DIMENSIONALITY - PIEZOELECTRICIY

To avoid confusion about dimensionality of quantities, I list the ones important for piezoelectricity. One can easily verify the correctness of the wave equations and the consistency of other relations. All quantities are given in SI units. Some useful relations to check out the consistency:

• C = A   s
• Joule = V A s =N m
• $N~={kg~m \over{s^2}}$

• ${\bf \sigma_1}$ in $[{N\over{m^2}}]$
• ${\bf \sigma_2}$ in $[{V\over{m}}]$
• ${\bf \epsilon_1}$ dimensionless
• ${\bf \epsilon_2}$ in $[{C\over{m^2}}]$
• $$\mathop{\mbox{${\bf c_{11}}$}}_{\mbox{$\sim$}}$$ in $[{N\over{m^2}}]$
• $$\mathop{\mbox{${\bf c_{22}}$}}_{\mbox{$\sim$}}$$ in $[{V~m\over{C}}]$
• $$\mathop{\mbox{${\bf c_{21}}$}}_{\mbox{$\sim$}}$$ in $[{V\over{m}}]$ (piezoelectric stress constants times inverse electrical permittivity)
• $$\mathop{\mbox{${\bf c_{12}}$}}_{\mbox{$\sim$}}$$ =$$\mathop{\mbox{${\bf c_{21}^T}$}}_{\mbox{$\sim$}}$$
• eps₀ in $[{C \over{Vm}}]$
• $$\mathop{\mbox{${\bf \kappa_e}$}}_{\mbox{$\sim$}}$$ in $[{A \over{Vm}}]$
• $$\mathop{\mbox{${\bf \mu}$}}_{\mbox{$\sim$}}$$ in $[{Vs \over{Am}}]$
• $\omega$ in $[{1\over s}]$
• $$\mathop{\mbox{${\bf K}$}}_{\mbox{$\sim$}}$$ in $[{1\over m}]$
• $$\mathop{\mbox{${\bf R}$}}_{\mbox{$\sim$}}$$ in $[{1\over m}]$

For example, we can verify the dimensions of the system of equations 18, omitting thermal effects, by substituting from the list as follows:

$$\pmatrix{ {m^2\over{kg}} & 0 \cr & & \cr 0 & {A\over{V~m~s... ...cr} ~=~{1\over {s^2}} \pmatrix{ - \cr \cr {C \over{m^2}}\cr}$$ (35)

If the whole system is divided by $\vert{\bf k}\vert^2$ then the eigenvalues have the dimension $[{m \over s}]$, which is the propagation velocity.

APPENDIX D

The parameters I used to calculate the slowness surfaces are as follows: