Advanced elasticity/Motion and displacement

Motion

Let the undeformed (or reference) configuration of the body be ${\mathrm{\Omega }}_0$ and let the undeformed boundary be ${\mathrm{\Gamma }}_0$. Let the deformed (or current) configuration be $\mathrm{\Omega }$ with boundary $\mathrm{\Gamma }$. Let $\mathit{\phi }(𝐗,t)$ 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

$𝐱=\mathit{\phi }(𝑿,t)$

where $𝐱$ is the position of material point $𝑿$ at time $t$.

In index notation,

$x_i={\phi }_i(X_j,t),i,j=1,2,3.$

Displacement

The displacement of a material point is given by

$𝐮(𝑿,t)=\mathit{\phi }(𝑿,t)-\mathit{\phi }(𝑿,0)=\mathit{\phi }(𝑿,t)-𝑿=𝐱-𝑿.$

In index notation,

$u_i={\phi }_i(X_j,t)-X_i=x_i-X_i.$

Velocity

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

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

Now,

$𝐮(𝑿,t)=\mathit{\phi }(𝑿,t)-𝑿.$

Therefore, the material time derivative of $𝐮$ is

$\dot{𝐮}=\frac{\partial }{\partial t}[𝐮(𝑿,t)]=\frac{\partial }{\partial t}[\mathit{\phi }(𝑿,t)-𝑿]=\frac{\partial }{\partial t}[\mathit{\phi }(𝑿,t)]=𝐯(𝑿,t).$

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

$𝐮(𝐱,t)=𝐮(\mathit{\phi }(𝑿,t),t).$

Then the material time derivative of $𝐮(𝐱,t)$ is

$\frac{D}{Dt}[𝐮(𝐱,t)]=\frac{\partial 𝐮}{\partial t}+\frac{\partial 𝐮}{\partial 𝐱}\frac{\partial 𝐱}{\partial t}=\frac{\partial 𝐮}{\partial t}+\frac{\partial 𝐮}{\partial 𝐱}\frac{\partial }{\partial t}[\mathit{\phi }(𝑿,t)]=𝐯(𝐱,t)+\frac{\partial 𝐮}{\partial 𝐱}𝐯(𝑿,t).$

Acceleration

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

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