High-strength cellular ceramic composites with3D microarchitectureJens Bauera,1, Stefan Hengsbachb, Iwiza Tesaria, Ruth Schwaigera, and Oliver Krafta

aInstitute for Applied Materials and bKarlsruhe Nano Micro Facility, Karlsruhe Institute of Technology, D-76131 Karlsruhe, Germany

Edited* by William D. Nix, Stanford University, Stanford, CA, and approved January 9, 2014 (received for review August 12, 2013)

To enhance the strength-to-weight ratio of a material, one may tryto either improve the strength or lower the density, or both. Thelightest solid materials have a density in the range of 1,000 kg/m3;only cellular materials, such as technical foams, can reach consid-erably lower values. However, compared with corresponding bulkmaterials, their specific strength generally is significantly lower.Cellular topologies may be divided into bending- and stretching-dominated ones. Technical foams are structured randomly andbehave in a bending-dominated way, which is less weight effi-cient, with respect to strength, than stretching-dominated behav-ior, such as in regular braced frameworks. Cancellous bone andother natural cellular solids have an optimized architecture. Theirbasic material is structured hierarchically and consists of nanometer-size elements, providing a benefit from size effects in the materialstrength. Designing cellular materials with a specific microarch-itecture would allow one to exploit the structural advantages ofstretching-dominated constructions as well as size-dependentstrengthening effects. In this paper, we demonstrate that suchmate-rials may be fabricated. Applying 3D laser lithography, we producedand characterized micro-truss and -shell structures made from alu-mina–polymer composite. Size-dependent strengthening of aluminashells has been observed, particularly when appliedwith a character-istic thickness below 100 nm. The presented artificial cellular materi-als reach compressive strengths up to 280 MPa with densities wellbelow 1,000 kg/m3.

The suitability of a material for lightweight applications isdetermined mainly by two properties: the specific strength

and the specific stiffness, here defined as the strength and stiff-ness of a material divided by its density. In the past century,major advancements have been made in optimizing classicallightweight materials, such as aluminum alloys or compositematerials, with respect to these properties. However, the lightestsolid materials have a density in the range of 1,000 kg/m3 (1).Natural lightweight materials, such as bone and wood, are notfully dense and may exhibit considerably lower values (1). Theycontain several levels of hierarchical structuring down to thenanometer scale (1–3), leading to remarkable specific mechani-cal properties (1–4). For instance, cancellous bone is built oftruss- or shell-like framework architectures grown adaptively tothe loading situation (5, 6). The material thickness and the ori-entation of the individual structural elements depend on themagnitude and orientation of loading. This leads to an optimizedtopology, in which each structural element is aligned with theprincipal stress trajectories (5, 6).Technical foams are materials with open- or closed-cell po-

rosity of comparable low density and are used in lightweightcomponents, such as foam-core sandwich panels (1, 7). However,their specific strength and stiffness are limited by their charac-teristic stochastic architecture. Typically, considerably lowervalues of specific strength and stiffness compared with the cor-responding bulk materials are reached (1, 7). In addition to thematerial properties, the architecture strongly affects the me-chanical behavior of such cellular solids (1, 8, 9). Buckling, in-homogeneity, and local stress concentrations (10) occur, becausefoams cannot be considered only as materials but also as struc-tures (1, 7).

Cellular topologies may be divided into bending- and stretch-ing-dominated ones (8). Foams generally behave in a bending-dominated manner (1, 8, 9). When abstracted to pin-jointedframeworks, open-cell foams consist of unit cells that are stati-cally indeterminate (8), e.g., cubic cells. Their topology wouldallow the struts to rotate around the joints leading to a collapseunder loading (11). However, the joints of foams are frozenrather than pin-jointed, causing the struts to bend. Gibson andAshby (1) showed that the mechanical properties of such bend-ing-dominated foams depend on the relative density, ρp=ρs, whereρp and ρs are the densities of the foam and the correspondingsolid material, respectively. The compressive strength of the foamscales with ðρp=ρsÞ1:5 or even with higher exponents, dependingon the failure mechanism.Stretching-dominated structures are considered to have much

