Mechanical cloak design by direct lattice transformationTiemo Bückmanna,1, Muamer Kadica, Robert Schittnya, and Martin Wegenera,b

aInstitute of Applied Physics and bInstitute of Nanotechnology, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany

Edited by David A. Weitz, Harvard University, Cambridge, MA, and approved March 16, 2015 (received for review January 20, 2015)

Spatial coordinate transformations have helped simplifying math-ematical issues and solving complex boundary-value problems inphysics for decades already. More recently, material-parametertransformations have also become an intuitive and powerfulengineering tool for designing inhomogeneous and anisotropicmaterial distributions that perform wanted functions, e.g.,invisibility cloaking. A necessary mathematical prerequisite forthis approach to work is that the underlying equations are forminvariant with respect to general coordinate transformations. Un-fortunately, this condition is not fulfilled in elastic–solid mechanicsfor materials that can be described by ordinary elasticity tensors.Here, we introduce a different and simpler approach. We directlytransform the lattice points of a 2D discrete lattice composed of asingle constituent material, while keeping the properties of the el-ements connecting the lattice points the same. After showing thatthe approach works in various areas, we focus on elastic–solid me-chanics. As a demanding example, we cloak a void in an effectiveelastic material with respect to static uniaxial compression. Corre-sponding numerical calculations and experiments on polymer struc-tures made by 3D printing are presented. The cloaking quality isquantified by comparing the average relative SD of the strain vec-tors outside of the cloaked void with respect to the homogeneousreference lattice. Theory and experiment agree and exhibit verygood cloaking performance.

mechanical metamaterials | cloaking | coordinate transformations |direct lattice transformation

Taking advantage of spatial coordinate transformations has along tradition in science. For example, conformal mappings

have helped scientists to analytically solve complex boundary-value problems in hydromechanics (1) or nanooptics (2). Theidea underlying “transformation optics” or material-parametertransformations (3, 4) is distinct. Here, one starts from a(homogeneous) material distribution, performs a coordinatetransformation, and then equivalently maps this coordinatetransformation onto an inhomogeneous and anisotropic mate-rial-parameter distribution. This together forms a first part. In asecond part, one needs to find a microstructure that approxi-mates the properties of the wanted material-parameter distri-bution. This second part is a difficult inverse problem that has nogeneral explicit solution. However, the extensive literature onartificial materials (or metamaterials) can often be used as alook-up table (5–8).Material-parameter transformations have successfully been

applied experimentally in many different areas of physics, espe-cially regarding the functionality of cloaking (9–15). In a generalcontext including but not limited to optics, cloaking means thatone makes an arbitrary object that is different from its sur-rounding with respect to some physical observable appear justlike the surrounding by adding a cloak around the object.A necessary mathematical prerequisite for material-parameter

transformations to work is that the underlying equations mustbe form invariant with respect to general coordinate trans-formations. This form invariance is given for the Maxwellequations (3, 4), the time-dependent heat conduction equa-tion (15), stationary electric conduction (14) and diffusion(16), and for the acoustic wave equation for gases/liquids (12).Unfortunately, for usual elastic solids, the continuum–mechanics

equations, derived from Newton’s law and generalized Hooke’slaw, do not pass this hurdle, neither in the dynamic nor in thestatic case (17). Mathematically, the continuum–mechanicsequations are form invariant (18) for the more general class ofCosserat materials (19, 20), but little is known how to actu-ally realize specific anisotropic Cosserat tensor distributionsexperimentally by concrete microstructures. This situationhas hindered experimental realizations of cloaking in elasto-mechanics with the exception of a few notable special cases (13,21). Ref. 21 does not use coordinate transformations at alland is restricted to the limit of small shear moduli. Notably,other facets of mechanical metamaterials have lately alsoattracted considerable attention (22–27).

