Spalling due to dynamic action on bodies with plane-parallel faces

Abstract

The neighborhood of the free surface of a medium and the interfaces between different media are very liable to spalling fracture when they interact with stress waves. At present, owing to the increased popularity of drilling and blasting, it is becoming more and more urgent to study the breaking action of stress waves in a medium and to investigate the stability of the free surfaces of the solid rock -the contours of workings, the sides of quarries and pillars, etc. It is very important to know how to estimate the character and extent of the state of stress of the solid rock near such surfaces: this will enable us to preserve the contours of workings or to estimate correctly the possible fractures. The main features of these phenomena can be investigated by taking as an example a body with plane-parallel faces, the simplest case being a plate. The spallation of the backs of plates is of independent interest. The literature contains much experimental material on the laws of variation of the state of stress in a real material and on the actual pattem of possible breakage; there are various approaches to the analytical solution of the problem -the acoustic versions, quasistatic estimation, and exact solutions of the stress field [1][2][3]. This article gives a numerical investigation of the stress fields in an elastic plate and discusses the nature of the spallation in it.For a numerical solution of the problem it is convenient to adopt the following scheme [4,5]. From the equations of motion in the Cauchy form in cylindrical coordinates (r, O, z), ~9ar (rr --O0 Orrz := p c)~'u 0-7 + ~ + Oz ~7 ~' a6 z a~ tJTrz Tr....~z _--~ :: p Or + r or" (1) and from Hooke's law, 2it ()it. o~ --=).e + ' "o7' oo ..... ; oz-=...; Ou u am e:=a7 + -7-+ -aT, (au rr,=F -bE + 0r]'(3)(where X, p, and p are the Lam6 parameters and the density of the medium, o r , %, o z, rrz, and u and w are the components of the stress tensor and the displacement vector respectively, and t is the time) we eliminate the displacements. The equations must be differentiated with respect to r or z so that on the fight-hand sides of the superposition of the resulting relations we can formulate the second derivative with respect to time of any component of the stress tensor, as we find from Eqs.(2), (3), and (1). As a result we get the equation (T-' Crz = (~, -l-" " a2frz ~, o2ffr ~, O (o" r --o'0) [, ~ z~t)~ +_ Or" + 7 Or ~. do r 02rrz 2 (~, ;-It) Orrz (4) + r-~7 + 2 (~ + P) b-7"~ + r oz and similar ones for the other two normal components of the stresses. A distinguishing feature of Eq. (4) is that, as well as the second derivativeofo z with respect to time, it contains only the derivative of the same function with Institute of Mining, Novosibirsk.

A numerical investigation of the state of stress and fracture of a plate subjected to dynamic action

“…Relating the possibility of fracture of the medium to the appearance of tensile stress zones, we can expect the appearance of different fracture mechsnisms depending on the properties of the material and the influence of the different waves [4,7]. These are as follows: fractures near the front surface behind the s' and k' wave fronts, internal fractures near the symmetry line behind the transverse wave front, rear scabbing fractures on reflection of the longitudinal wave triangular profile from the free benurl~ry, and, finally, fractures of the splitting type on the rear side, due to tension in the horizontal component.…”