Introduction to Elasticity/Kinematic admissibility

Kinematically admissible displacement field

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

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

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

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

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

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.