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 (Fig. 14.4.1 c-d). Since the intemodal links also transmit shear, moment equilibrium of the nodes needs to be considered, while for the pin-jointed truss it need not. Therefore, this model has three degrees-of-freedom per node (two displacements and one rotation, with corresponding two force components and one moment) for planar lattices, while the pin-jointed model has only two degre. es-of-freedom per node. In the spatial case, the model ofZubelewicz and Bazant requires six degrees-of-freedom per node, i. e., three displacements and three rotations, while the pin-jointed The simplest model truss requires only three degrees-of-freedom per node. There is an additional important advantage of shear transmissionThe simplest model —it makes it possible to obtain with the lattice any Poisson ratio, while a random or regular pin-jointed lattice (truss) has Poisson ratio always 1/3 in two dimensions and 1/4 in three dimensions.
The simplest model In the model of Bazant, Tabbara et al. (1990) and Zubelewicz. and Bazant (1987), the major particles in the material (large aggregate pieces) are imagined as circular and interacting through links as shown in Fig. 14.4.2. In the initial work ofZubelewicz. and Bazant, the link between particles was assumed to transmit both axial forces and shear forces, the latter based on the rotations of particles. In the subsequent model by Bazant, Tabbara et al. (1990), the particle rotations and transmission of shear were neglected and only axial forces were assumed to be transmitted through the links. In such a case, the system of particle links is equivalent to a truss. As pointed out before, the penalty to pay for this simplification is that the Poisson ratio of a random planar truss is always 1/3 (and for a spatial truss 1/4). Another consequence of ignoring particle rotations and interparticle shears is that the fracture process zone obtained becomes narrower. But this can be counteracted by assuming a smaller postpeak softening slope for the interparticle stress displacement law, and also by introducing a greater random scatter in the link properties, both of which tend to widen the fracture process zone.
A random particle configuration must be statistically homogeneous and isotropic on the macroscale. In the simulation of concrete, the configuration must meet the required granulometric distribution of the particles of various sizes, as prescribed for the mix of concrete. The problem of generation of random configurations of particles in contact under such constrains involves some difficult and sophisticated aspects (see, e. g., Plesha and Aifantis 1983).
However, the problem becomes much simpler when the particles do not have to be in contact, as is the case for aggregate pieces in concrete. In that case, a rather simple procedure (Bazant, Tabbara et al. 1990) can proceed as follows: (1) using a random number generator, coordinate pairs of particle centers (nodes)
arc generated one after another, assuming a uniform probability distribution of the coordinates within the area of the specimens; (2) for each generated pair a check for possible overlaps of the particles is made, and if the generated particle overlaps with some previously generated one, it is rejected; (3) the random generation of coordinate pairs proceeds until the last particle of the largest size has been placed within the specimen, (4) then the entire random placement process is repeated for the particles of the next smallest size, and then again for the next smallest size, etc. (The number of particles of each size is determined in advance according to the prescribed mix ratio and granulometry.)
To determine which particles interact, a circle of radius i>r, is drawn around each particle i (with ft ~ 5/3) as shown by the dashed lines in Fig. 14.4.2a: Two particles interact if their dashed circles intersect each other. See Bazant, Tabbara et al. (1990) for the details of the assignment of the dimensions of the truss element, particularly the cross-section Am and length Lm of the deformable portion (labeled with subscript m for matrix). In a later study by Jirasek and Bazant (1995a), a uniform stiffness of all the links was assumed.
Fig. 14.4.2b shows a typical computer-generated random particle arrangement resembling concrete, and the corresponding truss (random lattice). Fig. 14.4.2c shows the stress-strain relation for the interparlicle links, characterized by the elastic modulus Em, tensile strength limit /4m, and the postpeak softening slope Es (or alternatively by strain су at complete failure, or by G™, each of which is related to the foregoing three parameters). The microscopic fracture energy of the material, G™, is represented by the area under the stress-strain curve in Fig. 14.4.2c, multiplied by the length of the link. The ratio of є/ to the strain єр at the peak stress may be regarded as the microductility of the material.
The lattices in Fig. 14.4.1a–d attempt to directly simulate the major inhomogeneities in the microstructure of concrete. By contrast, the model introduced by Schlangen and van Mier (1992) takes a lattice (in the early versions regular, but later randomized) that is much finer than the major inhomogeneities. Its nodal locations and links are not really reflections of the actual microstructure (Fig. 14.4. le-f). Rather, the microstructure is simulated by giving various links different properties, which is done according to the match of the lattice to a picture of a typical aggregate arrangement.
Van Mier and Schlangen take advantage of the available simple computer programs for frames and assume the lattice to consist of beams which resist not only axial forces but also bending. Due to bending, the internodal links (beams), of course, also transmit shear, same as in the model of Zubelewicz and Bazant (Figs. 14.4.1c-d and e-f). This feature is useful, because shears are indeed transmitted between adjacent aggregate pieces and across weak interfaces in concrete, and because arbitrary control of the Poisson ratio is possible. However, the idea of bending of beams is a far-fetched idealization that has nothing to do with reality. No clear instances of bending in the microstructures of concrete can be identified.
