High-Strength Cellular Ceramic Composites with 3D Microarchitecture

High-strength cellular ceramic composites with 3D microarchitecture

Jens 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 try Cellular topologies may be divided into bending- and stretch- to either improve the strength or lower the density, or both. The ing-dominated ones (8). Foams generally behave in a bending- lightest solid materials have a density in the range of 1,000 kg/m3; dominated manner (1, 8, 9). When abstracted to pin-jointed only cellular materials, such as technical foams, can reach consid- frameworks, open-cell foams consist of unit cells that are stati- erably lower values. However, compared with corresponding bulk cally indeterminate (8), e.g., cubic cells. Their topology would materials, their specific strength generally is significantly lower. allow the struts to rotate around the joints leading to a collapse Cellular topologies may be divided into bending- and stretching- under loading (11). However, the joints of foams are frozen dominated ones. Technical foams are structured randomly and rather than pin-jointed, causing the struts to bend. Gibson and behave in a bending-dominated way, which is less weight effi- Ashby (1) showed that the mechanical properties of such bend- ρp=ρ cient, with respect to strength, than stretching-dominated behaving-dominated foams depend on the relative density, s, where ρp ρ ior, such as in regular braced frameworks. Cancellous bone and and s are the densities of the foam and the corresponding other natural cellular solids have an optimized architecture. Their solid material, respectively. The compressive strength of the foam basic material is structured hierarchically and consists of nanometer- ðρp=ρ Þ1:5 scales with s or even with higher exponents, depending size elements, providing a benefit from size effects in the material on the failure mechanism. strength. Designing cellular materials with a specific microarch- Stretching-dominated structures are considered to have much itecture would allow one to exploit the structural advantages of better mechanical properties (8, 9, 12). The struts of a frame- stretching-dominated constructions as well as size-dependent work, which is rigid when regarded as pin-jointed, are loaded in strengthening effects. In this paper, we demonstrate that such mate- tension or compression largely without bending (8). In two rials may be fabricated. Applying 3D laser lithography, we produced dimensions, a triangle is the only statically determinate polygon. and characterized micro-truss and -shell structures made from alu- In three dimensions, fully triangular structures, such as tetrahe- mina–polymer composite. Size-dependent strengthening of alumina dral truss constructions as initially developed by Bell (13), reach shells has been observed, particularly when applied with a character- a maximum of rigidity and stretching-dominated behavior (8). istic thickness below 100 nm. The presented artificial cellular materi- Designing foam materials in such a manner facilitates a linear als reach compressive strengths up to 280 MPa with densities well scaling behavior of the strength and the stiffness with the relative 3 below 1,000 kg/m . density (8, 9). However, the specific properties of bulk material still are not quite reached (9). he suitability of a material for lightweight applications is Bone and other biological materials with a similar funda- Tdetermined mainly by two properties: the specific strength mental structure, such as shells (14) and teeth (15), achieve and the specific stiffness, here defined as the strength and stiff- improved strength of their basic material as a result of the ap- ness of a material divided by its density. In the past century, pearance of mechanical size effects (16). On the lowest level of major advancements have been made in optimizing classical hierarchy, bone consists of mineral crystal platelets with thickness lightweight materials, such as aluminum alloys or composite materials, with respect to these properties. However, the lightest Significance solid materials have a density in the range of 1,000 kg/m3 (1). Natural lightweight materials, such as bone and wood, are not It has been a long-standing effort to create materials with low fully dense and may exhibit considerably lower values (1). They density but high strength. Technical foams are very light, but contain several levels of hierarchical structuring down to the compared with bulk materials, their strength is quite low be- – nanometer scale (1 3), leading to remarkable specific mechani- cause of their random structure. Natural lightweight materials, – cal properties (1 4). For instance, cancellous bone is built of such as bone, are cellular solids with optimized architecture. truss- or shell-like framework architectures grown adaptively to They are structured hierarchically and actually consist of ENGINEERING the loading situation (5, 6). The material thickness and the ori- nanometer-size building blocks, providing a benefit from me- entation of the individual structural elements depend on the chanical size effects. In this paper, we demonstrate that magnitude and orientation of loading. This leads to an optimized materials with a designed microarchitecture, which provides topology, in which each structural element is aligned with the both structural advantages and size-dependent strengthening principal stress trajectories (5, 6). effects, may be fabricated. Using 3D laser lithography, we Technical foams are materials with open- or closed-cell po- produced micro-truss and -shell structures from ceramic–poly- rosity of comparable low density and are used in lightweight mer composites that exceed the strength-to-weight ratio of all components, such as foam-core sandwich panels (1, 7). However, engineering materials, with a density below 1,000 kg/m3. their specific strength and stiffness are limited by their characteristic stochastic architecture. Typically, considerably lower Author contributions: J.B., I.T., and O.K. designed research; J.B., S.H., and R.S. performed values of specific strength and stiffness compared with the cor- research; J.B., S.H., I.T., and O.K. analyzed data; and J.B. wrote the paper. responding bulk materials are reached (1, 7). In addition to the The authors declare no conflict of interest. material properties, the architecture strongly affects the me- *This Direct Submission article had a prearranged editor. chanical behavior of such cellular solids (1, 8, 9). Buckling, in- Freely available online through the PNAS open access option. homogeneity, and local stress concentrations (10) occur, because 1To whom correspondence should be addressed. E-mail: [email protected]. foams cannot be considered only as materials but also as struc- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. tures (1, 7). 1073/pnas.1315147111/-/DCSupplemental.

