For our research, we are motivated by dynamic simulations of 3D fiber-reinforced materials in lightweight structures. In such materials, the material reinforcement is performed by fiber rovings with a separate bending stiffness, which can be modelled by a second order gradient of the deformation mapping. Therefore, we extend a thermo-viscoelastic Cauchy continuum for fiber-matrix composites with single fibers by an independent field for the gradient of the right Cauchy-Green tensor. On the other hand, we focus on numerically stable dynamic long-time simulations with locking free meshes, and thus use higher-order accurate energy-momentum schemes emanating from mixed finite element methods. Hence, we adapt the variational-based space-time finite element method to the new material formulation, and additionally include independent fields to obtain well-known mixed finite elements. As representative numerical example, Cook’s cantilever beam is considered. We primarily analyze the influence of the fiber bending stiffness, as well as the spatial and time convergence up to cubic order. Furthermore, we look at the influence of the physical dissipation in the material.

Rotor-dynamical systems made of 3D-fiber-reinforced composites which are subjected to dynamical loads exhibit an increased fiber bending stiffness in numerical simulations. We propose a numerical modeling approach of fiber-reinforced composites that treats this behaviour accurately. Our model uses a multi-field mixed finite element formulation based on a dynamic variational approach, as demonstrated in [1], to perform long-term dynamic simulations that yield numerical solutions with increased accuracy in efficient CPU-time.We extend a Cauchy continuum with higher-order gradients of the deformation mapping as an independent field in the functional formulation, as suggested in [2], to model the bending stiffness of fibers accurately. This extended continuum also takes into account the higher-order energy contributions including the fiber curvature along with popular proven approaches that avoid the numerical locking effect of the fibers efficiently.We apply the proposed approach on Cook’s cantilever beam with a hyperelastic, transversely isotropic, polyconvex material behavior in a transient dynamic analysis. The beam is subjected to bending loads with a strong dependence of the overall stiffness on the fiber orientation. The spatial and temporal convergence as well as the conservation properties are analyzed. It is observed that the model needs an improved numerical treatment to conserve total momenta as well as total energy.REFERENCES M. Groß and J. Dietzsch, "Variational-based locking-free energy–momentum schemes of higher-order for thermo-viscoelastic fiber-reinforced continua", Computer Methods in Applied Mechanics and Engineering, (2019), 631-671, 343. T. Asmanoglo and A. Menzel, “A multi-field finite element approach for the modelling of fibre-reinforced composites with fibre-bending stiffness”, Computer Methods in Applied Mechanics and Engineering, (2017), 1037-1067, 317.

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Investigating the bending stiffness of fibers in fiber-reinforced composites for rotor-dynamical systems which are subjected to dynamical loads are essential in the development of system design. The proposed numerical modeling approach of fiberreinforced composites uses a multi-field mixed finite element formulation based on a dynamic variational approach to perform long-term dynamic simulations with CPU-time efficient and increased accurate numerical solutions. To model the bending stiffness of fibers accurately, we extend a Cauchy continuum with higher-order gradients of the deformation mapping as an independent field in the functional formulation. This extended continuum also takes into account the higher-order energy contributions including the fiber curvature along with popular proven approaches that efficiently avoid the numerical locking effect of the fibers. The effect of the proposed approach is demonstrated through transient dynamic analysis on a cantilever beam with a hyperelastic, transversely isotropic, polyconvex material behavior. The beam is subjected to bending load with a strong dependence of the overall stiffness on the fiber orientation. The material and conservation properties are analyzed.

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