Mechanical Cloak Design by Direct Lattice Transformation

Mechanical cloak design by direct lattice transformation

Tiemo 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- equations, derived from Newton’s law and generalized Hooke’s ematical issues and solving complex boundary-value problems in law, do not pass this hurdle, neither in the dynamic nor in the physics for decades already. More recently, material-parameter static case (17). Mathematically, the continuum–mechanics transformations have also become an intuitive and powerful equations are form invariant (18) for the more general class of engineering tool for designing inhomogeneous and anisotropic Cosserat materials (19, 20), but little is known how to actu- material distributions that perform wanted functions, e.g., ally realize specific anisotropic Cosserat tensor distributions invisibility cloaking. A necessary mathematical prerequisite for experimentally by concrete microstructures. This situation this approach to work is that the underlying equations are form has hindered experimental realizations of cloaking in elasto- invariant with respect to general coordinate transformations. Un- mechanics with the exception of a few notable special cases (13, fortunately, this condition is not fulfilled in elastic–solid mechanics 21). Ref. 21 does not use coordinate transformations at all for materials that can be described by ordinary elasticity tensors. and is restricted to the limit of small shear moduli. Notably, Here, we introduce a different and simpler approach. We directly other facets of mechanical metamaterials have lately also transform the lattice points of a 2D discrete lattice composed of a attracted considerable attention (22–27). single constituent material, while keeping the properties of the elements connecting the lattice points the same. After showing that Results and Discussion – the approach works in various areas, we focus on elastic solid me- In this report, we introduce and exploit the direct lattice-trans- chanics. As a demanding example, we cloak a void in an effective formation approach, which is simpler than and conceptually elastic material with respect to static uniaxial compression. Corre- distinct from established material-parameter transformations. sponding numerical calculations and experiments on polymer struc- Instead from material parameters, we start from a discrete lat- tures made by 3D printing are presented. The cloaking quality is tice, which can be seen as an artificial material or metamaterial. quantified by comparing the average relative SD of the strain vec- As an example, we consider 2D hexagonal lattices (Fig. 1A), like tors outside of the cloaked void with respect to the homogeneous graphene, with lattice constant a (square lattices lead to similar reference lattice. Theory and experiment agree and exhibit very results). The situation is immediately clear for electric current good cloaking performance. conduction (15). Upon transforming the lattice points (black dots) and keeping the resistors connecting the lattice points the mechanical metamaterials | cloaking | coordinate transformations | same, the hole in the middle and the distortion around it cannot direct lattice transformation be detected from the outside because all resistors and all connections between them are the same (Fig. 1B). This means that aking advantage of spatial coordinate transformations has a we have built a cloak in a single simple step. We have previously Tlong tradition in science. For example, conformal mappings mentioned this possibility to provide an intuitive understanding have helped scientists to analytically solve complex boundary- of cloaking (15, 28). Importantly, we here translate the concep- value problems in hydromechanics (1) or nanooptics (2). The “ ” tual resistor networks (upper halves of Fig. 1) into concrete idea underlying transformation optics or material-parameter microstructured metamaterials (lower halves) composed of only transformations (3, 4) is distinct. Here, one starts from a (homogeneous) material distribution, performs a coordinate Significance transformation, and then equivalently maps this coordinate transformation onto an inhomogeneous and anisotropic material-parameter distribution. This together forms a first part. In a Calculating the behavior or function of a given material mi- second part, one needs to find a microstructure that approxi- crostructure in detail can be difficult, but it is conceptually mates the properties of the wanted material-parameter distri- straightforward. The inverse problem is much harder. Herein, bution. This second part is a difficult inverse problem that has no one searches for a microstructure that performs a specific targeted function. For example, one may want to guide a wave general explicit solution. However, the extensive literature on or a force field around some obstacle as though no obstacle artificial materials (or metamaterials) can often be used as a were there. Such function can be represented by a coordinate look-up table (5–8). transformation. Transformation optics maps arbitrary co- Material-parameter transformations have successfully been ordinate transformations onto concrete material-parameter applied experimentally in many different areas of physics, espe- – distributions. Unfortunately, mapping this distribution onto cially regarding the functionality of cloaking (9 15). In a general a microstructure poses another inverse problem. Here, we context including but not limited to optics, cloaking means that suggest an alternative approach that directly maps a co- one makes an arbitrary object that is different from its sur- ordinate transformation onto a concrete one-component rounding with respect to some physical observable appear just microstructure, and we apply the approach to the case of like the surrounding by adding a cloak around the object. static elastic–solid mechanics. A necessary mathematical prerequisite for material-parameter transformations to work is that the underlying equations must Author contributions: T.B. and M.W. designed research; T.B. performed research; T.B. be form invariant with respect to general coordinate trans- analyzed data; and T.B., M.K., R.S., and M.W. wrote the paper. formations. This form invariance is given for the Maxwell The authors declare no conflict of interest. equations (3, 4), the time-dependent heat conduction equa- This article is a PNAS Direct Submission. tion (15), stationary electric conduction (14) and diffusion 1To whom correspondence should be addressed. Email: [email protected]. (16), and for the acoustic wave equation for gases/liquids (12). This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. Unfortunately, for usual elastic solids, the continuum–mechanics 1073/pnas.1501240112/-/DCSupplemental.

