Beam Finite Element Model Modification Considering Shear Stiffness: Octet-Truss Unit Cell with Springs

Abstract

This study investigates the effects of modifying a beam model for octet-truss lattice structures to calculate the homogenized material properties using the average stress method. While alignment is observed at low relative densities, the unmodified beam model derives underestimated results at higher relative densities, reaching up to 40% and 30% for elastic and shear modulus values, respectively, for a relative density of 0.5. Beam model modification achieved by increasing strut stiffness at the joints is investigated in detail, and we conclude that both modulus values cannot fit the solid model’s results with this type of modification. This study proposes a novel modification method involving seven spring elements with two constants to capture both the elastic and shear moduli. This study concludes by compensating differences between the solid and beam models’ moduli with the inserted springs, providing an analytical solution for the linear elastic system. The performance of the unit cell models is tested by solving two lattice structures at which the elastic modulus and shear modulus were dominant, respectively, on the mechanical behavior. The results converge to a constant value when the number of unit cells is six, and the beam with a spring model achieved a performance that was close to that of the solid model for the shear-modulus-dominant lattice structure.

1. Introduction

Lattice materials manufactured using additive manufacturing (AM) technology provide advantages: ultra-lightweight, high energy absorption, heat isolation and dissipation, and high specific strength and stiffness. They are thus applicable in a very wide range of fields, such as the aerospace, automotive, architecture, biomedical, packaging, and sports industries. These technologies bring an opportunity to design material constants by changing the geometries of a material’s structure [1,2]. The strongest evidence of the superiority of lattice materials is their existence in nature, such as in bones, honeycombs, and plant stems.

Non-stochastic lattice materials are a class of cellular solids formed through the periodic repetition of identical unit cells [1,3]. According to cell topology, they can be classified as strut-based, plate-based, and triply periodic minimal surfaces (TPMSs)-based, and can have both open- and closed-cell forms. Strut-based lattice structures are called truss lattice materials. The mechanical behavior of truss-based lattice structures is distinguished as either bending-dominated or stretch-dominated using their Maxwell number M = b − 3j + 6, where b is the number of struts and j is the number of joints within a unit cell [4]. When a material has an M < 0, it has a bending-dominated structure, characterized by its relatively high modulus and initial collapse strength. This makes it well suited for lightweight load-bearing applications. When a material has an M ≥ 0, it has a stretch-dominated structure, and it exhibits high compliance and a prolonged flat plateau in the post-yield regime, making it desirable for applications that are focused on energy absorption [5]. Octet-truss materials are strut-based open-cell non-stochastic lattice materials which are products of AM technology. As demonstrated in Figure 1, for an octet-truss unit cell with b = 36 and j = 14, the Maxwell number is zero. However, although the octet-truss structure is generally classified in the literature as having a stretch-dominated lattice structure based on its Maxwell number, there exists a specific threshold of relative density that limits this classification. Extensive research has been conducted on the octet-truss lattice material due to its high specific properties [4,6,7,8].

Homogenization methods are used to replace the detailed geometry of the lattice structure with an equivalent homogeneous medium for which the effective material properties are derived. The purpose of homogenization is to reduce the time and effort spent on the large-scale lattice structures that are made of these types of materials. On the other hand, a micro-model is a physically exact simulation of a material that includes all the material and geometry details, such as the different materials, voids, and interface properties. In order to characterize the material, tensile, compression, and shear tests are conducted on the micro-model, which may be a unit cell (UC) or may contain multiple cells. The preferred micro-model is called a representative volume element (RVE).

Multiple techniques have been proposed to address the homogenization problem [9], including beam theory approach [10], strain energy equivalence [11], asymptotic homogenization (AH) [8,12,13], the multi-scale homogenization method (also known as the FE2 method) [14,15], the FFT-based homogenization method [16], and the average stress method [17], highlighting the diversity of approaches in this area found in the literature. By means of the beam theory and strain energy equivalence approaches, closed analytical formulas of material constants can be determined for a single cell, which is limited with small deformations and low relative density values. On the other hand, AH-, FE2-, and FFT-based methods and average stress methods are numerical homogenization methods. AH is not restricted with the value of relative density. Both AH and FE2 give the most accurate results, as verified by the experiments, but their computational cost is relatively expensive. The computational cost of the FFT-based homogenization method is low and it is efficient for large-scale structures. The average stress method is not limited with the relative density value and has very low computational cost for elastic large-scale structures, but the accuracy of this method is moderate, similar to the beam theory approach [9].

In this study, the average stress numerical homogenization method is used to obtain effective material properties, commonly used for composite materials in the literature [17]. In order to determine the stiffness matrix of the material, the required average stresses are calculated for the composite material, using the expression enclosed in the parentheses in Equation (1). However, in the case of lattice materials, which contain void spaces, the average ${\sigma }_{ij}^o$ stress is modified, as follows:

$${\sigma }_{ij}^o=\left({\displaystyle \frac{1}{V_{st}}}{\int }_V{\sigma }_{ij}dV\right){\displaystyle \frac{V_{st}}{V_{cell}}}=\left({\displaystyle \frac{1}{V_{st}}}{\sum }_{n=1}^N{\sigma }_{ij}^n{V}_n\right){\displaystyle \frac{V_{st}}{V_{cell}}}$$

(1)

Here, N is the number of finite elements within the model, V_n is the volume of each finite element, V_st is the total volume of the struts, V_cell = L_c³ is the volume of the cell including the void space and the struts,${\sigma }_{ij}^n$ is the stress of each finite element. Alternatively, the average stress can also be calculated by dividing the sum of reaction forces at all nodes located on the related surface of the unit cell by the area of this surface.

Deshpande et al. [4] have published the elastic modulus, E_D, and the shear modulus, G_D, of the octet-truss unit cell with Equation (2a,b) in terms of the aspect ratio, R/L. Here, R and L are the radius and the length of struts, as depicted in Figure 1a, respectively. These equations are derived from the analytical solution of the unit cell truss system at which the struts are two-force members connected by hinges at the joints. Thus, bending of the struts are ignored and these equations are adequate for small aspect ratio values.

$${\displaystyle \frac{E_D}{E_b}}={\displaystyle \frac{2\pi \sqrt{2}}{3}}{\left({\displaystyle \frac{R}{L}}\right)}^2$$

(2a)

$${\displaystyle \frac{G_D}{E_b}}={\displaystyle \frac{\pi }{\sqrt{2}}}{\left({\displaystyle \frac{R}{L}}\right)}^2$$

(2b)

Here, E, G, and ν are the elastic modulus, shear modulus and Poisson’s ratio, respectively. The subscript b designates the properties of bulk material. The Poisson ratio of the octet-truss lattice material has been reported as constant ν_D = 1/3, independently from the relative density [7]. Also, the G_D/G_b expression can be rewritten by introducing the relationship E_b = 2(1 + ν_b)G_b into Equation (2b). These expressions can be obtained by either applying unit strain in one direction provided that the strains vanish in other directions, or by applying unit stress in one direction, provided that the stresses vanish in other directions.

