Keywords: visco-plastic behaviour, creep, polycrystalline materials, long-term behaviour, material model.

Time-dependent behaviour of geomaterials plays an important role in stability of deep underground structures as trans-alpine tunnels, open underground galleries, waste disposal structures, etc. The conditions that may cause the loss of stability are, among others, viscous rock, deviatoric stress state, water income resulting in swelling, etc. The examples of existing tunnels in Europe built in creeping rocks and problems they display, provide a motivation for more sophisticated study of the time-dependent phenomena in rock environment for future tunnel projects.

The long-term behaviour of a crystalline rock has been modelled as a visco-plastic problem. The Perzyna types of material models which are based on so-called overstress concept have been used. Particularly, the visco-plastic model proposed by Lemaitre and Chaboche for metals and Bingham model have been exploited. The Lemaitre-Chaboche model showed to be suitable also for other materials, as polycrystalline rocks as anhydrite or rate sensitive geomaterials as clays. The model contains three parameters (viscosity parameter, stress exponent and strain hardening parameter), the Bingham model is based on one viscosity parameter.

The finite element method has been used for solving mentioned problem. Numerical solution is based on velocity formulation of equilibrium conditions because the presented problem is time-dependent. Explicit method has been applied to the equilibrium conditions and results into three basic phases. The overstress, the increment directions, the value of viscous function and the right hand side are computed in the first phase of the algorithm. The second phase contains solution of equation system. The last phase contains same steps as the first one plus computation of strains, stresses and internal variables increments. All computations were carried out on software developed at our department.

The Lemaitre viscoplastic model was tested on simple one dimensional problem. The effect of unloading on onedimensional problem was observed as the stress drops below the yield limit. The simulation of a laboratory condition creep test was an object of the second example. The example is computed as a plane strain problem. The twodimensional quadrilateral finite elements with bilinear approximation functions have been used. Response of the structure has been obtained by proposed algorithm.

The Bingham viscous model proved to be very efficient tool for testing the algorithm for viscoplastic problems. The sensitivity on viscous parameter was shown. The Lemaitre model for long-term viscoplastic behaviour of rocks was implemented into finite element code. The example of specimen subjected to constant load was simulated and axial displacement evolution in time obtained. The Lemaitre model is less stable that Bingham model. It is very sensitive on hardening parameter and stress exponent. The sensitivity on viscous parameter was found similar to that for Bingham model.

1

Plesek, J., Korous, J., "Integration Method with Time Step Control for Viscoplasticity and Creep", In B.H.V. Topping, editor, Computational Techniques for Material, Composites and Composite Structures, pages 127-135, Civil-Comp Press, Edinburgh, 2000. doi:10.4203/ccp.67.1.15

2

Cormeau, I., "Numerical Stability in Quasi-static Elasto/visco-plasticity", Int. J. Numer. Methods Eng., 9, 109-127, 1975. doi:10.1002/nme.1620090110

3

Olszak, E., Perzyna, P., "On elastic-viscoplastic soils, Rhéologie et mécanique des sols", Symposium Franco-Polonais, Grenoble, Springer-Verlag, 1964, pp.47-57

4

Lemaitre, J., Chaboche, J.-L., "Mechanics of solid materials", Cambridge University Press, U.K., 1990, 556 pp.

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