A multi-scale process for mechanical characterization of ceramic materials produced by Direct Ink Writing

Mechanical characterization of ceramics is challenging, as the statistical nature of their strength requires numerous specimens to extract reliable distribution parameters. Here, we first propose a high-throughput testing procedure that allows extraction of statistical information on the strength of ceramic materials. We process large numbers of bending specimens from low volumes of material via Direct Ink Writing (DIW), and rapidly characterize them to extract Weibull parameters for bending strength. After investigating five ceramics and down-selecting two formulations, we develop a multi-scale procedure to explore the impact of printing-induced defects on the strength distribution of DIW-processed ceramics. Finally, we demonstrate that a judicious choice of the printing strategy produces porous architected structures which can significantly exceed the strength of fully dense DIW-produced monolithic materials. While the results are presented on DIW-processed alumina-based ceramics, the approach is versatile and can provide rapid statistical data on the strength of many ceramic materials.


Introduction
Ceramics have exceptional thermo-mechanical properties, such as high stiffness, high theoretical strength, high melting point and low density.However, their sensitivity to defects can lead to crack initiation and catastrophic failure during mechanical loading (particularly under tension), severely limiting their applications in mission-critical structural components.Improving the fracture toughness of ceramics entails increasing the energy required to drive a crack in the material.Over the past five decades, three strategies have primarily been pursued towards this goal [1]: (1) increasing the crack tortuosity during crack growth, thus exploiting non-mode I fracture toughness [2][3][4], (2) introducing energy dissipation mechanisms around the crack tip, e.g.frictional losses during pull-out in fiber-reinforced ceramic matrix composites (CMCs) [5][6][7][8][9], or (3) introducing compressive stresses near the crack tip by virtue of stress-induced phase transformations that are accompanied by substantial specific volume increase, e.g., tetragonal-to-monoclinic phase transformations in stabilized zirconia, which gave this exceptional material the moniker 'ceramic steel' [10][11][12].
More recently, the development of ceramic architected materials (e. g., porous materials with generally periodic and topologically optimized unit cells) has provided new strategies to increase the tensile strength of ceramics.The first one consists of fabricating metamaterials where the constituent ceramic is locally never thicker than a few hundred nanometers, thus limiting the size of processing induced flaws, ensuring that the material locally achieves theoretical strength.While exceptionally promising, this technique requires scalable manufacturing of metamaterials with an enormously large number of unit cells, a currently impossible task.The second approach involves careful design of the material topology to impede fast growth of a continuous crack through the entire component [13][14][15][16].In our previous study [17], we demonstrated that 50 % dense alumina metamaterials with an architected woodpile topology possessed similar compressive strength as fully dense bulk alumina samples printed with the same technology [17].This study strongly suggested that manufacturing of ceramic systems with complex internal architectures presents an intriguing and promising pathway to produce a new class of strong and light materials for thermo-structural applications.
The complex topologies of architected materials, combined with the reduction in processing-induced internal flaws that arises from scale reduction at the sub-unit cell level, necessitate the use of highly precise additive manufacturing (AM) methods.Among the multitude of AM techniques, Direct Ink Writing (DIW) offers unique combinations of properties, namely resolution at the hundreds of microns scale, ability to print components at virtually any overall size, and an extremely versatile materials palette, allowing printing of virtually any ceramic material with minimal internal porosity [18].DIW involves the layer-by-layer deposition of a slurry, whereby control of the rheology is crucial to achieve good printability and thus viable parts.
The development of DIW strategies for printing of ceramic materials and structures with optimal mechanical properties involves formulation of an ink with the desired Hershey-Buckley rheological behavior, optimization of the printing parameters (pressure, speed and printing strategy), and identification of the optimal sintering cycle [17,19,20].In the conventional process [20][21][22][23][24][25], performance is ascertained by microstructural analysis and mechanical testing (generally via compression) on centimeter-scale samples, often in bulk form or woodpile architecture.In typical DIW systems, this involves synthesis, rheological characterization of multiple batches of ink, printing parameter optimization to minimize porosity, and several sintering cycles, a long and tedious process that is not well suitable to test the large number of samples required for proper statistical analysis of the strength of novel ceramic materials.Furthermore, tensile or compressive fracture of a ceramic sample containing a multi-scale distribution of defects (pores, cracks, potential inter-layer delamination …) is a complex process, not easily amenable to the extraction of meaningful and well-defined mechanical properties.By contrast, in a three-point bend test, the highest stress region is highly localized, minimizing the effect of printing defects (in particular, pores) located outside of that region.When single lines are tested, the effects of other printing defects (e.g., delaminations, inter-line porosity) are also eliminated, thus enabling a more accurate measurement of the intrinsic material strength.
Here we propose a novel high-throughput testing procedure that allows extraction of statistical information on the strength of DIW ceramic components.The method only requires processing and testing of single-line specimens, thus using a very small amount of material.We employ this method to rapidly characterize five different ceramics, and downselect two for further study.Subsequently, we develop a multiscale procedure to explore and quantify the impact of multi-scale defects introduced by the printing process on the strength distribution of DIW-processed ceramic components, and use the results of this investigation to produce architected structures with ~20% porosity which can exceed the strength of fully dense DIW-produced monolithic 3D printed materials by as much as 45% (albeit with large scatter in the data).While all results are presented on alumina-based ceramics in the context of DIW, the high-throughput single-line-based approach proposed herein is general and can provide rapid statistical data on the strength of a wide range of ceramic materials, regardless of the processing technology ultimately used to produce the final components.

