The theory of Cosserat points applied to the analyses of wrinkled and slack membranes

Abstract

Numerical simulations of wrinkling and slacking of geometrically nonlinear membrane structures are considered using planar Cosserat points. The finite element method (FEM) solves the problem by weakly projecting the governing PDEs and thus requires numerical integration. This is contrasted with Cosserat point elements wherein governing equations are solved in an averaged sense at a point. The point is equipped with a few directors and can describe the deformation kinematics of a finite region containing itself. Numerical modeling through the Cosserat point provides freedom from numerical integration and locking. Presently a plane stress quadrilateral Cosserat point element is used to study the wrinkling and slacking of isotropic membranes. The approach by Roddeman et al. (ASME J Appl Mech 54: [884][885][886][887][888][889][890][891][892] 1987) is exploited to detect wrinkled/slack elements in the membrane structure. Here stretching parameters are employed to modify the deformation tensor to represent a fictive non-wrinkled surface. A variation of the algorithm to detect spatial variations of the stretching parameters within a point element is also described. Several numerical examples on static deformations of wrinkled/slack membranes are presented. Limited comparisons with a reported experiment and with results via the FEM as well as a mesh-free approach are provided to assess the performance of the approach.

Influence of local wrinkling on membrane behaviour: A new approach by the technique of slowly variable Fourier coefficients

“…There is a considerable amount of literature about membrane wrinkling, see Banerjee et al (2009) and the references in this article but none with the same approach as in this paper. A common point between classical wrinkling models and the present one is the introduction of additional kinematic variables, as the inhomogeneous strains used in Banerjee et al 2009) or the Fourier coefficients of the displacements u 1 , v 1 , u 2 , v 2 in the present paper.…”

“…A common point between classical wrinkling models and the present one is the introduction of additional kinematic variables, as the inhomogeneous strains used in Banerjee et al 2009) or the Fourier coefficients of the displacements u 1 , v 1 , u 2 , v 2 in the present paper. A second common point is the introduction of additional stress variables so that the model is a generalized continuum: in Banerjee et al (2009), the intrinsic director couple of Cosserat theory, here the Fourier coefficient of the stress.…”

“…A common point between classical wrinkling models and the present one is the introduction of additional kinematic variables, as the inhomogeneous strains used in Banerjee et al 2009) or the Fourier coefficients of the displacements u 1 , v 1 , u 2 , v 2 in the present paper. A second common point is the introduction of additional stress variables so that the model is a generalized continuum: in Banerjee et al (2009), the intrinsic director couple of Cosserat theory, here the Fourier coefficient of the stress. Nevertheless the present paper is based on a quite different point of view, the macroscopic potential energy (39) being completely deduced from the basic model as in computational homogenization (Kouznetsova et al, 2004), while the classical wrinkling models are phenomenological and postulate the expression of the additional potential energy.…”

Membrane wrinkling revisited from a multi-scale point of view

“…Then the final bifurcation Equation 32includes an internal length, which permits to retrieve the multi-scale instability analysis of Part 2. Finally this model can be qualitatively compared with the models of Banerjee et al [29,30], where an internal length is introduced via Cosserat theory. Body forces and boundary forces can be introduced easily by the same procedure.…”

“…The partial differential equations deduced from these membrane models are not elliptic (or hyperbolic in the dynamical case) in the presence of a non-positive principal stress so that this problem is mathematically ill-posed. A well-posed problem can be obtained if the macroscopic model includes an internal length, for instance within Cosserat theory [29,30]. However this regularisation may be not necessary for an explicit dynamic computation.…”