Mechanical cloak design by direct lattice transformation

Significance

Calculating the behavior or function of a given material microstructure in detail can be difficult, but it is conceptually straightforward. The inverse problem is much harder. Herein, one searches for a microstructure that performs a specific targeted function. For example, one may want to guide a wave or a force field around some obstacle as though no obstacle were there. Such function can be represented by a coordinate transformation. Transformation optics maps arbitrary coordinate transformations onto concrete material-parameter distributions. Unfortunately, mapping this distribution onto a microstructure poses another inverse problem. Here, we suggest an alternative approach that directly maps a coordinate transformation onto a concrete one-component microstructure, and we apply the approach to the case of static elastic–solid mechanics.

Abstract

Spatial coordinate transformations have helped simplifying mathematical issues and solving complex boundary-value problems in physics for decades already. More recently, material-parameter transformations have also become an intuitive and powerful engineering tool for designing inhomogeneous and anisotropic material distributions that perform wanted functions, e.g., invisibility cloaking. A necessary mathematical prerequisite for this approach to work is that the underlying equations are form invariant with respect to general coordinate transformations. Unfortunately, this condition is not fulfilled in elastic–solid mechanics for materials that can be described by ordinary elasticity tensors. Here, we introduce a different and simpler approach. We directly transform the lattice points of a 2D discrete lattice composed of a single constituent material, while keeping the properties of the elements connecting the lattice points the same. After showing that the approach works in various areas, we focus on elastic–solid mechanics. As a demanding example, we cloak a void in an effective elastic material with respect to static uniaxial compression. Corresponding numerical calculations and experiments on polymer structures made by 3D printing are presented. The cloaking quality is quantified by comparing the average relative SD of the strain vectors outside of the cloaked void with respect to the homogeneous reference lattice. Theory and experiment agree and exhibit very good cloaking performance.

Taking advantage of spatial coordinate transformations has a long tradition in science. For example, conformal mappings have helped scientists to analytically solve complex boundary-value problems in hydromechanics (1) or nanooptics (2). The idea underlying “transformation optics” or material-parameter transformations (3, 4) is distinct. Here, one starts from a (homogeneous) material distribution, performs a coordinate transformation, and then equivalently maps this coordinate transformation onto an inhomogeneous and anisotropic material-parameter distribution. This together forms a first part. In a second part, one needs to find a microstructure that approximates the properties of the wanted material-parameter distribution. This second part is a difficult inverse problem that has no general explicit solution. However, the extensive literature on artificial materials (or metamaterials) can often be used as a look-up table (5–8).

Material-parameter transformations have successfully been applied experimentally in many different areas of physics, especially regarding the functionality of cloaking (9–15). In a general context including but not limited to optics, cloaking means that one makes an arbitrary object that is different from its surrounding with respect to some physical observable appear just like the surrounding by adding a cloak around the object.

A necessary mathematical prerequisite for material-parameter transformations to work is that the underlying equations must be form invariant with respect to general coordinate transformations. This form invariance is given for the Maxwell equations (3, 4), the time-dependent heat conduction equation (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’s law, do not pass this hurdle, neither in the dynamic nor in the static case (17). Mathematically, the continuum–mechanics equations are form invariant (18) for the more general class of Cosserat materials (19, 20), but little is known how to actually realize specific anisotropic Cosserat tensor distributions experimentally by concrete microstructures. This situation has hindered experimental realizations of cloaking in elastomechanics with the exception of a few notable special cases (13, 21). Ref. 21 does not use coordinate transformations at all and is restricted to the limit of small shear moduli. Notably, other facets of mechanical metamaterials have lately also attracted considerable attention (22–27).

