Mathematical Physics - Volume II - Numerical Methods

Chapter 6. Introduction to Computational Mechanics of Discontinua

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Figure 6.17: Schematic illustration of a prediction of crack path in shear test: (a) the geometries of the experimental setup of the Nuru-Mohammed shear-load test specimen with a double-edge-notched specimen and corresponding crack propagation patterns in a concrete slab (red). Crack propagation patterns were obtained by computational simulations using a geometrically regular triangular lattice with: (b) central interaction (Chapter 6.4.2), (c) central-angular interaction (Chapter 6.4.3), and (d) beam interaction (Chapter 6.4.4). (Reproduced based on [93]). As far as computer implementation procedures are concerned, lattice models simulate the process of damage and fracture by performing an analysis for a given load with the removal from the network of those bonds that meet the prescribed fracture criteria. In the case of beam lattices, the forces and moments are calculated using the appropriate beam theory. A global stiffness matrix is formed for the whole lattice, the corresponding inverse matrix is calculated which is then multiplied by the load vector to obtain the displacement vector. The heterogeneity of the material structure can be taken into account in different ways by: (i) assigning to the beams different tensile strengths, (ii) assuming a random distribution of cross-sectional dimensions and/or beam lengths, or (iii) mapping to beams different material properties (aggregates, cement matrix, interface layers,...; Figure 6.3b). These various types of disorder (geometrical, topological, chemical,...) could be introduced by using statistical distributions. When lattice models are used for fracture analysis, the breaking rule of the basic one dimensional structural element (that is, the rupture criterion on the micro- or meso-scale) must be defined in advance. Again, depending on the type of material that is the subject of modeling, several bond-removal criteria can be applied based on strength theory, energy dissipation, fracture mechanics. The simplest examples include those used for the lattices with central interactions

f ( b ) = f

( b ) = ε

( b ) = E

cr , ε

cr , E

(6.62)

cr ,

where the critical parameter—as indicated by subscript ( cr ) — refers to the bond axial force, elongation, or elastic strain energy, respectively. Naturally, more complex bond rupture criteria are commonly used for beam lattices, characterized by more complex stress states. This complexity necessitates the failure definition in terms of the failure envelopes that take into account contributions of all relevant deformation types. For example, van Mier and co-workers [65], [66] established