Mechanical Performance of a Node-Reinforced Body-Centered Cubic Lattice Structure: An Equal-Strength Concept Design

Abstract

Lattice structures are characterized by a light weight, high strength, and high stiffness, and have a wide range of applications in the aerospace field. Node stress concentration is a key factor affecting the mechanical performance of lattice structures. In this paper, a new equal-strength body-centered cubic (ES-BCC) lattice structure was additively manufactured using 316L stainless steel via selective laser melting (SLM). The results of a mechanical compression test and finite element analysis revealed that the failure location of the ES-BCC structure changed from the nodes to the center of the struts. At the same density, the energy absorption, elastic modulus, and yield strength of the ES-BCC structure increased by 11.89%, 61.80%, and 53.72% compared to the BCC structure, respectively. Furthermore, the change in angle of the ES-BCC structure achieves significant changes in strength, stiffness, and energy absorption to meet different design requirements and engineering applications. The equal-strength concept design can be applied as a general design method to the design of other lightweight energy-absorbing lattice structures.

1. Introduction

Lattice structures are structures constructed by combining one or more structural cytons in a periodic array. Due to their superior characteristics of light weight, high strength, high stiffness, energy absorption, vibration damping, and heat and noise insulation, they are widely used in lightweight design, aerospace, biomedicine, and automotive ships [1,2,3]. The advancement of additive manufacturing technology has advantages in the design and processing of lattice structures, which has greatly promoted the development of such structures and made them a hotspot for lightweight structure research in the aerospace and aviation fields [4].

The current reported lattice structures include body-centered cubic (BCC), face-centered, simple cubic, octet, diamond, rhombic dodecahedron, pyramidal, octet-truss, triply periodic minimal surface, and hybrid structures [5,6,7,8,9,10,11]. Besides the materials used for manufacturing structures, the shape and geometrical parameters of the lattice structure determine its mechanical properties, i.e., the number, orientation, and diameter of the struts [12]. Lattice structures have been classified into tensile-dominated and bending-dominated types [13,14,15]. Therefore, the deformation mechanism of the struts under load has been considered as determining the mechanical performance of the structure [16,17]. However, almost all of the currently reported stress concentration is often initiated at the nodes, which leads to faster crack initiation and failure [18,19,20].

Some studies improve the mechanical properties of lattice structures through the node design. A novel BCC lattice structure with tapered struts was designed to reduce the stress concentration at the nodes [6]. The results show that the structure fracture location changed from the nodes to the tapered strut center, and the elastic modulus was increased by 67%. A new graded-strut body-centered cubic structure was obtained by replacing node right angles with node fillets [21,22]. In the filleted structure, strut failure initiated near the end of the fillet, and the compressive modulus and yield strength were increased by at least 38.20% and 34.12%, respectively. Taking inspiration from a crystal structure, a BCC structure with atoms as reinforced nodes and atomic bonds as connected struts has been manufactured [23], and the yield strength and collapse strength were increased by 12.31% and 31.19%, respectively. Although experience-based node reinforcement design improves the mechanical performance of structures, it lacks theoretical support and is difficult to widely apply.

In the present work, a new equal-strength body-centered cubic (ES-BCC) lattice structure was proposed. The material is additively manufactured with 316L stainless steel via the SLM process. The deformation mode, elastic modulus, yield strength, and energy absorption of the ES-BCC structure were investigated using numerical simulation and compression tests and compared with the BCC structure. The equal-strength design of the struts can effectively improve the mechanical properties of the structure, which provides a new idea for the design of lightweight energy-absorbing structures in the aerospace field.

2. Materials and Methods

2.1. Design of ES-BCC Structure

The ES-BCC structure was designed based on the equal-strength concept of the BCC structure. Figure 1 shows the schematic diagram of the BCC structure under axial compression: the shaded portion indicates that the bottom of the structure is a fixed constraint. The BCC lattice structure, as shown in Figure 1a, contains unit cells with bottom dimensions of L × L and the height H. The eight struts surrounding the node are taken as the analysis module, as shown in Figure 1b. Assuming that the compression load is F, since the structure is geometrically symmetric, any strut near the node can be taken for force analysis, which is simplified into a cantilever beam structure. The strut AB is taken for force analysis, as shown in Figure 1c.

Decomposing the load F, the axial and tangential forces on the strut can be obtained as:

$$\left\{\begin{array}{c}{F}_x=F\sin \theta \\ F_y=F\cos \theta \end{array}\right.$$

(1)

where $\theta$ is the angle between the bottom surface and the structural strut.