Ink preparation
All materials were produced by DIW of an appropriately designed viscoelastic ink, followed by sintering.The ink is composed of four materials: (1) Ammonium polyacrylate (NHPA) (Darvan 821 A, R.T. Vanderbilt Company, USA), used as a dispersant, (2) 1-ethenyl-2-pyrrolidinone (PVP) homopolymer (Sigma-Aldrich, USA), used as a rheology modifier, (3) DI water, used as solvent, and (4) ceramic powder.The only chemical interactions occur between the particles and the dispersant; NHPA has been shown to work well with oxides [26][27][28], although alternative dispersants may perform better with non-oxide ceramics [29,30].The rheological modifier is a hydrophilic and chemically inert polymer (PVP has low absorption onto the surface of ceramic particles, and particularly oxides), which stabilizes the suspension and tunes the rheology of the ink by controlling depletion interactions [31][32][33].The result is an ink with the desirable Herschel-Bulkley rheological response (i.e., shear thinning behavior with a defined yield strength), which is optimal for printing.The fact that the rheology is controlled by an inert polymer (and not by ingredients which chemically interact with the ceramic particles) ensures general applicability to a wide range of ceramics.As formulated, the ink is expected to display very similar rheology with a wide range of oxide powders (with tweaks of the volume fractions of constituents possibly required to compensate for differences in particle size distribution); extension to non-oxide ceramic powders is also possible, with the caveat that NHPA may need to be replaced with a different dispersant.
The base alumina ink was carefully optimized for printability in our earlier study [17] and was used here with no modification.The ink preparation process consisted of three steps: (1) the alumina powder was mixed with a solution of DI water and dispersant in a centrifugal planetary mixer; (2) the resulting ink was subsequently placed in a vacuum chamber to remove bubbles while the viscosity remained low; (3) the rheological modifier was added and the ink was mixed again in the planetary mixer.The weight of the ink was measured before and after each step, and DI water was added as needed to compensate for evaporation.For the other inks, the secondary ceramics were added before the primary alumina powder to ensure a good homogenization of the minority phase while the viscosity of the ink is low.The new chemical compositions were created by removing the same volume percent of alumina as the added volume percent of secondary ceramic.As the difference in size, shape and physical properties (e.g., hygroscopicity) among the different powders may alter the rheology of the ink, the volume ratio of ceramic powder to DI water was slightly adjusted for each ceramic powder composition, until we observed a similar consistency and printing parameters as for the previously optimized base ink.