The idealization of the links as beams subject to bending implies that a bending moment applied at one node is transmitted to the adjacent node with the сапу-over factor 0.5, as is well known from the theory of frames. This value of the carry-over factor is arbitrary and cannot not have anything in common with real behavior. In the model of Zubelewicz. and Bazant (Fig. 14.4.1 c-d), the shear resistance also causes a transmission of moments from node to node, however, the carry-over factor is not 0.5 and can have different values. The transmission of moments is, in that model, due to shear in contact layers between particles, which is a clearly identifiable mechanism. In consequence of this analysis, it would seem better to consider the carry-over factor in the lattices of van. Mier and Schlangen to be an arbitrary number, determined either empirically or by some microstructural analysis. This means that the 6×6 stiffness matrix for the element of the lattice, relating the 6 generalized displacement and force components of a beam sketched in Fig. 14.4.If, should be considered to have general values in its off diagonal members, not based on the bending solutions for a beam but on other considerations. In fact, the use of such a stiffness matrix would require only an elementary change in the computer program for a frame (or lattice with bending). Of course, if the need for such a modification is recognized, the model and van Mier and Schlangen becomes essentially equivalent to that of Zubelewicz and Bazant, except that the nodes do not represent actual particles and the lattice is much finer than the particles.
The beams’in the lattice model of Schlangen and van Mier are assumed to be elastic-brittle, and so, when the failure criterion is met at one of the beams, the link may be removed. This means that at each step the computation is purely elastic. This is computationally efficient, but makes the model predict a far too brittle behavior, even for three-dimensional lattices (van Mier, Vervuurt and Schlangen 1994).
Another important aspect in lattice models is the size of the links. Unlike the lattice of Zubelewicz and Bazant, which directly reflects the particle configurations and thus cannot (and should not) be refined, the
Figure 14.4.3 Dependence of the load-displacement curve on the size of Ihe lattice links: (a) basic element geometry, (b) load-displacement curves for various lattice spacings (adapted from Schlangen 1995).
lattice of van Mier and Schlangen lias an undetermined nodal spacing, which raises additional questions. First, as is well known, frames or lattices with bending are on a large scale asymptotically equivalent to the so-called micro-polar (or Cosserat) continuum (e. g., Baz. ant and Ccdolin 1991, Sec. 2.10-2.11). Pin-jointed trusses, on the other hand, asymptotically approach a regular continuum on a large scale. The micropolar continuum is a continuum with nonsymmetric shear stresses and with couple stresses. It possesses a characteristic length, which is essentially proportional to the typical nodal spacing of the lattice approximated by the micro-polar continuum. While, in principle, the presence of a characteristic length is a correct property for a model of concrete, the characteristic length should not be arbitrary but should be of the order of the spacing of the major aggregate pieces. In this regard, the lattice of van Mier and Schlangen appears to be too refined. Moreover, as transpired from recent researches and the previous chapter on nonlocal concepts, the micro-polar character or the presence of characteristic length should refer only to the fracturing behavior and not to the elastic part of its bonds. The model of Schlangen and van Mier goes against this conclusion, since even the elastic response of the lattice is asymptotically approximated by a micro-polar rather than regular continuum.
Furthermore, a question arises about the dependence of the response on the lattice spacing. A recent study of Schlangen (1995) shows that the crack pattern is not strongly affected by the size of the beams, but the load-displacement is affected in much the same way as mesh refinement in local strain-softening models: the finer the lattice, the less the inelastic displacement and the dissipated energy, as illustrated in the load-crack opening curves in Fig. 14.4.3 for a square specimen subjected to pure tension. Indeed, it is easy to imagine that upon infinite refinement the stresses in a beam close to a crack lip must tend to infinity and thus a precracked specimen must fail for a vanishingly small load (roughly proportional to the square root of the beam siz. e). Also, the shorter the beams, the smaller the dissipated energy, because the volume of material affected by the crack is smaller the smaller the elements. Note that the lattice analyzed by Schlangen in Fig. 14.4.3 has random strength in all the cases with identical probabilistic distribution. Therefore, randomness does not relieve mesh sensitivity as sometimes claimed.
The mesh-sensitivity of Schlangen and van Mier’s model can probably be artificially alleviated or eliminated by taking a beam strength inversely proportional to the square root of the beam size, similar to the equivalent strength method described in Section 8.6.4 for crack band analysis. However, this is purely speculative and a more sound basis should be built for the lattice models before they can be confidently used as predictive (rather than just descriptive) models. A nonlocal fracture criterion may serve as an alternative solution to the problem, but this would break the computational efficiency of the elastic-brittle beam lattice model. Note that nonlocality (i. e., interaction at finite distance) is automatically implemented in the particle models because the particle distances are finite and fixed, so the lattice siz. e should also be fixed as dictated by the microstruclurc.