Nonlinear Cosserat Continuum Research Articles

Influence of internal stresses on stability of multilayer micropolar tubes

Within the nonlinear Cosserat continuum, the stability of a compressed multilayer tube under internal and external hydrostatic pressures is studied. The tube is formed by attaching to each other N hollow cylinders that have been subjected to preliminary deformations of axial extension-compression and contains internal stresses. For the model of a physically linear micropolar material, the system of linearized equilibrium equations is derived, which describes the behavior of a multilayer cylindrical tube in a perturbed state. Using a special substitution, the stability analysis is reduced to solving a linear homogeneous boundary-value problem for a system of 6N ordinary differential equations. In the case of a three-layer tube made of dense polyurethane foam, the stability regions were constructed in the planes of loading parameters (the relative axial compression and the relative external or internal pressure) for different preliminary deformations of inner and outer layers (coatings). According to the results obtained, the preliminary extension of both the outer and inner coatings generally stabilizes the considered deformations of the three-layer tube, while the effect of their preliminary compression is negative. It should be noted here that the pre-stressed outer coating has a more significant influence on the tube stability than the pre-stressed inner coating. In addition, the described effects of preliminary deformations are more pronounced for tubes with thicker coatings.

On rotational instability within the nonlinear six-parameter shell theory

Within the six-parameter nonlinear shell theory we analyzed the in-plane rotational instability which occurs under in-plane tensile loading. For plane deformations the considered shell model coincides up to notations with the geometrically nonlinear Cosserat continuum under plane stress conditions. So we considered here both large translations and rotations. The constitutive relations contain some additional micropolar parameters with so-called coupling factor that relates Cosserat shear modulus with the Cauchy shear modulus. The discussed instability relates to the bifurcation from the static solution without rotations to solution with non-zero rotations. So we call it rotational instability. We present an elementary discrete model which captures the rotational instability phenomenon and the results of numerical analysis within the shell model. The dependence of the bifurcation condition on the micropolar material parameters is discussed.

On vectorially parameterized natural strain measures of the non-linear Cosserat continuum

The natural Lagrangian stretch and wryness tensors of the non-linear Cosserat continuum are expressed in terms of the general finite rotation vector. These expressions are then specialized for seven particular definitions of the rotation vectors known in the literature. It is expected that some of the vectorially parameterized strain measures derived here may be more convenient than others in specific applications.

A non-linear Cosserat continuum-based formulation and moving least square approximations in computations of size-scale effects in elasticity

Abstract The Cosserat continuum falls into the group of so-called generalized continua which have the capacity to consider internal lengths and so describe a certain type of size effects. The paper addresses some aspects of the non-linear formulation of the Cosserat continuum and its meshfree approximation. The use of moving least square approximations [P. Lancaster, K. Salkauskas, Mathematics of Computations 37(155) (1981) 141–158] in a non-linear Cosserat continuum -based formulation gives rise to certain implications which are related to the essential boundary condition enforcement as well as to the updating of the rotation field. Here, the enforcement of the displacement boundary conditions is accomplished by modifying the initial variational principle or weak form of equations such that the essential boundary conditions appear as Euler–Lagrange equations. The updating of the rotational degrees of freedom in the meshfree code adopts a multiplicative scheme which is based on the spinor theory and has been already successfully applied to finite elements by the first author. The suitability of the proposed methodology is exemplified in modelling size-scale effects in elasticity. The impact of the Cosserat continuum -based formulation on small-scale structures is demonstrated and the comparison with a classical Green strain tensor -based approach reveals significant differences.

Constitutive modeling and FEM for a nonlinear Cosserat continuum

AbstractThe choice of non‐standard material parameters in the presented micropolar continuum model leads to a highly nonlinear problem which calls also for a nonlinear numerical treatment. The model benefits in describing length scale effects as really non‐linear effects. The numerical torsion test is one example for the possibilities of our model. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)