Under combined compression-bending deformation, the stress can be represented by:

$$\sigma \left(x\right)=\frac{F_x}{A\left(x\right)}+\frac{M\left(x\right)}{W_z\left(x\right)}$$

(2)

For circular cross-section struts, $A\left(x\right)=\frac{\pi d^2\left(x\right)}{4}$ is the cross-sectional area at x, $M\left(x\right)=F_{y}x$ is the bending moment at x, and $W\left(x\right)=\frac{\pi d^3\left(x\right)}{32}$ is the bending section coefficient.

Letting the stress at any x be equal to the allowable stress of the base material, we have:

$${\sigma }_{\max }=\left[\sigma \right]=\left\{\begin{array}{c}\frac{4F\sin \theta }{\pi d_0{}^2}\left(x=0\right)\\ \frac{4F\sin \theta }{\pi d_1{}^2}+\frac{32F\cos \theta l}{\pi d_1{}^3}\left(x=l\right)\end{array}\right.$$

(3)

where d₀ is the initial diameter (diameter at the center of struts), and d₁ is the maximum diameter (diameter at nodes), $l=\sqrt{2L^2+H^2}/4$. From Equation (3), we have:

$$F=\frac{\pi d_0^2\left[\sigma \right]}{4\sin \theta }$$

(4)

Substituting Equation (4) into $\sigma \left(x\right)=\left[\sigma \right]$, the function of the strut diameter can be given as:

$$d^3\left(x\right)-d_0^{2}d\left(x\right)=8\cot \theta d_0^{2}x$$

(5)

Equation (5) is the design formula for the ES-BCC structure. Obviously, the diameter of the strut is related to d₀ and $\theta$.

As shown in Figure 1b, $\varphi$ is the angle between the struts in the front view of the structure. From the geometric relationship, the relationship between $\varphi$ and $\theta$ can be obtained:

$$\sin \theta =\frac{\tan \frac{\varphi }{2}}{\sqrt{2+{\tan }^2\frac{\varphi }{2}}}$$

(6)

ES-BCC unit cell structures with d₀ = 0.3 mm are shown in

Table 1. The ES-BCC unit cell structure with d₀ = 0.3 mm.

$\mathit{\varphi }$	$\mathit{\theta }$	d (x)	Strut Profile	Unit Cell Model
75°	27.37°	$d^3\left(x\right)-0.09d\left(x\right)=\frac{18\sqrt{2}}{25\left(\sqrt{3}-1\right)}x$
90°	35.26°	$d^3\left(x\right)-0.09d\left(x\right)=\frac{18\sqrt{2}}{25}x$
105°	42.66°	$d^3\left(x\right)-0.09d\left(x\right)=\frac{18\sqrt{2}}{25\left(2+\sqrt{6}-\sqrt{2}-\sqrt{3}\right)}$
120°	50.77°	$d^3\left(x\right)-0.09d\left(x\right)=\frac{18\sqrt{6}}{75}x$

. Since a non-smooth transition at the center of the strut would result in stress concentration, a rounded corner transition with R = 3 mm was performed in the actual modeling. As shown in Table 1, the shapes of the strut profiles determined using Equation (5) are similar, with differences only in the size of the diameters at each location. For structures with $\varphi$ of 75°, 90°, 105°, and 120°, the minimum diameters of the struts after adding rounded corners are 0.86, 0.76, 0.69, and 0.63 mm when d₀ is 0.30 mm, respectively.

2.2. Relative Density

The ratio of the solid volume of the structure to the volume of the quadrilateral filled by the structure is the relative density of the lattice structure. When filling the lattice structure into the cube, the ends of each strut are partially truncated in the x, y, and z directions, and a schematic diagram of the intercepted structural strut dimensions is shown in Figure 2.

The distance between the nodes of the force can be expressed by 2($l-\Delta l$). If the strut is uniform, then $\Delta l=\frac{a+b}{2}=\frac{d}{4}\left(\tan \theta +\frac{1}{\tan \theta }\right)$, where d is the strut diameter. For the new design structures, the diameter d is a function of x, and d is approximated as $\left(d_0+d_1\right)/2$. Thus, we obtain:

$$\Delta l=\frac{d_0+d_1}{8}\left(\tan \theta +\frac{1}{\tan \theta }\right)$$

(7)

here, d₀ is the minimum diameter after adding rounded corners and d₁ is the maximum diameter.

