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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

 (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]

 (4.55)

which leads to the components discretization

 (4.56)

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

 (4.57)

where is the matrix of the derivatives of the shape functions for the node i [151]

 (4.58)

and the displacement matrix

 (4.59)

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

 (4.60)

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

 (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

 (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

 (4.63)

or

 (4.64)

which can be conveniently expressed as

 (4.65)

where

 (4.66)

is the so-called stiffness matrix, and

 (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