Gradient operators
The reference coordinates of a Cartesian coordinate system can be expressed as: X=XeX+YeY+ZeZ. The current coordinates can be expressed as: x=xex+yey+zez, and the underlying motion is x(X,Y,Z)y(X,Y,Z)z(X,Y,Z)=X+ux(X,Y,Z),=Y+uy(X,Y,Z),=Z+uz(X,Y,Z).
The gradient operator (with respect to the reference coordinates) is given as (⋅),X=(⋅),XeX+(⋅),YeY+(⋅),ZeZ, and the deformation gradient is given as F= (1+ux,X)exeX+ux,YexeY+ux,ZexeZ+uy,XeyeX+(1+uy,Y)eyeY+uy,ZeyeZ+uz,XezeX+uz,YezeY+(1+uz,Z)ezeZ.
Components of the gradient operator are ∇xϕx∇yϕy∇zϕz=ϕx,XexeX+ϕx,YexeY+ϕx,ZexeZ,=ϕy,XeyeX+ϕy,YeyeY+ϕy,ZeyeZ,=ϕz,XezeX+ϕz,YezeY+ϕz,ZezeZ.
The reference coordinates of an axisymmetric cylindrical coordinate system can be expressed in terms of the radial coordinate R, the axial coordinate Z, and unit vectors eR and eZ: X=ReR+ZeZ. The current coordinates can be expressed in terms of coordinates and unit vectors in the current (displaced) configuration: x=rer+zez, and the underlying motion is r(R,Z)z(R,Z)θ(Θ)=R+ur(R,Z),=Z+uz(R,Z),=Θ+uθ(Θ). Notice that the motion is assumed to be torsionless and ∇Θx=0.
In axisymmetric cylindrical coordinates, the gradient operator (with respect to the reference coordinates) is given as (⋅),X=(⋅),ReR+(⋅),ZeZ+R1(⋅),ΘeΘ, and the deformation gradient is given as F=(1+ur,R)ereR+ur,ZereZ+(1+Rur)eθeΘ+uz,RezeR+(1+uz,Z)ezeZ
Components of the gradient operator are ∇rϕr∇zϕz=ϕr,RereR+ϕr,ZereZ+RϕreθeΘ,=ϕz,RezeR+ϕz,ZezeZ.
The reference coordinates of a centrosymmetric spherical coordinate system can be expressed in terms of the radial coordinate R and the unit vector eR: X=ReR. The current coordinates can be expressed in terms of the radial coordinate and the unit vector in the current (displaced) configuration: x=rer, and the underlying motion is r(R,Z)θ(Θ)ϕ(Φ)=R+ur(R,Z),=Θ+uθ(Θ),=Φ+uϕ(Φ). Notice that the motion is torsionless.
In centrosymmetric spherical coordinates, the gradient operator (with respect to the reference coordinates) is given as (⋅),X=(⋅),ReR+R1(⋅),ΘeΘ+R1(⋅),ΦeΦ, and the deformation gradient is given as F=(1+ur,R)ereR+(1+Rur)eθeΘ+(1+Rur)eϕeΦ.
Components of the gradient operator are ∇rϕr=ϕr,RereR+RϕreθeΘ+RϕreϕeΦ.