www.pnas.org/cgi/doi/10.1073/pnas.1315147111 PNAS | February 18, 2014 | vol. 111 | no. 7 | 2453–2458 Downloaded by guest on September 30, 2021 on the order of a few nanometers integrated in a collagen matrix with design control down to the nanometer scale. We describe the (17). The strength of both ductile (18) and brittle (19–21) materials design, processing, and mechanical characterization of several ce- typically increases with decreasing dimensions. In the early 20th ramic–polymer composite structures with submicron feature size. century, Griffith (22) proposed the relationship Although to date 3D-DLW is strongly limited in achievable sample volume, it allows for production of almost arbitrary polymeric ge- σ ∝ p1ffiffiffi [1] ometries (26). In conjunction with coating techniques, such as f c atomic layer deposition (ALD) (27), multimaterial composites (28),aswellasmetallic(29)orceramic(30)structuresinwhichthe between the fracture strength, σf , and the critical size of a flaw, c, polymer has been removed, may be fabricated. We present truss for brittle materials such as ceramics. A flaw cannot be larger constructions with different structuralpropertiesaswellasashape- than the component in which it is located. Assuming c correlates optimized honeycomb design (Fig. 1). All structures were fabri- with the material thickness, t, of a structural element (16), Eq. 1 cated from polymer (IP-Dip; Nanoscribe GmbH) by 3D-DLW and may be written as homogeneously coated with alumina (Al2O3)layersofdifferent thicknesses using ALD. The alumina coating carries tensile and 1 compressive forces, whereas the light polymeric core serves to σf ∝ pffiffi : [2] t prevent early face buckling and to improve toughness. For mechanical characterization, uniaxial in situ and ex situ compression Thus, the smaller the component, the higher is its fracture tests were performed. strength. It has been argued that when fabricated thin enough, When coatings were applied with a characteristic thickness materials might even exhibit strength values close to the below 100 nm, a substantial increase was observed in the mate- theoretical strength when a critical thickness in the nanometer rial strength of alumina. Surpassing all technical foam materials, range is reached (16). Assuming that failure no longer is gov- the trusses reach compressive strength values up to 55 MPa for erned by the Griffith criterion but by the strength of atomic an estimated density of 410 kg/m3. Shape-optimized honeycomb bonds in that regime, materials should become insensitive to structures achieve up to 280 MPa at 810 kg/m3, exceeding all flaws (16). natural and engineering materials with a density below 1,000 kg/m3. Designing cellular materials with a specific microarchitecture The specific compressive strengths obtained are higher than would allow one both to exploit the structural advantages of thoseofmostengineeringmetals and close to the ones of stretching-dominated constructions and to gain enhanced ma- technical ceramics. terial strength due to mechanical size effects, leading to superior As predicted theoretically (9, 23), the designed and minia- cellular materials (9, 23). Recent improvements in material turized architecture benefits from both structural advantages processing methods have led to cellular materials of extremely and size-dependent strengthening effects, facilitating values low density with a periodic microarchitecture (24). However, the of specific strength beyond the accessible range of standard freedom of design of these structures is limited by processing- cellular materials. related restrictions (25). The producible topologies are bending dominated and behave in a highly compliant manner. Results In this paper, we propose 3D direct laser writing (3D-DLW) (26) All fabricated structures were tested in compression, as illus- as a method for fabricating real 3D composite microarchitectures 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 as behaving fairly isotropically. Depending on the stiffness of junctions, collapse mechanisms are less urgent and topologies may be designed so that a maximum of 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. Downloaded by guest on September 30, 2021 Fig. 2. All cellular material designs were tested mechanically under uniaxial compressive loading (compare Movies S1–S4). (A) An in situ test of a polymeric truss structure (side view). (B) Stress–strain curves of fully triangular trusses (Fig. 1A). Alumina layers of the indicated thicknesses have been deposited onto a polymeric core structure. With increasing layer thickness, the compressive strength and stiffness (Young’s modulus E) increase strongly compared with bare polymeric structures. (C) Specific compressive strength as a function of the coating thickness t. Labels refer to the nomenclature of Fig. 1. The estimated stress at failure inside the alumina layer, σA (Eq. 5), increases with decreasing layer thickness. All data points correspond to at least three measurements. Error bars give the corresponding maximal upper and lower deviation.