A micro-model, in which a detailed geometry is built by brick-solid finite elements, provides the most accurate representation of a material. A macro model, comprising continuous medium with the attained homogenized material constants, minimizes the time and effort spent on large-scale structures, but some failure mechanisms like the availability to observe buckling is lost. On the other hand, modeling with beam finite elements is easier in processing and significantly reduces the required computer capacity and solution time compared to the micro-model. Beam elements also offer the advantage of directly plotting the resultants. Researchers have extensively used beam elements in the simulation of lattice structures [6,18,19,20,21,22,23,24,25,26]. Ling et al. [6] reported that their polymer resin octet-truss solid finite element model provided better predictions than their beam element model. Labeas and Sunaric [18] and Smith et al. [23] increased the cross-sectional area of the struts at the joint sections over a particular length to account for the strut contact. Luxner et al. [19] increased the elastic modulus of the material at the vicinity of the joints to account for the increased material aggregation. Ptochos and Labeas [21,22] presented solutions for body-centered cuboid beam models through both analytical and finite element methods and compared their results with the experimental data for a material with a relative density of 0.035. They concluded that the beam analytical solution can be considered satisfactory for strut aspect ratio R/L lower than 0.1, which corresponded to a relative density of 0.22. They suggested that higher-order beam theories should be used to obtain more accurate strut deflections for higher aspect ratios. In this study, the models simulated by using brick finite elements are referred to as solid models, and those simulated by beam finite elements are referred to as beam models.

In the literature, various modification approaches the beam models have focused on the compression testing of lattice materials, and they emphasize the determination of the elastic modulus. Generally, in most cases, the effect of shear stiffness is not significant in comparison with that of bending stiffness. However, for cases such as those involving tall beams with short spans, thin-walled cross-sections, and beams subjected to concentrated loads, the effect of shear stiffness on deflection becomes important [27]. Octet-truss lattice materials are cubic symmetric, and the shear modulus of a material is needed for a complete prediction of its homogenized behavior.

In Section 2, the geometric and material properties of an octet-truss unit cell are presented. By means of finite element simulations, the average stress method is applied on four different UC models to obtain the homogenized material constants in Section 3. In this study, one octet-truss unit cell is preferred for use as the RVE in the homogenization process. A case in which more than one cell is used as the RVE is also presented at the end of Section 3.1. Discrepancies between the geometrically most accurate solid UC model and the unmodified beam UC model are demonstrated in Section 3.1, Section 3.2 and Section 3.3. Section 3.4 presents a detailed discussion on the modification of the beam UC model, achieved by increasing the stiffness of the beam elements at the joints. In their study, Gholibeygi et al. [28] stated that, with parameters that align the elastic modulus value of the solid UC and beam model, the shear modulus of the beam UC model with stiff joints was always higher than that of the solid UC model with this kind of modification. The aim of this study was to modify the beam UC model through a consideration of both the elastic modulus and the shear modulus. This is why Section 3.5 proposes an alternative new beam UC model, modified with spring elements. For this new model, the two required spring constants are calculated; with these, both the elastic and shear modulus values of the solid UC model are captured. Finally, in Section 4, two example problems are solved in order to observe and compare the performance of these UC models. Lattice structures are built with three different UC models with an increasing number of unit cells. The first example included the compression of the lattice structure with fixed ends and the second example included the lateral loading of the lattice structure.

2. Properties of Octet-Truss Unit Cell Lattice Material

The octet-truss unit cell’s (UC’s) geometry is shown in Figure 1a. Here, L_c is the cubic unit cell side length. There are 36 struts of the same length, L, and radius, R. Joints are located at the face centers and at the corners. At each joint, 12 struts connect each other, resulting in a lattice connectivity of Z = 12. Octet-truss UC can be constructed in two ways, both are depicted in Figure 1a,b. In Figure 1a, the dark-colored cell inside, with circle-sectioned struts, is called the octahedral cell; the light-colored cell outside, with semi-circle sectioned struts, is called the all face-centered cubic cell (AFCC). So, the first way to build an octet-truss cell is constructed by inserting the octahedral cell inside the AFCC cell. In Figure 1b, the dark-colored cell, with semi-circle sectioned struts at the corner, is called a regular tetrahedral cell; the second way to build an octet-truss is through the replication of eight tetrahedrons through mirror symmetry.

The relative density of the octet-truss, $\overline{\rho }=\rho$/ρ_b, is defined as the ratio of the macroscopic density of a cellular material, ρ, and the density of the bulk material, ρ_b. The relative density of the octet-truss unit cell with cylindrical struts in terms of aspect ratio is given as follows [7].

$$\overline{\rho }=6\pi \sqrt{2}{\left({\displaystyle \frac{R}{L}}\right)}^2-{\displaystyle \frac{32}{2-\sqrt{2}}}{\left({\displaystyle \frac{R}{L}}\right)}^3$$

(3)

Here, the first term represents the material volume occupied by all of the struts, with the exception of half of the struts, which are located on the cell faces, as they belong to the adjacent cells. The second term in Equation (3) is subtracted to exclude the overlapping volumes of the struts at the joints. This term becomes significant for the high relative densities. The values of relative density, $\overline{\rho }$, in terms of aspect ratio, calculated with Equation (3), are tabulated in

Table 1. Octet-truss unit cell aspect ratio, $R/L$, and relative density, $\overline{\rho }$.

$\overline{\mathit{\rho }}$	$\mathit{R} \left(\mathbf{m}\mathbf{m}\right)$	$\mathit{R}/\mathit{L}$
$0.10$	$0.27939$	$0.0658$
$0.20$	$0.41041$	$0.0967$
$0.30$	$0.52013$	$0.1226$
$0.40$	$0.62119$	$0.1464$
$0.50$	$0.71928$	$0.1695$

.

