Nonlinear finite elements/Motion in Lagrangian form

Motion in Lagrangian form

The motion is given by

$x=\phi (X,t)=x(X,t),X\in [0,L_0].$

For the reference configuration,

$X=\phi (X,0)=x(X,0).$

The displacement is

$u(X,t)=\phi (X,t)-X=x-X.$

For the reference configuration,

$u_0=u(X,0)=\phi (X,0)-X=X-X=0.$

The deformation gradient is

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

For the reference configuration,

$F_0=F(X,0)=\frac{\partial }{\partial X}[\phi (X,0)]=\frac{\partial X}{\partial X}=1.$

The Jacobian determinant of the motion is

$J=\frac{A}{A_0}F.$

For the reference configuration,

$J_0=\frac{A_0}{A_0}{F}_0=1.$

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