As shown in Figure 2, the volume of the variable-section strut of the 2l segment is $V_1=2{\displaystyle {\int }_0^l\pi r^2\left(x\right)dx}$.

From the above analysis, the volume of the 2($l-\Delta l$)-segment strut can be obtained as:

$$V_2=2{\displaystyle {\int }_0^{l-\Delta l}\pi r^2\left(x\right)dx}$$

(8)

Since each strut of the structure is partially truncated, the volume of the $\Delta l$-segment strut is approximated as $V_{\Delta l}=\left(V_1-V_2\right)/8$. The unit cell volume can be obtained as follows:

$$V=8\left(V_2+2V_{\Delta l}\right)$$

(9)

The relative density can be obtained with rectification as follows:

$$\overline{\rho }=\frac{V}{L^{2}H}$$

(10)

2.3. Specimen Fabrication

The bottom surface dimensions of the ES-BCC unit cell structure are L × L, taking L = 6 mm, and the height H is related to $\varphi$. The ES-BCC structure is named according to $\varphi$ to facilitate later analysis. For example, a lattice structure with $\varphi$ = 75° is named an A075 structure.

The specimens were fabricated using a selective laser melting (SLM) process, and 316L stainless steel was selected as the fabrication material. To eliminate the influence of the specimen printing direction, all specimens were fabricated along the direction of the experimental load. The overall dimensions of the structures were 30 mm × 30 mm × 30 mm. To compare the mechanical properties of the ES-BCC structures with those of the original BCC lattice structure, the conventional BCC lattice structure (strut diameter d = 1.025 mm) was fabricated with the same dimensions and relative density as the A090 structure. In order to avoid errors in the experimental process, two specimens were prepared for all lattice structures. The specimens are shown in Figure 3.

Table 2. 316L stainless steel’s chemical composition (%) [24].

C	Si	Mn	S	P	Cr	Ni	Mo	Fe
0~0.03	0~1.0	0~2.0	0~0.03	0~0.045	16.5~18.5	10.0~13.0	2.0~3.0	Balance

shows 316L stainless steel’s chemical composition [24].

2.4. Experimental Procedure

Quasi-static tests were performed on the specimens using an Instron 5982 universal material testing machine, including tensile and compressive. The quasi-static compression test setup is shown in Figure 4a. The loading speed was set to 2 mm/min. For compression testing, the loading was stopped at the densification stage or 25 mm displacement. For tensile testing, the loading was stopped when the specimen fractured. The force and displacement are automatically recorded by the data acquisition system. A DSLR camera Canon 80 D was used to record the entire process of the experiment in real time. The tensile stress–strain curves of the additive manufacturing 316L stainless steel specimens are shown in Figure 4b. The mechanical properties shown in

Table 3. The mechanical property parameters of 316L stainless steel.

Elastic Modulus (GPa)	Density (g/cm3)	Poisson’s Ratio	Yield Stress (MPa)	Ultimate Stress (MPa)
67	7.59	0.3	443.63	551.63

are similar to those in reference [25,26], which were used in the subsequent numerical simulations.

2.5. Finite Element Modeling

The modeling, meshing, and solving were performed using the CAD, HyperMesh 2021, and LS-DYNA R11.1.0 software. The finite element model is shown in Figure 5a. The effect of the strain rate is not considered in the quasi-static compression tests. To simulate the upper and lower indenters in the compression test, rigid plates are placed and meshed with shell elements, and the lattice structure is meshed with solid elements (C3D8R). The top rigid plate was loaded with displacement at a constant rate of 1 m/s [27,28], which was to improve the computational efficiency of the numerical simulation [29,30,31,32], with a compression displacement of 25 mm, and the bottom rigid plate was completely fixed. The static friction coefficient between the lattice structure and indenters is 0.3 [33,34].

The mesh convergence test was conducted to reduce the influence of the mesh element size on the stress–strain curve, as shown in Figure 5b. The element sizes were 0.2, 0.3, 0.5, and 0.8 mm, respectively. The stress–strain curve of the finite element model with an element size of 0.8 mm had large deviations from those of the other sizes, and the error between the results of element sizes of 0.2, 0.3, and 0.5 mm was very small. Therefore, elements with a grid size of 0.3 mm were used for subsequent simulation calculations.

