Dyskin, AV & Caballero, A 2008, 'Mesh Scalability Concept for Explicit Simulation of Rock Failure', in Y Potvin, J Carter, A Dyskin & R Jeffrey (eds), SHIRMS 2008: Proceedings of the First Southern Hemisphere International Rock Mechanics Symposium
, Australian Centre for Geomechanics, Perth, pp. 185-194, https://doi.org/10.36487/ACG_repo/808_45
Rock failure is a phenomenon which involves the initiation and propagation of (multiple) fractures and the development of surfaces of strain localisation. Direct numerical simulation of these phenomena by introducing the relevant constitutive behaviour in finite elements (or similar computational elements, e.g. particles in the discrete element method) lead to mesh dependence: the computations with successively refined meshes do not stabilise. The mesh dependence is caused by the fact that theoretically, fractures and strain localisation surfaces have zero thickness, while in numerical calculations this thickness is finite controlled by the size of the finite element or particle, or by a characteristic size of the non-local functions in mesh-free methods. Two methods are currently used to overcome this problem. The first is the introduction of a generalised continuum which contributes its own characteristic lengths. The computations stabilise as soon as the size of the computational element becomes considerably smaller than the smallest characteristic length of the continuum. The use of generalised continua as numerical stabilisers leads however to hard questions as to what type of the continuum is to be chosen and how to determine its numerous parameters. In the second method a size of the mesh element is chosen which corresponds to the size of microstructural elements (if this size in known). In this approach the flexibility of choosing computationally efficient element size is lost if the microstructural elements are very small compared with the characteristic size of the geometry to be modelled (e.g. the excavation size) the modelling may become computationally prohibitive, exactly as in the case of particle methods. To deal with mesh dependence we propose the concept of mesh scalability. We use the assumption that, if the element size is sufficiently small, all mesh-dependent quantities should depend upon the size as power functions. The mesh with this property is called scalable and is regarded as sufficiently fine for the simulations. The obtained exponent and the pre-factor can be used to extrapolate the mesh dependent quantities to the actual microstructural sizes. Furthermore, the method permits the identification of stress singularities from finite element computations with different element sizes in similar meshes. The pre-factors are then called the mesh stress intensity factors and can be used to analyse failure instead of the ones referred to in the Linear Fracture Mechanics. We consider examples of application of this philosophy to modelling the stability of rock slopes.
Barenblatt, G.I. and Botvina, L.R. (1980) Application of the similarity method to damage calculation and fatigue crack growth studies. Defects and Fracture, G.C. Sih and H. Zorski (editors), Martinus Nijhoff Publishers, pp. 71–79.
Bažant, Z.P., Belytschko, T.B. and Chang, T.P. (1984) Continuum theory for strain softening J. Engg. Mech. ASCE, 110, p. 1666.
de Borst, R., Sluys, L.J., Mühlhaus, H.-B. and Pamin, J. (1993) Fundamental issues of finite element analyses of localisation of deformation, Engineering Computations 10, pp. 99–121.
Gelikman, M.B. and Pisarenko, V.F. (1989) About the self-similarity in geophysical phenomena. In: Discrete Properties of the Geophysical Medium, Nauka, Moscow, pp. 109–130.
Klein, P.A., Foulk, J.W., Chen, E.P. Wimmer, S.A. and Gao, H.J. (2001) Physics-based modelling of brittle fracture: cohesive formulations and the application of meshfree methods. Theoretical and Applied Fracture Mechanics, Vol. 37, pp. 99–166.
Pasternak, E., Dyskin, A.V. and Mühlhaus, H.-B. (2001) Stress and failure localization associated with sliding in layered materials, Bifurcation and Localization in Geomechanics. H.-B. Mühlhaus, A.V. Dyskin & E. Pasternak (editors), Swets & Zeitlinger, Lisse, pp. 271–278.
Regenauer-Lieb, K. and Yuen, D.A. (2000) Quasi-adiabatic instabilities associated with necking processes of an elasto-viscoplastic lithosphere. Physics of Earth and Planetary Interiors, Vol. 118, pp. 89–102.
Severyn, A. and Molski, K. (1996) Elastic stress singularities and corresponding generalized stress intensity factors for angular corners under various boundary conditions. Engineering Fracture Mechanics, 55, No. 4, pp. 529–556.
Sluys, L.J., de Borst, R. and Mühlhaus, H.-B. (1993) Wave propagation, localisation and dispersion in a gradient dependent medium, Int. J. of Solids and Structures, 30, pp. 1153–1171.
Tang C.-A. (1997) Numerical simulation of progressive rock failure and associated seismicity. Int. J. Rock Mech. Min. Sci. 34, No 2, pp. 249–261.
Zosimov, V.V. and Lyamishev, L.M. (1995) Fractals in wave processes. Physics-Uspekhi, 38 (4), pp. 347–384.