The octet-truss lattice material is cubic symmetric and its direction-dependent elastic behavior can be described through three independent material constants: the elastic modulus, E, Poisson’s ratio, ν, and the shear modulus, G. The stress and strain relation of the cubic symmetric material is described by the stiffness matrix, as follows.

$$\left\{\begin{array}{c}{\sigma }_1\\ {\sigma }_2\\ {\sigma }_3\\ {\tau }_{23}\\ {\tau }_{13}\\ {\tau }_{12}\end{array}\right\}=\left[\begin{array}{cccccc}{C}_{11}& C_{12}& C_{13}& 0& 0& 0\\ C_{21}& C_{22}& C_{23}& 0& 0& 0\\ C_{31}& C_{32}& C_{33}& 0& 0& 0\\ 0& 0& 0& C_{44}& 0& 0\\ 0& 0& 0& 0& C_{55}& 0\\ 0& 0& 0& 0& 0& C_{66}\end{array}\right]\left\{\begin{array}{c}{\epsilon }_1\\ {\epsilon }_2\\ {\epsilon }_3\\ {\gamma }_{23}\\ {\gamma }_{13}\\ {\gamma }_{12}\end{array}\right\}$$

(4)

Here, σ_i and τ_ij represent the engineering normal and shear stresses, while ε_i and γ_ij represent the engineering normal and shear strains of the unit cell, respectively. The cubic symmetry of the octet-truss unit cell induces specific relations between the stiffness matrix components: C₁₁ = C₂₂ = C₃₃, C₂₁ = C₂₃ = C₁₃, and C₄₄ = C₅₅ = C₆₆. Once the components of the stiffness matrix are obtained, the material constants can be calculated as follows:

$$E_1=E_2=E_3=\left(3C_{11}{C}_{21}^2-C_{11}^3-2C_{21}^3\right)/\left(C_{21}^2-C_{11}^2\right)$$

(5a)

$${\nu }_{21}={\nu }_{13}={\nu }_{23}=\left(C_{21}^2-C_{21}{C}_{11}\right)/\left(C_{21}^2-C_{11}^2\right)$$

(5b)

$$G_{21}=G_{13}=G_{23}=C_{44}$$

(5c)

3. Finite Element Simulations of Octet-Truss UC for Homogenization

All of the simulations in this study were executed on a computer with an Intel^® Core™ i9-11900K processor, 64 GB RAM system memory, and NVIDIA^® GeForce RTX 4060 Ti graphics card, and using Ansys Mechanical APDL Release 2025 R1 software on a Windows 64 platform.

3.1. Solid UC Model

Using the Parametric Design Language and Command references of the Ansys APDL 2025 R1 finite element program, the Command files for the test simulations were written in a text format. The solid model was simulated using 3D 8-node structural brick finite elements (SOLID185) [29]. This element has eight nodes and has three translational degrees of freedom at each node. One-eighth of the solid model is shown in Figure 2a and the full solid model is shown in Figure 2b. In order to understand the sensitivity of the model to mesh density, the number of finite elements in the strut longitudinal direction is incrementally increased, while the number of finite elements along the radius of the strut is kept constant, as n_R = 4, and the corresponding modulus values are monitored. It was observed that a higher mesh density resulted in reduced elastic and shear modulus values, but an increased Poisson’s ratio value. In other words, a finer mesh decreased the stiffness of the material. It was observed that, once the number of finite elements in the longitudinal direction of a strut exceeded fifteen, there was minimal difference in the last digit of the modulus values in MPa. Thus, it was decided to use 16 elements per strut in the simulations in this study, as shown in Figure 2a,b. The full model used here has 25,344 elements and 32,314 nodes.

The isotropic linear elastic material was defined, having an elastic modulus of E_b = 18 GPa and a Poisson’s ratio of ν_b = 0.3, inspired by the study of Qi et al. [8], which was conducted using Al-Si10-Mg. Here, the isotropic bulk material’s shear modulus can be calculated as G_b = E_b/2/(1 + ν_b) = 6923.08 MPa. The unit cell dimensions used in the simulations were cell length L_c = 6 mm and strut length L = 3√2 = 4.24 mm, as shown in Figure 1a. The simulations were repeated across the relative density range $\overline{\rho }$ = 0.1–0.5 to capture the trend of material constants. The strut radius values of some octet-truss cells used in this study are tabulated in Table 1 [28].

In order to derive the homogenized material properties, two tests were conducted on the unit cell. The first test was applied on one-eighth of the unit cell. The boundary conditions are tabulated in

Table 2. Boundary conditions used for tensile and shear tests for homogenization.

		Finite Element Type
	Surface Name	Solid UC Model (SOLID185)	Beam UC Models (BEAM188)
tensile test	Top, Bottom	$Y-sym$	$Y-sym$
	Front, Back	$Z-sym$	$Z-sym$
	Left	$X-sym$	$X-sym$
	Right	$U_X=L_c/2$	$U_X=L_c/2, {ROT}_Y={ROT}_Z=0$
shear test	Left, Right	$X-sym$	$X-sym$
	Front, Back	nodes are coupled	nodes are coupled
	Bottom	$U_Y=U_Z=0$	$U_Y=U_Z=0, RO{T}_X=RO{T}_Y=RO{T}_Z=0$
	Top	$U_Y=0, U_Z=L_c$	$U_Y=0, U_Z=L_c, RO{T}_X=RO{T}_Y=RO{T}_Z=0$

, with the surface names illustrated in Figure 2a [17].

The surface names in Figure 2 and Table 2 are labeled as left and right, bottom and top, and back and front; those are parallel to the YZ, ZX, and XY planes, respectively. Symmetric boundary conditions were applied at all surfaces, except the right surface, which was subject to the displacement of U_X = L_c/2. Here, L_c/2 is the side length of one-eighth of the unit cell used in this test. These boundary conditions led to the only non-zero strain of ε_X = U_X/L_c/2 = 1. The average stress components were calculated using Equation (1); those were directly equal to the first column components of the stiffness matrix, C_j1, as shown in Equation (4). Then, the elastic modulus and Poisson ratio were calculated using Equation (5a,b).

The second shear test simulation was conducted on the entire unit cell; the boundary conditions are illustrated in Figure 2b,c and tabulated in Table 2. The symmetric boundary conditions were applied on the left and right surfaces [17,28]. At the bottom surface, displacement was constrained in the Y and Z directions. The front and back surfaces were meshed to have identical node patterns, and they are coupled to have the same nodal displacements as those shown in Figure 2c [29]. At the top surface, displacement in the Y direction was constrained and this surface was subject to the displacement of U_Z = L_c in the Z direction. Here, L_c was the side length of the entire unit cell used in this test. These boundary conditions led to the only non-zero strain of γ_YZ = U_Z/L_c = 1. The average stress components were calculated using Equation (1); those were directly equal to the fourth column components of the stiffness matrix, C₄₄, as shown in Equation (4). Then, the shear modulus was calculated by Equation (5c).

