1. Model: Shear Bender

2. Introduction This is A 3D static piezoelectric benchmark problem
A sandwiched beam is modeled where a part of the core has been replaced by piezoelectric material.
The electric field is applied perpendicular to the poling direction, inducing a transverse shear strain.

3. Geometry
The shear bender consist of a 100 mm long sandwiched cantilever beam.
The 2 mm thick core is sandwiched by two 8 mm thick aluminum layers.
The rigid foam core is partially replaced by a 10 mm long piezoceramic actuator.

4. The constitutive equations for the piezoceramic
where
Ti Stress vector components
Di Electric displacement components
Si Strain vector components
Ei Electric field components
cij Elasticity matrix components
eij Piezoelectric matrix components
eij Permittivity matrix components

5. Constitutive equations continued
Specific notation in piezoelectric context :
Stress vector notation Strain vector notation

6. Constitutive equations continued
Elasticity matrix entered using the anisotropic material option in the solid application

7. Constitutive equations continued
Piezoelectric matrix components are added to the stress component definitions using the variable expression fields

8. Constitutive equations continued
The governing electrostatic equation is introduced
using the c-coefficient
The constitutive equations which defines the dependence of the electric displacement on strains and electric fields, are included in the governing equation

9. Domain equations
Weak formulation for the mechanical part: Internal virtual strain energy
Strong formulation for the electrostatic part: Gauss law

10. Boundary Conditions Structural mechanics application:
The cantilever beam is fixed at the surfaces at z=0.
Electrostatic application:
A 20 V potential difference is applied between the top and bottom surfaces of the piezoceramic subdomain.

11. Results
The deflection is calculated to 83 nm, which is in good agreement with Piefort and Benjeddou et al.