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Special Upscaling Theory, Int J Rock Mech , 45 (2008) 1102–1125.

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A Specific Upscaling Theory of Rock Mass Parameters Exhibiting Spatial Variability: Analytical relations and computational scheme

G. Exadaktylos[1] and M. Stavropoulou


Mining Engineering Design Laboratory, Department of Mineral Resources Engineering, Technical University of Crete, GR-73100 Chania, Greece.


Department of Dynamic, Tectonic and Applied Geology, Faculty of Geology and Geoenvironment, University of Athens, Greece






In this paper the aspect of the representation of geological conditions in a numerical simulation model is considered. By the expression “geological conditions” we mean the 3D volume geometry of the geological formations, the spatial variability exhibited by the rock parameters inside each of these geological volumes, and the necessary upscaling of the rock deformability and strength parameters that are determined in the laboratory from cores collected in the field. A specific theory is outlined of how to go from laboratory tests, geological information and field measurements and observations to the full-scale numerical or “ground model” that includes apart from initial and boundary conditions and ground water, the rock constitutive laws and associated material parameters for use in simulation models. The term “specific” used in the title of this paper stems from the fact that other possible approaches for the same problem may be applied; i.e. empirical rock mass classification systems, explicit modelling of joints in rock by the distinct element or finite element methods, homogenization techniques etc. The manner of taking into account the spatial variability of rock mass properties by virtue of Geostatistical Theory and 3D modelling tools is also outlined with an example case study.



[1] Corresponding author. Tel.: +30 28210 37690; fax: +30 28210 37891. E-mail address: This e-mail address is being protected from spambots. You need JavaScript enabled to view it (G. Exadaktylos).


Download this file (upscaling paper E&S paper #5.pdf)upscaling paper E&S paper #5.pdf[ ]2464 Kb

G2 theory of cracks, Computational Mechanics, 2009

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A G2 Constant Displacement Discontinuity Element for Analysis of Crack Problems





Mining Engineering Design Laboratory, Department of Mineral Resources Engineering, Technical University of Crete, GR-73100 Chania, Greece.



A new constant displacement discontinuity element is presented for the numerical solution of Mode I, II and III crack problems, based on the strain-gradient elasticity theory in its simplest possible Grade-2 (second gradient of strain or G2 theory) variant [1-6]. The accuracy of the proposed new element is demonstrated herein in a first attempt only for isolated straight cracks or for co-linear straight cracks for which closed form solutions exist. It is shown that the results based on this new element are in good agreement with the exact solutions. Moreover, the new method preserves the simplicity and hence the high speed of the constant displacement discontinuity method originally proposed by Crouch [7-9] with only one collocation point per element for plane crack problems, but it is far more efficient compared to it, especially close to the crack tips where the displacement and stress gradients are highest.


Keywords: Displacement Discontinuity, Strain-gradient Elasticity, Crack problems, Stress Intensity Factors.


[1]   Vardoulakis I, Exadaktylos G, Aifantis E (1996) Gradient elasticity with surface energy: Mode III crack problem. Int. J. Solids Structures, 33/30: 4531-4559.

[2]   Exadaktylos G, Vardoulakis I, Aifantis E (1996) Cracks in gradient elastic bodies with surface energy. Int. J. Fracture 79:107-119.

[3]   Exadaktylos G, Aifantis E (1996) Two and three dimensional crack problems in gradient elasticity. J. Mech. Beh. Mtls., 7/2:93-117.

[4]   Exadaktylos G (1998) Gradient elasticity with surface energy: Mode-I crack problem. Int. J. Solids Structures 35/5-6: 421-456 .

[5]   Exadaktylos GE (1999) Some Basic Half-Plane Problems of the Cohesive Elasticity Theory With Surface Energy. Acta Mechanica 133/1-4:175-198.

[6]   Exadaktylos GE, Vardoulakis I (2001) Microstructure in Linear Elasticity and Scale Effects: A Reconsideration of Basic Rock Mechanics and Rock Fracture Mechanics. Tectonophysics 335/1-2 :81-110.

[7]   Crouch SL (1976) Analysis of stresses and displacements around underground excavations:An application of the Displacement Discontinuity Method, University of Minnesota Geomechanics Report, Minneapolis, Minnesota, November, 1976.

[8]   Crouch SL (1976) Solution of plane elasticity problems by the displacement discontinuity method. Int. J. Numer. Meth. Engng 10:301-343.


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