Results and Discussion

In this report, we introduce and exploit the direct lattice-transformation approach, which is simpler than and conceptually distinct from established material-parameter transformations. Instead from material parameters, we start from a discrete lattice, which can be seen as an artificial material or metamaterial. As an example, we consider 2D hexagonal lattices (Fig. 1A), like graphene, with lattice constant a (square lattices lead to similar results). The situation is immediately clear for electric current conduction (15). Upon transforming the lattice points (black dots) and keeping the resistors connecting the lattice points the same, the hole in the middle and the distortion around it cannot be detected from the outside because all resistors and all connections between them are the same (Fig. 1B). This means that we have built a cloak in a single simple step. We have previously mentioned this possibility to provide an intuitive understanding of cloaking (15, 28). Importantly, we here translate the conceptual resistor networks (upper halves of Fig. 1) into concrete microstructured metamaterials (lower halves) composed of only one conductive material in vacuum/air. This is accomplished by replacing the resistors by double trapezoids. To fix their resistance $R$ and changing their length $L\to L\prime$, we adjust their width $W\to W\prime$ while fixing $w$ (Fig. 1 and Fig. S1A). The resistance of a wire is proportional to its length and inversely proportional to its cross-section. For fixed material yet varying cross-section along the wire axis, one needs to integrate the inverse cross-section along the length. This immediately leads to the relation

$$\frac{L\prime }{L}=\frac{W\prime /w-1}{W/w-1}\frac{\ln \left(W/w\right)}{\ln \left(W\prime /w\right)}.$$

In Fig. S1A, we depict $W^{\prime }/W$ versus $L^{\prime }/L$ for various fixed values of $w$. For decreasing unit cell size, the width $W^{\prime }$ can become so large that elements of adjacent cells start overlapping (not yet the case in Fig. 1B). We hence limit the width $W^{\prime }$ to $W_{\text{max}}^{\prime }$. For the coordinate transformation of a point to a circle (3) used throughout this work, i.e. (in polar coordinates and for $r_1\le r\le r_2$),

$$r\to r\prime =r_1+\frac{r_2-r_1}{r_2}r,$$

Fig. 1.

Direct lattice-transformation approach. (A) A hexagonal lattice with lattice constant $a$ composed of identical Ohmic resistors with resistance $R$. The lumped resistors (upper half) can equivalently be replaced by double-trapezoidal conductive elements (lower half) with length $L=a/\sqrt{3}$ 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 $L\text{'}$ and fixing $w$, the width $W$ is changed to $W\text{'}$ 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 $W\text{'}$ to keep the Hooke’s spring constant $D$ identical while changing the length from $L$ to $L\text{'}$ and fixing $w$.

the original lattice constant $a$ is compressed in the radial direction to $a(r_2-r_1)/r_2$. Leaving an extra margin of $10\%$, this leads to

$$\frac{W_{\text{max}}^{\prime }}{a}=0.9\frac{r_2-r_1}{r_2}.$$

For all of the below cases, this truncation concerns only a small fraction of the double-trapezoidal elements; thus, it is a reasonable approximation.

The construction procedure is strictly the same for electrostatics, magnetostatics, static diffusion, and static heat conduction, because the underlying equations are mathematically equivalent. By qualitative analogy, we suspect that we can also proceed similarly in mechanics by replacing the resistors by linear Hooke’s springs (28), which are then again translated into a microstructure (Fig. 1 and Fig. S1B). This structure should allow us to make a void in an effective material invisible in a mechanical sense. Notably, simple core-shell cloaking geometries (21) fundamentally do not allow for doing that (29). They do allow for the cloaking of stiff objects with respect to hydrostatic compression though (21).

We have used the phrase “qualitative analogy” because static electric current conduction and static mechanical elasticity are not equivalent mathematically. Precisely, one can describe a homogeneous isotropic electric conductor by a single scalar conductivity, whereas one needs a rank-4 elasticity tensor containing two independent scalars, the bulk modulus, and the shear modulus for describing an ordinary homogeneous isotropic elastic solid. Thus, how well the above qualitative analogy between electric conduction and mechanics actually works needs to be explored.

