Displacement field (mechanics)

A displacement field $\mathbf{u}(\mathbf{x})$ gives the change in position of each material point from a reference configuration.

The Lagrangian fluid displacement $\mathbf{\xi }$ compares neighbouring fluid trajectories. Linear velocity perturbations about a steady flow $\mathbf{U}$ obey $\delta \mathbf{u}=({\partial }_t+\mathbf{U}\cdot \nabla )\mathbf{\xi }-(\mathbf{\xi }\cdot \nabla )\mathbf{U}$. For a background scalar $f_0$, its material perturbation is $\Delta f=\delta f+\mathbf{\xi }\cdot \nabla f_0$. In an axisymmetric normal mode about cylindrical rotation, the radial and vertical velocities are $-i\omega {\xi }_r$ and $-i\omega {\xi }_z$. The displacement of a free surface is needed to impose the Lagrangian pressure perturbation condition.

// Target: astrophysics.bigb

The displacement gradient has components $(\nabla \mathbf{u}{)}_{ij}={\partial }_i{u}_j$ and records local changes of the displacement field.

This tensor is a local derivative of the displacement field; it enters the deformation gradient and strain measures.