It was investigated already that the stresses in the plastic region of ground around a circular tunnel are analysed by the equations of equilibrium of stresses and the yield condition. The two partial differential equations introduced by this analysis are of the hyperbolic kind, and yield the characterisic equations on the physical r, θ plane, and on the stress plane respectively. In this theory the elastic-plastic boundary around this tunnel was approximately determined by the plotting curve of the intersecting Points of the plastic shearing stress distribution curve and the elastic snearing stress distribution curve along the slip lines (Oda & Yamagami 1976: 847–858).
The authors consider the relations between the state of strain rate and velocity, and the relation among the variation in the Volume strain rate, the variation in the shear strain rate and the angle of dilation in the plastic region around an unlined circular tunnel in soft ground. We further consider the state of strain rate in the polar co-ordinates r, θ plane, whose origin is the center of the circular tunnel. The equations governing the distribution of velocity throughout the r, θ plane are obtained from the corresponding Mohr's circle of strain rate and the above-mentioned relations.
These equations are the two partial differential equations which yield the characteristic equations on "he physical r, θ Plane, and on the velocity plane respectively. The characteristic equations are re- Placed by the linear defference relations for numerical computation of the approximate step-by-step procedure.
The boundary conditions on the elastic- Plastic boundary are the deformations calculated by the elastic theory outside the Plastic region around the tunnel. It is proved that the results of deformations in the plastic region, which are obtained by this theory, are equal nearly to the experimental deformations of model tunnel tests (Oda & Horita 1980: 825–828).
The equations governing the distribution of stress around a circular tunnel in gravitating plastic ground with frictional resistance were obtained by the equations of equilibrium of stresses and yield condition. These equations are the two partial differential equations of the hyperbolic kind, which yield the characteristic equations on the physical plane (polar co-ordinates r, θ plane), and on the stress plane respectively. The characteristic equations for the stresses are replaced by the linear difference relations for numerical computation of the approximate step-by-step procedure (Oda & Yamagami 1976: 847–858).
In seeking the equations governing the distribution of velocity in the plastic region of ground deforming at a constant angle of dilation, in accordance with the concepts of classical plasticity theory a rate of strain may be interpreted as an increment of strain and a velocity as an increment of displacement.
Consider the state of strain rate and velocity on the element refered to polar coordinates r, θ whose origin is the center of the circular tunnel as illustrated in Fig. 1(a)