thickness of the deposited alumina layer, as selected stress–strain of only 15% alumina (50 nm) reach the same specific strength as curves show in Fig. 2B. Regardless of the coating thickness, those with 40% (200 nm) (compare Table S1). the size of all polymeric core structures remains unvaried. Based on the test data of the honeycomb structures, the stress σ 5 Starting from low values of about 2 MPa for pure polymeric at failure, A, inside the alumina layers (Eq. ) has been esti- structures, the compressive strength increases up to 34 MPa mated analytically. A significant increase may be observed for σ with the introduction of the alumina coatings. Simultaneously, a characteristic thickness in the range below 100 nm; A increases ’ from 1,180 to 3,900 MPa, a value noticeably higher than the typical

Young s modulus, E, increases from about 0.1 to 1.4 GPa. All ENGINEERING curves exhibit nonlinear behavior for lower strains, which is compressive strength of bulk alumina (1, 19). related to experimental issues, such as small misalignment and Failure mechanisms depend on both the architecture of the structures and the thicknesses of the coatings (Fig. 3). We roughness at the top surface of the structure. There is no sys- detected buckling and brittle fracture as the two major tematic dependency on the coating thickness (compare Mate- mechanisms. Bare polymeric and thinner-coated structures rials and Methods ). collapse by buckling. Whereas trusses A and B buckle locally Fig. 2C shows the specific strength of several designs in re- (Fig. 3A), designs C and D allow global buckling modes (Fig. lation to the layer thickness of alumina (all data are given in 3H; compare Movies S1–S3). With growing layer thickness of Supporting Information). Architectures A and B (compare Table the coatings, the failure mechanism shifts to brittle fracture. S1) achieve almost equal specific values. Topology C (compare Depending on the architecture, this transition occurs at dif- Table S1) behaves similar to design D but is less effective ferent layer thicknesses. For 50 nm, local buckling still may be throughout. It clearly is shown that the specific strength of all critical for designs A and B, but fracture due to high tensile structures increases once thin alumina layers are deposited, as stresses at the notches of the junctions may occur as fre- the strength of alumina is much higher than that of IP-Dip. quently (Fig. 3C) and dominates at 100 nm (Fig. 3D). For 200 However, with growing coating thickness, the increase becomes nm alumina thickness, compressive failure of the vertical less pronounced (design D) or even saturates (designs A and E). struts sets in (Fig. 3E). Compared with A and B, the honey- Optimized honeycomb structures with a solid-material fraction comb design resists buckling for thinner coatings and crushes

Bauer et al. PNAS | February 18, 2014 | vol. 111 | no. 7 | 2455 Downloaded by guest on September 30, 2021 Fig. 3. Failed samples of different coating thickness (see headings) and architecture (Upper, design A; Lower, designs C–E). With growing layer thickness of alumina, 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, and squamous 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 as close to the junctions, was observed. (D) Reaching 100 nm, fracture was detected to appear exclusively at the junctions. (E) For 200 nm, we found the structures 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-coated honeycomb designs burst in a brittle manner. (H) Designs C and D buckle globally and fracture without notable plastic deformation for both bare polymeric structures (compare Movie S3) and coated ones. (I) Reaching 100 nm, we observed cracking of the vertical compressive bars normal to the loading direction and fracture close to the junctions.

in a brittle manner (Fig. 3J). Truss structures C and D exhibit Fig. 4 shows a so-called Ashby chart (CES EduPack; Granta local buckling and beginning brittle fracture near the junc- Design Ltd.) for compressive strength vs. density. Compared with tions at 100-nm layer thickness (Fig. 3I). other materials with a density below 1,000 kg/m3, the presented

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 designs achieve strength-to-weight ratios comparable to those of technical ceramics and high-strength steels. The nomenclature refers to Fig. 1. Labels indicate the thicknesses of the deposited alumina layers.

