If there exists a conserved energy density then the modulus tensor Cijkl is equal to Cklij. The corresponding compliance elements have the same symmetry. If there does not exist a conserved energy density, these moduli or compliance values can be unequal. That is entirely distinct from the asymmetry of the stress tensor that occurs in Cosserat elasticity. Asymmetric physical property tensors have recently been called odd.
Asymmetry in the compliance tensor has been thoroughly observed experimentally in wood which is orthotropic and viscoelastic and has long been known. All of the 12 compliance components have been determined experimentally vs. time for various woods. The different time dependence of the non-diagonal elements of the compliance matrix reveals the odd nature of the viscoelastic character of wood though the word odd was not used at the time. The theoretical basis has also long been known.
Chiral, directionally isotropic gyroid lattices are observed to exhibit nonclassical thermal effects incompatible with an asymmetric ("odd'') second rank conductivity tensor but consistent with a third rank tensor property that provides a curl term. The lattices are passive materials so no driving torques are needed to obtain transverse flow. A method for determination of the length scale associated with chirality is provided. The polymer gyroid lattice made by 3D printing is a metamaterial which allows chirality that is tunable by geometry.
References
H. Neuhaus, Elastic behavior of spruce wood as a function of moisture content, Holz Roh-Werkst. 41 (1), 21-25 (1983).
https://doi.org/10.1007/BF02608449
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Ozyhar, T., S. Hering, and P. Niemz, 2013, Viscoelastic characterization of wood: Time dependence of the orthotropic compliance in tension and compression, Journal of Rheology 57(2), 699-717 (2013).
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S. Hering, D. Keunecke, and P. Niemz, Moisture-dependent orthotropic elasticity of beech wood, Wood Sci. Technol. 46, 927-938 (2012).
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A. C. Pipkin and G. Rogers, Asymmetric relaxation and compliance matrices in linear viscoelasticity, ZAMP, 14, 334-343 (1963).
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doi link https://doi.org/10.1007/BF01603090
R. S. Lakes, "Nonclassical heat flow in passive chiral solids is third rank, not odd", ArXiV 2412.08309 (2024).
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This is a preprint of
R. S. Lakes, "Nonclassical heat flow in passive chiral solids is third rank, not odd", Z. Angew. Math. Phys. (ZAMP) 76: 81 (2025).
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R. S. Lakes, Materials that exceed classical thermodynamic bounds on properties, Reviews of Modern Physics, 97, 021002 14 May, (2025).
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In many proofs of bounds, assumptions are made about the material. If those assumptions are relaxed, limits or bounds can be exceeded as is demonstrated by analysis and experiments. For example, heat capacity, compressibility, electrical capacitance and refractive index can be negative. Thermal expansion in composites can be larger or smaller than that of any constituent and can be negative. Materials and systems are known that are non-Hermitian or are non-reciprocal. Currently active fields of endeavor have arisen from such conceptual sources. Research efforts have led to the development of new materials and new classes of materials.
Asymmetric material property tensors have long been known and have recently been called "odd". Elastic materials by definition exhibit a relation between stress and strain that is independent of path. For elastic solids there exists a conserved energy density so that the modulus tensor Cijkl is equal to Cklij and is symmetric with respect to exchange of pairs of indices. Asymmetry of the tensor can occur in viscoelastic materials as has been demonstrated experimentally; also in active materials provided with an energy source.