The compatibility conditions for generalised continua are studied in the framework of differential geometry, in particular Riemann-Cartan geometry. We show that Vallee's compatibility condition in linear elasticity theory is equivalent to the vanishing of the three dimensional Einstein tensor. Moreover, we show that the compatibility condition satisfied by Nye's tensor also arises from the three dimensional Einstein tensor which appears to play a pivotal role in continuum mechanics not mentioned before. We discuss further compatibility conditions which can be obtained using our geometrical approach and apply it to the micro-continuum theories.

中文翻译：

---

使用 Riemann-Cartan 几何的连续体的相容性条件

在微分几何，特别是黎曼-嘉当几何的框架内研究了广义连续体的相容性条件。我们证明了线性弹性理论中的 Vallee 相容条件等价于三维爱因斯坦张量的消失。此外，我们表明，奈氏张量满足的相容性条件也源于三维爱因斯坦张量，该张量似乎在之前未提及的连续介质力学中发挥着关键作用。我们讨论了可以使用我们的几何方法获得的进一步兼容性条件，并将其应用于微连续理论。