Elasticity/Homogeneous and inhomogeneous displacements

Examples of pure strain

If ${\displaystyle \textstyle \mathbf {X} _{0}}$ is a given point, ${\displaystyle \textstyle \mathbf {p} _{0}(\mathbf {X} )=\mathbf {X} -\mathbf {X} _{0}}$, and ${\displaystyle \textstyle \{\mathbf {e} _{1},\mathbf {e} _{2},\mathbf {e} _{3}\}}$ is an orthonormal basis, then

For a simple extension ${\displaystyle \textstyle e}$ in the direction of the unit vector ${\displaystyle \textstyle \mathbf {n} }$

${\displaystyle \mathbf {u} =e({\mathbf {n} }\bullet {\mathbf {p} _{0}})\mathbf {n} }$

and

${\displaystyle {\boldsymbol {\varepsilon }}=e\mathbf {n} \otimes \mathbf {n} }$

If ${\displaystyle \textstyle \mathbf {n} =\mathbf {e} _{1}}$ and ${\displaystyle \textstyle \mathbf {X} _{0}=\{0,0,0\}}$, then (in matrix notation)

${\displaystyle \mathbf {u} =\{e,0,0\}}$

and

${\displaystyle {\boldsymbol {\varepsilon }}={\begin{bmatrix}e&0&0\\0&0&0\\0&0&0\end{bmatrix}}}$

The volume change is given by ${\displaystyle \textstyle {\text{Tr}}({\boldsymbol {\varepsilon }})=e}$.

For a uniform dilatation ${\displaystyle \textstyle e}$,

${\displaystyle \mathbf {u} =e~\mathbf {p} _{0}}$

and

${\displaystyle {\boldsymbol {\varepsilon }}=e~{\boldsymbol {\it {1}}}}$

If ${\displaystyle \textstyle \mathbf {X} _{0}=\{0,0,0\}}$ and ${\displaystyle \textstyle \mathbf {X} =\{X_{1},X_{2},X_{3}\}}$, then (in matrix notation)

${\displaystyle \mathbf {u} =\{eX_{1},eX_{2},eX_{3}\}}$

and

${\displaystyle {\boldsymbol {\varepsilon }}={\begin{bmatrix}e&0&0\\0&e&0\\0&0&e\end{bmatrix}}}$

The volume change is given by ${\displaystyle \textstyle {\text{Tr}}({\boldsymbol {\varepsilon }})=3e}$.

For a simple shear ${\displaystyle \textstyle \theta }$ with respect to the perpendicular unit vectors ${\displaystyle \textstyle \mathbf {m} }$ and ${\displaystyle \textstyle \mathbf {n} }$,

${\displaystyle \mathbf {u} =\theta [({\mathbf {m} }\bullet {\mathbf {p} _{0}})\mathbf {n} +({\mathbf {n} }\bullet {\mathbf {p} _{0}})\mathbf {m} ]}$

and

${\displaystyle {\boldsymbol {\varepsilon }}=\theta [{\mathbf {m} }\otimes {\mathbf {n} }+{\mathbf {n} }\otimes {\mathbf {m} }]}$

If ${\displaystyle \textstyle \mathbf {m} =\mathbf {e} _{1}}$, ${\displaystyle \textstyle \mathbf {n} =\mathbf {e} _{2}}$, ${\displaystyle \textstyle \mathbf {X} _{0}=\{0,0,0\}}$, and ${\displaystyle \textstyle \mathbf {X} =\{X_{1},X_{2},X_{3}\}}$, then (in matrix notation)

${\displaystyle \mathbf {u} =\{\theta X_{2},\theta X_{1},0\};{\boldsymbol {\varepsilon }}={\begin{bmatrix}0&\theta &0\\\theta &0&0\\0&0&0\end{bmatrix}}}$

The volume change is given by ${\displaystyle \textstyle {\text{Tr}}({\boldsymbol {\varepsilon }})=0}$.