better mechanical properties (8, 9, 12). The struts of a frame-work, which is rigid when regarded as pin-jointed, are loaded intension or compression largely without bending (8). In twodimensions, a triangle is the only statically determinate polygon.In three dimensions, fully triangular structures, such as tetrahe-dral truss constructions as initially developed by Bell (13), reacha maximum of rigidity and stretching-dominated behavior (8).Designing foam materials in such a manner facilitates a linearscaling behavior of the strength and the stiffness with the relativedensity (8, 9). However, the specific properties of bulk materialstill are not quite reached (9).Bone and other biological materials with a similar funda-

mental structure, such as shells (14) and teeth (15), achieveimproved strength of their basic material as a result of the ap-pearance of mechanical size effects (16). On the lowest level ofhierarchy, bone consists of mineral crystal platelets with thickness

Significance

It has been a long-standing effort to create materials with lowdensity but high strength. Technical foams are very light, butcompared with bulk materials, their strength is quite low be-cause of their random structure. Natural lightweight materials,such as bone, are cellular solids with optimized architecture.They are structured hierarchically and actually consist ofnanometer-size building blocks, providing a benefit from me-chanical size effects. In this paper, we demonstrate thatmaterials with a designed microarchitecture, which providesboth structural advantages and size-dependent strengtheningeffects, may be fabricated. Using 3D laser lithography, weproduced micro-truss and -shell structures from ceramic–poly-mer composites that exceed the strength-to-weight ratio of allengineering materials, with a density below 1,000 kg/m3.

Author contributions: J.B., I.T., and O.K. designed research; J.B., S.H., and R.S. performedresearch; J.B., S.H., I.T., and O.K. analyzed data; and J.B. wrote the paper.

The authors declare no conflict of interest.

*This Direct Submission article had a prearranged editor.

Freely available online through the PNAS open access option.1To whom correspondence should be addressed. E-mail: jens.bauer@kit.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1315147111/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1315147111 PNAS | February 18, 2014 | vol. 111 | no. 7 | 2453–2458

ENGINEE

RING

on the order of a few nanometers integrated in a collagen matrix(17). The strength of both ductile (18) and brittle (19–21) materialstypically increases with decreasing dimensions. In the early 20thcentury, Griffith (22) proposed the relationship

σf ∝1ffiffiffi

cp [1]

between the fracture strength, σf , and the critical size of a flaw, c,for brittle materials such as ceramics. A flaw cannot be largerthan the component in which it is located. Assuming c correlateswith the material thickness, t, of a structural element (16), Eq. 1may be written as

σf ∝1ffiffi

tp : [2]

Thus, the smaller the component, the higher is its fracturestrength. It has been argued that when fabricated thin enough,materials might even exhibit strength values close to thetheoretical strength when a critical thickness in the nanometerrange is reached (16). Assuming that failure no longer is gov-erned by the Griffith criterion but by the strength of atomicbonds in that regime, materials should become insensitive toflaws (16).Designing cellular materials with a specific microarchitecture

would allow one both to exploit the structural advantages ofstretching-dominated constructions and to gain enhanced ma-terial strength due to mechanical size effects, leading to superiorcellular materials (9, 23). Recent improvements in materialprocessing methods have led to cellular materials of extremelylow density with a periodic microarchitecture (24). However, thefreedom of design of these structures is limited by processing-related restrictions (25). The producible topologies are bendingdominated and behave in a highly compliant manner.In this paper, we propose 3D direct laser writing (3D-DLW) (26)

as a method for fabricating real 3D composite microarchitectures

with design control down to the nanometer scale. We describe thedesign, processing, and mechanical characterization of several ce-ramic–polymer composite structures with submicron feature size.Although to date 3D-DLW is strongly limited in achievable samplevolume, it allows for production of almost arbitrary polymeric ge-ometries (26). In conjunction with coating techniques, such asatomic layer deposition (ALD) (27), multimaterial composites(28), as well as metallic (29) or ceramic (30) structures in which thepolymer has been removed, may be fabricated. We present trussconstructions with different structural properties as well as a shape-optimized honeycomb design (Fig. 1). All structures were fabri-cated from polymer (IP-Dip; Nanoscribe GmbH) by 3D-DLW andhomogeneously coated with alumina (Al2O3) layers of differentthicknesses using ALD. The alumina coating carries tensile andcompressive forces, whereas the light polymeric core serves toprevent early face buckling and to improve toughness. For me-chanical characterization, uniaxial in situ and ex situ compressiontests were performed.When coatings were applied with a characteristic thickness

