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Nonlinear electro-mechanical behaviors of piezoelectric materials and viscoelastic nature of polymers result in the overall nonlinear and hysteretic responses of active polymeric composites. This study presents a hybrid-unit-cell model for obtaining the effective nonlinear and rate-dependent hysteretic electro-mechanical responses of hybrid piezocomposites. The studied hybrid piezocomposites consist of unidirectional piezoelectric fibers embedded in a polymeric matrix, which is reinforced with piezoelectric particles. The hybrid-unit-cell model is derived based on a unit-cell model of fiber-reinforced composites consisting of fiber and matrix subcells, in which the matrix subcells are comprised of a unit-cell model of particle-reinforced composites. Nonlinear electro-mechanical responses are considered for the piezoelectric constituents while a viscoelastic solid constitutive model is used for the polymer constituent. The hybrid-unit cell model is used to examine the effects of different responses of the constituents, microstructural arrangements, and loading histories on the overall nonlinear and hysteretic electro-mechanical responses of the hybrid piezocomposites, which are useful in designing active polymeric composites.

Piezoelectric fiber-reinforced composites have widely been used in aerospace, automobiles and medical industries due to their inherently large electro-mechanical coupling effects, compliant and lightweight characteristics. For example, a piezoelectric fiber-reinforced composite has a relatively high electro-mechanical coupling property ^{1}, as reported by [

Micromechanical models have been used to determine the overall electro-mechanical properties of piezocomposites, which focus mainly on the linear electro-mechanical responses of two-phase piezocomposites, e.g., [

In many applications, hybrid piezocomposites consisting of PZT inhomogeneities and polymeric matrix are often exposed to various mechanical and electrical stimuli. Large electric driving fields can cause significant nonlinear strain responses of polarized PZTs [

This study presents formulations of a hybrid unit-cell model for determining the effective nonlinear and hysteretic responses of hybrid piezocomposites, which consist of unidirectional piezoelectric fibers embedded in a viscoelastic polymeric matrix reinforced with piezoelectric particle fillers, subjected to high electric fields and mechanical stresses. In this paper, fibers and particles are made of PZTs; however, the unit-cell model formulation is general and can incorporate different piezoelectric materials for the different inhomogeneities. We consider both nonlinear electro-mechanical response of polarized PZTs and polarization switching behavior of PZTs under large cyclic electric field inputs. This article is organized as follows: Section 2 briefly discusses the constitutive models for the constituents followed by numerical methods for solving the coupled nonlinear electro- mechanical constitutive models in Section 3. Section 4 presents the formulation of the hybrid-unit-cell model for obtaining the effective nonlinear and rate-dependent hysteretic response of composites. Numerical results on the effective responses of the hybrid piezocomposites are discussed in Section 5. Section 6 is dedicated to conclusions.

PZTs are polarized by applying high electric field, above the coercive electric field at elevated temperature [

A nonlinear constitutive model proposed by [

where

A rate-dependent electro-mechanical constitutive model, incorporating polarization switching response, formulated by [

where _{3} direction chosen as the poling axis. The upper right superscript t indicates the current time. The piezoelectric constant

where

where

where

Both

The rate of the residual polarization during polarization switching response is:

where

The compressive stresses along the poling axis could significantly affect the hysteretic polarization switching response. In this study, it is assumed that the coercive electric field varies with the compressive stresses along the x_{3} direction:

where

where _{2} is a material parameter.

^{−9} F/m) | ^{−9} F/m) | ^{−6} F/m) | ^{−6} F/m) | ||||
---|---|---|---|---|---|---|---|

0.67 | 70 | 225 | 1 | 0.35 | 3 | 1.6 | 4 |

^{−12} m/V) | ^{−12} m/V) | ^{−9} F/m) | ^{−9} F/m) | ^{2}) | |
---|---|---|---|---|---|

1520 | −570 | 38 | 42 | 0.194 | 0.19 |

34.48 | 33.00 | 13.19 | 12.37 | 0.307 | 0.334 |

^{−6} F/m) | ^{−6} F/m) | ||||
---|---|---|---|---|---|

25 | 0.3 | 0.40 | 3 | 1.1 | 4 |

The polymeric matrix is assumed as an isotropic viscoelastic solid, which is:

where

where

Here D_{0} is the instantaneous (elastic) compliance and the transient compliance is expressed in terms of a series of exponential functions, where N is the number of terms, ^{th} coefficient of the time-dependent compliance and ^{th} reciprocal of retardation time.

