SUMMARYA simple computer implementation of membrane wrinkle behaviour is presented within the classical elastic plane stress constitutive model. In the present method, a projection technique is utilized for modelling of the wrinkle mechanisms, in which the total strains in wrinkled membranes are decomposed into elastic and zero-strain energy parts, and a projection matrix that extracts the elastic parts from the total strains is derived. The resulting modified elasticity matrix that represents the stress-strain relations in wrinkled membranes is thus obtained as product of the classical elasticity matrix and the projection matrix. The modified elasticity matrix is straightforward to implement within the context of the finite element method. Numerical examples are presented to demonstrate the accuracy and effectiveness of the proposed method.
A new modification scheme of the stress-strain tensor for wrinkled membranes is presented on the basis of the tension field theory. The scheme is applicable to finite element analysis of partly wrinkled membranes with arbitrary shapes. Derivation of the modification scheme rests on an introduction of the so-called "wrinkle strain" and a simplification of the virtual work equation of wrinkled membranes. Because all of the modifications required to account for wrinkling are totally confined within the stress-strain relations of membranes, the scheme can be easily implemented with existing finite element codes. Furthermore, the modified stress-strain tensor automatically leads to the consistent tangent stiffness matrix, where changes in both the wrinkling direction and the amount of wrinkliness are taken into account. Three numerical examples are treated to show the accuracy and effectiveness of the proposed modification scheme.
Nomenclature a, b= inner and outer radii of annular membrane (example 2) C = stress-strain tensor assumed between S and E C I = modified stress-strain tensor C II = modified stress-strain tensor in incremental form E = Green-Lagrange strain tensor E = Young's modulus E W = wrinkle strain tensor e i = orthonormal base vectors of Cartesian coordinate system F = deformation gradient tensor G α , g α = covariant base vectors of convected coordinate system in undeformed and deformed configurations G α , g α = contravariant base vectors of convected coordinate system in undeformed and deformed configurations H = width of rectangular membrane (example 1) h = width of wrinkled region of rectangular membrane (example 1) M = bending moment applied to rectangular membrane (example 1) P = axial load applied to rectangular membrane (example 1) R = radius of wrinkled region of annular membrane (example 2) r α = convected coordinates S = second Piola-Kirchhoff stress tensor T = twisting moment applied to annular membrane (example 2) t = thickness of membrane (examples 1-3) t = unit vector along to wrinkling direction w = unit vector transverse to wrinkling direction β = physical amount of wrinkliness γ = parameter associated with β, defined by Eq. (28) θ = parameter associated with wrinkling direction, introduced in Eq. (23) κ = overall curvature of rectangular membrane (example 1) ν = Poisson's ratio σ = Cauchy stress tensor σ 0 = uniform stress applied to membrane (examples 1 and 2) φ = angle of twist of rigid hub (example 2) { } = second-order tensor expressed in column vector form [ ] = fourth-order tensor expressed in matrix form Superscripts · = material time derivative = variables modified to account for wrinkling
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