An analysis of the conditions for rupture due to griffith cracks

Abstract

The solutions for the problem of an infinite isotropic elastic solid stressed under tension T₀ and containing a single internal crack of length c on the plane z=0 are given in a form suitable for the computation of the stresses and displacements at all points. These are used to find the stress distribution on, and the displacements of, the plane situated ½a from the plane containing the crack. The normal stress σ_z on z=½a (as found above) is plotted as a function f(2u_z) of the normal displacement u_z and τ_rz is small compared with σ_z. A model is used in which the crack is considered to be bounded by the atoms centred on the planes z=±½a, these planes being the boundaries of two semi-infinite elastic solids. Equilibrium is maintained by postulating that an attractive force, f(z), acts between the atoms of these bounding planes when they are z+a apart. It is found that f(z) approximates to the law of force expected from atomic considerations, and the condition for unstable equilibrium of the crack, i.e. a value T₀^c of T₀ such that for T₀<T₀^c the crack closes (c decreases), and for T₀>T₀^c the crack spreads (c increases), is found. The surface energy is calculated from the results and the equilibrium condition is found in a form similar to that of Griffith. Agreement is found with the experimental results of Griffith. In the absence of the tension T₀, the crack cannot be maintained without an inclusion to prevent closing. Possible physical models are discussed.