This section describes the formulation of the generalized
axisymmetric membrane elements. The formulation of the regular axisymmetric
membrane elements is a subset of this formulation.
Abaqus
includes two libraries of axisymmetric membrane elements, MAX and MGAX, whose geometry is axisymmetric (bodies of revolution) and that
can be subjected to axially symmetric loading conditions. In addition, MGAX elements support torsion loading and general material anisotropy.
Therefore, MGAX elements will be referred to as generalized axisymmetric membrane
elements, and MAX elements will be referred to as regular axisymmetric membrane
elements. In both cases the body of revolution is generated by revolving a line
that represents the membrane surface (a membrane has negligible thickness)
about an axis (the symmetry axis) and is described readily in cylindrical
coordinates r, z, and
.
The radial and axial coordinates of a point on this cross-section are denoted
by r and z, respectively. At
,
the radial and axial coordinates coincide with the global Cartesian
X- and Y-coordinates.
If the loading consists of radial and axial components that are independent
of
and the material is either isotropic or orthotropic, with
being a principal material direction, the displacement at any point will have
only radial ()
and axial ()
components. The only nonzero stress components are
and ,
where s denotes a length measuring coordinate along the
line representing the membrane surface on any
r–z plane. The deformation of any
r–z plane (or, more precisely, any
r–z line) completely defines the
state of stress and strain in the body. Consequently, the geometric model is
described by discretizing the reference cross-section at
.
If one allows for a circumferential component of loading (which is
independent of )
and general material anisotropy, displacements and stress fields become
three-dimensional. However, the problem remains axisymmetric in the sense that
the solution does not vary as a function of ,
and the deformation of the reference
r–z cross-section characterizes the
deformation in the entire body. The motion at any point will have—in addition
to the aforementioned radial and axial displacements—a twist
(in radians) about the z-axis, which is independent of
.
There will also be a nonzero in-plane shear stress, ,
as a result of the deformation.
Kinematic description
The coordinate system used with both families of elements is the cylindrical
system (r, z,
),
where r measures the distance of a point from the axis of
the cylindrical system, z measures its position along this
axis, and
measures the angle between the plane containing the point and the axis of the
coordinate system and some fixed reference plane that contains the coordinate
system axis. The order in which the coordinates and displacements are taken in
these elements is based on the convention that z is the
second coordinate. This order is not the same as that used in three-dimensional
elements in
Abaqus,
in which z is the third coordinate; nor is it the order
(r, ,
z) that is usually taken in cylindrical systems.
Let ,
,
and
be unit vectors in the radial, axial, and circumferential directions at a point
in the undeformed state, as shown in
Figure 1.
The reference position
of the point can be represented in terms of the original radius,
R, and the axial position, Z:
Likewise, let ,
,
and
be unit vectors in the radial, axial, and circumferential directions at a point
in the deformed state. As shown in
Figure 1,
the radial and circumferential base vectors depend on the
coordinate:
and .
The current position, ,
of the point can be represented in terms of the current radius,
r, and the current axial position, z,
as
The general axisymmetric motion at a point on the membrane surface can be
described by
As this description implies, the degrees of freedom
,
,
and
are independent of .
Moreover, the reference cross-section of interest is at
;
however, for the benefit of the mathematical analysis to follow, it is
important that
be considered an independent variable in the above expression for
.
Parametric interpolation and integration
The following isoparametric interpolation scheme is used for the motion:
where g is the isoparametric coordinate in the
reference r–z cross-section at
;
and ,
,
are the nodal degrees of freedom. The interpolation functions are identical to
those used for truss elements (see
Truss elements).
All elements use reduced integration.
Deformation gradient
For a material point the deformation gradient
is defined as the gradient of the current position, ,
with respect to the original position, ,
and is given by
The components of the deformation gradient require that two sets of
orthonormal basis vectors be defined. In the undeformed configuration the basis
vectors are defined by
where the
denote length measuring coordinates in the reference configuration along the
element length and the hoop direction, respectively. Thus,
In the current configuration
Abaqus
formulates the equations in terms of a fixed spatial basis with respect to the
axisymmetric twist degree of freedom. The basis vectors convect with the
material. However, because of the axisymmetry of the model in the deformed
configuration, these vectors can be defined at
as
where the
denote length measuring coordinates in the current configuration along the
element length and the hoop direction, respectively. Thus, the basis vectors in
the reference and current configurations can be written as
where S and s are length measuring
coordinates along the element length in the reference and current
configurations, respectively. The components of the deformation gradient in the
two sets of basis vectors may be computed as
Using the definitions of the basis vectors in
Equation 3,
the components of the deformation gradient tensor are
Virtual work
As discussed in
Equilibrium and virtual work,
the formulation of equilibrium (virtual work) requires the virtual velocity
gradient, which takes the form
where
represents the first variation of the deformation gradient tensor.
Alternatively, the virtual velocity gradient can be written as
Recall that
Abaqus
formulates the finite element equations in terms of a fixed spatial basis with
respect to the axisymmetric twist degree of freedom. Therefore, the desired
result for
does not simply follow from the linearization of
Equation 2.
Namely, the contributions from the variations
arising from the spin of the coordinate system must be canceled out. To this
end,
can be modified according to
where
instantaneously, but its variation is given by
where
is skew-symmetric with components
with respect to the basis ,
,
and
at .
With this modification, the corotational virtual deformation gradient is
given by
and the corotational virtual velocity gradient by
The individual components of
are given by
The components
are not determined by the kinematics.
Stiffness in the current state
The second variation has the usual contribution:
Moreover, there are additional contributions from ,
which are given by