4930–4934 | PNAS | April 21, 2015 | vol. 112 | no. 16 www.pnas.org/cgi/doi/10.1073/pnas.1501240112 Downloaded by guest on September 29, 2021 Fig. 1. Direct lattice-transformation approach. (A)A hexagonal lattice with lattice constant a composed of identical Ohmic resistors with resistance R.Thelumped resistors (upper half) can equivalently be replaced by double-trapezoidalpffiffiffi conductive elements (lower half) with length L = a= 3 and widths w and W as defined in the magnifying glass. (B) The lattice points (black dots) of the lattice in A are subject to a coordinate transformation. To keep the resistors R identical, while locally changing the length from L to L’ and fixing w, the width W is changed to W’ as indicated in the magnifying glass. One can proceed equivalently in heat conduction, particle diffusion, electrostatics, and magnetostatics. For elastic solids in mechanics, the resistors can be replaced by linear Hooke’s springs. The width W in the corresponding microstructure is again adjusted to W’ to keep the Hooke’s spring constant D identical while changing the length from L to L’ and fixing w.

one conductive material in vacuum/air. This is accomplished by inverse cross-section along the length. This immediately leads SCIENCES

replacing the resistors by double trapezoids. To fix their re- to the relation APPLIED PHYSICAL sistance R and changing their length L → L′, we adjust their L′ W′=w − 1 lnðW=wÞ W → W′ w A = . width while fixing (Fig. 1 and Fig. S1 ). The re- L W=w − 1 lnðW′=wÞ sistance of a wire is proportional to its length and inversely proportional to its cross-section. For fixed material yet varying In Fig. S1A, we depict W′=W versus L′=L for various fixed values cross-section along the wire axis, one needs to integrate the of w. For decreasing unit cell size, the width W′ can become so

Fig. 2. Calculated performance of a lattice-transformation cloak. Constant pressure is exerted from the left- and right- hand side in each case. We compare the response of a homogeneous reference lattice (first row), the same finite lattice 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 component of the strain at the lattice points in the middle column, and the y component of the strain at the lattice points 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 underneath. The corresponding average relative error Δ of the strain vectors outside the cloak (i.e., for radii r > r2) with respect to the reference case is given on the right-hand side. The hole in the reference leads to large strains at the inner radius as well as outside of the cloak. Both aspects are dramatically improved by the cloak, Δ decreases by a = factor of 34 from 738% topffiffiffi 22%. Parameters are: r1 30 mm, r2 = 60 mm, L = 4 mm, a = 3 × L, w = 0.4 mm, and W = 1mm.

