Abstract
Metamaterialbased cloaks make objects different from their surrounding appear just like their surrounding. To date, cloaking has been demonstrated experimentally in many fields of research, including electrodynamics at microwave frequencies, optics, static electric conduction, acoustics, fluid dynamics, thermodynamics and quasi twodimensional solid mechanics. However, cloaking in the seemingly simple case of threedimensional solid mechanics is more demanding. Here, inspired by invisible coreshell nanoparticles in optics, we design an approximate elastomechanical coreshell ‘unfeelability’ cloak based on pentamode metamaterials. The resulting threedimensional polymer microstructures with macroscopic overall volume are fabricated by rapid dipin direct laser writing optical lithography. We quasistatically deform cloak and control samples in the linear regime and map the displacement fields by autocorrelationbased analysis of recorded movies. The measured and the calculated displacement fields show very good cloaking performance. This means that one can elastically hide objects along these lines.
Introduction
Artificial materials called metamaterials^{1,2} or composites^{3} have significantly improved our ability to steer waves and energy fluxes in electromagnetism, optics, acoustics, thermodynamics and mechanics. Invisibility cloaks^{4,5,6,7} and counterparts thereof are demonstrations of this progress. Cloaks have been demonstrated experimentally in electromagnetism at microwave^{8} and at optical frequencies^{9,10,11}, in airborne^{12,13} and fluidborne acoustics^{14}, for fluid surface waves^{15}, for electrical^{16} and heat conduction^{17,18,19}, as well as for flexural waves^{20} in thin (quasi twodimensional) elastic membranes. In threedimensional (3D) solid mechanics, only unidirectional cloaking of stress waves in fibrous materials has been achieved^{21}.
What makes cloaking in 3D solid mechanics harder than cloaking in other systems? An isotropic dielectric optical material can be described by a simple scalar electric permittivity, an isotropic magnetic material by a scalar magnetic permeability, an isotropic electric (or thermal) conductor by a scalar electric (or heat) conductivity, an isotropic diffusive medium by a scalar diffusivity and so on. In contrast, in mechanics, regular 3D elastic solids can only be treated by a rank4 elasticity tensor^{22}. Even for isotropic media, it contains two independent elements, namely the material’s bulk modulus B and shear modulus G. Intuitively, these moduli describe the forces needed to change the volume and the shape of a linearly elastic material, respectively, while fixing the other. Mechanics becomes much simpler for the acoustics of gases/fluids^{2,3}, which are defined by strictly zero shear modulus, that is, G=0.
The theory underlying cloaking has either been based on analytical coordinate transformations^{3,4,5,6} or on numerical topology optimization^{13,23,24,25}. The analytical theory for mere invisibility and analogues thereof based on simple spherical or cylindrical coreshell geometries has been worked out a long time ago^{26,27,28}. We use the coreshell approach below. Whatever the approach, a cloak—as opposed to just a ‘neutral inclusion’ that is not necessarily a cloak—makes an arbitrary object within the cloak appear just like a given surrounding. To accomplish this goal, the object needs to be isolated from the surrounding for all conditions. In optics, isolation can be achieved by an opaque metal wall, in electrical conduction by an electrically insulating wall and in heat conduction by a thermally insulating wall. In mechanics, one needs a rigid (that is, immovable) wall. Next, a suitable structure is wrapped around these walls to make them and their interior appear identical to the surrounding. A similar coreshell approach has just recently been successfully used in thermal cloaking^{18,19}. In this paper, we experimentally demonstrate an approximate coreshell elastomechanical cloak.
Results
Elastomechanical cloak design
What function does such an ‘unfeelability cloak’ serve? For example, lying on an elastic bed mattress, you would normally feel a stiff object underneath the mattress (also see illustration Supplementary Fig. 1). In contrast, a cloaking mattress on top of the same stiff object would appear to the outside as a homogeneous isotropic elastic solid. In this fashion, one can elastically hide objects.