One cell of the octet-truss was used as the RVE in this study; but, in order to also inspect the effect of the number of unit cells in the RVE, more than one cell was also introduced through the homogenization process for $\overline{\rho }$ = 0.5. It was observed that the elastic modulus and Poisson’s ratio were 3189.60 MPa and 0.292, respectively, independent from the number of unit cells of RVE. Decreasing homogenized shear modulus values with increasing number of unit cells (n × n × n) in RVE and the computational cost of simulations in seconds for n = 1, 2, …, 9 are tabulated in

Table 3. Homogenized shear modulus versus number of unit cells in RVE for $\overline{\rho }=0.5$.

$\mathit{n}$	$\mathit{G} \mathbf{\left(}\mathbf{M}\mathbf{P}\mathbf{a}\right)$	$\mathbf{T}\mathbf{i}\mathbf{m}\mathbf{e} (\mathbf{s}\mathbf{e}\mathbf{c})$
$1$	$1797.58$	$14$
$2$	$1775.07$	$33$
$3$	$1767.32$	$46$
$4$	$1763.46$	$97$
$5$	$1761.16$	$224$
6	$1759.63$	$540$
7	$1758.53$	$1134$
8	$1757.71$	$2476$
9	$1757.08$	$7085$

and plotted in Figure 3. The execution time for n = 9 was almost 2 h. The hardware of the computer used in this study was not sufficient to obtain the solution for 10 × 10 × 10 unit cells. An RVE with n = 9 gives a 2.3% smaller shear modulus value than an RVE with n = 1.

3.2. Unmodified Beam UC Model

The beam model was constructed by beam finite elements perfectly connected at the joints. The struts of the octet-truss were simulated by 3D 2-node beam finite elements (BEAM188) [29]. This had six degrees of freedom at each node, including translational and rotational. This element was based on the Timoshenko beam theory, where transverse shear strain is constant through the cross-section. Within the options of this element, the one with linear shape functions is used in this study. Although 3D solid finite elements have high sensitivity to mesh density and quality, beam finite elements improve with element order more than density. The convergence of the beam model’s results can be achieved using a few elements along the strut longitudinal direction. In order to have a geometrically consistent comparison with the solid model, the same mesh was used along the struts with 16 elements. The mesh of the beam element’s circular cross-section was defined to have 16 divisions in circumferential direction and 4 divisions in the radial direction. The full beam model had 720 finite elements and 1578 nodes, corresponding to 5.4% and 3.1% of the solid model, respectively. The average stress method, as explained for the solid UC model, was also applied for the homogenization of beam UC models considered in this study.

SOLID185 brick and BEAM188 beam finite elements have three and six degrees of freedom at each node, respectively. In order to provide same boundary conditions for both models, the required rotation related constraints were added to the beam UC models as shown in Table 2.

3.3. Comparison of Solid UC and Unmodified Beam UC Models

The results obtained from the solid UC model, the beam UC model without any modification, and the truss model given by Equation (2a,b) [4] are tabulated in

Table 4. Octet-truss material constants obtained by average stress method (unit: MPa).

	Solid UC Model			Deshpande et al. (2001) [4]			Unmodified Beam UC Model
$\overline{\mathit{\rho }}$	$\mathit{E}$	$\mathit{G}$	$\mathit{\nu }$	${\mathit{E}}_{\mathit{D}}$	${\mathit{G}}_{\mathit{D}}$	${\mathit{\nu }}_{\mathit{D}}$	${\mathit{E}}_{\mathit{f}}$	${\mathit{G}}_{\mathit{f}}$	${\mathit{\nu }}_{\mathit{f}}$
$0.10$	$272.85$	$193.08$	$0.333$	$231.20$	$173.40$	$0.333$	$238.88$	$177.14$	$0.328$
$0.20$	$675.03$	$453.21$	$0.325$	$498.90$	$374.17$	$0.333$	$532.99$	$390.95$	$0.322$
$0.30$	$1248.05$	$792.62$	$0.315$	$801.32$	$600.99$	$0.333$	$884.90$	$642.56$	$0.316$
$0.40$	$2056.70$	$1232.17$	$0.304$	$1142.92$	$857.19$	$0.333$	$1303.99$	$938.21$	$0.310$
$0.50$	$3189.60$	$1797.58$	$0.292$	$1532.38$	$1149.28$	$0.333$	$1805.37$	$1288.26$	$0.304$

for comparison. Solid UC model results are presented without any subscript. Subscripts f and D are used to present the results obtained from the unmodified beam model and the results given by Deshpande et al. [4], respectively.

From the results plotted in Figure 4, it can be observed that the difference between the solid and beam UC models are less than 12% for low values of relative density, $\overline{\rho }$ < 0.09, but for higher values, the beam UC model underestimates the solid UC model’s results. Somnic and Jo [9] have reported that the beam theory approach is limited to the density value of $\overline{\rho }$ < 0.3. On the other hand, Ushijima et al. [20] reported that the finite element results using beam elements and solid elements are in good agreement if the strut aspect ratio is relatively small (d/L < 0.1), where d is the strut diameter. This aspect ratio corresponds to R/L < 0.05 and $\overline{\rho }$ < 0.1 for octet-truss lattice material. For higher values of relative density, the beam model suffers from the lack of material contact at the joints. Also, Gibson and Ashby [3] explained this as follows in their study about foams for higher relative densities: they stated that this happens if the state of material has changed from a structure of connected beams to a solid structure containing voids. The differences between the solid UC model and the unmodified beam UC model results can be calculated from Table 4. The differences E_k = E − E_f and G_k = G − G_f between the solid UC model and unmodified beam UC model are shown in Figure 4 and will be discussed in Section 3.5 later in this study. Elastic modulus values found with beam UC model for $\overline{\rho }$ = 0.1, 0.2, 0.3, 0.4, and 0.5 were 12%, 21%, 29%, 37%, and 43% smaller, respectively, than those found using the solid UC model. The same is correct for the shear modulus values with percentages of 8%, 14%, 19%, 24%, and 28% as well.

The variation in Poisson’s ratio with respect to relative density in this study was not extensive, as shown in Figure 5, but it is worth mentioning that the behavior of the material is observed to be noticeably different before and after the relative density, $\overline{\rho }$ = 0.1. Deshpande et al. [4] reported that Poisson ratio is constant at ν_D = 1/3 for octet-truss structures; they also mentioned that it is expected to be independent of relative density.

