A Discrete Direct Formulation for Nonlinear Structural Analysis

The paper deals with a discrete formulation that makes use of primal and dual meshes. While the first mesh is needed to introduce a compatible set of displacements, the second one is utilised to enforce equilibrium by taking into account the forces concerned with each polygon (dual cell) of this mesh: body forces acting on the cell, applied external surface loads and tractions exchanged with contiguous cells. Typical meshes can be generated by considering the Delaunay network (a set of triangles) and the Voronoi tessellation (a set of polygons). Possible applications to basic elastic-plastic problems are briefly discussed. Next, some emphasis is given to the possibility of modelling crack propagation through a nontraditional approach. In fact, the particula pattern of the Voronoi tessellation (dual mesh) provides a discrete number of zig-zag paths, since the interface course (where equilibrium conditions must be satisfied) tends to turn left and right, up and down, repeatedly in a sharp way. Therefore, interfaces can be recognized as potential crack paths and a convenient criterion can be introduced in order to define critical loads beyond which cracks start to propagate. Finally, some numerical examples are briefly discussed in order to show possible applications of the proposed procedure.