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4.2.4 Discretization of the Mechanical Equations
The deformation in a three-dimensional body is expressed by the displacement field
![$\displaystyle \vec d(\vec r) = \begin{bmatrix}u(\vec r)\\ v(\vec r)\\ w(\vec r)\end{bmatrix},$](img552.png) |
(4.54) |
where
,
, and
are the displacements in the x, y, and z direction, respectively.
The displacement is discretized on a tetrahedral element as [151]
![$\displaystyle \vec d(\vec r) = \sum_{i=1}^{4}\vec d_i \symShapeFun_i(\vec r),$](img554.png) |
(4.55) |
which leads to the components discretization
![$\displaystyle u(\vec r) = \ensuremath{\sum_{i=1}^{4}{u}}_{i}^n\symShapeFun_i(\v...
...),\quad w(\vec r) = \ensuremath{\sum_{i=1}^{4}{w}}_{i}^n\symShapeFun_i(\vec r).$](img555.png) |
(4.56) |
Applying this discretization in the strain-displacement relationship (3.79), the components of the strain tensor can be written as
![$\displaystyle \boldsymbol\symStrain = \mathbf{B}\mathbf{d} = \begin{bmatrix}\mathbf{B_1}\ \mathbf{B_2}\ \mathbf{B_3}\ \mathbf{B_4} \end{bmatrix} \mathbf{d},$](img556.png) |
(4.57) |
where
is the matrix of the derivatives of the shape functions for the node i [151]
![$\displaystyle \mathbf{B_i} = \begin{bmatrix}\displaystyle\ensuremath{\ensuremat...
...\frac{\partial \symShapeFun_i}{\partial x}}} \end{bmatrix},\quad i = 1,\dots,4,$](img558.png) |
(4.58) |
and the displacement matrix
![$\displaystyle \mathbf{d} = \begin{bmatrix}\vec d_1\\ \vec d_2\\ \vec d_3\\ \vec d_4 \end{bmatrix}.$](img559.png) |
(4.59) |
Using (4.57), the stress-strain equation (3.81) can written as a function of the displacements according to
![$\displaystyle \boldsymbol\symHydStress = \mathbf{C}(\boldsymbol\symStrain - \bo...
...in_{0}) = \mathbf{C}\mathbf{B}\mathbf{d} - \mathbf{C}\boldsymbol\symStrain_{0}.$](img560.png) |
(4.60) |
Applying the principle of virtual work, the work of internal stresses on a continuous elastic body is given by [151]
![$\displaystyle W_{in}=\int_{\symDomain} \boldsymbol\symStrain^{T}\boldsymbol\symHydStress\ d\symDomain,$](img561.png) |
(4.61) |
where
is the transposed strain tensor.
Combining (4.57), (4.60), and (4.61) the work on a finite element is written as
![$\displaystyle W_{in}^{el} = \mathbf{d}^T \int_{T} \left(\mathbf{B}^T\mathbf{C}\...
...mathbf{d} - \mathbf{B}^T\mathbf{C}\boldsymbol\symStrain_{0}\right) d\symDomain.$](img563.png) |
(4.62) |
From energy balance the internal work should be equal to the work done by external forces, i.e.
, and, since during electromigration there are no external forces (
), one obtains
![$\displaystyle \int_{T} \left(\mathbf{B}^T\mathbf{C}\mathbf{B}\mathbf{d} - \mathbf{B}^T\mathbf{C}\boldsymbol\symStrain_{0}\right) d\symDomain = 0,$](img566.png) |
(4.63) |
or
![$\displaystyle \int_{T} \mathbf{B}^T\mathbf{C}\mathbf{B}\mathbf{d}\ d\symDomain = \int_{T} \mathbf{B}^T\mathbf{C}\boldsymbol\symStrain_{0}\ d\symDomain,$](img567.png) |
(4.64) |
which can be conveniently expressed as
![$\displaystyle \mathbf{K}\mathbf{d} = \mathbf{f_{in}},$](img568.png) |
(4.65) |
where
![$\displaystyle \mathbf{K} = \int_{T} \mathbf{B}^T\mathbf{C}\mathbf{B}\ d\symDomain,% = \mathbf{B}^T\mathbf{C}\mathbf{B}V_e,
$](img569.png) |
(4.66) |
is the so-called stiffness matrix, and
![$\displaystyle \mathbf{f_{in}} = \int_{T} \mathbf{B}^T\mathbf{C}\boldsymbol\symStrain_{0}\ d\symDomain,% = \mathbf{B}^T\mathbf{C}\boldsymbol\symStrain_{0}V_e,
$](img570.png) |
(4.67) |
is the internal force vector.
Equation (4.65) forms a linear system of equations of 12 equations with 12 unknowns (the three displacement components
,
, and
for each tetrahedron node).
The inelastic strain
determines the internal force vector according to the electromigration induced strain given by (3.78).
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R. L. de Orio: Electromigration Modeling and Simulation