Theory Background for the Membrane Interface
A 3D membrane is similar to a shell but it has only translational degrees of freedom and the results are constant in the thickness direction.
The thickness of the membrane is d, which can vary over the element. The displacements are interpolated by Lagrange shape functions.
A 2D axisymmetric membrane is similar to the 3D membrane and it has a nonzero circumferential strain in the out-of-plane direction.
Local Coordinate Systems
Many quantities for a membrane can best be interpreted in a local coordinate system aligned to the membrane surface. Material data, initial stresses-strains, and constitutive laws are always represented in the local coordinate system.
This local membrane surface coordinate system is defined by the boundary coordinate system (t1, t2, n).
The quantities like stresses and strains are also available as results in the global coordinate system after a transformation from a local (boundary) system.
Strain-Displacement Relation
The kinematic relations of the membrane element are first expressed along the global coordinate axes. The strains are then transformed to the element local direction. Since the membrane is defined only on a boundary, derivatives in all spatial directions are not directly available. This makes the derivation of the strain tensor somewhat different from what is used in solid mechanics.
The deformation gradient F is in general defined as the gradient of the current coordinates with respect to the original coordinates:
In the Membrane interface, a tangential deformation gradient is computed as
Here is a displacement gradient computed using the tangential derivative operator, and N is the normal vector to the undeformed membrane. FT now contains information about the stretching in the plane of the membrane.
The Right Cauchy-Green tensor C is generally defined as
Since the tangential deformation gradient does not contain any information about the transversal stretch λn, it must be augmented when creating C.
Similar augmentation is also needed in the full deformation gradient tensor F.
From C, the Green-Lagrange strains are computed using the standard expression
The local tangential strains in the membrane are calculated by transformation of this strain tensor into the local tangent plane coordinate system.
The Jacobian J is the ratio between the current volume and the initial volume. In full 3D it is defined as
In the membrane, only the C tensor is available, so instead the following expression is used:
The area scale factor is also computed as
In the case of geometrically linear analysis, a linearized version of the strain tensor is used.
Constitutive Relation and Weak EXPRESSIONS
The constitutive relations for the membrane on the reference surface are similar to those used in the Solid Mechanics interface.
The thermal strains and initial stresses-strains (only for the in-plane directions of the membrane) are added in the constitutive relation in a similar manner as it is done in Solid Mechanics.
The weak expressions in the Membrane interface are similar to that of linear elastic continuum mechanics.
See also Analysis of Deformation in the documentation of the Solid Mechanics interface.
External Loads
Contributions to the virtual work from the external load are of the form
where the forces (F) can be distributed over a boundary or an edge or be concentrated in a point. In the special case of a follower load, defined by its pressure p, the force intensity is where n is the normal in the deformed configuration.
For a follower load, the change in midsurface area is taken into account, and integration of the load is done in the spatial frame.
Stress Calculations
The stresses are computed by applying the constitutive law to the computed strains.
The membrane does not support transverse and bending forces so the only section forces it support is the membrane force defined as:
where is the local stress tensor and contains only in-plane stress components.