Models and Phenomena in Fracture Mechanics

A crack is made up of empty space, but it is so sharp that it easily cuts glass, rock and metal. Fracture mechanics studies just this phenomenon. The main question in fracture mechanics is whether the crack is stable or how does it grow under given conditions. Usually this question is answered by comparing the crack-tip asymptotes of the stresses with critical values in accordance with a criterion of fracture.

Geometrically, a crack is an inner material surface, the crack surface, split into two unconnected crack faces (Fig. 1.1) . Note the difference between the crack surface area, or simply the crack area, and the area of the crack faces which is twice as much. For twodimensional problems, the crack length is considered instead of the crack area. The crack faces can interact with each other by normal stresses to prevent interpenetration and by tangential stresses through frictional contact.

Usually the Fourier transform of a function, f (x), —∞ < x < ∞, is defined as This definition imposes a restriction on the original, f(x). The transformation in (2.1) exists if the original has a bounded variation, i.e. if a constant, C, exists, such that Idfl < C. However, this condition is too restrictive. In the problems considered in this book, unbounded functions will be met as, for instance, the crack opening displacement. Besides, generalized functions will also be encountered. To enable the use of Fourier transforms in these cases, it is necessary to adopt a generalized definition, a version of which is considered below.

Consider a linear partial differential or integral-differential equation with no explicit dependence on the x-coordinate and time, t. This uniformity also concerns homogeneous boundary conditions posted on a cylindrical surface of an arbitrary cross-section (if the cross-section of the considered waveguide is finite). In other words, the equation considered, together with additional conditions, admits translation symmetry respective to these variables (the substitutions , where a and b are arbitrary real constants, do not change the equation) . In the following, the boundary is assumed to be unable to accumulate energy. Otherwise, the boundary must be included in the bulk of the waveguide. For example, in a beam on a Winkler foundation, the energy of the wave is a sum of the strain and kinetic energies of the beam and the strain energy of the foundation, and this fact is reflected in the equation for the beam.

Although linear elasticity is not intended for a high strain gradient, that is high strain and rotation, it is the main model in fracture mechanics. Linear elasticity provides the basis for the determination of the global (macrolevel) energy release and of the stress field outside a vicinity of the crack tip where it is not valid. In linear elastic fracture mechanics, crack equilibrium and crack propagation are considered on the basis of linear elasticity comparing the global energy release rate or the stress intensity factors with the corresponding critical values. In turn, these latter values reflect the influence of the fracture process zone; they can be obtained experimentally or by calculations based on other material models. Here a homogeneous isotropic elastic medium is considered.

Two states of the body are considered. One is called the initial state (this usually means the unstressed body state), while the other is the actual state, that is the stressed body state. Note that from a geometrical point of view this is a conditional definition; however, it becomes important when physical relations are considered.

Some materials, or materials under some conditions, exhibit a time-dependent response to a given stress or strain. This can be caused by a change in the material’s properties and/or by viscosity. A linear one-dimensional stress-strain dependence is used to illustrate this.

Elastic-plastic fracture is characterized by a dramatic difference in the stress and strain fields corresponding to fixed and growing cracks. Plasticity results in a decrease of the body stiffness and hence it leads to increased strains under a given loading of a body with a fixed crack. However, for a growing crack the plasticity influence is just opposite. Roughly speaking, as the crack passes a material point, the stress circumscribes a closed trajectory in the space of stress components. When the crack is approaching the point, the stress increases and when the crack is leaving the point, the stress decreases and vanishes. Under such a stress cycle, the elastic-plastic strain circumscribes an open trajectory. In this process the plastic work can be as large as the strain amplitude is high. In other words, in contrast to the elastic case, crack growth in a stressed elastic-plastic body is accompanied by energy dissipation within a layer where plasticity occurs. This creates a resistance to crack growth. At least for a perfect elastic-plastic material the resistance appears to be completely defined by dissipation in the bulk of the body and there is no energy release through the moving singular point. As a result, solutions corresponding to fixed and moving cracks appear to be radically different.

In this Chapter, along with the unbounded elastic medium, the classical thin plate and its flexural interaction with an incompressible fluid is also considered. Some of the elastic dynamic relations used here are contained in Sect. 3.4.

Let a through-the-thickness crack arise in a plate subjected to a bending moment. In this case, crack closure arises with crack face interaction. This results in in-plane deformation in addition to the bending. Thus, in general, crack closure leads to a coupled plane-bending problem. Besides, the determination of the contact forces seems to be the subject of a three-dimensional problem.

In this Chapter, one-dimensional dynamic problems are considered related to the propagation of a phase transition wave with a negative jump in the elastic modulus at its front. From both physical and mathematical points of view, such a process in a structured material model has much in common with the crack propagation in lattices considered above.

Consider a conservative dynamic system suddenly loaded by a force, P, which then remains invariable: P = P ₀ H(t), where P ₀ = const. As a rule, the maximum dynamic displacement related to this force (the displacement in the force direction at the point where the force is applied), u _max , exceeds the static value, u _static which is assumed here to exist.