# 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 $\Omega _{0}$  and let the undeformed boundary be $\Gamma _{0}$ . Let the deformed (or current) configuration be $\Omega$  with boundary $\Gamma$ . Let ${\boldsymbol {\varphi }}(\mathbf {X} ,t)$  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

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

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

In index notation,

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

### Displacement

The displacement of a material point is given by

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

In index notation,

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

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

### Velocity

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

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

Now,

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

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

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

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

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

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

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

### Acceleration

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

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