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.