Gradient operators
3D Cartesian coordinates
The reference coordinates of a Cartesian coordinate system can be expressed as: The current coordinates can be expressed as: and the underlying motion is
The gradient operator (with respect to the reference coordinates) is given as and the deformation gradient is given as
Components of the gradient operator are
2D axisymmetric cylindrical coordinates
The reference coordinates of an axisymmetric cylindrical coordinate system can be expressed in terms of the radial coordinate , the axial coordinate , and unit vectors and : The current coordinates can be expressed in terms of coordinates and unit vectors in the current (displaced) configuration: and the underlying motion is Notice that the motion is assumed to be torsionless and .
In axisymmetric cylindrical coordinates, the gradient operator (with respect to the reference coordinates) is given as and the deformation gradient is given as
Components of the gradient operator are
1D centrosymmetric spherical coordinates
The reference coordinates of a centrosymmetric spherical coordinate system can be expressed in terms of the radial coordinate and the unit vector : The current coordinates can be expressed in terms of the radial coordinate and the unit vector in the current (displaced) configuration: and the underlying motion is Notice that the motion is torsionless.
In centrosymmetric spherical coordinates, the gradient operator (with respect to the reference coordinates) is given as and the deformation gradient is given as
Components of the gradient operator are