# Nonlinear finite elements/Kinematics - motion and displacement

## Continuum Mechanics

To understand the updated Lagrangian formulation and nonlinear finite elements of solids, we have to know continuum mechanics. A brief introduction to continuum mechanics is given in the following. If you find this handout difficult to follow, please read Chapter 2 from Belytschko's book and Chapter 9 from Reddy's book. You should also read an introductory text on continuum mechanics such as Nonlinear continuum mechanics for finite element analysis by Bonet and Wood.

### Motion

Let the undeformed (or reference) configuration of the body be ${\displaystyle \Omega _{0}}$ and let the undeformed boundary be ${\displaystyle \Gamma _{0}}$. Let the deformed (or current) configuration be ${\displaystyle \Omega }$ with boundary ${\displaystyle \Gamma }$. Let ${\displaystyle {\boldsymbol {\varphi }}(\mathbf {X} ,t)}$ be the motion that takes the body from the reference to the current configuration (see Figure 1).

 Figure 1. The motion of a body.

We write

${\displaystyle \mathbf {x} ={\boldsymbol {\varphi }}({\boldsymbol {X}},t)}$

where ${\displaystyle \mathbf {x} }$ is the position of material point ${\displaystyle {\boldsymbol {X}}}$ at time ${\displaystyle t}$.

In index notation,

${\displaystyle x_{i}=\varphi _{i}(X_{j},t)~,\qquad i,j=1,2,3.}$

### Displacement

The displacement of a material point is given by

${\displaystyle \mathbf {u} ({\boldsymbol {X}},t)={\boldsymbol {\varphi }}({\boldsymbol {X}},t)-{\boldsymbol {\varphi }}({\boldsymbol {X}},0)={\boldsymbol {\varphi }}({\boldsymbol {X}},t)-{\boldsymbol {X}}=\mathbf {x} -{\boldsymbol {X}}~.}$

In index notation,

${\displaystyle u_{i}=\varphi _{i}(X_{j},t)-X_{j}{\delta }_{ij}=x_{i}-X_{j}{\delta }_{ij}~.}$

where ${\displaystyle {\delta }_{ij}}$ is the Kronecker delta.

### Velocity

The velocity is the material time derivative of the motion (i.e., the time derivative with ${\displaystyle \mathbf {X} }$ held constant). This type of derivative is also called the total derivative.

${\displaystyle \mathbf {v} ({\boldsymbol {X}},t)={\frac {\partial }{\partial t}}\left[{\boldsymbol {\varphi }}({\boldsymbol {X}},t)\right]~.}$

Now,

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

Therefore, the material time derivative of ${\displaystyle \mathbf {u} }$ is

${\displaystyle {\dot {\mathbf {u} }}={\frac {\partial }{\partial t}}\left[\mathbf {u} ({\boldsymbol {X}},t)\right]={\frac {\partial }{\partial t}}\left[{\boldsymbol {\varphi }}({\boldsymbol {X}},t)-{\boldsymbol {X}}\right]={\frac {\partial }{\partial t}}\left[{\boldsymbol {\varphi }}({\boldsymbol {X}},t)\right]=\mathbf {v} ({\boldsymbol {X}},t)~.}$

Alternatively, we could have expressed the velocity in terms of the spatial coordinates ${\displaystyle \mathbf {x} }$. Let

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

Then the material time derivative of ${\displaystyle \mathbf {u} (\mathbf {x} ,t)}$ is

${\displaystyle {\cfrac {D}{Dt}}\left[\mathbf {u} (\mathbf {x} ,t)\right]={\frac {\partial \mathbf {u} }{\partial t}}+{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}{\frac {\partial \mathbf {x} }{\partial t}}={\frac {\partial \mathbf {u} }{\partial t}}+{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}{\frac {\partial }{\partial t}}\left[{\boldsymbol {\varphi }}({\boldsymbol {X}},t)\right]=\mathbf {v} (\mathbf {x} ,t)+{\frac {\partial \mathbf {u} }{\partial \mathbf {x} }}\mathbf {v} ({\boldsymbol {X}},t)~.}$

### Acceleration

The acceleration is the material time derivative of the velocity of a material point.

${\displaystyle \mathbf {a} ({\boldsymbol {X}},t)={\frac {\partial }{\partial t}}\left[\mathbf {v} ({\boldsymbol {X}},t)\right]={\dot {\mathbf {v} }}={\frac {\partial ^{2}}{\partial t^{2}}}\left[\mathbf {u} ({\boldsymbol {X}},t)\right]={\ddot {\mathbf {u} }}~.}$