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Pseudo-ductile fracture of 3D printed alumina triply periodic minimal surface structures
Journal Pre-proof Pseudo-ductile fracture of 3D printed alumina triply periodic minimal surface structures Lei Zhang, Stefanie Feih, Stephen Daynes, Shuai Chang, Michael Yu Wang, Jun Wei, Wen Feng Lu
PII:
S0955-2219(19)30667-3
DOI:
https://doi.org/10.1016/j.jeurceramsoc.2019.09.048
Reference:
JECS 12758
To appear in:
Journal of the European Ceramic Society
Please cite this article as: Zhang L, Feih S, Daynes S, Chang S, Wang MY, Wei J, Lu WF, Pseudo-ductile fracture of 3D printed alumina triply periodic minimal surface structures,
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Pseudo-ductile fracture of 3D printed alumina triply periodic minimal surface structures Lei Zhanga, b, Stefanie Feihb, Stephen Daynesb, Shuai Changa, Michael Yu Wangc, Jun Weib, and Wen Feng Lua,*
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Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore, 117575 b
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Singapore Institute of Manufacturing Technology (SIMTech), 2 Fusionopolis Way, Singapore 138634 c
Department of Mechanical and Aerospace Engineering, and Department of Electronic and Computer Engineering of Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
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Abstract
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Additive manufacturing enables the fabrication of periodic ceramic lattices with controllable micro-architectures. Many studies reported their catastrophic brittle fracture behaviour. However, ceramic lattices may fail by a layer-by-layer pseudo-ductile fracture mode, by controlling micro-architectures and porosities. Moreover, their fracture behaviour can be optimised by introducing strut/wall thickness gradients. This paper investigates the fracture behaviour and the fracture mode transition of ceramic triply periodic minimal surface (TPMS) structures. Alumina TPMS structures with relative densities of 0.14-0.37 are fabricated by ceramic stereolithography. Quasi-static compression tests validate a transition density range for non-graded samples: low (<0.21) and moderate (>0.25) relative density samples show layer-by-layer pseudo-ductile and catastrophic brittle fracture modes, respectively. The pseudo-ductile failure mode increases the energy absorption performance, enabling loadbearing capacity for a compressive strain up to 50%. With appropriate thickness gradients, graded structures exhibit significant increase of energy absorption without a decrease of fracture strength compared to their non-graded counterparts.
Key words: Additive manufacturing; Porous ceramics; Fracture behaviour; Energy absorption; Triply periodic minimal surface (TPMS)
To be submitted to: Journal of the European Ceramic Society
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1. Introduction
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Porous ceramics consisting of periodic micro-architectures and open pores are widely used for multifunctional applications including structural components, thermal management devices, filters, and tissue engineering scaffolds [1, 2]. Nowadays, additive manufacturing (AM) of this group of ceramics has attracted an increasing interest due to the enhanced design freedom and hence of controllable micro-architectures and pore geometries, which are generally not achievable by traditional fabrication techniques [2, 3]. A variety of AM technologies have been developed for porous ceramics [2-4], such as robocasting (extrusion based methods) [5-8], powder based 3D printing [9] and stereolithography (SLA) [1, 2, 10]. The latter technique is suitable for periodic porous ceramics since it can produce complex and nearly dense ceramic parts (possible to reach 99% of theoretical density) with high precision, which is favourable for high mechanical strength [2, 10]. Ceramic SLA fabricates each layer of a part by laser or UV light scanning of a photopolymerizable suspension filled with ceramic powders, followed by de-binding and sintering processes.
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A range of periodic porous ceramics with highly complex micro-architectures have been studied. One of the well-studied structures is a mesh of ceramic rods; their compressive strength and failure characteristics were investigated through simulations and experiments [6, 11]. The influence of pore orientation on the mechanical properties of the ceramic scaffolds was investigated on SLA printed ceramic meshes [12]. Hierarchical porous ceramics that included circular rods in the mm-range and randomly distributed micro pores in the µm-range were fabricated using the robocasting technique [7, 8, 13]. 3D lattices with various ceramic materials and architectures such as cubic and tetradecahedron architectures have also been fabricated [10, 14]. However, considering the structure-property relationship of porous structures, shell-like and plate-like architectures generally outperform strut-based lattices on stiffness and strength. This has been observed through analytical, numerical and experimental methods [15-18]. Recently, multifunctional triply periodic minimal surface (TPMS) structures have drawn significant attention because of their smooth shell-like geometries, open cell topology and large surface area [19-22]. Simulations and compressive tests on 3D printed metallic TPMS structures validated their superior energy absorption abilities compared to body-centred cubic strut-based lattices [20, 22]. Bonatti and Mohr investigated smooth shell lattices having similar geometries to TPMS, showing their higher mechanical properties than truss lattices of equal density [16, 17]. Therefore, the combination of ceramic materials and TPMS structures is expected to achieve high mechanical properties for porous ceramics. However, due to the increased geometric complexity of creating TPMS structures via ceramic AM technologies, few studies have investigated ceramic TPMS structure fabrication and their performance. AlKetan et al. [23] fabricated alumina TPMS structures as catalytic substrates by ceramic SLA, and studied their mechanical and flow characteristics. The failure behaviour of porous ceramics is dominated by material fracture due to the brittle nature of ceramic materials. Experimental studies on porous ceramics have shown two typical 2
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fracture modes under compression. In the moderate and high relative density range, the compressive stress linearly increases with the compressive strain up to a sudden rupture, at which point the samples fail catastrophically, similar to monolithic ceramics [11, 12, 24]. In this case global cracks generally initiate and grow along the loading direction. In contrast, for highly porous ceramics, the stress-strain curves are characterized by a linear region followed by a fluctuating stress-strain response or a stress plateau due to a layer-by-layer fracture mode [5, 24-26]. During this failure process, successive layer-wise cracks initiate and propagate. The transition relative density (porosity) at which the failure mode changes from brittle fracture to layer-by-layer collapse was estimated by experiments [25, 26] and empirical formulas [25] for stochastic ceramic foams. This failure mode for highly porous ceramics leads to significantly improved energy absorption capacity. However, according to analytical models for mechanical behaviour of cellular solids [27-30], reducing the relative density also reduces failure strength. The challenge of improving the energy absorption performance for porous ceramics is to increase the relative density while maintaining the gradual failure behaviour.
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The introduction of graded designs is a feasible solution to avoid catastrophic failure for porous materials. This strategy has been used for metallic and polymeric lattice structures to change their failure behaviour from a global shear mode to a localized layer-by-layer mode [31-36]. One of the commonly used graded designs is to gradually vary their strut/wall thickness or relative density along a linear direction. By designing strut thickness graded structures along the compressive loading direction, experimental results showed that a layerby-layer deformation mode was obtained for graded lattice structures, such as Gyroid structures [31] and body-centred-cubic lattices [34]; made of different materials including Ti-6Al-4V [3133], Al-Si10-Mg [34] and nylon [35]. In contrast, their non-graded counterparts failed by a global shear mode throughout the structure [32-34]. While this has not been previously validated for ceramic materials, graded TPMS ceramic structures are expected to fail in a gradual manner and therefore to have improved energy absorption performance. In this work, we fabricate Diamond (D)-type TPMS structures with relative densities ranging from = 0.14 to 0.37, with and without wall thickness gradients, using the SLA
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technique and alumina material. Alumina is one of the widely used structural ceramics due to its relatively high strength, hardness and maximum service temperature among ceramic materials [37]. The failure behaviour, fracture strength and energy absorption capacity are studied by quasi-static compression tests. The influences of the relative density and thickness gradient on fracture modes, as well as the fracture mode change and the transition relative density are investigated. Finally, an empirical formula is used to predict the transition relative density for TPMS structures.
