Introduction to Elasticity/Homogeneous and inhomogeneous displacements

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

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