3. Results and Discussion

3.1. Morphological Characteristics

The masses of all specimens were obtained by weighing them, and the relative densities were calculated. Figure 6 shows the design values, theoretical values, and measured values of the relative density of the ES-BCC structures. The relative density of all specimens was smaller than the design value, the maximum error was 6.2%, and the relative density error between the design and theoretical values was less than 6.9%. The main reason for errors would have been rounded transitions, approximately calculated volumes, defects, and residual powder particles.

Scanning electron microscopy (SEM) was used to assess the manufacturing quality of the experimental specimens. As shown in Figure 7, the surface of each strut was rough. The adhesion of partially melted powder particles can be observed, especially at the nodes and below the struts. Moreover, porosity defects were observed on the struts. At the intersection of the struts, small-angle structures had more residual powder particles than large-angle structures. Furthermore, porosity defects might also exist inside the struts of the lattice structure.

3.2. Mechanical Properties

The compressive stress–strain curves of the A090 structure and the BCC lattice structure were compared, as shown in Figure 8. The simulation curves are in good agreement with the experimental results, especially in the initial phases. Under quasi-static compression, the lattice structure exhibited the following deformation responses: an initial linear elastic phase, a plastic plateau phase where the stress remained constant, and a densification phase where the stress sharply grew. The pentagram in the Figure 8 represents the starting point of the structure into the densification phase. The compressive stress–strain curves of the A090 and BCC lattice structures had the same trend, but at the same strain, the former’s stress exceeded the latter.

The deformation behavior can directly reflect the damage failure mode. Figure 9 depicts the deformation process of the lattice structure in compression tests and simulations. When the strain reached 40%, both the A090 and BCC lattice structures had significant deformation on the diagonal struts, and the center of the structure showed local densification. Struts in various areas exhibited different deformation patterns, with the external struts producing tensile deformation and the internal struts producing buckling deformation. The BCC lattice structure was deformed uniformly in the compression process. The overall stress distribution of the structure was not uniform, and at the nodes of the struts, stress concentration was observed. However, the A090 structure was the first to show bending deformation and compaction at the center of the strut where it made contact with the upper and lower plates. The rest of the layers deformed uniformly. During the compression process, the stress distribution of the A090 structure was more uniform, and the damage failure location was at the center of the strut.

To quantitatively describe the mechanical properties and energy absorption characteristics of the ES-BCC structures, the elastic modulus (E), yield strength (${\sigma }_y$), and energy absorption (EA) were compared. The elastic modulus was defined as the slope of the initial linear elastic segment (about 1–3% of the strain). The yield strength was defined as the stress at the point where the stress–strain curve intersected a straight line whose slope equaled the elastic modulus and passed through the 0.2% strain [35]. The energy absorption of a lattice structure can be represented by the energy absorption per unit volume as the area under the compressive stress–strain curve:

$$EA={\displaystyle {\int }_0^{{\epsilon }_d}\sigma \left(\epsilon \right)d\epsilon }$$

(11)

where $\sigma \left(\epsilon \right)$ and $\epsilon$ are the stress and strain during compression, respectively, and ${\epsilon }_d$ is the maximum strain when the structure is compressed to the densification stage [36].

Table 4. Structural parameters of lattice structures.

Lattice Type		BCC	A075	A090	A105	A120
Relative density	Exp.	0.1265	0.1757	0.1307	0.1023	0.0841
	Sim.	0.1344	0.1842	0.1344	0.1070	0.0880
Densification strain	Exp.	0.6288	0.4492	0.4848	0.5356	0.5864
	Sim.	0.6075	0.4327	0.5142	0.5283	0.5708

shows the densification strains of the experimental and simulated compression stress–strain curves. It can be observed that the densification strain increases with $\varphi$. The ${\epsilon }_d$ of the BCC lattice structure exceeded that of the A090 structure, consistent with the compressive stress–strain curves shown in Figure 9.

Table 5. The elastic modulus, yield strength, and energy absorption of lattice structures.

Lattice Type	Elastic Modulus (MPa)			Yield Strength (MPa)			Energy Absorption (J/cm3)
	Exp.	Sim.	DEVES	Exp.	Sim.	DEVES	Exp.	Sim.	DEVES
A090	373.58	345.46	−7.53%	6.69	6.61	−1.20%	4.68	4.80	2.56%
BCC	238.34	213.51	−10.42%	4.28	4.30	0.47%	4.21	4.29	1.90%

Note: Deviation DEV_ES = (Sim. − Exp.)/Exp. × 100%.

compares the mechanical properties of the structures under study in compression tests and simulations. The discrepancies among the elastic moduli, yield strengths, and energy absorptions were within 10.5%, 1.2%, and 2.6%, respectively, proving the reliability of the numerical simulation. The elastic modulus, yield strength, and EA of the A090 structure outperformed those of the BCC lattice structure by 61.80%, 53.72%, and 11.89%, respectively.