2. Materials and methods 2.1 Design and modelling 3
One of the most widely used methods to describe TPMS structures is the level-set approximation. Among numerous types of TPMS structures, we select the Diamond (D) type surface because of the high mechanical properties based on the studies of structure-property relationships of TPMS structures for metallic alloys [22, 38]. The D-type surface is approximated by D c with
D ( x, y, z ) sin( x)sin( y)sin( z ) sin( x) cos( y) cos( z ) cos( x)sin( y) cos( z) cos( x) cos( y )sin( z )
(1)
where x, y, z are spatial coordinates; ω = 2π/l and l is the unit cell length. c is a constant controlling the shape of the surface. Since the equation D c only defines periodic surfaces
(D )2 c 2
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without thicknesses, solid models can be created by uniformly thickening the surfaces. However, using this method to create graded TPMS structures by non-uniform thickening is generally not straightforward. To address this difficulty, Eq. (1) is modified as: (2)
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This modified function is able to describe a solid structure instead of a surface, by taking the regions with 0 as voids, and the regions with 0 as solids. In other words, the sheetlike TPMS structures are bounded by the two surfaces of D c . Using Eq. (2), non-graded
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TPMS structures, as defined by
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structures are generated with a constant c (Fig. 1 (a)). It should be noted that a constant c in Eq. (2) generates non-uniform thicknesses throughout a unit cell as the distance between the two surfaces D c is not constant. Therefore an average thickness, tavg, is used to characterise tavg Vs / S
(3)
of a unit cell.
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where Vs is the volume of the solids of a unit cell and S is the area of the mid-surface ( D 0 )
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In this work, Matlab scripting is used to generate TPMS sheet structures with/without thickness gradients based on Eq. (2), including 4 × 4 × 4 repeating unit cells and a cell size of 4 mm, see Fig. 1 (a, b). This results in a total sample size of 16 × 16 × 16 mm. Non-graded structures with nominal average wall thicknesses of 150 - 300 µm (relative densities ranging 0.15 - 0.3) are created by keeping c constant (see Table 1). For the graded structures, wall thickness gradients can be introduced by replacing the constant c with a spatially varying function c(x, y, z) (Fig. 1 (b)). In this work, the thickness gradient is introduced by a linear function of c(z) = b + mz (0 ≤ z ≤ 16) where b and m are prescribed parameters, in the building direction (z), and later also the testing direction. In order to control the wall thickness effectively in the design stage, the average wall thickness as well as relative density, , are investigated with respect to c (see Fig. 1 (c)). Since the relative density is proportional to the average wall thickness, the two curves are the same but with 4
slightly different axis scales. Fig. 1 (c) shows that the average thickness and relative density of a TPMS structure created by Eq. (2) are nearly linear (coefficient of determination R2 = 0.99995) with respect to variable c, so the linear function c(z) introduces a constant gradient of the average thickness along the z-direction. In this work, functions c(z) used to generated graded TPMS structures are shown in Table 2, and all graded structures have a same wall thickness gradient of 6.25 µm/mm.
2.2 Ceramic stereolithography and post-processing
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The ceramic TPMS structures were fabricated by the Lithoz CeraFab 8500 stereolithography printer (Lithoz GmbH, Vienna), which operates according to the lithographybased ceramic manufacturing (LCM) process. A commercial alumina (Al2O3) suspension LithaLox HP 500 (Lithoz GmbH, Vienna) was used to fabricate TPMS structures. The lateral resolution of the printer is 40 µm, and the z-resolution (layer thickness) was set to 25 µm for printing. A blue light source with a wavelength of 465 nm and a digital light processing projector is used to cure the suspension throughout the glass bottom of the rotating vat. The green parts with TPMS structures were built in a bottom-up manner.
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2.3 Density measurement
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The cleaning fluid of Lithoz was used to clean the green parts. The non-cured suspension sticking on the printed green parts was removed with the assistance of compressed air. The debinding and sintering post-processing were done in ambient air. During the de-binding cycle, the binder leaves the green parts. De-binding was performed with different dwell times at 75, 115, 205 and 430 °C, followed by being pre-sintered at 900 °C (2 hours) to improve the handling strength. The sintering process was completed at a temperature of 1600 °C with a heating rate of 1°C/min and a dwell time of 2 hours.
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Since ceramic AM technologies generally introduce porosities into the printed structure, the densities for both the solids and the structures are of interest. At the micro scale, the solid density of the ceramic walls was measured using Archimedes’ principle. The Archimedes’ method is able to account for the porosity within the walls. The structural densities of the ceramic samples were measured by the dry weight method, including the porosities introduced by both AM processes at the microscale and the TPMS architectures. The structural densities and their ratios to the theoretical density of alumina are referred to as densities and relative densities, respectively, of TPMS structures in the following text. It should be noted that the relative densities of the geometric models only consider the TPMS architectures, assuming the walls of TPMS structures are fully solid (Table 1). 2.4 Compression testing Compression tests with displacement control were performed using an Instron 5982 universal test machine with a 100 kN load cell. For testing of the quasi-static fracture strength, 5
a strain rate of 5.2 × 10-5 /s (crosshead speed of 0.05 mm/min) was applied. After 5 % compressive strain, a higher strain rate of 5.2 × 10-4 /s (crosshead speed of 0.5 mm/min) was applied to investigate the layer-by-layer failure behaviour. Since the non-graded TPMS structures with moderate relative densities (> 0.25) failed catastrophically, tests were stopped at a compressive strain of about ε = 2 %, while the low relative density (< 0.21) non-graded
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and all graded structures were tested up to a compressive strain of ε = 50 %. For samples with relative densities within the transition range, tests were terminated depending on the observed failure behaviour. The loading direction coincided with the building direction, as well as the thickness grading direction for graded structures. The crosshead movement of the Instron 5982 system was used to measure the compressive displacement Δu, which is then converted to compressive strain by u / h , where h is the sample height. Compressive stresses are calculated by F / A , where F and A are compressive load and the area of samples’ top or bottom faces, respectively. The fracture modes were captured via a camera during the tests, operating at a frequency of 0.5 Hz.
3. Results and discussions
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3.1 Sample morphology
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The solid density of ceramic samples measured by the Archimedes’ method is 3.796 ± 0.175 g/cm3, which is above 95% of the theoretical density of alumina (3.985 g/cm3 [39]). The measured material densities were slightly smaller than the theoretical density of alumina because of the micro pores/cracks generated during the AM process inside the walls. Ceramic SLA is able to fabricate porous ceramic structures with high solid density because of the fine powder size and the high solid loading of the slurry [40]. Ketan et al. [23] also reported a high solid density of 3.72 g/cm3 of alumina TPMS structures fabricated by ceramic SLA. The relative densities measured by the dry weight method are shown in Tables 1 and 2. The relative density includes both the micro pores/cracks and the voids introduced by TPMS architectures, and is therefore significantly smaller than 1. The optical microscope images show no macroscopic pores or cracks on the walls with different thicknesses; and the ceramic walls are uniform for non-graded structures and gradually vary for the graded designs without distortions (see Fig. 2). Some of the samples’ faces are not perfectly flat and include a few flaws due to the cleaning processes. The measured relative densities of ceramic TPMS structures are higher than the designs as shown in Tables 1 and 2, which is consistent with measured wall thickness, calculated by Eq. (3) based on density measurements. This could be attributed to the usage of high energy input and high shrinkage coefficient compensation. No further efforts were made to optimise the processing parameters. For later results, the measured (relative) densities are used. 3.2 Fracture characteristics of non-graded alumina TPMS structures 6
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All alumina TPMS structures were tested under quasi-static compression loading along the building direction, i.e., the z-direction. The non-graded samples in the moderate relative densities (> 0.25) show a stress-strain behaviour consisting of a linear region followed by a sudden stress drop (see Fig. 3 (a)). The linear region includes a few minor stress fluctuations with applied loads. This is caused by the local cracks forming at the faces of TPMS samples contacting with the rigid compression plates. Since the faces are not perfectly flat and flaws exist on the faces of samples, stress concentrations are formed and small cracks grow along the faces, leading to the small pieces breaking off the samples’ contacting surface. These minor cracks and broken pieces do not cause catastrophic failure, so the stress continues to rise. When the compressive loads reaching its maximum, global cracks formulate globally along the loading direction (see Fig. 3 (b, c)). The global cracks grow throughout the samples and the TPMS structures fail by brittle rupture at a relatively small compressive strain typical for brittle materials (ε ≤ 0.02), resulting in a dramatic stress drop. As can be seen in the deformation images during testing (Fig. 3 (b, c)), the TPMS samples break apart completely and lose all load-bearing capacity.
