# Elasticity/Homogeneous and inhomogeneous displacements

## Homogeneous and inhomogeneous displacements

### Homogeneous Displacement Field

A displacement field ${\displaystyle \textstyle \mathbf {u} (\mathbf {X} )}$  is called homogeneous if

${\displaystyle \mathbf {u} (\mathbf {X} )=\mathbf {u} _{0}+{\boldsymbol {A}}\bullet [\mathbf {X} -\mathbf {X} _{0}]}$

where ${\displaystyle \textstyle \mathbf {X} _{0},\mathbf {u} _{0},{\boldsymbol {A}}}$  are independent of ${\displaystyle \textstyle \mathbf {X} }$ .

#### Pure Strain

If ${\displaystyle \textstyle \mathbf {u} _{0}=0}$  and ${\displaystyle \textstyle {\boldsymbol {A}}={\boldsymbol {\varepsilon }}}$ , then ${\displaystyle \textstyle \mathbf {u} }$  is called a pure strain from ${\displaystyle \textstyle \mathbf {X} _{0}}$ , i.e.,

${\displaystyle \mathbf {u} (\mathbf {X} )={\boldsymbol {\varepsilon }}\bullet [\mathbf {X} -\mathbf {X} _{0}]}$

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

##### Simple Extension

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 ${\displaystyle \textstyle {\text{Tr}}({\boldsymbol {\varepsilon }})=e}$ .

##### Uniform Dilatation

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 ${\displaystyle \textstyle {\text{Tr}}({\boldsymbol {\varepsilon }})=3e}$ .

##### Simple Shear

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 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 ${\displaystyle \textstyle \mathbf {X} _{0}}$ . Every pure strain ${\displaystyle \textstyle \mathbf {u} }$  can be decomposed into the the sum of three simple extensions in mutually perpendicular directions, ${\displaystyle \textstyle \mathbf {u} =\mathbf {u} _{1}+\mathbf {u} _{2}+\mathbf {u} _{3}}$ . 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 ${\displaystyle \textstyle \mathbf {p} _{0}=\mathbf {X} -\mathbf {X} _{0}}$ . 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 ${\displaystyle \textstyle {\frac {1}{\sqrt {2}}}(\mathbf {m} \pm \mathbf {n} )}$ . Every simple shear is isochoric. Every isochoric pure strain is the sum of simple shears.

### Inhomogeneous Displacement Field

Any displacement field that does not satisfy the condition of homogeneity is inhomogenous. Most deformations in engineering materials lead to inhomogeneous displacements.

Properties of inhomogeneous displacement fields

#### Average strain

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 ${\displaystyle \textstyle dA}$ .

#### Korn's Inequality

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,

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