# Continuum mechanics/Motion and displacement: Wikis

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# Study guide

Up to date as of January 14, 2010

## 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 Ω0 and let the undeformed boundary be Γ0. Let the deformed (or current) configuration be Ω with boundary Γ. 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 δij 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.

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