Inspired by invisible coreshell geometries in optics^{29}, we use the mechanical coreshell approach illustrated in Fig. 1. Theory for elastic coreshell geometries was published by Kerner^{26} and later by Hashin and Shtrikman^{28}. The textbook of Milton^{3} comprehensively reviews the theoretical work published since then, much of which uses the notion of ‘neutral inclusions’. More recently, a theoretical optimization study has been published^{30}. In Fig. 1, we aim at statically cloaking an arbitrary object located inside a rigid (that is, ideally incompressible and not deformable) hollow cylinder with inner radius R_{i}, outer radius R_{1}, bulk modulus B_{1}→∞ and shear modulus G_{1}→∞. As pointed out above, this rigid wall already isolates and protects any object inside with respect to the outside. By additionally wrapping around the cylinder a homogeneous isotropic shell (with outer radius R_{2}, bulk modulus B_{2} and shear modulus G_{2}), we wish to make this coreshell geometry appear elastically as its isotropic compliant homogeneous surrounding (with bulk modulus B_{0} and shear modulus G_{0}).
To accomplish this goal, we need to make the coreshell appear as the surrounding with respect to compression as well as with respect to shear. It is the combination of these two aspects—compression and shear—that makes cloaking in mechanics harder than cloaking in other areas of physics. In regard to shear, Hashin and Shtrikman^{28} showed that this necessarily requires G_{1}=G_{0}, from which the condition G_{2}=G_{1}=G_{0} follows^{28}. Mathematically, this condition is immediately fulfilled by any three liquids with zero shear moduli. The liquids would, however, rapidly intermix and flow away. Hence, the cloak would not be stable. A rigid isolating wall (with B_{1}→∞ and G_{1}→∞) requires a solid. This immediately leads to a surrounding with G_{0}→∞, which is not deformable as well, and would hence not represent an impressive demonstration of cloaking at all.
We conclude that the mechanical coreshell approach does not allow for perfect elastomechanical cloaking using a rigid wall for isolation and a compliant solid as surrounding. Nevertheless, cloaking^{24,31} may still be possible upon simply ignoring the condition G_{2}=G_{1}=G_{0}, in which case cloaking is no longer expected to be strict. Our approximate approach is to search for solutions based on a rigid wall, whereas the surrounding as well as the shell exhibit relatively small shear moduli. In 1995, corresponding pentamode metamaterials have independently been suggested by Sigmund^{32} and by Milton and Cherkaev^{33}. Bendsøe and Sigmund^{32} used numerical topology optimization^{25}; Milton and Cherkaev’s^{33} considerations were analytical and included anisotropic versions. Pentamode materials have a pseudoelasticity tensor with one nonzero eigenvalue that is of the pure pressure type. Polymeric realizations of pentamode metamaterials with ratios of bulk to shear modulus as large as B/G=1,000 have recently become available experimentally in microscopic^{34} and macroscopic^{35} form. The pentamode microstructure^{33,34} is illustrated in Fig. 1.
In regard to static compression, assuming an incompressible and not deformable hollow cylinder (that is, B_{1}/B_{0}≫1, B_{1}/B_{2}≫1, G_{1}/G_{0}≫1 and G_{1}/G_{2}≫1) as well as pentamode metamaterials for the surrounding (that is, B_{0}/G_{0}≫1) and for the shell (that is, B_{2}/G_{2}≫1), one gets the simple result^{3} for the relative bulk modulus of the shell (upon using in the nomenclature of ref. 3)
Numerical continuummechanics calculations (not depicted) indicate that this formula remains valid as long as the B_{2}/G_{2} ratio is larger than some 100. For example, choosing a thin shell with R_{2}/R_{1}=4/3, the formula leads to a relative bulk modulus of the shell of B_{2}/B_{0}=7/16≈0.44.
Intuitively, the operation with respect to compression can easily be understood by analogy to a onedimensional periodic chain of identical Hooke springs, each with finite spring constant D_{0}. One spring shall now be replaced by a very stiff one, that is, D_{1}→∞. To compensate for that, the two neighbouring springs with spring constant D_{2} must be made softer. To become undetectable with respect to compression, the effective spring constant of the three springs must be identical to three identical springs (each with spring constant D_{0}) in series. One immediately gets D_{2}/D_{0}=2/3. In three dimensions, however, the situation is more complex because a compression of an elastic solid in one direction is accompanied by an expansion/contraction in the two orthogonal directions. This makes the problem nontrivial.