Either the relative density, $\overline{\rho },$ or the aspect ratio, R/L, can be used as metrics for the characterization of the mechanical behaviors of lattice materials [3,19]. According to Gibson and Ashby [3], these expressions take the form of E = A₁${\overline{\rho }}^{n_1}$E_b and G = A₂${\overline{\rho }}^{n_2}$E_b, where A_i and n_i represent fitted constants and exponents, respectively. It is mentioned that the exponent n_i will be around 2 for bending-dominated lattice materials and around unity for stretch-dominated lattice materials. Gholibeygi et al. [29] adapted this idea for the material constants of octet-truss lattice material in their study and gave the relations in Equation (6a,b) by means of curve-fitting on the solid model results for $\rho$ = 0.1–0.5, as follows:

$${\displaystyle \frac{E}{E_b}}=17.56 {\left({\displaystyle \frac{R}{L}}\right)}^{2.61}$$

(6a)

$${\displaystyle \frac{G}{G_b}}=16.30 {\left({\displaystyle \frac{R}{L}}\right)}^{2.35}$$

(6b)

The maximum error was 6.3% with Equation (6a) and that derived with Equation (6b) was 3.2% in the range of ρ = 0.1–0.5. Both the normalized solid UC model results and those from Equation (6a,b) are plotted in Figure 6.

In order to present the relation between the results of unmodified beam UC model and the solid UC model, the expressions in Equation (7a,b) were obtained by means of curve-fitting in terms of aspect ratio for ρ = 0.1–0.5, as follows:

$${\displaystyle \frac{E}{E_f}}=5 {\left({\displaystyle \frac{R}{L}}\right)}^{0.58}$$

(7a)

$${\displaystyle \frac{G}{G_f}}=2.6 {\left({\displaystyle \frac{R}{L}}\right)}^{0.35}$$

(7b)

The error percentages derived using the unmodified beam UC model (E_f, G_f), Equation (6a,b), and Equation (7a,b) are compared to those derived using the solid UC model (E, G), as presented in Figure 7. From this figure, we can say that, if the unmodified beam UC model is used for $\overline{\rho }$ < 0.09 or for R/L = 0.0622, and Equation (6a,b) or Equation (7a,b) are used for $\overline{\rho }$ > 0.09, then the error will be less than 12%.

3.4. Investigation of the Modified Beam UC Model Found in the Literature

In this section, the modification method of the beam UC model presented in [18,19,20,21,22,23,24,25,26] is investigated. The modification involves increasing the stiffness of the struts at the joints over a particular length. An increase in either the cross-sectional area or the elastic modulus can be introduced. In this study, the elastic modulus is increased at the joints, as shown with the purple color in Figure 8.

Modification of this beam UC model is based on two parameters. The first parameter is le, that is multiplied with R, to assign the length of the strut at which the modification is applied, measured from the joint. The second parameter is ne, that is multiplied with the elastic modulus of the bulk material to increase the stiffness over this length, as shown in Figure 8.

The octet-truss structure’s half-strut volume, shown in Figure 9, is represented by the beam finite element in the beam model that has an axis on the x-y plane. Twelve beam elements intersect at joint point O. The darker region in the figure is the common area shared with the other two adjacent struts. The distance measured from the corner point, O, and the point where the struts separate from each other ranges from 1.41 R to 2 R in space. The projection of this distance on the beam axis ranges from R to 1.73 R. Thus, it is reasonable to explore a suitable value for the modification parameter $le$ around these values.

The objective is to determine the modification parameters, le and ne, to be used in the beam UC model, so that the results of the solid UC model can be obtained. The procedure carried out here is explained in detail, using $\overline{\rho }$ = 0.5 as an example. First, for a constant le value, the tensile test was repeated to observe the E_le (ne) variation. This procedure was repeated for other le values, as shown in Figure 10. For ne = 1, all the curves corresponded to the unmodified beam UC model’s elastic modulus value, E_f = 1805.4 MPa, as presented in Table 4. Notice that the ne axis is plotted in logarithmic scale to ensure a better view. All the curves exhibit a consistent trend, where the elastic modulus increases continuously by ne until it approaches an asymptotic value. A slight scattering of about 1 MPa in magnitude starts around ne = 10⁶–10⁷, as indicated by the cross symbol, ×, in Figure 10. Subsequently, a substantial scattering takes place at about ne = 10¹¹ due to numerical instability. The dotted horizontal line corresponds to the solid UC model’s elastic modulus value, E = 3189.6 MPa, for $\overline{\rho }$ = 0.5. Thus, the intersection of this horizontal line with any curve gives the $ne$–$le$ parameter pairs, to be used for the modification of the beam UC model and to match the elastic modulus value of the solid UC model.

It is noteworthy that only the curves for le > 1.0654 intersect with the horizontal solid UC model target value for $\overline{\rho }$ = 0.5. For example, the intersection points for le = 1.1, 1.2, and 1.3 are marked by circle symbols in Figure 10. Numerous additional parameter pairs are computed enough to see the le-ne relation, as tabulated in

Table 5. The ne–le parameters pairs with E_le/E = 1 for $\overline{\rho }$ = 0.5.

$\mathit{l}\mathit{e}$	$\mathit{n}\mathit{e}$	${\mathit{G}}_{\mathit{l}\mathit{e}}/\mathit{G}$	${\mathit{\nu }}_{\mathit{l}\mathit{e}}$
$1.06540$	$N/A$	$-$	$-$
$1.06544$	$\mathrm{100,000}$	$1.20793$	$0.27845$
$1.0656$	$\mathrm{10,000}$	$1.20793$	$0.27845$
$1.0670$	1000	$1.20803$	$0.27847$
$1.0789$	$100$	$1.20841$	$0.27873$
$1.2137$	$10$	$1.21390$	$0.28158$
$1.3$	$6.743$	$1.21733$	$0.28332$
$1.4$	$5.066$	$1.22099$	$0.28526$
$1.5$	$4.159$	$1.22450$	$0.28710$
$1.6$	$3.589$	$1.22784$	$0.28885$
$1.7$	$3.197$	$1.23095$	$0.29050$
$1.8$	$2.910$	$1.23382$	$0.29206$
$1.9$	$2.690$	$1.23642$	$0.29351$
$2.0$	$2.516$	$1.23876$	$0.29487$

and shown in Figure 11.

Notice that the ne values are presented on the primary vertical axis, in logarithmic scale, on the left in Figure 11. It is important to note that there is an infinite number of parameter pairs which can be used to match the elastic modulus of the modified beam UC model with that of the solid UC model. Using these ne–le parameter pairs, shear modulus values G_le are calculated by running the shear test simulation. These values are normalized with the solid UC model shear modulus G and presented on the secondary vertical axis on the right side in Figure 11. Notice that G_le/G = 1 would indicate a successful match between the shear modulus of beam UC model and the solid UC model, but all curves are larger than unity.

The same procedure described here for $\overline{\rho }$ = 0.5 is applied for all the other relative densities and the results are plotted in Figure 11. For lower relative density values, the curves approach towards the le and ne axis, indicating a reduced necessity for modification, as expected.