Fig. 2 depicts calculated results for the untransformed mechanical lattice, the same lattice with a circular hole with radius $r_1$ in the middle, and for the transformed lattice designed as described above. Similar to the nonmechanical cases, we have adjusted the width $W\to W\prime$ to obtain the same Hooke’s spring constant $D$ while varying the length $L\to L\prime$ (Fig. S1B). Regarding $W_{\text{max}}^{\prime }/a$, we proceed as above, leading to $W_{\text{max}}^{\prime }/W=3.12$. The smallest occurring ratio is $W_{\text{min}}^{\prime }/W=0.35$.

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 $r_1$ in the middle (second row), and the elastic cloak with inner radius $r_1$ and outer radius $r_2$ (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 $\mathrm{\Delta }$ of the strain vectors outside the cloak (i.e., for radii $r>r_2$) 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, $\mathrm{\Delta }$ decreases by a factor of 34 from $738\%$ to $22\%$. Parameters are: $r_1=30\hspace{0.17em}\text{mm}$, $r_2=60\hspace{0.17em}\text{mm}$, $L=4\hspace{0.17em}\text{mm}$, $a=\sqrt{3}\times L$, $w=0.4\hspace{0.17em}\text{mm}$, and $W=1\hspace{0.17em}\text{mm}$.

Upon exerting the same constant pressure (≠ constant displacement) of $33\text{kPa}$ on all of these lattices from the left- and right-hand-side boundaries, while imposing sliding boundary conditions at the top and bottom, the hole leads to a very different behavior than the reference. First, the hole shrinks significantly as a result of the stress at its boundaries. Second, the strain field in the surrounding for $r>r_2$ is very different as well.

To quantify this behavior, we compute the dimensionless relative error, $\mathrm{\Delta }$, via

$$\mathrm{\Delta }=\frac{\sqrt{{{\displaystyle \sum }}_i{\left({\overrightarrow{u}}_i-{\overrightarrow{u}}_i^0\right)}^2}}{\sqrt{{{\displaystyle \sum }}_i{\left({\overrightarrow{u}}_i^0\right)}^2}}.$$

Here, ${\overrightarrow{u}}_i$ is the displacement vector field of either the hole or the cloak structure and ${\overrightarrow{u}}_i^0$ is that of the reference. The index $i$ in the sums runs over all lattice sites outside of the cloak with outer radius $r_2$ (as any cloak is supposed to work with respect to its outside, but not necessarily with respect to its inside). Obviously, the average relative error $\mathrm{\Delta }$ depends on the size of the surrounding. For an infinitely extended surrounding, $\mathrm{\Delta }$ tends toward 0. We use the surrounding as shown in Fig. 2 and in the other corresponding figures. Obviously, within the regime of linear elasticity, the relative error $\mathrm{\Delta }$ does not depend on the absolute magnitude of the strain or stress at all.

For the void in Fig. 2, we find an error as large as $\mathrm{\Delta }=738\%$. Open boundary conditions (Fig. S2) and pure shearing of the structure (Fig. S3) lead to much smaller effects. In presence of the cloak, the average relative error is reduced by a factor of 34 to $\mathrm{\Delta }=22\%$, indicating an excellent performance of the cloak. Correspondingly, the stress field shown in the left-hand-side column of Fig. 2 shows almost no stress at the inner boundary at $r=r_1$. Consequently, the inner elements can be eliminated without much change in performance (Fig. S4). Rotating the pushing direction by 90 degrees with respect to Fig. 2 leads to minor changes, too (Fig. S5). This fact, together with the sixfold rotational symmetry of the lattice, means that we have investigated the cloaking performance for $0°,30°,60°,\dots ,360°$.