Next, we map these parameters onto a pentamode microstructure (see Fig. 1). The bulk modulus can be tailored via the ratio of the small connection diameter d and the facecentered cubic (fcc) unit cell lattice constant a (see Fig. 1). For simplicity, we also fix the diameter D of the thick ends of the double cones throughout the entire structure as this quantity hardly enters into the mechanical properties^{34}. We have previously shown^{35} that the effective pentamode bulk (shear) modulus scales approximately according to B∝d/a and G∝(d/a)^{3}. We choose a cylinder rather than a sphere because this geometry allows us to observe the displacement close to the surface of the cloaking shell as directly as possible at the samples’ boundaries.
For any specific pushing direction in Fig. 1, e.g., along the vertical direction, by symmetry, the problem is equivalent to a half of it. This geometry somewhat resembles the 3D carpet cloak demonstrated experimentally in optics a few years ago^{11}. For simplicity, we use this carpet or mattress geometry (see Supplementary Fig. 1) in our experiments as it is clear by symmetry that the complete freespace structure is omnidirectional.
Experimental results
Altogether, our fabricated pentamode cloak microstructure depicted in Fig. 2 has the following design parameters: a=125 μm and D/a=8% (or D=10 μm) throughout, R_{i}/a=2 (not critical because the wall is rigid), R_{1}/a=3, R_{2}/a=4, d_{0}/a=5.3% (or d_{0}=6.6 μm) and d_{2}/a=2.4% (or d_{2}=3.0 μm). Within the surrounding, this leads to B_{0}/G_{0}=120, in the cloaking shell to B_{2}/G_{2}=908. The overall polymer microstructure shown in Fig. 2 contains 16 × 8 × 8=1,024 extended fcc pentamode unit cells and has a total volume of V=2 mm^{3}. Within measurement accuracy, the structure parameters a, D and d_{0} are identical to their design values. Owing to fabrication constraints, only the diameter d_{2} in Fig. 2f is slightly larger than its design value. The fabrication of each sample by fast galvoscanner dipin 3D direct laser writing (DLW) optical lithography has taken about 12 h (for details see Methods). We are not aware of any other approach capable of yielding structures with such small internal and such large overall dimensions.
In our quasistatic characterization experiments, a hard silicon stamp aligned parallel to the sample surface (see Fig. 3) pushes onto the sample via a motorized translational stage. The glass substrate at the bottom is fixed. We optically image the entire structure from the side, that is, in a plane perpendicular to the stamp, and record movies while varying the strain. To quantitatively analyse these movies, we use autocorrelation software to track individual points in the unit cells, delivering a spatial resolution beyond that of the individual camera pixels (for details see Methods). This analysis provides us with the displacement field or localstrain field directly from the experiment with good signaltonoise ratio, even at maximum vertical strains (=vertical component of the displacement vector divided by the overall sample height) as low as 1%.
Figure 3 shows the results for the homogeneous pentamodemetamaterial structure (the ‘reference’), the homogeneous pentamode structure with a rigid hollow cylinder (the ‘obstacle’) and the cloak including an additional pentamodemetamaterial shell. Data are shown for selected movie frames corresponding to the same strain of 2.8% at the top of the structures. Complete movies for the reference, the obstacle and the cloak are shown in Supplementary Movies 1, 2 and 3, respectively. There (see Supplementary Fig. 2), we also show data for a different choice of d_{2}/a=2.3% (instead of the above 2.4%) for a cloaking structure represented just like in Fig. 3. These data provide information on the sensitivity of fabrication errors. Figure 3 displays the vertical strain along three selected horizontal lines. Most relevant is cut #2 located just slightly above the cloaking shell (indicated by the dashed semicircle), where the displacement strongly varies for the obstacle structure: in the center, the modulus of the strain is nearly three times smaller than on the sides. This also means that one cannot simply cloak the obstacle by a homogeneous material, for example, by a block of some foam. In contrast, the displacement for the spatially inhomogeneous cloak structure is similar to that for a homogeneous environment without cylinder and without shell. Both displacement curves are just slightly bent towards the open sample sides. The other cut #1 further away from the rigid cylinder shows a similar behaviour, however, as to be expected, with much less pronounced differences between the homogeneous pentamode metamaterial and the obstacle in the first place, that is, there is less to be cloaked. This also means that the height of the structure is sufficiently large to avoid artifacts owing to finite sample height. Furthermore, from the fact that the above design process is based on an infinite depth of the cloak, whereas our fabricated structure is finite in depth as well, we estimate that artifacts due to measuring on the surface of the sample (actually about the first unit cell) rather than in the bulk are on the order of relative 10%. Likewise, the effects of the rigid cylinder are small for cut #3 as well. These overall observations mean that we have succeeded in approaching an ideal elastomechanical cloak.
