Gradient operators

3D Cartesian coordinates

The reference coordinates of a Cartesian coordinate system can be expressed as: X=XeX+YeY+ZeZ. \boldsymbol{X} = X \textbf{e}_X + Y \textbf{e}_Y + Z \textbf{e}_Z. The current coordinates can be expressed as: x=xex+yey+zez, \boldsymbol{x} = x \textbf{e}_x + y \textbf{e}_y + z \textbf{e}_z, and the underlying motion is x(X,Y,Z)=X+ux(X,Y,Z),y(X,Y,Z)=Y+uy(X,Y,Z),z(X,Y,Z)=Z+uz(X,Y,Z). \begin{aligned} x(X,Y,Z) &= X + u_x(X,Y,Z), \\ y(X,Y,Z) &= Y + u_y(X,Y,Z), \\ z(X,Y,Z) &= Z + u_z(X,Y,Z). \end{aligned}

The gradient operator (with respect to the reference coordinates) is given as (),X=(),XeX+(),YeY+(),ZeZ, (\cdot)_{,\boldsymbol{X}} = (\cdot)_{,X}\textbf{e}_X + (\cdot)_{,Y}\textbf{e}_Y + (\cdot)_{,Z}\textbf{e}_Z, 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. \begin{aligned} \boldsymbol{F} = &\ (1+u_{x,X}) \textbf{e}_x\textbf{e}_X + u_{x,Y} \textbf{e}_x\textbf{e}_Y + u_{x,Z} \textbf{e}_x\textbf{e}_Z \\ &+ u_{y,X} \textbf{e}_y\textbf{e}_X + (1+u_{y,Y}) \textbf{e}_y\textbf{e}_Y + u_{y,Z} \textbf{e}_y\textbf{e}_Z \\ &+ u_{z,X} \textbf{e}_z\textbf{e}_X + u_{z,Y} \textbf{e}_z\textbf{e}_Y + (1+u_{z,Z}) \textbf{e}_z\textbf{e}_Z. \end{aligned}

Components of the gradient operator are xϕx=ϕx,XexeX+ϕx,YexeY+ϕx,ZexeZ,yϕy=ϕy,XeyeX+ϕy,YeyeY+ϕy,ZeyeZ,zϕz=ϕz,XezeX+ϕz,YezeY+ϕz,ZezeZ. \begin{aligned} \nabla^x \phi_x &= \phi_{x,X}\textbf{e}_x\textbf{e}_X + \phi_{x,Y}\textbf{e}_x\textbf{e}_Y + \phi_{x,Z}\textbf{e}_x\textbf{e}_Z, \\ \nabla^y \phi_y &= \phi_{y,X}\textbf{e}_y\textbf{e}_X + \phi_{y,Y}\textbf{e}_y\textbf{e}_Y + \phi_{y,Z}\textbf{e}_y\textbf{e}_Z, \\ \nabla^z \phi_z &= \phi_{z,X}\textbf{e}_z\textbf{e}_X + \phi_{z,Y}\textbf{e}_z\textbf{e}_Y + \phi_{z,Z}\textbf{e}_z\textbf{e}_Z. \end{aligned}

2D axisymmetric cylindrical coordinates

The reference coordinates of an axisymmetric cylindrical coordinate system can be expressed in terms of the radial coordinate RR, the axial coordinate ZZ, and unit vectors eR\textbf{e}_R and eZ\textbf{e}_Z: X=ReR+ZeZ. \boldsymbol{X} = R \textbf{e}_R + Z \textbf{e}_Z. The current coordinates can be expressed in terms of coordinates and unit vectors in the current (displaced) configuration: x=rer+zez, \boldsymbol{x} = r \textbf{e}_r + z \textbf{e}_z, and the underlying motion is r(R,Z)=R+ur(R,Z),z(R,Z)=Z+uz(R,Z),θ(Θ)=Θ+uθ(Θ). \begin{aligned} r(R,Z) &= R + u_r(R,Z), \\ z(R,Z) &= Z + u_z(R,Z), \\ \theta(\Theta) &= \Theta + u_\theta(\Theta). \end{aligned} Notice that the motion is assumed to be torsionless and Θx=0\nabla_{\Theta} \boldsymbol{x} = 0.

