CHAPTER 2. INTRODUCTION

2.1 Covering new industry requirements: Additive Manufacturing Techniques of polymers

Since the last century, transport industry “moves the world”. They are in constant development of lightweight structures for reducing fuel consumption while maintaining safety and comfort standards. There are two typical ways to manage them. On one hand using lighter materials which maintain the performance reached with heavier ones. An example of this evolution can be seen in racing car chassis, which has evolved from steel frames used in the early years of the automotive to aluminium alloys introduced at 1950s, arriving at composite materials and high-performance polymers in use since 1980s. The second way is to modify the design of the structure, which could imply complex designs clearly dependent on the capability of the manufacturing to get the end products. Therefore, this way requires an evolution of the manufacturing processes.

Until about thirty years ago, the manufacturing techniques were classified into two categories. Formative manufacturing which models the material with heat and pressure. It includes techniques as forging, stamping, injection moulding or casting. The second category is subtractive manufacturing, which uses cutting tools to remove material from a blank to achieve the final design. Turning, milling and drilling are some of these techniques. It was not until the early 1980s when the development of a third category of manufacturing processes began: additive manufacturing.

According to ASTM F2792-12a standard definition, additive manufacturing (AM) technologies are “a set of processes of joining materials to make objects from 3D model data layer upon layer” [1]. These techniques were thought initially as rapid prototyping processes which reduce the cost and time during the development stage. Since beginning, they have suffered a speedy development translated in new and better techniques.

The first additive manufacturing technique was stereolithography and it was invented in 1983 by Hideo Kodama in Japan [2]. However, it was not commercialized until 1986. In addition, the selective laser sintering (SLS) technology was presented that year and since that time, the evolution of additive manufacturing caused a direct impact on the industry, as is shown in Figure 2-1 for the aerospace industry [3][4]. AM technologies have evolved from obtaining low quality parts, which were used in rapid prototyping, to produce parts with complex geometry in reduced time and with no tooling requirements in different technological and industrial sectors[5]–[7].

According to Wohler’s Report, the growth of the AM industry, consisting of all AM products and services worldwide, has surged from 21% in 2017 to 33.5% in 2018, overcoming nowadays $9.8 billion US [8]. This rising trend is expected to continue during the next years.

A report published recently by SmarTech shows the growing trend of automotive AM market for the next decade (Figure 2-2). As Figure 2-2 shows, AM market does not consist just in manufacturing final parts. Although this part hoards most of the production, AM techniques are also used for prototyping and producing specific tools. This figure also shows the general material trend: the AM techniques are focused on polymeric materials, whose final properties could be obtained directly in the manufacturing process in contrast with metallic, ceramic and composite materials, which obtain their final properties after a secondary process such as sintering or infiltration [9]. Volkswagen and The Royal Netherlands air force are two examples of the use of AM tools made of polymer on their processes, saving time and cost [10], [11]. Total automotive AM market is estimated to reach $12.4 billion US in revenues by 2028 [12]. In addition, considering the development of these techniques, revenues of fabrication of final parts is thought to surpass the revenues of prototyping, tooling and hardware.

For this reason, the principal automotive Original Equipment Manufacturers, OEMs, cooperate with AM hardware producers to integrate this technology in their processes.

On the other hand, it is important to know the level of competitiveness of these techniques. Figure 2-3 shows how unitary cost of a general part varies depending on the manufacturing process chosen. In economic terms, formative manufacturing is the most cost-effective way for manufacturing high number of parts, being followed by the subtractive processes. For these two manufacturing ways, higher number of parts implies lower cost per unit, while the unitary price for additive manufacturing remains constant without relying on the number of manufactured pieces. Thus, AM takes a key role when a short number of parts is required or when the geometry of the part is complicated or impossible to obtain with the traditional techniques. However, there are other limiting factors for choosing AM, like the mechanical behaviour of AM pieces or processing times.

Despite the general point of view, AM is not only linked with industry. There are some relevant events which help to popularize AM outside the industry (Figure 2-1). Among others, the RepRap project that began in 2005, whose aim was to develop a low-cost 3D printer capable to print its own parts, and the expiration of the Fuse Deposition Modelling (FDM) patent in 2009, helped the development of desktop 3D printers. Wohler’s Report informs that more than 500.000 desktop 3D printers have been sold between 2015 and 2017 [8].

There are several common steps in all the wide variety of AM techniques: all of them begin with the creation of a 3D file by Computer Aided Design (CAD) which contains the desired part. This file is converted into a simplified computer-readable format file compatible with the AM equipment. Once this Stereolithography (STL) file is obtained, it is divided by the software into slices of specific thickness, which gives to the AM equipment the manufacturing information of each layer of the part. Then, the AM equipment can start with the manufacture of the piece layer by layer until the piece is completed. When the part is finished, it is removed from the machine and usually subjected to post-processing procedures to improve the final quality of the piece [13]. This general procedure is summarized in Figure 2-4.

