A novel continuum mechanical framework for decoupled material behavior in thickness and in-plane directions

Abstract

A novel continuum mechanical framework for decoupling the material behavior in in-plane and thickness directions is presented that is motivated by the macroscopic behavior of paper and paperboard. Due to the manufacturing process of these composite materials and the underlying microstructure resulting thereof, the material response is highly anisotropic and can be divided into in-plane and out-of-plane behavior that is independent of each other. This decoupling is mostly incorporated in existing material models by setting the corresponding material parameters to zero which limits the models to an extension to nonlinear problems and is restrictive from a continuum mechanical point of view.Therefore, the aim of this paper is to derive a continuum mechanical framework that captures the in-plane and out-of-plane decoupling in a thermodynamically consistent manner for large deformations. Thus, the right Cauchy–Green deformation tensor was split into in-plane and out-of-plane deformations by use of structural tensors that were aligned with the preferential material directions. Therefore, the Helmholtz free energy was formulated in terms of modified invariants of the deformation tensor for in-plane and out-of-plane deformations separately, which lead to a decoupled framework that could easily be modified due to the flexible formulation of the Helmholtz free energy terms. The extension to inelasticity, in particular viscoelasticity here, was straightforward due to the split of the deformation tensor. Finally, the decoupling of the in-plane and out-of-plane response of the presented framework was demonstrated by structural examples for elastic and inelastic material behavior.The presented model incorporates the split into in-plane and out-of-plane behavior in a continuum mechanical manner, and thus, is a flexible constitutive framework for decoupled material responses that can easily be adjusted to different material behavior.