Results and DiscussionIn this report, we introduce and exploit the direct lattice-trans-formation approach, which is simpler than and conceptuallydistinct from established material-parameter transformations.Instead from material parameters, we start from a discrete lat-tice, which can be seen as an artificial material or metamaterial.As an example, we consider 2D hexagonal lattices (Fig. 1A), likegraphene, with lattice constant a (square lattices lead to similarresults). The situation is immediately clear for electric currentconduction (15). Upon transforming the lattice points (blackdots) and keeping the resistors connecting the lattice points thesame, the hole in the middle and the distortion around it cannotbe detected from the outside because all resistors and all con-nections between them are the same (Fig. 1B). This means thatwe have built a cloak in a single simple step. We have previouslymentioned this possibility to provide an intuitive understandingof cloaking (15, 28). Importantly, we here translate the concep-tual resistor networks (upper halves of Fig. 1) into concretemicrostructured metamaterials (lower halves) composed of only

Significance

Calculating the behavior or function of a given material mi-crostructure in detail can be difficult, but it is conceptuallystraightforward. The inverse problem is much harder. Herein,one searches for a microstructure that performs a specific tar-geted function. For example, one may want to guide a waveor a force field around some obstacle as though no obstaclewere there. Such function can be represented by a coordinatetransformation. Transformation optics maps arbitrary co-ordinate transformations onto concrete material-parameterdistributions. Unfortunately, mapping this distribution ontoa microstructure poses another inverse problem. Here, wesuggest an alternative approach that directly maps a co-ordinate transformation onto a concrete one-componentmicrostructure, and we apply the approach to the case ofstatic elastic–solid mechanics.

Author contributions: T.B. and M.W. designed research; T.B. performed research; T.B.analyzed data; and T.B., M.K., R.S., and M.W. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. Email: tiemo.bueckmann@kit.edu.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1501240112/-/DCSupplemental.

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one conductive material in vacuum/air. This is accomplished byreplacing the resistors by double trapezoids. To fix their re-sistance R and changing their length L→L′, we adjust theirwidth W →W ′ while fixing w (Fig. 1 and Fig. S1A). The re-sistance of a wire is proportional to its length and inverselyproportional to its cross-section. For fixed material yet varyingcross-section along the wire axis, one needs to integrate the

inverse cross-section along the length. This immediately leadsto the relation

L′L=W ′=w− 1W=w− 1

lnðW=wÞlnðW ′=wÞ.

In Fig. S1A, we depict W ′=W versus L′=L for various fixed valuesof w. For decreasing unit cell size, the width W ′ can become so

Fig. 1. Direct lattice-transformation approach. (A) Ahexagonal lattice with lattice constant a composed ofidentical Ohmic resistors with resistance R. The lumpedresistors (upper half) can equivalently be replaced bydouble-trapezoidal conductive elements (lower half)with length L= a=

ffiffiffi3

pand widths w and W as defined

in the magnifying glass. (B) The lattice points (blackdots) of the lattice in A are subject to a coordinatetransformation. To keep the resistors R identical, whilelocally changing the length from L to L’ and fixing w,the width W is changed to W ’ as indicated in themagnifying glass. One can proceed equivalently in heatconduction, particle diffusion, electrostatics, and mag-netostatics. For elastic solids in mechanics, the resistorscan be replaced by linear Hooke’s springs. The widthWin the corresponding microstructure is again adjustedto W ’ to keep the Hooke’s spring constant D identicalwhile changing the length from L to L’ and fixing w.

Fig. 2. Calculated performance of a lattice-transformationcloak. Constant pressure is exerted from the left- and right-hand side in each case. We compare the response of ahomogeneous reference lattice (first row), the same finitelattice with a hole of radius r1 in the middle (second row),and the elastic cloak with inner radius r1 and outer radius r2(third row) designed by direct lattice transformation. The(von Mises) stress is shown in the left column, the x com-ponent of the strain at the lattice points in the middlecolumn, and the y component of the strain at the latticepoints in the right column. We use a highly saturated false-color representation to exhibit all data on the same scale.The metamaterial structures (cf. Fig. 1) are shown un-derneath. The corresponding average relative error Δ ofthe strain vectors outside the cloak (i.e., for radii r > r2) withrespect to the reference case is given on the right-handside. The hole in the reference leads to large strains at theinner radius as well as outside of the cloak. Both aspectsare dramatically improved by the cloak, Δ decreases by afactor of 34 from 738% to 22%. Parameters are: r1 = 30mm,r2 = 60mm, L= 4 mm, a=

ffiffiffi3

p× L, w = 0.4mm, and W = 1mm.

