56
4 Nonlinear Constitutive Relations
After the discovery of piezoelectricity, due to the high operational frequency
(in the range of one megaHertz) of quartz and Rochelle-salt plates, many applications were implemented ranging from radio transmitter, underwater detection,
pressure measurement to many kinds of electrical measurements, microphones, and
accelerometers. Around the Second World War, polycrystalline piezoceramic materials was discovered, which has high dielectric constants and could be manufactured in
high volumes. The low piezoelectric effect of nature materials limits the applications
until ferroelectric material was found during the Second World War. The man-made
ferroelectrics exhibited piezoelectric effects many times higher than those found
in natural materials. In 1940s, Arthur von Hippel and coworkers at MIT discovered barium titanate (BaTiO 3 ) that has the capability of repolarization under a high
electric field. However, the Curie Temperature takes only 120
◦ , which means that
the piezoelectric effect disappears when the temperature is above 120
◦ . In 1950s,
with the discovery of piezoelectric effects in lead metaniobate (PbNb 2 O 6 ) and lead
zirconate titanate [Pb(Ti,Zr)O 6 ], the Curie Temperature increases to 250
◦ . Different
from ceramic, a soft piezoelectric material, polymer polyvinylidenefluoride (PVDF),
was discovered by Kawai [1]. Due to the flexibility, PVDF is frequently manufactured in thin films, which is easy to fit curved geometries. However, the low stiffness
makes the material usually used as sensors.
There are several practical limitations in the applications of piezoceramic materials, for example the brittle nature of ceramics which makes them susceptible to
fracture during handling and bonding procedures, and their extremely limited ability
to fit with curved surfaces [2]. Even though the PVDF material is soft and flexible, but
with low stiffness, which are only used for sensors. To overcome the limitations existing in conventional piezoelectric materials, piezo composite materials were proposed
and developed by some researchers in the 1990s. The first type of piezo composite is referred to as 1-3 composite invented at the Fraunhofer Research Facility in
Germany [2]. The second one is an active fiber composite (AFC) initially developed by MIT, which were the first composite actuators primarily used on structural
actuation [2]. The third one is a macro-fiber composite (MFC) proposed by NASA
Langley Research Center [3] in 1999. The flexible nature of MFC allows the material
conforming to a curved surface easily. Additionally, an MFC patch even has larger
actuation forces than a PZT patch, since the d 33 effect dominates the actuation mode
in MFCs. For more detailed information of active piezoelectric fiber composites, we
refer to [4–6].
4.1.2 Piezoelectric Effects
The raw piezoceramics illustrate electrically neutral, without piezoelectric effect.
They need to be polarized by applying strong electric field. In most cases the piezoelectric materials are also ferroelectric, the piezoelectric phase can be transformed to
a symmetric non-piezoelectric state at a certain high temperature, which here refers
to the Curie temperature, as shown in Fig. 4.1. The ion Ti
4+ in the center will be
4 Nonlinear Constitutive Relations
After the discovery of piezoelectricity, due to the high operational frequency
(in the range of one megaHertz) of quartz and Rochelle-salt plates, many applications were implemented ranging from radio transmitter, underwater detection,
pressure measurement to many kinds of electrical measurements, microphones, and
accelerometers. Around the Second World War, polycrystalline piezoceramic materials was discovered, which has high dielectric constants and could be manufactured in
high volumes. The low piezoelectric effect of nature materials limits the applications
until ferroelectric material was found during the Second World War. The man-made
ferroelectrics exhibited piezoelectric effects many times higher than those found
in natural materials. In 1940s, Arthur von Hippel and coworkers at MIT discovered barium titanate (BaTiO 3 ) that has the capability of repolarization under a high
electric field. However, the Curie Temperature takes only 120
◦ , which means that
the piezoelectric effect disappears when the temperature is above 120
◦ . In 1950s,
with the discovery of piezoelectric effects in lead metaniobate (PbNb 2 O 6 ) and lead
zirconate titanate [Pb(Ti,Zr)O 6 ], the Curie Temperature increases to 250
◦ . Different
from ceramic, a soft piezoelectric material, polymer polyvinylidenefluoride (PVDF),
was discovered by Kawai [1]. Due to the flexibility, PVDF is frequently manufactured in thin films, which is easy to fit curved geometries. However, the low stiffness
makes the material usually used as sensors.
There are several practical limitations in the applications of piezoceramic materials, for example the brittle nature of ceramics which makes them susceptible to
fracture during handling and bonding procedures, and their extremely limited ability
to fit with curved surfaces [2]. Even though the PVDF material is soft and flexible, but
with low stiffness, which are only used for sensors. To overcome the limitations existing in conventional piezoelectric materials, piezo composite materials were proposed
and developed by some researchers in the 1990s. The first type of piezo composite is referred to as 1-3 composite invented at the Fraunhofer Research Facility in
Germany [2]. The second one is an active fiber composite (AFC) initially developed by MIT, which were the first composite actuators primarily used on structural
actuation [2]. The third one is a macro-fiber composite (MFC) proposed by NASA
Langley Research Center [3] in 1999. The flexible nature of MFC allows the material
conforming to a curved surface easily. Additionally, an MFC patch even has larger
actuation forces than a PZT patch, since the d 33 effect dominates the actuation mode
in MFCs. For more detailed information of active piezoelectric fiber composites, we
refer to [4–6].
4.1.2 Piezoelectric Effects
The raw piezoceramics illustrate electrically neutral, without piezoelectric effect.
They need to be polarized by applying strong electric field. In most cases the piezoelectric materials are also ferroelectric, the piezoelectric phase can be transformed to
a symmetric non-piezoelectric state at a certain high temperature, which here refers
to the Curie temperature, as shown in Fig. 4.1. The ion Ti
4+ in the center will be