Finite element analysis of the wrinkling of orthotropic membranes Igor P. Oliveira (1) Eduardo M. B. Campello (1) Paulo M. Pimenta (2) (1) Polytechnic School at the University of São Paulo, Brazil (2) On leave from University of São Paulo to Institut für Bau- und Numerische Mechanik at the University of Hannover, Germany ECCM 2006 Lisbon

Topics Introduction Underlying shell model The constitutive equation: isotropic part orthotropic part FE implementation: triangular shell element Numerical examples

Introduction: motivation Finite elements from shell theories Rotation as DOF, moments as stress resultants Fully nonlinear theories, no kinematical approximation unless those of the shell theory Simple finite element that doesn’t lock: our T6-3i

Introduction: our recent work on shells Shear-flexible (1st order or Reissner-Mindlin) models constant thickness shell model variable-thickness shell model Soon: shear- rigid and higher order shell models

Introduction: common features of our shell models Fully nonlinear shell models: large strains, displacements and rotations Shell kinematical assumption is the only geometrical approximation (geometrically-exact approach) Basic quantities: 1st Piola-Kirchhoff stress tensor and deformation gradient tensor

Introduction: common features of our shell models Cross-sectional resultant formulations (variational consistent) Variational formulation of the equilibrium equations Variational obtainment of the boundary conditions Consistent derivation of the tangent bilinear weak form, which is symmetric for hyperelastic materials and conservative loads

Introduction: common features of our shell models Constant-thickness shell models: consistent plane stress condition based on 1st Piola-Kirchhoff stresses (transversal nominal stress) thickness variation described at the shell theory level by 2 parameters (no thickness locking) variable-thickness shell models: 3D material description (von Mises elastoplasticity, 3D effects, pressure due to contact) Initially curved models: initial curvatures described by means of an initial stress-free configuration (exact description of initial geometry is possible)

Introduction: benchmark examples

Introduction: our goal To apply our shell models and finite element to the analysis of the wrinkling of orthotropic membranes

Underlying shell model current configuration reference configuration Reference configuration: Current configuration:

Underlying shell model Shell director on current configuration: where: with

Underlying shell model Deformation gradient: where: and (Notation:

Underlying shell model 1st Piola-Kirchhoff stress tensor: Internal virtual work: where:

Underlying shell model External virtual work: where:

Underlying shell model Shell equilibrium (weak form):

Underlying shell model Linearization of weak form (tangent operator): where:

Underlying shell model where: (constitutive contribution) (geometric contribution of internal forces) (geometric contribution of external loading)

The constitutive equation Specific strain energy function: shell behavior: isotropic membrane behavior: orthotropic Second Piola-Kirchhoff stress tensor:

The constitutive equation First Piola-Kirchhoff stress tensor: where: Total 1st Piola-Kirchhoff stresses:

Isotropic part Isotropic strain energy function: where: 2nd Piola-Kirchhoff stress tensor:

Isotropic part 1st Piola-Kirchhoff stresses: plane stress assumption where:

Orthotropic part Orthotropic strain energy function: we adopt: where: ( are “structural tensors” of orthotropy directions )

Orthotropic part 2nd Piola-Kirchhoff stress tensor: where:

Orthotropic part 1st Piola-Kirchhoff stresses:

The total constitutive equation Total stresses: Total constitutive tangent stiffness:

Finite element implementation  u 2 1 reference configuration current configuration T6-3i triangular element: complete quadratic (all nodes) linear, nonconforming (mid-sides only) Full Gauss integration (3 integration points)

Numerical Examples Biaxial stretch test of orthotropic membrane: Analyses: Only isotropic behavior with F1 = F2. Isotropic + Orthotropic behaviors with F1 = F2. Isotropic + Orthotropic behaviors with F1:F2 = 2:1 Isotropic + Orthotropic behaviors with F1:F2 = 1:2

Numerical Examples Biaxial stretch test of orthotropic membrane: Experimental data from [Shelter Rite©, Seaman Corp., 2005] was available Curve-fitting adjustment with only 2 graph points in each analysis: (all in N/m)

Numerical Examples Biaxial stretch test of orthotropic membrane: Case 1 Case 2 Case 3 Case 4 Load versus longitudinal strains at center point A in warp and weft directions

Numerical Examples Stretching of square orthotropic membrane:

Numerical Examples Stretching of square orthotropic membrane: weft warp weft

Concluding remarks Membrane motion described with thin shell kinematics Small bending stiffness is always present Orthotropy: changes are only on the constitutive equation No fictitious non-wrinkled states are introduced Deformed configurations are possible to be visualized Performance of the T6-3i is excellent

Thank you for your attention