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

$$\mathbf{K}\mathbf{d} = \mathbf{f_{in}},$$ (4.96)

with

$$\mathbf{K} = \int_{T} \mathbf{B}^T\mathbf{C}\mathbf{B}\ d\symDomain,$$ (4.97)

$$\mathbf{f_{in}} = \int_{T} \mathbf{B}^T\mathbf{C}\boldsymbol\symStrain_{0}\ d\symDomain,$$ (4.98)

Since $\mathbf{B}$, $\mathbf{C}$, and $\boldsymbol\symStrain_{0}$ are constant within an element, the assembly of (4.96) is performed in FEDOS using

$$\mathbf{K} = \mathbf{B}^T\mathbf{C}\mathbf{B},$$ (4.99)

and

$$\mathbf{f_{in}} = \mathbf{B}^T\mathbf{C}\boldsymbol\symStrain_{0}.$$ (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 $n$, the internal force vector is determined by the electromigration induced strain given in (3.82), so that

$$\boldsymbol\symStrain_{0} = \frac{1}{3}\symStrain^{v,n}\mathbf{I}.$$ (4.101)

From (3.78), the trace of the electromigration strain for each node of the tetrahedron is calculated by

$$\symStrain_i^{v,n} = \symStrain_i^{v,n-1} -\symAtomVol(1-\symVacR... ...ht) + \symAtomVol\left(C_{im,i}^{n} - C_{im,i}^{n-1}\right),\qquad i=1,\dots,4,$$ (4.102)

resulting in the element strain which is set in (4.101),

$$\symStrain^{v,n} = \sum_{i=1}^{4}\symStrain_i^{v,n}.$$ (4.103)

The solution of (4.96) yields the interconnect line deformation due to electromigration. Once the displacement field $\mathbf{d}$ is determined, the electromigration induced stress vector for an element is obtained using (4.60),

$$\boldsymbol\symHydStress = \mathbf{C}\mathbf{B}\mathbf{d} - \mathbf{C}\boldsymbol\symStrain_{0}.$$ (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 $p \in T_h(\symDomain)$ and defining the set of all elements which contain the node as $T(p)$, the mechanical stress at the node $p$ is given by

$$\boldsymbol\symHydStress_p = \frac{1}{\displaystyle\sum_{T\in T(p)} V_T}\ \sum_{T\in T(p)} V_T \boldsymbol\symHydStress_T,$$ (4.105)

where $V_T$ is the volume, and $\boldsymbol\symHydStress_T$ is the stress calculated by (4.104) for the element $\T$.