Direct Ink Writing procedure
All samples were printed using a custom Direct Ink Writing system, composed of a 3-axis motion stage (Aerotech, USA) and a pneumatic dispenser (Ultimus V, Nordson EFD, USA).More details can be found in the previous study [17].The substrate was coated with Teflon spray to help the release of the samples after printing.All parts were printed using a 580 μm nozzle diameter.A new nozzle was used for each print to ensure consistent quality of the printed parts and to avoid premature clogging of the nozzle.Additionally, the distance between the nozzle and the substrate/previous layer was set to be 85 % of the nozzle diameter to ensure a good adhesion between the printed layers and to allow the lines to relax after deposition.For the study of the single lines (section 3.2), the samples were printed in batches of 25 lines with lengths of 25 mm.A rim was printed around each batch to prevent the warping of the lines during drying.After printing, silicone oil with low viscosity was applied to the samples to slow down the drying process and allow the sample to dry homogenously, further reducing the warping.Finally, at the end of the ink transition from gel to dry solid, but before the samples were completely dry, a small weight was applied to the samples to minimize warping.For the multi-line study (section 3.3), the printing process was similar to the single-line samples, except that the batches were reduced to 5 to 10 samples.In section 3.4, the multi-track samples were scaled up to bulk samples and architected sandwich structures, with as-printed dimensions of approximately 42 mm × 4 mm × 3 mm.The printing direction for the bulk sample was the same as the multi-track samples to ensure consistency.The sandwich structures consisted of flat single-layer face sheets and an architected core with woodpile topology.The face sheets were printed similarly to the multi-track samples.For the bulk samples, the lines were printed with an overlap of 20% to ensure good inter-line contact.For the first (bottom) layer of the sandwich samples, the inter-line overlap was set to 0%, to provide the ink with sufficient time to stabilize, and the layer height was set smaller than the nozzle diameter, resulting in wider lines.However, for the final (top) layer, it was observed that an overlap of 20% would result in a similar layer quality as the bottom layer.This change was critical to ensure symmetry and avoid sample warping upon drying.Even though the pitch and diameter of the lines were designed to be identical for all sandwich structures, small fluctuations in the rheology of each batch modified these geometrical parameters from sample to sample, resulting in a range of relative densities in the printed sandwich structures.Though unexpected, this result allowed investigation of the relationship between strength and relative density in architected structures.

Sintering
All samples were sintered using the same conditions, inside a box furnace (Thermcraft, USA) in air environment.The sintering cycle consisted of an initial ramp to 700 • C at 5 • C/min, followed by a dwell for 1 h to burn off the polymer.Subsequently, a second ramp was performed up to 1100 • C at 5 • C/min, followed by a dwell for 1 h.Finally, the temperature was increased in 100 • C increments at a rate of 1 • C/ min followed by a 1 h dwell, until reaching 1500 • C for a last 1 h dwell.This multi-step sintering procedure was chosen to avoid warping of the samples, which was observed to occur at temperatures higher than 1100 • C under faster heating.After the final dwell, the furnace was cooled down to room temperature at a rate of 5 • C/min.Some weight was applied onto the samples during the sintering cycle to further prevent warping upon sintering.

Material characterization
The fracture surfaces of the samples were imaged using an Olympus DSX10-UZH digital microscope (Olympus, Japan).The grain size of the alumina and ZTA samples were investigated from fracture surfaces using a FEI Magellan 400 XHR scanning electron microscope (SEM), operated at 3 kV.Computed tomography (CT) scans were performed using VJ Technologies micro-CT scan with a voltage of 150 kV, a power of 10 W and a resolution of about 10 μm.The focus of material characterization was placed on the alumina and ZTA10, as these materials were downselected for the following part of the study (see Sec. 3.3).

Mechanical measurements
Three-point bending tests were performed on the single tracks and single-layer samples using a Q800 dynamic mechanical analyzer (DMA) (TA instruments, USA), with an 18 N load cell.The DMA was used in displacement control with a rate of 60 μm/min.The remaining samples were tested using an Instron ElectroPuls load frame (Instron, USA) with the same displacement rate.The flexural strength was calculated from beam theory as σ f = My I , with y the distance from the center of mass to the tensile face of the sample, M = FL 4 the moment at the center of the beam, L the inter-span distance and I the moment of inertia.As the shape of the cross section of the samples is irregular, I and y were extracted from binarized images of the fracture surface, obtained from optical images via ImageJ processing, through a MATLAB code implementing the method proposed by Borislav et al. [34] For the studies of the single tracks, multi-tracks and bulk/structure samples, 20, 10 and 6 samples were tested, respectively.The distribution of flexural strength from nominally identical samples was quantified with Weibull statistics [35].According to Weibull theory, the probability of failure, P, of a brittle material will scale with the applied stress as with σ 0 the strength parameter (expressing the stress value at which  S3.A large value of m denotes a narrow distribution of strength.As commonly done in the ceramic literature, here we take the strength parameter, σ 0 as a measure of tensile strength.As this study focused on statistical characterization of multiple materials through multiple length scales, a compromise had to be made between guaranteeing statistical significance and maintaining a reasonable scope of work: as a result, while the data collected on single-line specimens (20 samples per material) is expected to provide statistically accurate extractions of the Weibull parameters, the results on multi-line multi-layer specimens (10 and 6 samples per material, respectively) should be interpreted as general trends only.Nonetheless, the approach is general and further studies with larger sample populations can certainly be conducted.
For the analysis of the fracture surface, the side of the sandwich samples was imaged.Using those images, the fracture angle was measured by taking the angle between the bottom and the fracture surface of the samples as shown in Figs.11 and 12. Furthermore, the same images were used to measure the length of the fracture by tracking of the edge of fracture surface.Then, the angle and the length were correlated to the load at breaking.

