Effective properties of the octet-truss lattice material

TL;DR: In this article, the effective mechanical properties of the octet-truss lattice structured material have been investigated both experimentally and theoretically, and the intervention of elastic buckling of the struts is also analysed in an approximate manner.

Abstract: The effective mechanical properties of the octet-truss lattice structured material have been investigated both experimentally and theoretically. Analytical and FE calculations of the elastic properties and plastic yielding collapse surfaces are reported. The intervention of elastic buckling of the struts is also analysed in an approximate manner. Good agreement is found between the predictions of the strength and experimental observations from tests on the octet-truss material made from a casting aluminium alloy. Moreover, the strength and stiffness of the octet-truss material are stretching-dominated and compare favourably with the corresponding properties of metallic foams. Thus, the octet-truss lattice material can be considered as a promising alternative to metallic foams in lightweight structures.

Summary

1. Introduction

• Over the past few years, a variety of metallic and polymeric foams have been produced for a wide range of potential applications such as the cores of sandwich panels and various automotive parts.
• The aim of this study is to investigate the mechanical ∗ Corresponding author.
• It identiFes several classes of unit cells from which stretching-dominated cellular materials (referred to as lattice materials in the following) can be synthesised.
• Analytical studies on the properties of full 3D lattice materials are lacking.

1.1. Description of microstructure

• A unit cell of the lattice structure is sketched in Fig. 1 and clearly shows its FCC nature.
• Octahedral cells can be stacked to synthesise the octet-truss structure, with each strut of an octahedral cell shared between two neighbouring cells.
• This formula is a Frst-order approximation and overestimates the relative density due to double counting of the volume of the nodes.

2. E ective elastic properties

• Note that an isotropic material has only two independent elastic constants with s3 = 2(s1 + s2).
• For small a=l, the contribution to overall sti&ness from the bending of the struts is negligible compared to stretching of the struts.
• Young’s modulus in the x- and 3-directions will be denoted by Exx and E33, respectively, while the shear modulus in the x–y direction will be referred to as Gxy.

2.1. Comparison with the FE predictions

• The accuracy of the approximate analytical expressions for the moduli was checked against FE calculations performed using the general purpose Fnite element package ABAQUS (HKS, 1997).
• In these FE calculations the pin-jointed strut assumption was relaxed.
• The displacements of the nodes at the vertices of the cell were constrained so as to prevent rigid-body translation and rotation of the cell.
• Thus, for the sake of brevity only results for s = 0:3 are presented.
• A comparison between the analytical and FE predictions of the moduli Exx, Gxy and E33 is shown in Fig. 3 for ; ranging from 0.01 to 0.5.

3. Collapse criteria

• The octet-truss lattice material can fail either by plastic yielding or elastic buckling of the struts.
• In this section the collapse of the lattice material by these two competing mechanisms is explored.
• The authors shall calculate plastic collapse surfaces of the material under various combinations of loading and then proceed to propose an anisotropic yield criterion.

3.1. Plastic collapse

• In the analytical calculations it is assumed that the struts are pin-jointed and made from a rigid, ideally plastic solid.
• The macroscopic collapse stress is calculated by equating the external work with the plastic dissipation in stretching the struts for kine- matically admissible modes of collapse; that is, an upper bound approach is adopted.
• The elastic Poisson’s ratio s of the material was assumed to be 0.3, the yield strain Y of the strut material was taken equal to 0.1% and the hardening co-eEcient m = 80 (this small degree of hardening was required to get convergence of the FE calculations).
• For the calculations presented in this section was taken equal to 0.01.
• The displacements of the nodes at the vertices of the cell were constrained so as to prevent rigid-body translation and rotation of the cell; and the rotations of the nodes at the vertices were set to zero.

3.1.1. Collapse surface calculations

• The overall yield surface in macroscopic stress space consists of intersecting collapse surfaces which are associated with particular collapse modes.
• The collapse modes for this combination of macroscopic stressing are sketched in side views of the octahedral cell in Fig.
• The plastic dissipation at the hinges is neglected in comparison with the dissipation in axial stretching of the yielded struts.
• FE calculations show that the shear strength xz varies by approximately 10% as the octet-truss is rotated about the z-axis with the shear strength being minimum at a 45◦ rotation.

3.2. Anisotropic yield criterion

• While the collapse surfaces presented in the previous section are useful for displaying the yield stress under speciFc load paths, a closed-form expression for the yield surface would be advantageous in summarising the collapse response of the octet-truss lattice material.
• With respect to the principal axes of anisotropy (x; y; z), Hill’s yield criterion has the form ≡.
• Thus, to fully describe the state of anisotropy of the octet-truss lattice material the authors must know the orientations of the principal axes of anisotropy and the measured uniaxial yield strength in the principal directions.
• Comparisons between the calculated collapse surfaces and the predictions of the above yield criterion are shown in Figs.
• No improvements in the accuracy of the predictions were found.

