Introduction to Elasticity/Homogeneous and inhomogeneous displacements

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[edit] Homogeneous and inhomogeneous displacements

[edit] 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}.

[edit] 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

[edit] 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.

[edit] 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.

[edit] 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

  1. 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.
  2. 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.
  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.
  4. 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}).
  5. Every simple shear is isochoric. Every isochoric pure strain is the sum of simple shears.

[edit] 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

[edit] 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.

[edit] 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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