Bückmann et al. PNAS | April 21, 2015 | vol. 112 | no. 16 | 4931 Downloaded by guest on September 29, 2021 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 the connections between lattice points are nearly bars. Compared with Fig. 2, the average relative error of the hole of Δ = 244% is smaller here because the reference 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%.

large that elements of adjacent cells start overlapping (not yet the conductivity, whereas one needs a rank-4 elasticity tensor con- case in Fig. 1B). We hence limit the width W′ to Wmax′ .Forthe taining two independent scalars, the bulk modulus, and the coordinate transformation of a point to a circle (3) used throughout shear modulus for describing an ordinary homogeneous iso- this work, i.e. (in polar coordinates and for r1 ≤ r ≤ r2), tropic elastic solid. Thus, how well the above qualitative anal- r − r ogy between electric conduction and mechanics actually works r → r′ = r + 2 1 r 1 r , needs to be explored. 2 Fig. 2 depicts calculated results for the untransformed me- the original lattice constant a is compressed in the radial direction chanical lattice, the same lattice with a circular hole with radius to aðr2 − r1Þ=r2. Leaving an extra margin of 10%, this leads to r1 in the middle, and for the transformed lattice designed as W′ r − r described above. Similar to the nonmechanical cases, we have max = 2 1 adjusted the width W → W′ to obtain the same Hooke’s spring a 0.9 r . 2 constant D while varying the length L → L′ (Fig. S1B). Regarding W′ =a W′ =W = For all of the below cases, this truncation concerns only a small max , we proceed as above, leading to max 3.12. The W′ =W = fraction of the double-trapezoidal elements; thus, it is a rea- smallest occurring ratio is min 0.35. sonable approximation. Upon exerting the same constant pressure (≠ constant dis- The construction procedure is strictly the same for electro- placement) of 33kPa on all of these lattices from the left- and statics, magnetostatics, static diffusion, and static heat conduc- right-hand-side boundaries, while imposing sliding boundary tion, because the underlying equations are mathematically conditions at the top and bottom, the hole leads to a very different equivalent. By qualitative analogy, we suspect that we can also behavior than the reference. First, the hole shrinks significantly as a result of the stress at its boundaries. Second, the strain field in proceed similarly in mechanics by replacing the resistors by lin- r > r ear Hooke’s springs (28), which are then again translated into a the surrounding for 2 is very different as well. B To quantify this behavior, we compute the dimensionless rel- microstructure (Fig. 1 and Fig. S1 ). This structure should allow Δ us to make a void in an effective material invisible in a me- ative error, , via chanical sense. Notably, simple core-shell cloaking geometries rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P 2 (21) fundamentally do not allow for doing that (29). They do 0 i ~ui − ~ui allow for the cloaking of stiff objects with respect to hydrostatic Δ = rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi . P 2 compression though (21). ~u0 We have used the phrase “qualitative analogy” because static i i electric current conduction and static mechanical elasticity are not equivalent mathematically. Precisely, one can describe Here, ~ui is the displacement vector field of either the hole or the 0 a homogeneous isotropic electric conductor by a single scalar cloak structure and ~ui is that of the reference. The index i in the

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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 have reduced 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. For the hole, we find Δ = 2,368%. In presence of the cloak (third row), the relative error decreases by a factor of 45 to 56%.