For convenience in analyzing the time-dependent and nonlinear electro-mechanical behavior, we present a linearized incremental form of the constitutive relations, i.e., Equations (1), (2), (3), (4), (15) and (16). A recursive time-integration algorithm presented in [

where superscript

The linearized constitutive relation can be expressed in a single equation, which follows a conventional indicial notation with lower case subscripts range from 1 to 3 while upper case subscripts range from 1 to 4:

where

The components of

From Equations (1) and (2), the resulting components of

From Equations (3) and (4), the resulting components of

where

where

Using the rate of residual polarization in Equation (12), the incremental residual polarization at current time t is approximated by:

Finally,

where the history variable related to the polarization is:

and the incremental polarization is determined by:

From Equations (15) and (16), the resulting components of

where the history variables related to the deviatoric and volumetric strains are:

This section presents formulations of a hybrid-unit-cell model for obtaining the overall responses of hybrid piezocomposites whose constituents experience nonlinear electro-mechanical and viscoelastic behaviors. The microstructures of a hybrid piezocomposite are idealized with periodically distributed fibers of square cross section in a matrix medium and the microstructures of the matrix are idealized with periodically distributed cubic particles in a homogeneous viscoelastic matrix. Here, we consider a unit cell as the smallest representative microstructures and each unit cell is divided into several subcells.

The first subcell of the fiber unit cell is the piezoelectric fiber constituent and the rest of the subcells represent the matrix, whose response is determined from a homogenized active composite of the particulate unit cells. The first subcell of the particulate unit cell is the piezoelectric particle constituent and the remaining subcells in the particulate unit cell indicate the homogeneous viscoelastic matrix. The fibrous and particulate unit cells lead to rather simple micromechanical relations by satisfying equilibrium condition and displacement compatibility among all subcells. The time-integration algorithm for the rate-dependent PZT (Equation (9)) and viscoelastic matrix (Equation (15)) is nested to the hybrid-unit-cell model in order to obtain approximate solutions of the overall nonlinear and time-dependent responses of the hybrid piezocomposites. The cross section of the fiber is assumed to be square and the one of the particle is taken to be a cube, which is done to simplify the micromechanics formulation. In the previous work by one of the author ( [

For the derivation of the hybrid-unit-cell model, we start with the fibrous unit cell. Using a volume-average scheme, the effective field variable, denoted by an overbar, of the fibrous unit cell at current time t is written as:

The superscript

and also for the subcell

In order to relate the effective incremental independent field variables in the fibrous unit cell to the corresponding incremental field variables in its subcells, a concentration matrix

Substituting

Substituting

From Equations (43) and (39), the effective electro-mechanical property and history variable of the fibrous unit cell are:

The linearized constitutive model for the fiber subcell I is obtained directly from Equation (22). The matrix subcells II, III, and IV in the fiber unit cell consist of piezoelectric fillers dispersed in the polymeric matrix. The electro-mechanical properties of these subcells are determined using the particle unit-cell model, comprising of eight subcells (

The superscript

It is also necessary to determine the concentration matrix for the particulate unit-cell

The above equation relates the incremental independent field variables of the matrix subcells II, III and IV to the corresponding incremental field variables of the particulate and polymer subcells. Substituting Equation (48) into Equation (47) and using the volume-average scheme in Equation (46), the corresponding dependent field variables for the matrix subcells are:

Comparing Equation (49) to Equation (40) gives the overall electro-mechanical properties and history variables of the matrix subcells:

Finally, in order to evaluate the concentration matrices and history variables

where

and

where

Once

This section presents numerical analyses of the hybrid-unit-cell model. We first compare the predictions of the effective properties of hybrid composites with existing experimental data, which is limited to linear elastic moduli. We then conduct parametric studies on investigating the effects of constituent compositions, boundary conditions and loading history on the overall performance of hybrid piezocomposites.

Available experimental data for hybrid composites were primarily focused on the overall mechanical properties. The presented nonlinear hybrid-unit-cell model should be capable of predicting the overall properties of the hybrid composites without electro-mechanical coupling effect. Reference [_{22} and E_{33}) for the carbon fiber and the modulus (E) for the epoxy by using the experimental data on the FRP composite with the fiber volume fraction 0.41 shown in ^{2}. The constituent properties used in the simulation are listed in ^{3}. The calibrated material properties are then used to evaluate the effective longitudinal and transverse moduli of the hybrid composite with different fiber volume contents, as shown in

We first examine the effect of constituent compositions on the overall nonlinear electro-mechanical responses of a hybrid piezocomposite subjected to large electric fields but lower than coercive electric field (the constitutive relations in Equations (1) and (2) are used for polarized PZT fibers and particles). The matrix of the hybrid piezocomposite is first considered as elastic solid such as Araldite D while the polarized PZT-G1195 is used for the inhomogeneities^{4}. The material properties of the Araldite D and polarized PZT-G1195 used for simulations reported in [

_{3} axis) up to 1 MV/m^{5} for a fully constrained displacement of the PZT-G1195/[PZT-G1195/Araldite D] hybrid piezocomposite with PZT-G1195 fiber volume fraction (VF) = 0.4 and several PZT-G1195 particle VFs = 0 - 0.5. The linear response for the composite with zero content of PZT-G1195 fillers is also shown for comparison.

