The key conclusions follow:
(i) Circular nodal fillets in octet trusses enhance the bending stiffness of the strut
ends. In turn, the degree of softening due to strut bending is reduced and the
stress for buckling is increased (by about 20%).
(ii) The axial strut strain for tensile rupture in the material used in the current
study does not appear to be affected significantly by the presence of fillets.
Whether this result is a general one, applicable to other classes of materials
(iii) Boundary effects in octet trusses lead to elevations in local strut strains, a result
of reduced local nodal connectivity. Most importantly, the peak tensile strains
in type III struts are elevated almost 2-fold relative to nominal values.
(iv) Boundary effects persist to depths comparable to the unit cell size. This result
is consistent with previous findings [19, 32, 35].
In light of the findings regarding tensile strut fracture and effects of external
boundaries, two potential strategies for enhancing the performance of the current
truss system are envisioned. One would involve modifications to truss geometry in
the near-boundary region, especially along the truss edges. These could include in-
creases in the diameter of type III struts and the fillet radii. A second strategy would
involve changes in local truss topology. For example, the octet truss could be locally
augmented by layers or columns of simple cubic (SC) trusses, interlaced with the
octet truss and sharing the same nodes. In effect this would comprise a local com-
pound truss of the form {F CC}|{SC}. Assessment of the efficacy of these strategies
would require consideration of the increased mass of near-boundary enhancements.
For example, if modifications were made at constant mass, the strut diameter in the
truss interior would need to be reduced to offset the additional mass allocated to
the near-boundary regions. A problem of this nature would likely lend itself well
Figure 4.1: (a-d) Photographs of printed trusses with and without filleted nodes viewed in two orientations. (e,f) Higher magnification views of the nodal regions in computer models (left image in each pair) and in the printed parts (right image in each pair).
Figure 4.2: Compressive stress-strain responses of trusses with and without fil- leted nodes and from FE calculations. Dotted and dashed lines represent mea- sured responses of individual test specimens whereas the solid colored lines rep- resent averages. (See also Videos 1 and 2 in Supplementary information at at
Figure 4.3: Evolution of measured axial strut strains in the three strut types (I, II and III) in trusses with and without filleted nodes as well as results from FE calculations (assuming rigid nodes). Results are further sorted on the basis of strut locations relative to free boundaries. Line colors correspond to strut colors in insets in (a). Dotted lines (with slopes of 1/3 and −1/3) represent analytic predictions for periodic boundary conditions and infinitesimal nodes. Arrows in bottom row indicate type III edge struts that experience the greatest axial strain and are the ones that rupture first.
Figure 4.4: Evolution of measured bending strut strains in the three strut types (I, II and III) in trusses with and without filleted nodes as well as results from FE calculations. Line colors correspond to strut colors in insets in (a). Arrows in top row show strut that experience the greatest bending strain and are most prone to buckling.
Figure 4.5: Three-dimensional renderings of the computed axial strut strains in the {2F CC}2 truss at the macroscopic compressive strains indicated. (See also Video 3
in Supplementary information at doi.org/10.1016/j.actamat.2017.12.060.)
Figure 4.6: Summary of maximum and minimum principal strut strains at macro- scopic compressive strains of 0.02 and 0.038. Central lines represent means, boxes contain the middle two quartiles of data, and outlying hash marks represent mini- mum and maximum values.
Figure 4.7: Images of trusses (a-c) without and (d-f) with fillets, essentially at the load maximum, immediately before and after strut rupture begins. In both cases, fracture of an edge type III strut leads to secondary fractures, a consequence of the dynamic nature of the fracture process. Close-up views (c, f) show strut fracture locations (arrows) and struts that had been ejected (indicated by dashed lines).
Figure 4.8: Transverse displacement profiles of two co-linear struts emanating from the bottom left corner of the truss: (a) experimental results and (b) finite element simulations. Each corresponds to the respective maximum stress for that truss.
Figure 4.9: a, b) Nodal rotations on the external faces in the two truss at their respective peak stress and (c) those obtained from rigid-node FE calculations. Line colors correspond to node colors in the inset in (a).
Figure 4.10: Images of specimens after compression tests. Fracture locations in- dicated by arrows in (c) and (d). In the specimen without filleted nodes, fracture occurs at the nodes; in contrast, in the specimen with filleted nodes, fracture occurs a short distance from the nodes, close to the end of the node fillet.
Figure 4.11: Compressive stress-strain response of the {nF CC}3 truss for n = 2,
5 and 11.
Figure 4.12: Three-dimensional renderings of the computed axial strut strains in the {5F CC}3truss at the macroscopic compressive strains indicated. (See also Video
Figure 4.13: Effects of truss size on macroscopic response and strut strains in {nF CC}3 trusses. (a) Minimum and maximum strut strains prior to strut buckling
(at ǫ1 = 0.01). Central lines represent means, boxes contain the middle two quartiles
of data, and outlying hash marks represent minimum and maximum values. (b) Locations of struts with minimum and maximum principal strains (also at ǫ1 = 0.01).
Figure 4.14: Axial and bending strains for struts with the maximum and minimum strut strains within each strut type as well as struts in the bulk of the {5F CC}3
truss. Dotted lines (with slopes of 1/3 and −1/3) represent analytic predictions for periodic boundary conditions and infinitesimal nodes.