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When reducing relative densities within or below the transition range, non-graded TPMS samples exhibit a completely different failure behaviour. The failure modes change from brittle fracture to pseudo-ductile layer-by-layer failure. This failure mode is dominated by numerous local cracks and fractures initiating at relatively small compressive loads. Therefore, the stressstrain response of these structures shows high-frequency stress fluctuations throughout (Fig. 4 (a)). The stress of the TPMS structure with a relative density of = 0.25 gradually reduces
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with non-regular fluctuations after the first peak (Fig. 4 (a)). During this region, local fractures form at the top and bottom faces (Fig. 4 (b)). Once a compressive strain around ε = 0.32 is reached, a global crack forms and grows through the structure, leading to the loss of load-
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bearing capacity. When further decreasing relative densities below a threshold of = 0.21, the
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TPMS structures display a clear layer-by-layer failure mode with semi-regular serrations spanning a large strain range (Fig. 4 (a)). The period of each stress serration corresponds to the height of a quarter of a unit cell, namely a strain of ε = 0.0625. This implies that each layer collapse is within a quarter of a unit cell, and other layers remain mostly intact during each stress serration. This is validated by the deformation images, showing the cracks propagating within the layer with a height of a quarter of a unit cell near to the compressive plates, while the rest of the structures remains intact (see Fig. 4 (c, d)). During each layer collapse, there are numerous high-frequency stress fluctuations, corresponding to the local cracks growing along the TPMS walls. However, the local cracks grow mostly within a layer, and there is no global crack formation during the stress serrations (see Fig. 4 (c, d)). The stress peak magnitude and the number of stress serrations depend on the relative density; reducing relative densities reduces the stress magnitude and increases the number of serrations. This trend for non-graded TPMS structures with relative densities less than = 0.21 is shown in Table 3. One exception
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regarding the stress serrations number of the structure with = 0.156 may be attributed to stochastic printing imperfections. Experimental results for non-graded samples show that the transition relative density at which the fracture modes change is around = 0.21 - 0.25. Within this transition relative density range, the fracture modes of alumina TPMS structures are sensitive to geometric imperfections and flaw distributions, as can be seen by the mixture of fracture modes existing within this range.
3.3 Fracture characteristics of graded alumina TPMS structures Fig. 5 shows typical stress strain curves and deformed geometries of the graded structures
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for relative densities in the range = 0.26 - 0.32. The fracture modes of the graded TPMS structures are a mixture of brittle fracture and layer-by-layer collapse. Firstly, the stress increases linearly with applied load with only a few minor fluctuations. When reaching the maximum compressive strength, the stresses of the graded samples with relative densities
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around = 0.26 drop gradually, but do not approach zero stress (Fig. 5 (a)). The deformation
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images reveal that at the first major facture, a global crack is formed along the direction at a 45º angle to the loading direction (see Fig. 5 (b)). Although global cracks are formed, the structures do not completely lose the load-bearing capacity. Following the first major fracture, the stresses rise again with non-regular fluctuations, until a strain of around ε = 50 % is reached. The following stress peaks are able to reach 30 - 50 % of the first peak. In contrast, the graded structures with higher relative densities (around = 0.31) display a more brittle fracture
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behaviour. A rapid stress drop after the first peak can be seen in Fig. 5 (a), with the formation of a few global cracks. These cracks are formed of a small angle respective to the loading direction. After the first major fracture, the stress-strain response continues to experiences additional stress peaks, but their magnitude are mostly less than 20 % of the compressive strength of the undamaged structures.
3.4 Mechanical properties
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The compressive fracture strength and energy absorption performance are investigated in this section. The fracture strength is considered as the maximum peak stress of each stressstrain curve under compression loading, which is not necessarily the first peak as can be seen in Fig. 4 (a). Since the alumina TPMS structures do not show a distinct densification point, the maximum stress is used to represent the load-carrying capacity before complete collapse. Fig. 6 (a) shows that the fracture strength of all alumina TPMS structures decreases with reducing relative density. The fracture strength of porous ceramics is sensitive to flaws as the micro defects on the top and bottom faces of the samples are likely to initiate local cracks, which is 8
the cause of minor stress fluctuations in the linear region and reduce the fracture strength. Samples with similar relative densities may therefore exhibit a large variation in fracture strength. A general trend line of the strength-density relation is shown in Fig. 6 (a). The graded TPMS structures studied in this paper exhibit the strength comparable to the non-graded ones. The strength of D-type TPMS structures made of stainless steel [22] is also plotted in Fig. 6 (a) for comparison. The strength of alumina TPMS structures with density above the transition shows a similar trend to that of the stainless steel structures. Below the transition density range, the fracture strength of alumina structures is seen to reduce rapidly due to the numerous local cracks formed at small strains.
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The energy absorbed per volume of each TPMS structure is calculated up to compressive strains of 0.02 and 0.5 for the samples failing by brittle and layer-by-layer fracture, respectively. The energy absorbed per volume of all TPMS structures is plotted in Fig. 6 (b). The data points of non-graded TPMS ceramics are separated into two distinct groups. In the range of moderate densities (> 1 g/cm3), the samples exhibit low energy per volume decreasing with reducing relative densities. However, when further decreasing relative densities, a remarkable increase of energy absorption capacity is shown. The two groups of the data points below and above the transition density have similar gradients, but with a spacing more than one order of magnitude. The transition relative density range of = 0.21 - 0.25 (structural densities in the range of ρ =
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0.84 - 1.00 g/cm3) can also be identified in the energy absorption plot as the shaded region in Fig. 6 (b). A density range instead of a distinct value is identified because the fracture behaviour of TPMS ceramics is sensitive to flaws, which introduce uncertainties and deviations of fracture modes as well as energy absorption performance. When compared with the energy absorption performance of stainless steel D-type TPMS structures [22], the gap between the two base materials is significantly reduced at the low density range (see the trend lines in Fig. 6 (b)) due to the pseudo-ductile failure mode. This pseudo-ductile failure mode possesses either a sustained region of elevated stress with applied strain or semi-regular stress serrations. Both of these responses result in improved energy absorption performance. Considering the effects of micro-architectures on mechanical properties, previous experimental data of stochastic alumina foams [25] is plotted in Fig. 6 for comparison. Alumna TPMS structures outperform foams in terms of fracture strength and energy absorption, indicating the mechanical efficiency of TPMS architectures. In order to obtain improved energy absorption performance while maintain the compressive stiffness and strength, we introduced thickness gradients into TPMS ceramics at moderate relative densities ( = 0.26 - 0.32). The selected gradient (6.25 μm/mm) provides more than a 10-fold increase of energy absorption for the graded structures with relative densities of = 0.26 and 0.27. However, this gradient is not large enough to change the fracture mode for the structures with densities around = 0.32; the fracture modes tend to be more brittle and the increase of energy absorption is only around 2-fold. Therefore, larger thickness gradients are 9
expected to shift the fracture mode to be more gradual at higher relative densities. Since graded structures include thin-walled regions, a decrease of fracture strength is expected if larger gradients are introduced. The optimization of gradient sharpness is recommended for future research studies to achieve the optimal balance between the increase of energy absorption abilities and the reduction of the compressive modulus and strength.
3.5 Prediction of the transition relative density
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The failure mechanisms of brittle porous solids with spherical holes under compression stresses were previously predicted using an analytical model developed by Sammis and Ashby [41]. The failure modes of brittle porous solids are affected by the stable crack length, the wall thickness to pore size ratio, initial porosity, as well as a confining pressure. This model predicts that brittle porous solids fail in a pseudo-ductile mode instead of a sudden and catastrophic fracture when reducing relative densities (increasing porosities). In addition, applying a confining pressure in the lateral direction can also stabilize the compressive failure behaviour, which is comprehensively analysed [41, 42]; however, the confining pressure is considered outside the scope of this paper. Meille et al. [25] investigated the fracture behaviour of stochastic alumina foams in terms of relative densities and estimated the transition relative density of = 0.4 with an empirical
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formula, which agreed well with their experimental results. The transition relative density of alumina TPMS structures in our work is lower than that of foams. In the work of [25], it is assumed that the fracture mode changes when the average wall thickness equals to the stable crack length and this hypothesis holds for different pore sizes. Through a theoretical modelling method of crack growth from a spherical hole in [41], the stable crack length is related to the pore diameter by a factor of 0.14, stated by l 0.14 p
(4)
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where l and p are the stable crack length and pore diameter, respectively. The pore diameter of the D-type TPMS architecture can be simply estimated by the distance of the ridges as shown in Fig. 2, p
2 a tavg 4
(5)
where a is the cell size. According to the relationship of the average wall thickness and relative density in Fig. 1 (c), and the given cell size a = 4 mm, we have tavg in mm by tavg 1.04
(6)
Taking the assumption that the average wall thickness equals to the stable crack length, and substituting Eqs. (4 - 6) into tavg = l, we calculate the transition relative density of 0.17 for alumina TPMS structures, which is slightly smaller than our experimental findings. The 10
discrepancy may be explained by the fact that the pore geometries of TPMS structures are nonspherical and highly complex. The formula adopts a simplified pore size estimation and does not consider the pores within TPMS walls, and thus provides an estimation of the transition density. Additional experiments on other TPMS-type structures need to be conducted to confirm or modify the predicted transition density. Nevertheless, the applied empirical formula indicates that the transition density depends on the micro-architectures of porous ceramics, and alumina TPMS structures have a lower transition density than stochastic foams due to the pore/micro-architecture geometries.