A multi-scale procedure for statistical analysis of ceramic strength
The testing procedure, schematically depicted in Fig. 1, consists of the following steps: (1) a ceramic particle-loaded polymeric ink is formulated, synthesized, rheologically characterized, and loaded in a syringe; (2) three families of multiple beam-shaped specimens are DIW printed: single lines, multi-line and multi-line multi-layer samples; (3) this large collection of small samples is sintered; (4) 3-point bending tests are performed on all specimens to extract the fracture load; (5) fracture surfaces of all specimens are optically imaged and binarized, to extract detailed information on the cross-sectional shape and enable calculation of the flexural strength (modulus of rupture); (6) Weibull parameters are extracted to quantify the statistics of strength.
The application of this procedure to single-line specimens enables high-throughput and accurate screening of ceramic materials.A large number of specimens can be fabricated from the same batch of ink, sintered and rapidly characterized mechanically.The lower-throughput extension of this approach to multi-line multi-layer samples allows a quantitative assessment of the role of printing defects at multiple scales, and can be used to inform the development of printing strategies for large scale complex components.
Here we demonstrate this multi-scale procedure by (1) characterizing the single-line strength distribution for five different alumina-based ceramic materials, downselecting the two most interesting formulations; (2) investigating the strength distribution of multiline multi-layer specimens from these two formulations, identifying the effect of printing-induced defects; and (3) designing a sandwich-type structure that leverages the printing strategies to achieve bending strengths larger than bulk specimens at a fraction of the weight.For this study, about 1500 samples have been printed, over 300 of which have been characterized in detail.

High-throughput statistical analysis of material strength at the singleline scale
Multiple material systems were synthesized and characterized to verify the validity of our proposed high-throughput approach, namely: (1) pure alumina (Al 2 O 3 ); (2-3) zirconia-toughened alumina (ZTA10 and ZTA20), with 10 and 20 vol% zirconia (ZrO 2 ), respectively, and (4-5) titania (TiO 2 ) and magnesia (MgO)-doped alumina (TDA and MDA), with 0.4 wt% and 0.25 wt% of dopant, respectively.These material systems have been heavily investigated and their mechanical behavior is well established.Rheologically optimized slurries were prepared for each composition as described in Sec.2.2-2.3.More than 20 single-line samples were printed for each composition.All samples were printed and sintered under similar conditions, regardless of composition, as described in Sec.2.3-2.4.
Fig. 2 shows (a) the Weibull distribution of flexural strength for each material system, and (b) the corresponding Weibull coefficient, m and strength parameter, σ 0 .The line fits for the Weibull analysis of the single-line specimens are provided in Fig. S3.The R-square values confirm that a simple Weibull analysis performed on 20 samples captures the statistical behavior of strength reasonably well for single-line specimens.As commonly done, we interpret the strength parameter as a proxy for the averageaverage flexural strength of the material.The alumina lines have a strength of 432 MPa, in good agreement with the literature and approaching some of the highest values reported for alumina [20,23,[36][37][38][39][40][41].The value m = 5.9 is toward the lower values of m found in the literature [42][43][44].The addition of zirconia significantly strengthens the material, as expected from the literature [45][46][47][48][49][50], with strength values for ZTA10 and ZTA20 averaging 779 MPa and 677 MPa, respectively.Although ZTA20 has a higher strength than pure alumina, it displays a lower strength parameter than ZTA10.A decline in ZTA strength with increasing ZrO 2 content above ~10 wt% has been observed previously in the literature and is often attributed to grain size effects.If the zirconia grain size reaches a critical value, premature phase transformation from tetragonal to monoclinic zirconia may occur, reducing the potential for transformation toughening and hence depressing sample strength [51][52][53].MgO is a well-known sintering aid for alumina [54,55].The MDA samples reach a strength of 501 MPa, a 16 % increase from pure alumina, in good agreement with the literature [56], while no significant change in the Weibull modulus was observed.Finally, the addition of TiO 2 as a dopant in alumina reduces the strength of the material system compared to undoped alumina, with our TDA sample strength averaging 357 MPa.A reduction in the strength of Al 2 O 3 with the addition of TiO 2 has been observed previously and is commonly attributed to abnormal grain growth and the formation of trapped porosity during sintering [57,58].
Single printed lines of ceramic material will inherently exhibit higher strength than a large bulk sample, thanks to the confinement of defects and critical flaws.As ZTA10 exhibits the highest flexural strength, we downselect this material for further studies, with pure Al 2 O 3 used as reference.Fig. 3a shows a fracture surface of a representative alumina sample, clearly showing intergranular fracture, and revealing a dense microstructure with an average grain size of 2.2 ± 1.25 μm.The microstructure of the ZTA10 fracture surface is shown in Fig. 3b: notice that a bimodal grain structure can be observed, with large alumina grains (1.1 μm) surrounded by smaller zirconia grains [59,60].
CT scans were performed on a single line for the alumina and ZTA10 in order to characterize the sample porosity and explore its connection to sample strength.Fig. 4 shows micrographs and CT scans of a representative pure alumina sample (the other two materials showed similar features).The observed pores originate from the bubbles present inside the ink.Quantitative results are reported in Table 1.The average pore size is similar across the two different material systems, indicating that the porosity differences are not responsible for the differences in strength.Additionally, the absence of clear stress concentration regions due to their rounded shapes indicates that those pores are unlikely to control the failure of the lines [61][62][63].This claim was verified by performing fractographic analysis on the surface of the single-line specimens for every ceramic system under consideration.The number and size of pores on the fracture surface was extracted by image analysis.Fig. S5 depicts the strength of single-line samples as a function of the diameter of the largest pore identified on the fracture surface.Fracture surfaces which were free of pores are denoted with a diameter of 0 μm.
To capture the impact of pore location, we denote the pores in the tensile