Chapter 5
Defect Sensitivity of Truss
Strength
5.1
Introduction
The present chapter addresses one specific aspect of truss performance: that of
defect sensitivity of compressive strength. It is motivated by numerous studies show-
ing that truss properties often fall short of theoretical predictions, a consequence of
defects and imperfections introduced during manufacturing. In one study, effects of
strut waviness in woven metal trusses were found to produce a knock-down in com-
pressive stiffness and compressive strength of about 20% relative to those obtained in
corresponding structures with straight struts [46]. Such effects are well-predicted by
analytical and finite element models that account for deviations in strut orientations
within these struts. In another, variations in strut geometry in hollow-microtube
truss structures were measured and the results used to build a stochastic model of
geometric imperfections [52]. In turn, through Monte Carlo simulations and finite
element analyses, the critical buckling loads were computed for many instantiations
of strut geometry. The results were used to rationalize large deviations in strength
from the theoretical values for perfectly uniform trusses as well as large statistical
strength variations from sample to sample. Yet other studies have found imperfec-
tions in the form of progressive changes in strut diameter and local strut properties
in polymeric trusses made by a self-propagating photocuring method [50, 51]. Anal-
ogous effects of geometric imperfections on the elastic response of solid-strut Ti-alloy
trusses fabricated by selective electron beam melting have also been reported [6]. In
another computional study, the stiffness and yield strength of an octet-truss panel
were found to decrease approximately linearly with the fraction of struts removed
from the truss [63]. Similar results have been reported for the effects of missing wall
members on the modulus, elastic buckling strength and plastic collapse strength of
hexagonal honeycombs [55].
Defect sensitivity of truss properties may also be affected by the predominant de-
formation mode: that is, whether the truss is stretch-dominated or bend-dominated.
fected by random removal of struts [57]. This is because the starting truss is stretch-
dominated and, in the presence of a small number of defects, remains essentially
stretch-dominated for all loading states. In contrast, the behavior of 2-dimensional
hexagonal trusses depends on the nature of the macroscopic stress state. When
loaded in shear, the truss is bend-dominated and therefore its shear modulus is ex-
tremely low; removing struts only reduces the shear modulus slightly [57]. But, when
loaded hydrostatically, it deforms entirely by strut stretching. Here, removal of even
a small number of struts triggers a transition from stretch- to bend-dominated defor-
mation and a precipitous drop in the bulk modulus. For example, when 10% of struts
are randomly removed from such a truss, the computed bulk modulus decreases by
nearly three orders of magnitude [57].
Viewed from a different perspective, the extent to which defects affect truss prop-
erties may be influenced by the nature of the failure mechanism. For example, if the
load-bearing capacity is dictated by elastic strut buckling, the presence of a small
number of missing struts should not significantly affect truss strength. Although
missing struts may cause strain elevations in neighboring struts and lead to prema-
ture buckling of the affected struts, eventually all remaining compressive struts also
buckle, each supporting nominally the same load. In this case, the truss strength
trast, if failure occurs by brittle fracture of struts that experience tensile stresses, the
strain elevations around a single missing strut may initiate fracture of neighboring
struts, possibly leading to a cascade of further strut fractures and ultimately com-
plete truss failure. Here, the load-bearing capacity of the truss would be determined
by extreme values of tensile stresses within the struts and would follow weakest link
scaling laws. Failure via plastic strut yielding is likely to exhibit an intermediate sen-
sitivity to strut defects. That is, local strain elevations may trigger strut yielding in
regions adjacent to strut defects which, in turn, may lead to plastic buckling before
the remaining struts have yielded. Because of the strain softening inherent to plastic
buckling, the process is likely not as benign as elastic buckling (where buckled struts
sustain essentially a constant load); but it is likely to spread in a more progressive
manner relative to that associated with strut fracture (where failed struts have no
load-bearing capacity).
Free surfaces of trusses are, themselves, defects. Because of reduced nodal con-
nectivity at surfaces, strut strains may be elevated relative to those in the bulk [19].
Finite element simulations for the {F CC} (octet) truss have shown that struts ori- ented perpendicular to the loading direction and situated along the edges experience
strains that are as much as 50% greater than those of equivalent struts in the bulk
of strut strains using digital image correlation [33]. These measurements also con-
firm that the strains in the affected tensile struts are almost entirely due to axial
deformation; the contributions from bending are negligible, as predicted by the sim-
ulations. They further show that the strains in compressive struts situated at the
truss corners experience significant bending, with bending strains comprising up to
50% of the peak principal strut strains.
The goal of the present study is to determine the effects of individual strut defects
and free surfaces, both separately and together, on strains in neighboring struts
and the effects of strain elevations on the strength of three elastic-brittle, stretch-
dominated truss structures. The article is organized in the following way. The truss
topologies and defect types are described in Section5.2. Finite element (FE) models
are described in Section5.3. Results for strut strain distributions in the elastic (pre-
buckling) domain are summarized in Section 5.4. The nonlinear responses of the
trusses, wherein struts buckle and/or fail in tension, are presented in Section 5.5.