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3.1 Strain

In this section the basic expressions and notations for strain in cubic crystalline solids are established. A more detailed analysis can be found in textbooks [Bir74,Kittel96,Singh93]. Starting point for the definition of strain in a system is a set of orthonormal vectors ${\ensuremath{\mathitbf{x}}},{\ensuremath{\mathitbf{y}}}$ and ${\ensuremath{\mathitbf{z}}}$ embedded in an unstrained solid. These vectors are distorted to ${\ensuremath{\mathitbf{x}}}',{\ensuremath{\mathitbf{y}}}'$ , and ${\ensuremath{\mathitbf{z}}}'$ under the influence of a uniform deformation

$\displaystyle {\ensuremath{\mathitbf{x}}}'$	$\displaystyle = (1+\epsilon_{xx}) {\ensuremath{\mathitbf{x}}} + \epsilon_{xy} {\ensuremath{\mathitbf{y}}} + \epsilon_{xz} {\ensuremath{\mathitbf{z}}}\ ,$
$\displaystyle {\ensuremath{\mathitbf{y}}}'$	$\displaystyle = \epsilon_{yx} {\ensuremath{\mathitbf{x}}} + (1+\epsilon_{yy}) {\ensuremath{\mathitbf{y}}} + \epsilon_{yz} {\ensuremath{\mathitbf{z}}}\ ,$
$\displaystyle {\ensuremath{\mathitbf{z}}}'$	$\displaystyle = \epsilon_{zx} {\ensuremath{\mathitbf{x}}} + \epsilon_{zy} {\ensuremath{\mathitbf{y}}} + (1+\epsilon_{zz}) {\ensuremath{\mathitbf{z}}}\ ,$	(3.1)

where the coefficients $\epsilon_{ij}$ define the deformation of the system.

For a uniform deformation of a body point originally located at ${\ensuremath{\mathitbf{r}}}=x{\ensuremath{\mathitbf{x}}} + y{\ensuremath{\mathitbf{y}}} + z{\ensuremath{\mathitbf{z}}}$ the displacement to ${\ensuremath{\mathitbf{r}}}'=x{\ensuremath{\mathitbf{x}}}' + y{\ensuremath{\mathitbf{y}}}' + z{\ensuremath{\mathitbf{z}}}'$ is defined as

$\displaystyle {\ensuremath{\mathitbf{R}}}\equiv {\ensuremath{\mathitbf{r}}}' - {\ensuremath{\mathitbf{r}}}$	$\displaystyle ={} x({\ensuremath{\mathitbf{x}}}'-{\ensuremath{\mathitbf{x}}}) +... ...h{\mathitbf{y}}}) + z({\ensuremath{\mathitbf{z}}}'-{\ensuremath{\mathitbf{z}}})$
	$\displaystyle = (x \epsilon_{xx} + y \epsilon_{yx} + z \epsilon_{zx}) {\ensuremath{\mathitbf{x}}}$
	$\displaystyle +\, (x \epsilon_{xy} + y \epsilon_{yy} + z \epsilon_{zy}) {\ensuremath{\mathitbf{y}}}$
	$\displaystyle +\, (x \epsilon_{xz} + y \epsilon_{yz} + z \epsilon_{zz}) {\ensuremath{\mathitbf{z}}}\ .$	(3.2)

More generally, the displacement for a non-uniform deformation can be defined by introducing a position dependent vector function ${\ensuremath{\mathitbf{u}}}({\ensuremath{\mathitbf{r}}})$ ,

$\displaystyle {\ensuremath{\mathitbf{R}}}({\ensuremath{\mathitbf{r}}}) = u_x({\... ...mathitbf{y}}} + u_z({\ensuremath{\mathitbf{r}}}) {\ensuremath{\mathitbf{z}}}\ .$

(3.3)

The vector function ${\ensuremath{\mathitbf{u}}}$ can be related to the local strain, when taking the origin of ${\ensuremath{\mathitbf{r}}}$ close to the region of interest. A Taylor series of ${\ensuremath{\mathitbf{R}}}$ up to the first order, using ${\ensuremath{\mathitbf{R}}}(\mathbf 0)=\mathbf 0$ , and a comparison of the coefficients yields:

$\begin{displaymath}\begin{array}{lll} \epsilon_{xx} = \frac{\partial u_x}{\parti... ... \quad \quad \epsilon_{ij} = \frac{\partial u_j}{\partial x_i} \end{displaymath}$

(3.4)

**Figure:** Deformation acting on a orthonormal coordinate system ${\ensuremath{\mathitbf{x}}}, {\ensuremath{\mathitbf{y}}}, {\ensuremath{\mathitbf{z}}}$ yielding the strained axes ${\ensuremath{\mathitbf{x}}}',{\ensuremath{\mathitbf{y}}}'$ , and ${\ensuremath{\mathitbf{z}}}'$ .
$\includegraphics[scale=1.0, clip]{inkscape/strainAxes.eps}$

Thus, the displacements can be written as $\epsilon_{ij} = \frac{\partial u_j}{\partial x_i}$ .

Usually, instead of the displacements $\epsilon_{ij}$ , the strain tensor $\varepsilon_{ij}$ is used to quantify the the deformation of an body in three dimensions. In the case of small deformations, the strain tensor is known as the Green tensor or Cauchy's infinitesimal strain tensor, which is defined as

$\displaystyle {\ensuremath{\varepsilon_{ij}}}=\frac{1}{2}(\epsilon_{ij} + \epsilon_{ji})\ .$

(3.5)

The diagonal coefficients $\varepsilon_{ii}$ of this symmetric tensor define the relative change in length in direction ${\ensuremath{\mathitbf{x}}}_i$ , and the off-diagonal terms $\varepsilon_{ij}$ $(i\neq j)$ are the shear strains, which are related to the angular distortions.

Frequently, in literature the engineering strains $e_{ij}$ are used which are related to the components of the strain tensor via

$\displaystyle \begin{pmatrix}e_{xx} & e_{xy} & e_{xz}\\ e_{xy} & e_{yy} & e_{yz... ...{yz}\\ 2\varepsilon_{xz} & 2\varepsilon_{yz} & \varepsilon_{zz}\end{pmatrix}\ .$

(3.6)

Here, sometimes the off-diagonal terms of the engineering strains are denoted as $\gamma_{ij} = e_{ij} = 2 {\ensuremath{\varepsilon_{ij}}}$ . Using the one suffix Voigt notation the six components of the strain tensor can be arranged in a vector

$\displaystyle ({\ensuremath{\varepsilon_{xx}}}, {\ensuremath{\varepsilon_{yy}}}... ...on_{xz}}}, 2{\ensuremath{\varepsilon_{xy}}}) =(e_1, e_2, e_3, e_4, e_5, e_6)\ .$

(3.7)

This notation is convenient when writing Hooke's law in cubic semiconductors as will be seen later.

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