According to Noor [11] lattices can be virtually analysed by direct, discrete field, periodic structure or substitute methods. Direct method involves discrete finite element analysis where the structure is divided into simpler elements. The contributions of each element are evident in global matrix assembled from the properties of the element. This method is computationally expensive for lattices in large domains, which is the case for lattices embedded in mechanical parts. In the discrete field methods, the repetitive nature of the lattice is used to construct compatibility and equilibrium equations which are then solved. This works best for lattices with simple topology and is unsuitable for relatively complex lattices. When analysing lattices with periodic approaches, finite element and matrix methods are used to reduce the size of the lattice. Again, this method requires the exact representation of the behaviour of a member which might not be easily available for complex lattices. The most versatile methods are substitute methods since they allow the replacement of the lattice by a suitable continuum model with similar characteristics to the original lattice. There is significant discrepancies in literature on how best to substitute these lattices; it is therefore paramount that the methods are validated. Advancement in computational resources has allowed the use of direct approaches on relatively smaller lattices. This therefore offers a way to validate efficient substitution methods on large lattices. Luxner et al. [12] used beam elements to investigate elastic and plastic deformation in simple cubic lattices. Highly ordered simple cubic lattices were found to perform better than their disordered counterpart, though they are susceptible to localized damage. Meguid et al. [13] showed that shell elements within a representative unit cell model can represent the crushing behaviour of metallic lattice. Giorgi et al. [14] modelled closed cell aluminium lattice with shell elements and Smith et al. [15] modelled the body centred cubic lattice and its hybrid with continuum three dimensional and beam elements; demonstrating the effects of cellular architecture on SLM built lattices. Some of this work include aspect of material non-linearity, little has been done to include non-linear deformation caused by changes in the boundary and geometry. Material non-linearity is associated with the existence of stresses greater that the yield as such the stress-strain relationship is not linear [16]. Geometric non-linearity is caused by the changing shape of the loaded lattice which affects the stiffness of the structure. The stiffness matrix of the lattice is therefore iteratively updated for each load increment. Also, the deformation of the lattice could cause the boundary conditions to change in a non-linear manner. Non-linear contact analysis is therefore a powerful technique that allow better insight into the deformation mechanism of AM lattices. In this paper, four self-supporting lattice are subjected to compressive loads via non-linear contact analysis. The lattice are explicitly modelled as the computational expense is tolerable for the size of the domain considered. 7.2 Contact Finite Element Study 7.2.1 Lattice models The non-linear behaviour of the four lattice types shown in Fig. 7.1 was investigated. The BCC, BCCZ and FCC unit cells shown in Fig. 7.1a, b, d belong to a family of lattices with cylindrical member connected at joints. The truncated members form the cylindrical members of the lattices when the cells are tessellated in a Cartesian coordinate system. The radius of the member controls the density of the densities of cell and their corresponding lattices. The inclinations of the members to a horizontal plane (which is usually parallel to a build platform) is greater than or equal to 45 ∘ , thereby minimizing the need for supports. Previous work on AM lattice has been largely focused on this sort of structures. The double gyroid cell shown in Fig. 7.1c belongs to a family of relatively unexplored lattices called minimal surfaces. These are surfaces with a mean curvature of zero and topologies that locally minimize their total surface area. The double gyroid lattice is structured in a manner that do not require support when manufactured via an AM technique. It is governed by a trigonometric equation which can be seen in [17]. The four cells were embedded in a 18 mm cubic domain within a Cartesian coordinate system shown in Fig. 7.2. The unit cell size was set at 4.5 mm, thereby giving lattices with four cells in the x, y, and z directions. Also, a common density, 0.23, was enforced across the lattices to ensure a fair comparison can be made. Such a constraint imply that lattice with more cylindrical member would have smaller radii. 7.2.2 Problem Formulation and Mesh Generation Hexahedral meshes were constructed for the four lattices which experience compressive displacement from two rigid bodies as illustrated in Fig. 7.2. The meshes were constituted with approximately 200,000 elements as preliminary 56 A. Aremu et al.
RkJQdWJsaXNoZXIy MTMzNzEzMQ==