3.3.3 Stress-Strain Relation

In general, in addition to the influence of the electromigration induced strain, the total strain ε of the line has contributions from the thermal strain ε^th and the elastic strain ε^σ [20]. Thus, the total strain is given by

\[\begin{equation} \epsilon=\epsilon^\sigma+\epsilon^\text{th}+\epsilon^\text{v}. \end{equation}\] (3.59)

A temperature change of ΔT, with respect to a reference temperature T₀ for which thermal strains are assumed to be zero, produces a thermal strain expressed in the form

\[\begin{equation} \epsilon^\text{th}_\text{ik} =\alpha_\text{th}\Delta T\delta_\text{ik}, \end{equation}\] (3.60)

where α_th is the coefficient of thermal expansion. Assuming that the metal line is linearly elastic and isotropic, the mechanical behavior of the material is described by a constitutive equation, which relates the stress to an imposed history of strain and other sources which cause inelastic strains, such as material transport due to electromigration and temperature [161]. Hooke's law applies to elastic strains, so that

\[\begin{equation} \sigma_\text{ik}=C_\text{iklm}\epsilon^\sigma_\text{lm}=C_\text{iklm}(\epsilon_\text{lm}-\epsilon^\text{th}_\text{lm}-\epsilon^\text{v}_\text{lm}), \end{equation}\] (3.61)

where C_iklm are the components of the stiffness tensor defined by

\[\begin{equation} C_\text{iklm}=\lambda\delta_\text{ik}\delta_\text{lm}+\mu(\delta_\text{il}\delta_\text{km}+\delta_\text{im}\delta_\text{kl}), \end{equation}\] (3.62)

where λ and μ are the Lame parameters expressed in terms of the Young's modulus E and the Poisson ratio ν using

\[\begin{equation} \lambda=\cfrac{\nu \text{E}}{(1+\nu)(1-2\nu)}, \enspace \mu=\cfrac{\text{E}}{2(1+\nu)}. \end{equation}\] (3.63)

For linear isotropic materials, the stress-strain relation simplifies to

\[\begin{equation} \sigma_\text{ik}=\lambda\delta_\text{ik}(\epsilon_\text{ll}-\epsilon^\text{th}_\text{ll}-\epsilon^\text{v}_\text{ll})+2\mu(\epsilon_\text{ik}-\epsilon^\text{th}_\text{ik}-\epsilon^\text{v}_\text{ik}). \end{equation}\] (3.64)

In matrix notation, Hooke's law for isotropic materials can be written as

\[\begin{equation} \begin{bmatrix} \sigma_\text{xx} \\ \sigma_\text{yy} \\ \sigma_\text{zz} \\ \tau_\text{xy} \\ \tau_\text{yz} \\ \tau_\text{zx} \\ \end{bmatrix} = \begin{bmatrix} \lambda+2\mu & \lambda & \lambda & 0 & 0 & 0 \\ \lambda & \lambda+2\mu & \lambda & 0 & 0 & 0 \\ \lambda & \lambda & \lambda+2\mu & 0 & 0 & 0 \\ 0 & 0 & 0 & \mu & 0 & 0 \\ 0 & 0 & 0 & 0 & \mu & 0 \\ 0 & 0 & 0 & 0 & 0 & \mu \\ \end{bmatrix} \begin{bmatrix} \epsilon_\text{xx} \\ \epsilon_\text{yy} \\ \epsilon_\text{zz} \\ \gamma_\text{xy} \\ \gamma_\text{yz} \\ \gamma_\text{zx} \\ \end{bmatrix} -B(\epsilon^\text{th}+\epsilon^\text{v}) \begin{bmatrix} 1 \\ 1 \\ 1 \\ 0 \\ 0 \\ 0 \\ \end{bmatrix}, \end{equation}\] (3.65)

where τ_ik are the shear stresses, τ_ik=ε_ik+ε_ki=2ε_ik (i≠k) are the "engineering" shear strains, and B is the bulk modulus expressed in the form

\[\begin{equation} B=\cfrac{3\lambda+2\mu}{3}=\cfrac{\text{E}}{3(1-2\nu)}. \end{equation}\] (3.66)