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 $$(\mathbf{u})$$ is kinematically admissible if
 * $$\mathbf{u}$$ satisfies the displacement boundary conditions  $$\mathbf{u} = \tilde{\mathbf{u}}$$ on $$\partial B^u$$.
 * $$\mathbf{u}$$ is continuously differentiable, i.e. $$\mathbf{u} \in C^3(\mathcal{R})$$.
 * $$|\boldsymbol{\nabla}{\mathbf{u}}| \ll 1$$.

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.