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large that elements of adjacent cells start overlapping (not yet thecase in Fig. 1B). We hence limit the width W ′ to Wmax′ . For thecoordinate transformation of a point to a circle (3) used throughoutthis work, i.e. (in polar coordinates and for r1 ≤ r≤ r2),

r→ r′= r1 +r2 − r1r2

r,

the original lattice constant a is compressed in the radial directionto aðr2 − r1Þ=r2. Leaving an extra margin of 10%, this leads to

Wmax′a

= 0.9r2 − r1r2

.

For all of the below cases, this truncation concerns only a smallfraction of the double-trapezoidal elements; thus, it is a rea-sonable approximation.The construction procedure is strictly the same for electro-

statics, magnetostatics, static diffusion, and static heat conduc-tion, because the underlying equations are mathematicallyequivalent. By qualitative analogy, we suspect that we can alsoproceed similarly in mechanics by replacing the resistors by lin-ear Hooke’s springs (28), which are then again translated into amicrostructure (Fig. 1 and Fig. S1B). This structure should allowus to make a void in an effective material invisible in a me-chanical sense. Notably, simple core-shell cloaking geometries(21) fundamentally do not allow for doing that (29). They doallow for the cloaking of stiff objects with respect to hydrostaticcompression though (21).We have used the phrase “qualitative analogy” because static

electric current conduction and static mechanical elasticityare not equivalent mathematically. Precisely, one can describea homogeneous isotropic electric conductor by a single scalar

conductivity, whereas one needs a rank-4 elasticity tensor con-taining two independent scalars, the bulk modulus, and theshear modulus for describing an ordinary homogeneous iso-tropic elastic solid. Thus, how well the above qualitative anal-ogy between electric conduction and mechanics actually worksneeds to be explored.Fig. 2 depicts calculated results for the untransformed me-

chanical lattice, the same lattice with a circular hole with radiusr1 in the middle, and for the transformed lattice designed asdescribed above. Similar to the nonmechanical cases, we haveadjusted the width W →W ′ to obtain the same Hooke’s springconstant Dwhile varying the length L→L′ (Fig. S1B). RegardingWmax′ =a, we proceed as above, leading to Wmax′ =W = 3.12. Thesmallest occurring ratio is Wmin′ =W = 0.35.Upon exerting the same constant pressure (≠ constant dis-

placement) of 33kPa on all of these lattices from the left- andright-hand-side boundaries, while imposing sliding boundaryconditions at the top and bottom, the hole leads to a very differentbehavior than the reference. First, the hole shrinks significantly asa result of the stress at its boundaries. Second, the strain field inthe surrounding for r> r2 is very different as well.To quantify this behavior, we compute the dimensionless rel-

ative error, Δ, via

Δ=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPi

�~ui − ~u0i

�2rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiP

i

�~u0i

�2r .

Here,~ui is the displacement vector field of either the hole or thecloak structure and~u0i is that of the reference. The index i in the

Fig. 3. Calculated performance of the lattice-transformation cloak as in Fig. 2, but for w = 0.8 mm. This value is nearly as large as W = 1.0 mm, such that theconnections between lattice points are nearly bars. Compared with Fig. 2, the average relative error of the hole of Δ= 244% is smaller here because thereference lattice (first row) is stiffer. Correspondingly, the strains are generally smaller here. The cloak (third row) reduces Δ by a factor of 13 from 244% to 19%.

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sums runs over all lattice sites outside of the cloak with outerradius r2 (as any cloak is supposed to work with respect to itsoutside, but not necessarily with respect to its inside). Obviously,the average relative error Δ depends on the size of the surround-ing. For an infinitely extended surrounding, Δ tends toward 0.We use the surrounding as shown in Fig. 2 and in the othercorresponding figures. Obviously, within the regime of linearelasticity, the relative error Δ does not depend on the absolutemagnitude of the strain or stress at all.For the void in Fig. 2, we find an error as large as Δ= 738%.