4. Conclusion
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This paper studies the fracture behaviour of ceramic TPMS structures made of alumina via ceramic SLA with and without thickness gradients at relative densities ranging from = 0.14
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to 0.37. We validate the existence of a transition density at which brittle fracture changes to pseudo-ductile failure for alumina TPMS sheet structures. Non-graded structures experience brittle fracture at moderate relative densities and pseudo-ductile layer-by-layer fracture at low relative densities. Experimental results indicate that the transition relative density for the Dtype TPMS structures is between = 0.21 and 0.25 (ρ = 0.84 - 1.00 g/cm3). The compressive
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fracture strength of alumina structures is mainly determined by relative density while the energy absorption capacities are determined by relative densities and thickness gradients. Structures with moderate relative densities (> 0.25) provide high compressive strength, but very low energy absorption capacity. In contrast, low relative density (< 0.21) structures have significantly higher energy absorption capacity, but low compressive strength. We demonstrate that the introduction of thickness gradients into ceramic TPMS structures can be used to tailor the fracture behaviour for alumina TPMS structures with moderate densities. With appropriate gradients, graded structures show significant increase in energy absorption performance and comparable fracture strength compared to the non-graded structures of similar density. This work therefore provides a guideline for the design of ceramic cellular structures with pseudoductile failure behaviour, making porous ceramics more suitable for a wide range of applications requiring fracture resistance.
Acknowledgments This work is supported by a National University of Singapore (NUS) scholarship for Dr. Lei Zhang. The authors further acknowledge the support from the Agency for Science, Technology and Research and the Science and Engineering Research Council of Singapore through the Additive Manufacturing Centre Initiative (SERC Grant no. 142 68 00088).
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[11] C. Petit, S. Meille, E. Maire, L. Gremillard, J. Adrien, G.Y. Lau, A.P. Tomsia, Fracture behavior of robocast HA/β-TCP scaffolds studied by X-ray tomography and finite element modeling, Journal of the European Ceramic Society 37(4) (2017) 1735-1745. [12] J.-B. Lee, W.-Y. Maeng, Y.-H. Koh, H.-E. Kim, Porous Calcium Phosphate Ceramic Scaffolds with Tailored Pore Orientations and Mechanical Properties Using LithographyBased Ceramic 3D Printing Technique, Materials 11(9) (2018) 1711. [13] C. Minas, D. Carnelli, E. Tervoort, A.R. Studart, 3D printing of emulsions and foams into hierarchical porous ceramics, Advanced Materials 28(45) (2016) 9993-9999.
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[14] P. Jana, O. Santoliquido, A. Ortona, P. Colombo, G.D. Sorarù, Polymer‐derived SiCN cellular structures from replica of 3D printed lattices, Journal of the American Ceramic Society 101(7) (2018) 2732-2738. [15] J. Berger, H. Wadley, R. McMeeking, Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness, Nature 543(7646) (2017) 533. [16] C. Bonatti, D. Mohr, Mechanical performance of additively-manufactured anisotropic and isotropic smooth shell-lattice materials: Simulations & experiments, Journal of the Mechanics and Physics of Solids 122 (2019) 1-26. [17] C. Bonatti, D. Mohr, Smooth-shell metamaterials of cubic symmetry: Anisotropic elasticity, yield strength and specific energy absorption, Acta Materialia 164 (2019) 301-321. [18] T. Tancogne‐Dejean, M. Diamantopoulou, M.B. Gorji, C. Bonatti, D. Mohr, 3D Plate‐
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[23] O. Al‐Ketan, M. Pelanconi, A. Ortona, R.K. Abu Al‐Rub, Additive manufacturing of architected catalytic ceramic substrates based on triply periodic minimal surfaces, Journal of the American Ceramic Society (2019). [24] J. Hubálková, C. Voigt, A. Schmidt, K. Moritz, C.G. Aneziris, Comparative Phenomenological Study of Fracture Behavior of Ceramic and Glass Foams under
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Compressive Stress Using In Situ X‐Ray Microtomography, Advanced Engineering Materials 19(9) (2017) 1700286.
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[25] S. Meille, M. Lombardi, J. Chevalier, L. Montanaro, Mechanical properties of porous ceramics in compression: On the transition between elastic, brittle, and cellular behavior, Journal of the European Ceramic Society 32(15) (2012) 3959-3967. [26] M. Banda, D. Ghosh, Effects of porosity and strain rate on the uniaxial compressive response of ice-templated sintered macroporous alumina, Acta Materialia 149 (2018) 179-192. [27] L.J. Gibson, M.F. Ashby, Cellular solids: structure and properties, Cambridge university press1999. [28] M. Ashby, The properties of foams and lattices, Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 364(1838) (2006) 1530. 13
[29] D. Jang, L.R. Meza, F. Greer, J.R. Greer, Fabrication and deformation of threedimensional hollow ceramic nanostructures, Nature materials 12(10) (2013) 893. [30] L.R. Meza, G.P. Phlipot, C.M. Portela, A. Maggi, L.C. Montemayor, A. Comella, D.M. Kochmann, J.R. Greer, Reexamining the mechanical property space of three-dimensional lattice architectures, Acta Materialia 140 (2017) 424-432. [31] F. Liu, Z. Mao, P. Zhang, D.Z. Zhang, J. Jiang, Z. Ma, Functionally graded porous scaffolds in multiple patterns: New design method, physical and mechanical properties, Materials & Design 160 (2018) 849-860. [32] X.-Y. Zhang, G. Fang, L.-L. Xing, W. Liu, J. Zhou, Effect of porosity variation strategy on the performance of functionally graded Ti-6Al-4V scaffolds for bone tissue engineering, Materials & Design 157 (2018) 523-538.
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[36] C. Han, Y. Li, Q. Wang, S. Wen, Q. Wei, C. Yan, L. Hao, J. Liu, Y. Shi, Continuous functionally graded porous titanium scaffolds manufactured by selective laser melting for bone implants, Journal of the mechanical behavior of biomedical materials 80 (2018) 119-127.
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[37] D. de Faoite, D.J. Browne, F.R. Chang-Díaz, K.T. Stanton, A review of the processing, composition, and temperature-dependent mechanical and thermal properties of dielectric technical ceramics, Journal of Materials Science 47(10) (2012) 4211-4235.
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[38] O. Al ‐ Ketan, R. Rezgui, R. Rowshan, H. Du, N.X. Fang, R.K. Abu Al ‐ Rub, Microarchitected Stretching ‐ Dominated Mechanical Metamaterials with Minimal Surface Topologies, Advanced Engineering Materials (2018) 1800029.
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[39] Lithoz GmbH, Available materials for the LCM technology, 2019. http://www.lithoz.com. [40] J.W. Halloran, Ceramic stereolithography: additive manufacturing for ceramics by photopolymerization, Annual Review of Materials Research 46 (2016) 19-40.
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[41] C. Sammis, M. Ashby, The failure of brittle porous solids under compressive stress states, Acta metallurgica 34(3) (1986) 511-526. [42] M. Ashby, C. Sammis, The damage mechanics of brittle solids in compression, Pure and Applied Geophysics 133(3) (1990) 489-521.
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Fig. 1. Generation of TPMS structures using Eq. (2). (a) The non-graded and (b) graded designs of Dtype TPMS structures. (c) The average thickness and relative density of D-type TPMS structures with respect to c. The linear fitting line (dashed line) with a coefficient of determination R2=0.99995 indicates a nearly linear relationship.
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Fig. 2. SLA printed (a) non-graded (measured relative density of = 0.37) and (b) graded (measured
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relative density of = 0.26) alumina TPMS structures. Arrows indicate the building direction. The
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definition of pore size is illustrated in (a).
