An investigation into reinforced and functionally graded lattice structures

Lattice structures are regarded as excellent candidates for use in lightweight energy-absorbing applications, such as crash protection. In this paper we investigate the crushing behaviour, mechanical properties and energy absorption of lattices made by an additive manufacturing process. Two types of lattice were examined: body-centred-cubic (BCC) and a reinforced variant called BCC z . The lattices were subject to compressive loads in two orthogonal directions, allowing an assessment of their mechanical anisotropy to be made. We also examined functionally graded versions of these lattices, which featured a density gradient along one direction. The graded structures exhibited distinct crushing behaviour, with a sequential collapse of cellular layers preceding full densification. For the BCC z lattice, the graded structures were able to absorb around 114% more energy per unit volume than their non-graded counterparts before full densification, 1371 ± 9 kJ/m3 versus 640 ± 10 kJ/m3. This highlights the strong potential for functionally graded lattices to be used in energy-absorbing applications. Finally, we determined several of the Gibson–Ashby coefficients relating the mechanical properties of lattice structures to their density; these are crucial in establishing the constitutive models required for effective lattice design. These results improve the current understanding of additively manufactured lattices and will enable the design of sophisticated, functional, lightweight components in the future.


Introduction
Porous metal foams and, more recently, regularly repeating lattices, have been investigated for use in applications including structural lightweighting, thermal transfer, and impact and blast protection. [1][2][3][4][5][6][7] Additive manufacturing (AM) now provides a means to produce lattices with almost complete geometric freedom, and with a level of control over the volume fraction and repeating cell size which is unachievable for foams. Also, through the range of AM processes available, these structures can be made in a wide range of materials, including polymers and metal alloys, and at a range of length scales from sub-millimeter to several meters.
This makes AM an attractive route to a new generation of lightweight functional components that incorporate lattices based on multi-objective topology optimisation (MTO). [8][9][10][11] Latticed AM components designed in this way will be material-efficient and will offer superior functionality over those they replace; an optimised component can benefit from enhanced convective cooling thanks to the large surface area of an embedded lattice, 12,13 and the same lattice can absorb the impact energy of a projectile, for example in protection equipment such as armour. 14,15 For a combined lattice and MTO design approach to be used effectively, it must incorporate constitutive models relating the distribution of the lattice material and the resulting physical performance. These models must be informed, and validated, by experiment. The purpose of the research laid out here is to gain insight into the performance of two variants of AM lattice, body-centred-cubic (BCC) 1 Additive Manufacturing & 3D Printing Research Group, Faculty of Engineering, University of Nottingham, Nottingham NG7 2RD, UK and z-reinforced body-centred-cubic (BCC z ), and assess this with the pre-existing models of Gibson and Ashby. 1 Our investigation also includes lattice structures featuring a density gradient. These graded structures are representative of those we can expect from a combined lattice and MTO design approach, where spatially varying material properties are required. Understanding the deformation and energy absorption processes of these graded structures, and how they compare to those of non-graded lattices, provides the main motivation for this work and will inform the future design process for lightweight functionally graded parts.
Previous investigations of graded density cellular structures have focussed mainly on graded foams [16][17][18][19][20] and honeycombs. 21 In this paper, we build on previous investigations into graded structures by examining video recordings of their deformation, and correlating the collapse processes with features in the stress-strain curves. We compare the energy absorption of graded and non-graded lattices, and provide the energy absorption per unit volume up to densification, W V D , which can be a key criterion in the selection of a lattice for a given impact protection application. Lastly, through the use of the Gibson-Ashby relationships, we empirically determine several parameters for BCC and BCC z lattices that enable informed decisions about their density to be made in future designs.

