Linear viscoelastic transversely isotropic model based on the spectral decomposition of elasticity tensors

Abstract

The linear viscoelasticity is still a useful model in the engineering for studying the behavior of materials loaded with different loading rates (frequencies). Certain types of materials reveal also an anisotropic behavior: fiber reinforced composites, asphalt concrete mixtures, or wood, to name a few. In general, researchers try to identify experimentally the dependence of engineering constants like: directional Young’s moduli and Poisson’s ratios on loading velocity by means of creep or harmonic oscillatory tests. This approach is appealing from the experimental point of view. However, from the modeling perspective, this is not the case. The engineering constants emerge in nonlinear manner in the relationship between the strain and stress via fourth order stiffness tensor components. This is especially true in higher order anisotropies, yet even in isotropy Poisson’s ratio appears nonlinearly in the stiffness tensor. Several models for the linear viscoelasticity of anisotropic materials already exist in the literature that try to tackle this issue. In this paper, we propose a linear viscoelasticity model for anisotropic materials based on the spectral decomposition of the stiffness tensor. The proposed model offers several advantages: the natural choice of the stiffness tensor eigenvalues as time-dependent variables, state variables with clear interpretation as creep strains, and reduced burden of storage utilizing the orthogonality of the eigenspaces. We implemented the model in the finite-element method system AceGen/AceFEM, and calibrated parameters of the model with the experimental data available in the literature, thus proving the adequacy of the proposed model to describe anisotropic viscoelastic materials.