below 100 nm, a substantial increase was observed in the mate-rial strength of alumina. Surpassing all technical foam materials,the trusses reach compressive strength values up to 55 MPa foran estimated density of 410 kg/m3. Shape-optimized honeycombstructures achieve up to 280 MPa at 810 kg/m3, exceeding allnatural and engineering materials with a density below 1,000 kg/m3.The specific compressive strengths obtained are higher thanthose of most engineering metals and close to the ones oftechnical ceramics.As predicted theoretically (9, 23), the designed and minia-

turized architecture benefits from both structural advantagesand size-dependent strengthening effects, facilitating valuesof specific strength beyond the accessible range of standardcellular materials.

ResultsAll fabricated structures were tested in compression, as illus-trated in Fig. 2A. Their bulk strength and stiffness depend on the

Fig. 1. Computer-aided design models (Upper) and SEM images (Lower) of examined cellular microarchitectures (scale bars: 10 μm). Design C is an ortho-tropic construction with nonrigid cubic unit cells (C); therefore it generally is considered to behave in a rather bending-dominated manner. We also in-troduced global diagonal bracings to obtain a more stretching-dominated and collapse-resistant behavior (B). When applied to all faces of every unit cell (A),the structural stability is enhanced further. Stretching domination is maximized because the structure is fully triangular. The design may be regarded asbehaving fairly isotropically. Depending on the stiffness of junctions, collapse mechanisms are less urgent and topologies may be designed so that a maximumof structural elements are arranged in loading direction, without risking early global buckling. We realized that approach with a hexagonal truss structure(D), whose unit cell geometry is not rigid, just like that of design C, and with a shape-optimized honeycomb design (E). Both constructions behave aniso-tropically.

2454 | www.pnas.org/cgi/doi/10.1073/pnas.1315147111 Bauer et al.

thickness of the deposited alumina layer, as selected stress–straincurves show in Fig. 2B. Regardless of the coating thickness,the size of all polymeric core structures remains unvaried.Starting from low values of about 2 MPa for pure polymericstructures, the compressive strength increases up to 34 MPawith the introduction of the alumina coatings. Simultaneously,Young’s modulus, E, increases from about 0.1 to 1.4 GPa. Allcurves exhibit nonlinear behavior for lower strains, which isrelated to experimental issues, such as small misalignment androughness at the top surface of the structure. There is no sys-tematic dependency on the coating thickness (compare Mate-rials and Methods).Fig. 2C shows the specific strength of several designs in re-

lation to the layer thickness of alumina (all data are given inSupporting Information). Architectures A and B (compare TableS1) achieve almost equal specific values. Topology C (compareTable S1) behaves similar to design D but is less effectivethroughout. It clearly is shown that the specific strength of allstructures increases once thin alumina layers are deposited, asthe strength of alumina is much higher than that of IP-Dip.However, with growing coating thickness, the increase becomesless pronounced (design D) or even saturates (designs A and E).Optimized honeycomb structures with a solid-material fraction

of only 15% alumina (50 nm) reach the same specific strength asthose with 40% (200 nm) (compare Table S1).Based on the test data of the honeycomb structures, the stress

at failure, σA, inside the alumina layers (Eq. 5) has been esti-mated analytically. A significant increase may be observed fora characteristic thickness in the range below 100 nm; σA increasesfrom 1,180 to 3,900 MPa, a value noticeably higher than the typicalcompressive strength of bulk alumina (1, 19).Failure mechanisms depend on both the architecture of the

structures and the thicknesses of the coatings (Fig. 3). Wedetected buckling and brittle fracture as the two majormechanisms. Bare polymeric and thinner-coated structurescollapse by buckling. Whereas trusses A and B buckle locally(Fig. 3A), designs C and D allow global buckling modes (Fig.3H; compare Movies S1–S3). With growing layer thickness ofthe coatings, the failure mechanism shifts to brittle fracture.Depending on the architecture, this transition occurs at dif-ferent layer thicknesses. For 50 nm, local buckling still may becritical for designs A and B, but fracture due to high tensilestresses at the notches of the junctions may occur as fre-quently (Fig. 3C) and dominates at 100 nm (Fig. 3D). For 200nm alumina thickness, compressive failure of the verticalstruts sets in (Fig. 3E). Compared with A and B, the honey-comb design resists buckling for thinner coatings and crushes