Despite these general steps, ASTM F2792-12a [1] separates AM technologies in different categories according to the raw polymer used and the processing method. The most important ones are contained in Figure 2-5 [14]–[16]. In VAT techniques, a photopolymer resin is used and selectively cured with ultraviolet (UV) light, so they are limited to very specific polymers.

In the case of material extrusion, the polymer is melted or semimelted and is selectively dispensed with a nozzle. The raw material for powder bed fusion and for material and binder jetting is a powder bed of polymer, with the difference that in jetting processes a binder moves across the layers forming the final part, and in the powder bed fusion, the melted polymer links layer by layer creating the entire piece. This possibility of using just the material powder without any addition, the varieties of suitable powders, the no need for structural support, the excess powder recyclability and the low-cost quality of the parts, make powder bed fusion processes the most popular and applied AM technique [8], [16]. Among the powder bed fusion systems, the Selective Laser Sintering (SLS) is the most firmly established technique for polymers.

2.2 Selective Laser Sintering of Polymers

Selective Laser Sintering (SLS) is one of the AM processes which has evolved the most during the last decade. This technique allows building polymeric parts with mechanical properties very close to those obtained from specimens processed by conventional manufacturing processes such as injection moulding [8], [12], [13], [16]. This process is presented schematically in Figure 2-6. It begins when a thin layer of powder, preheated somewhat below the melting temperature, is spread in the building area positioned in an elevated temperature chamber. A high power laser (CO₂) beam scans the surface of the powders following a 3D computer model pattern. The energy applied by the laser melts and fuses small powder particles upon contact, which finish adhering the previous sintered layer. When a layer is finished, another fresh layer of powder is swept along the build platform which moves downwards a pre-set amount, commonly one layer thickness. The next layer is traced out and the process repeats layer by layer till the model is fully grown. The unsintered powder remains during the process to support the piece.

This technique can generate high quality and complex 3D parts using the major variety of materials of all the AM processes. However, this variety of polymeric powders is limited due to the requirements that have to be fulfilled. Figure 2-7 contains representative thermoplastics, indicating their SLS availability, their microstructure and their mechanical performance. They are divided into three groups: commodity polymers, which are not suitable for structural applications; engineering polymers, which have good mechanical properties; and finally, high performance polymers, usually are used in very demanding tasks like in aerospace industry [18].

The polymeric powders must accomplish some requirements to be used in the SLS process. Schmid et al. [19] defined five main aspects, divided into intrinsic (thermal, optical and rheological) and extrinsic properties (particle and powder).

The intrinsic properties are defined by the molecular structure of the polymer, which cannot be usually modified. Thermal properties are one of the most crucial factors because SLS aims to obtain a full coalescence of the polymeric particles when the CO₂ laser melts selectively the powder. Therefore, temperature during sintering (T_s) must be precisely controlled and optimized attending to material thermal properties. For semi-crystalline polymers, T_s must be chosen attending to the crystallization temperature (T_c) and the melting temperature (T_m). Ts should be higher but near T_c to maintain the crystallization rate as slow as possible, at least few sintered layers, to avoid residual stresses and warping of the piece. On the other hand, T_s must be lower than T_m for keeping dimensional resolution. With these requirements, the temperature range between T_c and T_m could be defined as the sintering window as is outlined in Figure 2-8. Although this figure shows the sintering window obtained from the Differential Scanning Calorimetry (DSC), the practice one in SLS is narrowed due to the difficult control of heating and cooling temperatures during processing [19], [20].

Other important point of the thermal properties is the resistance to thermal degradation. The exposition of the polymeric powders to temperatures near Tc during long periods of time results in ageing. So, the powder used in one sintering process that remains unmolten cannot be used directly in other sintering processes. It has to be mixed with new powder up to 50 %wt [21].

Other intrinsic factors are the viscosity and the surface tension, which must be kept as low as possible to attain full dense parts. Otherwise, a complete coalescence of polymeric particles is not produced during sintering, obtaining parts with reduced mechanical properties. Due to the low viscosity requirement, SLS of amorphous polymers, whose viscosity maintains high values above the glass transition temperature (T_g), are not widespread used because of the poor coalescence during sintering, resulting in brittle materials.

Optical properties are also of some concern as the material must have the ability to absorb the energy provided by the CO₂ laser used in SLS (wavelength of 10.6 μm). Most polymers consist of aliphatic compounds (C-H) in their molecules, which can absorb relevant portions of the laser radiation.

On the other hand, the extrinsic factors are the ones which can be modified attending to the powder production procedures. One of these factors is related with the shape of particles. To obtain an optimum powder flow and slip between particles in the processing bed, the best particle geometry is spherical. In addition, a smooth surface of the particle also improves the flow and the final roughness of the piece [22]. Commercial powders are not usually spherical due to the manufacturing process, showing irregular shapes. Also, the particle size distribution of the powder is important. To improve the flowability of the powder, it is important to have particles of different sizes. If the powder is composed of particles with the same size, the slip between particles in the bed is difficult. When particles are too small, an excessive adhesion between particles also reduces the flowability. A commercial distribution usually has a particle size distribution between 20 μm and 80 μm and a low percentage of particles with size below 10 μm [22]. The porosity of the sintered piece is reduced optimizing both factors.

