Modelling anisotropic viscoelasticity of fibre-reinforced polymer laminates for general loading conditions

Polymer materials typically show a viscoelastic material response which is commonly described using spring-damper systems to capture the time and rate-dependent characteristics. To model the damping behaviour of fibre-reinforced polymers, formulations for anisotropic viscoelasticity are needed. Some recent publications have addressed this issue for the special case of relaxation due to static long-time loading [Pettermann (2021), An (2022)]. There, each entry in the stiffness matrix E is defined as a function of time, i.e. each entry of the stiffness matrix is considered to be such a viscoelastic spring-damper system. For a general anisotropic material, viscoelasticity is therefore modelled using 21 functions of time. These were computed from micro-mechanical unit cell models of uni-directional composites subjected to unit load cases. In our contribution, this idea is adopted and extended for general loading conditions in the time domain such that any kind of stress or strain history can be captured, including creep, relaxation, and cyclic loading. To do so, the response of a spring-damper system not only depends on a given strain history but also on the strain rate, therefore, the entries of the stiffness matrix are defined as E_ij = f(ε(t),ε ̇(t)). These entries are determined by a unit cell loaded in a relaxation scenario. From these relaxation functions, the individual spring- and damper properties for each system of the various directions are back computed which allows for the computation of the full viscoelastic response for general loading. The whole procedure is automated by Python scripts which generate a VUMAT user subroutine for ABAQUS explicit containing all the spring and damper properties from which the viscoelastic material response is then computed for the current time increment. The applicability of the model is evaluated by comparing its predictions to experimental tests of laminated plates subjected to bending vibration. For this, two fibre-reinforced epoxy materials (glass and carbon fibres) are considered using uni-directional 0° and 90° specimens as well as several symmetric angle-ply laminates of varying layup angle.