# 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 transition from the finite element simulations, with the classical hypothesis of continuum mechanics, to discrete particle models and lattice models in which the continuum is approximated a priori by a system of discrete elements: particles, trusses, or frames.

An extreme example of the continuum approach —in view of the fineness of material subdivision— is the numerical concrete of Roclfstra, Sadouki and Wittmann (1985), Witlmann, Roelfstra and Kamp (1988) and Roelfstra (1988), in which the mortar, the aggregates, and their interfaces are independently modeled by finite elements. This requires generation of the geometry of the material (random placement of aggregates within the mortar) and the detailed discretization of the elements to adequately reproduce the geometry of the interfaces. With a completely different purpose, but with the same kind of analysis, Rossi and Richer (1987) and Rossi and Wu (1992) developed a random finite element model in which the microstruclure is not directly modeled, but is taken into account by assigning random properties to the element interfaces. The common feature of these approaches is that, before cracking starts, the displacement field is approximated by a continuous function.

The particle and lattice models do not model the material continuously, but substitute the continuum by an array of discrete elements in the form of particles in contact, trusses, or frames, in such a manner that the displacements are defined only at the centers of the particles, or at the nodes of the truss or frame.

The origin of the particle approach can generally be traced to the development of the so-called distinct element method by Cundall (1971, 1978), Serrano and Rodriguez-Ortiz (1973), Rodrigucz-Ortiz (1974), Kawai (1980), and Cundall and Struck (1979) in which the behavior of particulate materials (originally just cohesionless soils and rock blocks) was analyzed simulating the interactions of the particles in contact. This kind of analysis, which deals with a genuine problem of discrete particle systems, used highly simplified contact interaction laws permitted by the fact that the overall response is controlled mainly by kinematic restrictions (grain interlock) rather than by the details of the force-deformation relation at the contacts. However, although the kinematics of the simulations appeared very realistic, the quantitative stress-strain (averaged) response was not quite close to the actual behavior. This shortcoming, which still

persists in many modem particle and lattice models, is largely caused by the fact that the simulations are usually two-dimensional while a realistic simulation ought to be three-dimensional.

The basic idea of the particle model can be extended to simulate the particular structure of composite materials, for example, the configuration of the large aggregate pieces of concrete, as done by Zubelewicz (1980,1983), Zubelewicz and Mroz(1983), and Zubelewicz and BaXant (1987), or the grains in a rock (Ple – sha and Aifantis 1983). In these cases the model requires defining the force interaction between particles (aggregates or grains) which are caused mainly by the relative displacements and rotations of neighboring particles. Although, for computational purposes, the problem is reduced to a truss (Fig. 14.4. la-b) or to a frame (Fig. 14.4. lc-d), the basic ingredient of such models is that the geometry (size) of the truss or frame elements and their properties (stiffness, strength, etc.) arc dictated by the geometry of the physical structure of the material (stiffness, size, shape, and relative position of aggregates or grains).

In contrast to this, the pure lattice models replace the actual material by a truss or frame whose geometry and element sizes are not related to the actual internal geometry of the material, but are selected freely by the analyst. The truss approach to elasticity, elementary atomistic representations of the physics of elasticity (i. e., arrays of atoms linked by springs shown in textbooks of solid state physics), was already proposed as early as 1941 by Hrennikoff. The lattice models have been championed by theoretical physicists for the simulation of fracturing in disordered materials (Herrmann, Hansen and Roux 1989; Channel, Roux and Guyon 1990; Herrmann and Roux 1990; Herrmann 1991) and have been developed to analyze concrete fracture by Schlangen and van Mier at Delft University of Technology (Schlangen and van Mier 1992; Schlangen 1993, 1995; van Mier, Vervuurt and Schlangen 1994). In their approach, a regular triangular frame of side length less than the dimensions of the smallest aggregates, is laid over the actual material structure (Fig. 14.4.1 e—F) and the properties of each beam are assigned according to the material the beam lies over, mortar, aggregate or interface. However, to eliminate directional bias of fracture, the lattice must be random (see Section 14.4.2).

This section presents a brief overview of the main concepts and results of the particle and lattice models as far as concrete fracture is concerned.

Figure 14.4.2 Random particle model of Bazant, Jirasek et al. (1994): (a) two adjacent circular particles with radii ті and and corresponding truss member ij; (b) typical randomly generated specimen and its corresponding mesh of truss elements; (c) constitutive law for matrix. (Adapted from Bazant, Tabbara et al. 1990.)