Flexural elasticity of woodpile lattice beams

Abstract

Flexure of slender structures, composed of filaments in a woodpile arrangement, is theoretically studied. Expressions for the apparent bending stiffness are derived. The model is validated experimentally using three-point bending. Computer simulations show that bending is accompanied by lattice shear for increased porosity. A shear-inclusive micromechanical model for two different stacking arrangements is developed. This is further refined by including shear deformation within the filaments, which is supported by numerical simulations for relatively short filament overhang. The apparent flexure and shear of lattice beams are attributed primarily to stretch and flexure of the filaments, respectively. Asymptotic formulas for the effective bending stiffness in terms of the relative density are presented. The apparent bending stiffness scales linearly with the volume fraction, whereas the apparent shear stiffness scales with the cube of the volume fraction. Exclusion of lattice shear leads to errors for by over an order of magnitude in the extreme, unlike void-free solid beams where shear is a higher order effect. Excellent quantitative agreement with numerical results is obtained for a wide range of porosity and external aspect ratio, when lattice shear is included in the model.

Characterization of the Mechanical Properties of FFF Structures and Materials: A Review on the Experimental, Computational and Theoretical Approaches

“…Two analytical models reviewed in this section [95,96] were developed for fully-filled FFF structures. One of the few works that have related the effective properties of the printed part to the structure formed in the presence of voids in a FFF part was done by Cuan-Urquizo et al [67,98]. Cuan-Urquizo considered the FFF raster structure as a cellular material [99] and treated FFF parts as being composed of lattice materials.…”

“…In [67], the flexural properties of FFF beams were studied for a structure of rasters with angles and sequence of 0/90°. The primary deformation mechanisms identified for individual filaments were stretching and bending.…”

“…For any value of low infill density, the response is more sensitive to filament orientation in the outermost layers. For θ = 0°, the rasters that run axially withstand the most loading [67]. Somireddy et al [68] studied the flexural stiffness of ABS samples with the following stacking sequences: 0°/90°, 15°/−75°, 30°/−60°, and 45°/−45.…”

“…Nevertheless, as the air gap used approached zero, the flexural stiffness resulted with maximum variation of 33% among all the combinations of print orientation, layer thickness and feed rate. The effect of air gaps greater than zero was analyzed by Cuan-Urquizo and Bhaskar [67]. They studied the structure–property relationship for the 0°/90° stacking sequence of raster angles of PLA samples, but to several raster densities.…”

“…They studied the structure–property relationship for the 0°/90° stacking sequence of raster angles of PLA samples, but to several raster densities. The approach in [67] is mainly theoretical, hence it is reviewed in Section 3. Although it is beyond the scope of this paper, it is worth mentioning that recently Kuznetsov et al [71] reported the ultimate fracture strength for tubular (0% infill) samples under three-point bending.…”