## Tangential Stiffness Tensor Via Solution of a Body with Many Growing Cracks

The power of the microplane model is limited by the assumption of a kinematic (or static) constraint, which is a simplification of reality. To avoid this simplification, one needs to tackle the boundary value problem of the growth of many statistically unifonnly distributed cracks in an infinite elastic body. This problem is not as difficult […]

## 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 […]

## Examples of Results of Particle and Lattice Models

Bazant, Tabbara et al. (1990) used the random particle system described before (Figs. 14.4.la-b and 14.4.2) to simulate tensile tests and bending tests on notched specimens. A similar model was used by Jirasek and Bazant (1995a, b) to relate the microscopic features of the model (such as the softening curve and the statistics of strength […]

## Directional Bias

An important aspect of the model is the generation of the lattice configuration. In many works regular lattices have been used. However, recently Jirasek and Bazant (1995a), and also Schlangen (1995) Figure 14.4.5 Failure patterns for various values of a: (a) 0°, (b) 22.5°, (c) 45° using a regular lattice with random strength, elastic stiffness […]

## Truss, Frame, and Lattice Models

The simplest model is a pin-jointed truss, in which only the center-to-center forces between the particles are considered (Fig. 14.4.la-b, Bazant, Tabbara et al. 1990). A more refined model is that ofZubelewicz. and Bazant (1987), which imagines rigid particles separated by deformable thin contact layers of matrix that respond primarily by thickness extension-contraction and shear […]

## Particle and Lattice Models

A large amount of research, propitiated by the advent of powerful computers, has been devoted to the simulation of material behavior based directly on a realistic but simplified modeling of the microstructure —its particles, phases, and the bonds between them. A spectrum of diverse approaches can be found in the literature spanning an almost continuous […]

## Nonlocal Adaptation of Microplane Model or Other Constitutive Models

In unconfined straining, the microplane model displays softening. Therefore, localization limiters of some kind must be used to avoid spurious localization and nresh sensitivity, as for all other models with strain softening. This can be easily implemented using a nonlocal adaptation of the microplane model in which the inelastic stress increment is made nonlocal following […]

## Other Aspects

To check for the limit of stability and for bifurcations of the response path, the tangential stiffness matrix is needed. The microplane model does not provide it directly, but it can always be computed by incrementing the strain components (or the displacements) one by one and solving for the corresponding stress changes with the microplane […]

## Vertex Effects

There is another important property that is exhibited by the microplane model, and not, for example, by macroscopic plasticity models. For a nonproportional path with an abrupt change of direction such that the load increment in the ay space is directed parallel to the yield surface, the response of a plasticity model is perfectly elastic, […]

## Calibration of Microplane Model and Comparison with Test Data

The microplane model we described has been calibrated and compared to the typical test data available in the literature (Bazant, Xiang et al. 1996). They included: (1) uniaxial compression tests by van Mier (1984, 1986; Fig. 14.2.2a), for different specimen lengths and with lateral strains and volume changes measured, and by Hognestad, Hanson and McHenry […]