Irregular lattice model of cohesive cracking

Fracture model formulation

The irregular lattice model of fracture (Bolander and Berton, 2003; Berton, 2003) is based on the crack band concept of Bazant and Oh (1983). In general, loading of an element will be skew to the element axis ij, as shown in Figure 1. The normal and tangential springs are activated and, for tensile loading of the material, the normal spring will be in tension.

The fracture criterion is based on the following measure of stress

where FR is the resultant of the spring forces acting on the element facet and denominator is the projection of the facet area on a plane perpendicular to the direction of FR (Fig. 1b). Within each computational cycle, the ratio sR/s(w) is computed for all of the elements, where the cohesive stress s(w) is a bilinear function of the crack opening displacement w (Fig. 1c). A prismatic crack band develops within the element with max(sR/s(w)) > 1. The width of this crack band is equal to the element length multiplied by cos q, where q is the angle between the facet normal and the resultant force FR. Crack opening displacement is related to the fracture strain over the crack band:

For a critical element, fracture involves an isotropic reduction of the spring stiffnesses and an associated release of spring forces, so that sR follows the material softening relation. The release of spring forces causes an imbalance between the external and internal nodal force vectors, which is corrected through subsequent iterations. A maximum of one element is modified per iteration cycle, as is common for lattice models (Herrmann and Roux, 1990; Schlangen and van Mier, 1992). The modeling of fracture differs from that of conventional lattice models in several respects: 1) fracture can form at angle q to the element axis; and 2) element breaking is a gradual process that conserves energy in association with a cohesive softening relation.

Three-point bend test simulation

A three-point bend test of a notched concrete beam is simulated to show the fracture properties of the irregular lattice network (Bolander and Berton, 2003; Berton, 2003). The beam test was part of a series of round robin tests, which were performed to evaluate a method for determining the tension softening properties of concrete (Kitsutaka et al. 2001). Figure 2 shows a Voronoi discretization of the typical configuration for the test series. Here, the batch C test carried out at Gifu University is used for the comparison. The strength and stiffness characteristics of the concrete are as follows: fc = 40.9 MPa and E = 31.6 GPa. The notch has a depth of half the height of the beam (50 mm).

Two different meshes are used to model the three-point bend test of the notched-beam specimen. The models are similar in terms of number of nodes and elements. However, a different approach has been used to discretize the ligament zone above the notch. In the first model, shown in Fig. 2, a random procedure has been employed to generate the entire mesh. For the other case, a semi-random discretization is used, such that a predefined planar surface is created above the notch. Because of the imposed load applied to the specimen during the simulated test, the crack is likely to develop and propagate along this predefined flat surface. During the generation of both meshes, several nodes were prepositioned to accurately model the locations for supports and impose displacements. The prenotch was modeled by assigning zero stiffness values to the elements crossing the prenotch area.

The four parameters defining the bilinear softening relation (Fig. 1c) were determined using an inverse analysis procedure (Thomure et al. 2001), based on a Levenberg-Marquardt minimization algorithm. Starting with an assumed softening curve, the softening curve parameters are gradually adjusted to reduce the error between each successive computed load-CMOD curve and the experimental load-CMOD curve. To speed up this process, a planar RBSN model of the three-point bend test was used for the inverse analyses. Based on the experimental load-CMOD curve shown in Fig. 3, the inverse procedure gave the following values for the softening parameters: ft = 4.119 MPa, b = 0.247, h = 0.118 and wc = 0.1540 mm.

The analyses are carried out by imposing, in small increments, a downward displacement at the top mid-span nodes to simulate the action of the loading device. The following figure compares the load-CMOD curves obtained from the simulations with the experimental curve. The two simulated curves agree well with the experimental result. The higher load capacity of the numerical models in the tail region of the curves is due to the rather coarse discretization of the ligament region. The uppermost elements in that region remain in compression and therefore limit the advance of the fracture process zone.

