3.2.8 Axisymmetric elements

Products: ABAQUS/Standard ABAQUS/Explicit

ABAQUS includes two libraries of solid elements, CAX and CGAX, whose geometry is axisymmetric (bodies of revolution) and which can be subjected to axially symmetric loading conditions. In addition, CGAX elements support torsion loading. As a result, CGAX elements will be referred to as generalized axisymmetric elements, and CAX elements as torsionless axisymmetric elements. In both cases, the body of revolution is generated by revolving a plane cross-section about an axis (the symmetry axis) and is readily described in cylindrical polar coordinates , , and . The radial and axial coordinates of a point on this cross-section are denoted by and , respectively. At , the radial and axial coordinates coincide with the global Cartesian - and -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 only have radial () and axial () components and the only stress components that will be nonzero are , , , and . Moreover, the deformation of any – plane completely defines the state of strain and stress 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 for general material anisotropy, displacements and stress fields become three-dimensional, but the problem remains axisymmetric in the sense that the solution does not vary as a function of and the deformation of the reference – cross-section still 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 -axis, which is independent of .

This section describes the formulation of the generalized axisymmetric elements. The formulation of the torsionless axisymmetric elements is a subset of this formulation.

Kinematic description

The coordinate system used with both families of elements is the cylindrical system (, , ), where measures the distance of a point from the axis of the cylindrical system, 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 is the second coordinate. This order is not the same as that used in three-dimensional elements in ABAQUS, in which is the third coordinate, nor is it the order (, , ), 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 3.2.8–1.

Figure 3.2.8–1 Cylindrical coordinate system and definition of position vectors.

The reference position

of the point can be represented in terms of the original radius

and the axial position

Likewise, let

, and

be unit vectors in the radial, axial, and circumferential directions at a point in the deformed state. As shown in Figure 3.2.8–1, 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

and the current axial position

The general axisymmetric motion at a point can be described by

As the above description implies, the degrees of freedom

, and

are independent of

. Moreover, the reference cross-section of interest is at

, but for the benefit of the mathematical analysis to follow it is important that

be nonzero in the above expression for

Parametric interpolation and integration

The following isoparametric interpolation scheme for the motion is used:

where

are isoparametric coordinates in the reference

–

cross-section at

, and

are the nodal degrees of freedom. The interpolation functions

are those described in “Solid isoparametric quadrilaterals and hexahedra,” Section 3.2.4, where the integration scheme of isoparametric solid elements is also discussed.

Deformation gradient

For a material point in space, the deformation gradient is defined as the gradient of the current position with respect to the original position :

The current position

is given by Equation 3.2.8–1, and the gradient operator can be described in terms of partial derivatives with respect to the cylindrical coordinates:

Since the radial and circumferential base vectors depend on the original circumferential coordinate

, the partial derivatives of these base vectors with respect to

are nonvanishing:

Thus, the chain rule allows us to write

With these results, the deformation gradient is obtained as

Alternatively, it can be written in matrix form as

where the motion given by Equation 3.2.8–2 has been used explicitly.

Similarly, the inverse deformation gradient is readily obtained as

Virtual work

As discussed in “Equilibrium and virtual work,” Section 1.5.1, the formulation of equilibrium (virtual work) requires the virtual velocity gradient , which is the variation in the gradient of the position with respect to the current state. This tensor is given by

where

is the linearized deformation gradient.

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 in Equation 3.2.8–4 does not simply follow from the linearization of Equation 3.2.8–3. Namely, it is necessary to cancel out the contributions from the variations

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

With this modification the corotational virtual deformation gradient is given by

and the corotational virtual velocity gradient by

The modified virtual rate of deformation tensor is simply

Stiffness in the current state

As shown in “Procedures: overview and basic equations,” Section 2.1.1, the contribution of the internal work terms to the Jacobian of the Newton method that is used in ABAQUS/Standard for solid element formulations is