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.

Elasticity