# Homogeneous and inhomogeneous displacements

## Homogeneous Displacement Field

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

$\mathbf{u}(\mathbf{X}) = \mathbf{u}_0 + \boldsymbol{A}\bullet[\mathbf{X} - \mathbf{X}_0]$

where $\textstyle \mathbf{X}_0, \mathbf{u}_0, \boldsymbol{A}$ are independent of $\textstyle \mathbf{X}$.

### Pure Strain

If $\textstyle \mathbf{u}_0 = 0$ and $\textstyle \boldsymbol{A} = \boldsymbol{\varepsilon}$, then $\textstyle \mathbf{u}$ is called a pure strain from $\textstyle \mathbf{X}_0$, i.e.,

$\mathbf{u}(\mathbf{X}) = \boldsymbol{\varepsilon}\bullet[\mathbf{X} - \mathbf{X}_0]$

Examples of pure strain

If $\textstyle \mathbf{X}_0$ is a given point, $\textstyle \mathbf{p}_0(\mathbf{X}) = \mathbf{X} - \mathbf{X}_0$, and $\textstyle \{\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3\}$ is an orthonormal basis, then

#### Simple Extension

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

$\mathbf{u} = e ({\mathbf{n}}\bullet{\mathbf{p}_0}) \mathbf{n}$

and

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

If $\textstyle \mathbf{n} = \mathbf{e}_1$ and $\textstyle \mathbf{X}_0 = \{0,0,0\}$, then (in matrix notation)

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

and

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

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

#### Uniform Dilatation

For a uniform dilatation $\textstyle e$,

$\mathbf{u} = e~ \mathbf{p}_0$

and

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

If $\textstyle \mathbf{X}_0 = \{0,0,0\}$ and $\textstyle \mathbf{X} = \{X_1,X_2,X_3\}$, then (in matrix notation)

$\mathbf{u} = \{e X_1, e X_2, e X_3\}$

and

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

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

#### Simple Shear

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

$\mathbf{u} = \theta[({\mathbf{m}}\bullet{\mathbf{p}_0}) \mathbf{n}+({\mathbf{n}}\bullet{\mathbf{p}_0})\mathbf{m}]$

and

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

If $\textstyle \mathbf{m} = \mathbf{e}_1$, $\textstyle \mathbf{n} = \mathbf{e}_2$, $\textstyle \mathbf{X}_0 = \{0,0,0\}$, and $\textstyle \mathbf{X} = \{X_1,X_2,X_3\}$, then (in matrix notation)

$\mathbf{u} = \{\theta X_2, \theta X_1, 0\} ; \boldsymbol{\varepsilon} = \begin{bmatrix}0&\theta&0\\\theta&0&0\\0&0&0 \end{bmatrix}$

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

 Properties of homogeneous displacement fields If $\textstyle \mathbf{u}$ is a homogeneous displacement field, then $\textstyle \mathbf{u} = \mathbf{w} + \widehat{\mathbf{u}}$, where $\textstyle \mathbf{w}$ is a rigid displacement and $\textstyle \widehat{\mathbf{u}}$ is a pure strain from an arbitrary point $\textstyle \mathbf{X}_0$. Every pure strain $\textstyle \mathbf{u}$ can be decomposed into the the sum of three simple extensions in mutually perpendicular directions, $\textstyle \mathbf{u} = \mathbf{u}_1 + \mathbf{u}_2 + \mathbf{u}_3$. Every pure strain $\textstyle \mathbf{u}$ can be decomposed into a uniform dilatation and an isochoric pure strain, $\textstyle \mathbf{u} = \mathbf{u}_d + \mathbf{u}_c$ where $\textstyle \mathbf{u}_d = \cfrac{1}{3} ~\text{Tr}(\boldsymbol{\varepsilon}) ~\mathbf{p}_0~~$, $\textstyle \mathbf{u}_c = [\boldsymbol{\varepsilon} - \cfrac{1}{3}~\text{Tr}(\boldsymbol{\varepsilon})~ \boldsymbol{\it{1}}]\bullet\mathbf{p}_0$, and $\textstyle \mathbf{p}_0 = \mathbf{X}-\mathbf{X}_0$. Every simple shear $\textstyle \mathbf{u}$ of amount $\textstyle \theta$ with respect to the direction pair ($\textstyle \mathbf{m},\mathbf{n}$) can be decomposed into the sum of two simple extensions of the amount $\textstyle \pm \theta$ in the directions $\textstyle \frac{1}{\sqrt{2}}(\mathbf{m}\pm\mathbf{n})$. 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 $\textstyle \mathbf{u}$ be a displacement field, $\textstyle \boldsymbol{\varepsilon}$ be the corresponding strain field. Let $\textstyle \mathbf{u}$ and $\textstyle \boldsymbol{\varepsilon}$ be continuous on B. Then, the mean strain $\textstyle \overline{\boldsymbol{\varepsilon}}$ depends only on the boundary values of $\textstyle \mathbf{u}$.

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

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

### Korn's Inequality

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

$\int_B |\boldsymbol{\nabla}\mathbf{u}|^2 ~dV \le 2 \int_B |\boldsymbol{\varepsilon}|^2 ~dV$