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Pentamode Materials: From Underwater Cloaking to Cushioned Sneakers

By Andrej Cherkaev, Muamer Kadic, Graeme W. Milton, and Martin Wegener

The fundamental fields of classic three-dimensional linear elasticity are the strain \(\bf{\epsilon}(\bf{x})\)—the symmetrized gradient of the displacement field \(\bf{u}(x)\), \(\epsilon = [\triangledown \bf{u} + (\triangledown u)\)\(^{T}]/2\)—and the stress \(\bf{\sigma}(x)\). The stress \(\bf{\sigma}(x)\) is defined so that removing a small tetrahedron of material at \(\bf{x}\) and loading this cavity’s four faces by surface traction forces \(\bf{\sigma} \cdot n\)\(_i\)—where \(\bf{n}\)\(_i\), \(i=1,2,3,4\) are the normals to the cavity walls—means that the surrounding displacement field is essentially unchanged by the cavity’s insertion. One can represent both \(\bf{\epsilon}(\bf{x})\) and \(\bf{\sigma}(x)\) in cartesian coordinates with symmetric \(3\times 3\) matrix-valued fields, also labeled \(\bf{\epsilon}(\bf{x})\) and \(\bf{\sigma}(x)\). In a homogeneous medium, these fields are linked by the constitutive law \(\bf{\sigma}(x)=C \epsilon({x})\), where \(\bf{C}\) (the elasticity tensor) characterizes the material response and is a linear self-adjoint map on the space of symmetric \(3\times 3\) matrices. This space is six-dimensional. Thus, choosing a basis in the space of symmetric \(3\times 3\) matrices allows a symmetric \(6\times 6\) matrix to represent \(\bf{C}\) and six-dimensional vector fields to represent \(\bf{\sigma}(x)\) and \(\bf{\epsilon}(\bf{x})\). Nonnegativity of the elastic energy \(\bf{\epsilon}({x}) \cdot C\epsilon(x)/\)\(2\) then forces the matrix representing \(\bf{C}\) to be positive semidefinite. Under special conditions, such as with prescribed displacements at a body’s boundary, the elastic energy need only be quasiconvex rather than nonnegative.

Is this inclusive of all the constraints on \(\bf{C}\), or might there be some hidden restrictions? We first addressed this question in 1995, ultimately demonstrating that given a positive definite symmetric matrix \(\bf{C}\), one could find a microstructure built from two isotropic materials—one sufficiently stiff and the other sufficiently compliant—such that \(\bf{C}\) is the matrix representing its effective elasticity tensor [8]. There are consequently no hidden restrictions on \(\bf{C}\). The introduction of a new class of elastic materials called pentamode materials, for which \(\bf{C}\) is essentially a rank-one tensor, was key to this result. The “null space” of \(\bf{C}\) is five-dimensional, hence the name “pentamode.” Five independent strains \(\bf{\epsilon}\)\(_j\) exist in \(\bf{C}\)'s null space and cost negligible elastic energy when applied to the material. We can write \(\bf{C} \approx A \otimes A,\), where the \(3\times 3\) matrix \(\bf{A}\) represents the stress that the structure supports.

The simplest pentamodes are fluids with \(\bf{A} = \sqrt{\kappa}I\), where \(\kappa\) is the bulk modulus and gels—being close to fluids—display almost the same ideal behavior. Ole Sigmund used topology optimization to numerically identify other constructions with “fluid-like” effective elasticity tensors of this form [10]. Independently, we designed a complete family of pentamodes that allow for any matrix \(\bf{A}\) [8]; like fluids, they only support one loading (up to a multiplicative constant). Unlike with fluids, this loading is not necessarily hydrostatic, but can be arbitrary and thus expressible as a combination of a hydrostatic and shear loading [5, 8].

Figure 1. Visualization of pentamodes. 1a-1c. These computer-generated images each show four primitive unit cells of pentamodes in varying degrees of anisotropy, as dictated by the position of point P in the unit cell. 1b corresponds to the pentamode that supports a loading A proportional to I. 1d. An electron micrograph of a pentamode corresponding to 1b, built with three-dimensional laser lithography. Image courtesy of Tiemo Bückmann.
Our pentamode design consists of double-cone elements arranged in a diamond-like structure (see Figure 1). The tips of precisely four double-cone elements meet at any vertex in the structure. By balance of forces, the tension in one double-cone element thus uniquely determines the tension in the other three that meet it — and by induction, the tension in every double-cone element in the structure. The material’s stress is hence uniquely determined (up to a multiplicative constant). Moving the vertex \(\bf{P}\) where four double-cone elements meet within the unit cell—and applying an affine transformation to the structure if needed—lets one obtain any desired matrix \(\bf{A}\). Even pentamodes with \(\bf{A}\) proportional to \(\bf{I}\) are interesting, as they can be more “rubbery than rubber” in the sense that they may have a greater ratio of bulk to shear modulus. When we began this work in 1995, we never dreamed that pentamodes would actually be built. 17 years later, in collaboration with Tiemo Bückmann, Nicolas Stenger, and Michael Thiel, we managed to construct them via three-dimensional laser lithography techniques [6] (see Figure 1d). Now they attract considerable attention (see, for example, the references in [7]).

