Summary and Limitations

The lattice models or particle systems are computationally very demanding and a very efficient compu­tational algorithm must be used. A highly efficient algorithm, which was applied to simulation of sea ice fracture, is presented in Jirasek and Bazant (1995a). It is an explicit algorithm for fracture dynamics, but it can also be used for static analysis in the sense of dynamic relaxation method. (Fracturing in particle systems with more than 120,000 degrees-of-freedom was simulated with this algorithm on a desktop 1992 work station.) Computational effectiveness will be particularly important for three dimensional lattices, the use of which is inevitable for obtaining a fully realistic, predictive model.

As it now stands, the lattice or particle models can provide a realistic picture of tensile cracking in concrete in two-dimensional situations. However, solution of significant three-dimensional problems or nonlinear triaxial behavior as well as simulation of behavior in which compression and shear fracturing is important is still beyond reach. Thus, the lattice or particle models have not attained the degree of generality already available with the finite element approach.

Although computer programs for lattices are attractive by their simplicity, it must nevertheless be recognized that a lattice modeling of a continuum is far less powerful than the finite element method because the stress and strain tensors cannot be simulated by the elements of the lattice and, thus, nonlinear tensorial behavior cannot be directly described. From this viewpoint, the numerical concrete of Roelfstra and Wittmann, seems preferable to the lattice model of van Mier and Schlangen because the particles and mortar are discretized by a much finer mesh of finite elements.

To sum up, lattice models or particle systems have proven to be a useful tool for understanding fracture process’and clarifying some relationship between the micro – and macro – characteristics of quasibrittle heterogeneous materials. These models appear to be particularly suited for failures due to tensile fracturing

and capture well the distributed nature of such fracturing and its localization. However, one must keep in mind that these models, in their present form, cannot simulate three-dimensional situations, larger structures (even two-dimensional), compressive and shear fractures, and nonlinear triaxial stress-strain relations. Overall, these models are still far inferior to finite element models and do not really have a predictive capability. No doubt significant improvements may be expected in the near future.