This article proposes a mortar type finite element formulation for consistently embedding curved, slender beams, i.e. 1D Cosserat continua, into 3D solid volumes. A consistent 1D-3D coupling scheme for this problem type is proposed, which enforces both positional and rotational constraints. Since Boltzmann continua exhibit no inherent rotational degrees of freedom, suitable definitions of orthonormal triads are investigated that are representative for the orientation of material directions in the 3D solid. The rotation tensor defined by the polar decomposition of the deformation gradient is demonstrated to represent these material directions in a L2-optimal manner. Subsequently, objective rotational coupling constraints between beam and solid are formulated and enforced in a variationally consistent framework. Eventually, finite element discretization of all primary fields results in an embedded mortar formulation for rotational and translational constraint enforcement. Based on carefully chosen numerical test cases, the proposed scheme is demonstrated to exhibit a consistent spatial convergence behavior and to offer the up-scaling potential for studying real-life engineering applications such as fiber-reinforced composite materials.

翻译：由于Boltzmann continua没有固有的旋转自由度,因此对正态三角形的合适定义进行了调查,这些定义代表了3D固体物质方向的方向。极地变形梯度所定义的旋转振幅以L2-最优化的方式代表了这些物质方向。随后,在变式一致的框架内,制定并执行了横线和固态之间的客观旋转组合制约。最终,所有主要野地的有限分解要素在固定的迫击炮配制中产生结果,用于旋转和翻译约束执行。根据仔细选择的数字测试案例,所拟议的办法证明显示了一种一致的空间趋同行为,并提供了研究诸如纤维弹性复合材料等实际生命工程应用的升级潜力。