1.3.7 Loading of piezoelectric elements

Product: ABAQUS/Standard  

I. Plane stress and plane strain piezoelectric elements

Elements tested

CPS3E    CPS4E    CPS6E    CPS8E    CPS8RE    CPE3E    CPE4E    CPE6E    CPE8E    CPE8RE   

Problem description

Material:

Linear elastic, Young's modulus = 30 × 106, Poisson's ratio = 0.3, no piezoelectric coupling, isotropic dielectric constant 1.0 × 10–3.

Boundary conditions:

0, 0.

Loading:

Distributed pressure of 1000/length on each edge.

Equivalent concentrated shear forces corresponding to distributed shear loading of 1000/length on each edge in the directions shown.

Distributed charges of 1000/length on each edge.

Concentrated charges at each node to negate the distributed charges, except for the distributed charge of 1000/length on the top surface.

Reference solution

Stresses

Both plane stress and plane strain elements,

–1000;

and for plane strain elements,

–600.

Strains

Plane strain elements,

–1.7333 × 10–5, –8.6667 × 10–5.

Plane stress elements,

–2.3333 × 10–5, –8.6667 × 10–5.

Electrical fluxes

Both plane stress and plane strain elements, 0, –1000.

Electrical potential gradients

Both plane stress and plane strain elements, 0, –1.0 × 106.

Displacements

, .

Potentials

.

Results and discussion

Elements using reduced integration may have additional boundary conditions to those specified above. All elements yield exact solutions.

The *SECTION FILE and *SECTION PRINT output requests are used in some of the input files to output accumulated quantities on the face in the xy plane.

Input files

ecs3efs1.inp

CPS3E elements.

ecs4efs1.inp

CPS4E elements.

ecs6efs1.inp

CPS6E elements.

ecs8efs1.inp

CPS8E elements.

ecs8ers1.inp

CPS8RE elements.

ece3efs1.inp

CPE3E elements.

ece4efs1.inp

CPE4E elements.

ece6efs1.inp

CPE6E elements.

ece8efs1.inp

CPE8E elements.

ece8ers1.inp

CPE8RE elements.

II. Three-dimensional piezoelectric elements

Elements tested

C3D4E    C3D6E    C3D8E    C3D10E    C3D15E    C3D20E    C3D20RE   

Problem description

Material:

Linear elastic, Young's modulus 30 × 106, Poisson's ratio 0.3, no piezoelectric coupling, isotropic dielectric constant 1.0 × 10–3.

Boundary conditions:

, , , , .

Loading:

Distributed pressure of 1000/area on each face, and equivalent concentrated forces for shear loading, defined such that all three shear stresses are of magnitude –1000.

Distributed charges of 1000/area on each face.

Concentrated charges at each node to negate the distributed charges, except for the distributed charge of 1000/area on the top surface.

Reference solution

Stresses

–1000.

Strains

–1.3333 × 10–5, –8.6667 × 10–5.

Electrical fluxes

0, 0, –1000.

Electrical potential gradients

0, 0, –1.0 × 106.

Displacements

.

Potentials

.

Results and discussion

Elements using reduced integration may have additional boundary conditions to those specified above. All elements yield exact solutions.

The *SECTION FILE and *SECTION PRINT output requests are used in some of the input files to output accumulated quantities on the face in the xy plane.

Input files

ec34efs2.inp

C3D4E elements.

ec36efs2.inp

C3D6E elements.

ec38efs2.inp

C3D8E elements.

ec3aefs2.inp

C3D10E elements.

ec3fefs2.inp

C3D15E elements.

ec3kefs2.inp

C3D20E elements.

ec3kers2.inp

C3D20RE elements.

III. Axisymmetric piezoelectric elements

Elements tested

CAX3E    CAX4E    CAX6E    CAX8E    CAX8RE   

Problem description

Material:

Linear elastic, Young's modulus 30 × 106, Poisson's ratio 0.3, no piezoelectric coupling, isotropic dielectric constant 1.0 × 10–3.

Boundary conditions:

.

Loading:

Distributed pressure of 1000/area on each face.

Distributed charges of 1000/area on each face.

Concentrated charges at each node to negate the distributed charges, except for the distributed charge of 1000/area on the top surface.

Reference solution

Stresses

–1000, 0.

Strains

–1.3333 × 10–5, 0.

Electrical fluxes

0, –1000.

Electrical potential gradients

0, –1.0 × 106.

Displacements

= –1.33 × 10–2 along 1000, –1.33 × 10–5z.

Potentials

.

Results and discussion

Elements using reduced integration may have additional boundary conditions to those specified above. All elements yield exact solutions.

The *SECTION FILE and *SECTION PRINT output requests are used in some of the input files to output accumulated quantities on the face in the xy plane.

Input files

eca3efs3.inp

CAX3E elements.

eca4efs3.inp

CAX4E elements.

eca6efs3.inp

CAX6E elements.

eca8efs3.inp

CAX8E elements.

eca8ers3.inp

CAX8RE elements.