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An Analytical Solution for Axisymmetric Tunnel Problems in Elasto-Viscoplastic Media


An analytical solution is presented for the time-dependent stresses and displacements around a circular hole when it is loaded by an axisymmetric internal and far-field pressure. The material is assumed to be elasto-viscoplastic with dilatant plastic deformations according to a non-associated flow rule. Strain softening is considered by a modified St. Venant slider which is characterized by MohrCoulomb yield conditions for both the peak and the residual strengths.
The derivations outlined apply for a constant internal pressure pi. However, a flexible viscoelastic lining which is installed after a certain time may be taken into account by subdividing the time-domain into a number of discrete time intervals. After each interval the internal pressure is increased according to the increase in deformation at the excavation boundary of which the lining takes part. Likewise, the stresses must be adapted corresponding to the actual value of pi(t). Experience shows that the inaccuracies introduced with this procedure will be of the order of magnitude of a few per cent.


Fritz, Pit

Index Terms:

tunnel; rotational symmetry; analytical; rock; TunnelingGroup; rheology; analysis


It may be shown that for the elasto-viscoplastic material model presented, an optimum time of installation for the lining does not exist. This is because for an infinitely small, but positive internal pressure, numerically a stable equilibrium will be reached, even without a lining. The introduction of such a small, fictitious internal pressure for numerical reasons only seems always to be justified. Disregarding big deformations, stability will therefore, theoretically, always be guaranteed.
It seems that for any material model consisting of an arbitrary combination of Hooke, Newton and modifind St. Venant elements, no optimum time of installation can he found.

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Date published: 1984