Fig. 2. All cellular material designs were tested mechanically under uniaxial compressive loading (compare Movies S1–S4). (A) An in situ test of a polymerictruss structure (side view). (B) Stress–strain curves of fully triangular trusses (Fig. 1A). Alumina layers of the indicated thicknesses have been deposited ontoa polymeric core structure. With increasing layer thickness, the compressive strength and stiffness (Young’s modulus E) increase strongly compared with barepolymeric structures. (C) Specific compressive strength as a function of the coating thickness t. Labels refer to the nomenclature of Fig. 1. The estimated stressat failure inside the alumina layer, σA (Eq. 5), increases with decreasing layer thickness. All data points correspond to at least three measurements. Error barsgive the corresponding maximal upper and lower deviation.

Bauer et al. PNAS | February 18, 2014 | vol. 111 | no. 7 | 2455

ENGINEE

RING

in a brittle manner (Fig. 3J). Truss structures C and D exhibitlocal buckling and beginning brittle fracture near the junc-tions at 100-nm layer thickness (Fig. 3I).

Fig. 4 shows a so-called Ashby chart (CES EduPack; GrantaDesign Ltd.) for compressive strength vs. density. Compared withother materials with a density below 1,000 kg/m3, the presented

Fig. 3. Failed samples of different coating thickness (see headings) and architecture (Upper, design A; Lower, designs C–E). With growing layer thickness ofalumina, the failure mechanism changes from buckling to brittle fracture. (A) Polymeric truss designs A and B buckle with large plastic deformations. (B–E)Once coated with Al2O3, plastic deformation is reduced significantly (compare Movies S1 and S2). (B) Face buckling, fine networks of surface cracks, andsquamous flaking occur. (C) In the range of 50-nm thick coatings, brittle fracture of the vertical compressive bars, normal to the loading direction as well asclose to the junctions, was observed. (D) Reaching 100 nm, fracture was detected to appear exclusively at the junctions. (E) For 200 nm, we found thestructures and especially their vertical compressive bars to burst into small pieces. (F) Polymeric honeycomb structures buckle with large plastic deformations.(G) For 10-nm thick coatings, buckling causes fine networks of surface cracks, leading to vertical crack propagation (compare Movie S4). (J) Thicker-coatedhoneycomb designs burst in a brittle manner. (H) Designs C and D buckle globally and fracture without notable plastic deformation for both bare polymericstructures (compare Movie S3) and coated ones. (I) Reaching 100 nm, we observed cracking of the vertical compressive bars normal to the loading directionand fracture close to the junctions.

Fig. 4. Compressive strength–density Ashby chart showing the cellular ceramic composite materials described in this report compared with other materials(compare CES EduPack, Granta Design Ltd.). The truss structures A, B, and D outperform all technical foam materials. The optimized honeycomb designsachieve strength-to-weight ratios comparable to those of technical ceramics and high-strength steels. The nomenclature refers to Fig. 1. Labels indicate thethicknesses of the deposited alumina layers.

2456 | www.pnas.org/cgi/doi/10.1073/pnas.1315147111 Bauer et al.

designs obtain outstanding values. Truss structures A, B, and Dexceed all technical foam materials (31–33). Their specific com-pressive strength is in the range of bone material and advancedmetallic alloys (1, 32). Designs with a solid material fraction ofonly 13% alumina (A and B at 50 nm) already reach the highestspecific strength of pure alumina foam (32, 33). A maximumcompressive strength of 55 MPa at 410 kg/m3 is achieved (D at200 nm). Shape-optimized honeycomb designs surpass all nat-ural and technical cellular materials. Specific ratios up to 280MPa at 810 kg/m3, in the range of technical ceramics and high-strength steels, not far below bulk alumina, are reached (32).For a summary of the numerical values, see Supporting In-formation (Table S1).

