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3.4  Continuum Mechanical Model

The vacancy flux due to the mechanisms described above and the generation and annihilation of vacancies has an influence on the stress inside the structure, as the volume is smaller for vacancies then for atoms. The ratio of the atom volume and the vacancy volume is represented by \( f \). Therefore the inelastic strain due to vacancy flow is given by

(3.35) \{begin}{gather} \displaystyle \frac {\partial \epsilon _{\mathrm {f}\mathrm {l}\mathrm {o}\mathrm {w},ij}^{\mathrm {v}}}{\partial t}=\Omega (1-f)\nabla \cdot \mathrm {\b {J}}_{\mathrm {v}}\delta _{ij}, \mathref {(3.35)} \{end}{gather}

where the negative divergence of the vacancy flux, representing an accumulation of vacancies, leads to a contraction of the material. For the generation/annihilation term inelastic strain is built up due to the creation of a lattice position where no atom is present, occupying a volume of \( f\Omega \) leading to the tensor components of the inelastic strain

(3.36) \{begin}{gather} \displaystyle \frac {\partial \epsilon _{\mathrm {G},ij}^{\mathrm {v}}}{\partial t}=\Omega fG\delta _{ij}. \mathref {(3.36)} \{end}{gather}

These equations connect the solid mechanics to the vacancy dynamics. The solid mechanics model is given by Hook’s law

(3.37) \{begin}{gather} \pmb {\sigma }=\mathrm {\pmb {S}} : (\pmb {\epsilon }-\pmb {\epsilon }^{\mathrm {v}}-\pmb {\epsilon }^{\mathrm {T}}), \mathref {(3.37)} \{end}{gather}

where the inelastic strain due to thermal expansion is already included and \( \mathrm {\b {S}} \) stands for the elastic tensor with its components given by

(3.38) \{begin}{gather} S_{ijkl}=\lambda \delta _{ij}\delta _{kl}+\mu (\delta _{il}\delta _{jk}+\delta _{ik}\delta _{jl}) \mathref {(3.38)} \{end}{gather}

where \( \mu \) and \( \lambda \) are the Lamé parameters. The components of the strain tensor of the thermal expansion are given by

(3.39) \{begin}{gather} \epsilon _{ij}^{\mathrm {T}}=\alpha _{\mathrm {T}}(T-T_{\mathrm {r}\mathrm {e}\mathrm {f}})\delta _{ij}, \mathref {(3.39)} \{end}{gather}

where \( \alpha _{\mathrm {T}} \) is the thermal expansion coefficient. The connection between the strain tensor and the displacement field is given component-wise by

(3.40) \{begin}{gather} \displaystyle \epsilon _{ij}=\frac {1}{2}\ \Bigl (\frac {\partial u_{i}}{\partial x_{j}}+\frac {\partial u_{j}}{\partial x_{i}}\Bigr ) . \mathref {(3.40)} \{end}{gather}

The momentum conservation is expressed by

(3.41) \{begin}{gather} \displaystyle \rho \frac {\partial ^{2}\mathrm {\b {u}}}{\partial t^{2}}-\nabla . \pmb {\sigma }=\mathrm {\b {f}}_{\mathrm {b}\mathrm {o}\mathrm {d}\mathrm {y}}, \mathref {(3.41)} \{end}{gather}

where \( \mathrm {\b {f}}_{\mathrm {b}\mathrm {o}\mathrm {d}\mathrm {y}} \) is the body force density, the first term stands for the acceleration according to Newton’s law, and the second is the force due to the stress in the structure. For EM calculations the only body force is gravity and can be neglected. Furthermore, the accelerations in the structure are low and negligible leading to

(3.42) \{begin}{gather} \nabla \ \pmb {\sigma }=\b {0}. \mathref {(3.42)} \{end}{gather}

Putting (3.36), (3.37), and (3.39) into the equation above leads to the mechanical problem for the displacement field \( \mathrm {\b {u}}. \)

(3.43) \{begin}{gather} \displaystyle \mu \nabla ^{2}u_{i}+(\lambda +\mu )\frac {\partial }{\partial x_{i}}(\nabla \cdot \mathrm {\b {u}})= \displaystyle \frac {\mu (3\lambda +2\mu )}{\lambda +\mu }\frac {\partial }{\partial x_{i}} \mathrm {T}\mathrm {r} (\pmb {\epsilon }^{\mathrm {v}}+\pmb {\epsilon }^{\mathrm {T}}) \mathref {(3.43)} \{end}{gather}

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