Static analysis for piezoelectric materials

This problem contains basic test cases for one or more Abaqus elements and features.

This page discusses:

Elements tested
Features tested
Problem description
Results and discussion
Input files
References

ProductsAbaqus/Standard

Elements tested

CAX4E

Features tested

The static analysis capability for materials that include piezoelectric coupling is discussed and illustrated. Both mechanical loads and electrical surface charges are applied. In Mercer, Reddy, and Eve (1987) a problem subjected to a sinusoidal load is analyzed. The model definition from that problem is used to illustrate the static response due to a constantly applied load. In the following sections the applicable linear dynamics capabilities are discussed.

Problem description

A cylinder of piezoelectric ceramic is subjected to both a pressure load and a distributed charge load. The cylinder is 20 mm thick with an inner radius of 5 mm and an outer radius of 25 mm. The cylinder is subjected in the first step to a pressure load on the top surface. The second step applies a distributed electrical charge on the top surface. Both the top and bottom surfaces have electrodes. The potentials on the bottom surface are prescribed to zero. The electrodes are generated by using equations that set all the potentials to the same value.

The cylinder is modeled as an axisymmetric problem using only one CAX4E element. The material properties for the PZT4 material are given as

Elasticity matrix:: $[\begin{matrix} 139.0 & 74.28 & 77.84 & 0 & 0 & 0 \\ 74.28 & 115.4 & 74.28 & 0 & 0 & 0 \\ 77.84 & 74.28 & 139.0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 25.64 & 0 & 0 \\ 0 & 0 & 0 & 0 & 25.64 & 0 \\ 0 & 0 & 0 & 0 & 0 & 25.64 \end{matrix}] GPa$
Piezoelectric coupling matrix (stress coefficients):: $[\begin{matrix} 0 & - 5.207 & 0 \\ 0 & 15.08 & 0 \\ 0 & - 5.207 & 0 \\ 12.710 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 12.710 \end{matrix}] coulomb/m ^{2}$
Dielectric matrix:: $[\begin{matrix} 6.752 & 0 & 0 \\ 0 & 5.872 & 0 \\ 0 & 0 & 6.752 \end{matrix}] 10^{- 9} farad/meter$

The material is poled in the 2-direction.

Results and discussion

In the first step the $σ_{22}$ value should be equal and opposite to the applied vertical pressure. It is correctly computed as −1.0 × 10⁶. The stresses in the other directions are negligible. The stress is computed as

σ_{i j} = D_{i j k l} ε_{k l} - e_{m k l}^{φ} E_{m} .

The $σ_{22}$ term is calculated (neglecting the zero terms) as

σ_{22} = D_{2211} ε_{11} + D_{2222} ε_{22} + D_{2233} ε_{33} - e_{222}^{φ} E_{2}

σ_{22} = (7.428 \times 10^{10}) ε_{11} + (11.54 \times 10^{10}) ε_{22} + (7.428 \times 10^{10}) ε_{33} - (15.08) E_{2} .

This relationship can be verified from the results. The electrical flux density is negligible in both directions for the pressure loading. This is correct, considering the flux conservation equation. The potential gradient is constant in the vertical direction. The maximum vertical displacement, −1.65 × 10⁻⁷, occurs at the top surface.

In the second step instead of the pressure load, a distributed electrical charge is applied to the top surface of the model. The $q_{2}$ value should be equal and opposite to the charge density applied to the top surface. It is correctly computed as −1.0 × 10⁻³. The flux density in the other direction is negligible. The flux densities are computed as

q_{i} = e_{i j k}^{φ} ε_{j k} + D_{i j}^{φ} E_{j} .

The $q_{2}$ term is calculated (neglecting the zero terms) as

q_{2} = e_{211}^{φ} ε_{11} + e_{222}^{φ} ε_{22} + e_{233}^{φ} ε_{33} + D_{22}^{φ} E_{2}

q_{2} = (- 5.207) ε_{11} + (15.08) ε_{22} + (- 5.207) ε_{33} + (5.872 \times 10^{- 9}) E_{2} .

This relationship can be verified from the results. This problem, from equilibrium considerations, should produce a stress-free state. The strain field is such that the equation given above for the stress gives a negligible value.

Input files

ppzostat.inp: Input file for this analysis.

References

Mercer, C. D., B. D. Reddy, and R. A. Eve, “Finite Element Method for Piezoelectric Media,” UCT/CSIR Applied Mechanics Research Unit Technical Report, no. 92, April 1987.