2456 | www.pnas.org/cgi/doi/10.1073/pnas.1315147111 Bauer et al. Downloaded by guest on September 30, 2021 designs obtain outstanding values. Truss structures A, B, and D in relation to the diameter of the struts. For a given coating exceed all technical foam materials (31–33). Their specific com- thickness, further downscaling of the polymeric cores would pressive strength is in the range of bone material and advanced improve both the buckling strength and the ratio of alumina to metallic alloys (1, 32). Designs with a solid material fraction of polymer, allowing more efficient access to the observed strong only 13% alumina (A and B at 50 nm) already reach the highest increase of the material strength below 50 nm. However, the specific strength of pure alumina foam (32, 33). A maximum minimal producible length scale of architecture is limited by compressive strength of 55 MPa at 410 kg/m3 is achieved (D at the lithography process (26). 200 nm). Shape-optimized honeycomb designs surpass all nat- How efficient a design is in profiting from size effects depends ural and technical cellular materials. Specific ratios up to 280 on the actual architecture. Designing foam topologies as tri- MPa at 810 kg/m3, in the range of technical ceramics and high- angular braced frameworks is an expedient way to maximize strength steels, not far below bulk alumina, are reached (32). stretching-dominated behavior independently of the loading For a summary of the numerical values, see Supporting In- situation. However, for thicker coatings, we observed design D to formation (Table S1). reach higher values of specific strength than the braced truss Discussion structures A and B. A general approach in lightweight design is to approximate a truss topology corresponding to the orthogonal Macroscopic properties of cellular materials are determined by network of the acting principal tensile and compressive stresses both material and structural properties. Reaching the size scale (34), as in cancellous bone (5, 6). Although such a structure may discussed in this paper, structural characteristics and material not be triangular at all, it is stretching dominated, because struts strength are coupled (23). aligned with the principal stress direction experience no bending σ Based on the strength of the tested honeycomb structures, A moment (6). A construction is most weight efficient when as estimates the size dependency of the material strength of alu- many structural elements as possible are oriented in the loading mina. The observed behavior (Fig. 2C) is in good agreement with direction while guaranteeing sufficient structural stability, e.g., by the considerations of Gao et al. (16), as well as recently pub- avoiding slender bars and thin shells under compressive load (6). lished effects in hollow ceramic architectures of comparable Under axial loading, the honeycomb structure attains the highest dimensions (30), supporting the theory of decreasing flaw size specific strength, once its walls are thick and stiff enough to resist and size-dependent strengthening below a certain length scale. buckling, because the entire solid material is aligned in the Data points in Fig. 2C approximately correlate with the de- 2 loading direction. The braced framework of designs A and B pendency given by Eq. . Because buckling before material causes global rigidity and enforces localized failure (compare failure, multiaxial stress states, and local stress concentrations Movies S1 and S2), whereas the rigidity of designs C and D are neglected, the estimation may be regarded as conservative. depends on the stiffness of their junctions (8, 11). Those were Before brittle crushing sets in, all designs are expected to frac- observed to behave relatively compliantly when coatings were ture initially because of high local tensile stresses (compare Fig. thin (compare Movie S3). Acting as elastic–plastic hinges, they 3G: crack formation due to transversal tension). Therefore, we allow global buckling modes that start without rupture or notable expect σ to depend on the tensile strength rather than repre- A deformation of single struts. With increasing coating thickness, sent the actual compressive strength, especially when coatings junctions become stiffer and designs C and D more rigid. When become thinner. their resistance to global buckling reaches the range of that of local Brittle cellular solids fail when buckling occurs or when local stresses attain the tensile or compressive strength of the solid (1). buckling or material failure, the benefit of diagonal bracings, as Therefore, buckling always occurs at lower stresses than material applied in designs A and B, decreases. Thus, constructions such as failure (1, 6). The buckling load of a strut is proportional to designs C and D, which generally are considered bending domi- EI=l2, where E is Young’s modulus, I is the second moment of nated (8) may, for a particular load case, behave largely in area, and l is the strut length (6). When the alumina layers be- a stretching-dominated manner (6) and become more weight come thicker, the modulus, E, of the composite and, I, of the efficient than braced frameworks, once their mechanical be- struts increase. The buckling strengths of the structures improve, havior is not determined by global instability. and at a certain point, failure by fracture of the material becomes Architecture designed on a length scale that allows one to take dominant. advantage of size-dependent material strengthening effects is the The observed behavior of the specific strength in relation to key to developing superior cellular materials. In Fig. 4, all data the coating thickness (Fig. 2C) may be explained by the interplay points along one line with a slope of 1 (dotted guidelines) have of mechanical size effects and the failure mode. Within our tests, the same specific strength. The influence of different architec- the transition from buckling to material fracture (Fig. 3) roughly tural approaches in relation to the failure mechanism can be seen clearly. Keeping in mind that structures with thinner coat- correlates with the sections of beginning saturation of the spe- ENGINEERING cific strength. The fraction of alumina in the solid material ings actually should have a disadvantageous material composi- increases from only a few percent at 10 nm up to 40% at 200 nm. tion compared with thicker-coated ones, but reach the same Because the specific strength of alumina is much higher than that values of specific strength, again reflects the impact of the me- of IP-Dip, one would expect the specific strength of the struc- chanical size effect. tures to increase simultaneously. However, when coatings be- A more quantitative description of the relationship among come thinner while still being thick enough to resist buckling, material composition, failure mode, and attained specific the observed mechanical size effect compensates for the de- strength of such composite architectures, as well as the influence creasing fraction of alumina in the composite. When the of mechanical size effects, requires detailed modeling and sim- thickness is reduced further, buckling occurs before material ulations. In classical foam theory, the slope of a line through one failure and the specific strength decays. set of data (compare Fig. 4) allows detection of the governing Microarchitecture allows the presented cellular materials to failure mechanism, applying the according relations of Gibson benefit from the observed size effect, whereas self-similar mac- and Ashby (1, 7). However, these relations are not applicable roscopic constructions would be unable to do so. To induce here, because they require a homogeneous and unvaried base strong mechanical size effects in ceramics, material thicknesses material. For different coating thicknesses, the presented struc- are required to be in the range of a few nanometers (16) (compare tures correspond to bulk materials with different effective Fig. 2C: trend of σA). However, buckling tends to dominate the strengths and Young’s moduli and, therefore, cannot be scaled mechanical behavior for decreasing thickness of the ceramic shell with the relative or absolute density.

