166 
MR. A. A. GRIFFITH ON 
There is, however, an important exception to this statement. If the body is such 
that a crack forms part of its surface in the unstrained state, it is not to be expected 
that the spreading of the crack, under a load sufficient to cause rupture, will result in 
any large change in the shape of its extremities. If, further, the crack is of such a 
size that its width is greater than the radius of molecular action at all points except 
very near its ends, it may be inferred that the increase of surface energy, due to the 
spreading of the crack, will be given with sufficient accuracy by the product of the 
increment of surface into the surface tension of the material. 
The molecular attractions across such a crack must be small except very near its 
ends ; it may therefore be said that the application of the mathematical theory of 
elasticity on the basis that the crack is assumed to be a traction-free surface, must 
give the stresses correctly at all points of the body, with the exception of those near 
the ends of the crack. In a sufficiently large crack the error in the strain energy so 
calculated must be negligible. Subject to the validity of the other assumptions involved, 
the strength of smaller cracks calculated on this basis must evidently be too low. 
The calculation of the potential energy is facilitated by the use of a general theorem 
which may be stated thus : In an elastic solid body deformed by specified forces applied 
at its surface, the sum of the potential energy of the applied forces and the strain energy 
of the body is diminished or unaltered by the introduction of a crack whose surfaces are 
traction-free. 
This theorem may be proved* as follows : It may be supposed, for the present purpose, 
that the crack is formed by the sudden annihilation of the tractions acting on its surface. 
At the instant following this operation, the strains, and therefore the potential energy 
under consideration, have their original values ; but, in general, the new state is not 
one of equilibrium. If it is not a state of equilibrium, then, by the theorem of minimum 
energy, the potential energy is reduced by the attainment of equilibrium ; if it is a state 
of equilibrium the energy does not change. Hence the theorem is proved. 
Up to this point the theory is quite general, no assumption having been introduced 
regarding the isotropy or homogeneity of the substance, or the linearity of its stress- 
strain relations. It is necessary, of course, for the strains to be elastic. Further 
progress in detail, however, can only be made by introducing Hooke’s law. 
If a body having linear stress-strain relations be deformed from the unstrained state 
to equilibrium by given (constant) surface forces, the potential energy of the latter is 
diminished by an amount equal to twice the strain energy.f It follows that the net 
reduction in potential energy is equal to the strain energy, and hence the total decrease 
in potential energy due to the formation of a crack is equal to the increase in strain 
energy less the increase in surface energy. The theorem proved above shows that the 
former quantity must be positive. 
* The proof is due to Mr. C. Wigley, late of the Royal Aircraft Establishment. 
I A. E. H. Love, ‘Mathematical Theory of Elasticity,’ 2nd ed., p. 170. 
