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1. Odd elasticity

   https://doi.org/10.1038/s41567-020-0795-y  

   Scheibner, Colin; Souslov, Anton; Banerjee, Debarghya; Surówka, Piotr; Irvine, William T.; Vitelli, Vincenzo (April 2020, Nature Physics)
2. Quadratic-stretch elasticity

   https://doi.org/10.1177/10812865211022417  

   Vitral, E.; Hanna, J. A. (March 2022, Mathematics and Mechanics of Solids)

   A nonlinear small-strain elastic theory is constructed from a systematic expansion in Biot strains, truncated at quadratic order. The primary motivation is the desire for a clean separation between stretching and bending energies for shells, which appears to arise only from reduction of a bulk energy of this type. An approximation of isotropic invariants, bypassing the solution of a quartic equation or computation of tensor square roots, allows stretches, rotations, stresses, and balance laws to be written in terms of derivatives of position. Two-field formulations are also presented. Extensions to anisotropic theories are briefly discussed. 

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3. Elasticity in Apery Sets

   https://doi.org/10.1080/00029890.2020.1790926  

   Autry, Jackson; Gomes, Tara; O'Neill, Chris; Ponomarenko, Vadim (October 2020, The American mathematical monthly)
4. Prestressed elasticity of amorphous solids

   https://doi.org/10.1103/PhysRevResearch.4.043181  

   Zhang, Shang; Stanifer, Ethan; Vasisht, Vishwas V.; Zhang, Leyou; Del_Gado, Emanuela; Mao, Xiaoming (December 2022, Physical Review Research)

   Prestress in amorphous solids bears the memory of their formation and plays a profound role in their mechanical properties. Here we develop a set of mathematical tools to investigate mechanical response of prestressed systems, using stress rather than strain as the fundamental variable. This theory allows microscopic prestress to vary for the same bond or contact configuration and is particularly convenient for nonconservative systems, such as granular packings and jammed suspensions, where there is no well-defined reference state, invalidating conventional elasticity. Using prestressed nonconservative triangular lattices and a computational model of amorphous solids, we show that drastically different mechanical responses can show up in amorphous materials at the same density, due to nonconservative interactions which evolve over time, or different preparation protocols. In both cases, the information is encoded in the prestress of the network and not visible at all from the configurations of the network in the case of nonconservative interactions. 

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5. Odd Viscosity and Odd Elasticity

   https://doi.org/10.1146/annurev-conmatphys-040821-125506  

   Fruchart, Michel; Scheibner, Colin; Vitelli, Vincenzo (March 2023, Annual Review of Condensed Matter Physics)

   Elasticity typically refers to a material's ability to store energy, whereas viscosity refers to a material's tendency to dissipate it. In this review, we discuss fluids and solids for which this is not the case. These materials display additional linear response coefficients known as odd viscosity and odd elasticity. We first introduce odd viscosity and odd elasticity from a continuum perspective, with an emphasis on their rich phenomenology, including transverse responses, modified dislocation dynamics, and topological waves. We then provide an overview of systems that display odd viscosity and odd elasticity. These systems range from quantum fluids and astrophysical gases to active and driven matter. Finally, we comment on microscopic mechanisms by which odd viscosity and odd elasticity arise. 

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