3.4.1 Membrane elements

Products: ABAQUS/Standard ABAQUS/Explicit

Membrane elements are sheets in space that can carry membrane force but do not have any bending or transverse shear stiffness, so the only nonzero stress components in the membrane are those components parallel to the middle surface of the membrane: the membrane is in a state of plane stress.

At any time we use a local orthonormal basis system , where and are in the surface of the membrane and is normal to the membrane. The basis system is defined by the standard convention used in ABAQUS for a basis on a surface in space. In this section Greek indices take the range 1, 2, and Latin indices take the range 1, 2, 3. Greek indices are used to refer to components in the first two directions of the local orthonormal basis (in the surface of the membrane).

Equilibrium

The virtual work contribution from the internal forces in a membrane element is

where

is the Cauchy stress,

is the virtual rate of deformation (

, where

is the virtual velocity field), and V is the current volume of the membrane.

We assume that only the membrane stress components in the surface of the membrane are nonzero: . Then Equation 3.4.1–1 simplifies to

where

and

, where t is the current thickness of the element and A is its current area.

Jacobian

The consistent Jacobian contribution from the element is

Since we assume that

, the first term in the integrand is

We also assume that there is no transverse shear strain of the element:

, and, hence,

. Thus, the second term in the integrand is

We can write this out as

Thickness change

In geometrically nonlinear analyses the cross-section thickness changes as a function of the membrane strain with a user-defined “effective section Poisson's ratio,” .

In plane stress ; linear elasticity gives

Treating these as logarithmic strains,

where A is the area on the membrane's reference surface. This nonlinear analogy with linear elasticity leads to the thickness change relationship:

For

the material is incompressible; for

the section thickness does not change.

Total deformation

The deformation gradient is . Since we take normal to the current membrane surface and assume no transverse shear of the membrane,

By the thickness change assumption above, the direct out-of-plane component of the deformation gradient is

To calculate the deformation gradient at the end of the increment, first we calculate the two tangent vectors at the end of the increment defined by the derivative of the position with respect to the reference coordinates:

where

is obtained by interpolation with the shape function derivatives from the nodal coordinates and the change of coordinate transformation

is based on the reference geometry. The deformation gradient components are defined

To choose the element basis directions , we do the following. Find any pair of in-plane orthonormal vectors (by the standard ABAQUS projection). Then find the angle such that the element basis vectors , defined

satisfy the symmetry condition

Using the definitions of

in terms of

in the above equation, the rotation angle

is found to be

where

The deformation gradient then follows immediately.

For elastomers we work directly in terms of and . For inelastic material models we need measures of incremental strain and average material rotation, which we compute from defined by , where is the deformation gradient at the start of the current increment (at increment “n”):