Theory and Implementation of Time‐Dependent Fibre Reorientation in Transversely Isotropic Materials

Abstract

Transversely isotropic materials are commonly described by one characteristic direction in the material configuration, which commonly is assumed to be constant. However, for some applications it makes sense to consider a reorientation of the characteristic direction, as for instance the orientation of biological materials adapts to the mechanical loading, see e.g. KUHL ET AL. [4]. Other examples are piezoelectric materials, which reorient due to electric or mechanical loading, as described in SMITH [7], and liquid crystals changing their orientation on account of electric or magnetic fields, see ERICKSEN [1]. Furthermore simulations with a reorientation of the characteristic direction can be used in the context of optimisation of composites. In this contribution we restrict ourselves to the modelling of hyper-elasticity and assume a time dependent reorientation of the characteristic direction with respect to the principal strain directions, see for instance MENZEL [5]. Thereby we concentrate on the numerical implementation into a finite element code. Constitutive EquationsTransverse isotropy can be described by the characteristic direction n A or respectively by the structural tensor A = n A ⊗ n A . In this contribution we assume a reorienting characteristic direction n A . Thus, insertion of the standard free energy density ψ 0 = ψ 0 (C, A) and the Piola-Kirchhoff stresses S = 2∂ C ψ 0 into the dissipation inequality yields the reduced dissipation inequalitywith the extra entropy source S 0 satisfying the second law of thermodynamics, see for instance HIMPEL ET AL. [2]. Analogous to the director in the theory of shells a rotation of the characteristic direction n A is assumed, so that its evolution becomeṡFor the definition of the angular velocity ω an alignment with the principal strain directions, see e.g. MENZEL [5], the principal stress directions, see e.g. HARITON ET AL.[3], the gradients of the electric or magnetic field or other characteristic directions is thinkable. In this contribution we concentrate on a reorientation in terms of the principal strain direction. It can be shown that the critical state of free energy can be reached for coaxial stress and strain tensors. The Piola-Kirchhoff stresses for transverse isotropy can be depicted asdepending on the scalars S i=1,...5 = ∂ Ii ψ. Thus it can easily be observed, that for an alignment of n A with one of the principal strain directions, the Piola-Kirchhoff stresses S and the right Cauchy-Green tensor C become coaxial. Hence, following to MENZEL [5] we assume an alignment of n A with the maximum principal strain directionTo avoid drilling rotation the angular velocity is defined orthogonal to n A and n Cwith the material parameter t acting as time relaxation parameter. Implementation into a Finite Element CodeFor the implementation into a finite element code the characteristic direction has been introduced as internal variable. Due to the EULER-theorem exp(−ε · ω ∆t) is a rotation about ω by the angle ω ∆t. Thus we apply an implicit exponential map fo...