Loss of uniqueness of numerical solutions of the borehole problem modelled with enhanced media

Abstract

It is well known that, initial boundary value problems involving constitutive equations modeling the degradation of the strength of materials are not well posed, which renders computations questionable. To overcome this issue it is necessary to enhance the models by incorporating some internal length. It has been shown that such an enhancement restores the objectivity of the computation as spurious mesh dependency is avoided. However, at least for simple problems (e.g. one dimensional ones), it has been proven that uniqueness of the underlying mathematical problem itself is not restored. Moreover numerical modeling of element tests yields several solutions. This paper demonstrates that several numerical solutions can be obtained also for less simple problems, namely the borehole problems. Even when a defect is introduced in the computed problems, different numerical solutions are found. Contrary to the one dimensional problem there is no proof that this loss of uniqueness comes from the underlying mathematical problem. It is our opinion that this is an inherent property of initial boundary value problems where, broadly speaking, strong degradation of the mechanical properties is modeled. In any case, it is necessary to be aware of this issue. For problems involving constitutive equation modeling strength degradation, it is important to try to find other solutions than the one obtained by using routinely a numerical code. The failure patterns of the different solutions found are however similar to experimental observations. This possible loss of uniqueness can then be seen as a counterpart of the difficulties encountered when attempting to reproduce experiments. This is crucial when dealing with geomaterials.