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Fig. 3. Compressive responses of non-graded alumina TPMS structures failed by brittle fracture. (a) Stress-strain curves of samples. The brittle fracture mode of the samples with relative densities of (b) 0.32 and (c) 0.21 at indicated points on the stress-strain curves. Dashed ellipses indicate the formation of global cracks.
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Fig. 4. Compressive responses of non-graded alumina TPMS structures failed by layer-by-layer failure. (a) Stress-strain curves of three representative samples. The layer-by-layer fracture mode of the samples with relative densities of (b) 0.25, (c) 0.20, and (d) 0.14 at the strains labelled on the stress-strain curves. Dashed ellipses indicate the formation of global cracks. Dashed arrows indicate the crashed layers.
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Fig. 5. Compressive behaviour of graded alumina TPMS structures. (a) Stress-strain curves. The fracture modes of the samples with relative densities of (b) 0.26 and (c) 0.31 at the strains labelled on the stress-strain curves. Wall thicknesses decrease with height in the z-direction. Dashed ellipses indicate the formation of global cracks.
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Fig. 6. The (a) fracture strength, σs, and (b) energy absorbed per volume, W, of TPMS structures. The transition (relative) density range is highlighted by the shaded region. The power fit trend lines for the respective TPMS structures are shown by dashed lines.
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Table 1. Details of non-graded TPMS structures studied in this work. Two samples are fabricated for non-graded designs with c = 0.24 and 0.18; one sample is fabricated for others.
0.35 0.29 0.24 0.18 0.16 0.14 0.12
, design
, measured by tavg, design (mm) tavg,
(-)
dry weight (-)
0.30 0.25 0.20 0.15 0.12 0.10 0.09
0.37 0.32 0.27, 0.25 0.21, 0.12 0.18 0.15 0.14
measured
(mm) 0.30 0.25 0.20 0.15 0.13 0.11 0.10
0.38 0.33 0.28, 0.26 0.22, 0.20 0.18 0.16 0.15
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Table 2. Details of graded TPMS structures studied in this work. Two samples are fabricated for all graded designs. tavg (z) (mm), 0 ≤ z ≤ 16, design -3
, measured by
dry weight (-)
-3
0.2 + 6.25z × 10 0.24 0.15 + 6.25z × 10-3 0.19
0.32, 0.31 0.27, 0.26
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0.24 + 6z × 10 0.18 + 6z × 10-3
, design (-)
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c(z), 0 ≤ z ≤ 16
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Table 3. Number of stress serrations and magnitude of stress peaks of non-graded low relative density (less than 0.21) structures. Stress peaks less than 30% of the Max peak do not count. , measured by Number of stress Average
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22.5 12.9 8.7 8.1
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0.20 0.18 0.16 0.14
serrations (-)
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dry weight (-)
magnitude of stress peaks (MPa)
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PII:
S0955-2219(19)30667-3
DOI:
https://doi.org/10.1016/j.jeurceramsoc.2019.09.048
Reference:
JECS 12758
To appear in:
Journal of the European Ceramic Society
Please cite this article as: Zhang L, Feih S, Daynes S, Chang S, Wang MY, Wei J, Lu WF, Pseudo-ductile fracture of 3D printed alumina triply periodic minimal surface structures,
This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.
Pseudo-ductile fracture of 3D printed alumina triply periodic minimal surface structures Lei Zhanga, b, Stefanie Feihb, Stephen Daynesb, Shuai Changa, Michael Yu Wangc, Jun Weib, and Wen Feng Lua,*
a
Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore, 117575 b
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Singapore Institute of Manufacturing Technology (SIMTech), 2 Fusionopolis Way, Singapore 138634 c
Department of Mechanical and Aerospace Engineering, and Department of Electronic and Computer Engineering of Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
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Abstract
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Additive manufacturing enables the fabrication of periodic ceramic lattices with controllable micro-architectures. Many studies reported their catastrophic brittle fracture behaviour. However, ceramic lattices may fail by a layer-by-layer pseudo-ductile fracture mode, by controlling micro-architectures and porosities. Moreover, their fracture behaviour can be optimised by introducing strut/wall thickness gradients. This paper investigates the fracture behaviour and the fracture mode transition of ceramic triply periodic minimal surface (TPMS) structures. Alumina TPMS structures with relative densities of 0.14-0.37 are fabricated by ceramic stereolithography. Quasi-static compression tests validate a transition density range for non-graded samples: low (<0.21) and moderate (>0.25) relative density samples show layer-by-layer pseudo-ductile and catastrophic brittle fracture modes, respectively. The pseudo-ductile failure mode increases the energy absorption performance, enabling loadbearing capacity for a compressive strain up to 50%. With appropriate thickness gradients, graded structures exhibit significant increase of energy absorption without a decrease of fracture strength compared to their non-graded counterparts.
Key words: Additive manufacturing; Porous ceramics; Fracture behaviour; Energy absorption; Triply periodic minimal surface (TPMS)
To be submitted to: Journal of the European Ceramic Society
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1. Introduction
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Porous ceramics consisting of periodic micro-architectures and open pores are widely used for multifunctional applications including structural components, thermal management devices, filters, and tissue engineering scaffolds [1, 2]. Nowadays, additive manufacturing (AM) of this group of ceramics has attracted an increasing interest due to the enhanced design freedom and hence of controllable micro-architectures and pore geometries, which are generally not achievable by traditional fabrication techniques [2, 3]. A variety of AM technologies have been developed for porous ceramics [2-4], such as robocasting (extrusion based methods) [5-8], powder based 3D printing [9] and stereolithography (SLA) [1, 2, 10]. The latter technique is suitable for periodic porous ceramics since it can produce complex and nearly dense ceramic parts (possible to reach 99% of theoretical density) with high precision, which is favourable for high mechanical strength [2, 10]. Ceramic SLA fabricates each layer of a part by laser or UV light scanning of a photopolymerizable suspension filled with ceramic powders, followed by de-binding and sintering processes.
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A range of periodic porous ceramics with highly complex micro-architectures have been studied. One of the well-studied structures is a mesh of ceramic rods; their compressive strength and failure characteristics were investigated through simulations and experiments [6, 11]. The influence of pore orientation on the mechanical properties of the ceramic scaffolds was investigated on SLA printed ceramic meshes [12]. Hierarchical porous ceramics that included circular rods in the mm-range and randomly distributed micro pores in the µm-range were fabricated using the robocasting technique [7, 8, 13]. 3D lattices with various ceramic materials and architectures such as cubic and tetradecahedron architectures have also been fabricated [10, 14]. However, considering the structure-property relationship of porous structures, shell-like and plate-like architectures generally outperform strut-based lattices on stiffness and strength. This has been observed through analytical, numerical and experimental methods [15-18]. Recently, multifunctional triply periodic minimal surface (TPMS) structures have drawn significant attention because of their smooth shell-like geometries, open cell topology and large surface area [19-22]. Simulations and compressive tests on 3D printed metallic TPMS structures validated their superior energy absorption abilities compared to body-centred cubic strut-based lattices [20, 22]. Bonatti and Mohr investigated smooth shell lattices having similar geometries to TPMS, showing their higher mechanical properties than truss lattices of equal density [16, 17]. Therefore, the combination of ceramic materials and TPMS structures is expected to achieve high mechanical properties for porous ceramics. However, due to the increased geometric complexity of creating TPMS structures via ceramic AM technologies, few studies have investigated ceramic TPMS structure fabrication and their performance. AlKetan et al. [23] fabricated alumina TPMS structures as catalytic substrates by ceramic SLA, and studied their mechanical and flow characteristics. The failure behaviour of porous ceramics is dominated by material fracture due to the brittle nature of ceramic materials. Experimental studies on porous ceramics have shown two typical 2
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fracture modes under compression. In the moderate and high relative density range, the compressive stress linearly increases with the compressive strain up to a sudden rupture, at which point the samples fail catastrophically, similar to monolithic ceramics [11, 12, 24]. In this case global cracks generally initiate and grow along the loading direction. In contrast, for highly porous ceramics, the stress-strain curves are characterized by a linear region followed by a fluctuating stress-strain response or a stress plateau due to a layer-by-layer fracture mode [5, 24-26]. During this failure process, successive layer-wise cracks initiate and propagate. The transition relative density (porosity) at which the failure mode changes from brittle fracture to layer-by-layer collapse was estimated by experiments [25, 26] and empirical formulas [25] for stochastic ceramic foams. This failure mode for highly porous ceramics leads to significantly improved energy absorption capacity. However, according to analytical models for mechanical behaviour of cellular solids [27-30], reducing the relative density also reduces failure strength. The challenge of improving the energy absorption performance for porous ceramics is to increase the relative density while maintaining the gradual failure behaviour.
