Nonlinear finite elements/Motion in Lagrangian form

The motion is given by

${\displaystyle {x=\varphi (X,t)=x(X,t)~,~~\qquad X\in [0,L_{0}]~.}}$

For the reference configuration,

${\displaystyle X=\varphi (X,0)=x(X,0)~.}$

The displacement is

${\displaystyle {u(X,t)=\varphi (X,t)-X=x-X~.}}$

For the reference configuration,

${\displaystyle u_{0}=u(X,0)=\varphi (X,0)-X=X-X=0~.}$

The deformation gradient is

${\displaystyle {F(X,t)={\frac {\partial }{\partial X}}[\varphi (X,t)]={\frac {\partial x}{\partial X}}~.}}$

For the reference configuration,

${\displaystyle F_{0}=F(X,0)={\frac {\partial }{\partial X}}[\varphi (X,0)]={\frac {\partial X}{\partial X}}=1~.}$

The Jacobian determinant of the motion is

${\displaystyle {J={\cfrac {A}{A_{0}}}F~.}}$

For the reference configuration,

${\displaystyle J_{0}={\cfrac {A_{0}}{A_{0}}}F_{0}=1~.}$