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 $(\mathbf{u})$ 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.

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