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 or Serendipity 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
Boundaries
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 (tl, 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.
Local Edge System
Many features, such as an edge load, allow input in an edge local coordinate system. The orthogonal local edge coordinate system directions xl, yl, and zl are defined so that:
The first direction (xl) is along the edge. This direction can be visualized by selecting the Show edge directions arrows check box in the View settings.
The third direction (zl) is the same as the membrane normal direction of the adjacent boundary.
The second direction (yl) is in the plane of the shell and orthogonal to the edge. It is formed by the cross product of zl and xl; yl = zl × xl.
The local edge system can be visualized by plotting the components of the local edge transformation matrix with an Arrow Line plot. The matrix components are defined per feature. For instance, the variable name for the xx-component is <interface>.<feature_tag>.TleXX.
When an edge is shared between two or more boundaries, the directions may not always be unique. It is then possible to use the control Face Defining the Local Orientations to select from which boundary the normal direction zl should be picked. The default is Use face with lowest number.
If the geometry selection contains several edges, the only available option is Use face with lowest number, since the list of adjacent boundaries would then be different for each edge. For each edge in the selection, the face with the lowest number attached to that edge is then used for the definition of the normal orientation.
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 tu is the displacement gradient computed using the tangential gradient on the membrane surface, and N is the normal vector to the undeformed membrane. The tangential deformation gradient Ft then contains information about the stretching in the membrane plane.
Since the tangential deformation gradient does not contain any information about the transverse (out-of-plane) stretch λn, it must be augmented by the normal deformation gradient Fn to define the full deformation gradient. It is given by
where n is the normal vector to the deformed membrane. For anisotropic materials, the shear deformation gradient Fs is also needed to define the full deformation gradient. It is given by
where t1 and t2 are the tangent vectors on the deformed membrane surface. The full deformation gradient F is then computed from the sum of tangential, shear, and normal deformation
Note that Fs is only nonzero for anisotropic materials, otherwise Fs = 0.
The right Cauchy–Green tensor C is then defined as
The Green–Lagrange strains are computed using the standard expression
The local tangential strains are calculated by transforming this tensor into the local coordinate system.
The Jacobian J is the ratio between the current volume and the initial volume. In full 3D it is defined as
and the area scale factor is computed from
In the case of geometrically linear analysis, a linearized version of the strain tensor is used.
Constitutive Relation and Weak Contributions
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.
Wrinkling
Due to the zero bending stiffness assumption, the numerical treatment of thin structures is much simpler with the membrane theory as compared to shell theory. However, this assumption is disadvantageous in some cases such as wrinkling, which happens when the membrane is subjected to negative principal stresses.
A thin shell will wrinkle when the compressive stress reaches a critical level defined by its bending stiffness, which is a local buckling phenomenon. When such thin structures are modeled within the membrane theory, wrinkles appear at the onset of the compressive stresses as the bending stiffness is assumed to be zero. Due to zero bending stiffness such states can be represented by continuously distributed infinitesimal wrinkles.
When using the traditional membrane theory, which does not incorporates a wrinkling model, negative principal stresses result in an equilibrium instability. In order to overcome this instability, the wrinkling model within the framework of the tension field theory can remove compressive stresses from wrinkled regions resulting in a correct stress distribution, Ref. 13.
The modified membrane theory, which incorporates a wrinkling model to the traditional membrane theory, disregards the out-of-plane deformation in wrinkling; so wrinkles including details like their amplitude and wavelength are determined on the mean surface of the membrane (Ref. 13).
There are two approaches to embed the tension field theory into the framework of classical membrane theory, but essentially both approaches are equivalent (Ref. 3). One approach is to replace the strain energy density with a relaxed variant of it, while the other approach is to modify the deformation gradient (Ref. 1). While the first approach can only be used for isotropic membranes, the second approach is more general (Ref. 1), and it is the method implemented in COMSOL Multiphysics.
The undeformed configuration of the membrane is described by the material coordinates X, Y, and Z. The uniaxial stress occurs in the X direction; Y is the wrinkling direction and Z is the normal to the membrane plane (Figure 7-1). The deformed configuration is represented by the x, y, and z coordinates. After deformation, the membrane can be in one of three possible states:
As shown in Figure 7-1 there are three different kinematic descriptions (Ref. 1):
The tensor F* maps the reference configuration to the true wrinkled configuration. This mapping is not suitable to describe the strain field in wrinkled membranes.
The tensor F maps the reference configuration to the mean configuration, where the current area is smaller than the actual wrinkled area. Hence, this mapping is also not suitable to describe the strain field in wrinkled membranes.
Figure 7-1: Kinematics of wrinkling.
When n1 is the direction of uniaxial extension, and assuming that wrinkling occurs in the n2 direction, the modified deformation tensor is written as
(7-1)
Here, β is the wrinkling parameter, so β = 0 represents a taut condition. According to the orthogonality condition in tension field theory, these two vectors satisfy
(7-2)
where σ is the Cauchy stress tensor written in terms of the second Piola–Kirchhoff stress tensor
Assuming that the mean configuration F is known, the only unknowns in Equation 7-1 and Equation 7-2 are β and n2.
Since the material properties and membrane theory itself are given in the reference configuration, Equation 7-1 and Equation 7-2 are transformed to the reference configuration (Ref. 2). The fictive Green–Lagrange strain tensor is then written with the help of a vector in the reference configuration which corresponds to the wrinkling direction n2, so g = n2F,
This can be written as
where βm and N2 are the wrinkling parameter and wrinkling direction in the reference configuration. The two unknowns β and n2 in the deformed configuration are replaced by the two unknowns βm and N2 in the reference configuration.
The membrane surface is spanned by a coordinate system having two in-plane orthogonal unit vectors e1 and e2. Thus, N1 and N2 can be written with the help of the wrinkling angle αm
So the scalar wrinkling angle αm is sufficient to define the wrinkling vector N2. The two nonlinear equations in Equation 7-2 are then solved for the two unknowns αm and βm with the Newton-Raphson method. Once the parameters αm and βm are determined, the deformation gradient and the second Piola–Kirchhoff stress tensor are modified to get the correct stress distribution and to define the wrinkled regions.
External Loads
The contributions to the virtual work from an external load is given by
where the force F can be distributed over a boundary, an edge, or it can be concentrated in a point. In the special case of a follower load, defined by a pressure p, the force intensity is F = pn, where n is the normal vector to the membrane surface 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 deformed configuration.
Stress Calculations
The stresses are computed by applying the constitutive law to the computed strains.
Membranes do not support transverse nor bending forces, and the only section forces are defined as:
Here, is the local stress tensor which contains in-plane stress components only.
References
1. D. Roddeman, J. Drukker, C. Oomens, and J. Janssen, “The Wrinkling of Thin Membranes: Part 1: Theory,” J. Appl. Mech.: Transactions of the ASME, vol  54, pp. 884–887, 1987.
2. H. Schoop, L.Taenzer, and J. Hornig, “Wrinkling of Nonlinear Membranes,” Comp. Mech., vol. 29, pp. 68–74, 2002.
3. A. Patil, “Inflation and Instabilities of Hyperelastic Membranes,” PhD thesis, KTH, 2016.