Comparison with numerical calculations
Finally, we compare our experimental results with numerical continuummechanics calculations (for details see Methods) for the 3D pentamode microstructure to see how much possible experimental imperfections influence the results. Owing to memory and computation time constraints, we have not been able to perform calculations for the complete microstructure with 16 × 8 × 8 extended fcc unit cells. However, we have been able to obtain converged results for a reduced structure with 16 × 8 × 1 extended fcc unit cells and all of the above design parameters. In these calculations, we have employed fixed boundary conditions at the bottom substrate, have assumed zero horizontal component of the displacement vector at the top stamp and openboundary conditions on the left and on the right—just like in the experiments discussed above and shown in Figs 2 and 3. In the zdirection, we have used openboundary conditions on one side (like in the experiment) and slidingboundary conditions on the other. These mixed boundary conditions are an attempt to model the finite structure given tight numerical constraints. The numerical results in Fig. 4 are in good agreement with the experiments shown in Fig. 3. In particular, the vertical component of the displacement vector just outside the cloaking shell in cut #2 already approaches a horizontal straight line—just like for a homogeneous pentamodemetamaterial environment.
Discussion
In 2006, Milton et al.^{36} pointed out that the equations of solid elastostatics (and dynamics) are not forminvariant under arbitrary curvilinear coordinate transformations. In sharp contrast, the Maxwell equations, the acoustic wave equation and the equations of heat conduction and diffusion are forminvariant. Their^{36} mathematical finding suggests that perfect elastostatic cloaking may not be possible. Here, we have designed, fabricated and characterized an approximate elastostatic coreshell cloak based on 3D pentamode metamaterials. By using an autocorrelationbased image analysis to directly measure the displacementvector field, we find very good cloaking performance under uniaxial pushing conditions.
The fabrication of the underlying intricate cloaking microstructures with as many as 1,024 extended unit cells and with 2 mm^{3} overall volume has only become possible by virtue of 3D dipin galvoscannerbased DLW optical laser lithography. Our results raise hopes that further mechanical metamaterial architectures may become accessible experimentally in the near future.
Methods
Fabrication
For the fabrication of the mechanical cloak as well as the reference structures, we used the commercially available DLW system Photonic Professional GT (Nanoscribe GmbH, Germany). In this setup, a liquid photoresist (IPS resist, Nanoscribe GmbH) was polymerized via multiphoton absorption using a frequencydoubled Erbium fibre laser with a center wavelength of 780 nm and with a pulse duration of about 90 fs. The 3D exposure pattern was addressed by laser scanning using a set of galvomirrors and mechanical stages. The samples were prepared by dropcasting the negativetone photoresist on a glass cover slip (22 × 22 × 0.17 mm). To avoid depthdependent aberrations, the objective lens ( × 25, numerical aperture=0.8, Carl Zeiss) was directly dipped into the resist. Structural data were created in STL file format using the opensource software Blender and COMSOL Multiphysics. The software package Describe (Nanoscribe GmbH) was used to compile the CAD data into machine code. The scan raster was set to 0.5 μm laterally and 1 μm axially. The structure was laterally split into 8 scan fields of about 500 × 500 μm^{2} footprint each that were stitched together. The writing speed was set to 5 cm s^{−1}. After the DLW of the preprogrammed pattern, the exposed sample was developed for 20 min in mrDev 600 and acetone. The process was finished in a supercritical point dryer to avoid capillary forces during drying.
Sample characterization
The images used for the extraction of the strain field were recorded with a camera (Sony GigE Vision XCG5005CR) attached to a stereo microscope (Leica Mz 125 and a 0.5 × adapter from Leica mount to CMount). To reduce data, the images were then cropped to show only the structure and its close vicinity. For each picture taken, a linear stage induced a different predefined strain into the sample. The strain was successively increased in 50 steps towards the maximum value and afterwards decreased in 50 steps back to the initial value with a strain rate of 2% per minute. The glass substrate with the sample was attached to a goniometer and a micrometre stage to allow for positioning and aligning the sample with respect to the rest of the setup. The stamp was moved with a linear stage to which part of a silicon wafer with welldefined surface was attached.
