4.2.4 Discretization of the Mechanical Equations

The deformation in a three-dimensional body is expressed by the displacement field

$$\vec d(\vec r) = \begin{bmatrix}u(\vec r)\\ v(\vec r)\\ w(\vec r)\end{bmatrix},$$ (4.54)

where $u(\vec r)$, $v(\vec r)$, and $w(\vec r)$ are the displacements in the x, y, and z direction, respectively. The displacement is discretized on a tetrahedral element as [151]

$$\vec d(\vec r) = \sum_{i=1}^{4}\vec d_i \symShapeFun_i(\vec r),$$ (4.55)

which leads to the components discretization

$$u(\vec r) = \ensuremath{\sum_{i=1}^{4}{u}}_{i}^n\symShapeFun_i(\v... ...),\quad w(\vec r) = \ensuremath{\sum_{i=1}^{4}{w}}_{i}^n\symShapeFun_i(\vec r).$$ (4.56)

Applying this discretization in the strain-displacement relationship (3.79), the components of the strain tensor can be written as

$$\boldsymbol\symStrain = \mathbf{B}\mathbf{d} = \begin{bmatrix}\mathbf{B_1}\ \mathbf{B_2}\ \mathbf{B_3}\ \mathbf{B_4} \end{bmatrix} \mathbf{d},$$ (4.57)

where $\mathbf{B_i}$ is the matrix of the derivatives of the shape functions for the node i [151]

$$\mathbf{B_i} = \begin{bmatrix}\displaystyle\ensuremath{\ensuremat... ...\frac{\partial \symShapeFun_i}{\partial x}}} \end{bmatrix},\quad i = 1,\dots,4,$$ (4.58)

and the displacement matrix

$$\mathbf{d} = \begin{bmatrix}\vec d_1\\ \vec d_2\\ \vec d_3\\ \vec d_4 \end{bmatrix}.$$ (4.59)

Using (4.57), the stress-strain equation (3.81) can written as a function of the displacements according to

$$\boldsymbol\symHydStress = \mathbf{C}(\boldsymbol\symStrain - \bo... ...in_{0}) = \mathbf{C}\mathbf{B}\mathbf{d} - \mathbf{C}\boldsymbol\symStrain_{0}.$$ (4.60)

Applying the principle of virtual work, the work of internal stresses on a continuous elastic body is given by [151]

$$W_{in}=\int_{\symDomain} \boldsymbol\symStrain^{T}\boldsymbol\symHydStress\ d\symDomain,$$ (4.61)

where $\boldsymbol\symStrain^{T}$ is the transposed strain tensor. Combining (4.57), (4.60), and (4.61) the work on a finite element is written as

$$W_{in}^{el} = \mathbf{d}^T \int_{T} \left(\mathbf{B}^T\mathbf{C}\... ...mathbf{d} - \mathbf{B}^T\mathbf{C}\boldsymbol\symStrain_{0}\right) d\symDomain.$$ (4.62)

From energy balance the internal work should be equal to the work done by external forces, i.e. $W_{in} = W_{ext}$, and, since during electromigration there are no external forces ($W_{ext}=0$), one obtains

$$\int_{T} \left(\mathbf{B}^T\mathbf{C}\mathbf{B}\mathbf{d} - \mathbf{B}^T\mathbf{C}\boldsymbol\symStrain_{0}\right) d\symDomain = 0,$$ (4.63)

or

$$\int_{T} \mathbf{B}^T\mathbf{C}\mathbf{B}\mathbf{d}\ d\symDomain = \int_{T} \mathbf{B}^T\mathbf{C}\boldsymbol\symStrain_{0}\ d\symDomain,$$ (4.64)

which can be conveniently expressed as

$$\mathbf{K}\mathbf{d} = \mathbf{f_{in}},$$ (4.65)

where

$$\mathbf{K} = \int_{T} \mathbf{B}^T\mathbf{C}\mathbf{B}\ d\symDomain,% = \mathbf{B}^T\mathbf{C}\mathbf{B}V_e,$$ (4.66)

is the so-called stiffness matrix, and

$$\mathbf{f_{in}} = \int_{T} \mathbf{B}^T\mathbf{C}\boldsymbol\symStrain_{0}\ d\symDomain,% = \mathbf{B}^T\mathbf{C}\boldsymbol\symStrain_{0}V_e,$$ (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 $u$, $v$, and $w$ for each tetrahedron node). The inelastic strain $\boldsymbol\symStrain_{0}$ determines the internal force vector according to the electromigration induced strain given by (3.78).