A passive solid cannot do work on its surroundings through any quasistatic cycle of deformations. This property places strong constraints on the allowed elastic moduli. In this Article, we show that static elastic moduli altogether absent in passive elasticity can arise from active, non-conservative microscopic interactions. These active moduli enter the antisymmetric (or odd) part of the static elastic modulus tensor and quantify the amount of work extracted along quasistatic strain cycles. In two-dimensional isotropic media, two chiral odd-elastic moduli emerge in addition to the bulk and shear moduli. We discuss microscopic realizations that include networks of Hookean springs augmented with active transverse forces and non-reciprocal active hinges. Using coarse-grained microscopic models, numerical simulations and continuum equations, we uncover phenomena ranging from auxetic behaviour induced by odd moduli to elastic wave propagation in overdamped media enabled by self-sustained active strain cycles. Our work sheds light on the non-Hermitian mechanics of two- and three-dimensional active solids that conserve linear momentum but exhibit a non-reciprocal linear response.

被动实体不能通过任何准静态变形循环对其周围环境做功。此属性对允许的弹性模量有很强的限制。在本文中，我们表明，被动弹性中完全不存在的静态弹性模量可能来自主动的、非保守的微观相互作用。这些活动模量进入静态弹性模量张量的反对称（或奇数）部分，并量化沿准静态应变循环提取的功量。在二维各向同性介质中，除了体积模量和剪切模量外，还会出现两个手性奇弹性模量。我们讨论了微观实现，包括增加了主动横向力和非互易主动铰链的胡克弹簧网络。使用粗粒度微观模型、数值模拟和连续方程，我们发现了各种现象，从奇模量引起的拉胀行为到通过自持主动应变循环实现的过阻尼介质中的弹性波传播。我们的工作揭示了二维和三维活性固体的非厄米力学，它们保持线性动量但表现出非互易线性响应。