In addition to the thermoplastics contained in Figure 2-7, Ligon et al. [23] registered 31 different commercial powders on their work in 2017 including the recently added thermoplastic elastomers. They summarized the mechanical properties of those materials in a stiffness/toughness balance diagram which is shown in Figure 2-9. Although this number has increased, there are just a few material suppliers and most of those materials are made by 3D Systems, Electro Optical Solutions (EOS) or Advanced Laser Materials. For this reason, the different materials have their origin in one of these companies and they can be mainly divided into thermoplastics and thermoplastic elastomers.

Despite this variety, Polyamide (PA)-based thermoplastic are the most used polymers in SLS with 95% market share of total SLS manufacturing [18], [23], [24]. Polyamide 11 (PA-11) and Polyamide 12 (PA-12) are semi-crystalline thermoplastics which fulfil all the critical requirements previously mentioned. Moreover, PA-12 has high mechanical properties, high toughness and high fatigue resistance compared to other thermoplastics, good tribological performance and high chemical resistance [25], [26]. Figure 2-9 shows the prevalence of these polymers when high stiffness is needed. Different types of PA-11 (Primepart DC, EOS GmbH) and PA-12 (PA2200 by EOS GmbH, Duraform^® by 3D systems or PA 250 by ALM) stand out against others. This figure also included PAs with inorganic fillers added for increasing mechanical properties, resulting in higher strength and thermal stability but lower elongation at break. Some of the used fillers are glass beads, alumina, aluminium particles, carbon and carbon nanotubes and titanium whiskers [23].

Although PA-11 is cheaper than PA-12 and provide parts with higher ductility, it is more difficult to process due to its narrower sintering window and a fast recrystallization and degradation during preheating processes, giving as result poor accuracy and higher geometrical distortions in the pieces [18], [27]. For these manufacturing difficulties, PA-12 and PA-12 composites take the majority of the SLS market [8], [23], [24].

2.2.1 Mechanical properties of PA-12 processed via Selective Laser Sintering

The mechanical performance of PA-12 processed by SLS has been evaluated mainly by tensile tests and fracture toughness tests. Table 2-1 collects the tensile properties such as the ultimate tensile strength, the elongation at break and the Young’s modulus of SLS PA-12 measured by different authors. The processing parameters, when provided by the authors, are also included, as the mechanical performance is strongly dependent on the resulting microstructure which is governed by the values of the fabrication parameters. The table also incorporates the values given by the two main manufactures of PA-12 for SLS as PA 2200 by EOS [28] and DuraForm PA [29], as well as those obtained from the conventional technique as injection moulding (IM). In general trends, the values of the tensile strength and the Young’s modulus of SLS PA-12 are very similar to those of IM PA-12, except for the elongation at break. In this case, the elongation at break of IM PA-12 is almost one order of magnitude higher than that of SLS PA-12. The main factors behind these differences are the processing parameters that control the SLS technique, the intrinsic anisotropy of the layer wise SLS manufacturing process or the hygroscopic nature of the polyamide.

Among the manufacturing parameters of the SLS technique, the supplied energy density (dependent on the laser power, laser scan spacing, laser beam displacement velocity and laser radius) has shown a great impact on the microstructure and crystalline features and consequently, on the mechanical performance of SLS PA-12. Dupin et al. [37] showed that the increase in the energy density decreases the porosity and nascent polyamide particles and increases the recrystallized fraction, implying higher mechanical properties. Similar results were obtained by Caulfield et al. [33] and by Stichel et al. [34].

The latter evaluated the tensile parameters of PA-12 sintered by different machines and then, different parameter sets optimum for the respective machine.

Another relevant factor is the intrinsic anisotropy induced by the stacking layer SLS process and there are many authors that have evaluated the mechanical response as a function of the applied load direction with respect to the layered structure. Hooreweder et al. [30] compared the tensile results of PA-12 manufactured by SLS obtained when the applied load was applied along the scanning direction and perpendicular to the scanning direction with the results obtained from IM PA-12. For SLS parts, the main difference between the two orientations was found in the elongation at break which was lower when the load was applied perpendicularly to the layered structure. Regarding the effect of the manufacturing technique, the SLS specimens showed a more brittle behaviour than IM but higher Young’s modulus. The same trend was obtained by Caulfield et al. [33], Stichel et al. [34] and Lammens et al. [36].The latter also analysed the effect of the different machine cross-head displacement rates (5, 50 and 500 mm/min) on the tensile properties, registering higher stiffness at higher displacement rates accompanied with lower elongation at break. On the other hand, the results obtained by Starr et al. [35], which tested six different orientations, revealed small differences in the mechanical properties among the distinct orientations, although the lower elongation at break was also found when the load was applied perpendicularly. In all the cases, the low elongation at break along the building direction was related to layer adhesion, very sensitive to process conditions. Some authors have analysed the effect of the hygroscopic nature of the SLS PA-12 on the mechanical performance. Seltzer et al. [31] studied the effect of humidity on SLS PA-12, obtaining an adverse effect of the hydrothermal aging on the mechanical properties, specifically, for water saturated specimens, with a reduction in strength, in the Young’s modulus, and in ductility. Similar results were obtained by Goodridge et al. [32] in their study comparing the effect of the storage conditions and length of time of SLS PA-12 and IM PA-12, except for the elongation at break which in water conditioned samples presented higher values than those of the dry specimens. In addition to the detrimental effect of humidity in PA-12, the SLS PA-12 showed better water resistance than IM PA-12. Besides, tensile testing at temperatures that ranged from -40 °C to 140 °C were performed on specimens with distinct humidity conditions and in all the cases, SLS samples held the strength better than IM specimens.

