In the Coupled-Field Guide problem 2.15, "Sample Piezoelectric Analysis," the piezoelectric strain matrix is input with the commands


How do constants d31, d33, and d32 correlate to positions 2, 5, and 8 in the piezoelectric strain matrix?

Over the years, different authors may have used different formats for piezoelectric coefficients. When IEEE published Standard 176, they used a format that they found to be common in literature, which was a 3x6 matrix with shear stresses in the order of yz, zx, xy. Since that time, other authors have remained fairly consistent with IEEE Standard 176, so today, you will most likely find piezo material data in this format. Unfortunately, this format did not line up with the input used by ANSYS, which is a 6x3 matrix with shear stresses in the order of xy, yz, xz, so we need to convert from "Standard 176" format into "ANSYS format."

We also have to make sure that the polling direction is along the appropriate axis. A common convention is to denote the polling direction as the z (or 3) direction; however, 2D analysis in ANSYS typically uses the Y (or 2) direction as the polling direction. Consequently, if we keep the x (or 1) direction consistent, we need to exchange the 1,2,3 directions for 1,3,2, so d31, d32, d33 becomes d21, d23, d22. Furthermore, because of the issues noted above, we must also transpose this piezoelectric matrix, so d21, d23, d22 becomes d12, d32, d22. Now, we have

d12 = d31 (TBDATA,2)
d22 = d33 (TBDATA,5)
d32 = d32 (TBDATA,8)

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