Crack kinetics in an isotropic plate of variable thickness

O. Dolgov1, I. Dolgova2, D. Kolosov1

Dnipro University of Technology, Dnipro, Ukraine

2 Prydniprovska State Academy of Civil Engineering and Architecture,Dnipro, Ukraine

Coll.res.pap.nat.min.univ. 2020, 60:207-216


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Purpose. Study of the thickness variability effect of a plate with crack on a brittle fracture kinetics.

Research methodology. An isotropic plate with a crack is modeled by a homogeneous double-cantilever beam, split by the forces applied at its ends. Griffith's theory is used as a destruction criterion.  By replacing variables, the equation of energy balance of a moving crack is represented as a dovetail catastrophe. In the case of a resting crack, the canonical fold-type catastrophe is investigated. Further study of the kinetics of destruction carried out by using the elementary theory of catastrophes. For a special case of constant thickness of a plate or crystal, the conclusions obtained are compared with the known results of other authors.

Research results. In the framework of the linear theory of brittle fracture, the Griffiths energy balance equation is obtained for a homogeneous isotropic beam of linearly variable thickness with a crack in the middle surface, taking into account its edges motion kinetic energy in the direction perpendicular to the front of crack. The influence of changes in the beam thickness on the critical length of a quasistatic and unstable crack is estimated. The dependence of the initial dynamic crack length on the thickness change of the double cantilever beam is obtained.

Scientific novelty is in establishing and analyzing the dependencies of the stationary and dynamic brittle cracks critical parameters in a plate of variable stiffness (thickness), modeled by a double cantilever beam. New results based on analytical dependencies suitable for practical use are obtained.

Practical value. The results can be used to assess the strength of elements of real structures and their safe operation under quasistatic loading. Relatively simple final formulas for determining the critical lengths of cracks can be useful in the analysis of crack growth resistance characteristics, experimentally determined on the samples of variable cross section.

Keywords: brittle fracture, crack, stability, Griffith’s criterion, double cantilever beam, Mott theory, variable thickness, catastrophe theory.


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