The research on the fracture behaviour of SLS PA-12 using the Fracture Mechanics approaches are not so prolific as those dealing with the tensile properties. Table 2-2 collects some of the most representative works found in the literature addressing the influence of different factors, such as the anisotropy due to the layered structure, the hygroscopic nature of polyamides or the processing parameters, on the fracture toughness of PA-12. Together with the values of the fracture parameters in terms of the mode I critical stress intensity factor, K_IC, or the energy at crack growth initiation, J_IC, the values of the manufacturing parameters are also included as their knowledge can shed more light on the results. Hitt et al. [38] compared the fracture behaviour of SLS and IM PA-12 parts as a function of the specimen thickness. The results were in contradiction in the sense that while for IM specimens, the energy at crack growth initiation decreased as the specimen thickness increased, the opposite trend occurred for SLS samples. The reason of this anomaly could be in the computation of the fracture parameters through the Linear Elastic Fracture Mechanics (LEFM) approach, despite the well-defined non-linear behaviour shown by the materials. Seltzer et al. [31] analysed the effect of water conditioning on the fracture behaviour of neat PA-12 and PA-12 composites reinforced with glass beads and short ceramic fibres. The reinforcement implied both a better fracture behaviour and a better water resistance. Salazar et al. [39], [40] investigated the effect of the temperature (23 °C and -50 °C) and the hygrothermal ageing on the fracture toughness of neat PA-12 and PA-12 filled with short ceramic fibres. In the case of the neat PA-12, the fracture toughness was not affected by the testing temperature but in case of the composite, the fracture toughness values at -50 °C were higher than at room temperature. In the specimens saturated in water, the fracture toughness showed an important impoverishment of up to 50%. Brugo et al. [41] evaluated the influence of the notch sharpening method and of the load direction with respect to the building direction. Regarding the notching technique, no differences on the fracture toughness values were obtained between the specimens notched by pressing a razor blade into the notch and those with a 0.3 mm in thickness notch generated during the SLS manufacturing process. Nevertheless, none of these techniques provided a sharp crack with no damage at the crack front which guaranteed quality fracture toughness values [42], [43]. In case of evaluating the orientation, the fracture toughness of the specimens loaded perpendicularly to the layers was lower than when the load was applied along the layers.

The effect of the load direction with respect to the layered structure on the fracture parameters of SLS PA-12 was also studied by Crespo et al [44], [45], Linul et al. [46], Schneider and Kumar [47] and Stoia et al. [48]. The former inferred the fracture toughness from the fracture of notched samples through the application of the theory of the critical distances. The fracture toughness of the specimens loaded parallel to the layered structure was also higher than for the specimens loaded perpendicularly. The same authors [44], [45] analysed the influence of the strain rate on the fracture toughness following the same methodology, carrying out tests that ranged from quasi-static conditions in electromechanical machines to very high strain rates attained by Hopkinson bar tests. On the other hand, Linul et al. [46] inspected the effect of the process energy density on the mode I and mode II fracture toughness evaluated on specimens loaded at different directions with respect to the layered structure. As expected, the fracture behaviour of the specimens processed with higher applied energy and loaded parallel to the layered structure was the best, unequivocal sign that the higher applied energy implied higher density and consequently, less defects. Moreover, the interlayer adhesion is still a limiting factor as the authors observed the crack growth tended to occur along the layer bonding interface. Schneider and Kumar [47] determined under plane stress conditions the fracture toughness as a function of the load direction with respect to the layered structure, finding no relevant differences between the parallel and perpendicular orientations. Finally, Stoia et al. [48] studied the effect of the geometrical defects introduced during sintering as well as the load direction on the fracture toughness. In case of defect-free specimens, there was an important anisotropy in the fracture toughness, with values of the specimens tested with the load applied perpendicular to the building direction more than twice those obtained from tests with the load applied along the building direction.

