Elasticity/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.,


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

{| cellspacing="0" cellpadding="0" style="margin:0em 0em 1em 0em; width:80%"  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
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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 $$\textstyle \text{Tr}(\boldsymbol{\varepsilon}) = 0$$.
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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.

{| cellspacing="0" cellpadding="0" style="margin:0em 0em 1em 0em; width:80%"  Properties of inhomogeneous displacement fields
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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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