Chapter 4
Nonlinear Constitutive Relations
Abstract This chapter deals with constitutive relations for piezoelectric materials
with isotropy or orthotropy. Firstly, piezoelectricity is introduced for brief understanding of piezoelectric materials. To deep understand the basic principles of
piezo materials, the fundamental theory of piezoelectricity is discussed in the threedimensional case. In order to deal with fiber based piezo materials or fiber reinforced composites, coordinate transformation law between fiber coordinates (material coordinates) and convective coordinates (structural coordinates) is introduced.
Afterwards, the constitutive relations for two typical configurations of macro-fiber
composite piezo materials are developed, where multi-layered structures are considered. Finally, an electro-mechanically coupled nonlinear constitutive relations for
piezoelectric with either isotropy or orthotropy are constructed.
4.1 Piezoelectricity
4.1.1 History of Piezoelectricity
The phenomenon piezoelectricity was first discovered by the brothers Pierre Curie
and Jacques Curie in 1880. They found that some crystals, e.g. quartz, Rochelle salt,
cane sugar, etc., will produce positive or negative electric charges under a compressive load. The amount of electric charges were found to be proportional to the applied
compressive load. This effect of generation of electric charges because of compressive load is referred to as “direct effect”. In contrast to “direct effect”, piezoelectric
material has “converse effect”. The converse effect sometimes is referred to as reciprocal or inverse effect, which describes that an additional strain or deformation will
be caused by an electric field. Again, the induced strain is proportional to the applied
electric charges. The converse effect of piezoelectric material was first mathematically proved through fundamental thermo-dynamic principles by Gabriel Lippmann
in 1881. In the same year, the complete reversibility of electroelastic deformations
in piezoelectric crystals was experimentally demonstrated by the Curie brothers.
© The Editor(s) (if applicable) and The Author(s), under exclusive license
to Springer Nature Singapore Pte Ltd. 2021
S.-Q. Zhang, Nonlinear Analysis of Thin-Walled Smart Structures, Springer Tracts
in Mechanical Engineering, https://doi.org/10.1007/978-981-15-9857-9_4
55
Nonlinear Constitutive Relations
Abstract This chapter deals with constitutive relations for piezoelectric materials
with isotropy or orthotropy. Firstly, piezoelectricity is introduced for brief understanding of piezoelectric materials. To deep understand the basic principles of
piezo materials, the fundamental theory of piezoelectricity is discussed in the threedimensional case. In order to deal with fiber based piezo materials or fiber reinforced composites, coordinate transformation law between fiber coordinates (material coordinates) and convective coordinates (structural coordinates) is introduced.
Afterwards, the constitutive relations for two typical configurations of macro-fiber
composite piezo materials are developed, where multi-layered structures are considered. Finally, an electro-mechanically coupled nonlinear constitutive relations for
piezoelectric with either isotropy or orthotropy are constructed.
4.1 Piezoelectricity
4.1.1 History of Piezoelectricity
The phenomenon piezoelectricity was first discovered by the brothers Pierre Curie
and Jacques Curie in 1880. They found that some crystals, e.g. quartz, Rochelle salt,
cane sugar, etc., will produce positive or negative electric charges under a compressive load. The amount of electric charges were found to be proportional to the applied
compressive load. This effect of generation of electric charges because of compressive load is referred to as “direct effect”. In contrast to “direct effect”, piezoelectric
material has “converse effect”. The converse effect sometimes is referred to as reciprocal or inverse effect, which describes that an additional strain or deformation will
be caused by an electric field. Again, the induced strain is proportional to the applied
electric charges. The converse effect of piezoelectric material was first mathematically proved through fundamental thermo-dynamic principles by Gabriel Lippmann
in 1881. In the same year, the complete reversibility of electroelastic deformations
in piezoelectric crystals was experimentally demonstrated by the Curie brothers.
© The Editor(s) (if applicable) and The Author(s), under exclusive license
to Springer Nature Singapore Pte Ltd. 2021
S.-Q. Zhang, Nonlinear Analysis of Thin-Walled Smart Structures, Springer Tracts
in Mechanical Engineering, https://doi.org/10.1007/978-981-15-9857-9_4
55