3.3. The elastic buckling strength

• When a strut buckles, the rotation of its ends is opposed by the bending of the other struts: they exert a restoring moment and it is this that determines the factor n2 in (19).
• The cells of the octet-truss lattice material may buckle in many di&erent modes and the resulting problem is very complicated to analyse completely.
• It is recalled that the buckling load of an axially loaded strut is strongly dependent on the end constraints, and so the calculations presented below should be viewed as lower bounds to the buckling strength.
• As in (8), is the imperfection level and x is the axial co-ordinate along the strut measured from one end.

3.3.1. Buckling collapse surfaces

• The authors now proceed to detail the collapse surfaces due to elastic buckling, as computed by analytical and FE methods for the combinations of macroscopic stressing considered earlier, ( zz; xz), ( xx; yy) and ( 33; 13).
• The sketches in Fig. 7 show the possible buckling modes in side views of the octahedral cell, with the dashed lines representing the buckled struts.
• The appropriate plastic collapse planes are included in the Fgure.
• Good agreement is seen between the FE and analytical calculations in support of the inFnitesimal deformation assumption made in the analytical calculations.
• Under biaxial tension, the collapse mode is always by plastic yield irrespective of the value of Y. 3.3.1.3. Collapse surface in ( 33; 13) space for elastic buckling.

3.4. E6ect of geometric imperfections

• In general, the elastic buckling and plastic collapse surfaces overestimate the collapse stresses of an elasto-plastic lattice material; interactions between the elastic buckling and plastic yielding of the struts substantially knock-down the collapse stresses.
• The collapse stresses for both levels of imperfections are substantially lower than those for the perfect structure.
• In fact, for = 0:1 the collapse load of the imperfect strut is about half that of the perfect strut which results in mode B-III becoming active and truncating the tensile side of the plastic collapse surface.
• The authors suggest here that the elasto-plastic collapse stresses of an imperfect octet-truss lattice material idealised as a pin-jointed structure can be estimated by re-calibrating the buckling collapse planes against the axial collapse load of an imperfect pin-ended strut.

4. Comparison between measured and predicted modulus and yield strength

• The authors proceed by comparing the predictions detailed in the previous sections with the measured uniaxial compression strength of the octet-truss lattice material made from a casting aluminium alloy (LM25) of composition Al–Si 7–Mg 0.3 (wt%).
• Triangulated layers with locating holes at the nodes, and tetrahedral cores with locating pins at the nodes, were injection moulded in polystyrene.
• These bedding-in e&ects occur at the nodes in the lattice material: the pins of the tetrahedral core bed into the holes of the triangulated layers during the initial stages of deformation.
• The analytical and FE predictions of the strength of the lattice material are shown in Fig. 13a: they are in good agreement with the experimental data (note that ( a=2l)2(1= Y) ≈ 5 and thus the collapse of this material is not expected to be imperfection sensitive).

5.1. Octet-truss lattice material vs. metallic foams

• Here, the sti&ness and strength of the octet-truss lattice material are compared in Fig. 14 with those of metallic foams, for relative densities ; in the range 0.01–0.1.
• Fig. 14 clearly shows that the sti&ness and strength of the octet-truss lattice material exceed the corresponding values for metallic foams by a factor between 3 and 10.

5.2. Octet-truss lattice material versus optimal microstructures

• The design aim in the development of the octet-truss lattice material is to maximise the strength (or sti&ness) to weight ratio of a nearly isotropic cellular material.
• A number of classes of two-phase composites are known to attain the H–S bounds on the bulk and shear moduli.
• Rank laminates are obtained by a sequential process where at each stage the previous laminate is laminated again with a single lamina (always the same) in a new direction.
• Thus, there exist a variety of multi-length-scale microstructures with extremal values of the bulk and shear moduli.
• Thus, the octet-truss lattice material represents a relatively cheap and weight-eEcient structural material with potential multi-functional applications.

6. Concluding remarks

• The e&ective mechanical properties of the octet-truss lattice material have been investigated through analytical and FE calculations.
• Thus, these calculations are expected to be underestimate the collapse stresses.
• Further, the FE calculations conFrmed that shape imperfections of the struts knock-down the collapse stresses only when the elastic buckling and plastic yielding loads of the struts are approximately equal.
• Good agreement is seen between the analytical and FE calculations of the strength and the experimental data.
• In fact, the sti&ness and strength values of the octet-truss material are about half the theoretical maximum values for isotropic voided materials: its high strength-to-weight ratio, relative ease of manufacture and potential for multi-functional applications makes the octet-truss lattice material an attractive alternative to metallic foams.

