This test computes deformations of a piezoelectric bimorph beam (L x B x t), which consists of two identical PVDF uniaxial beams with opposite polarities is considered for both an actuator and a sensor test. For the actuator test, an external voltage distribution of 1V is applied across the thickness of the beam. As a result the bimorph beam bends according to the converse piezoelectric effect. For the sensor test, the tip of the bimorph beam is deflected 10 mm and the sensor voltage due to the direct piezoelectric effect across the thickness of each element is determined. The static deflections at the nodes are calculated for different element types and different number of elements in length direction. The beam is modelled with shell elements.

The beam has a length L = 100.0mm, a width B = 5mm and a thickness t = 1.0mm

Material is linear isotropic, with modulus of elasticity E set to
2.0E9N/m^{2}, Poisson number
^{3}.

The di-electric properties are: pe_31 =
0.046c/m^{2}. de_11 = de_22 = de_33 =
0.1062E-9F/m.

The edge at x=0 is fully mechanically clamped. All others are free. DOF 6 is locked for all nodes. DOF 7 (electric potential) is free for all nodes.

Two loading conditions will be used, one for the actuator test and one for the sensor test. Actuator loading: A voltage of 1 V across the thickness of the beam. Because the beam consists of two PVDF layers, the applied voltage on all nodes must be 0.5[V]. Sensor loading: A prescribed tip displacement of 10 mm.

Please observe how the shell mesh and the piezo-electric element mesh
are created. Since the piezo-electric element mesh is an overlay element
mesh the elements must be defined with the same connectivity as the
underlying shell mesh. This is achieved with the new
**epatch** option `usepatch`

, which creates
'clone' meshes, creating new elements with the same element connectivity as
the underlying shell mesh. To this end the overlay element patch 2 must be
defined with options

start_node_id 1 usepatch 1 body

which will copy the element connectivity from
patch 1 and replace all other element attributes with the ones defined for
the current patch 2. The `start_node_id`

option will number
the new nodes starting from 1.

The static deflection of the beam can be calculated with:

w(x)=1.5 e_31 V/E (x/t)^{2}

The results are listed in the table below:

**Table 15. Piezoelectric beam actuator: Actuator loading results**

Element | Mesh | w(0.2) | w(0.4) | w(0.6) | w(0.8) | w(1.0) |

analytical | - | 0.138E-07 | 0.552E-07 | 1.242E-07 | 2.208E-07 | 3.450E-07 |

Q4.S.MITC.E4 | coarse | 0.1336E-07 | 0.5340E-07 | 0.1223E-06 | 0.2182E-06 | 0.3417E-06 |

Q4.S.MITC.E4 | fine | 0.1345E-7 | 0.5455E-7 | 0.1232E-6 | 0.2195E-6 | 0.3433E-06 |

Q9.S.MITC | coarse | 0.1352E-07 | 0.5463E-07 | 0.1233E-06 | 0.2197E-06 | 0.3436E-06 |

Q9.S.MITC | fine | 0.1366E-07 | 0.5492E-07 | 0.1238E-06 | 0.2202E-06 | 0.3443E-06 |

The sensor voltage for beam bending is dependent on the first derivative of the bending strain, i.e. the second derivative of flexural displacement (see Hwang, W.S., Park, H.C., "Finite element modelling of piezoelectric sensors and actuators", AIAA Journal, Vol. 31, No. 5, pp. 930-937, 1993). The total voltage from this derivation can be calculated analytically. The voltage per layer (as calculated in the numerical model) is half the total voltage. The results are listed in the table below.

node 1: V(0.0) = 300 V -> per layer: V(0.0) = 150 V; node 2: V(0.2) = 240 V -> per layer: V(0.2) = 120 V; node 3: V(0.4) = 180 V -> per layer: V(0.4) = 90 V; node 4: V(0.6) = 120 V -> per layer: V(0.6) = 60 V; node 5: V(0.8) = 60 V -> per layer: V(0.8) = 30 V; node 6: V(1.0) = 0 V -> per layer: V(1.0) = 0 V.

**Table 16. Piezoelectric beam actuator: Sensor voltage results**

Element | Mesh | V(0.0) | V(0.2) | V(0.4) | V(0.6) | V(0.8) | V(1.0) |

Q4.S.MITC.E4 | - | 150. | 120. | 90. | 60. | 30. | 0. |

Q4.S.MITC.E4 | coarse | 154.96 | 138.99 | 103.76 | 69.173 | 32.045 | 10.256 |

Q4.S.MITC.E4 | fine | 157.63 | 136.73 | 102.04 | 67.950 | 34.271 | 5.1256 |

Q9.S.MITC | coarse | 160.68 | 136.83 | 102.20 | 67.855 | 33.886 | 0.80464 |

Q9.S.MITC | fine | 160.21 | 135.59 | 101.22 | 67.455 | 33.730 | 0.12225 |