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The introduction of graded designs is a feasible solution to avoid catastrophic failure for porous materials. This strategy has been used for metallic and polymeric lattice structures to change their failure behaviour from a global shear mode to a localized layer-by-layer mode [31-36]. One of the commonly used graded designs is to gradually vary their strut/wall thickness or relative density along a linear direction. By designing strut thickness graded structures along the compressive loading direction, experimental results showed that a layerby-layer deformation mode was obtained for graded lattice structures, such as Gyroid structures [31] and body-centred-cubic lattices [34]; made of different materials including Ti-6Al-4V [3133], Al-Si10-Mg [34] and nylon [35]. In contrast, their non-graded counterparts failed by a global shear mode throughout the structure [32-34]. While this has not been previously validated for ceramic materials, graded TPMS ceramic structures are expected to fail in a gradual manner and therefore to have improved energy absorption performance. In this work, we fabricate Diamond (D)-type TPMS structures with relative densities ranging from = 0.14 to 0.37, with and without wall thickness gradients, using the SLA
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technique and alumina material. Alumina is one of the widely used structural ceramics due to its relatively high strength, hardness and maximum service temperature among ceramic materials [37]. The failure behaviour, fracture strength and energy absorption capacity are studied by quasi-static compression tests. The influences of the relative density and thickness gradient on fracture modes, as well as the fracture mode change and the transition relative density are investigated. Finally, an empirical formula is used to predict the transition relative density for TPMS structures.
2. Materials and methods 2.1 Design and modelling 3
One of the most widely used methods to describe TPMS structures is the level-set approximation. Among numerous types of TPMS structures, we select the Diamond (D) type surface because of the high mechanical properties based on the studies of structure-property relationships of TPMS structures for metallic alloys [22, 38]. The D-type surface is approximated by D c with
D ( x, y, z ) sin( x)sin( y)sin( z ) sin( x) cos( y) cos( z ) cos( x)sin( y) cos( z) cos( x) cos( y )sin( z )
(1)
where x, y, z are spatial coordinates; ω = 2π/l and l is the unit cell length. c is a constant controlling the shape of the surface. Since the equation D c only defines periodic surfaces
(D )2 c 2
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without thicknesses, solid models can be created by uniformly thickening the surfaces. However, using this method to create graded TPMS structures by non-uniform thickening is generally not straightforward. To address this difficulty, Eq. (1) is modified as: (2)
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This modified function is able to describe a solid structure instead of a surface, by taking the regions with 0 as voids, and the regions with 0 as solids. In other words, the sheetlike TPMS structures are bounded by the two surfaces of D c . Using Eq. (2), non-graded
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TPMS structures, as defined by
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structures are generated with a constant c (Fig. 1 (a)). It should be noted that a constant c in Eq. (2) generates non-uniform thicknesses throughout a unit cell as the distance between the two surfaces D c is not constant. Therefore an average thickness, tavg, is used to characterise tavg Vs / S
(3)
of a unit cell.
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where Vs is the volume of the solids of a unit cell and S is the area of the mid-surface ( D 0 )
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In this work, Matlab scripting is used to generate TPMS sheet structures with/without thickness gradients based on Eq. (2), including 4 × 4 × 4 repeating unit cells and a cell size of 4 mm, see Fig. 1 (a, b). This results in a total sample size of 16 × 16 × 16 mm. Non-graded structures with nominal average wall thicknesses of 150 - 300 µm (relative densities ranging 0.15 - 0.3) are created by keeping c constant (see Table 1). For the graded structures, wall thickness gradients can be introduced by replacing the constant c with a spatially varying function c(x, y, z) (Fig. 1 (b)). In this work, the thickness gradient is introduced by a linear function of c(z) = b + mz (0 ≤ z ≤ 16) where b and m are prescribed parameters, in the building direction (z), and later also the testing direction. In order to control the wall thickness effectively in the design stage, the average wall thickness as well as relative density, , are investigated with respect to c (see Fig. 1 (c)). Since the relative density is proportional to the average wall thickness, the two curves are the same but with 4
slightly different axis scales. Fig. 1 (c) shows that the average thickness and relative density of a TPMS structure created by Eq. (2) are nearly linear (coefficient of determination R2 = 0.99995) with respect to variable c, so the linear function c(z) introduces a constant gradient of the average thickness along the z-direction. In this work, functions c(z) used to generated graded TPMS structures are shown in Table 2, and all graded structures have a same wall thickness gradient of 6.25 µm/mm.
2.2 Ceramic stereolithography and post-processing
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The ceramic TPMS structures were fabricated by the Lithoz CeraFab 8500 stereolithography printer (Lithoz GmbH, Vienna), which operates according to the lithographybased ceramic manufacturing (LCM) process. A commercial alumina (Al2O3) suspension LithaLox HP 500 (Lithoz GmbH, Vienna) was used to fabricate TPMS structures. The lateral resolution of the printer is 40 µm, and the z-resolution (layer thickness) was set to 25 µm for printing. A blue light source with a wavelength of 465 nm and a digital light processing projector is used to cure the suspension throughout the glass bottom of the rotating vat. The green parts with TPMS structures were built in a bottom-up manner.
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2.3 Density measurement
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The cleaning fluid of Lithoz was used to clean the green parts. The non-cured suspension sticking on the printed green parts was removed with the assistance of compressed air. The debinding and sintering post-processing were done in ambient air. During the de-binding cycle, the binder leaves the green parts. De-binding was performed with different dwell times at 75, 115, 205 and 430 °C, followed by being pre-sintered at 900 °C (2 hours) to improve the handling strength. The sintering process was completed at a temperature of 1600 °C with a heating rate of 1°C/min and a dwell time of 2 hours.
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Since ceramic AM technologies generally introduce porosities into the printed structure, the densities for both the solids and the structures are of interest. At the micro scale, the solid density of the ceramic walls was measured using Archimedes’ principle. The Archimedes’ method is able to account for the porosity within the walls. The structural densities of the ceramic samples were measured by the dry weight method, including the porosities introduced by both AM processes at the microscale and the TPMS architectures. The structural densities and their ratios to the theoretical density of alumina are referred to as densities and relative densities, respectively, of TPMS structures in the following text. It should be noted that the relative densities of the geometric models only consider the TPMS architectures, assuming the walls of TPMS structures are fully solid (Table 1). 2.4 Compression testing Compression tests with displacement control were performed using an Instron 5982 universal test machine with a 100 kN load cell. For testing of the quasi-static fracture strength, 5
a strain rate of 5.2 × 10-5 /s (crosshead speed of 0.05 mm/min) was applied. After 5 % compressive strain, a higher strain rate of 5.2 × 10-4 /s (crosshead speed of 0.5 mm/min) was applied to investigate the layer-by-layer failure behaviour. Since the non-graded TPMS structures with moderate relative densities (> 0.25) failed catastrophically, tests were stopped at a compressive strain of about ε = 2 %, while the low relative density (< 0.21) non-graded
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and all graded structures were tested up to a compressive strain of ε = 50 %. For samples with relative densities within the transition range, tests were terminated depending on the observed failure behaviour. The loading direction coincided with the building direction, as well as the thickness grading direction for graded structures. The crosshead movement of the Instron 5982 system was used to measure the compressive displacement Δu, which is then converted to compressive strain by u / h , where h is the sample height. Compressive stresses are calculated by F / A , where F and A are compressive load and the area of samples’ top or bottom faces, respectively. The fracture modes were captured via a camera during the tests, operating at a frequency of 0.5 Hz.
