A novel mathematical model for fiber-reinforced materials is proposed. It is based on a 1-dimensional beam model for the thin fiber structures, a flexible and general 3-dimensional elasticity model for the matrix and an overlapping domain decomposition approach. From a computational point of view, this is motivated by the fact that matrix and fibers can be easily meshed independently. Our main interest is in fiber-reinforced polymers where the Young modulus is quite different. Thus the modeling error from the overlapping approach is of no significance. The coupling conditions acknowledge both, the forces and the moments of the beam model and transfer them to the background material. A suitable static condensation procedure is applied to remove the beam balance equations. The condensed system then forms our starting point for a numerical approximation in terms of isogeometric analysis. The choice of our discrete basis functions of higher regularity is motivated by the fact, that as a result of the static condensation, we obtain second gradient terms in fiber direction. Eventually, a series of benchmark tests demonstrate the flexibility and robustness of the proposed methodology. As a proof-of-concept, we show that our new model is able to capture bending, torsion and shear dominated situations.

提出了一种新型的纤维增强材料数学模型。它基于用于细纤维结构的一维梁模型，用于矩阵的灵活且通用的三维弹性模型以及重叠域分解方法。从计算的角度来看，这是由于基质和纤维可以很容易地独立啮合这一事实而引起的。我们的主要兴趣是杨氏模量完全不同的纤维增强聚合物。因此，来自重叠方法的建模误差不重要。耦合条件承认梁模型的力和力矩，并将它们传递到背景材料。应用适当的静态冷凝程序以消除光束平衡方程。浓缩的系统形成了我们的起点等几何分析方面的数值近似。选择较高规则性的离散基函数的原因是这样的事实：由于静态凝聚，我们获得了纤维方向上的第二个梯度项。最终，一系列基准测试证明了所提出方法的灵活性和鲁棒性。作为概念验证，我们证明了我们的新模型能够捕获弯曲，扭转和剪切为主的情况。