Figures

Fig. 3. Comparison of the analytical and FE predictions of the elastic moduli and strength for the octet-truss lattice material.

Fig. 8. Comparison between the analytical and FE predictions of the buckling governed collapse surface in ( xx; yy) space, for Y = 0:05 and 0.1 (a=l = 0:1).

Fig. 6. Comparison between the analytical, FE and proposed yield criterion predictions of the plastic collapse surface in ( 33; 13) space (a=l = 0:1). In the sketches of the collapse modes, the dashed and solid lines represent struts at yield and in a rigid state, respectively.

Fig. 7. Comparison between the analytical and FE predictions of the buckling governed collapse surface in ( zz ; xz) space, for Y = 0:05 and 0.1 (a=l = 0:1).

Fig. 1. Structure of the octet-truss lattice material. The darkened struts represent a octahedral cell while the nodes labelled p1–p4 form a tetrahedral cell.

Fig. 11. E&ect of imperfections on the collapse surface in ( zz ; xz) space, for Y =0:001 and 0.1 (a=l=0:1).

Fig. 4. Comparison between the analytical, FE and proposed yield criterion predictions of the plastic collapse surface in ( zz ; xz) space (a=l = 0:1). In the sketches of the collapse modes, the dashed and solid lines represent struts at yield and in a rigid state, respectively.

Fig. 2. Isometric sketches of the octahedral and tetrahedral cells with the associated co-ordinate systems.

Fig. 12. Photograph of the octet-truss lattice material made from a casting aluminium alloy, LM25.

Fig. 13. (a) Comparison between the experimental observations and predictions of the 33 versus 33 response of the LM25 lattice material. (b) Uniaxial tensile response of the as-cast LM25 aluminium alloy.

Fig. 10. E&ect of imperfections on the collapse surface in ( zz ; xz) space, for Y = 0:025 and a=l = 0:1.

Fig. 5. Comparison between the analytical, FE and proposed yield criterion predictions of the plastic collapse surface in ( xx; yy) space (a=l = 0:1). In the sketches of the collapse modes, the dashed and solid lines represent struts at yield and in a rigid state, respectively.

Fig. 14. Comparison between the sti&ness and strength of the octet-truss lattice material, metallic foams and the upper bounds for isotropic voided materials.

Fig. 9. Comparison between the analytical and FE predictions of the buckling governed collapse surface in ( 33; 13) space, for Y = 0:05 and 0.1 (a=l = 0:1).

Frequently asked questions

Q1. What are the contributions in "E&ective properties of the octet-truss lattice material" ?

Analytical and FE calculations of the elastic properties and plastic yielding collapse surfaces are reported. Thus, the octet-truss lattice material can be considered as a promising alternative to metallic foams in lightweight structures. 

Q2. What is the collapse load of an imperfect octet-truss la?

In fact, for = 0:1 the collapse load of the imperfect strut is about half that of the perfect strut which results in mode B-III becoming active and truncating the tensile side of the plastic collapse surface. 

Q3. How many Timoshenko beam elements were used in the FE analysis?

In the FE analysis each cylindrical strut was again modelled by between 20 and 40 Timoshenko beam elements (B32 element of ABAQUS) depending on its length. 

Q4. What is the shear strength of the octet-truss?

The shear strength 13 of the octet-truss is periodic with respect to rotations of period 60◦ about the three-axis; FE calculations reveal that the shear strength 13 varies by less than 10% as the octet-truss is rotated about the three-axis, with the shear strength a maximum for a 30◦ rotation. 

Q5. What is the FE method used to calculate the macroscopic collapse stress?

The macroscopic collapse stress is calculated by equating the external work with the plastic dissipation in stretching the struts for kine-matically admissible modes of collapse; that is, an upper bound approach is adopted. 

Q6. How is the collapse stress of a pin-ended strut calculated?

As expected the collapse surface is reasonably insensitive to the imperfection level, with the collapse stresses decreasing by less than 10% for =0:1. 

Q7. What is the condition for the structure to be stretching dominated?

For this case they showed that the necessary and suEcient condition for the structure to be stretching dominated is that the connectivity at each node is Z = 12 (or Z = 6 if the material is two dimensional). 

Q8. What is the post-buckling load-shortening relation for a pin-?

The post-buckling load-shortening relation for an inextensional pin-ended strut of length l is given by (Budiansky, 1974)P ≈ PE ( 1 +" 2l) (20)for small axial displacements ".