Explicit modeling of individual fibers -- Hybrid fiber composites

Consider an individual fiber superimposed upon a planar RBSN (lattice) element, as shown in Fig. 1a. The fiber is inclined to the general loading direction by angle q. The lattice nodes and their degrees of freedom are indicated in the figure. Two approaches have been developed for both the pre- and post-cracking contributions of such individual fibers within the composite (Bolander and Yip, 2003).

Discrete fiber model
The fiber is modeled as a discrete entity using ordinary frame elements (Fig. 1b). This model has the following characteristics:

• Frame elements are assigned the elastic, plastic, and rupture properties of the fiber;
• Each frame element has six degrees of freedom for this two-dimensional case;
• Frame element nodes are related to the lattice DOF via bond link elements and rigid arm constraints; and
• Link tangential spring stiffness, Kt, is based on a local bond stress-slip relation (Fig. 2).

The bond stress-slip relation represents the micromechanics of the fiber-matrix interface and can accommodate frictional decay (as shown) or slip hardening during fiber pullout. The bonded length associated with a given spring set is lb.

Semi-discrete fiber model
The strength and stiffness contributions of individual fibers are lumped at the point of boundary crossing (Fig. 1c). That is, a zero-length spring element, aligned with the fiber direction, connects the two cells at the point where the fiber crosses the common boundary. The semi-discrete model has the following characteristics:

• The elastic, plastic, and rupture properties of the fiber are modeled by a zero-length spring spanning the boundary crossing;
• No additional degrees of freedom are introduced into the system;
• Pre-cracking contributions of the fiber to the lattice element stiffness are based on elastic shear lag theory, such as that derived by Cox (1952);
• The micromechanics of interfacial debonding and frictional slip guide the post-cracking response of the nonlinear spring. The effects of geometrical deformations of the fiber can be included in the post-cracking response of the fiber (Sujivorakul et al., 2000).

Figure 3 shows localized fracture in a model of a FRCC tensile specimen and the resulting load versus displacement relations. The peak strength (i.e. P/Pe >1.0) is associated with brittle matrix fracture; the post-cracking strength is due to fiber debonding and pullout. Both fiber models consume nearly the same amount of energy to completely fail the specimen. The semi-discrete approach exhibits the mathematically expected post-cracking strength, Pe, for a semi-random distribution of fibers, where n is the number of fibers bridging the crack and d is the fiber diameter. The small difference in post-cracking strength (i.e. P/Pe is just less than 1.0) is attributable to statistical variation in expected average pullout length, l/4. Increasing the fiber content beyond a critical volume fraction induces multiple cracking along the specimen length, in agreement with mixture rule predictions that account for length and orientation efficiencies of the fibers.

Hybrid fiber composites, which are composed of multiple fiber types, can provide excellent strength and toughness at potentially reasonable cost. The results shown in Fig. 4a are based on flexural testing of extruded hybrid fiber composite specimens, with dimensions 100 x 25 x 4mm (Cyr et al., 2001). The fiber types (length/diameter) were: polypropylene (13 mm/0.050 mm); glass (6 mm/0.014 mm); and PVA (2 mm/0.014mm).

The following preliminary modeling of the Cyr et al. specimens illustrates some of the challenges in simulating hybrid fiber composites. Figure 4b shows the simulated distribution of fibers within just a small portion of a computational mesh (which is similar to that shown in Fig. 3a). The polypropylene fibers appear white in the figure, while the other two types appear as a shade of gray. The numerous fibers seem like a fine fabric that stitches together the Voronoi cells, which are shown in the background. Figure 4c indicates the number of cell boundary crossings for each fiber type in the hybrid mixture. The figure also indicates the number of fibers needed to reach the specified volume fractions for the given specimen dimensions. Due to the large number of fibers present in the material, we propose a hybrid computational model to represent the hybrid composite material:

• The semi-discrete approach is used to model the microfibers, since fiber addition does not increase the number of degrees of freedom of the model.
• The discrete approach is used to model the macrofibers (i.e. the polypropylene fibers in this case). Due to the longer fiber length, relative to the average cell size, these fibers are likely to bridge multiple cracks during quasi-strain hardening of the composite.

For both modeling approaches, the fiber routines are completely uncoupled making parallel computation attractive. On route to simulating the specimen response to loading, there are a number of issues that need to be addressed, including the modeling of microcracking.

References

1. J.E. Bolander and M. Yip, "Numerical modeling of fiber reinforced cement composites subjected to drying." In Fourth International Workshop on High Performance Fiber Reinforced Cement Composites - HPFRCC4, eds. A.E. Naaman and H.W. Reinhardt, RILEM, 2003, pp. 7-20.
2. Cox, H.L., 'The elasticity and strength of paper and other fibrous materials', British J. Applied Physics, 3 (1952) 72-79.
3. Sujivorakul, C., Waas, A.M. and Naaman, A.E., 'Pullout response of a smooth fiber with an end anchorage', ASCE J. Engng. Mech. 126(9) (2000) 986-993.
4. Cyr, M.F., Peled, A. and Shah, S.P., 'Extruded hybrid fiber reinforced cementitious composites', In 'Seventh International Symposium on Ferrocement and Thin Reinforced Cement Composites', Eds. M.A. Mansur and K.C.G. Ong (National University of Singapore, 2001) 199-207.