Fractal methods in analysis of fatigue fracture dynamics

M.A.Artamonov

State Center of Flight Safety

box 54, Sh.-1, Khimki, MO, 141426, Russia

A.N.Markin

"MATI" - K.E.Tsiolkovskiy Russian State Technological University, Moscow, Russia

Fractography research of destroyed materials fractures allows receiving the important information on origin and dynamics of development of fatigue cracks, on the factors influencing development of fracture in a material, and on the mechanism of interaction of a material with environment.

However, the analysis of fracture images has mainly qualitative character. It is associated with the fact that the fracture relief is formed under influence of a set of factors including the random processes display. This complicates determination of the space-topological signs which meet a certain fracture mechanism. The qualitative analysis of fractures does not allow receiving full information on development of fracture in a material or this information is subjective. Thus for high-strength steels a relief is formed, too complex for definition of the signs responsible for crack propagation for one cycle of fatigue load of material. An operator needs a wide experience to define the dynamics of a fatigue crack growth for such materials, and reliability of received result can be prejudiced. At certain fracture mechanisms allowing for real properties of materials (creep, heat-fatigue, corrosion, etc.), there are no techniques for definition of crack growth dynamics. Complications arise also at defining the fracture mechanism.

One should also note a strong dependence of kinds of the received images on a choice of research scale. Taking pictures of a certain fracture part at different scales, we sometimes observe a drastic change of the received image. Similar dependence of a kind of the image on a choice of scale leads to application of a technique based on fractal mathematics for analysis. A fractal is a self-similar object for which a unique dependence is defined between a parameter describing an object and a scale at which this parameter was defined.

It is especially convenient to use for the analysis of fractured materials the images received with the help of a raster electronic microscope, a law of the normalized amplitude or Hurst law. Hurst demonstrated that for many natural processes a dependence between the period of analyzed process and the normalized amplitude of parameter fluctuation describing the given process was observed.

Hurst parameter, as is generally known, is associated with local fractal dimension. Thus it is possible to investigate (using Hurst law) dependencies between different physical quantities (including dependence of intensity level on coordinates) which are ensured by the digital photo of the destroyed material fracture. A special feature of Hurst law is that its application allows analyzing not only a fractal, but also a self-affine object. If for fractal object the factor of similarity for all coordinates is identical, then for a self-affine object it can differ for different coordinates. At the same time, it is possible to expect that the fractures formed by such mechanisms (as for example fatigue) are self-affine objects, but not fractal. The crack growth with such fracture mechanism is anisotropic in spatial coordinates and depends on stress distribution in a material which affects the formed fracture relief.

For the fracture images analysis a technique was developed which became a basis for a program written in programming language, working according to the following algorithm.

Thus, a technique for the quantitative description of a fracture relief is developed, based on the Hurst law of the normalized amplitude, and power dependence between Hurst parameter and the size of the fracture area, where Hurst parameter is calculated, is established. The power dependence can be used for definition of dynamics of fatigue crack growth.