A large variety of solid materials - rocks, asphalt, plastics, biomaterials etc. – possess rheological/viscoelastic characteristics, which causes delayed and damped tending towards the equilibrium state that may last for years around an opened tunnel. Rheology also provides explanation why dynamic deformation constants of rocks are larger, sometimes 100% larger, than static ones, which influences evaluation of laboratory measurements as well. A practical utilization of rheology is the Anelastic Strain Recovery (ASR) method, which determines underground in situ stress via measuring the rheological relaxation of borehole rock samples and extrapolating back to the initial in situ state.
In classical continuum theory, Volterra's principle is a long-known method to solve linear rheological problems derived from the corresponding elastic ones. Here, we introduce and present another approach – an exact analytical method – that is simpler to apply (no operator inverse is required to compute but only linear ordinary differential equations to solve). Our method is based on the corresponding elastic solutions assumed to be already known: the elasticity coefficients are replaced with time dependent functions, which are determined from the rheological equations. After introducing the methodology, we show concrete examples, which not only illustrate the method but also demonstrate its power and limitations, and reveal that rheology is not only about delay and damping but may cause unexpected temporary behaviour, too. Then we discuss how our solutions contribute to both qualitative and quantitative understanding of time dependencies around tunnels.
In discretisation based numerical methods for rock mechanical problems, solutions may considerably depend on the resolution of the applied discretization, and calculation times are large for complex problems. Analytical solution of a simplified version of the problem may provide a reasonable first approximation, can give useful insight, and can be utilized for validating numerical methods.
Rheological behaviour of solid media is well-known in civil engineering and in mine industry. A hollow opened in an underground stone block often takes its eventual shape only years after the drilling (see Fig. 1). Accordingly, rheological behaviour must be taken into account when designing technical devices and facilities. This means not only disadvantages and problems but also benefits: for example, one can rely on its effect of damping and absorbing vibrations. Similarly, a freshly opened underground tunnel may need initially only a temporary – relatively weak – support, and the eventual support is enough to be established only later, when most of displacements have already been occured, allowing thus a much cheaper eventual support. All these, as well as the ASR method, need reliable calculations, nevertheless.
