Talk:Nonlinear finite elements/Lagrangian finite elements
Add topic1. To clarify Jacobian determinant
Jacobian determinant is the same as the determinant of the deformation gradient
It's often simply called the Jacobian as well (see here a good explanation why is called Jacobian, because
"This is such an important result that is given a special symbol, , and a special name, the Jacobian."
2. It's also a bit difficult to understand how this equation is derived. Since it we start from the definition of Jacobian, it will be
which is different from the presented equation .
But I found an explanation from Ted Belytschko’s book, p.22, explaining how is derived.
The Jocobian is usually defined by 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, , to the corresponding volume of the segment in the undeformed body :
If we substitute into above equation, we will get ( as it is a scalar for this 1D problem)
which is consistent to the interpretation of as volume ratio.
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