Table 1
Average diameter of the pores within the lines for pure alumina and ZTA10 (alumina/10 vol% zirconia).The measurements were performed using CT scanning.The density of the sample was measured using the Archimedes method.region of the cross-section (i.e., the most dangerous pores from a fracture perspective) with red markers.No trend was detected (Fig. S5), confirming that DIW-induced pores do not play a major effect on the fracture strength for all materials systems under consideration.As discussed in Sec.2.2, the ink formulation we present here is compatible with a wide range of ceramic materials, as long as a proper  dispersant is identified and subject to the choice of an appropriate sintering cycle.Sintering small-scale single-line samples is generally very easy and results in high-density ceramics.The important implication is that, although this study focuses on the characterization of DIW-processed alumina-based ceramics, the high-throughput single-line process described here is an extremely fast, versatile, and economical approach for extraction of Weibull parameters for multiple ceramic systems.While this DIW-based approach clearly cannot capture the impact of processing-specific defects on the strength of ceramic components fabricated with non-DIW technologies, it can still provide useful preliminary data to enable rapid screening and downselection of ceramic compositions.

From single line to DIW component: statistical analysis of strength of multi-line multi-layer samples
The single lines investigated in the previous section can be considered as a building bock, each of which containing a limited number of defects.When a part is printed, these building blocks are replicated horizontally and vertically to create multi-line multi-layer structures.As a result, the final component will contain two families of defects: those contained within individual building blocks and those formed between building blocks.
In this section, we print progressively larger samples (1-10 lines and 1-2 layers) and investigate the effect of sample size on flexural strength and Weibull modulus.Flexural strength is extracted from 3-point bending experiments, as in sec.3.2.Fig. 5 illustrates (a) the 3D model of the ideal geometry of the samples and (b) micrographs of some fracture surfaces of a few representative samples with different numbers of lines and layers.Pores are present within the prints, which is consistent with our observations on single line specimens.The fracture surfaces of the single line specimens (Fig. 4a) and the scaled-up samples appear similar.Not surprisingly, the discrepancy between the 3D model geometry and the shape of the as-sintered samples is significant.The top and bottom surfaces of the samples are significantly flattened, which we attribute both to the viscoelastic relaxation of the ink after deposition and the tendency to reduce surface energy upon sintering.This shape change is useful, as it significantly reduces stress concentration.Nonetheless, some areas of stress concentration still remain, as shown in Fig. 5b (dashed red circle).Clearly the effect of stress concentrations on strength measurements collected by 3-point bending depends not only on the size and shape of the defects, but also on their distance from the neutral axis of the specimen.
As emphasized in Sec.2.6, the number of multi-line multi-layer specimens (10 and 6 samples per material, respectively) is probably insufficient to extract statistically rigorous values of the Weibull parameters.Furthermore, the inherently multi-scale population of defects in these samples is likely to defy application of a statistical description of strength as simplistic as the Weibull theory.Nonetheless, the extraction of Weibull parameters for these larger-scale specimens will allow us to draw useful conclusions on the impact of scale on the strength of DIW structures.The Weibull diagrams and the evolution of average strength and Weibull modulus with the number of lines and layers is depicted in Figs. 6 and 7, for the pure alumina and the ZTA10 materials, respectively.Bulk samples consisting of 10 lines and 6 layers were also printed for comparison.For pure alumina (Fig. 6), the average strength decreases with increasing number of lines and number of layers, until a plateau in strength is reached at 5 lines (Fig. 6c).While single-line specimens are clearly the strongest, all multiple line samples have relatively similar strength (10-line single-layer specimens have a strength of 375 MPa, 13% lower than the single line value), indicating the quick emergence of a plateau.The effect of number of layers is more pronounced, with bulk specimens (6-layers) exhibiting a strength of only 200 MPa, over 54% lower than single-layer specimens with the same number of lines.With the exception of a few outliers, the Weibull modulus generally increases with the number of lines for single-layer specimens (from m=6 to m=9) and decreases with the number of lines for 2-layer specimens (from m=6 to m=4) (Fig. 6d).These results can be rationalized as follows.In single-layer specimens, printing multiple lines introduces interline defects in the tensile region (see Fig. 5b), which act as stress intensification points and reduce the strength of the sample.At the same time, spreading the load over a wider beam reduces the deleterious impact of the occasional critical pore within singe lines (see Fig. 5b), reducing scatter and increasing the Weibull modulus.For 2layer samples, the probability of introducing deleterious defects between layers (in particular at the diamond-shaped intersections at two lines and two layers) increases with the number of lines, causing a reduction in both the average strength and the Weibull modulus.The further significant decrease in strength from a 2-layer sample to a 6-layer (bulk) sample can be explained by the nature of the bending test: as the number of layers increases, the interlayer surface defects (see red circled regions on the side in Fig. 5b) move progressively closer to the region of maximum tensile stress in the sample (the bottom surface), playing a more dominant role on controlling sample strength.
For ZTA10 (Fig. 7), the strength drops significantly when a second line is added, but the results stabilize thereafter, to 482-529 MPa for the 1-layer samples and 385-454 MPa for the 2-layer samples, respectively (Fig. 7c).The 10-line 2-layer sample is still 38 % stronger than the bulk sample.We note that the overall trend between strength and line number in the ZTA10 samples is similar to what is observed in our alumina samples.However, there is a significant drop in the strength between 1 line and 2-line ZTA10.We attribute this to the presence of the transformation toughening ZrO 2 phase in the ZTA10 material, as well as the lack of critical defects in the single line structure.However, the ZTA structures with two or more lines possess interline defects as well as regions of potential stress concentration at the surface, leading to mechanical failure dominated by a critical defect.No clear trend is observed with the Weibull modulus, which oscillates between m=4 and m=10 for all sample geometries (Fig. 7d).
It is worth investigating whether the strength/size relationships described above could simply be explained by Weibull-type size effects.According to the Weibull theory, if the strength of a single-line specimen of volume V E1 is σ 1 , the strength σ 2 of a specimen of volume V E2 is given by Ref. [64]:  Clearly, the large deviation between the model and the data (particularly for ZTA10) confirms that a simple Weibull scaling approach is inadequate to explain the trends observed in this study.