DiscussionMacroscopic properties of cellular materials are determined byboth material and structural properties. Reaching the size scalediscussed in this paper, structural characteristics and materialstrength are coupled (23).Based on the strength of the tested honeycomb structures, σA

estimates the size dependency of the material strength of alu-mina. The observed behavior (Fig. 2C) is in good agreement withthe considerations of Gao et al. (16), as well as recently pub-lished effects in hollow ceramic architectures of comparabledimensions (30), supporting the theory of decreasing flaw sizeand size-dependent strengthening below a certain length scale.Data points in Fig. 2C approximately correlate with the de-pendency given by Eq. 2. Because buckling before materialfailure, multiaxial stress states, and local stress concentrationsare neglected, the estimation may be regarded as conservative.Before brittle crushing sets in, all designs are expected to frac-ture initially because of high local tensile stresses (compare Fig.3G: crack formation due to transversal tension). Therefore, weexpect σA to depend on the tensile strength rather than repre-sent the actual compressive strength, especially when coatingsbecome thinner.Brittle cellular solids fail when buckling occurs or when local

stresses attain the tensile or compressive strength of the solid (1).Therefore, buckling always occurs at lower stresses than materialfailure (1, 6). The buckling load of a strut is proportional toEI=l2, where E is Young’s modulus, I is the second moment ofarea, and l is the strut length (6). When the alumina layers be-come thicker, the modulus, E, of the composite and, I, of thestruts increase. The buckling strengths of the structures improve,and at a certain point, failure by fracture of the material becomesdominant.The observed behavior of the specific strength in relation to

the coating thickness (Fig. 2C) may be explained by the interplayof mechanical size effects and the failure mode. Within our tests,the transition from buckling to material fracture (Fig. 3) roughlycorrelates with the sections of beginning saturation of the spe-cific strength. The fraction of alumina in the solid materialincreases from only a few percent at 10 nm up to 40% at 200 nm.Because the specific strength of alumina is much higher than thatof IP-Dip, one would expect the specific strength of the struc-tures to increase simultaneously. However, when coatings be-come thinner while still being thick enough to resist buckling,the observed mechanical size effect compensates for the de-creasing fraction of alumina in the composite. When thethickness is reduced further, buckling occurs before materialfailure and the specific strength decays.Microarchitecture allows the presented cellular materials to

benefit from the observed size effect, whereas self-similar mac-roscopic constructions would be unable to do so. To inducestrong mechanical size effects in ceramics, material thicknessesare required to be in the range of a few nanometers (16) (compareFig. 2C: trend of σA). However, buckling tends to dominate themechanical behavior for decreasing thickness of the ceramic shell

in relation to the diameter of the struts. For a given coatingthickness, further downscaling of the polymeric cores wouldimprove both the buckling strength and the ratio of alumina topolymer, allowing more efficient access to the observed strongincrease of the material strength below 50 nm. However, theminimal producible length scale of architecture is limited bythe lithography process (26).How efficient a design is in profiting from size effects depends

on the actual architecture. Designing foam topologies as tri-angular braced frameworks is an expedient way to maximizestretching-dominated behavior independently of the loadingsituation. However, for thicker coatings, we observed design D toreach higher values of specific strength than the braced trussstructures A and B. A general approach in lightweight design isto approximate a truss topology corresponding to the orthogonalnetwork of the acting principal tensile and compressive stresses(34), as in cancellous bone (5, 6). Although such a structure maynot be triangular at all, it is stretching dominated, because strutsaligned with the principal stress direction experience no bendingmoment (6). A construction is most weight efficient when asmany structural elements as possible are oriented in the loadingdirection while guaranteeing sufficient structural stability, e.g., byavoiding slender bars and thin shells under compressive load (6).Under axial loading, the honeycomb structure attains the highestspecific strength, once its walls are thick and stiff enough to resistbuckling, because the entire solid material is aligned in theloading direction. The braced framework of designs A and Bcauses global rigidity and enforces localized failure (compareMovies S1 and S2), whereas the rigidity of designs C and Ddepends on the stiffness of their junctions (8, 11). Those wereobserved to behave relatively compliantly when coatings werethin (compare Movie S3). Acting as elastic–plastic hinges, theyallow global buckling modes that start without rupture or notabledeformation of single struts. With increasing coating thickness,junctions become stiffer and designs C and D more rigid. Whentheir resistance to global buckling reaches the range of that of localbuckling or material failure, the benefit of diagonal bracings, asapplied in designs A and B, decreases. Thus, constructions such asdesigns C and D, which generally are considered bending domi-nated (8) may, for a particular load case, behave largely ina stretching-dominated manner (6) and become more weightefficient than braced frameworks, once their mechanical be-havior is not determined by global instability.Architecture designed on a length scale that allows one to take