Bauer et al. PNAS | February 18, 2014 | vol. 111 | no. 7 | 2457 Downloaded by guest on September 30, 2021 Materials and Methods strains is the result of varying alignment inaccuracy and the appearance of To manufacture polymeric microarchitectures, we applied the commercial small particles in the contact area between the structures and the test setup, 3D-DLW system Photonic Professional by Nanoscribe GmbH (26), in the Dip-in leading to differences from experiment to experiment when the contact Laser Lithography configuration (28). The photoresist IP-Dip (Nanoscribe is made. GmbH) has been used. The unit cells of designs A, B, and C are 10 μm × 10 μm × Based on the test data of the optimized honeycomb structures, we esti- 10 μm. The hexagonal cells of construction D have an edge length of 5 μm mated the size dependency of the material strength of alumina. For the and are 10 μm high. Two different sets of honeycomb structures were ex- honeycomb structure, it may be assumed that the two materials simply are σ amined, with edge lengths of 3 and 1.5 μm at heights of 10 and 5 μm, re- loaded in parallel with a uniaxial stress state. The compressive strength C of – spectively. Honeycomb walls and vertical struts have a slightly curved shape the polymer ceramic composite is given by to increase the axial buckling resistance (35). We implemented structural A elements with both rectangular and circular cross-sections. Rectangular σ = s σ , [3] C A S struts are 900–950 nm high and wide, with an edge radius of 190–250 nm. c Circular cross-sections of the vertical struts have diameters of 1,000–1,070 where σS is the compressive strength of the structure, As is the nominal cross- nm near the junctions and maximally 1,550–1,600 nm at the free length. sectional area, and Ac is the material cross-sectional area. σC is defined as Honeycomb walls are 580 and 290 nm thick, respectively.