Open boundary conditions (Fig. S2) and pure shearing of thestructure (Fig. S3) lead to much smaller effects. In presence ofthe cloak, the average relative error is reduced by a factor of 34to Δ= 22%, indicating an excellent performance of the cloak.Correspondingly, the stress field shown in the left-hand-sidecolumn of Fig. 2 shows almost no stress at the inner boundaryat r= r1. Consequently, the inner elements can be eliminatedwithout much change in performance (Fig. S4). Rotating thepushing direction by 90 degrees with respect to Fig. 2 leads tominor changes, too (Fig. S5). This fact, together with the sixfoldrotational symmetry of the lattice, means that we have investi-gated the cloaking performance for 0°, 30°, 60°, . . . , 360°.To further test the lattice-transformation approach, we have also

considered other radii r1, namely r2=r1 = 1.5 (Fig. S6) and r2=r1 = 4(Fig. S7), while fixing r2=a. This again leads to excellent cloaking.Our design approach fixes the Hooke’s spring constants but

does not independently control the shear force constants. It is thusinteresting to compare the effective shear modulus G of the ho-mogeneous hexagonal reference lattice with its bulk modulus B.From independent phonon band structure calculations and for theparameters of Fig. 2, we obtain B=G= 40. Changing the width w

from w= 0.4 mm to w= 0.8 mm (w= 0.05 mm) leads to B=G= 13(B=G= 1300). For w= 0.05 mm and w= 0.4 mm, the initial shearmodulus is rather small compared with the bulk modulus; whereasfor w= 0.8 mm, the ratio of bulk to shear modulus of 13 is com-parable to that of ordinary materials. Note that the absolutecloaking quality as measured by the relative error Δ= 19% is bestfor w= 0.8 mm (Fig. 3), compared with Δ= 22% for w= 0.4 mm(Fig. 2) and Δ= 56% for w= 0.05 mm (Fig. 4). This behavior in-dicates that cloaking is not restricted to the limit of small shearmoduli G, in sharp contrast to ref. 21. The improvement factor ofcloak versus hole gets larger for smaller w (cf. Figs. 2–4).We have also fabricated a polymer version of the structure

shown in Fig. 2 by using a 3D printer (Fig. S8). Constant pressureis applied from the left- and right-hand side via Hooke’s springs.The displacement vectors (and hence the strain) of the latticepoints are directly measured using an autocorrelation approach.Details are given in ref. 30. The results are depicted in Fig. 5 inthe same representation and on the same scales as in Fig. 2. Theagreement between Figs. 2 and 5 is very good, again confirmingthe validity of our approach. Minor deviations at the top andbottom edges are due to imperfect realization of the targetedsliding boundary conditions there.In conclusion, we have presented an approach that directly

maps coordinate transformations onto realizable mechanical mi-crostructures. This approach is applied to cloaking of a void. Wefind very good cloaking performance for different loading condi-tions, although cloaking will not be perfect. To be fair, however,one should be aware that mathematically perfect cloaking is un-avoidably connected with singular material parameters that justcannot be achieved in reality.

Fig. 4. Calculated performance of the lattice-transformation cloak as in Fig. 2, but for w = 0.05 mm. This lattice is much more compliant, thus we havereduced the pressure (hence the stress) by factor 1,000 to obtain reasonably large strains. This reduction does not affect the average relative errors Δ at all. Forthe hole, we find Δ= 2,368%. In presence of the cloak (third row), the relative error decreases by a factor of 45 to 56%.

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What are possible practical implications of the presenteddirect lattice-transformation approach? A tunnel underneath ariver is subject to significant stress peaks at the tunnel walls.The cloak described in this report allows civil engineers todistribute the stress around the tunnel, while also separatingthe stress maximum from the tunnel walls. Using material-parameter transformations, such practical mechanical designshave not been possible previously. Our simple-to-use design

recipe could also be applied to construct support structuresfor buildings or bridges. Although we have shown 2D exam-ples, the extension to three dimensions appears straightfor-ward. It remains to be seen whether our approach can also beextended to dynamic wave problems.

ACKNOWLEDGMENTS. We acknowledge support by the Karlsruhe School ofOptics & Photonics and the Hector Fellow Academy.

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Fig. 5. Measured performance of a lattice-transformation cloak. Same as Fig. 2, but measured directly on polymer structures fabricated by a 3D printer.Photographs of the structures are shown in the left-hand-side column. Again, the large distortions introduced by the hole in the homogeneous lattice aredramatically reduced in presence of the cloak, i.e., the average relative error with respect to the reference case, Δ, decreases from 714% to 26% in goodagreement with theory shown in Fig. 2. Parameters are: r1 =30 mm, r2 = 60 mm, L= 4 mm, a=

ffiffiffi3

p× L, w = 0.4 mm, and W = 1 mm.

4934 | www.pnas.org/cgi/doi/10.1073/pnas.1501240112 Bückmann et al.