In axisymmetric cylindrical coordinates, the gradient operator (with respect to the reference coordinates) is given as (),X=(),ReR+(),ZeZ+1R(),ΘeΘ, (\cdot)_{,\boldsymbol{X}} = (\cdot)_{,R}\textbf{e}_R + (\cdot)_{,Z}\textbf{e}_Z + \frac{1}{R}(\cdot)_{,\Theta}\textbf{e}_\Theta, and the deformation gradient is given as F=(1+ur,R)ereR+ur,ZereZ+(1+urR)eθeΘ+uz,RezeR+(1+uz,Z)ezeZ \begin{aligned} \boldsymbol{F} &= (1+u_{r,R}) \textbf{e}_r\textbf{e}_R + u_{r,Z} \textbf{e}_r\textbf{e}_Z + \left(1+\frac{u_r}{R}\right)\textbf{e}_\theta\textbf{e}_\Theta + u_{z,R} \textbf{e}_z\textbf{e}_R + (1+u_{z,Z}) \textbf{e}_z\textbf{e}_Z \end{aligned}

Components of the gradient operator are rϕr=ϕr,RereR+ϕr,ZereZ+ϕrReθeΘ,zϕz=ϕz,RezeR+ϕz,ZezeZ. \begin{aligned} \nabla^r \phi_r &= \phi_{r,R}\textbf{e}_r\textbf{e}_R + \phi_{r,Z}\textbf{e}_r\textbf{e}_Z + \frac{\phi_{r}}{R}\textbf{e}_\theta\textbf{e}_\Theta, \\ \nabla^z \phi_z &= \phi_{z,R}\textbf{e}_z\textbf{e}_R + \phi_{z,Z}\textbf{e}_z\textbf{e}_Z. \end{aligned}

1D centrosymmetric spherical coordinates

The reference coordinates of a centrosymmetric spherical coordinate system can be expressed in terms of the radial coordinate RR and the unit vector eR\textbf{e}_R: X=ReR. \boldsymbol{X} = R \textbf{e}_R. The current coordinates can be expressed in terms of the radial coordinate and the unit vector in the current (displaced) configuration: x=rer, \boldsymbol{x} = r \textbf{e}_r, and the underlying motion is r(R,Z)=R+ur(R,Z),θ(Θ)=Θ+uθ(Θ),ϕ(Φ)=Φ+uϕ(Φ). \begin{aligned} r(R,Z) &= R + u_r(R,Z), \\ \theta(\Theta) &= \Theta + u_\theta(\Theta), \\ \phi(\Phi) &= \Phi + u_\phi(\Phi). \\ \end{aligned} Notice that the motion is torsionless.

In centrosymmetric spherical coordinates, the gradient operator (with respect to the reference coordinates) is given as (),X=(),ReR+1R(),ΘeΘ+1R(),ΦeΦ, (\cdot)_{,\boldsymbol{X}} = (\cdot)_{,R}\textbf{e}_R + \frac{1}{R}(\cdot)_{,\Theta}\textbf{e}_\Theta + \frac{1}{R}(\cdot)_{,\Phi}\textbf{e}_\Phi, and the deformation gradient is given as F=(1+ur,R)ereR+(1+urR)eθeΘ+(1+urR)eϕeΦ. \begin{aligned} \boldsymbol{F} &= (1+u_{r,R}) \textbf{e}_r\textbf{e}_R + \left(1+\frac{u_r}{R}\right)\textbf{e}_\theta\textbf{e}_\Theta + \left(1+\frac{u_r}{R}\right)\textbf{e}_\phi\textbf{e}_\Phi. \end{aligned}

Components of the gradient operator are rϕr=ϕr,RereR+ϕrReθeΘ+ϕrReϕeΦ. \nabla^r \phi_r = \phi_{r,R}\textbf{e}_r\textbf{e}_R + \frac{\phi_{r}}{R}\textbf{e}_\theta\textbf{e}_\Theta + \frac{\phi_{r}}{R}\textbf{e}_\phi\textbf{e}_\Phi.