Size Effect in LEFM
Consider now a family of geometrically similar plane cracked structures loaded in mode I. I. et do be the initial crack length and ao — 0jD the initial relative crack length. From (2.3.11), the crack growth condition, К = Кіс is fulfilled when reaches a value (initiation stress) given by
Obviously, if k(a) increases with a, then decreases after the crack starts to grow and the peak load coincides with the onset of crack growth. If, on the other hand, k(a) decreases with a, then егдг increases after the crack starts to grow and, eventually, reaches a maximum when k(a) reaches a minimum. The first case corresponds to the so-called positive geometries (Planas and Elices 1989a) and for them
CTNu – <?Ni ^ – /Ki f" 4 if fc,(a°) > 0 (2.3.16)
where k'(ao) stands for the derivative of k(a) at a — ao – For negative geometries, the peak load occurs when the crack length reaches a value ам for which k(a) goes through a minimum, thus,
In any case, since both сад and ft. u are constant for geometrically similar. structures, it turns out that the nominal strength is always inversely proportional to the square root of the size. Hence, for similar precracked structures of sizes D] and D — AD, the nominal strengths are related by
Thus, it has been generally proved that geometrically similar structures following LEFM exhibit the inverse square root size effect.