The large scale behaviour of the ground is assumed to be transversely isotropic and viscoelastic. The seam is supposed to be deep and sufficiently thin, in comparison with other significant measurements, that it can be considered to be of infinitesimal thickness. The face advances at a constant rate. The resulting stress distribution and displacement field in the surrounding rockmass are computed with the aid of an electronic computer for a number of viscoelastic materials of one simple type.


On suppose qu'en grand le caractère du terrain est horizontalement isotrope et viscoelastique. La couche est d'une grande profondeur et est assez mince, en comparaison avec d'autres mesures significatives, pour qu'on puisse la considerer d'une epaisseur infinitesimale. La taille s'avance à une vitesse constante. On calcule la distribution de la contrainte et le deplacement qui en resultent dans les roches environnantes à l'aide d'un calculateur electronique pour plusieurs materiaux visco-elastiques d'un seul genre simple.


Das allgemeine Benehmen des Gebirges ist als transversal isotropisch und viskoelastisch angenornrnen. Es wird angenommen, dass das Flöz tief liegt und duenn genug ist im Vergleich mit anderen zutreffenden Abmessungen, um als unendlich duenn angenommen zu werden, Der Streb schreitet gleichmassig vorwàrts. Die daraus folgende Spannungsverteilung und das Verschiebungsgebiet in dem umgebenden Gebirge werden mit Hilfe einer elektronischen Rechenmaschine fuer eine Anzahl viskoelastischer Materialien einfacher Art berechnet.

1. Introduction

The work of the theoretical rock mechanics group at Nottingham has been directed chiefly towards the solution of problems of large-scale ground movement by the methods of continuum mechanics. The earlier work was based on the assumption that rock masses behaved elastically in most of the region of interest, but the results were compatible with the degree of subsidence found in British coalfields only if anisotropy were assumed [1, 2]. It is often observed that the full effects of underground work do not appear until sometime later and it is possible that a phenomenological explanation can be given in terms of viscoelasticity theory. If the viscoelastic solutions tend to static conditions with increasing time, the corresponding elastic solutions are indistinguishable from the final viscoelastic states. This means that elastic and viscoelastic analyses are not necessarily incompatible, but rather that one supplements the other. A first approach to the problem of an advancing longwall face in a viscoelastic medium was made by ASTIN [3], who considered the limiting case of immediate and complete closure of roof and floor behind the face.

2. Formulation of the problem

The face advances at a steady rate, c, in the x-directionand we consider a cross-section far enough from the ribsides that we may assume plane-strain. The rear of the excavation is so far behind that it does not affect the behaviour in the region of the face. This means that the system is practically in a steady state in which there is no change with time if all dependent variables are referred to axes (x - ct, y) moving with a velocity c in the x-direction. As in previous papers [1, 2, 4, 5], the seam thickness is assumed to be infinitesimal in comparison with all other dimensions of the problem, so that the convergence of roof and floor may be represented as a discontinuity, - Δ v, in the displacement in the y-direction. In the present paper the excavation is assumed to be sufficiently deep that the influence of the ground surface can be ignored. The stresses involved in the constitutive equations (1) are the changes in stress induced by the change in conditions at the level of the excavation, y= 0. Over the open interval, length d, immediately behind the face, an induced vertical stress σ y =p is required to cancel the initial (primitive) stress σ y = - p and thus leave the roof and floor traction free. There are two constants rand 8 common to each of the creep functions postulated in equation (3).


A powerful tool for the solution of viscoelastic problems is the correspondence principle[6], but it is not possible to use it directly in the present problem because the conditions on y= 0 are part stress and part displacement with the boundary between them changing with time. The problem is solved in two steps. First we find the solution for a known constant discontinuity moving with velocity c in the x-direction. Then we superpose a number of elementary solutions of this kind in order to find an approximation to the required solution which satisfies the stress condition σ y=P at a number of points on the open part of y= 0. The creep functions Skl/il(t) are ordinarily defined for t > 0 only, but for convenience we extend the definition so that dSkl/il (t)/dt= 0 for t < 0.

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