Mechanical properties of a new type of plate–lattice structures

Highlights

• Plate lattice is constructed by placing a plate between two adjacent trusses.

• The equivalent stiffness of cubic-plate is twice higher than that of cubic-truss at the same relative density.

• The mechanical properties of plate-lattice attains the theoretical Hashin-Shtrikman (HS) upper bound.

Abstract

In this study, a new type of lattice structure, namely plate–lattice, was investigated. The plate–lattice structure was constructed by placing a plate between two adjacent trusses. Theoretical analysis of Young's modulus on a simple cubic–truss and cubic–plate revealed that the Young's modulus of the plate–lattice was twice that of the lattice composed of trusses. Subsequently, further studies on more complex structures, such as octet–trusses and octet–plates, were conducted using the finite element method (FEM). Furthermore, the periodic boundary condition (PBC) was applied to a unit cell to reflect the response of an infinite structure. Additionally, equivalent Young's modulus, strength, and shear modulus were compared at the same density. The results indicated that the stiffness of the plate–lattices was more likely to realize the Hashin–Shtrikman theoretical upper bounds. The plate–lattice structure exhibited 2–3 times higher stiffness values, including Young's modulus and shear modulus, than those of the truss lattice structure. Furthermore, the Ashby charts of relative compressive modulus (E/E_s) and relative strength (σ/σ_y) were plotted as a function of relative density, and the data indicated that the plate–lattice structure was a new low density structure, which could utilize materials to the maximum extent.

Introduction

Low density cellular solids appear widely in nature and have been studied in mathematicians, mathematics, physics, engineering and biomedicine. The mechanical properties of lightweight cellular structures are a function of the relative density and have attracted tremendous attention in theoretical research and engineering applications. Furthermore, they have exhibited potential in a wide range of multifunctional fields, including mechanical, acoustic, electrical, thermal, and energy absorption [1], [2], [3], [4]. Additionally, recent advanced manufacturing techniques, such as additive manufacturing (AM) [5], [6], [7], [8], have improved the efficiency in fabrication of extremely complex designs. Based on these advantageous features, hierarchical cellular structures in the nanometer to centimeter length scales are being increasingly manufactured and studied [9], [10], [11].

Among all the lightweight cellular structures, metal foams, which are based on the stochastic distribution of cells, are the first generation of artificial structures [12], [13], [14]. They exhibit bending–dominated deformation under elastic loading, and the equivalent stiffness values of the foam scales correspond to ${\overline{\rho }}^2$, where $\overline{\rho }$ is the relative density. The second class is the truss lattice structure, which is composed of slender struts and exhibits a significantly higher specific strength and specific stiffness than those of the stochastic foams [2]. In truss lattice structures, the struts are mainly deformed in tension or compression, while shear and bending are neglected. This deformation mechanism leads to a stiffness and strength scale of $\overline{\rho }$, which significantly enhances the utilization of materials in load bearing structures. To date, emerging additive manufacturing technology has promoted the investigation of various lattices, i.e., pyramidal [15,16], tetrahedral [17], Kagome [18], octet [19], [20], [21], rhombic dodecahedron [22], and X–type [23,24] via theories and experiments. Perhaps, the octet–truss structure is considered as the best lattice and exhibits superior mechanical properties among all the truss lattices. However, the octet–truss structure is still not considered as optimal. The third type is termed as “Shellular,” which is a combination of the words “shell” and “cellular.” In this type of lattice, the cells are composed of continuous and smooth–curved shells [25]. The triply periodic minimal surface (TPMS) [26], [27], [28] is a typical example of a shellular structure. These types of surfaces exhibit zero mean curvature and usually exist as interfaces that separate two sub–volumes. Furthermore, TPMSs have strict mathematical significance; however, they do not perform well as load–bearing structures. The search for an optimal microstructure has been a topic of research for a long time. Interestingly, Sigmund [29] observed that the optimal structures are close–walled as opposed to open–walled (i.e., truss–like) during pure stiffness optimization. In his study, Sigmund reported that the closed box with a thin–wall microstructure exhibited 2–3 times higher stiffness than the open cell with 12 trusses at the edges of a cube with a low volume fraction. Hence, the results indicate that the closed–walled lattice with plates should be investigated further. To distinguish it from the smoothed shellular, we term it as a plate–lattice in this study. Berger [30] first introduced the plate–lattice, which can store more strain energy than the truss lattice structures.

With respect to porous lattice structures, the current popular method involves the use of additive manufacturing for production. However, the plate–lattice structure mentioned in this paper still has some limitations. Firstly, the panel parallel to the horizontal plane cannot be printed owing to the overhangs with zero inclination angle [31,32]. Secondly, given that the structure is closed, the excess powder inside cannot be removed. In this case, the production engineer recommends making a small hole in the panel to remove the excess powder. However, the effect of the hole on the mechanical properties of the structure is unclear. Therefore, the finite element method (FEM) is widely used to predict the mechanical behavior of cellular structures. Qi [21] compared the octet–truss and truncated–octahedron unit cells using a commercial FEM software. The failure modes of TPMS have been derived using FEM [33]. Mahbod [34] proposed the plastic behavior of uniform and graded lattice structures via explicit dynamic FEM. A direct numerical simulation can be performed by using an extremely fine spatial solid finite element mesh to capture the mechanical response in micro scale. The resulting system of equations contains billions of numerical unknowns. However, computations with billions of elements are beyond the capacity of computing machines to date. Hence, two simplified methods are considered to solve this problem. The first method involves modeling with a simplified beam/shell element [35], [36], [37]. Hundreds or thousands of elements are required to model a lattice structure. However, the geometry of the joint is not considered because all the truss elements exhibit a uniform thickness. Furthermore, only the macro strain and stress of the entire structure can be realized, and there is almost no detailed stress field information in the strut. Second, a more common representative volume element (RVE) homogenization method [27,38] is performed to improve the computational efficiency. For lattice structures that exhibit repeating patterns, the representative/effective response can be obtained from a single repeating unit cell via the application of periodic boundary conditions (PBCs). The key point in this method involves the application of PBCs [39], which can reflect the interaction between the neighboring unit cells and calculated unit cell.

In the present study, we first introduce the design concept of a plate–lattice structure with simple cubic–truss and cubic–plate structures in Section 2. From theoretical analyses, it is revealed that the stiffness of a cubic–plate structure is twice that of a cubic–truss structure when both are of the same mass. The octet–plate exhibits higher specific stiffness and strength than the octet–truss. In Section 3, the FEM analysis on the complex octet–truss and octet–plate structures is discussed. The results, including Young's modulus, shear modulus, and anisotropy, are compared and discussed in Section 4.