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4.3.4 Calculation of the Mechanical Stress
The discretization of the mechanical problem presented in Section 4.2.4 yields the linear system of equations
![$\displaystyle \mathbf{K}\mathbf{d} = \mathbf{f_{in}},$](img568.png) |
(4.96) |
with
![$\displaystyle \mathbf{K} = \int_{T} \mathbf{B}^T\mathbf{C}\mathbf{B}\ d\symDomain,$](img626.png) |
(4.97) |
![$\displaystyle \mathbf{f_{in}} = \int_{T} \mathbf{B}^T\mathbf{C}\boldsymbol\symStrain_{0}\ d\symDomain,$](img627.png) |
(4.98) |
Since
,
, and
are constant within an element, the assembly of (4.96) is performed in FEDOS using
![$\displaystyle \mathbf{K} = \mathbf{B}^T\mathbf{C}\mathbf{B},$](img629.png) |
(4.99) |
and
![$\displaystyle \mathbf{f_{in}} = \mathbf{B}^T\mathbf{C}\boldsymbol\symStrain_{0}.$](img630.png) |
(4.100) |
The mechanical problem has to be solved each time step after the solution of the vacancy dynamics problem, as shown in Figure 4.3.
Thus, for the time step
, the internal force vector is determined by the electromigration induced strain given in (3.82), so that
![$\displaystyle \boldsymbol\symStrain_{0} = \frac{1}{3}\symStrain^{v,n}\mathbf{I}.$](img631.png) |
(4.101) |
From (3.78), the trace of the electromigration strain for each node of the tetrahedron is calculated by
![$\displaystyle \symStrain_i^{v,n} = \symStrain_i^{v,n-1} -\symAtomVol(1-\symVacR...
...ht) + \symAtomVol\left(C_{im,i}^{n} - C_{im,i}^{n-1}\right),\qquad i=1,\dots,4,$](img632.png) |
(4.102) |
resulting in the element strain which is set in (4.101),
![$\displaystyle \symStrain^{v,n} = \sum_{i=1}^{4}\symStrain_i^{v,n}.$](img633.png) |
(4.103) |
The solution of (4.96) yields the interconnect line deformation due to electromigration.
Once the displacement field
is determined, the electromigration induced stress vector for an element is obtained using (4.60),
![$\displaystyle \boldsymbol\symHydStress = \mathbf{C}\mathbf{B}\mathbf{d} - \mathbf{C}\boldsymbol\symStrain_{0}.$](img635.png) |
(4.104) |
The mechanical stress at each node is obtained by an extrapolation from the stress calculated for the elements, given by (4.104). The stress at a particular node is calculated by performing a weighted average of the stress on all elements connected to the node. Considering a node
and defining the set of all elements which contain the node as
, the mechanical stress at the node
is given by
![$\displaystyle \boldsymbol\symHydStress_p = \frac{1}{\displaystyle\sum_{T\in T(p)} V_T}\ \sum_{T\in T(p)} V_T \boldsymbol\symHydStress_T,$](img639.png) |
(4.105) |
where
is the volume, and
is the stress calculated by (4.104) for the element
.
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R. L. de Orio: Electromigration Modeling and Simulation