As a result, it was concluded that there is an infinite number of ne–le parameter pairs that are capable of aligning the elastic modulus values of the solid UC model and those of the modified UC beam model. However, the shear modulus of the modified beam UC model is always larger than that of the solid UC model. And the extent to which the ne value increased does not help: the shear modulus of the modified beam UC model fails to reach the desired solid UC model’s shear modulus value. For a constant ne = 10⁴ value, the required le parameters for E_le/E = 1 and G_le/G values for other relative densities are listed in

Table 6. The parameter le values for ne = 10⁴ and E_le/E = 1 for $\overline{\rho }=0$.1–0.5.

$\overline{\mathit{\rho }}$	$\mathit{l}\mathit{e}$	${\mathit{G}}_{\mathit{l}\mathit{e}}/\mathit{G}$
$0.1$	$0.43604$	$1.0428$
$0.2$	$0.66304$	$1.0774$
$0.3$	$1.01876$	$1.1165$
$0.4$	$1.04959$	$1.1586$
$0.5$	$1.06556$	$1.2079$

. It can be concluded from the third column of Table 4 that the shear modulus value obtained from the modified beam UC model for $\overline{\rho }$ = 0.5 was at least 20.79% higher than the solid UC model’s shear modulus value, with this modification method found in the literature. For relative densities of $\overline{\rho }=$0.1, 0.2, 0.3, and 0.4, the error percentages were 4.28%, 7.74%, 11.65%, and 15.86%, respectively, as shown in Table 6.

3.5. A Novel Beam UC Model Modification Using Spring Elements

Firstly, an unmodified UC beam model was constructed, following exactly the same procedure that was explained in Section 3.2. Then, seven springs with two different constants, k₁ and k₂, were introduced within the unit cell. The proposed modification of the beam UC model is illustrated in Figure 12. These springs are simulated using the spring finite element (COMBIN14), which has two nodes and provides uniaxial tension–compression capabilities in 3D applications [29]. The spring constant is the only required data, and it is denoted as the real constant to define the behavior of this element. In the first test of homogenization conducted on the 1/8 of the unit cell, twice the constants were defined for the transverse cut springs.

In Figure 12, the beam finite elements are visually represented by grey color, without a section display. There are three spring elements with the constant k₁ and each connects to the opposite face centers (labeled by C) of the unit cell, which is aligned in parallel with the cell side length, as shown by the blue color in Figure 12. These CC springs were stretched in the tensile test, but did not change the length of the shear test during the application of the average stress method used in this study. Additionally, four spring finite elements, with the constant k₂, were introduced into the unit cell, as shown by the red color in Figure 12. Each of these springs connected to the opposite corners (labeled by D) of the unit cell. These DD springs changed in length and contributed not only to the tensile but also to the shear test. The modification proposed here also has two parameters: the spring constants k₁ and k₂. Inserting the spring elements into a finite element model for modification is a straightforward and practical approach for the modeling of truss lattice structures.

Since the CC springs do not affect shear test, k₂ can be determined first, which is independent from k₁. So, the shear test was applied on this modified beam UC model iteratively until the required constant, k₂, of the DD springs was found to give the desired shear modulus value of the solid UC model. Next, this k₂ value was used as input data in the tensile test simulation, which was repeated until the required spring constant, k₁, was reached to obtain the desired elastic modulus value of the solid UC model. This procedure was applied within the range of relative density, $\overline{\rho }$ = 0.01–0.5, and all the required spring constant values, k₁ and k₂, were calculated. These k₁ and k₂ values were normalized with E_b L_c and G_b L_c, respectively, and they are presented in

Table 7. The parameters k₁ and k₂ calculated for the modification of beam model with springs.

$\overline{\mathit{\rho }}$	$\mathit{R}/\mathit{L}$	${\mathit{k}}_1(\mathbf{N}/\mathbf{m}\mathbf{m})$	$\frac{{\mathit{k}}_1}{{\mathit{E}}_{\mathit{b}}{\mathit{L}}_{\mathit{c}}}$	${\mathit{k}}_2(\mathbf{N}/\mathbf{m}\mathbf{m})$	$\frac{{\mathit{k}}_2}{{\mathit{G}}_{\mathit{b}}{\mathit{L}}_{\mathit{c}}}$
$0.10$	$0.0659$	$158.614$	$0.0015$	$71.777$	$0.0017$
$0.20$	$0.0967$	$668.002$	$0.0062$	$280.371$	$0.0067$
$0.30$	$0.1226$	$1719.652$	$0.0159$	$675.800$	$0.0163$
$0.40$	$0.1464$	$3588.791$	$0.0332$	$1323.556$	$0.0319$
$0.50$	$0.1695$	$6647.467$	$0.0616$	$2293.321$	$0.0552$

and plotted in Figure 13 and Figure 14.

The closed expressions of the spring constants in terms of aspect ratio are obtained by means of curve-fitting on this data within the range of relative density, $\overline{\rho }$ = 0.1–0.5, as follows.

$${\displaystyle \frac{k_1}{E_b{L}_c}}=89 {\left({\displaystyle \frac{R}{L}}\right)}^{4.1}$$

(8a)

$${\displaystyle \frac{k_2}{G_b{L}_c}}=39 {\left({\displaystyle \frac{R}{L}}\right)}^{3.7}$$

(8b)

Here, the coefficients of determination from Equation (8a) and Equation (8b) are 0.99977 and 0.99992, respectively. These expressions are also illustrated in Figure 14. For the particular relative density of the octet-truss lattice material, after calculating the aspect ratio from Equation (3), Equation (8a,b) can be used to calculate the required spring constants to apply this modification method for $\overline{\rho }$ = 0.1–0.5.

This proposed modified beam model provides a straightforward procedure for calculating the required parameters, because k₂ is independent from k₁. Once k₂ is obtained from the shear test, k₁ can be calculated from the tensile test.

In this modification method, the springs were attached to the joints at the faces’ centers, and at the corners, those were present in the octet-truss unit cell. Although the same spring arrangement was not possible for the unit cells, which do not have joints at both the face centers and at the corners, using more than one unit cell in the RVE can overcome this discrepancy.

Heretofore, the spring constants were determined by running the beam UC model with springs in “do loops” to match the solid UC model results. Analytical expressions for spring constants can also be determined in terms of E_k and G_k. The reason for introducing the springs was to cover the difference between the moduli of the solid UC model and the unmodified beam UC model. The differences were denoted as E_k = E − E_f and G_k = G − G_f for the elastic and shear modulus values, respectively, as has been shown in Figure 4. In order to find the spring constants, the average stress method is applied on the inserted linear spring system alone, as shown in Figure 15; the homogenized material constants of this spring system can be directly equated to E_k and G_k.

