We consider mechanics of composite materials in which thin inclusions are modeled by lower-dimensional manifolds. By successively applying the dimensional reduction to junctions and intersections within the material, a geometry of hierarchically connected manifolds is formed which we refer to as mixed-dimensional. The governing equations with respect to linear elasticity are then defined on this mixed-dimensional geometry. The resulting system of partial differential equations is also referred to as mixed-dimensional, since functions defined on domains of multiple dimensionalities are considered in a fully coupled manner. With the use of a semi-discrete differential operator, we obtain the variational formulation of this system in terms of both displacements and stresses. The system is then analyzed and shown to be well-posed with respect to appropriately weighted norms. Numerical discretization schemes are proposed using well-known mixed finite elements in all dimensions. The schemes conserve linear momentum locally while relaxing the symmetry condition on the stress tensor. Stability and convergence are shown using a priori error estimates and confirmed numerically.

我们考虑了复合材料的力学，其中薄夹杂物是由低维流形建模的。通过依次将降维应用于材料内的交点和相交处，形成了层次连接的歧管的几何形状，我们称其为混合尺寸。然后在这种混合尺寸的几何体上定义关于线性弹性的控制方程。所得的偏微分方程组也称为混合维，因为以完全耦合的方式考虑了在多维域上定义的函数。通过使用半离散微分算子，我们从位移和应力两个方面获得了该系统的变分公式。然后对该系统进行分析，并显示出相对于适当加权的规范而言是正确的。数值离散化方案是在各个维度上使用众所周知的混合有限元提出的。该方案局部保留线性动量，同时放松应力张量上的对称条件。使用先验误差估计来显示稳定性和收敛性，并通过数值进行确认。