On a formulation for anisotropic elastoplasticity at finite strains invariant with respect to the intermediate configuration

The paper is concerned with a formulation of anisotropic finite strain inelasticity based on the multiplicative decomposition of the deformation gradient F=F_eF_p. A major feature of the theory is its invariance with respect to rotations superimposed on the inelastic part of the deformation gradient. The paper motivates and shows how such an invariance can be achieved. At the heart of the formulation is the mixed-variant transformation of the structural tensor, defined as the tensor product of the privileged directions of the material as given in a reference configuration, under the action of F_p. Issues related to the plastic material spin are discussed in detail. It is shown that, in contrast to the isotropic case, any flow function formulated purely in terms of stress quantities, necessarily exhibits a non-vanishing plastic material spin. The possible construction of spin-free rates is discussed as well, where it is shown that the flow rule must then depend not only on the stress but on the strain as well.     

A formulation of anisotropic continuum elastoplasticity at finite strains. Part I: Modelling

A constitutive model for anisotropic elastoplasticity at finite strains is developed together with its numerical implementation. An anisotropic elastic constitutive law is described in an invariant setting by use of structural tensors and the elastic strain measure C_e. The elastic strain tensor as well as the structural tensors are assumed to be invariant in relation to superimposed rigid body rotations. An anisotropic Hill-type yield criterion, described by a non-symmetric Eshelby-like stress tensor and further structural tensors, is developed, where use is made of representation theorems for functions with non-symmetric arguments. The model also considers non-linear isotropic hardening. Explicit results for the specific case of orthotropic anisotropy are given. The associative flow rule is employed and the features of the inelastic flow rule are discussed in full. It is shown that the classical definition of the plastic material spin is meaningless in conjunction with the present formulation. Instead, the study motivates an alternative definition, which is based on the demand that such a quantity must be dissipation-free, as the plastic material spin is in the case of isotropy. Equivalent spatial formulations are presented too. The full numerical treatment is considered in Part II.     

A finite element approach to the chaotic motion of geometrically exact rods undergoing in-plane deformations

The paper is concerned with a hybrid finite element formulation for the geometrically exact dynamics of rods with applications to chaotic motion. The rod theory is developed for in-plane motions using the direct approach where the rod is treated as a one-dimensional Cosserat line. Shear deformation is included in the formulation. Within the elements, a linear distribution of the kinematical fields is combined with a constant distribution of the normal and shear forces. For time integration, the mid-point rule is employed. Various numerical examples of chaotic motion of straight and initially curved rods are presented proving the powerfulness and applicability of the finite element formulation.     

A nonlinear generalized continuum approach for electro-elasticity including scale effects

Materials characterized by an electro-mechanically coupled behaviour fall into the category of so-called smart materials. In particular, electro-active polymers (EAP) recently attracted much interest, because, upon electrical loading, EAP exhibit a large amount of deformation while sustaining large forces. This property can be utilized for actuators in electro-mechanical systems, artificial muscles and so forth. When it comes to smaller structures, it is a well-known fact that the mechanical response deviates from the prediction of classical mechanics theory. These scale effects are due to the fact that the size of the microscopic material constituents of such structures cannot be considered to be negligible small anymore compared to the structure's overall dimensions. In this context so-called generalized continuum formulations have been proven to account for the micro-structural influence to the macroscopic material response. Here, we want to adopt a strain gradient approach based on a generalized continuum framework [Sansour, C., 1998. A unified concept of elastic-viscoplastic Cosserat and micromorphic continua. J. Phys. IV Proc. 8, 341-348; Sansour, C., Skatulla, S., 2007. A higher gradient formulation and meshfree-based computation for elastic rock. Geomech. Geoeng. 2, 3-15] and extend it to also encompass the electro-mechanically coupled behaviour of EAP. The approach introduces new strain and stress measures which lead to the formulation of a corresponding generalized variational principle. The theory is completed by Dirichlet boundary conditions for the displacement field and its derivatives normal to the boundary as well as the electric potential. The basic idea behind this generalized continuum theory is the consideration of a micro- and a macro-space which together span the generalized space. As all quantities are defined in this generalized space, also the constitutive law, which is in this work conventional electro-mechanically coupled nonlinear hyperelasticity, is embedded in the generalized continuum. In this way material information of the micro-space, which are here only the geometrical specifications of the micro-continuum, can naturally enter the constitutive law. Several applications with moving least square-based approximations (MLS) demonstrate the potential of the proposed method. This particular meshfree method is chosen, as it has been proven to be highly flexible with regard to continuity and consistency required by this generalized approach.     

Nonlinear Dynamics of Shells: Theory,Finite Element Formulation,and Integration Schemes

The paper is concerned with a dynamical formulation of a recently established shell theory capable to catch finite deformations and falls within the class of geometrically exact shell theories. A basic aspect is the design of time integration schemes which preserve specific features of the continuous system such as conservation of momentum, angular momentum, and energy when the applied forces allow to. The integration method differs from the one recently proposed by Simo and Tarnow in being applicable without modifications to shell formulations with linear as well as nonlinear configuration spaces and in being independent of the nonlinearities involved in the strain-displacement relations. A finite element formulation is presented and various examples of nonlinear shell dynamics including large overall and chaotic motions are considered.