Design of an architected structure with exceptional bending strength
The results in sec.3.2-3 clearly illustrated that (1) individual alumina and ZTA10 lines display excellent strength with fairly high Weibull modulus, in spite of the residual porosity, and that (2) a decrease in bending strength is associated with DIW sample scale-up, particularly related to increasing the sample thickness (number of layers).Here we capitalize on these results to design ceramic structures that sandwich an architected single-line woodpile core between singlelayer-thick face sheets.These structures are tested in 3-point bending and compared to monolithic bulk samples (10 lines wide and 6 layers thick).Fig. 8 displays the flexural strength of all samples as a function of their relative density.The relative density is defined simply as volume of ceramic material divided by the volume of the structure (clearly, for bulk samples, the relative density is equal to 1).While all sandwich samples were designed to have the same line diameter and pitch (and hence the same relative density), small variations in rheological properties from batch to batch resulted in appreciable differences in the relative density of the printed specimens, spanning the range 0.55-0.8).A number of interesting results emerge.
1.Both the monolithic bulk samples and the sandwich structures show significant scatter in their strength, varying between 120 and 320 MPa for bulk structures and between 120 and 460 MPa for the sandwich structures.sheet response controls the mechanical behavior of the sandwich structure.5. Two ZTA10 sandwich structures show ~45% higher bending strength than their fully dense counterparts, despite being 20% lighter.While the scatter on the data is too large to draw precise quantitative conclusions, this unexpected result demonstrates that judicious introduction of a significant amount of porosity in a ceramic sample may improve mechanical properties, in contrast with generally established design principles.
To correlate these results with differences in microstructure, in Fig. 9 we report optical images of representative fracture surfaces for sandwich and bulk structures, for both materials; Fig. 10 shows 3D computed tomography (CT) reconstructions.Notice that while all structures clearly show the presence of near-circular pores (introduced during ink preparation and/or extrusion of individual lines), the bulk structures clearly also show smaller and sharper pores between the lines and large cracks and delamination fronts that span multiple printing lines, likely introduced during the sintering process.This clearly correlates with the superior mechanical performance of the sandwich structures and demonstrates that architecting a ceramic material so that it is locally only one line-thick improves sintering and densification, avoids the occurrence of sharp deleterious cracks and results in significantly higher strength.
The failure behavior of the sandwich structures was further investigated.Fig. 11a shows load-displacement curves for the alumina samples extracted from the 3-point bending experiments, juxtaposed with optical images of the post-mortem samples (color coded) in Fig. 11b.The differences in sample stiffness simply correlate with differences in relative density (Fig. S1).Notice that the strongest samples display a straight crack propagation path, while the weaker samples fractured at a lower angle.These observations are quantified in Fig. 11c and d, where bending strength is plotted against fracture surface angle and crack path length.We interpret these results in terms of the location of critical flaws.If no major flaw is present, the sample will fail starting from the largest tensile stress point, immediately below the center loading anvil, and proceed vertically toward the loading anvil.Conversely, if a major flaw is present on the tensile surface near (but not immediately below) the anvil, fracture can initiate prematurely at that location; once the crack leaves the face sheet, it needs to repeatedly stop and restart as it travels through the porous woodpile core; in doing so, it will naturally move towards the center anvil, following the regions of maximum stress.As a result, stronger samples fail with a straight vertical crack, while weaker samples will fail along a diagonal crack path, consistently with the results in Fig. 11.