At each step of the numerical simulation, the energy consumed by a fracturing element can be derived as the difference between the work done by the external loads and the internal strain energy. Since only one element is allowed to fracture during each computational cycle, the difference between the incremental values of external work and strain energy must be equal to the energy consumed by that critical element. Therefore, at any stage during the numerical simulation, the total energy consumed by one element can be determined by summing all its contributions up to that point. By dividing this energy by the projected area of the element facet on the cracking plane, the corresponding energy density is calculated. If an element fractures completely (i.e. w > wc), the total energy consumed by that element should be equal to the fracture energy GF, which is the area under the softening curve given in Fig. 1c

.
Figures 4a and 5a show plots of the energy consumed by the fractured elements for the semi-random mesh simulation. The three-dimensional representations of the energy consumption are plotted alongside the cracked surface on the corresponding mesh view. Here, the local energy consumption, gF, is normalized with respect to the fracture energy GF = 115.9 N/m, calculated from the bilinear softening relation shown in Fig. 1c. The lateral view of the plot shows that gF/GF is approximately equal to unity for the fully fractured elements (i.e. the elements located in the bottom part of the ligament zone). For these elements, the differences between gF and GF are within 1.9%. In the upper part of the ligament, the values of energy consumption decrease since the elements in that location have only partially fractured prior to reaching the final CMOD value of 0.45 mm (Fig. 3).

Figures 4b and 5b show plots of the fracture energy density obtained from the random mesh simulation. The profile of the energy density plot exhibits more variation relative to that of the semi-random mesh model due, in part, to the various facet inclinations with respect to the vertical plane. Although the maximum value of the normalized energy density is about 45% larger than GF, the average value for the fully fractured elements is 118.6 N/m, which is only 2.3% greater than GF. As for the semi-random mesh case, the normalized energy consumption is fairly uniform and around unity for the elements in the lower part of the ligament. The energy values decrease in the upper part where the elements have not completely fractured by the end of the loading history.

These results demonstrate the performance of the irregular lattice model for arguably the most basic of material types and loading conditions: isotropic, homogeneous materials subjected to tensile loading. Nonetheless, the fundamental properties of lattice models are most evident under such conditions. For the lattice model described here, the scaling of element stiffness terms is based on a Voronoi discretization of the material domain and provides an elastically uniform description of the material under uniform modes of straining. The energy dissipation mechanisms active at finer scales are lumped into a macroscopic cohesive crack relation. In contrast to classical lattice models, element breaking is gradual and governed by rules that provide an energy conserving representation of fracture through the irregular lattice. The objective representation of fracture in homogeneous materials is prerequisite for modeling fracture of multiphase particulate materials, which is ongoing work within this project.

References

1. Bolander, J.E. and Berton, S., "Cohesive zone modeling of fracture in irregular lattices." In Fracture Mechanics of Concrete Structures, eds, V.C. Li et al., Ia-FraMCos, 2004, pp. 989-994.
2. Berton, S. (2003) Numerical simulation of the durability mechanics of cement-based materials, Ph.D. thesis, Civil and Environmental Engineering, University of California, Davis.
3. Bazant, Z.P. and Oh, B.H., "Crack band theory for fracture of concrete." Materials and structures, RILEM, 16(93), 1983, 155-177.
4. Herrmann, H.J. and Roux, S., editors. Statistical models for the fracture of disordered media, Amsterdam: Elsevier/North Holland, 353 pp., 1990.
5. Schlangen, E. and van Mier, J.G.M. Experimental and numerical analysis of micro-mechanisms of fracture of cement- based composites, Cement Concrete Composites 14, 105-118, 1992.
6. Kitsutaka, Y., Uchida, Y., Mihashi, H., Kaneko, Y., Nakamura, S. and Kurihara, N. 2001. Draft on the JCI standard test method for determining tension softening properties of concrete. In R. de Borst et al. (eds.), Fracture Mechanics of Concrete Structures. Lisse: Swets & Zeitlinger, 371-376.
7. Thomure, J.L., Bolander, J.E. and Kunieda, M., "Reducing mesh bias on fracture within Rigid-Body-Spring-Networks." JSCE Structural Engineering/Earthquake Engineering (18)2, 2001, 95-103.