advantage of size-dependent material strengthening effects is thekey to developing superior cellular materials. In Fig. 4, all datapoints along one line with a slope of 1 (dotted guidelines) havethe same specific strength. The influence of different architec-tural approaches in relation to the failure mechanism can beseen clearly. Keeping in mind that structures with thinner coat-ings actually should have a disadvantageous material composi-tion compared with thicker-coated ones, but reach the samevalues of specific strength, again reflects the impact of the me-chanical size effect.A more quantitative description of the relationship among

material composition, failure mode, and attained specificstrength of such composite architectures, as well as the influenceof mechanical size effects, requires detailed modeling and sim-ulations. In classical foam theory, the slope of a line through oneset of data (compare Fig. 4) allows detection of the governingfailure mechanism, applying the according relations of Gibsonand Ashby (1, 7). However, these relations are not applicablehere, because they require a homogeneous and unvaried basematerial. For different coating thicknesses, the presented struc-tures correspond to bulk materials with different effectivestrengths and Young’s moduli and, therefore, cannot be scaledwith the relative or absolute density.

Bauer et al. PNAS | February 18, 2014 | vol. 111 | no. 7 | 2457

ENGINEE

RING

Materials and MethodsTo manufacture polymeric microarchitectures, we applied the commercial3D-DLW system Photonic Professional by Nanoscribe GmbH (26), in the Dip-inLaser Lithography configuration (28). The photoresist IP-Dip (NanoscribeGmbH) has been used. The unit cells of designs A, B, and C are 10 μm × 10 μm ×10 μm. The hexagonal cells of construction D have an edge length of 5 μmand are 10 μm high. Two different sets of honeycomb structures were ex-amined, with edge lengths of 3 and 1.5 μm at heights of 10 and 5 μm, re-spectively. Honeycomb walls and vertical struts have a slightly curved shapeto increase the axial buckling resistance (35). We implemented structuralelements with both rectangular and circular cross-sections. Rectangularstruts are 900–950 nm high and wide, with an edge radius of 190–250 nm.Circular cross-sections of the vertical struts have diameters of 1,000–1,070nm near the junctions and maximally 1,550–1,600 nm at the free length.Honeycomb walls are 580 and 290 nm thick, respectively.

The polymeric structures were homogeneously coated with Al2O3 by ALDat 90 °C (Savannah 100; Cambridge NanoTech). We deposited layers of 10,50, 100, or 200 nm nominal thickness (compare Fig. S1). Considering anapproximated densification of up to 15% from liquid to solid, correlatingwith the observed shrinkage during development (36), we estimated thedensity of solid IP-Dip to be 1,190–1,370 kg/m3 (37). Depending on the purity(above 99%) and the porosity, that of bulk alumina is between 3,750 and4,000 kg/m3 (20, 38), whereas ALD layers generally are a little less dense (27).Based on these values, we calculated the density of all structures using thedesign parameters and optical measurements of cross-sections. The tensilestrength of bulk alumina is in the range of 250 MPa (19), and the com-pressive strength is on the order of 1–3 GPa (1, 20), both strongly dependenton processing conditions.

