The authors present a closed-form solution for determining the plastic zone around a circular tunnel in an elastoplastic halfspace obeying the Mohr–Coulomb criterion. The elastic stress field is first derived in the bipolar coordinate system, and the problem then reduced to find the elastic–plastic interface by imposing the continuity of stresses between the elastic and plastic zones. However, we have a number of comments on the theoretical analysis used in the paper.
First the authors claim that at the elastic–plastic interface the normal stress in the α direction can be expressed by equation (14b), which is of the same form as equation (13), with ri and di replaced by rc and dc, respectively. This statement is not correct because equation (13) is derived from the fact that the shear stress ταβ = 0 at the tunnel periphery α = αi and hence, the Mohr–Coulomb yield criterion at this point is satisfied by the two principal stresses σα and σβ as shown in equation (11). At the elastic–plastic interface αc, where αc = f(β) is a function of β to be determined, however, there is no guarantee that the shear stress will also vanish. In this case the Mohr–Coulomb criterion should be written as
For the same reason, the stress in the elastic region can no longer be obtained from equations (9) and (10) by simply replacing αi and Pi with αc and Pc respectively. Actually, the coefficients of the stress function for the elastic zone, after taking the shear stress into account, become
Obviously A1, B1, C1 and B0 are relevant to both the pressure Pc(αc, β) (Pc(αc, β) = −σα(αc, β)) and the shear stress τc(αc, β) (τc(αc, β) = ταβ(αc, β)) along the elastic–plastic interface. Now substituting equations (25a)–(25d) back into the expression for the stress function S/J, the normal stress σβ at the elastic–plastic interface can be expressed as a function of Pc, τc, αc and β, say σβ(αc,β)=F(Pc,τc,αc,β). By combining this with the yield criterion equation (24), one obtains
This is the desired equation governing the normal and shear stresses along the elastic–plastic interface that should be used to replace the incorrect one, equation (14b), in the paper.
The second point is the derivation of the stress components in the plastic zone. In the paper the authors present two simple expressions for σα and σβ, equations (15a) and (15b), which contain only one unknown A, but they have not given the general differential equation of equilibrium in the bipolar coordinate system. According to the formula for the tensor divergence in orthogonal curvilinear coordinates (Malvern, 1969), the equilibrium equations in the plastic zone can be written as follows
These equations, together with the yield criterion (24), are in principle sufficient to determine σα, σβ and ταβ in the plastic zone. Unfortunately, they will be very difficult, if not impossible, to solve mathematically. One certainly could not expect that the solutions for the plastic stresses take the particularly simple form like equations (15a) and (15b). Note that equations (24), (27) and (28) constitute a system of first-order partial differential equations; the general solutions of σα, σβ and ταβ thus should include at least two arbitrary constants.
Third, the authors assert that their closed-form solution should be confirmed by way of comparison with the analytical solution for the limiting case of a very deep tunnel. It is true that the plastic zone and plastic stress distribution around the tunnel calculated using the authors' method will reduce to the well-known formula corresponding to a circular tunnel in an infinite space, when the parameters κ and d approach infinite values. However, such accordance occurs just by chance and may not have a deeper physical cause. As is evident, as κ, d → ∊fty, the bipolar coordinate system (α, β) actually reduces to the conventional polar coordinate system (r, θ). Whereas in this special case, the shear stress ταβ = τrθ will always be equal to zero owing to the symmetry of the problem. It is thus not surprising that the general solution proposed by the authors comprises the case of a very deep tunnel. However, as discussed above, the whole solution procedure in the paper must be incorrect as the authors have completely missed the shear stress ταβ in both the elastic and plastic regions.
Finally, we think that the last term (A1cos h2α+ C1sin h2α) (1−2sin hαcos β) given for κσβ in equation (10) should be A1cos h2α+C1sin h2α−2(A1sin h2α+C1cos h2α)sin hαcos β.
M. El Tani, Lombardi SA, Minusio, Switzerland
On the ahead-of-print web page of Géotechnique, a paper by Massinas & Sakalleriou retained my interest. The authors have found an analytic solution for an excavated circular tunnel in an elastoplastic material using the Mohr–Coulomb law. They use the Mohr–Coulomb law without verifying that the stresses should be principal stresses, which is incorrect. I have found that the shear stress is not zero and is
This means that the stresses they are using are not the principal stresses. It is possible that I have missed something, but I cannot find in the paper any indication that the authors are using the principal stresses. Unless I am wrong in this respect, their paper has no value.
