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This article proposes a mortar type finite element formulation forconsistently embedding curved, slender beams, i.e. 1D Cosserat continua, into3D solid volumes. A consistent 1D-3D coupling scheme for this problem type isproposed, which enforces both positional and rotational constraints. SinceBoltzmann continua exhibit no inherent rotational degrees of freedom, suitabledefinitions of orthonormal triads are investigated that are representative forthe orientation of material directions in the 3D solid. The rotation tensordefined by the polar decomposition of the deformation gradient is demonstratedto represent these material directions in a L2-optimal manner. Subsequently,objective rotational coupling constraints between beam and solid are formulatedand enforced in a variationally consistent framework. Eventually, finiteelement discretization of all primary fields results in an embedded mortarformulation for rotational and translational constraint enforcement. Based oncarefully chosen numerical test cases, the proposed scheme is demonstrated toexhibit a consistent spatial convergence behavior and to offer the up-scalingpotential for studying real-life engineering applications such asfiber-reinforced composite materials.

Consistent coupling of positions and rotations for embedding 1D Cosserat beams into 3D solid volumes