The forces carried by the DD springs during the shear test are shown in Figure 15a; the shear test was used to calculate the fourth column of the stiffness matrix, given in Equation (4). In this test, two diagonal springs, denoted by D_tD_t, were under tension, while the other two, denoted by D_cD_c, were under compression. The components of all the spring forces shown in Figure 15a had the same magnitude and were equal to k₂ L_c/3. The CC springs did not carry any force in the shear test. The only non-zero average stresses can be calculated as (τ₂₃)_ave = C₄₄ = 4 k₂/3/L_c. Using G₂₃ = G_k and Equation (5c), G_k can be determined as follows:

$$G_k={\displaystyle \frac{4k_2}{3L_c}}$$

(9)

The forces carried by the CC and DD springs in the tensile test are shown in Figure 15b; the tensile test was used to calculate the first column of the stiffness matrix, given in Equation (4). The CC spring in 1-direction, and all the DD springs were under tension, as shown in Figure 15b. The magnitude of all the force components at the corners was equal to k₂L_c/3 and the force at the right face center was equal to k₁L_c. The non-zero average stresses can be calculated as (σ₁)_ave = C₁₁ = 4 k₂/3/L_c + k₁/L_c and (σ₂)_ave = C₂₁ = (σ₃)_ave = C₃₁ = 4 k₂/3/L_c. Using E₁ = E_k and Equation (5a), E_k can be calculated as follows:

$$E_k={\displaystyle \frac{3k_1\left(k_1+4k_2\right)}{L_c\left(3k_1+8k_2\right)}}$$

(10)

The parameter pairs, k₁ and k₂, can be solved from Equations (9) and (10) as follows:

$$k_1={\displaystyle \frac{1}{2}}\left[E_k-3G_k+\sqrt{E_k^2+2E_k{G}_k+9G_k^2}\right]L_c$$

(11a)

$$k_2={\displaystyle \frac{3}{4}}{G}_k{L}_c$$

(11b)

4. Octet-Truss Lattice Structure Examples

Two example problems are solved in this section in order to investigate the performances of the UC models, with which the lattice structures were constructed. The cubic lattice structures of n³ unit cells were solved with increasing number of unit cells; here, n is the number of unit cells in one direction. Both examples were solved only for the relative density value of $\overline{\rho }$ = 0.5, at which the maximum difference between the solid and beam UC models were observed. Depictions of the lattice structure volumes for n = 3 for both examples are shown in Figure 16a,b, where the symmetry of the problems is used. Four UC models were applied for each example problem.

The types of finite elements and the material types with numerical data, used for these four models, are presented in

Table 8. Finite element type and material data used in structure models of examples 1–2.

Model:	Solid Structure	Homogenized Bulk Structure	Beam Structure with Stiff Joints	Beam Structure with Springs
Finite element:	SOLID185	SOLID185	BEAM188	BEAM188
Material type:	Linear/isotropic	Linear/Anisotropic (cubic symmetric)	Linear/isotropic	Linear/isotropic
Material data:	$E=18 \mathrm{G}\mathrm{P}\mathrm{a}$ $v=0.3$	Stiffness matrix data: $C_{11}=4206.76 \mathrm{M}\mathrm{P}\mathrm{a}$ $C_{12}=1738.92 \mathrm{M}\mathrm{P}\mathrm{a}$ $C_{44}=1797.58 \mathrm{M}\mathrm{P}\mathrm{a}$	$E=18 \mathrm{G}\mathrm{P}\mathrm{a}$ $v=0.3$ $ne=\mathrm{10,000}$ $le=1.06556$	$E=18 \mathrm{G}\mathrm{P}\mathrm{a}$ $v=0.3$ $k_1=6647.47 \mathrm{N}/\mathrm{m}\mathrm{m}$ $k_2=2293.32 \mathrm{N}/\mathrm{m}\mathrm{m}$

. The solid lattice structure model was constructed with the solid UC models, as explained in Section 3.1. The homogenized bulk model comprised homogenous continuum media with anisotropic material properties. The stiffness matrix data given in Table 8 were obtained from the homogenization of the solid UC model. For the stiffness matrix components, E = 3189.60 MPa, G = 1797.58 MPa, and ν = 0.292 were the material constants for $\overline{\rho }$ = 0.5, calculated using Equation (5a–c), as shown in Table 4. The finite element mesh of the homogenized bulk structure model was L_c/2. The beam model with the stiff-joint-based structure was built with the beam UC model that was explained in Section 3.4. The beam models with a spring lattice structure consisted of the unit cells proposed in Section 3.5. If the number of cells in a model is odd, then the springs in the symmetry plane are cut. If the spring is cut transversely, twice the spring constant is defined; if the spring is cut lengthwise, then half the spring constant is attained. The first example is solved for the number of unit cells from 1 to 10. But, since the physical memory of the computer used in this study (as mentioned in Section 3) was not enough to solve the second example for 10 unit cells, the results for 1 to 9 unit cells are presented.

4.1. Example 1: Compressed Lattice Structure

The first example is the compression of a lattice structure with an increasing number of unit cells, $\overline{\rho }$ = 0.5. Due to x-, y-, and z-symmetry, 1/8 of the lattice model was used in the simulations. The simulation for the model with three unit cells is shown in Figure 16a. This first example was solved by fixing the top surface nodes in the lateral direction. The front and right surfaces were left free of conditions. The boundary conditions are tabulated in

Table 9. Boundary conditions of example 1 for the laterally fixed top surface.

	Solid Structure Models (Solid, Homogenized Bulk)	Beam Structure Models (with Stiff Joints, with Springs)
Left	X-sym	X-sym
Bottom	Y-sym	Y-sym
Back	Z-sym	Z-sym
Top	$U_X=U_Z=0,U_Y=-L_m/4$	$U_X=U_Z=0,RO{T}_X=RO{T}_Z=0,U_Y=-L_m/4$

for the laterally fixed top surface. This example was also solved for the laterally free top surface (U_X and U_Z are left free of conditions). With the increasing number of unit cells in the models, Ratio₁ = F_ytop L_m/A_top/U_ytop/2 was traced. Here, L_m is the lattice structure side length (n L_c), U_ytop is the displacement loading applied to the compressed surface A_top = L_m²/4, and F_ytop is the postprocessed force required for this displacement. The change in Ratio₁ with respect to the number of unit cells in the lattice material is plotted in Figure 17. The deformed shapes of the lattice material for the model with three unit cells are presented in Figure 18 for the laterally fixed ends.

4.2. Example 2: Laterally Loaded Lattice Structure

The second example involved the simulation of half of a beam built in at both ends and loaded at the mid-point. Due to z-symmetry, a half lattice structure model was used in the simulations. The one for three unit cells is shown in Figure 16b. The boundary conditions are tabulated in

Table 10. Boundary conditions of example 2.