To further test the lattice-transformation approach, we have also considered other radii $r_1$, namely $r_2/r_1=1.5$ (Fig. S6) and $r_2/r_1=4$ (Fig. S7), while fixing $r_2/a$. 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 thus interesting to compare the effective shear modulus $G$ of the homogeneous hexagonal reference lattice with its bulk modulus $B$. From independent phonon band structure calculations and for the parameters of Fig. 2, we obtain $B/G=40$. Changing the width $w$ from $w=0.4\hspace{0.17em}\text{mm}$ to $w=0.8\hspace{0.17em}\text{mm}$ ($w=0.05\hspace{0.17em}\text{mm}$) leads to $B/G=13$ ($B/G=1300$). For $w=0.05\hspace{0.17em}\text{mm}$ and $w=0.4\hspace{0.17em}\text{mm}$, the initial shear modulus is rather small compared with the bulk modulus; whereas for $w=0.8\hspace{0.17em}\text{mm}$, the ratio of bulk to shear modulus of $13$ is comparable to that of ordinary materials. Note that the absolute cloaking quality as measured by the relative error $\mathrm{\Delta }=19\%$ is best for $w=0.8\hspace{0.17em}\text{mm}$ (Fig. 3), compared with $\mathrm{\Delta }=22\%$ for $w=0.4\hspace{0.17em}\text{mm}$ (Fig. 2) and $\mathrm{\Delta }=56\%$ for $w=0.05\hspace{0.17em}\text{mm}$ (Fig. 4). This behavior indicates that cloaking is not restricted to the limit of small shear moduli $G$, in sharp contrast to ref. 21. The improvement factor of cloak versus hole gets larger for smaller $w$ (cf. Figs. 2–4).

Fig. 3.

Calculated performance of the lattice-transformation cloak as in Fig. 2, but for $w=0.8\hspace{0.17em}\text{mm}$. This value is nearly as large as $W=1.0\hspace{0.17em}\text{mm}$, such that the connections between lattice points are nearly bars. Compared with Fig. 2, the average relative error of the hole of $\mathrm{\Delta }=244\%$ is smaller here because the reference lattice (first row) is stiffer. Correspondingly, the strains are generally smaller here. The cloak (third row) reduces $\mathrm{\Delta }$ by a factor of $13$ from $244\%$ to $19\%$.

Fig. 4.

Calculated performance of the lattice-transformation cloak as in Fig. 2, but for $w=0.05\hspace{0.17em}\text{mm}$. This lattice is much more compliant, thus we have reduced the pressure (hence the stress) by factor $1\text{,}000$ to obtain reasonably large strains. This reduction does not affect the average relative errors $\mathrm{\Delta }$ at all. For the hole, we find $\mathrm{\Delta }=2\text{,}368\%$. In presence of the cloak (third row), the relative error decreases by a factor of 45 to $56\%$.

We have also fabricated a polymer version of the structure shown in Fig. 2 by using a 3D printer (Fig. S8). Constant pressure is applied from the left- and right-hand side via Hooke’s springs. The displacement vectors (and hence the strain) of the lattice points are directly measured using an autocorrelation approach. Details are given in ref. 30. The results are depicted in Fig. 5 in the same representation and on the same scales as in Fig. 2. The agreement between Figs. 2 and 5 is very good, again confirming the validity of our approach. Minor deviations at the top and bottom edges are due to imperfect realization of the targeted sliding boundary conditions there.

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 to the reference case, $\mathrm{\Delta }$, decreases from $714\%$ to $26\%$ in good agreement with theory shown in Fig. 2. Parameters are: $r_1=30\hspace{0.17em}\text{mm}$, $r_2=60\hspace{0.17em}\text{mm}$, $L=4\hspace{0.17em}\text{mm}$, $a=\sqrt{3}\times L$, $w=0.4\hspace{0.17em}\text{mm}$, and $W=1\hspace{0.17em}\text{mm}$.

In conclusion, we have presented an approach that directly maps coordinate transformations onto realizable mechanical microstructures. This approach is applied to cloaking of a void. We find very good cloaking performance for different loading conditions, although cloaking will not be perfect. To be fair, however, one should be aware that mathematically perfect cloaking is unavoidably connected with singular material parameters that just cannot be achieved in reality.

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