Vol 22, No 1 (2019)

This brief review takes a look at our joint research with Prof. G.I. Barenblatt and at some outcomes of his interdisciplinary initiatives. The research covers the issues of self-similarity in fatigue fracture, jump-like growth of fatigue cracks, and damage accumulation on different scales.

The famous Wieghardt, Griffith and Irwin criteria predict the onset of fracture in linear elastic materials. They have to be supplemented by appropriate criteria for predicting the path that the fracture will follow until the failure of the structure. These require the knowledge of the stress and displacement fields at the front of propagating fracture which depend on the actual loading on the structure and its boundary conditions. In this paper we shall review these fields in brittle and quasi-brittle materials. In the latter materials, a traction-free fracture front often has a large process zone ahead of it in which the material experiences progressive softening. Such a mixed traction-free and process zone is also called a cohesive crack. Over the process zone the material is able to transfer some tractions across the crack faces depending upon how much the faces have separated or slid relative to each other. In the famous Barenblatt model the process zone was very small in comparison with the traction-free crack so that the actual traction-separation relationship in the process zone was not explicitly involved. However, in real quasi-brittle materials the size of the process zone can be commensurate or even larger than the traction-free crack. It is therefore necessary to know this relationship explicitly in order to determine the corresponding stress and displacement fields at the front of the propagating cohesive crack. The asymptotic fields at the front of a crack in brittle materials were obtained by Williams and those for quasi-brittle materials by Xiao and Karihaloo.

As limiting behaviors of Eshelby ellipsoidal inclusions with transformation strain, crack solutions can be obtained both in statics and dynamics (for self-similarly expanding ones). Here is presented the detailed analysis of the static tension and shear cracks, as distributions of vertical centers of eigenstrains and centers of antisymmetric shear, respectively, inside the ellipse being flattened to a crack, so that the singular external field is obtained by the analysis, while the interior is zero. It is shown that a distribution of eigenstrains that produces a symmetric center of shear cannot produce a crack. A possible model for a Barenblatt type crack is proposed by the superposition of two elliptical inclusions by adjusting their small axis and strengths of eigenstrains so that the singularity cancels at the tip.

Fracture nanomechanics is the study of the interconnected process of the growth and birth of cracks and dislocations in the nanoscale. In this paper, it is applied to superplasticity and fatigue of metals and other polycrystalline materials in order to derive the basic equations describing some main features of these phenomena, namely, the fatigue threshold and the enormous neck-free superplastic elongation. It is shown that in most metals and alloys the fatigue threshold is greater than one per cent of the value of fracture toughness. Using the concepts of fracture nanomechanics, we study the superplastic deformation and fracturing of polycrystalline materials under uniaxial extension and calculate the neck-free elongation to failure in terms of strain rate, stress and temperature. Then, we determine the optimum strain rate of the maximum superplastic elongation in terms of temperature, creep index and other material constants. Further, we estimate the critical size o f ultrafine grains necessary to stop the growth of microcracks and open way to the superplastic flow, and find the superplastic deformation of grains, their maximum-possible elongation and the activation energy of superplastic state. Also, we introduce the dimensionless A-number in order to characterize the capability of different materials in yielding the superplastic flow. A t a very high elongation the alloying boundary of grains proves to be broken by a periodical system o f dead fractures of some definite period. It is shown that experimental results of the testing of the Pb-62% Sn eutectic alloy and Zn-22% A l eutectoid alloy at T = 473 K have substantially supported the theory of superplasticity advanced herewith.

The main focus of the paper is on similarity methods in application to solid mechanics and author's personal development of Barenblatt's scaling approaches in solid mechanics and nanomechanics. It is argued that scaling in nanomechanics and solid mechanics should not be restricted to just the equivalence of dimensionless parameters characterizing the problem under consideration. Many of the techniques discussed were introduced by Professor G.I. Barenblatt. Since 1991 the author was incredibly lucky to have many possibilities to discuss various questions related to scaling during personal meetings with G.I. Barenblatt in Moscow, Cambridge, Berkeley and at various international conferences as well as by exchanging letters and electronic mails. Here some results of these discussions are described and various scaling techniques are demonstrated. The Barenblatt- Botvina model of damage accumulation is reformulated as a formal statistical self-similarity of arrays of discrete points and applied to describe discrete contact between uneven layers of multilayer stacks and wear of carbon-based coatings having roughness at nanoscale. Another question under consideration is mathematical fractals and scaling of fractal measures with application to fracture. Finally it is discussed the concept of parametrichomogeneity that based on the use of group of discrete coordinate dilation. The parametric-homogeneous functions include the fractal Weierstrass-Mandelbrot and smooth log-periodic functions. It is argued that the Liesegang rings are an example of a parametric-homogeneous set.