3.3. Influence of $\varphi$

At the initial diameter d₀ = 0.3 mm, the effect of $\varphi$ on the mechanical properties of the ES-BCC structures was studied, as shown in Figure 10. The stress was initially concentrated at the center of the struts of the top and bottom cell layers and propagated to the adjacent layers with the ongoing crush. At a strain of 40%, significant densification occurred at the center of the structure. The smaller $\varphi$ was, with more crystal cells and a higher relative density resulting in faster densification of the structure during compression, the more uniform the stress distribution was. It was also noticed that the collapse began at the top and bottom cell layers, followed by the middle layer, and finally the adjacent two layers began to collapse. The specimens did not produce fracture damage during compression.

The compressive stress–strain curves of ES-BCC structures with different $\varphi$ are presented in Figure 11. It demonstrates the typical stress–strain behavior of lattice structures with three characteristic regions consisting of a linear elastic region, plateau region, and densification region. The change in $\varphi$ did not affect the general trend of the stress–strain curve but at the same strain, the stress decreased with an increasing $\varphi$. The smaller the angle, the greater the elastic modulus, yield strength, and energy absorption, but the influence of mass is not taken into account.

The mechanical properties of ES-BCC structures at different $\varphi$ values were further evaluated by calculating their relative elastic modulus ($E_{\overline{\rho }}$), relative yield strength (${\sigma }_{\overline{\rho }}$), and specific energy absorption (SEA) [37,38]. The calculation formula is as follows:

$$E_{\overline{\rho }}=\frac{E}{\overline{\rho }}$$

(12)

$${\sigma }_{\overline{\rho }}=\frac{{\sigma }_y}{\overline{\rho }}$$

(13)

$$SEA=\frac{EA}{\rho }=\frac{{\displaystyle {\int }_0^{{\epsilon }_d}\sigma \left(\epsilon \right)d\epsilon }}{\rho }$$

(14)

where $\rho$ is the density of the lattice structure.

Figure 12 shows the mechanical properties of ES-BCC structures with different $\varphi$. The larger $\varphi$, the greater the relative elastic modulus, relative yield strength, and specific energy absorption. When the $\varphi$ of the structure increased from 75° to 120°, the relative elastic modulus grew from 2.19 to 3.64 GPa (i.e., by 66.21%), the relative yield strength grew from 44.34 to 65.87 MPa (i.e., by 48.56%), and the SEA increased from 3.89 to 5.59 J/g (i.e., by 43.70%). Therefore, the A120 structure had a superior elastic modulus, yield strength, and EA at the same relative density. Mainly due to the increase in angle, the compression force of the structure increases and the bending force decreases.

3.4. Gibson–Ashby Model Prediction

The Gibson–Ashby (GA) model [39] is a classical model theory based on cubic porous structures simplified by cubic struts. It uses the geometric mechanics method to link porous materials’ mechanical properties and can effectively predict their tensile and compressive properties, such as the elastic modulus, tensile and compressive strength, and energy absorption. The normalized elastic modulus and normalized yield strength of the lattice structure can be characterized by power functions of the relative density via the GA equations as follows:

$$\frac{E}{E_s}=C_1{\overline{\rho }}^{n_1}$$

(15)

$$\frac{{\sigma }_y}{{\sigma }_s}=C_2{\overline{\rho }}^{n_2}$$

(16)

where E and ${\sigma }_y$ are the compressive modulus and yield strength of the lattice structure, respectively, $E_s$ and ${\sigma }_s$ are the elastic modulus and yield strength of the matrix material, respectively, while the coefficient C and the exponent n characterize the geometry of the unit cell. The exponent reflects the mechanical properties of the lattice structure better than the coefficient because the functional relationship follows power function characteristics. Structures with different relative densities were analyzed with a d₀ of 0.15, 0.20, 0.25, and 0.30 mm. The fitted curves of the GA model for the E and ${\sigma }_y$ of the lattice structures are shown in Figure 13.

