On the brittle fracture theory by Ya.~Frenkel and A.~Griffith.

V. M. Markochev ; M. I. Alymov

Čebyševskij sbornik, Tome 18 (2017) no. 3, pp. 381-393.

Résumé

The analysis of the theory of brittle fracture Frenkel. The analysis is based on the theory of catastrophes. By replacing the variables in the equation of potential energy Frenkel of the canonical reduced form of the equation of catastrophe folds. A state variable in the resulting equation of the fold is the crack length. Equating to zero first and second derivatives of the energy on the crack length, obtained critical force and critical length of crack. Critical crack length and critical load at Frenkel are independent from each other. Their values depend only on the internal of the system operating parameters — modulus of elasticity, surface energy and opening of the crack tip. It is shown that the length of the initial crack grows in the process of approach to the critical state. The resulting equation linking the length of a steadily growing crack with the external load and control parameters of the system. An attempt of modernization theories of brittle fracture Griffith based on the ideas of Frenkel. To do this in a well-known energy equation in Griffiths introduced the third member. The energy of this member is inversely proportional to the crack length. Equating to zero first and second derivatives on the crack length, obtained a system of equations. Solving this system of equations, obtained formulae for critical crack length and critical stress. The estimation of permanent member, the third member of the modernized equations Griffiths. The length of the critical crack for upgrade equation is 20% smaller than the crack length according to the classical equation of Griffith. The stable crack length in Frenkel and modernized Griffiths equation corresponds to the local minimum of potential energy. This fact virtually eliminates the singularity at zero crack length. The third member in the Frenkel equation can be interpreted as the energy of the crack opening. Thus Frankel joined the force and deformation criteria modern fracture mechanics. The Frenkel equation, which describes the critical state of a solid body with a crack that precedes the appearance of modern catastrophe theory in general and in relation to the mechanics of brittle fracture, in particular.

Détail

Keywords: fracture, brittle fracture, Griffiths theory, the theory of Frenkel, fracture mechanics, the theory of catastrophes.

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