3. Results and discussions
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3.1 Sample morphology
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The solid density of ceramic samples measured by the Archimedes’ method is 3.796 ± 0.175 g/cm3, which is above 95% of the theoretical density of alumina (3.985 g/cm3 [39]). The measured material densities were slightly smaller than the theoretical density of alumina because of the micro pores/cracks generated during the AM process inside the walls. Ceramic SLA is able to fabricate porous ceramic structures with high solid density because of the fine powder size and the high solid loading of the slurry [40]. Ketan et al. [23] also reported a high solid density of 3.72 g/cm3 of alumina TPMS structures fabricated by ceramic SLA. The relative densities measured by the dry weight method are shown in Tables 1 and 2. The relative density includes both the micro pores/cracks and the voids introduced by TPMS architectures, and is therefore significantly smaller than 1. The optical microscope images show no macroscopic pores or cracks on the walls with different thicknesses; and the ceramic walls are uniform for non-graded structures and gradually vary for the graded designs without distortions (see Fig. 2). Some of the samples’ faces are not perfectly flat and include a few flaws due to the cleaning processes. The measured relative densities of ceramic TPMS structures are higher than the designs as shown in Tables 1 and 2, which is consistent with measured wall thickness, calculated by Eq. (3) based on density measurements. This could be attributed to the usage of high energy input and high shrinkage coefficient compensation. No further efforts were made to optimise the processing parameters. For later results, the measured (relative) densities are used. 3.2 Fracture characteristics of non-graded alumina TPMS structures 6
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All alumina TPMS structures were tested under quasi-static compression loading along the building direction, i.e., the z-direction. The non-graded samples in the moderate relative densities (> 0.25) show a stress-strain behaviour consisting of a linear region followed by a sudden stress drop (see Fig. 3 (a)). The linear region includes a few minor stress fluctuations with applied loads. This is caused by the local cracks forming at the faces of TPMS samples contacting with the rigid compression plates. Since the faces are not perfectly flat and flaws exist on the faces of samples, stress concentrations are formed and small cracks grow along the faces, leading to the small pieces breaking off the samples’ contacting surface. These minor cracks and broken pieces do not cause catastrophic failure, so the stress continues to rise. When the compressive loads reaching its maximum, global cracks formulate globally along the loading direction (see Fig. 3 (b, c)). The global cracks grow throughout the samples and the TPMS structures fail by brittle rupture at a relatively small compressive strain typical for brittle materials (ε ≤ 0.02), resulting in a dramatic stress drop. As can be seen in the deformation images during testing (Fig. 3 (b, c)), the TPMS samples break apart completely and lose all load-bearing capacity.
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When reducing relative densities within or below the transition range, non-graded TPMS samples exhibit a completely different failure behaviour. The failure modes change from brittle fracture to pseudo-ductile layer-by-layer failure. This failure mode is dominated by numerous local cracks and fractures initiating at relatively small compressive loads. Therefore, the stressstrain response of these structures shows high-frequency stress fluctuations throughout (Fig. 4 (a)). The stress of the TPMS structure with a relative density of = 0.25 gradually reduces
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with non-regular fluctuations after the first peak (Fig. 4 (a)). During this region, local fractures form at the top and bottom faces (Fig. 4 (b)). Once a compressive strain around ε = 0.32 is reached, a global crack forms and grows through the structure, leading to the loss of load-
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bearing capacity. When further decreasing relative densities below a threshold of = 0.21, the
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TPMS structures display a clear layer-by-layer failure mode with semi-regular serrations spanning a large strain range (Fig. 4 (a)). The period of each stress serration corresponds to the height of a quarter of a unit cell, namely a strain of ε = 0.0625. This implies that each layer collapse is within a quarter of a unit cell, and other layers remain mostly intact during each stress serration. This is validated by the deformation images, showing the cracks propagating within the layer with a height of a quarter of a unit cell near to the compressive plates, while the rest of the structures remains intact (see Fig. 4 (c, d)). During each layer collapse, there are numerous high-frequency stress fluctuations, corresponding to the local cracks growing along the TPMS walls. However, the local cracks grow mostly within a layer, and there is no global crack formation during the stress serrations (see Fig. 4 (c, d)). The stress peak magnitude and the number of stress serrations depend on the relative density; reducing relative densities reduces the stress magnitude and increases the number of serrations. This trend for non-graded TPMS structures with relative densities less than = 0.21 is shown in Table 3. One exception
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regarding the stress serrations number of the structure with = 0.156 may be attributed to stochastic printing imperfections. Experimental results for non-graded samples show that the transition relative density at which the fracture modes change is around = 0.21 - 0.25. Within this transition relative density range, the fracture modes of alumina TPMS structures are sensitive to geometric imperfections and flaw distributions, as can be seen by the mixture of fracture modes existing within this range.
3.3 Fracture characteristics of graded alumina TPMS structures Fig. 5 shows typical stress strain curves and deformed geometries of the graded structures
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for relative densities in the range = 0.26 - 0.32. The fracture modes of the graded TPMS structures are a mixture of brittle fracture and layer-by-layer collapse. Firstly, the stress increases linearly with applied load with only a few minor fluctuations. When reaching the maximum compressive strength, the stresses of the graded samples with relative densities
-p
around = 0.26 drop gradually, but do not approach zero stress (Fig. 5 (a)). The deformation
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images reveal that at the first major facture, a global crack is formed along the direction at a 45º angle to the loading direction (see Fig. 5 (b)). Although global cracks are formed, the structures do not completely lose the load-bearing capacity. Following the first major fracture, the stresses rise again with non-regular fluctuations, until a strain of around ε = 50 % is reached. The following stress peaks are able to reach 30 - 50 % of the first peak. In contrast, the graded structures with higher relative densities (around = 0.31) display a more brittle fracture
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behaviour. A rapid stress drop after the first peak can be seen in Fig. 5 (a), with the formation of a few global cracks. These cracks are formed of a small angle respective to the loading direction. After the first major fracture, the stress-strain response continues to experiences additional stress peaks, but their magnitude are mostly less than 20 % of the compressive strength of the undamaged structures.
3.4 Mechanical properties
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The compressive fracture strength and energy absorption performance are investigated in this section. The fracture strength is considered as the maximum peak stress of each stressstrain curve under compression loading, which is not necessarily the first peak as can be seen in Fig. 4 (a). Since the alumina TPMS structures do not show a distinct densification point, the maximum stress is used to represent the load-carrying capacity before complete collapse. Fig. 6 (a) shows that the fracture strength of all alumina TPMS structures decreases with reducing relative density. The fracture strength of porous ceramics is sensitive to flaws as the micro defects on the top and bottom faces of the samples are likely to initiate local cracks, which is 8
the cause of minor stress fluctuations in the linear region and reduce the fracture strength. Samples with similar relative densities may therefore exhibit a large variation in fracture strength. A general trend line of the strength-density relation is shown in Fig. 6 (a). The graded TPMS structures studied in this paper exhibit the strength comparable to the non-graded ones. The strength of D-type TPMS structures made of stainless steel [22] is also plotted in Fig. 6 (a) for comparison. The strength of alumina TPMS structures with density above the transition shows a similar trend to that of the stainless steel structures. Below the transition density range, the fracture strength of alumina structures is seen to reduce rapidly due to the numerous local cracks formed at small strains.
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The energy absorbed per volume of each TPMS structure is calculated up to compressive strains of 0.02 and 0.5 for the samples failing by brittle and layer-by-layer fracture, respectively. The energy absorbed per volume of all TPMS structures is plotted in Fig. 6 (b). The data points of non-graded TPMS ceramics are separated into two distinct groups. In the range of moderate densities (> 1 g/cm3), the samples exhibit low energy per volume decreasing with reducing relative densities. However, when further decreasing relative densities, a remarkable increase of energy absorption capacity is shown. The two groups of the data points below and above the transition density have similar gradients, but with a spacing more than one order of magnitude. The transition relative density range of = 0.21 - 0.25 (structural densities in the range of ρ =
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0.84 - 1.00 g/cm3) can also be identified in the energy absorption plot as the shaded region in Fig. 6 (b). A density range instead of a distinct value is identified because the fracture behaviour of TPMS ceramics is sensitive to flaws, which introduce uncertainties and deviations of fracture modes as well as energy absorption performance. When compared with the energy absorption performance of stainless steel D-type TPMS structures [22], the gap between the two base materials is significantly reduced at the low density range (see the trend lines in Fig. 6 (b)) due to the pseudo-ductile failure mode. This pseudo-ductile failure mode possesses either a sustained region of elevated stress with applied strain or semi-regular stress serrations. Both of these responses result in improved energy absorption performance. Considering the effects of micro-architectures on mechanical properties, previous experimental data of stochastic alumina foams [25] is plotted in Fig. 6 for comparison. Alumna TPMS structures outperform foams in terms of fracture strength and energy absorption, indicating the mechanical efficiency of TPMS architectures. In order to obtain improved energy absorption performance while maintain the compressive stiffness and strength, we introduced thickness gradients into TPMS ceramics at moderate relative densities ( = 0.26 - 0.32). The selected gradient (6.25 μm/mm) provides more than a 10-fold increase of energy absorption for the graded structures with relative densities of = 0.26 and 0.27. However, this gradient is not large enough to change the fracture mode for the structures with densities around = 0.32; the fracture modes tend to be more brittle and the increase of energy absorption is only around 2-fold. Therefore, larger thickness gradients are 9
expected to shift the fracture mode to be more gradual at higher relative densities. Since graded structures include thin-walled regions, a decrease of fracture strength is expected if larger gradients are introduced. The optimization of gradient sharpness is recommended for future research studies to achieve the optimal balance between the increase of energy absorption abilities and the reduction of the compressive modulus and strength.
