Advanced elasticity/Motion and displacement

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).

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_i = x_i - X_i~. $$

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}} ~. $$