Fracture Mechanics

The mechanical design of engineering structures usually involves an analysis of the stress and displacement fields in conjunction with a postulate predicting the event of failure itself.

Fracture mechanics is based on the assumption that all engineering materials contain cracks from which failure starts.

The linear elastic analysis of the stress field in cracked bodies, dealt with in the preceding chapter, applies, strictly speaking, only to ideal brittle materials for which the amount of inelastic deformation near the crack tip is negligible. In most cases, however, there is some inelasticity, in the form of plasticity, creep or phase change in the neighborhood of the crack tip. A study of the local stress fields for the three modes of loading showed that they have general applicability and are governed by the values of three stress intensity factors. In other words, the applied loading, the crack length and the geometrical configuration of the cracked bodies influence the strength of these fields only through the stress intensity factors. We can have two cracked bodies with different geometries, crack lengths and applied loads with the same mode. The stress and deformation fields near the crack tip will be the same if the stress intensity factors are equal.

When a solid is fractured new surfaces are created in the medium in a thermodynamically irreversible manner. Material separation is caused by the rupture of atomic bonds due to high local stresses. The phenomenon of fracture may be approached from different points of view, depending on the scale of observation.

When a solid is fractured, work is performed to create new material surfaces in a thermodynamically irreversible manner. In Griffith’s theory of ideally brittle materials, the work of fracture is spent in the rupture of cohesive bonds. The fracture surface energy \( \gamma \), which represents the energy required to form a unit of new material surface, corresponds to a normal separation of atomic planes. For the fracture of polycrystals, however, the work required for the creation of new surfaces should also include: dissipation associated with nonhomogeneous slip within and between the grains; plastic and viscous deformation; and possible phase changes at the crack surfaces. The energy required for the rupture of atomic bonds is only a small portion of the dissipated energy in the fracture process. There are situations where the irreversible work associated with fracture is confined to a small process zone adjacent to the crack surfaces, while the remaining material is deformed elastically. In such a case the various work terms associated with fracture may be lumped together in a macroscopic term R (resistance to fracture) which represents the work required for the creation of a unit of new material surface. R may be considered as a material parameter. The plastic zone accompanying the crack tip is very small and the state of affairs around the crack tip can be described by the stress intensity factor.

A number of investigators have proposed the mathematical formulation of elastostatic conservation laws as path independent integrals of some functionals of the elastic field over the bounding surface of a closed region. For notch problems, Rice [1] introduced the two-dimensional version of the conservation law, a path independent line integral, known as the J-integral. The present chapter is devoted to the theoretical foundation of the path independent J-integral and its use as a fracture criterion. The critical value of the opening of the crack faces near the crack tip is also introduced as a fracture criterion.

So far, we have studied growth of a crack only for the case when the load is applied normal to the crack and such that the crack propagates in a self-similar manner.

The analysis of crack systems considered so far concerned only quasi-static situations in which the kinetic energy is relatively insignificant compared with the other energy terms and can be omitted. The crack was assumed either to be stationary or to grow in a controlled stable manner, and the applied loads varied quite slowly. The present chapter is devoted entirely to dynamically loaded stationary or growing cracks. In such cases rapid motions are generated in the medium and inertia effects become important.

It was first realized in the middle of the nineteenth century that engineering components and structures often fail when subjected to repeated fluctuating loads whose magnitude is well below the critical load under monotonic loading.

The phenomenon of fracture of solids may be approached from different viewpoints depending on the scale of observation. At one extreme is the atomic approach where the phenomena take place in the material within distances of the order of 10⁻⁷ cm; at the other extreme is the continuum approach which models material behavior at distances greater than 10⁻² cm. In the atomic approach, the problem is studied using the concepts of quantum mechanics; the continuum approach uses the theories of continuum mechanics and classical thermodynamics. A different approach should be used to explain the phenomena that take place in the material between these two extreme scales: movement of dislocations; formation of subgrain boundary precipitates, slip bands, grain inclusions and voids. The complex nature of the phenomenon of fracture prohibits a unified treatment of the problem, and the existing theories deal with the subject either from the microscopic or the macroscopic point of view. Attempts have been made to bridge the gap between these two approaches.

Fiber reinforced composite materials have gained popularity in engineering applications during the past decades due to their flexibility in obtaining the desired mechanical and physical properties in combination with lightweight components. For this reason they are now being widely used for aerospace and other applications where high strength and high stiffness-to-weight ratios are required. These materials are usually made of glass, graphite, boron, or other fibers embedded in a matrix. Modeling the mechanical and failure behavior of fiber composites is not a simple task. The materials are heterogeneous and have several types of inherent flaws. Failure of fiber composites is generally preceded by an accumulation of different types of internal damage. Failure mechanisms on the micromechanical scale include fiber breaking, matrix cracking, and interface debonding. They vary with type of loading and are intimately related to the properties of the constituents, i.e., fiber, matrix and interface/interphace. While the above failure mechanisms are common in most composites, their sequence and interaction depend on the type of the loading and the properties of the constituents. The damage is generally well distributed throughout the composite and progresses with an increasingly applied load. It coalesces to form a macroscopic fracture shortly before catastrophic failure. Study of the progressive degradation of the material as a consequence of growth and coalescence of internal damage is of utmost importance for the understanding of failure.

Thin layers of dissimilar materials are used in many modern technologies in order to achieve specialized functional requirements. Problem areas include protective coatings used for thermal protection, or for abrasion, oxidation and corrosion resistance, electronic packaging, magnetic recording media, multiplayer capacitors, layered structural composites and adhesive joints.

Nanoindentation constitutes a powerful method for measuring the mechanical properties and the interface fracture toughness of thin films on substrates. In the case of a brittle film weakly bonded to the substrate nanoindentation can be used to delaminate the film from the substrate. The interface toughness is obtained from measurements of the applied load, delamination length, film thickness and film/substrate material properties.

Linear elastic fracture mechanics (LEFM) has been successfully used since 1950s for the safe design of engineering materials and structures.

Experimental methods have extensively been used in fracture mechanics problems. A number of nondestructive testing methods for the detection, location and sizing of defects have been developed.

Numerical methods have extensively been used in solving fracture mechanics problems.