sums runs over all lattice sites outside of the cloak with outer from w = 0.4 mm to w = 0.8 mm (w = 0.05 mm) leads to B=G = 13 radius r2 (as any cloak is supposed to work with respect to its (B=G = 1300). For w = 0.05 mm and w = 0.4 mm, the initial shear outside, but not necessarily with respect to its inside). Obviously, modulus is rather small compared with the bulk modulus; whereas the average relative error Δ depends on the size of the surround- for w = 0.8 mm, the ratio of bulk to shear modulus of 13 is com- Δ ing. For an infinitely extended surrounding, tends toward 0. parable to that of ordinary materials. Note that the absolute We use the surrounding as shown in Fig. 2 and in the other cloaking quality as measured by the relative error Δ = 19% is best corresponding figures. Obviously, within the regime of linear w = Δ = w = Δ for 0.8 mm (Fig. 3), compared with 22% for 0.4 mm elasticity, the relative error does not depend on the absolute (Fig. 2) and Δ = 56% for w = 0.05 mm (Fig. 4). This behavior in- magnitude of the strain or stress at all. dicates that cloaking is not restricted to the limit of small shear For the void in Fig. 2, we find an error as large as Δ = 738%. moduli G, in sharp contrast to ref. 21. The improvement factor of Open boundary conditions (Fig. S2) and pure shearing of the w – structure (Fig. S3) lead to much smaller effects. In presence of cloak versus hole gets larger for smaller (cf. Figs. 2 4). We have also fabricated a polymer version of the structure the cloak, the average relative error is reduced by a factor of 34 Δ = shown in Fig. 2 by using a 3D printer (Fig. S8). Constant pressure to 22%, indicating an excellent performance of the cloak. ’ Correspondingly, the stress field shown in the left-hand-side is applied from the left- and right-hand side via Hooke s springs. column of Fig. 2 shows almost no stress at the inner boundary The displacement vectors (and hence the strain) of the lattice points are directly measured using an autocorrelation approach. at r = r1. Consequently, the inner elements can be eliminated without much change in performance (Fig. S4). Rotating the Details are given in ref. 30. The results are depicted in Fig. 5 in pushing direction by 90 degrees with respect to Fig. 2 leads to the same representation and on the same scales as in Fig. 2. The minor changes, too (Fig. S5). This fact, together with the sixfold agreement between Figs. 2 and 5 is very good, again confirming rotational symmetry of the lattice, means that we have investi- the validity of our approach. Minor deviations at the top and gated the cloaking performance for 0°, 30°, 60°, ..., 360°. bottom edges are due to imperfect realization of the targeted To further test the lattice-transformation approach, we have also sliding boundary conditions there. considered other radii r1, namely r2=r1 = 1.5 (Fig. S6)andr2=r1 = 4 In conclusion, we have presented an approach that directly (Fig. S7), while fixing r2=a. This again leads to excellent cloaking. maps coordinate transformations onto realizable mechanical mi- Our design approach fixes the Hooke’s spring constants but crostructures. This approach is applied to cloaking of a void. We does not independently control the shear force constants. It is thus find very good cloaking performance for different loading condi- interesting to compare the effective shear modulus G of the ho- tions, although cloaking will not be perfect. To be fair, however, mogeneous hexagonal reference lattice with its bulk modulus B. one should be aware that mathematically perfect cloaking is un- From independent phonon band structure calculations and for the avoidably connected with singular material parameters that just parameters of Fig. 2, we obtain B=G = 40. Changing the width w cannot be achieved in reality.

Bückmann et al. PNAS | April 21, 2015 | vol. 112 | no. 16 | 4933 Downloaded by guest on September 29, 2021 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 are dramatically reduced in presence of the cloak, i.e., the average relative error with respect top theffiffiffi reference case, Δ, decreases from 714% to 26% in good agreement with theory shown in Fig. 2. Parameters are: r1 = 30 mm, r2 = 60 mm, L = 4 mm, a = 3 × L, w = 0.4 mm, and W = 1 mm.

What are possible practical implications of the presented recipe could also be applied to construct support structures direct lattice-transformation approach? A tunnel underneath a for buildings or bridges. Although we have shown 2D exam- river is subject to significant stress peaks at the tunnel walls. ples, the extension to three dimensions appears straightfor- The cloak described in this report allows civil engineers to ward. It remains to be seen whether our approach can also be distribute the stress around the tunnel, while also separating extended to dynamic wave problems. the stress maximum from the tunnel walls. Using material- parameter transformations, such practical mechanical designs ACKNOWLEDGMENTS. We acknowledge support by the Karlsruhe School of have not been possible previously. Our simple-to-use design Optics & Photonics and the Hector Fellow Academy.

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