Mechanical equivalent diameter of single struts for the stiffness prediction of lattice structures produced by Electron Beam Melting
Graphical abstract
Introduction
Architectured materials received a great interest over the past few decades to achieve new requirements that monolithic material cannot reach, see e.g. [1], [2], [3], [4]. Among other properties, they can combine mechanical, thermal as well as acoustic properties and therefore fulfill multifunctional requirements. Cellular structures can be used as lightweight and stiff structures [5], [6], [7]. When optimized, their high thermal diffusivity can lead to suitable heat exchange media [4], [8]. Foams or random cellular structures can also be a suitable response to blast impact as their toughness is relatively high and their plastic plateau is almost constant [8].
Among cellular structures, lattice structures consist of a connected framework of struts [9]. If the struts are triangulated and satisfy the Maxwell criterion [10], the structure deforms by stretching of the struts. These structures are mainly used in lightweight applications. However the manufacturing of such structures using conventional techniques is rather complex because it requires multiple processing steps of perforating and folding from metal sheets [11], [12]. These steps are time-consuming and allow the production of a limited amount of lattice geometries. Additive Manufacturing (AM) technologies bring about a breakthrough for the production of such lattice structures. They enable the production of parts with a high degree of freedom in design. This work is focused on periodic lattice structures made by the Electron Beam Melting (EBM) technology.
In the last decade, lattice structures produced by EBM have been investigated in the literature [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23]. The properties of periodic lattices (consisting of hexagonal unit-cells) under compression and 3-points bending were first reported by Cansizoglu et al. [18]. They showed that geometrical discrepancies can arise between the designed structure and the fabricated one depending on the strut angle. Fatigue properties of rhombic dodecahedron lattices made by EBM have been studied by Li et al. [19]. Auxetic structures (with a negative Poisson's ratio) were investigated by several research groups, see e.g. [15], [20], [21]. Yang et al. [22] focused on a re-entrant lattice structure produced with the EBM technology. They reported a high variability in strut size and roughness. They proposed an effective strut size from a 2D scheme of a strut. The 3D imaging of struts produced by EBM was carried out by Hernández-Nava et al. [24]. They observed the roughness and the inner porosity but did not extract features to predict the mechanical properties of the produced lattice. In the case of struts produced by Direct Metal Deposition, Park et al. [25], [26] proposed a methodology to take into account the variability of the metal deposition of struts in the mechanical simulation. Their methodology was however based on strut size variability but do not rely on 3D characterizations.
The purpose of this paper is to predict the mechanical response of lattice structures produced by EBM, taking into account the constraints of the process. The outline is as follows.
The EBM process, the produced samples (single struts and octet-truss structure) and the experimental procedures are described in Section 2. Section 3 focuses on the geometrical characterization of 1 mm diameter single struts. The differences between the designed and produced struts are highlighted. The surface irregularities are also quantified and discussed. Section 4 suggests two concepts to link geometrical features and mechanical behavior. These approaches, based on the 3D images, aim at predicting the effective stiffness of the produced struts. Section 5 deals with the application to the stiffness prediction of the octet-truss lattice structure. The predicted values are compared with experimental results from compression tests. The concept of mechanical equivalent diameter is discussed and validated.
Section snippets
The Electron Beam Melting process
The EBM process is an additive manufacturing process that selectively melts metallic powders (in our case Ti–6Al–4V) using an electron beam [27], [28], [29]. The powder used is gas atomized with a median diameter of 77 μm (particle size distribution and other characteristics are detailed in a previous paper [30]). The process occurs under secondary vacuum, typically 10−4 mbar. The Ti–6Al–4V powder is deposited by layers of 50 μm on a stainless steel start plate, then slightly sintered with a
Geometrical characterization of 1 mm diameter single struts
3D geometrical characterization at the scale of a strut provides interesting insights to understand the mechanical behavior of lattice structures produced by EBM.
Concepts to link geometrical features and mechanical behavior
From a 3D binary image, every stack is projected on the plane perpendicular to the strut axis. The inscribed surface is defined as the area common to all the projected stacks.
A geometrical equivalent cylinder can be defined as a circular cylinder having the same length as the strut and a cross-section area equal to this inscribed surface area. Fig. 7 illustrates the nominal geometry of a 1 mm diameter strut (in blue) and compares it with the manufactured one (in green). The geometrical
Application to the stiffness prediction of the octet-truss lattice structure
This section is dedicated to the lattice structures consisting of octet-truss unit cells. Experimental results correspond to the four types of specimen illustrated in Fig. 2(a). After fabrication, their relative density was evaluated by measuring size and mass of each structure. For these structures, the ratio between the fabricated relative density (, ρS is the density of the titanium alloy) and the designed one () was quantified as 63%. This value is due to the
Conclusions
Single struts with a 1 mm diameter and different orientations with respect to the build direction were produced using the Electron Beam Melting technology with standard process parameters. They were characterized using X-ray tomography. Strut porosity content was found to be lower than 0.1%.
Significant size differences were found between the designed struts and the manufactured ones. These discrepancies induced a decrease in the stiffness of the struts and by extension a decrease in the Young's
Acknowledgements
This work was performed within the framework of the Center of Excellence of Multifunctional Architectured Materials “CEMAM” n°AN-10-LABX-44-01 funded by the “Investments for the Future” Program. The authors would like also to thank Jérome Adrien and Eric Maire at INSA, Lyon for their help with the X-ray tomography experiments.
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