16
2 Literature Review
2.4 Electroelastic Materially Nonlinear Modeling
2.4.1 Linear Piezoelectric Constitutive Equations
In most of the studies, linear constitutive laws were employed in finite element
models of smart structures, which are only valid for structures under weak electric
field. There are two typical models of electric potential through the thickness, namely
first-order and higher-order variation. The former distribution of electric potential is
mostly used in modeling of piezoelectric materials, which yields constant electric
field through the thickness. Because of the assumption of linear variation of electric
potential, it is applicable only for thin piezoelectric patches. Almost all the above
mentioned models of smart structures implemented this variation of electric potential.
In the case of thick piezoelectric layers, higher-order variation of electric potential
should be considered [154, 155]. Linear models with quadratic electric potential variation through the thickness were developed based on the FOSD hypothesis [28, 156]
and zigzag hypothesis [157]. Using the MITC elements, proposed by Dvorkin and
Bather [90], Bathe [105], FE models with the assumption of second-order variation
of electric potential were proposed by Kögl and Bucalem [158]. Moreover, geometrically nonlinear FE models with electric potential quadratic distribution were
developed for static and dynamic analysis [159, 160].
2.4.2 Strong Electric Field Models
Linear piezoelectric constitutive equations are only used when the structures undergo
small strains and under weak electric potential. In piezoelectric material, it is assumed
that the stresses generated by electric field is always below the yield stress, meaning
that structures undergo only in small strains. However, sometimes strong electric field
is considered to be applied on piezoelectric material for large actuation forces. This
requires an electroelastic materially nonlinear relations. Therefore, for the case of
small strains and strong electric field, the nonlinear part of constitutive law includes
only the electroelastic part.
The constitutive equations with electroelastic nonlinearity were first proposed by
Nelson [161] and Joshi [162]. Afterwards, the constitutive equations were extended
and implemented into transversely isotropic materials like piezoelectric ceramics
and the class of mm
2 symmetry materials like PVDF [163]. Many researchers investigated irreversible piezoelectric nonlinearities, known as piezoelectric hysteresis,
e.g. [164–168] among many others. To validate the numerical models of piezoelectric
hysteresis, Li et al. [169], Masys et al. [170] investigated experimentally. In addition, Klinkel [171], Linnemann et al. [172] applied the irreversible phenomenological
constitutive model into finite element analysis using solid elements for piezoelectric
materials.
2 Literature Review
2.4 Electroelastic Materially Nonlinear Modeling
2.4.1 Linear Piezoelectric Constitutive Equations
In most of the studies, linear constitutive laws were employed in finite element
models of smart structures, which are only valid for structures under weak electric
field. There are two typical models of electric potential through the thickness, namely
first-order and higher-order variation. The former distribution of electric potential is
mostly used in modeling of piezoelectric materials, which yields constant electric
field through the thickness. Because of the assumption of linear variation of electric
potential, it is applicable only for thin piezoelectric patches. Almost all the above
mentioned models of smart structures implemented this variation of electric potential.
In the case of thick piezoelectric layers, higher-order variation of electric potential
should be considered [154, 155]. Linear models with quadratic electric potential variation through the thickness were developed based on the FOSD hypothesis [28, 156]
and zigzag hypothesis [157]. Using the MITC elements, proposed by Dvorkin and
Bather [90], Bathe [105], FE models with the assumption of second-order variation
of electric potential were proposed by Kögl and Bucalem [158]. Moreover, geometrically nonlinear FE models with electric potential quadratic distribution were
developed for static and dynamic analysis [159, 160].
2.4.2 Strong Electric Field Models
Linear piezoelectric constitutive equations are only used when the structures undergo
small strains and under weak electric potential. In piezoelectric material, it is assumed
that the stresses generated by electric field is always below the yield stress, meaning
that structures undergo only in small strains. However, sometimes strong electric field
is considered to be applied on piezoelectric material for large actuation forces. This
requires an electroelastic materially nonlinear relations. Therefore, for the case of
small strains and strong electric field, the nonlinear part of constitutive law includes
only the electroelastic part.
The constitutive equations with electroelastic nonlinearity were first proposed by
Nelson [161] and Joshi [162]. Afterwards, the constitutive equations were extended
and implemented into transversely isotropic materials like piezoelectric ceramics
and the class of mm
2 symmetry materials like PVDF [163]. Many researchers investigated irreversible piezoelectric nonlinearities, known as piezoelectric hysteresis,
e.g. [164–168] among many others. To validate the numerical models of piezoelectric
hysteresis, Li et al. [169], Masys et al. [170] investigated experimentally. In addition, Klinkel [171], Linnemann et al. [172] applied the irreversible phenomenological
constitutive model into finite element analysis using solid elements for piezoelectric
materials.