Why am I getting the error message below about negative permittivity for this piezoelectric property?
*** ERROR ***
The calculated permittivity at constant strain matrix [epS] for
material 1 has negative diagonal terms.
MATERIAL INPUT ASSOCIATED WITH TB,DPER COMMAND FOR MATERIAL NUMBER 1
Dielectric (relative) permittivity at constant strain
computed from piezoelectric strain and stress matrices.
EP11 EP22 EP33 EP12 EP23 EP13
-69.025 52.629 0.0000 0.0000 0.0000 0.0000
Looking at the manufacturer's data, the set of compliance coefficients is incomplete and is inconsistent:
- s12 and s13 are missing and are assigned small values
- s11 is measured at constant electric field (E), while s33 and s55 are measured at constant electric flux (D).
SE and SD are used in different sets of piezoelectric constitutive equations. You cannot intermix them in one input. The relationship between them is the following:
where [d] is the matrix of piezoelectric strain coefficients and [g] is another set of piezoelectric coefficients associated with another set of constitutive equations.
There are four sets of piezoelectric matrices in total: [e], [d], [h] and [g]. The latter piezoelectric matrix can be derived from [d] and the inverse of [epT] as follows:
where [epT] is the permittivity matrix at constant stress.
These input inconsistencies are probably the source of non-physical values of permittivity [epS].
Below is APDL to perform the calculation as it is done internally. It confirms the values.
d31= -0.18e-9 !-0.15e-9