Elasticity/Kinematic admissibility

Kinematically admissible displacement field

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

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

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

A displacement field $(𝐮)$ is kinematically admissible if

• $𝐮$ satisfies the displacement boundary conditions $𝐮=\widetilde{𝐮}$ on $\partial B^u$.
• $𝐮$ is continuously differentiable, i.e. $𝐮\in C^3(ℛ)$.
• $|\mathrm{\nabla }𝐮|\ll 1$.

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.