2.3 Fatigue of Polymers

2.3.1 Fatigue Failures

The fatigue performance of polymers due to cyclic mechanical loads is commonly characterized by the stress-life or the S-N curves [49]. The approaches established for metals have been used to define the mechanisms of fatigue initiation [50] and crack growth [51] in polymers. The reason is that the majority of fatigue data of stress-life curves and fatigue crack growth curves of polymeric materials appear significantly similar in shape to those found in metals, despite their different microstructure and chemical makeup. At present, there is no physical theory to explain the mechanical S-N behaviour of polymers. Notwithstanding, the S-N curves, where the cyclic stress amplitude is plotted versus the failure cycles, are fitted to Basquin’s empirical equation [52], useful for engineering practice although this empirical relationship does not represent well the entire fatigue data range.

Fatigue failure in polymers can occur by thermal fatigue or by mechanical fatigue [53]. In the former, failures occur by thermal softening and melting due to massive hysteretic heating while in the latter, fatigue crack initiation and stable crack propagation to fracture take place similarly to metals.

Thermal fatigue failure normally occurs due to extrinsic factors such as relatively high loading frequencies or strain rates, where thermal softening due to hysteretic heating arises from the viscoelasticity, high damping and low conductivity typical in polymers [54]–[56]. The hysteretic energy is dissipated as heat producing a rise in specimen temperature [57] and, consequently, loss of stiffness [58]–[60]. The temperature rise is very dependent on the frequency because the hysteresis arises from the phase lag between stresses and strains in the polymer, which is larger as the loading frequency is higher.

As a rule, the influence of test frequency on the fatigue response of polymers originates from the viscoelastic behaviour of polymers. An increase in frequency normally involves a greater energy dissipation and temperature rise. Nevertheless, in low frequency test conditions where thermal effects are not crucial, an increase in frequency leads to increase in modulus and strength due to increased strain rate, implying an increased fatigue life as observed in Polypropylene (PP) and Polypropylene reinforced with short glass fiber by Eftekhari and Fatemi [60] and in Polystyrene (PS), High Impact Polystyrene (HIPS) and Acrylonitrile Butadiene Styrene (ABS) by Sauer and Chen [61].

In fatigue crack growth behaviour of polymers, the thermal effects are localized at the crack tip, leading to crack tip blunting and consequently, increasing the fatigue resistance to crack growth. These effects were observed in PS [62], Polymethyl Methacrylate (PMMA) [63] and Polyvinyl Chloride (PVC) [64] but not in Nylon [65] and Polycarbonate (PC) [64] in which the crack growth curves were independent of frequency in studies performed in the range from 1 Hz to 100 Hz. Nevertheless, in these studies the crack growth behavior was correlated on the basis of Linear Elastic Fracture Mechanics (LEFM) approach, using as control parameter the stress intensity factor range, ΔK, even at frequency up to 100 Hz, doubtful parameter if excessive thermal softening occurs.

Regarding mechanical fatigue failure, Sauer and Richardson [51] stated that true mechanical fatigue failure, involving initiation and growth of fatigue crack under cyclic loading takes place in the range between one-fourth and one-half of the yield strength of the polymer. The macroscopic description of fatigue fracture in polymers is formed by the sequence of mechanical events similar to those in metals [49]–[51], [66], [67]. In glassy polymers, the initiation occurs by the development of a single craze at some surface location while in semicrystalline polymers, it takes place at spherulite boundaries after some fatigue cycles. Thereafter a fatigue crack is generated within the craze, by cyclic fracturing the fibrils farthest from the initial craze tip, and the advancement of the macroscopic crack growth, either continuously or discontinuously under cyclic loading, till the point of rupture is not any different in glassy or semicrystalline polymers. Purely mechanical fatigue failures normally occur in the absence of thermal effects, that is, at low frequency conditions.

Finally, fatigue behaviour of polymers is also influenced by the environmental temperature because some polymers undergo crystallization or brittle-ductile transition as the temperature is increased [49]-[51], [66], [67]. The fatigue resistance is reduced with increased temperature due to alteration of the crack tip deformation and fracture mechanisms [25], [68]-[70], the reduction of the tensile strength and the fatigue limit as the test temperature is increased [25], [60], [67], [71] or the temperature-dependent variation on the viscoelastic deformation of the polymer [25], [60], [72].

