1. 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

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

3. 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

4. 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

5. 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

6. 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

7. 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)

8. Introduction: benchmark examples

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

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

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

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

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

14. Underlying shell model
External virtual work:
where:

15. Underlying shell model
Shell equilibrium (weak form):

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

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

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

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

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

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

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

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

24. Orthotropic part
1st Piola-Kirchhoff stresses:

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

26. 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)

27. 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

28. 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)

29. 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

30. Numerical Examples
Stretching of square orthotropic membrane:

31. Numerical Examples Stretching of square orthotropic membrane: weft
warp
weft

32. 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

33. Thank you for your attention