For mechanical characterization, we performed loading-rate–controlleduniaxial in situ and ex situ compression tests by nanoindentation (ex situ:Nanoindenter G200, Agilent Technologies; in situ: InSEM, NanomechanicsInc.) with a diamond flat punch tip 100 μm in diameter (Fig. 2A). Load-dis-placement curves were recorded. By using the nominal cross-sectional areaand height of the whole structure, engineering stress and strain wereobtained. Compressive strength is defined as the maximum compressivestress before collapse. We determined Young’s modulus, E, as the maximumslope in the corresponding stress–strain curve. Nonlinear behavior at low

strains is the result of varying alignment inaccuracy and the appearance ofsmall particles in the contact area between the structures and the test setup,leading to differences from experiment to experiment when the contactis made.

Based on the test data of the optimized honeycomb structures, we esti-mated the size dependency of the material strength of alumina. For thehoneycomb structure, it may be assumed that the two materials simply areloaded in parallel with a uniaxial stress state. The compressive strength σC ofthe polymer–ceramic composite is given by

σC =As

AcσS, [3]

where σS is the compressive strength of the structure, As is the nominal cross-sectional area, and Ac is the material cross-sectional area. σC is defined as

σC = fPσP + fAσA, [4]

where σP and σA are the stresses at failure inside the polymer and alumina,respectively, and fP and fA are the corresponding fractions of the loadedarea (compare Fig. S2). For bare polymeric structures, fA is zero and σP isequal to σC , which is roughly 200 MPa (compare Fig. S3). Thus, Eq. 4 may bewritten as

σA =σC − fPσP

fA[5]

to calculate σA. The polymeric core is likely to fail before 200 MPa is reachedwhen structures are coated with alumina, because the failure strain of IP-Dipis much higher than that of alumina. However, we decided to base the es-timate on the maximum value of roughly 200 MPa to avoid an overestimateof σA, especially for thinner coatings.

ACKNOWLEDGMENTS. The authors thank Andreas Frölich (Institute ofApplied Physics) for his support in ALD processing, and Sven Bundschuh[Institute for Applied Materials (IAM)], Reiner Mönig (IAM), the biome-chanics department (IAM), and the Institute of Microstructure Technol-ogy staff (all from Karlsruhe Institute of Technology) for their kind assistance.Financial support by the Robert Bosch Foundation is gratefully acknowledged.

1. Gibson LJ, Ashby MF (1997) Cellular Solids: Structure and Properties (Cambridge UnivPress, Cambridge, UK), 2nd Ed.

2. Weiner S, Wagner HD (1998) The material bone: Structure-mechanical function re-lations. Annu Rev Mater Sci 28:271–298.

3. Dinwoodie JM (1981) Timber: Its Nature and Behaviour (Van Nostrand Reinhold,New York).

4. Spatz H-C, Köhler L, Speck T (1998) Biomechanics and functional anatomy of hollow-stemmed sphenopsids. I. Equisetum giganteum (Equisetaceae). Am J Bot 85(3):305–314.

5. Wolff J (1986) The Law of Bone Remodeling (Springer, Berlin); trans of the German1892 Ed.

6. Currey JD (2002) Bones: Structure and Mechanics (Princeton Univ Press, Princeton),2nd Ed.

7. Ashby MF, et al. (2000) Metal Foams: A Design Guide (Butterworth–Heinemann,Oxford).

8. Deshpande VS, Ashby MF, Fleck NA (2001) Foam topology bending versus stretchingdominated architectures. Acta Mater 49(6):1035–1040.

9. Fleck NA, Deshpande VS, Ashby MF (2010) Micro-architectured materials: past, pres-ent and future. Proc R Soc A 466(2121):2495–2516.

10. Mattheck C (2004) The Face of Failure in Nature and Engineering (Karlsruhe InstTechnology, Karlsruhe, Germany).

11. Maxwell JC (1864) On the calculation of the equilibrium and stiffness of frames. PhilosMag Series 4 27(182):294–299.

12. Evans AG, Hutchinson JW, Fleck NA, Ashby MF, Wadley HNG (2001) The topologicaldesign of multifunctional cellular metals. Prog Mater Sci 46(3-4):309–327.

13. Bell AG (1903) The tetrahedral principle in kite structure. Natl Geogr Mag 14(6):219–251.14. Currey JD, Taylor JD (1974) The mechanical behaviour of some molluscan hard tissues.