2.3.2 Difficulties in fatigue characterization

The proofs provided by the literature allows the use of the continuum fatigue behaviour tools of metals for describing the mechanical fatigue response of polymers. The S-N curves obtained under mechanical fatigue conditions usually exhibit the asymptotic approach toward the tensile strength at one fatigue cycle and the asymptotic flattening at the fatigue limit (the physical stress limit below which no fatigue failure occurs), as in the case of metals. Nevertheless, there are some gaps or difficulties in the fatigue characterization of polymers. In case of the stress-life, there is not an accepted physical theory capable of constructing the shape of the S-N curves. Some attempts were made by Baltenneck et al. [73], who derived a kinetic S-N curve based on the cumulative evolution of micro-defects till the point of fatigue rupture. The predicted curves agreed partly well with the experimental ones despite the physical basis assumed that the number of cycles to failure increased with initial defect concentration. Williams et al. [74] developed an equation for fatigue crack growth life considering a fatigue crack growing to reach the point of unstable failure. Although it predicted fatigue lives reasonably well, some parameters were chosen arbitrarily and was not applied to describe S-N curves of plain specimens. Shojaei and. Wedgewood [67] developed a model within the Continuum Damage Mechanics framework to enhance the life prediction capabilities of polymers. Considering that in the Low Cycle Fatigue regime, damage can be modelled by microvoid nucleation and propagation and that in High Cycle Fatigue regime, damage can be described by initiation and propagation of microcracks, plasticity and creep laws were coupled in fatigue Finite Element Analysis. Although the results were promising, the computation cost and the update of the damage fatigue history was an important difficulty. The most elaborate theory is that of Ravi Chandran [25] who, in the absence of thermal effects and starting from the macroscopic crack growth mechanisms, derived a continuum-based fatigue theory capable of predicting the effect of the mean stress and the temperature on the shape of S-N curves significantly well. He developed the constitutive equation for polymer fatigue assuming that the increase in stress in the uncracked ligament is responsible for the acceleration of the crack growth during fatigue, consistent with the experimental findings in metals of Frost, Dugdale and Weibull [75]-[77], whose works predate that of Paris [78]. However, more experiments with controlled changes in the structural and microstructural parameters as well as in the mechanical properties of polymers are needed to fully validate the proposed theory.

Regarding the fatigue crack growth behaviour of both glassy and semicrystalline polymers, LEFM approach has been mostly used to describe the crack growth rate as a function of the stress intensity factor range, ΔK = K_max − K_min [49], [79]-[86]. However, some works have questioned the validity of ΔK when dealing with the R-ratio effect [50], [87]-[94]. Rink et al. [92], Furmanski and Pruitt [93] and Boonyaookana et al. [94] observed that the crack growth propagation curves seemed to be unaffected by R-ratio when the crack growth rate was correlated with K_max. In contrast, Takemori [50] and Radon [89] showed an uncommon trend as the decrease in the crack growth rate, $\frac{da}{dN}$, with the increase in the R-ratio when using either ΔK or K_max as crack driving parameters. This fact motivated the choice of energy release rate range, ΔG, [88], [95], [96] or the maximum energy release rate, G_max [97] as the crack driving force when brittle crack growth propagation was observed. Particularly, Sutton [88] employed ΔG in the analysis of fatigue crack growth propagation in epoxy polymers achieving the collapse onto one single master curve of the fatigue crack growing data obtained at different R-ratios. However, he also realized that this behaviour could not be extrapolated to all polymers.

The uncertainties on the predominant process involved at different R-ratios in polymers has driven to some researchers to consider two different crack driving forces to describe the fatigue crack growth rate as a function of the mean stress, applying the crack closure concept introduced by Elber [98] at low R-ratio and taking into account the creep process that seems to occur at high R-ratios [87], [89], [91], [93].

Cano et al. [99] proved the validity of $\mathrm{\Delta }\sqrt{G}=\sqrt{G_{max}}-\sqrt{G_{min}}$ with G_max as the maximum energy release rate and G_min as the minimum energy release rate, as a valid similitude parameter for describing the crack growth behaviour of glassy polymers, overhauling the gaps previously mentioned. $\mathrm{\Delta }\sqrt{G}$ relies on the same basis for similitude as ΔK, and is defined as:

$$\mathrm{\Delta }\sqrt{G}=\sqrt{G_{max}}-\sqrt{G_{min}}=\mathrm{\Delta }P\sqrt{\frac{1}{2B}\frac{dC}{da}}\mathrm{ (2-1)}$$

where ΔP is the amplitude load of the cycle, B is the specimen’s thickness, C is the sample’s compliance and a is the crack length. Therefore, the crack growth rate, $\frac{\mathit{d}\mathit{a}}{\mathit{d}\mathit{N}}$, was correlated with this similitude parameter through a Paris-type relationship:

$$\frac{da}{dN}=A{\left(\mathrm{\Delta }\sqrt{G}\right)}^n\mathrm{ (2-2)}$$

with N the number of elapsed cycles and A and n material’s constants.

$\mathrm{\Delta }\sqrt{G}$ parameter was proposed by Rans et al. [100] for adhesive joints and composites. The difficulty in the fatigue crack growth behaviour of these materials was that ΔK describes locally the stress field and this makes it unsuitable in fibre-reinforced polymer matrix composites and bonded joints. That was the reason of the widespread use of ΔG as crack driving force parameter [101]-[106].

