Appendix A
Elasticity Equations in Special
Coordinate Systems
This appendix provides the deformation gradient and equations of motion for
general 3D deformation in Cartesian, cylindrical, and spherical coordinates.
A.1 Cartesian Coordinates
x = x(X, Y, Z), y = y(X, Y, Z), z = z(X, Y Z)
∇ = e X
∂
∂X
+ e Y
∂
∂Y
+ e Z
∂
∂Z
∇ = e x
∂
∂x
+ e y
∂
∂y
+ e z
∂
∂z
A.1.1 Deformation Gradient
⎡
⎢
⎢
⎣
F xx F xy F xz
F yx F yy F yz
F zx F zy F zz
⎤
⎥
⎥
⎦ =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
∂x
∂X
∂x
∂Y
∂x
∂Z
∂y
∂X
∂y
∂Y
∂y
∂Z
∂z
∂X
∂z
∂Y
∂z
∂Z
⎤
⎥
⎥
⎥
⎥
⎥
⎦
© Springer Nature Switzerland AG 2020
L. A. Taber, Continuum Modeling in Mechanobiology,
https://doi.org/10.1007/978-3-030-43209-6
519