A similar analysis was performed on the ZTA sandwich structures (Fig. 12).The stiffness of all samples is fairly similar, consistent with smaller deviations in relative density compared to the alumina samples (Fig. S2).Interestingly, the fracture surfaces of all samples are also very similar, with most cracks following a nearly vertical line.However, the measured load at breaking values generally increases with increasing fracture length.We attribute this to a crack growth resistance effect in our ZTA samples.ZTA is well known to exhibit crack growth resistance due to the presence of the transformation toughening ZrO 2 phase [71].The observed crack length will be determined by the distance between the center loading anvil and the critical defect.The further the critical defect is from the center loading anvil, the longer the crack length will be.As the crack path gets longer, the transformation toughening, and crack growth resistance will lead to an increase in the observed strength of the structures [72,73].

Conclusions
A novel multi-scale procedure for statistical analysis of ceramic strength was introduced, consisting of preparation of a ceramic-loaded polymeric ink with printable rheology, DIW printing of multiple single-line, multi-line and multi-line/multi-layer samples followed by sintering, and rapid mechanical testing of all samples by 3-point bending followed by post-mortem geometric characterization of the cross-section to extract average strength and Weibull modulus.
When applied to single-line specimens, this approach is accurate and high-throughput, and enables rapid downselection of ceramic materials.We demonstrated this procedure on 5 well-known ceramic formulations and identified pure alumina and alumina/10 vol% zirconia (ZTA10) as the most interesting materials.ZTA is consistently stronger (779 MPa) than alumina (432 Mpa) in single line form, while having a lower Weibull modulus, in good agreement with some of the best literature data.Application of this procedure to multi-line multi-layer specimens is lower-throughput and requires a much larger amount of material and effort to extract statistically significant strength values.But even with a relatively small number of specimens, important conclusions on the effect of multi-scale defects on the mechanical properties of DIW-printed structures emerge: in particular, while multi-line and multi-layer samples were consistently weaker than single lines, increasing the number of layers resulted in a more dramatic decline in properties, whereas the effect of increasing the number of lined reached a plateau very quickly (2-3 lines).
These results were exploited to design a ceramic structure with single-layer face sheets sandwiching a woodpile core, capitalizing on the strength of individual single-line building blocks.Remarkably, two ZTA10 sandwich structures show 45% higher bending strength than their fully dense counterparts, despite being 20% lighter.While the scatter on the data is too large to draw precise quantitative conclusions, this simple result shows that judicious introduction of porosity in ceramic materialscombined with the exploitation of size effects, may be a viable approach for the design of high-strength low-density materials.
While this work has focused on DIW-processed alumina-based structures, the ink formulation proposed herein is remarkably versatile, and compatible with a wide range of ceramic powders (with the caveat that non-oxides will likely require a different dispersant).Additionally, DIW produces small samples that easily sinter to near-full densification; in this sense, the data collected with this high-throughput single-line approach can provide an upper bound on strength and be used for initial rapid screening of dozens of different formulations with minimal time and cost, regardless of the processing technology ultimately used to produce the final components.