As shown in Figure 13, the E and ${\sigma }_y$ of the lattice structures increased as $\overline{\rho }$ increased. The mechanical properties improved significantly with increasing $\varphi$, while the E and ${\sigma }_y$ of the A120 structure significantly exceeded those of other lattice structures with the same relative density. This substantiates the design concept of high-strength and high-stiffness lattice structures.

The fitted parameters of the GA model for the mechanical properties of ES-BCC structures are listed in

Table 6. The fitting parameters of the GA model for the mechanical properties of ES-BCC structures.

Lattice Type	Normalized Elastic Modulus			Normalized Yield Strength
	C1	n1	R2	C2	n2	R2
A075	0.16	2.00	0.999	0.32	1.87	0.996
A090	0.33	2.06	0.986	0.58	1.93	0.999
A105	0.50	2.08	0.999	1.01	2.03	0.999
A120	0.68	2.03	0.998	1.23	1.94	0.999

. Due to the large difference in the exponents of lattice structures with various geometric configurations [40], according to the Maxwell criterion [41], lattice structures are classified as tensile- or bending-dominated based on their deformation forms. Bending- and tensile-dominant deformation structures have a normalized elastic modulus index n₁ close to 2 and 1, respectively, and a normalized yield strength index n₂ close to 1.5 and 1, respectively. Notably, the indices n₁ and n₂ of the ES-BCC structures were close to 2 and 1.5, respectively. Therefore, the ES-BCC structures under study exhibited bending-dominated deformation under quasi-static compression, which were more stable in the plateau phase and had higher energy absorption [42].

3.5. Theoretical Analysis

3.5.1. Equivalent Elastic Modulus

Assume that the lattice structure is only subjected to an axial compression load F, as shown in Figure 14. The load F is decomposed into the axial force F_x and tangential force F_y, and the strut displacement can be decomposed into axial displacement ${\delta }_x$ and tangential displacement ${\delta }_y$.

The axial displacement ${\delta }_x$ of the strut is:

$${\delta }_x={\displaystyle {\int }_0^l\frac{F_x}{EA\left(x\right)}dx}={\displaystyle {\int }_0^l\frac{4F\sin \theta }{E\pi d^2\left(x\right)}dx}$$

(17)

where E is the elastic modulus of the structural material.

Tangential displacement ${\delta }_y$ is caused by the combined action of the bending moment and shear. The tangential displacement can be obtained using the unit load method as:

$${\delta }_y={\displaystyle {\int }_0^l\frac{\overline{M}M\left(x\right)}{EI\left(x\right)}dx}+{\displaystyle {\int }_0^l\frac{K_s{F}_y}{GA\left(x\right)}dx}$$

(18)

where $\overline{M}=x$ is the bending moment under unit load; K_s = 10/9 is the shear coefficient for a circular section; $G=E/2(1+\mu )$ is the shear modulus; and $I\left(x\right)=\frac{\pi d^4\left(x\right)}{64}=\frac{\pi r^4\left(x\right)}{4}$ is the moment of inertia at x.

Let the displacement of the unit cell in the compression direction be $\delta$. Projecting the axial and tangential displacements onto the force direction, we can obtain:

$$\delta =4\left({\delta }_x\sin \theta +{\delta }_y\cos \theta \right)$$

(19)

The equivalent elastic modulus can be expressed as:

$$E_e=\frac{{\sigma }_e}{{\epsilon }_e}$$

(20)

where ${\sigma }_e=\frac{4F}{L^2}$ is the equivalent stress, and ${\epsilon }_e=\frac{\delta }{H}$ is the equivalent strain, collated to obtain:

$$E_e=\frac{4FH}{L^2\delta }$$

(21)

3.5.2. Equivalent Yield Strength

When the lattice structure is subjected to axial compression, the combined action of the axial force and bending moment is applied to the variable-section struts, resulting in combined compression-bending deformation, eventually causing the damage failure of the structure. Struts are prone to buckling under compressive loading, but the struts in the ES-BCC structure studied in this paper do not satisfy the slender rod condition, so the buckling effect is not considered. According to the compression test, it is found that the plastic yielding damage of the structure was first produced on the tensile side, so taking the tensile stress surface as the plastic damage failure criterion surface, the stress expression at x is as follows:

$$\sigma \left(x\right)=\frac{M\left(x\right)}{W\left(x\right)}-\frac{F_x}{A\left(x\right)}$$

(22)

At a maximum x, i.e., when x = l, the maximum value of stress is observed. Making $\sigma {\left(x\right)}_{\max }$ equal to the yield strength of the substrate, we obtain:

$$F_{\max }=\frac{\sigma \left(l\right)}{\frac{32\cos \theta l}{\pi d^3\left(l\right)}-\frac{4\sin \theta }{\pi d^2\left(l\right)}}$$

(23)

Then, the plastic damage strength is:

$${\sigma }_y=\frac{4F_{\max }}{L^2}$$

(24)

When the lattice structure undergoes plastic failure, it can be regarded as reaching the yield state, and the plastic damage strength at this time is the equivalent yield strength.