J Zool 173(3):395–406.15. Tesch W, et al. (2001) Graded microstructure and mechanical properties of human

crown dentin. Calcif Tissue Int 69(3):147–157.16. Gao H, Ji B, Jäger IL, Arzt E, Fratzl P (2003) Materials become insensitive to flaws at

nanoscale: Lessons from nature. Proc Natl Acad Sci USA 100(10):5597–5600.17. Jäger I, Fratzl P (2000) Mineralized collagen fibrils: A mechanical model with a stag-

gered arrangement of mineral particles. Biophys J 79(4):1737–1746.18. Kraft O, Gruber PA, Mönig RM, Weygand D (2010) Plasticity in confined dimensions.

Annu Rev Mater Res 40:293–317.19. Richerson DW (1992) Modern Ceramic Engineering (Dekker, New York), 2nd Ed, pp

169–186.20. Riley F (2009) Structural Ceramics Fundamentals and Case Studies (Cambridge Univ

Press, Cambridge, UK), pp 137–145.

21. Chantikul P, Bennison SJ, Lawn BR (1990) Role of grain size in the strength andR-curve properties of alumina. J Am Ceram Soc 73(8):2419–2427.

22. Griffith AA (1921) The phenomena of rupture and flow in solids. Phil Trans R Soc LondA 221(582-593):163–198.

23. Valdevit L, Jacobsen AJ, Greer JR, Carter WB (2011) Protocols for the optimal designof multi-functional cellular structures: From hypersonics to micro-architected mate-rials. J Am Ceram Soc 94(S1):S15–S34.

24. Schaedler TA, et al. (2011) Ultralight metallic microlattices. Science 334(6058):962–965.

25. Jacobsen AJ, Barvosa-Carter W, Nutt S (2007) Micro-scale truss structures formed fromself-propagating photopolymer waveguides. Adv Mater 19(22):3892–3896.

26. von Freyman G, et al. (2010) Three-dimensional nanostructures for photonics. AdvFunct Mater 20(7):1038–1052.

27. George SM (2010) Atomic layer deposition: An overview. Chem Rev 110(1):111–131.28. Bückmann T, et al. (2012) Tailored 3D mechanical metamaterials made by dip-in

direct-laser-writing optical lithography. Adv Mater 24(20):2710–2714.29. Gansel JK, et al. (2009) Gold helix photonic metamaterial as broadband circular po-

larizer. Science 325(5947):1513–1515.30. Jang D, Meza LR, Greer F, Greer JR (2013) Fabrication and deformation of three-

dimensional hollow ceramic nanostructures. Nat Mater 12(10):893–898.31. Ashby MF, et al. (2000) Metal Foams: A Design Guide (Butterworth–Heinemann,

Oxford), pp 40–54.32. CES EduPack (2012) MaterialUniverse:\Hybrids: composites, foams, honeycombs, natural

materials\Foams (Granta Design Ltd, Cambridge, UK). Available at www.grantadesign.com/.Accessed October 10, 2013.

33. Ahmad R, Ha J-H, Song I-H (2014) Enhancement of the compressive strength of highlyporous Al2O3 foam through crack healing and improvement of the surface conditionby dip-coating. Ceram Int 40(2):3679–3685.

34. Michell AGM (1904) The limits of economy of material in frame-structures. Philos Mag8(47):589–597.

35. Wiedemann J (2007) Leichtbau Elemente und Konstruktion (Springer, Berlin), 3rd Ed,pp 63–71, pp 562–573.

36. Farsari M, Vamvakaki M, Chichkov BN (2010) Multiphoton polymerization of hybridmaterials. J Opt 12(12):124001.

37. Nanoscribe GmbH (2013) Safety Data Sheet: IP-Dip Photoresist (Nanoscribe GmbH,Karlsruhe, Germany).

38. Auerkari P (1996) Mechanical and Physical Properties of Engineering AluminaCeramics (VTT Offsetpaino, Espoo, Finland), p 7.

2458 | www.pnas.org/cgi/doi/10.1073/pnas.1315147111 Bauer et al.