The software used to extract the strain field was based on a freely available package^{37}. Here, selected markers with a set size of 15 × 15 image pixels were crosscorrelated with the images from the measurement. The initial marker positions were fixed in a square grid with a spacing of 15 pixels in both dimensions spanning the entire size of the sample. This resulted in 67 markers along the horizontal direction and about 35 in the vertical. The tracking algorithm was set to a precision of 1/1,000 pixel. After crosscorrelation, the position of each marker was known for each image. By subtracting the current marker positions from those of the reference frame, the displacement vector field was calculated for each image. Small movements of the glass substrate were corrected for. Movies of the reference, the obstacle and the cloak sample are given as Supplementary Movies 1–3. There, the full displacement vectors, multiplied by a factor of 4, are depicted. Additional colour coding of the modulus of the displacement vector helps to identify gradients. Colour coding and scales are identical for the three movies.
Numerical calculations
We used the commercial software package COMSOL Multiphysics to numerically solve the static equations for linear elasticity. This means that neither a nonlinearity of the constituent material nor of the structure was accounted for. The geometry with the design parameters described in the main text was built using the internal kernel of COMSOL Multiphysics. The mesh consisted of about 640,000 tetrahedral elements (in COMSOL nomenclature: maximum element size=0.2 × a, minimum element size=0.05 × a, maximum element growth rate=16, resolution of curvature=0.7 and resolution of narrow regions=0.4) corresponding to 3–4 × 10^{6} degrees of freedom. We used the direct solver MUMPS with a convergence tolerance of 10^{−3}. As constituent material, we set an isotropic polymer with Young’s modulus=1 GPa , Poisson’s ratio v=0.4 and mass density ρ=1,200 kg m^{−3}. Owing to the scalability of the underlying equations, Young’s modulus and mass density did not even enter into the final results. The Poisson’s ratio was not actually important^{28,29}. To deduce the displacements depicted in Fig. 4, we have tracked the connections with diameter d in the middle of the extended fcc unit cell with respect to the zdirection. Further data processing was done like in the experiment.
Additional information
How to cite this article: Bückmann, T. et al. An elastomechanical unfeelability cloak made of pentamode metamaterials. Nat. Commun. 5:4130 doi: 10.1038/ncomms5130 (2014).
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Acknowledgements
We thank Jonathan Müller (KIT) for discussions, Johannes Kaschke for taking the electron micrographs, Johann Westhauser for help in constructing the measurement setup and Tobias Frenzel for providing us with the drawing of the princess on the pea. We acknowledge support by the DFGCenter for Functional Nanostructures (CFN) at KIT through subprojects A1.04 and A1.05 and by the Karlsruhe School of Optics & Photonics (KSOP) at KIT. We also thank the Hector Fellow Academy for support.
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T.B. has performed the measurements and the data processing. M.T. has fabricated the samples. M.K. has helped in the simulations and discussions. R.S. has helped in the discussions. M.W. has led the effort and has written the first draft of the paper. All authors have contributed to the final version of the manuscript.
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Supplementary information
Supplementary Figures
Supplementary Figures 12 (PDF 597 kb)
Supplementary Movie 1
Displacement measurement on the "reference" sample. Optical movie taken while pushing onto the reference structure (see Fig. 2). The arrows indicate the local displacement vector multiplied with a factor of 4 for better visibility. The green data in Fig. 3 are taken from this movie. (AVI 6584 kb)
Supplementary Movie 2
Displacement measurement on the "obstacle" sample. Optical movie taken while pushing onto the obstacle structure (see Fig. 2). The arrows indicate the local displacement vector multiplied with a factor of 4 for better visibility. The blue data in Fig. 3 are taken from this movie. (AVI 6798 kb)
Supplementary Movie 3
Displacement measurement on the "cloak" sample. Optical movie taken while pushing onto the cloak structure (see Fig. 2). The arrows indicate the local displacement vector multiplied with a factor of 4 for better visibility. The red data in Fig. 3 are taken from this movie. (AVI 7035 kb)
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Bückmann, T., Thiel, M., Kadic, M. et al. An elastomechanical unfeelability cloak made of pentamode metamaterials. Nat Commun 5, 4130 (2014). https://doi.org/10.1038/ncomms5130
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DOI: https://doi.org/10.1038/ncomms5130
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