3.5.3. Elastic Modulus and Yield Strength Analysis

The parameters E and ${\sigma }_y$ of ES-BCC structures with different $\varphi$ can be obtained by applying the theoretical prediction equation. As shown in Figure 15, the theoretical values of the A075 and A090 structures are smaller than the experimental and simulated values because in the theoretical prediction, the calculation object is the unit cell structure with direct contact between the top layer and the indenter, but in the actual compression process, the top and bottom layers of the structure are the first to produce damage failure, while the inner unit cell is more constrained and bears part of the load. The theoretical values of the A105 and A120 structures are larger than the experimental and simulated values, probably because the struts are longer. The struts are prone to buckling and combined bending and torsional deformation at the nodes, while the influence of strut buckling is not considered in the theoretical calculations.

Table 7. Mechanical properties of ES-BCC structures: experimental, simulated, and theoretical.

Lattice Type	Elastic Modulus (MPa)					Yield Strength (MPa)
	Exp.	Sim.	Theor.	DEVES	DEVET	Exp.	Sim.	Theor.	DEVES	DEVET
A075	384.84	360.05	342.66	−6.44%	−10.96%	7.75	7.43	7.17	−4.13%	−7.48%
A090	373.58	345.46	340.66	−7.52%	−8.81%	6.69	6.61	6.47	−1.20%	−3.29%
A105	285.85	319.38	339.95	11.73%	18.93%	5.19	5.93	5.88	14.26%	13.29%
A120	306.21	323.60	340.09	5.68%	11.06%	5.55	6.07	5.39	9.37%	−2.88%

Note: DEV_ES = (Sim. − Exp.)/Exp. × 100%; DEV_ET = (Theor. − Exp.)/Exp. × 100%.

shows the E and ${\sigma }_y$ values of ES-BCC structures with different $\varphi$ values. The discrepancies in E between the experimental and simulated results ranged from −7.52% in the A090 structure to 11.73% in the A105 structure, while those in ${\sigma }_y$ ranged from −4.13% for the A075 structure to 14.26% for the A105 structure. The deviations between the experimental and simulated or theoretical values of the A105 structure were the largest because of the unavoidable processing defects in the specimens. However, its simulated results were more consistent with the theoretical ones. The discrepancies between the experimental and theoretical values of E and ${\sigma }_y$ for the ES-BCC structures were within 18.93 and 13.29%, proving that the theoretical model could predict E and ${\sigma }_y$ of the ES-BCC structures effectively.

4. Conclusions

In this study, an equal-strength concept was proposed for building novel lattice structures, and a new BCC lattice structure, equal-strength body-centered cubic (ES-BCC), is designed, which can be applied as a lightweight structure in the aerospace field. The mechanical properties of the designed structure were analyzed using experiments and finite element analysis, and the main conclusions are as follows:

(1) ES-BCC lattice structures based on the equal-strength concept were designed to reduce the node stress concentration effectively. The yield position of the BCC lattice structures changed from the node region to the center of strut. The elastic modulus, yield strength, and energy absorption of the A090 structure were improved by 61.80%, 53.72%, and 11.89% compared to the BCC structure, respectively.

(2) The ES-BCC lattice structures all exhibited bending-dominated deformation. A larger $\varphi$ corresponded to a higher relative elastic modulus, relative yield strength, and SEA. Those of the A120 structure reached 3.64 GPa, 65.87 MPa, and 5.59 J/g, respectively, outperforming those of the A075 structure by 66.21, 48.56, and 43.70%, respectively.

(3) By changing $\varphi$ from 75° to 120°, substantial variations in strength, stiffness, and energy absorption were achieved to meet different design requirements and engineering applications. Thus, small-angle ES-BCC lattice structures can be used in energy-buffering devices, and large-angle ones can meet high-strength requirements.