Numerical modeling of single-layered piezoelectric elements

The focus of this report is on numerical simulation of piezoelectric materials. One example of such a material is quartz. By applying a pressure load to a piezoelectric material, electricity is generated – the so-called direct piezoelectric effect. In the opposite case, by applying an electric field, the material deforms (mechanically) – the so-called inverse piezoelectric effect. A mathematical model for linear piezoelectricity is described. This includes a model for linear elasticity, as well as a model for linear electrostatics. Piezoelectricity is expressed mathematically by combining the two above mentioned models. The explicit coupling between the elasticity problem and the electrostatics problem is through (modified) constitutive laws. Two simulators are implemented; one in Diffpack and one in MSC.Marc. In this way we are able to compare numerical results, and then verify the implementation in both software packages. In addition, this increases the general understanding of the problem. Simulation in Diffpack requires some low-level programming, but gives large possibilities for adjustments and assessments. Moreover, it provides access to all programming details needed. MSC.Marc is on the other hand to a larger extent an application software package. An advantage of MSC.Marc is that one is able to use robust software for advanced problems relatively fast. A main disadvantage is that most of the implementation details are hidden, which gives, among other things, uncertainty regarding the choice of solution method, what presumptions are made for the calculations, and the interpretation of quantities that are returned (results). For such a complex and comprehensive software package as MSC.Marc, detailed documentation, that easily can be searched, is a major challenge to work out. Two test examples are simulated in both software tools for different piezoelectric materials. In the first case we simulate the piezoelectric effect, whereas in the second case we consider the inverse piezoelectric effect. Comparison of the numerical results shows very good agreement in the two software packages for the material model cases included in this report.