General analysis procedures for piezoelectric materials

The applied electrical potential of 1 volt results in a potential gradient of 1 volt/m. The piezoelectric constants $d_{\mathrm{3\hspace{0.17em}33}}^{\phi }$ and $g_{\mathrm{3\hspace{0.17em}33}}^{\phi }$ can be used to estimate the electrical charge per unit area. In the case of an unconstrained material

and

where ${\text{E}}_3$ is the potential gradient and $q_3$ is the charge density in the local 3-direction. So the charge density $q_3$ is equal to $d_{\mathrm{3\hspace{0.17em}33}}^{\phi }/g_{\mathrm{3\hspace{0.17em}33}}^{\phi }$= 3.01 × 10⁻⁸. The area to which the voltage is applied is 10; therefore, the static reaction charge should be about 3.01 × 10⁻⁷. The results of ppzodyn1.inp confirm this reaction charge. In the input file ppzodyn2.inp a concentrated nodal electrical charge of 3.01 × 10⁻⁷ is applied instead of a potential value of 1 at the top surface. This results in a potential of 1 volt on the top surface.

The critical buckling load for the beam is

where E is the Young's modulus in the longitudinal direction and I is the appropriate moment of inertia for the beam section. The analysis shows a critical compressive force of 773 N. The compressive force converges to the analytical buckling load with mesh refinement.

The applied potential gradient remains constant in magnitude but rotates appropriately with the element.

In the first step the potentials at the two opposite sides in the local 3-direction of the material are prescribed. Sufficient boundary conditions are applied to prevent rigid body motions, but the model is otherwise unconstrained. The piezoelectric constants $d_{\mathrm{3\hspace{0.17em}33}}^{\phi }$= 593 × 10⁻¹² and $g_{\mathrm{3\hspace{0.17em}33}}^{\phi }$ = 19.7 × 10⁻³ can be expressed in terms of the strain ${\epsilon }_{33}$, the potential gradient ${\text{E}}_3$, and the charge density $q_3$ as

and

The piezoelectric constants $d_{\mathrm{3\hspace{0.17em}11}}^{\phi }$, $d_{\mathrm{3\hspace{0.17em}22}}^{\phi }$, $d_{\mathrm{3\hspace{0.17em}33}}^{\phi }$, $g_{\mathrm{3\hspace{0.17em}11}}^{\phi }$, $g_{\mathrm{3\hspace{0.17em}22}}^{\phi }$, and $g_{\mathrm{3\hspace{0.17em}33}}^{\phi }$ are verified by using the numerically obtained values of the strains ${\epsilon }_{11}$, ${\epsilon }_{22}$, and ${\epsilon }_{33}$. The dielectric constant $D_{33}^{\phi \left(\sigma \right)}$in the local 3-direction for an unconstrained material is given by

The numerical results for $q_3$ and ${\text{E}}_3$ confirm the above relationships. In Steps 2–4 the model is charged in different ways verifying the same piezoelectrical material parameters as in Step 1. In Step 2 the potentials of the bottom and the top surface are switched. In Step 3 a nodal concentrated electrical charge is applied, and in Step 4 a distributed electrical charge is applied instead of prescribing the potentials. In Step 5 a potential gradient is applied in the local 1-direction to verify the piezoelectric properties $d_{\mathrm{1\hspace{0.17em}13}}^{\phi }$, $g_{\mathrm{1\hspace{0.17em}13}}^{\phi }$, and $D_{11}^{\phi \left(\sigma \right)}$.

In Steps 6–7 an open circuit condition is applied (the potential gradient is not prescribed by voltage boundary conditions), which results in reaction charges that are equal to zero. The piezoelectric constitutive equations can be written in different forms. In particular, the strain can be expressed in terms of either the potential gradient $\mathbf{E}$ or the charge density $\mathbf{q}$. If the constitutive relation is expressed in terms of the potential gradient, the compliance data (typically denoted as ${\mathbf{S}}^E$ in the piezoelectric literature) define the mechanical behavior at zero potential gradient. In Abaqus the stiffness data at zero potential gradient are used to specify the mechanical behavior. If the constitutive relation is expressed using the charge density, the compliance matrix (typically denoted as ${\mathbf{S}}^D$ in the piezoelectric literature) defines the mechanical behavior at zero charge density. The compliance ${\mathbf{S}}^D$ can be obtained from the compliance ${\mathbf{S}}^E$ and the electrical properties. For the PZT-5H material, $S_{11}^D$ = 14.05 × 10⁻¹², $S_{12}^D$ = −7.27 × 10⁻¹², $S_{13}^D$ = −3.05 × 10⁻¹², $S_{31}^D$ = −3.05 × 10⁻¹², and $S_{33}^D$ = 8.99 × 10⁻¹². By loading the model at zero charge (in open circuit condition), these elastic compliances are verified.