Conditions which are decisive for the stability of deep underground excavations are investigated in a numerical study. First a nonassociated elasto-plastic constitutive relation for brittle materials is derived. It is shown that an originally load controlled boundary value problem is equivalent to a displacement Controlled boundary value problem. As an application circular - and rectangular cavities in an isotropic stress field are investigated.
Les conditions decicives pour la stabilite des cavites très profondes ont ete etudiees par les etudes numeriques. D'abord on decrive une loi des conditions elasto-plastiques non associatives pour des materiaux cassants. On demontre comment un probleme aux conditions aux limites originalement controlle par la charge est equivalent à problème aux conditions aux limites controlle par le deplacement. Comme application on etude des cavites circulaires et rectangulaires dans un champ de tension isotropique.
Anhand numerischer Studien wird ueberprueft, unter welchen Bedingungen tiefliegende Hohlraume in sprödem Gestein versagen können. Zunachst wird ein nichtassoziiertes elasto-plastisches Stoffgesetz fuer sprödes Gestein hergeleitet. Es wird gezeigt, wie urspruenglich lastparametergeregelte Randwertprobleme in aquivalenter Weise verschiebungsgeregelt formuliert werden können. Als Anwendungen werden kreisförmige und rechteckige Untergrundhohlraume in einem isotropen Spannungsfeld behandelt.
One is frequently confronted with the problem of stability of deep underground excavations in brittle rock in connection with geophysical explorations or energy production.
In this paper constitutive relationships are derived for the desription of the mechanical behaviour of strongly coherent rocks such as granite, sandstone etc. On the basis of this constitutive model the stability of circular and rectangular cavities is studied by means of the finite element method.
It was pointed out by EGGER (1973) in an analytical study that the most important aspect of the material behaviour with respect to the stability of cavities is the law that describes how the material looses coherence. In the present paper this property will be considered in an elasto-plastic constitutive relation which allows the representation of frictional hardening and simultaneous cohesion softening. In finite element solutions in softening material behaviour one is confronted with two major problems:
Load control in the incremental solution is not possible because the load-displacement relation is not monotoneous in general. This means in a tunnel problem that the intensity of the support pressure cannot be used as a control parameter in a finite element solution because in softening material the support pressure first decreases to a certain threshold level and has to be increased to maintain stability then. By applying the so called arc-length algorithm for Solution of nonlinear systems of equations this problem can be overcome (RIKS, 1979; CRISFIELD, 1985; de BORST, VERMEER, 1984). In arc-length algorithm the increments in externally prescribed loads or displacements do not have to be definded by the user. An increase or decrease of the loads is determined by the algorithm in such a way that e.g. a certain norm of the incremental displacements do not exceed a specified value. Alternatively, as will be done here, the surface deformation which is conjugate in energy to the surface pressure can be used as a control parameter (provided this surface deformation is monotoneously increasing). In an tunnel problem this means that the volume change of the tunnel is prescribed instead of the support pressure.
The finite element solution becomes strongly mesh dependent. One of the reasons for this is the tendency of the displacement gradients to localize in thin shear bands. However, if the classical continuum theory is underlain the thickness of the shear bands is undetermined. To remedy this desease, it has been proposed to make the softening modulus dependent on the element size (PIETRUSCZAK, MROZ, 1981; WILLAH et al. 1984, 1986). A physically more evident approach is to underlay a generalized continuum theory as e.g. the Cosserat theory. In such a continuum the thickness of shear bands is defined by certain material parameters depending on the dimension of length (Grain diameter etc.) (BESDO 1985; MUHLHAUS 1986a, b; MUHLHAUS, VARDOULAKIS, 1986).
In this article we content with the classical continuum description. It will be shown that in the case of the boundary value problems considered here the essential properties of the solution are mesh independent. In the following section a constitutive relation is defined. Subsequently it is shown that an originally load controlled boundary value problem can be made displacement controlled. In the fourth section the stability of the surface of a rectangular cavity in an infinite plate is considered. The in-plane principle stresses at infinity are assumed to be equal. To get an idea of the influence of the shape of the cavity on the stability, the case of a circular cavity is also considered.
