Rigid Body Motion

Translation

A translation is a homogeneous deformation of the form

$$\mathbf{\phi }(\mathbf{X})=\bar{\mathbf{u}}+\mathbf{X}.$$

This deformation occurs when the displacement $\mathbf{u}(\mathbf{X},t)$ is a constant $\bar{\mathbf{u}}$, and thus not a function of reference position $\mathbf{X}$ or time $t$.

For the concrete example in the figure below, let $\bar{\mathbf{u}}=⟨{\bar{u}}_1,{\bar{u}}_2{⟩}^{\top }=⟨3,2{⟩}^{\top }$. In this case, we see the placement $\mathbf{\phi }$ moves right and up on the page, relative to the reference configuration $\mathbf{X}$, by an amount of 3 and 2, respectively. The reference configuration is shown in blue. The current configuration is shown in red.