The polymeric structures were homogeneously coated with Al2O3 by ALD σC = fP σP + fAσA, [4] at 90 °C (Savannah 100; Cambridge NanoTech). We deposited layers of 10, σ σ 50, 100, or 200 nm nominal thickness (compare Fig. S1). Considering an where P and A are the stresses at failure inside the polymer and alumina, approximated densification of up to 15% from liquid to solid, correlating respectively, and fP and fA are the corresponding fractions of the loaded σ with the observed shrinkage during development (36), we estimated the area (compare Fig. S2). For bare polymeric structures, fA is zero and P is 3 σ density of solid IP-Dip to be 1,190–1,370 kg/m (37). Depending on the purity equal to C , which is roughly 200 MPa (compare Fig. S3). Thus, Eq. 4 may be (above 99%) and the porosity, that of bulk alumina is between 3,750 and written as 4,000 kg/m3 (20, 38), whereas ALD layers generally are a little less dense (27). σ − σ Based on these values, we calculated the density of all structures using the C fP P σA = [5] design parameters and optical measurements of cross-sections. The tensile fA strength of bulk alumina is in the range of 250 MPa (19), and the com- to calculate σ . The polymeric core is likely to fail before 200 MPa is reached pressive strength is on the order of 1–3 GPa (1, 20), both strongly dependent A when structures are coated with alumina, because the failure strain of IP-Dip on processing conditions. is much higher than that of alumina. However, we decided to base the es- For mechanical characterization, we performed loading-rate–controlled timate on the maximum value of roughly 200 MPa to avoid an overestimate uniaxial in situ and ex situ compression tests by nanoindentation (ex situ: of σ , especially for thinner coatings. Nanoindenter G200, Agilent Technologies; in situ: InSEM, Nanomechanics A Inc.) with a diamond flat punch tip 100 μm in diameter (Fig. 2A). Load-dis- ACKNOWLEDGMENTS. The authors thank Andreas Frölich (Institute of placement curves were recorded. By using the nominal cross-sectional area Applied Physics) for his support in ALD processing, and Sven Bundschuh and height of the whole structure, engineering stress and strain were [Institute for Applied Materials (IAM)], Reiner Mönig (IAM), the biome- obtained. Compressive strength is defined as the maximum compressive chanics department (IAM), and the Institute of Microstructure Technol- stress before collapse. We determined Young’s modulus, E, as the maximum ogy staff (all from Karlsruhe Institute of Technology) for their kind assistance. slope in the corresponding stress–strain curve. Nonlinear behavior at low Financial support by the Robert Bosch Foundation is gratefully acknowledged.

