Introduction to Elasticity/Kinematic admissibility

Kinematically admissible displacement field

Consider a body $B$ with a boundary $\partial B$ with an applied body force field $\tilde{\mathbf{f}}$.

Suppose that displacement BCs $\mathbf{u} = \tilde{\mathbf{u}}$ are prescribed on the part of the boundary $\partial B^u$.

Suppose also that traction BCs $\widehat{\mathbf{n}}{}\bullet\boldsymbol{\sigma} = \tilde{\mathbf{t}}$ are applied on the portion of the boundary $\partial B^t$.

A displacement field $(\mathbf{v})$ is kinematically admissible if

• $\mathbf{v}$ satisfies the displacement boundary conditions $\mathbf{v} = \tilde{\mathbf{u}}$ on $\partial B^u$.
• $\mathbf{v}$ is continuously differentiable, i.e., $\mathbf{v} \in C^3(\mathcal{R})$ and $|\boldsymbol{\nabla}{\mathbf{v}}| << 1$.

A kinematically admissible displacement field needs only to satisfy compatibility condition and the displacement boundary conditions - but not the traction boundary conditions or equilibrium.

Recall that a kinematically admissible displacement field is used to define the principle of minimum potential energy.