А.С. Вьюн

АНАЛИЗ ЧИСЛЕННОГОРЕШЕНИЯ ЗАДАЧИ РАСТЯЖЕНИЯ ПЛАСТИНЫ
С ТРЕЩИНОЙ НА ОСНОВЕ АНАЛИТИЧЕСКОГО РЕШЕНИЯ

an interesting puzzle for fun:

Lame’s classical solution for an elastic 2D plate, with a hole of radius a and uniform tensile stress applied at the far field, gives a stress concentration factor (SCF) of two at the edge of the hole. This SCF=2 is independent of the hole radius.

Consider what happened to this concentration factor if the radius a approaches infinitely small. The SCF is independent of a, so it remains equal to two even when the hole disappears.

This is inconsistent with what one would expect physically, namely that the limit a->0 should be the same as when the plate is whole without a hole and has no stress concentration.

Henry.

Using the Griffith energy method for analysis of cavitation under hydrostatic tension we conclude that the critical tension tends to infinity when the cavity radius approaches zero (IJSS, 2006, doi: 10.1016/j.ijsolstr.2006.12.022). The conclusion is physically meaningless, of course. Moreover, if we assume that the failure process occurs at the edge of the cavity then the critical tension should be length-independent for small but finite cavities while the Griffith analysis always exhibits length-dependence. The main Griffith idea - introduction of the surface energy - is controversial because it sets up the characteristic length, say, surface energy over volume energy. By no means is this approach in peace with the length-independent classical continuum mechanics.

GRIFFITH AA, The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London, Sereis A, 221:163-198, 1921.

This is the foundational paper of fracture mechanics, and foreshadows much of the subsequent development. I urge all my students to start reading it when they take the course of fracture mechanics, and return to it for illumination later in their careers. In class, I spend several lectures just talking about this paper, uncluttered by the later refinements.