Properties of homogeneous displacement fields

1. If ${\displaystyle \textstyle \mathbf {u} }$ is a homogeneous displacement field, then ${\displaystyle \textstyle \mathbf {u} =\mathbf {w} +{\widehat {\mathbf {u} }}}$, where ${\displaystyle \textstyle \mathbf {w} }$ is a rigid displacement and ${\displaystyle \textstyle {\widehat {\mathbf {u} }}}$ is a pure strain from an arbitrary point ${\displaystyle \textstyle \mathbf {X} _{0}}$.
2. Every pure strain ${\displaystyle \textstyle \mathbf {u} }$ can be decomposed into the sum of three simple extensions in mutually perpendicular directions, ${\displaystyle \textstyle \mathbf {u} =\mathbf {u} _{1}+\mathbf {u} _{2}+\mathbf {u} _{3}}$.
3. Every pure strain ${\displaystyle \textstyle \mathbf {u} }$ can be decomposed into a uniform dilatation and an isochoric pure strain, ${\displaystyle \textstyle \mathbf {u} =\mathbf {u} _{d}+\mathbf {u} _{c}}$ where ${\displaystyle \textstyle \mathbf {u} _{d}={\cfrac {1}{3}}~{\text{Tr}}({\boldsymbol {\varepsilon }})~\mathbf {p} _{0}~~}$, ${\displaystyle \textstyle \mathbf {u} _{c}=[{\boldsymbol {\varepsilon }}-{\cfrac {1}{3}}~{\text{Tr}}({\boldsymbol {\varepsilon }})~{\boldsymbol {\it {1}}}]\bullet \mathbf {p} _{0}}$, and ${\displaystyle \textstyle \mathbf {p} _{0}=\mathbf {X} -\mathbf {X} _{0}}$.
4. Every simple shear ${\displaystyle \textstyle \mathbf {u} }$ of amount ${\displaystyle \textstyle \theta }$ with respect to the direction pair (${\displaystyle \textstyle \mathbf {m} ,\mathbf {n} }$) can be decomposed into the sum of two simple extensions of the amount ${\displaystyle \textstyle \pm \theta }$ in the directions ${\displaystyle \textstyle {\frac {1}{\sqrt {2}}}(\mathbf {m} \pm \mathbf {n} )}$.
5. Every simple shear is isochoric. Every isochoric pure strain is the sum of simple shears.

Properties of inhomogeneous displacement fields

Let ${\displaystyle \textstyle \mathbf {u} }$ be a displacement field, ${\displaystyle \textstyle {\boldsymbol {\varepsilon }}}$ be the corresponding strain field. Let ${\displaystyle \textstyle \mathbf {u} }$ and ${\displaystyle \textstyle {\boldsymbol {\varepsilon }}}$ be continuous on B. Then, the mean strain ${\displaystyle \textstyle {\overline {\boldsymbol {\varepsilon }}}}$ depends only on the boundary values of ${\displaystyle \textstyle \mathbf {u} }$.

${\displaystyle {\overline {\boldsymbol {\varepsilon }}}={\frac {1}{V}}\int _{B}{\boldsymbol {\varepsilon }}~dV={\frac {1}{V}}\int _{\partial B}({\mathbf {u} }\otimes {\mathbf {n} }+{\mathbf {n} }\otimes {\mathbf {u} })~dA}$

where ${\displaystyle \textstyle \mathbf {n} }$ is the unit normal to the infinitesimal surface area ${\displaystyle \textstyle dA}$.

Let ${\displaystyle \textstyle \mathbf {u} }$ be a displacement field on B that is ${\displaystyle \textstyle C^{2}}$ continuous and let ${\displaystyle \textstyle \mathbf {u} =\mathbf {0} }$ on ${\displaystyle \textstyle \partial B}$. Then,

${\displaystyle \int _{B}|{\boldsymbol {\nabla }}\mathbf {u} |^{2}~dV\leq 2\int _{B}|{\boldsymbol {\varepsilon }}|^{2}~dV}$