On the numerical implications of multiplicative inelasticity with an anisotropic elastic constitutive law

Abstract

SUMMARYThe statement that theories of inelasticity at ÿnite strains have arrived at a high level of development is only true in conjunction with isotropic material behaviour. From both points of view (theoretical and computational), the extension to anisotropic material behaviour seems to be a complicated task. The statement is especially true when the multiplicative decomposition of the deformation gradient is considered a basis for the formulation. Of special interest are questions related to the mathematical form of the stored energy function or, equivalently, of the constitutive relation for the material stress tensor as the thermodynamical force. This paper deals with the above issues. The anisotropic formulation is accomplished using the notion of structural tensors. Here we suggest that the privileged directions of the material should be transformed in a speciÿc way under the action of the inelastic part of the deformation gradient. The inelastic behaviour is assumed to be governed by evolution equations of the uniÿed type.Numerically, we deal with the full multiplicative structure of the theory. The numerical treatment is developed in full detail. Expressions concerning the local iteration as well as the tangent operator are derived. Various numerical examples with applications to shells are presented which demonstrate the in uence of anisotropy and the applicability of the theory.

Anisotropic finite strain viscoelasticity based on the Sidoroff multiplicative decomposition and logarithmic strains

“…Other proposed transformations perform a push-forward of the preferred structural tensors such as for velocity gradients, i.e. using the gradient X v and its inverse, see [68]. As in Ref.…”

An anisotropic fibre-matrix material model at finite elastic-plastic strains

“…In this regard a clear distinction can be drawn between models based on additive decompositions of suitable strain measures as in [8], [9], or [10], and those which are based on the multiplicative decomposition of the deformation gradient into elastic and inelastic parts such as in [11], [12], [13], or [14]. An important issue in relation to these different models is the effort to be considered with regard to numerical implementations.…”

An anisotropic viscoelastic fibre–matrix model at finite strains: Continuum formulation and computational aspects

“…Recently, a considerable effort has been devoted into the extension of isotropic finite strain inelastic models to the anisotropic case from the theoretical and the numerical points of view. We refer for example to the anisotropic elastoplastic formulations proposed by Spencer (2001) [32], Reese (2003) [20], Sansour and Bocko (2003) [23], and more recently by Klinkel, Sansour and Wagner (2005) [11]. And for anisotropic viscoelasticity, we refer for example to the recent formulation by Holzapfel and Gasser (2001) in [8], see also [9].…”