# FRACTURE MECHANICS

Fracture mechanics deals with the behavior of materials and structural members when cracks and failures develop in them, and with the development of methods for calculating crack growth rate and reducing residual strength. Fracture mechanics determines the critical size of cracks and cavities in a material admissible at given loads, and the time of crack growth from a certain initial size up to the critical one.

A material fails by either rupture or shear. The theoretical estimation of ultimate stress of the material for a simultaneous rupture or shear of the neighboring atomic crystal planes yields the quantity of the order of K/2π or G/2π, respectively, where K is the compression modulus and G is the shear modulus. The real strength of crystal solids is about one or two orders of magnitude lower. This fact was clarified by A.A. Griffith in 1921 in terms of an energy approach. Comparing the increment in the body elastic energy and the crack surface energy with increasing crack length, he derived the stability criterion of the crack in a brittle material exposed to tensile stress

where E is Young's modulus, a the crack length, and g1c the constant known as the critical velocity of energy release. At a stress higher then σc the crack in the body becomes unstable and catastrophically grows. Thus, the low strength of materials is accounted for by Griffith's cracks of micron size.

The energy fracture criterion (Griffith's criterion) is a necessary, but insufficient condition for the crack growth. In particular, if the curvature radius at the crack tip exceeds by far an atomic radius, the stress concentration appears insufficient for the rupture of atomic bonds. This crack grows at higher stresses than follows from the energy balance. Measuring the work of developing a new crack surface, we arrive, even for brittle materials, at the values at least an order of magnitude higher than the specific surface energy. The analysis shows that the crack development is practically always brought about by plastic deformation at its tip. The work of plastic deformation is so great that the energy fracture criterion is determined by this value rather than the specific surface energy.

Fracture is significantly affected by various stress concentrators such as pits and grooves in the material and the environmental factors, e.g., temperature and chemical processes, in corrosion. It has been revealed that the characteristics of material strength such as yield limit and ultimate stress are adequate for calculating structural members, but insufficient in case there is the probability of crack nucleation.

The analysis of the stress field in an elastic or an elastoplastic body with a crack reveals that in the vicinity of the crack tip stresses grow. The stress field around the crack tip is expressed in the generalized form in terms of the so-called stress intensity factor. For instance, for a through-thickness cleavage crack 2a in length in an infinite elastic plate

Here τ and θ are the polar coordinates. The stress field at the crack tip is completely determined given the stress intensity factor K1. Attainment of the critical value K1c causes failure. If K1c is known from the results of tests of the specimen with a preset size of a crack, it is possible to calculate the strength of the same material with cracks of any size or to determine the limit of the admissible crack size at a given load. K1c is a measure of a material's crack resistance known as a fracture toughness in the planar-strained state. The lower the fracture toughness, the lower is the admissible crack size in a material.

High-strength materials commonly possess a low fracture toughness. Fracture in these materials is studied by the methods of linear elastic fracture mechanics (LEFM). The materials with a low yield limit are commonly characterized with a high viscosity. The failure of these materials leads to a large-size plastic zone at the crack tip in comparison to the crack size and LEFM cannot be used in this case. The concept of crack opening is used for these materials. The crack is assumed to propagate if plastic deformation at the crack tip achieves the maximum admissible value. The deformation at the crack tip can also be expressed in terms of opening which is a measurable quantity.

A subcritical slow growth of a cavity or an incipient crack can occur under cyclic loading (a fatigue crack) or as a result of corrosion cracking under stress and by other mechanisms. The rate of growth of fatigue and corrosion cracks and, hence, the time of fracture are determined by the stress intensity factor.

An important role in physics and fracture mechanics is played by cold brittleness, which consists in transition of materials from a plastic to a brittle state and involves a drastic decrease in fracture toughness. The cold-brittle metals capable of turning brittle at low temperatures include in the first place metals and alloys with a bcc crystal lattice, e.g., iron, α—Fe-based steels, tungsten, and molybdenum, and with a hcp cristal lattice, e.g., cadmium and magnesium. Metals with a fee crystal lattice such as copper, aluminum, and nickel and austenitic steels, do not show any signs of cold brittleness due to a weak dependence of yield limit on temperature and due to a large difference between yield limit and ultimate stress.

When structural members are under load, specifically at elevated temperatures, there is a temporal dependence of strength: a long-term load application reduces the strength of solids the more, the longer the time of loading. In the moderate temperature range the relation between the life (the time until fracture) tD and the stress σ is described by the empirical equation

where A and a are the temperature-dependent constants. The expression

is used at higher temperatures for high-temperature alloys. The widely known thermofluctuation concept of strength treats the fracture of solids as a successive rupture of atomic bonds due to thermal motion of atoms. The role of stresses in this model is to endow the bonds in the dissipation processes, reversible in an unloaded body, a preferred orientation resulting in accumulation of ruptures. The energy of thermal fluctuations is expended on overcoming the potential barriers in a crystal lattice. The life of the material under loading is described by

where U0 is the magnitude of the energy barrier at σ = 0, the activation volume γ depends on local overstresses, and the preexponential factor t0 is of the order of 1012 – 10−13 s.

The material's strength in a microsecond range of load durations is studied by analizing cleavages. On reflection of the shock compression momentum from the free surface of the body it generates tensile stresses which may give rise to a failure or to a cleavage in the body. Under conditions of specimen loading by plane shock waves deformation is one-dimensional and the stressed state in cleavage approaches the state of three-dimensional tension. A short-term loading restricts the spread of information on failure development in separate portions of the specimen. In distinction from static rupture, failure under these conditions is not initiated in a single weakest site, but in a large number of sites, it is scattered throughout the body and develops through crack and void growth and merging. The development of cleavages until complete fracture of the body into fragments may last for a relatively long time compared to the loading time. The process can stop, depending on the duration of initial loading, at different phases: nucleation of microdiscontinuities on inclusions, their growth and joining together, development of the main crack, and separation of the cleavage element. The resistance to fracture in cleavage is commonly 1.5—3-fold higher than the real rupture stress under quasistatic conditions.