Authors' reply
The authors are grateful to Chen & Abousleiman and to El Tani for their comments, which we respond to as follows. In order to derive a closed-form solution for determining the plastic zone around a circular tunnel in an elastoplastic half-space obeying the Mohr–Coulomb criterion, a bipolar coordinate system (α, β) was adopted and the differential equations of equilibrium were derived in the α and β directions. In order to derive a solution for the specific geometric problem of the half-space we made the assumption that the trajectories of the principal stresses coincided with the bipolar coordinate system inside the plastic region, and thus the shear stress ταβ was taken to be equal to zero. In order to satisfy the continuity of stress components at the elastic–plastic interface the shear stress ταβ must also be taken as zero.
The discussion article by Chen & Abousleiman has given us the opportunity to restate our hypothesis and, therefore to avoid any misconceptions about the validity of our solution. Others have previously used a bipolar coordinate system, and assumed that the principal stresses' trajectories and the bipolar coordinate system coincide inside the plastic regions (e.g. Grigoriev, 1968) in order to derive closed-form solutions for plasticity problems in a half-space. The existence of the straight boundary on the half-space seems to be responsible for the co-identity of the principal stresses' trajectories and the bipolar coordinate system. Using this assumption we have been able to derive a solution that gives very good results in comparison with the results from numerical analysis using the finite difference method (see finite difference examples that are shown in the original paper).
The scope of our work was to compare the derived solution with finite difference examples, as well as to examine the overburden-effect problem in the plastic zone shape according to Bray's approach. In both cases the results proved to be efficient. The hypothesis that was made about the principal stresses' trajectories is supported by the results of numerical analyses.
The fact that in the deep tunnel case, α and β circles of the bipolar coordinate system tend to be concentric circles (α) and straight radial lines (β), is due to our solution procedure and assumptions, and not by chance as the authors of the discussion article claim. The implementation of logical assumptions and hypotheses does not lead to random mathematical solutions. The best way to validate such solutions is either through experimental or computational methods (e.g. finite difference examples), which proved to be efficient.
In light of the above, it is clear that our solution is correct, and as such also valid for the deep tunnel case. In this case, where the bipolar coordinate system tends to be polar, the trajectories of the principal stresses inside the plastic zone tend to be perpendicular and tangential to the concentric circles of the polar system. Therefore the shear stress τaβ (α-concentric circles, β-straight radial lines) inside the plastic zone is equal to zero, a conclusion also reached by Kachanov (1971) and Hill (1950).
In summary, it is evident from our analyses results that the existence of a straight boundary on the half-space seems to affect the direction of the principal stresses inside the plastic zone.
We refer Chen & Abousleiman to the differential equation of equilibrium that we used in order to derive the plastic stresses. Please note that this differential equation is the presented equation (27) in the discussion article and for the case of ταβ=0 takes the following form
By combining this with the yield condition (11) and after solving, equations (15a) and (15b) are derived that give the plastic stresses σαpl and σβpl respectively. Please find below the solution procedure
We therefore cannot agree with Chen & Abousleiman's comments on our theoretical analysis. The results, for the case where the trajectories of the principal stresses coincide with the bipolar coordinate system inside the plastic region, are proved to be efficient. Thus our initial hypothesis is logical.
Please note that the last term
in equation (10) is incorrect in the paper. The correct term is the following
Turning now to the discussion by El Tani, we note that he has omitted the term after the fraction in equation (30). Before the initial yielding, the shear stress in the half-space is derived by differentiating equation (7) according to equation (6c), which gives the following expression (and not the equation that is presented by El Tani)
where for α=αi (at the tunnel's periphery) the shear stress is equal to zero and the stresses are principal. Furthermore, after initial yielding, where a plastic zone αc is formed, by adopting our hypothesis of coincidence of the principal stresses' trajectories and the bipolar coordinate system, the shear stress ταβ inside the plastic region is taken to be equal to zero. As a result of this, at the elastic part of the half-space the shear stress is given by the following equation
where Pc is given by equation (14b) and is the critical value that limits further extension of the plastic zone. At the elasic–plastic interface the above equation (31) gives τaβ=0, satisfying in that way our assumption about the principal stresses' trajectories inside the plastic zone.