However, the main concern of ΔG is that it could not be considered as a valid similitude parameter due to its dependence on the mean load of the cycle, P_med:

$$\Delta G\mathit{=}{G}_{\mathit{max}}\mathit{-}{G}_{\mathit{min}}\mathit{=}\frac{{\mathit{P}}_{\mathit{max}}^{\mathit{2}}\mathit{-}{\mathit{P}}_{\mathit{min}}^{\mathit{2}}}{\mathit{2}\mathit{B}}\frac{\mathit{d}\mathit{C}}{\mathit{d}\mathit{a}}\mathit{=}\frac{\mathit{\Delta }\mathit{P}{\mathit{P}}_{\mathit{m}\mathit{e}\mathit{d}}}{\mathit{2}\mathit{B}}\frac{\mathit{d}\mathit{C}}{\mathit{d}\mathit{a}}\mathrm{ (2-3)}$$

where P_max and P_min are the maximum and minimum loads of the cycle.

$\mathrm{\Delta }\sqrt{G}$ has been applied with more or less success in polymers, composites and adhesive joints as the R-ratio effect is still evident in many cases [99], [100], [107]–[111].

2.3.3 Fatigue behaviour of Polyamide 12 manufactured by Selective Laser Sintering.

As previously mentioned, SLS PA-12 stands out for covering specific requirements for various applications in the automotive and aeronautical industries [26], [112]-[115]. In these applications, parts made of this material must have sufficient fatigue resistance to meet the in-service loading and operational performance as they are submitted to dynamic loading either applied directly or indirectly through the interaction of the entire structure with the surrounding environment [26], [114], [115].

Regarding the stress-life performance of SLS PA-12, Table 2-3 collects some of the most representative works in which the processing parameters (when provided), testing conditions and main results are summed up. Van Hooreweder et al al. [30], [116], [117] analysed the influence of the frequency (1 Hz and 3 Hz) and the building direction on the S-N curves at a stress ratio, R, of -1. They obtained that the building orientation had no influence on the fatigue properties and the thermal fatigue failure was attained at certain combination of high stress amplitude and test frequency. Besides, they also analysed the fatigue behaviour of notched samples and compared the results obtained from specimens processed via SLS and IM. Regarding the notch effect, the fatigue resistance of notched samples was better than that obtained from plain specimens, attributable to the reduced thermal load in the former. In case of the effect of the processing technique, no difference in the fatigue resistance was observed between the two manufacturing techniques [30]. Munguia and Dalgarno [118] obtained the SLS PA-12 fatigue behaviour in both reversed and rotating bending (R=-1), showing an isotropic response (no influence of the load direction with respect to the layered structure) in terms of the fatigue behaviour for both testing configurations. They also analysed the influence of the frequency (30 Hz and 50 Hz) in four point rotating bending fatigue tests, evidencing lower fatigue response at high frequency [119]. Moreover, they estimated a fatigue limit of 14 MPa, although no standard or recognized methodology was followed to attain this value. Amel et al. [120] investigated the effect of the geometry (thickness) on the fatigue behaviour explored through S-N curves obtained under displacement-controlled tension-tension (R>0) and force-controlled fully reverse fatigue loading (R=-1). The tests carried out in Low Cycle Fatigue regime evidenced creep as the main mechanism of failure. Schob et al. [121] followed the damage evolution through X-ray refraction, computed tomography and temperature rise under cyclic loading (fatigue tests at 3 Hz and R=-1) with the aim of finding the parameters of the Chaboche material model and the Gurson-Tvergarard-Needlemen damage model. Finally, Kim et al. [122] studied the effect of the shape and size of geometrical defects induced during sintering on the fatigue life. Evidently, the fatigue response impoverished with the increase in size of the defects and the authors applied the Castillo-Canteli-Siegele model [123] extended with the cyclic J-integral to predict the fatigue life of the specimens with induced geometrical defects.

Concerning the fatigue crack growth behaviour of SLS PA-12, Table 2-4 gathers the main results of the papers found in the literature dealing with this issue. Once more, for a proper comparison of the results in the literature, the processing parameters, when provided by the authors, are included. As observed, there is scarce literature in this topic. Salazar et al. analysed the effect of the temperature and the short glass fibre reinforcement and the hygrothermal ageing on the fatigue crack growth response of SLS PA-12 [39], and also determined the differences in the fatigue crack growth behaviour of petrol-based SLS PA-12 and bio-based SLS PA-11 [40]. The fatigue behaviour of the SLS PA-12 was similar at room temperature and at -50 °C but the reinforced material presented an improved fatigue crack growth behaviour at low temperature in comparison with the neat PA-12. The reason was the predominant crack tip bridging mechanism in the composite against the brittle intergranular fracture in the neat PA-12 [39]. On the other side, when comparing the fatigue performance of the bio-based PA-11 versus the petrol based PA-12 processed via SLS, the former presented better fatigue crack growth behaviour than the latter at both 23 °C and -50 °C. Once again, the reason was the fatigue failure, while in PA-11 the mechanism was ductile through cavitation, void growth and coalescence, in PA-12 the dominant mechanism was brittle crazing [40]. Under water conditions, the fatigue response worsened abruptly due to the reduction in crystallinity and molecular weight by hydrolysis, resulting in embrittlement [40]. Despite the inherent anisotropic nature of the stratified SLS process can suppose a major limitation, only Blattmeier et al. [124] have treated this issue, attaining no concluding results. Moreover, in this work the fatigue crack growth behaviour of SLS and IM processing techniques obtained through a load increase method is compared and the influence of the surface finishing is tackled. In case of the processing technique, there seemed to be a slight better performance on the fatigue response of the SLS specimens in comparison with the IM samples while the surface finishing seemed to have no significant influence. Finally, Boukhili et al. [79] studied the fatigue crack propagation behaviour of injection moulded PA-12 and the influence of the test frequency, load waveform, specimen configuration and thickness and the material orientation, that is, the material response depending if the load direction was applied parallel or perpendicularly to the injection flow direction. The most interesting result obtained by these authors was that ΔK was not an appropriate crack driving force to describe the fatigue crack propagation of polyamides. They also concluded that the fatigue crack growth behaviour was evidently dependant on the load wave form, the specimen configuration and thickness and above all, of the frequency.