	Solid Structure Models (Solid, Homogenized Bulk)	Beam Structure Models (with Stiff Joints, with Springs)
Left	$U_X=U_Y=0$	$U_X=U_Y=0,RO{T}_Y=RO{T}_Z=0$
Back	Z-sym	Z-sym
Right	$U_X=0,U_Y=-L_m/4$	$U_X=0,RO{T}_Y=RO{T}_Z=0,U_Y=-L_m/4$

. The top, bottom, and front surfaces were left free of conditions. With the increasing number of unit cells in the models, Ratio₂ = F_yright L_m/A_right/U_yright was traced from the results. Here, U_yright is the displacement loading applied to the right surface, A_right = L_m²/2, and F_yright is the postprocessed force required for this displacement. The change in Ratio₂ with respect to the number of unit cells in the lattice material is plotted in Figure 19. The deformed shapes of the lattice material for the model with three unit cells are presented in Figure 20.

The times spent for Ansys to complete the structure simulations for example 1 with 10 cells and example 2 with 9 cells are given in

Table 11. Ansys 25.1 simulation execution time (s), number of elements, and nodes of the structure models of 10 UC in example 1 and 9 UC in example 2.

	Example 1: Compression (1/8 of 10 Cell Structure)			Example 2: Laterally Loaded (Half of 9 Cell Structure)
UC Model	Time (Sec)	Elements	Nodes	Time (Sec)	Elements	Nodes
Solid	$508$	$\mathrm{3,168,000}$	$\mathrm{3,798,886}$	$2891$	$\mathrm{9,237,888}$	$\mathrm{11,032,565}$
Homogenized bulk	$<1$	$1000$	$1331$	$1$	$2916$	$3610$
Beam—stiff joint	$98$	$\mathrm{75,900}$	$\mathrm{87,066}$	$758$	$\mathrm{216,108}$	$\mathrm{238,325}$
Beam—spring	$109$	$\mathrm{70,175}$	$\mathrm{79,266}$	$713$	$\mathrm{200,151}$	$\mathrm{217,022}$

. Here, the software was run by reading the text files, so the time values include not only the solution but also the modeling of the simulations. It is shown in Table 11 that the computational cost of the simulations of the beam models are clearly very effective compared to those of the solid models.

The elastic modulus is primarily dominant in governing the behavior of the first example. The Ratio₁ value of example 1 for the laterally free top surface of all models converged to the homogenized effective elastic modulus value of 3189.60 MPa, as expected. For both examples solved in this study, the Ratio values converged to a constant value when the number of unit cells, n, was larger than 6. In other words, the effect of unit cell size on the lattice structure was almost lost after six cells.

The second example is suitable for inspecting the effect of the shear modulus on the behavior of the lattice structure. Compared to the lattice structure, constructed with solid unit cells for n = 9, the error percentages of the beam with stiff joints, the homogenized bulk, and the beam with the spring unit cell models are 11.1%, 2.1%, and −1.5%, respectively. Both the beam models with stiff joints and the homogenized bulk models gave higher values, but the beam models with springs gave lower values than Ratio₂ of the lattice structure constructed with solid unit cells.

5. Conclusions

In this study, the average stress homogenization method was applied by means of a finite element method on four different UC models to obtain the material constants of octet-truss lattice materials. One octet-truss unit cell was used as the RVE in the homogenization process. The homogenized material constants of the most accurate, geometrically solid model and those of the unmodified beam model were compared in the relative density range of $\overline{\rho }$ = 0.01–0.5. The modification of beam model, achieved by increasing the stiffness of the beam elements at the joints, is discussed in detail. An alternative new beam model, modified with spring elements, is proposed. Finally, two example problems are solved in order to observe and compare the performances of these models with an increasing number of unit cells. The first example was the compression of the lattice structure with fixed ends and the second example was the laterally loaded lattice structure.

• The average stress homogenization method commonly applied to composite materials in the literature was modified and used for lattice materials.
• The homogenized material constants of octet-truss cells are presented with respect to both the strut aspect ratio and the relative density in the range of $\overline{\rho }$ = 0.01–0.5.
• The results of the octet-truss solid model and the unmodified beam model are compared; here, it is observed that the difference between the solid model and the beam model is less than 12% for low values of relative density, $\overline{\rho }$ < 0.09, but for higher values, the results of the beam model underestimate the results of the solid model. These differences reach up to approximately 40% and 30% for the elastic and shear moduli, respectively, for $\overline{\rho }$ = 0.5.
• Closed expressions for the moduli of the octet-truss lattice material are presented in terms of the aspect ratio by curve-fitting on the results of the solid model for $\overline{\rho }$ = 0.1–0.5. The moduli with respect to those of bulk material and unmodified beam model are presented by Equations (6a,b) and (7a,b), respectively. As a practical alternative method to be used at this level, the error will remain under 12% if these equations are used for $\overline{\rho }$ > 0.09.
• Beam model modification, achieved by increasing the stiffness of the struts at the joint region, is investigated in detail. It is concluded that both of the modulus values cannot be exactly fitted to the solid model results with this type of modification.
• A new modification method for the beam model is proposed by including seven spring elements, with two different spring constants, k₁ and k₂, with which not only the elastic but also the shear modulus can be captured. The required spring constants for various relative densities of the octet-truss lattice unit cell are presented. This proposed modified beam model provides a straightforward procedure to calculate the parameters.
• The closed expressions of the spring constants, k₁ and k₂, in terms of aspect ratio, are presented for relative density range of $\overline{\rho }$ = 0.1–0.5.
• In this model, the springs are attached to the joints at both the face centers and the corners of the octet-truss unit cell. The same spring arrangement can also be applied to unit cells that lack joints at both of the locations by using multiple unit cells within the RVE.
• An alternative method for obtaining the spring constants is presented with closed expressions in terms of the differences between the moduli of the solid model versus those of the unmodified beam model, in addition to the differences in the unit cell side lengths, which are very practical for linear elastic materials. The spring system can be mounted to any beam model of lattice cell for calibration, and the constants can be calculated easily using Equation (11a,b).
• The performance of the UC models forming the basis of lattice structures is evaluated through two example problems with increasing numbers of unit cells. The first example involves the compression of the lattice structure, at which the elastic modulus is dominant in mechanical behavior. It is observed that the results of all the models converged to a constant value when the number of unit cells is n = 6.
• In the second example, where the shear modulus is primarily dominant in governing the behavior of the lattice structure, the beam model with springs gave results that were closest to those of the solid model. The proposed beam model with springs for the octet-truss lattice is suitable for large-scale structures where shear modulus is dominant in dictating mechanical behavior.