An energy–momentum couple stress formula for variational-based macroscopic modelings of roving-matrix composites in dynamics

Abstract

In dynamical systems, fiber-reinforced materials with fiber rovings in a cured matrix material, woven or manufactured by the technology of tailored fiber placement (TFP), for instance, have gained great significance for engineers. Therefore, many finite element simulations are currently in progress during the design of composite parts. But, standard finite element formulations do not take into account dimensions and the direction-dependent flexure and twist stiffness of the rovings. In order to avoid computationally costly mesoscopic finite element models with a discretization of roving-matrix interior boundary interfaces, a macroscopic material model can be derived by introducing micropolar rotation degrees of freedom. Micropolar rotation degrees of freedom allow for a separate constitutive description of the direction-dependent stiffness and inertia of the rovings in the cured matrix material. This continuum model represents an extended continuum formulation with an extended set of balance laws. In its computational setting, the discrete preservation of the complete set of balance laws can be realized by a mixed principle of virtual power in connection with energy–momentum tension stress and couple stress formulas. The guaranteed algorithmic energy–momentum consistency then allows an one-off virtual parameter identification of the macroscopic material constants with the mesoscopic model based on the error of the balance functions. Therefore, a mixed finite element formulation with independent micropolar rotation degrees of freedom is recently published, which are able to simulate dynamical problems by taking into account flexure and twist of the rovings. In this previously published paper, the constitutive laws are at most quadratic such that energy–momentum time integrations are possible without a special couple stress approximation. If a nonlinear strain energy function for flexure and twist of the rovings is considered, a new special couple stress approximation is necessary. Therefore, the present paper completes this mixed finite element formulation with a new couple stress approximation and shows its application by means of that numerical experiments with unidirectional rovings, which are designed for the virtual parameter identification. These fully dynamical examples with large translations, rotations and deformations demonstrate the flexure and twist stiffness of the rovings with roving curvature–twist strain energy functions of Kauderer-type by parameter studies. The complete set of discrete balance laws are satisfied independent of boundary conditions in order to allow the upcoming virtual parameter identification.