Abstract

Cellular materials are characterized by a complex interconnected structure of struts or plates and shells which make up the cells edges and faces. Their structure can be advantageously engineered in order to tailor their properties according to the specific application. This aspect makes them particularly attractive for the manufacturing of bone prosthetics since, compared to traditional fully dense implants, although more complex to produce and with less predictable properties, implants with a highly porous structure can be manufactured to match the bone stiffness and at the same time favor bone ingrowth and regeneration. The development of Selective Laser Melting (SLM) made possible to obtain metallic cellular materials with highly complex structures characterized by a wide range of cell morphologies that allow to finely tune the mechanical properties of the implant to the patient needs. Titanium alloys such as Ti-6Al-4V have shown excellent biocompatibility combined with good mechanical properties and have also been successfully used in the manufacturing of lattice structures with minute details via SLM. Nevertheless, there are still several issues to consider. For instance, despite the static mechanical properties of such lattices being addressed by many studies, the fatigue behavior still remains little investigated, even though it is a critical aspect in load-bearing biomedical implants (consider, for example, the periodic nature of human gait in the case of hip implants). In this regard, increasing the fatigue resistance of cellular lattices by finely adjusting the geometry, for instance by adding fillets at the cell-wall joints, is a new interesting opportunity made possible by additive manufacturing technologies. On the other hand, a discrepancy between the as-designed and the as-built geometry in SLM parts is an issue that can be critically important for lattices with pore size and strut thicknesses of a few hundred microns, such as biomedical lattices. Indeed, any geometrical imperfection introduces a degree of uncertainty that can alter the mechanical properties of the as-built lattice. This work represents an attempt in the direction of building a deeper understanding of the effect of the fine geometrical details, such as the fillet radius at the joints and the thickness of the struts, on the elastic constants and on the fatigue resistance of Ti-6Al-4V SLM lattices, with the aim to develop analytical predictive models of the mechanical properties. Moreover, this work also aims at investigating the as-built/as-designed morphological discrepancy in lattices in relation to the their as-designed geometry and its effects on the elastic modulus and the fatigue resistance. In this regard, the purpose is to develop quantitative relationships between the as-designed and the as-built geometry in order to obtain design tools to predict the final morphology of the lattice by taking into account the manufacturing errors. This thesis covers a wide range of topics, therefore, in the interest of a better presentation, the results of the research have been devided into three independent Chapters. Each of them has been provided of an abstract and an Introduction and divided into a Materials and Methods (or Modelling) section, a Results and Discussion section and finally Conclusions and References. Naturally, the chapters are logically connected and coherent with the frame defined by the title of the thesis. Therefore, this thesis is organized into five chapters. In the first Chapter the backrground to the topics discussed in the subsequent chapters is provided and the relevant literature is reviewed, while in the fifth and last Chapter some conclusions are drawn, and future perspectives are discussed. The core of the work is contained in the three central chapters. In Chapter II, analytical models developed to predict the elastic constants and the stress concentration factors (SCF) of 2D lattices with variously arranged square cells and filleted junctions are presented. The effect of stretching and bending actions on the elastic constants of a single cell is identified by devising an analytical model based on classical beam theory and and periodic boundary conditions. Specifically, two spatial arrangements are considered: a honeycomb with regular square cells and a honeycomb with square cells staggered by a prescribed offset of half of the cell wall length. The theoretical beam model is fitted to the results of a 2D Finite Elements (FE) model based on plane elements via an extensive parametric analysis. In this way, semi-analytical formulas are proposed to calculate the stiffness in large domains of the geometric parameters (strut thickness t0 and fillet radius R). A numerical method is also proposed to estimate the SCFs at the cell wall junctions of a 2D regular square cellular lattice. The aim is to obtain a model capable of calculating the values of the SCF as a function of the unit cell geometrical parameters and consequently assess the stress state in the lattice, which is one of the main factors determining fatigue resistance. This was achieved by applying the FE method to the unit cell for wide intervals of t0 and R to calculate the SCF for each couple of the parameters. The values of the SCFs were then fitted with functions. The models developed in this Chapter are then used in the subsequent chapters as a support in the design of 3D regular square lattices and in the interpretation of the mechanical characterization. In Chapter III, the results of the mechanical and morphological characterization of different regular cubic open-cell cellular structures produced via SLM of Ti-6Al-4V alloy, all with the same nominal elastic modulus of 3 GPa that matches that of human trabecular bone, are presented. The fully reversed fatigue strength at 106 cycles and the elastic modulus were measured and an attempt was made to link them to the manufacturing defects (porosity and geometrical inaccuracies). Half of the specimens was subjected to a stress relief thermal treatment while the other half to Hot Isostatic Pressing (HIP), and the effect of the treatments on porosity and on the mechanical properties was assessed. The results of fatigue and quasi-static tests on regular cubic lattices were compared with FE calculations based on the as-designed geometry and on the as-built geometry reconstructed from micro X-ray computed tomography (µCT) scans. It was observed that the fatigue strength and, to a lesser extent, the elastic modulus are correlated with the number and severity of defects and that predictions on the mechanical properties based on the as-designed geometry are not accurate. The fatigue strength seems to be highly dependent on the surface irregularities and on the notches introduced during the manufacturing process. In fully reversed fatigue tests, the high performances of stretching dominated structures compared to bending dominated structures are not found. In fact, with thicker struts, such structures proved to be more resistant, even if bending actions were present. Given the small size of the unit cells (the unit cell size is 1.5 mm and the strut thickness is 0.26 mm) and the limitations in accuracy of the printer, the fillet radii at the junctions were highly irregular and somewhat hard to recognize. In order to investigate the real benefit of filleted junctions on the stress concentration effects at the junctions and to assess the manufacturability of such minute geometrical detail, a new experimental campaign was set up. In Chapter IV, a set of cubic lattice specimens with filleted junctions was designed and produced via SLM. The size of the unit cell is considerably larger than that of the previous specimens, being 8 mm, 6 mm and 4 mm with the rest of the geometrical parameters scaled accordingly. Thus, nine combinations of the geometrical parameters of the unit cell and three orientations with respect to the printing direction are considered. The aim is to investigate the relationship between the as-designed and the as-built geometry and to find the smallest radius which can be accurately reproduced by the printer. Moreover, a compensation strategy of the morphological defects is devised using the mathematical relationships obtained between the as-designed and the as-built strut thickness. This strategy consists in modifying the input CAD to compensate for the deviations introduced by the SLM process.