With the assumption that the material properties are given, hexagonal elements can be created and the linear or non-linear behavior in them can be considered. Since the elements are considered to be small enough, the isotropic case is taken into account, i.e. each element behaves homogeneously and isotropically; the material characteristics are prescribed by the modulus of elasticity and Poisson's ratio, for example, in the linear case. Possible dislocations obey the damage conditions being described by the generalized Mohr-Coulomb law and by the exclusion of tensile stress that exceeds the tensile strength along the interfaces.

The principal ideas of the classical DEM are adopted here: a two-dimensional domain defining the continuum (both the tunnel lining and surrounding rock) is covered by non-overlapping elements (grains, particles) of a hexagonal shape (generally various shapes of hexagons may be taken into account). In this paper, a generalized Hooke's law is used, involving eigenparameters. Hence, the free hexagon element method presented here involves coupled modeling of stress, pore pressure and a change of temperature, all of them being assigned to the particles, and other geotechnical material parameters (the angle of internal friction, the shear strength or cohesion, the tensile strength) are allocated on the interfacial boundary of the elements. It is worth noting that if most particles are of the same shape an algorithm leads to very powerful iteration procedures, as the stiffness matrix can be stored in the internal memory of the computer only once, and during the iteration process only unknown displacements, spring stiffnesses and tractions along the interfaces change. The number of particles can then be comparable with that appearing in the PFC or UDEC, for the same computer time consumption. Also parallel computation speeds up the iterations, by placing it on a distributed net of computers according to a chain rule.

The free hexagon method is established in Reference [1] and applied to the occurrence of bumps in deep mines. It appears that the method can be used for the solution of other problems in many other branches of civil and underground engineering.

1

P.P. Prochazka, "Application of discrete element methods to fracture mechanics of rock bursts", Engineering Fracture Mechanics, 71, 601-618, 2004. doi:10.1016/S0013-7944(03)00029-8

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