Pentamodes are the building blocks for obtaining any elasticity tensor; for this reason, we call them the grandfather of all linearly elastic materials [2]. Roughly speaking, one may superimpose six pentamode structures—deforming them to avoid collisions if necessary—each of which supports a stress \(\bf{A}\)\(_i,\) \(i=1,2, \ldots, 6\). Inserting a very compliant material (like foam) in the space between the six structures ensures that they all share the same average strain when the entire arrangement is deformed. As a result, the overall effective elasticity tensor is loosely the superposition of the elasticity tensors of the six pentamodes \(\bf{C}\)\(_*\approx\Sigma_i\) \(\bf{A}\)\(_i\otimes \bf{A}\)\(_i\). This achieves our objective, since any symmetric positive definite \(6 \times 6\) matrix can be represented in this form. In fact, one can go even further and theoretically obtain all possible nonlocal effective behaviors in linear elasticity [4].

Interest in pentamodes grew when the material was affiliated with underwater cloaking of acoustic waves [9]. Pentamodes can guide waves around an object if one chooses a suitable field \(\bf{A}(x)\) and uses the appropriate pentamode at point \(\bf{x}\). If \(\bf{C} \approx A \otimes A\) matched the elasticity tensor of water (itself a pentamode) and shared water’s overall density at the outer surface of the pentamode cloak, there would be no “impedance mismatch” to reflect  the waves at this interface. Researchers have also proposed pentamode-inspired constructions between stiff horizontal plates for seismic insulation [1]. Such constructions would allow the plates to slide with respect to each other and with little change in their spacing. The ground underneath could move horizontally while the top plate (above which lies the protected structure) would remain relatively still.

One can also use pentamodes to construct an “unfeelability cloak” for static elasticity [3]. In Hans Christian Andersen’s The Princess and the Pea, the “true princess” might have failed to feel the pea underneath her mattresses if the mattresses had an appropriate pentamode construction designed to shield the pea from compressive forces. Who knows, perhaps pentamode mattresses will be built; they would be much lighter than water beds. Interestingly, Adidas’ “Futurecraft” sneakers have a lattice construction in their soles, which perhaps may make a stone less discernable.

Acknowledgments: This work was supported by the National Science Foundation through grants DMS-9501025, DMS-94027763, and DMS-9307324, and by the French Investissements d’Avenir program, project ISITE-BFC (contract ANR-15IDEX-03).

References
[1] Amendola, A., Smith, C., Goodall, R., Auricchio, F., Feo, L., Benzoni, G., & Fraternali, F. (2016). Experimental response of additively manufactured metallic pentamode materials confined between stiffening plates. Comp. Struc., 142, 254-262.
[2] Bückmann, T., Stenger, N., Kadic, M., Kaschke, J., Frölich, A., Kennerknecht, T., Eberl, C., Thiel, M., & Wegener, M. (2012). Tailored 3D mechanical metamaterials made by dip-in direct-laser-writing optical lithography. Adv. Mat., 24, 2710-2714.
[3] Bückmann, T., Thiel, M., Kadic, M., Schittny, R., & Wegener, M. (2014). An elasto-mechanical unfeelability cloak made of pentamode metamaterials. Nat. Comm., 5, 4130.
[4] Camar-Eddine, M., & Seppecher, P. (2003). Determination of the closure of the set of elasticity functionals. Arch. Rat. Mech. Anal., 170, 211-245.
[5] Kadic, M., Bückmann, T., Schittny, R., & Wegener, M. (2013). On anisotropic versions of three-dimensional pentamode metamaterials. New Journ. Phys., 15, 023029.
[6] Kadic, M., Bückmann, T., Stenger, N., Thiel, M., & Wegener, M. (2012). On the practicability of pentamode mechanical metamaterials. Appl. Phys. Lett., 100, 191901.
[7] Milton, G.W., & Camar-Eddine, M. (2018). Near optimal pentamodes as a tool for guiding stress while minimizing compliance in 3d-printed materials: a complete solution to the weak G-closure problem for 3d-printed materials. J. Mech. Phys. Solids, 114, 194-208.
[8] Milton, G.W., & Cherkaev, A.V. (1995). Which elasticity tensors are realizable? ASME J. Eng. Mat. Tech., 117, 483-493.
[9] Norris, A.N. (2008). Acoustic Cloaking Theory. Proc. Royal Soc. A: Math., Phys., & Eng. Sci., 464, 2411-2434.
[10] Sigmund, O. (1995). Tailoring materials with prescribed elastic properties. Mech. Mat.: Int. J., 20, 351-368.

Andrej Cherkaev and Graeme W. Milton are mathematicians at the University of Utah. Muamer Kadic is a physicist at the FEMTO-ST Institute, associated with the French National Center for Scientific Research at the Université Bourgogne Franche-Comté in Besançon, France. Martin Wegener is a physicist at the Institute of Applied Physics and Institute of Nanotechnology at the Karlsruhe Institute of Technology in Germany.

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