1. Gibson LJ, Ashby MF (1997) Cellular Solids: Structure and Properties (Cambridge Univ 21. Chantikul P, Bennison SJ, Lawn BR (1990) Role of grain size in the strength and Press, Cambridge, UK), 2nd Ed. R-curve properties of alumina. J Am Ceram Soc 73(8):2419–2427. 2. Weiner S, Wagner HD (1998) The material bone: Structure-mechanical function re- 22. Griffith AA (1921) The phenomena of rupture and flow in solids. Phil Trans R Soc Lond lations. Annu Rev Mater Sci 28:271–298. A 221(582-593):163–198. 3. Dinwoodie JM (1981) Timber: Its Nature and Behaviour (Van Nostrand Reinhold, 23. Valdevit L, Jacobsen AJ, Greer JR, Carter WB (2011) Protocols for the optimal design New York). of multi-functional cellular structures: From hypersonics to micro-architected mate- 4. Spatz H-C, Köhler L, Speck T (1998) Biomechanics and functional anatomy of hollow- rials. J Am Ceram Soc 94(S1):S15–S34. stemmed sphenopsids. I. Equisetum giganteum (Equisetaceae). Am J Bot 85(3):305–314. 24. Schaedler TA, et al. (2011) Ultralight metallic microlattices. Science 334(6058): 5. Wolff J (1986) The Law of Bone Remodeling (Springer, Berlin); trans of the German 962–965. 1892 Ed. 25. Jacobsen AJ, Barvosa-Carter W, Nutt S (2007) Micro-scale truss structures formed from 6. Currey JD (2002) Bones: Structure and Mechanics (Princeton Univ Press, Princeton), self-propagating photopolymer waveguides. Adv Mater 19(22):3892–3896. 2nd Ed. 26. von Freyman G, et al. (2010) Three-dimensional nanostructures for photonics. Adv – 7. Ashby MF, et al. (2000) Metal Foams: A Design Guide (Butterworth Heinemann, Funct Mater 20(7):1038–1052. Oxford). 27. George SM (2010) Atomic layer deposition: An overview. Chem Rev 110(1):111–131. 8. Deshpande VS, Ashby MF, Fleck NA (2001) Foam topology bending versus stretching 28. Bückmann T, et al. (2012) Tailored 3D mechanical metamaterials made by dip-in dominated architectures. Acta Mater 49(6):1035–1040. direct-laser-writing optical lithography. Adv Mater 24(20):2710–2714. 9. Fleck NA, Deshpande VS, Ashby MF (2010) Micro-architectured materials: past, pres- 29. Gansel JK, et al. (2009) Gold helix photonic metamaterial as broadband circular po- ent and future. Proc R Soc A 466(2121):2495–2516. larizer. Science 325(5947):1513–1515. 10. Mattheck C (2004) The Face of Failure in Nature and Engineering (Karlsruhe Inst 30. Jang D, Meza LR, Greer F, Greer JR (2013) Fabrication and deformation of three- Technology, Karlsruhe, Germany). dimensional hollow ceramic nanostructures. Nat Mater 12(10):893–898. 11. Maxwell JC (1864) On the calculation of the equilibrium and stiffness of frames. Philos 31. Ashby MF, et al. (2000) Metal Foams: A Design Guide (Butterworth–Heinemann, Mag Series 4 27(182):294–299. Oxford), pp 40–54. 12. Evans AG, Hutchinson JW, Fleck NA, Ashby MF, Wadley HNG (2001) The topological 32. CES EduPack (2012) MaterialUniverse:\Hybrids: composites, foams, honeycombs, natural design of multifunctional cellular metals. Prog Mater Sci 46(3-4):309–327. materials\Foams (Granta Design Ltd, Cambridge, UK). Available at www.grantadesign.com/. 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. Accessed October 10, 2013. J Zool 173(3):395–406. 33. Ahmad R, Ha J-H, Song I-H (2014) Enhancement of the compressive strength of highly 15. Tesch W, et al. (2001) Graded microstructure and mechanical properties of human porous Al2O3 foam through crack healing and improvement of the surface condition – crown dentin. Calcif Tissue Int 69(3):147–157. by dip-coating. Ceram Int 40(2):3679 3685. 16. Gao H, Ji B, Jäger IL, Arzt E, Fratzl P (2003) Materials become insensitive to flaws at 34. Michell AGM (1904) The limits of economy of material in frame-structures. Philos Mag nanoscale: Lessons from nature. Proc Natl Acad Sci USA 100(10):5597–5600. 8(47):589–597. 17. Jäger I, Fratzl P (2000) Mineralized collagen fibrils: A mechanical model with a stag- 35. Wiedemann J (2007) Leichtbau Elemente und Konstruktion (Springer, Berlin), 3rd Ed, gered arrangement of mineral particles. Biophys J 79(4):1737–1746. pp 63–71, pp 562–573. 18. Kraft O, Gruber PA, Mönig RM, Weygand D (2010) Plasticity in confined dimensions. 36. Farsari M, Vamvakaki M, Chichkov BN (2010) Multiphoton polymerization of hybrid Annu Rev Mater Res 40:293–317. materials. J Opt 12(12):124001. 19. Richerson DW (1992) Modern Ceramic Engineering (Dekker, New York), 2nd Ed, pp 37. Nanoscribe GmbH (2013) Safety Data Sheet: IP-Dip Photoresist (Nanoscribe GmbH, 169–186. Karlsruhe, Germany). 20. Riley F (2009) Structural Ceramics Fundamentals and Case Studies (Cambridge Univ 38. Auerkari P (1996) Mechanical and Physical Properties of Engineering Alumina Press, Cambridge, UK), pp 137–145. Ceramics (VTT Offsetpaino, Espoo, Finland), p 7.

2458 | www.pnas.org/cgi/doi/10.1073/pnas.1315147111 Bauer et al. Downloaded by guest on September 30, 2021