the atomic level when the bonds between atoms are broken across a fracture plane, giving rise to new surfaces. This can occur by breaking the bonds perpendicular to the fracture plane, a process called cleavage, or by shearing bonds along a fracture plane, a process called shear. The theoretical tensile strength of a material will therefore be associated with the cleavage phenomenon (Tetelman and McEvily 1967; Knott 1973).

Generally, atoms of a body at no load will be at a fixed distance apart, that is, the equilibrium spacing a0 (Fig. 2.1). When the external forces are applied to break the atomic bonds, the required force/stress (σ) increases with distance (a or x) till the theoretical strength σ c is reached. Further displacement of the atoms can occur under a decreasing applied stress. The variation can be represented approximately by a sinusoidal variation as follows.

In this chapter, we derive the balance laws by requiring that the material volume, i.e., a volume that comprises a fixed set of particles, obeys the axioms of mass conservation, balance of linear and angular momentum, and the laws of thermodynamics. The important point to note is that the choice of this material volume is arbitrary; for example, any arbitrary subset of a given material volume also constitutes a valid material volume to which the above axioms can be applied. The original statements of all the axioms are in integral form. Using the arbitrariness of the material volume yields a set of differential equations governing the field variables. The solution of these differential equations subject to appropriate boundary and initial conditions then defines the variation of each field variable with space and time.

In fluid mechanics, it is useful to derive balance laws for a control volume, which is a region of space where various flow quantities are observed. The control volume approach has the advantage that, under certain conditions, it is possible to calculate quantities such as the force exerted or the power generated or dissipated, simply by having the appropriate information at the control surface. However, the weakness of this approach is that we do not obtain the details of the various fields at every point within the control volume. For obtaining detailed information, we need to solve the governing differential equations subject to appropriate boundary and initial conditions.

As emphasized in Chapter 1, we write the governing equations in tensorial form, because then it is immediately evident that such equations are independent of the choice of coordinate system. Only while solving specific problems, we choose a particular coordinate system, and write the tensor equations in component form in that particular coordinate system.