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

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DOI: 10.1007/s00466-008-0314-y

- Cite this article as:
- Banerjee, B., Shaw, A. & Roy, D. Comput Mech (2009) 43: 415. doi:10.1007/s00466-008-0314-y

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## 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 54:884–892, 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.

### Keywords

Cosserat pointWrinkled/slack membranesNonlinear elasticityTension field theory### List of symbols

**b**^{i}*i*th director couple corresponding to body force**f**^{i}*i*th external director couple*σ*_{1},*σ*_{2}principal Cauchy stresses

*T*_{1},*T*_{2}principal Cosserat stresses

- \({\overline {\bf F}}\)
auxiliary deformation gradient

**F**Cosserat deformation gradient

**F**^{*}3D deformation gradient

**σ**_{avg}average Cauchy stress tensor

**σ**Cauchy stress tensor

*d*^{1/2}**T**Cosserat stress tensor

- Π
homogeneous part of the strain energy

**Ψ**inhomogeneous part of the strain energy

**β**_{i}*i*th inhomogeneous strain corresponding to the*i*th director**t**^{i}*i*th intrinsic director couple**Φ**total strain energy

*γ*_{1},*γ*_{2}stretching (wrinkliness) parameters

*E, μ , ν*Young’s modulus, shear modulus and Poisson’s ratio

*K*_{1},*K*_{2},*K*_{3}inhomogeneous constitutive constants

- \({\overline {\bf t}^i,\overline {\bf f}^i,\overline {\bf b}^i}\)
*i*th nodal quantity corresponding to**t, f, b***θ*^{1},*θ*^{2},*θ*^{3}locally convected curvilinear coordinates

- \({\kappa_1^1 ,\kappa_1^2 ,\kappa_1^3}\)
normalized inhomogeneous strain measures

- \({\overline{\bf D}_{\rm i},\overline {\bf d}_{\rm i} }\)
*i*th nodal deformed and deformed position vectors- A
transformation matrix

*D*_{0},*D*initial and final thicknesses of the membrane element

**D**^{i},**d**^{i}*i*th undeformed and deformed contravariant director vector**D**_{i},**d**_{i}*i*th undeformed and deformed covariant director vectors**E**homogeneous strain measure

**G**_{i},**g**_{i}*i*th undeformed and deformed base vectors*H*,*L*width and length of the planar Cosserat point

**K**constitutive tensor

*m*mass of the Cosserat point

**n**_{1},**n**_{2}unit vectors representing principal stress directions

- \({{\bf n^*}_{1}, {\bf n^*}_{2}}\)
unit vectors representing principal stress directions in 3D setup

**N**_{3}unit surface normal

*V*, vvolumes of the Cosserat point in undeformed and deformed states

**X, x**undeformed and deformed position vectors of a material point