2.4 Design against fatigue

There are two different philosophies of the design against fatigue: the safe-life approach and the damage tolerance approach. The former, and also the oldest, has the sole objective of avoiding the fatigue failure during the design life and is focused on the crack initiation phenomenon. On the other hand, the damage tolerance approach assumes that defects in form of cracks will exist, caused by processing or fatigue, and that this damage can progress during service life. In this case, the focus is on ensuring that fatigue cracks do not reach a critical value through regular maintenance labours. So, the attention is paid to the crack propagation phenomenon.

The failure criteria in the safe-life approach are the nominal stress-life (S-N) or the local strain-life (ε-N) models, which are used in the calculations of finite-life design to guarantee the duration of the component at the maximum expected stress or load. On the other hand, the damage tolerance approach is supported by Fracture Mechanics, employed to determine whether the fatigue cracks will grow enough to produce failures when detected in periodic inspections. For infinite-life design, Kitagawa and Takahashi [125] noticed that the fatigue life data could be represented in a double logarithm stress range, Δσ, and crack length, a, diagram by two lines plotted in blue in Figure 2-10. One represents the plain specimen fatigue strength, given by the stress-based fatigue limit, Δσ_fl, of the safe-life approach (line AB). The other line (line BC) is the prediction of the Fracture Mechanics and given by the stress intensity threshold, ΔK_th [78], [126].

The slope of the straight-line BC of Figure 2-10 is defined by:

$$\mathrm{\Delta }\sigma \mathit{=}\frac{\mathit{\Delta }{\mathit{K}}_{\mathit{t}\mathit{h}}}{\sqrt{\mathit{\pi }\mathit{a}}}\mathrm{ (2-4)}$$

For any stress-crack length combination below the line ABC, a crack will not grow and the fatigue life can be assumed as infinite (area in green in Figure 2-10).

The most characteristic feature of the Kitagawa diagram is that it can be divided into three zones. The zone named as Microstructurally short cracks (MSC) is controlled by crack initiation damage phenomena, the Long Cracks (LC) zone is controlled by crack propagation phenomena and there is a transition zone between the two called the Physically Short Cracks (PSC) zone, where a competition between the two mechanisms occurs. In the PSC area neither the Fracture Mechanics nor the stress or strain life models provide proper predictions.

The empirical results obtained for different metals revealed that Kitagawa-Takahashi diagrams describe correctly the fatigue behaviour for MSC and LC regimes, but in the PSC regime, the diagram overestimates the real behaviour of the material, which is actually below the two lines AB and BC defined by the Δσ_fl and ΔK_th values. For this reason, in Figure 2-10 it is also represented in dashed red line an empirical model proposed by El Haddad which modifies the initial Kitagawa-Takahashi predictions [127].

El Haddad et al. proposed their model just two years later than Kitagawa and Takahashi one [125], as a result of test results on aluminium alloys and on steels [128]. This model is governed by the following expression:

$$\mathrm{\Delta }{\sigma }_{Hd}=\frac{\mathrm{\Delta }{K}_{th}}{\sqrt{\pi (a+I_0)}}\mathrm{ (2-5)}$$

where Δσ_Hd is the applied stress range given by the model and I₀ is a characteristic length which depends on the material and on the microstructure and is calculated as follows:

$$I_0=\frac{1}{\pi }{\left(\frac{\mathrm{\Delta }{K}_{th}}{\mathrm{\Delta }{\sigma }_{fl}}\right)}^2\mathrm{ (2-6)}$$

The line given by Δσ_fl of the Kitagawa-Takahashi diagram overestimates the real critical stress range, which is reduced by the presence of short cracks which are lightly larger than the microstructural barriers. On the other hand, the line BC given by the Fracture Mechanics approach does not match the experimental values due to the small size of the crack. The validity of the El Haddad model has been confirmed in aluminium alloys [129], [130], steels [125], [127], [131]-[133] and titanium alloys [134].

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