3.5 Prediction of the transition relative density
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The failure mechanisms of brittle porous solids with spherical holes under compression stresses were previously predicted using an analytical model developed by Sammis and Ashby [41]. The failure modes of brittle porous solids are affected by the stable crack length, the wall thickness to pore size ratio, initial porosity, as well as a confining pressure. This model predicts that brittle porous solids fail in a pseudo-ductile mode instead of a sudden and catastrophic fracture when reducing relative densities (increasing porosities). In addition, applying a confining pressure in the lateral direction can also stabilize the compressive failure behaviour, which is comprehensively analysed [41, 42]; however, the confining pressure is considered outside the scope of this paper. Meille et al. [25] investigated the fracture behaviour of stochastic alumina foams in terms of relative densities and estimated the transition relative density of = 0.4 with an empirical
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formula, which agreed well with their experimental results. The transition relative density of alumina TPMS structures in our work is lower than that of foams. In the work of [25], it is assumed that the fracture mode changes when the average wall thickness equals to the stable crack length and this hypothesis holds for different pore sizes. Through a theoretical modelling method of crack growth from a spherical hole in [41], the stable crack length is related to the pore diameter by a factor of 0.14, stated by l 0.14 p
(4)
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where l and p are the stable crack length and pore diameter, respectively. The pore diameter of the D-type TPMS architecture can be simply estimated by the distance of the ridges as shown in Fig. 2, p
2 a tavg 4
(5)
where a is the cell size. According to the relationship of the average wall thickness and relative density in Fig. 1 (c), and the given cell size a = 4 mm, we have tavg in mm by tavg 1.04
(6)
Taking the assumption that the average wall thickness equals to the stable crack length, and substituting Eqs. (4 - 6) into tavg = l, we calculate the transition relative density of 0.17 for alumina TPMS structures, which is slightly smaller than our experimental findings. The 10
discrepancy may be explained by the fact that the pore geometries of TPMS structures are nonspherical and highly complex. The formula adopts a simplified pore size estimation and does not consider the pores within TPMS walls, and thus provides an estimation of the transition density. Additional experiments on other TPMS-type structures need to be conducted to confirm or modify the predicted transition density. Nevertheless, the applied empirical formula indicates that the transition density depends on the micro-architectures of porous ceramics, and alumina TPMS structures have a lower transition density than stochastic foams due to the pore/micro-architecture geometries.
4. Conclusion
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This paper studies the fracture behaviour of ceramic TPMS structures made of alumina via ceramic SLA with and without thickness gradients at relative densities ranging from = 0.14
-p
to 0.37. We validate the existence of a transition density at which brittle fracture changes to pseudo-ductile failure for alumina TPMS sheet structures. Non-graded structures experience brittle fracture at moderate relative densities and pseudo-ductile layer-by-layer fracture at low relative densities. Experimental results indicate that the transition relative density for the Dtype TPMS structures is between = 0.21 and 0.25 (ρ = 0.84 - 1.00 g/cm3). The compressive
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fracture strength of alumina structures is mainly determined by relative density while the energy absorption capacities are determined by relative densities and thickness gradients. Structures with moderate relative densities (> 0.25) provide high compressive strength, but very low energy absorption capacity. In contrast, low relative density (< 0.21) structures have significantly higher energy absorption capacity, but low compressive strength. We demonstrate that the introduction of thickness gradients into ceramic TPMS structures can be used to tailor the fracture behaviour for alumina TPMS structures with moderate densities. With appropriate gradients, graded structures show significant increase in energy absorption performance and comparable fracture strength compared to the non-graded structures of similar density. This work therefore provides a guideline for the design of ceramic cellular structures with pseudoductile failure behaviour, making porous ceramics more suitable for a wide range of applications requiring fracture resistance.
Acknowledgments This work is supported by a National University of Singapore (NUS) scholarship for Dr. Lei Zhang. The authors further acknowledge the support from the Agency for Science, Technology and Research and the Science and Engineering Research Council of Singapore through the Additive Manufacturing Centre Initiative (SERC Grant no. 142 68 00088).
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Fig. 1. Generation of TPMS structures using Eq. (2). (a) The non-graded and (b) graded designs of Dtype TPMS structures. (c) The average thickness and relative density of D-type TPMS structures with respect to c. The linear fitting line (dashed line) with a coefficient of determination R2=0.99995 indicates a nearly linear relationship.
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Fig. 2. SLA printed (a) non-graded (measured relative density of = 0.37) and (b) graded (measured
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relative density of = 0.26) alumina TPMS structures. Arrows indicate the building direction. The
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definition of pore size is illustrated in (a).
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Fig. 3. Compressive responses of non-graded alumina TPMS structures failed by brittle fracture. (a) Stress-strain curves of samples. The brittle fracture mode of the samples with relative densities of (b) 0.32 and (c) 0.21 at indicated points on the stress-strain curves. Dashed ellipses indicate the formation of global cracks.
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Fig. 4. Compressive responses of non-graded alumina TPMS structures failed by layer-by-layer failure. (a) Stress-strain curves of three representative samples. The layer-by-layer fracture mode of the samples with relative densities of (b) 0.25, (c) 0.20, and (d) 0.14 at the strains labelled on the stress-strain curves. Dashed ellipses indicate the formation of global cracks. Dashed arrows indicate the crashed layers.
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Fig. 5. Compressive behaviour of graded alumina TPMS structures. (a) Stress-strain curves. The fracture modes of the samples with relative densities of (b) 0.26 and (c) 0.31 at the strains labelled on the stress-strain curves. Wall thicknesses decrease with height in the z-direction. Dashed ellipses indicate the formation of global cracks.
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Fig. 6. The (a) fracture strength, σs, and (b) energy absorbed per volume, W, of TPMS structures. The transition (relative) density range is highlighted by the shaded region. The power fit trend lines for the respective TPMS structures are shown by dashed lines.
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Table 1. Details of non-graded TPMS structures studied in this work. Two samples are fabricated for non-graded designs with c = 0.24 and 0.18; one sample is fabricated for others.
0.35 0.29 0.24 0.18 0.16 0.14 0.12
, design
, measured by tavg, design (mm) tavg,
(-)
dry weight (-)
0.30 0.25 0.20 0.15 0.12 0.10 0.09
0.37 0.32 0.27, 0.25 0.21, 0.12 0.18 0.15 0.14
measured
(mm) 0.30 0.25 0.20 0.15 0.13 0.11 0.10
0.38 0.33 0.28, 0.26 0.22, 0.20 0.18 0.16 0.15
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c
Table 2. Details of graded TPMS structures studied in this work. Two samples are fabricated for all graded designs. tavg (z) (mm), 0 ≤ z ≤ 16, design -3
, measured by
dry weight (-)
-3
0.2 + 6.25z × 10 0.24 0.15 + 6.25z × 10-3 0.19
0.32, 0.31 0.27, 0.26
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0.24 + 6z × 10 0.18 + 6z × 10-3
, design (-)
-p
c(z), 0 ≤ z ≤ 16
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Table 3. Number of stress serrations and magnitude of stress peaks of non-graded low relative density (less than 0.21) structures. Stress peaks less than 30% of the Max peak do not count. , measured by Number of stress Average
na 3 8 6 9
22.5 12.9 8.7 8.1
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0.20 0.18 0.16 0.14
serrations (-)
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dry weight (-)
magnitude of stress peaks (MPa)
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