Axisymmetric elements

The coordinate system used with both families of elements is the cylindrical system (r, z, $\theta$), where r measures the distance of a point from the axis of the cylindrical system, z measures its position along this axis, and $\theta$ 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, $\theta$, z), usually taken in cylindrical systems.

Let ${\mathbf{e}}_R$, ${\mathbf{e}}_Z$, and ${\mathbf{e}}_{\mathrm{\Theta }}$ be unit vectors in the radial, axial, and circumferential directions at a point in the undeformed state, as shown in Figure 1.

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

The reference position $\mathbf{X}$ of the point can be represented in terms of the original radius R and the axial position Z:

Likewise, let ${\mathbf{e}}_r$, ${\mathbf{e}}_z$, and ${\mathbf{e}}_{\theta }$ 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 $\theta$ coordinate: ${\mathbf{e}}_r={\mathbf{e}}_r\left(\theta \right)$ and ${\mathbf{e}}_{\theta }={\mathbf{e}}_{\theta }\left(\theta \right)$. The current position $\mathbf{x}$ of the point can be represented in terms of the current radius r and the current axial position z:

The general axisymmetric motion at a point can be described by

As the above description implies, the degrees of freedom $u_r$, $u_z$, and $\varphi$ are independent of $\mathrm{\Theta }$. Moreover, the reference cross-section of interest is at $\mathrm{\Theta }=0$, but for the benefit of the mathematical analysis to follow it is important that $\mathrm{\Theta }$ be nonzero in the above expression for $\theta$.

The following isoparametric interpolation scheme for the motion is used:

where g, $h$ are isoparametric coordinates in the reference $r$–z cross-section at $\mathrm{\Theta }=0$, and ${\overline{u_r}}^N$, ${\overline{u_z}}^N$, ${\overline{\varphi }}^N$ are the nodal degrees of freedom. The interpolation functions $N^N\left(g,h\right)$ are those described in Solid isoparametric quadrilaterals and hexahedra, where the integration scheme of isoparametric solid elements is also discussed.

For a material point in space, the deformation gradient $\mathbf{F}$ is defined as the gradient of the current position $\mathbf{x}$ with respect to the original position $\mathbf{X}$:

The current position $\mathbf{x}$ is given by Equation 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 $\theta$, the partial derivatives of these base vectors with respect to $\theta$ 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 2 has been used explicitly.

Similarly, the inverse deformation gradient ${\mathbf{F}}^{-1}$ is readily obtained as

As discussed in Equilibrium and virtual work, the formulation of equilibrium (virtual work) requires the virtual velocity gradient $\delta \mathbf{L}$, which is the variation in the gradient of the position with respect to the current state. This tensor is given by

where $\delta \mathbf{F}$ 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 $\delta \mathbf{F}$ in Equation 4 does not simply follow from the linearization of Equation 3. Namely, it is necessary to cancel out the contributions from the variations

To this end $\mathbf{F}$ can be modified according to

where $\mathbf{R}=\mathbf{I}$ instantaneously, but its variation is given by

where $\widehat{\delta \mathit{\varphi }}$ is skew-symmetric with components

with respect to the basis ${\mathbf{e}}_r$, ${\mathbf{e}}_z$, and ${\mathbf{e}}_{\theta }$ at $\mathrm{\Theta }=0$.

With this modification the corotational virtual deformation gradient is given by

and the corotational virtual velocity gradient by

or

The modified virtual rate of deformation tensor and spin are simply

As shown in About procedures and basic equations, 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

The second variation in $\delta \tilde{\mathbf{D}}$ is obtained as

where $d\tilde{\mathbf{L}}$ has the same form as $\delta \tilde{\mathbf{L}}$ in Equation 5. Moreover, in this formulation $d\delta \tilde{\mathbf{F}}$ is nonzero, and it can be shown with the aid of Rotation variables that $d\delta \tilde{\mathbf{F}}\cdot {\tilde{\mathbf{F}}}^{-1}$ has the form

In component form,

Introduction of the corotational stress rate

yields the more familiar form