Mixed finite element formulations and energy‐momentum time integrators for thermo‐viscoelastic fiber‐reinforced continua

Abstract

AbstractToday, fiber‐reinforced materials and their exact dynamic simulation play a significant role in the construction of lightweight structures. These materials are used in aircraft, automobiles and wind turbines, for instance. The low density and the high modulus of elasticity play a major role, but also the thermal properties should not be neglected. First of all, the thermal expansion of the matrix part and the ability to conduct the heat in a directional way with the fibers. For these materials, volumetric locking effects of an incompressible matrix material as well as locking effects due to stiff fibers can occur. On the one hand, there are combinations of well known mixed elements with an independent approximation of the volume dilatation and an independent approximation of the right Cauchy‐Green tensor for the anisotropic part of the strain energy function to reduce these effects. On the other hand, we have developed mixed elements where fields for the fourth and fifth invariants are added, as well as a version with the corresponding tensor fields. For long‐term simulations it is necessary to use higher order time integrators to perform an accurate dynamic simulation. Galerkin‐based time integrators offer a good option for this application. To eliminate a huge energy error these have to be extended to an energy‐momentum time integration scheme. It is logical to combine these methods and thus combine the advantages of these methods. We formulate the mixed elements using Hu‐Washizu functionals and combine this with the mixed principle of virtual power. By adding a thermo‐mechanical coupling part in the strain energy and introducing Fourier's heat conduction we obtain a thermo‐mechanical formulation for the different mixed elements and a higher order Galerkin‐based time integrator. Dirichlet boundary conditions in the form of Lagrange multiplier methods as well as Neumann boundary conditions in the mechanical and thermal context are also provided. In addition, we extend the continuum so that we can model different fiber families and the directional heat conduction of the fibers. As numerical examples serve cook's cantilever beam as well as a rotating heat pipe. We primarily analyze the spatial and time convergence, the conservation properties as well as the effect of the heat conduction of the fibers.