The Gibson-Ashby model of lattice deformation
Gibson and Ashby et al 1,2 examined the properties of cellular solids extensively, and provided a series of equations relating their design (principally their relative density, ρ * ) to their physical properties. Those relationships relevant to this work are reproduced in equations 1a-1c, while the associated nomenclature, used throughout this paper, is provided in table 1.
Conventional uniformly dense open-cell foams and lattices are known to undergo compressive deformation in three successive stages. The first is a linear elastic region, where the modulus, E latt. , is roughly proportional to the square of the relative density, as given in equation 1a. If the cell walls are composed of an elastic-plastic material, the structure will develop plastic hinges, and the next regime will be a long plateau at constant stress, σ pl. latt. . σ pl. latt.
It is clear from equations 1a, 1b and 1c that the prefactors C 1 , C 5 and α play a significant role in determining the mechanical properties and deformation behaviour of lattice structures. For applications demanding high modulus, high strength and a long plastic plateau for the purpose of energy absorption, it is preferable for C 1 and C 5 to take larger values and for α to take a low value. In practice, these values, and the exponents n and m, will be determined by the physical properties of the structure; it will therefore be the task of the Effective stress of the lattice structure σ pl. latt.
Plastic collapse strength, or plateau stress, of the lattice σ y sol.
Yield strength of the lattice material σ * Relative collapse strength of the lattice, equal to σ pl. latt. /σ y sol.
Effective strain of the lattice structure ε pl. latt.
Lattice strain at plastic collapse

Results and discussion
Before the lattice deformation and stress-strain data are presented, some additional nomenclature must be Uniform density lattice structures     (1) and (2) in the legend refers to two samples of the same type, i.e. repeat tests. From these, and the relative density of the lattices, which is 0.19, we can estimate the Gibson-Ashby coefficients C 1 ,

Graded density lattice structures
The deformation processes of the graded density BCC and BCC z lattice structures are illustrated in figure 5, which shows a series of video frames from the compressive tests.

Energy absorption
The cumulative energy absorption per unit volume, W V , of the lattice structures under compressive deformation were calculated by numerically integrating the stress-strain curves.
These are provided in figure 6 for both the BCC and BCC z lattices. The total energies per unit volume absorbed by the lattices up to densification were calculated and are presented, along with the densification strains for each structure, in table 5.
The W V behaviour of the non-graded BCC lattices in figure 6(a) show long linear regions that are directly proportional to the lattice strain. These correspond to the plastic plateaux seen in the stress-strain behaviour and so extend from the plastic collapse point, at around 8% strain, up to densification, at around 53% strain. After densification, W V exhibit turning points to steeper gradients; this can be attributed to the much increased structural stiffness after this point. As observed previously in the stress-strain curves, there was very little difference in the W V curves of the BCC structures loaded parallel and perpendicular to their build direction (the z direction). The total energies absorbed up to densification for these conditions were 529 ± 6 kJ/m 3 and 570 ± 10 kJ/m 3 , respectively.
In contrast to the non-graded BCC lattices, the graded structures exhibited non-linear W V behaviour, in which W V were roughly proportional to ε latt. 3 . They absorbed much less energy per unit volume than the non-graded structures at low strain, during the successive collapse of the weaker, low density, cells, but this increased rapidly so that at around 52% strain the energy absorbed by graded and non-graded structures was equal. This difference in W V behaviour, and the higher densification strain for graded structures, led to the graded lattice structures absorbing more energy before full densification. They absorbed 940 ± 50 kJ/m 3 , which is (80 ± 10)% more than the non-graded structures.
Very similar behaviour was observed for the energy absorption of the BCC z lattices, as shown in figure 6(b).   (1) and (2) in the legend refers to two samples of the same type, i.e. repeat tests.
than the non-graded structures, 1371 ± 9 kJ/m 3 vs. 640 ± 10 kJ/m 3 . This represents a (114 ± 4)% improvement in energy absorption, larger than the 80% seen for the BCC lattices. As for the BCC graded structures, the graded BCC z structures exhibited W V behaviour that was roughly proportional to ε latt.
3 . Figure 7 provides an alternative representation of energy absorption in the examined lattice structures. The cumulative energy per unit volume is normalised with the elastic modulus of the lattice strut material, E sol. , and this is plotted against the stress, also normalised with E sol. . This representation was used by Gibson and Ashby 1 to demonstrate the effect of relative density on the energy absorption processes of various foams. It is useful in allowing a designer to select a foam or lattice that minimises the stress while the required energy is absorbed.
Three regions, A, B and C, are denoted in figure 7. Region A corresponds to the initial elastic region of the non-graded structures, and also includes the collapse of the first two lowdensity layers of the graded structures. In this region only a small amount of the total energy is absorbed. In region B the non-graded structures enter their plastic plateaux,