Fig. 1 .
Fig. 1.Schematic of the multi-scale testing process for the investigation of the mechanical properties of ceramics.

Fig. 2 .
Fig. 2. Weibull analysis on single lines for alumina and alumina-based material systems.(a) Probability of failure for the single line specimens.The second moment of inertia and the distance from the center of mass to the bottom face were extracted from actual cross section images.Star symbols identify Al 2 O 3 and ZTA10, the two materials downselected for further study.(b) Weibull coefficient (m) and strength parameter (σ 0 ) for the five material systems, extracted from (a).

Fig. 4 .
Fig. 4. (a) Optical micrograph of the fracture surface of a single line alumina sample.(b) Cross-sectional slice from the CT scan of the same line sample.(c-d) Optical micrograph along the entire line and CT scan section of the same line.Scale bar: 0.5 mm.

Fig. 5 .
Fig. 5. (a) 3D model of a 3-line 2-layer sample as it was programmed for printing.(b) Fracture surfaces of the samples after testing.The voids between the lines are filled due to the relaxation of the ink after deposition.The dashed circles show areas of potential stress concentration.Scale bar: 500 μm.

Fig. 6 .
Fig. 6.Effect of the increase of lines and layers on the Weibull analysis of alumina samples.The relationship between the probability of failure and the flexural strength with the increase of the number of lines for (a) 1-layer samples and (b) 2-layer samples.The Weibull coefficients, (c) σ 0 and (d) m, versus number of printed lines are shown for 1-and 2-layer samples.The red star represents the average strength of the bulk sample measured in Sec 3.4.(For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

Fig. 7 .Fig. 8 .
Fig. 7. Effect of the increase of lines and layers on the Weibull analysis of ZTA10 samples.The relationship between the probability of failure and the flexural strength with increase of the number of lines for (a) 1-layer samples and (b) 2-layer samples.The Weibull coefficients, (c) σ 0 and (d) m, versus number of printed lines are shown for 1-and 2-layer samples.The red star represents the average strength of the bulk sample measured in Sec 3.4.(For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article.)

Fig. 9 .
Fig. 9. Fracture surface of (a) alumina and (b) ZTA bulk and sandwich structures.Several types of defects can be identified in the bulk samples, including cracks are pores between the lines, which are not present in the sandwich samples.Only pores from the ink preparation are present in all samples.Scale bar: 1 mm.
with m being the Wiebull modulus of the single-line specimens.Given that all single-line specimens have the same gauge length, V E2 / V E1 = A E2 /A E1 , with A E2 and A E1 the cross-sectional areas of two specimens.The effective area of a n-line sample is equal to n times the effective area of the single-line sample.This scaling effect was investigated for the alumina and ZTA10 1-layer samples.The results are shown in Fig.S4.

Fig. 10 .
Fig. 10.CT scans of alumina and ZTA10 bulk and sandwich structures, obtained from post-mortem samples.Scale bar: 1 mm.

Fig. 11 .
Fig. 11.Analysis of failure for the alumina sandwich structures.(a) Load versus displacement from 3-point bending tests.(b) Side views of the fracture surface of all tested samples.Load at breaking versus (c) angle and (d) length of the fracture surface in the plane shown in (b).Scale bar: 2 mm.

Fig. 12 .
Fig. 12. Analysis of failure for the ZTA10 sandwich structures.(a) Load versus displacement from 3-point bending tests.(b) Side views of the fracture surface of all tested samples.Load at breaking versus (c) angle and (d) length of the fracture surface in the plane shown in (b).Scale bar: 2 mm.