Talk:Nonlinear finite elements/Lagrangian finite elements

1. To clarify Jacobian determinant

Jacobian determinant is the same as the determinant of the deformation gradient

$J=\det {\boldsymbol {F}}$

It's often simply called the Jacobian as well (see here a good explanation why $\det {\boldsymbol {F}}$ is called Jacobian, because

"This is such an important result that $\det {\boldsymbol {F}}$ is given a special symbol, $J$, and a special name, the Jacobian."

2. It's also a bit difficult to understand how this equation ${J={\cfrac {A}{A_{0}}}F}$ is derived. Since it we start from the definition of Jacobian, it will be

$J=\det {\boldsymbol {F}}=\det {\frac {\partial x}{\partial X}}=\det {\frac {\partial ({\frac {L}{L_{0}}}X)}{\partial X}}=\det {\frac {L}{L_{0}}}={\frac {L}{L_{0}}}$

which is different from the presented equation ${J={\cfrac {A}{A_{0}}}F}$.

But I found an explanation from Ted Belytschko’s book, p.22, explaining how ${J={\cfrac {A}{A_{0}}}F}$ is derived.

The Jocobian is usually defined by $J=\det {\boldsymbol {F}}={\frac {\partial x}{\partial X}}~$ for one-dimensional maps. However, to maintain the consistency with multi-dimensional formulations, we will define the Jacobian as the ratio of an infinitesimal volume in the deformed body, $A\Delta x$, to the corresponding volume of the segment in the undeformed body $A_{0}\Delta X$:

${J={\frac {\partial x}{\partial X}}{\frac {A}{A_{0}}}={\cfrac {A}{A_{0}}}F~}$

If we substitute $\det {\boldsymbol {F}}={\frac {L}{L_{0}}}$ into above equation, we will get ($\det {\boldsymbol {F}}=F~$ as it is a scalar for this 1D problem)

${J={\cfrac {A}{A_{0}}}F={\cfrac {A}{A_{0}}}{\frac {L}{L_{0}}}={\frac {V}{V_{0}}}}$

which is consistent to the interpretation of $J$ as volume ratio.

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