We also consider a stress free boundary condition for a PZT-G1195/[PZT-G1195/Araldite D] hybrid piezocomposite with PZT-G1195 fiber VF = 0.4 and PZT-G1195 particle VF varies from 0 to 0.5, subjected to an applied electric field

free strains, and vice versa. Adding stiffer fillers, i.e., PZTs, into a relatively soft matrix, i.e., polymer, in a fiber- reinforced piezocomposite is done to improve the transverse blocked stress.

In order to study the time-dependent responses due a viscoelastic constituent in a hybrid piezocomposite, FM73 polymer whose dielectric constants are taken as

Next, we investigate the overall hysteretic polarization switching and butterfly strain responses of a hybrid piezocomposite with various constituent compositions and under different loading histories. The constitutive relations in Equations (3) and (4) are used for polarization switching response of PZT-51 fibers and particles. The matrix of the hybrid piezocomposite is considered as FM73 polymer.

Carbon fiber | ||
---|---|---|

Longitudinal Young’s modulus, E_{33} (GPa) | 198 | |

Transverse Young’s modulus, E_{22} (GPa) | 16 | |

Major Poisson’s ratio, v_{31} | 0.20 | |

In-plane Poisson’s ratio, v_{12} | 0.25 | |

Longitudinal shear modulus, G_{31} (GPa) | 28 | |

Epoxy | Alumina | |

Young’s modulus, E (GPa) | 3.4 | 416 |

Poisson’s ratio, v | 0.35 | 0.23 |

n^{a} | λ_{n} (sec^{−1}) | D_{n} (GPa^{−1}) |
---|---|---|

1 | 1 | 0.0210 |

2 | 10^{−1} | 0.0216 |

D_{0} = 0.369 (GPa^{−1}) | ||

v = 0.35 |

^{a}We only consider the first two terms of the series of exponential functions to the viscoelastic FM73 polymer. This simplification will not affect us to qualitatively understand the influence of the viscoelastic constituent to the overall responses of composites but it will dramatically reduce computational cost.

the longitudinal fiber direction (x_{3} direction) with the frequency f = 1 Hz. It is expected that the heights of the butterfly curves (

Next, we examine the effect of prescribing compressive stresses on the overall nonlinear rate-dependent hysteretic electro-mechanical responses of a PZT-51/[PZT-51/FM73 polymer] hybrid piezocomposite with PZT-51 fiber VF = 0.4 and PZT-51 particle VF = 0.2, subjected to a cyclic electric loading

We also study the effect of frequencies on the overall hysteretic electro-mechanical responses of a hybrid piezocomposite. We consider a stress free PZT-51/[PZT-51/FM73 polymer] hybrid piezocomposite with PZT-51 fiber VF = 0.4 and PZT-51 particle VF = 0.2 subjected to cyclic electric loadings

along the fiber axis with different frequencies f = 0.5, 1 and 10 Hz.

f = 1 Hz along the poling direction is used in the analysis. ^{6} at various cycles. The initial drop in the normalized effective strain amplitude is due to time-dependent polarization effect in the PZT-51 fibers and then the strain amplitude increases at later cycles because of the creep deformation effect in the FM73 polymer constituent. For further explanation, it is seen in

We have developed a hybrid-unit-cell model for predicting the effective nonlinear and rate-dependent hysteretic responses of active hybrid composites. The studied hybrid piezocomposites consist of unidirectional piezoelectric fibers embedded in a polymeric matrix, which is reinforced with piezoelectric particles. Nonlinear electro- mechanical constitutive models, including polarization switching response, are used for the active fibers and particles, while a viscoelastic solid-like model is used for the polymer. In order to predict the effective nonlinear rate-dependent electro-mechanical responses, linearized micromechanical relations are first imposed in order to provide trial solutions at each instant of time. An iterative scheme, i.e., fixed-point method, is then added to minimize errors from linearizing the nonlinear electro-mechanical and time-dependent responses.

We have performed several analyses on understanding the nonlinear electro-mechanical responses of hybrid piezocomposites using the above hybrid-unit-cell model. The results are summarized as follow: The hybrid unit-cell model is capable of capturing the linear elastic response of fiber-reinforced composites and hybrid composites, which are tested with limited experimental data. Adding PZT fillers significantly improve the blocked stress in the transverse fiber direction while insignificantly affects the overall electro-mechanical performance in the longitudinal fiber direction. This is because the matrix, whose properties change with adding the PZT fillers and dominate the transverse response. The free strains, however, significantly decrease in both transverse and longitudinal fiber directions as the amount of PZT fillers increase. This is due to the fact that adding stiffer PZT particles in a softer epoxy matrix results in a stiffer overall matrix. Thus, adding PZT fillers is useful for improving the blocked stress for active composites with 3 - 1 operating mode. Responses of the hybrid piezocomposites under cyclic electric field, with amplitude higher than the coercive electric field limit of the materials, and compressive stress loadings have been studied. Adding PZT fillers slightly reduces the hysteretic polarization response, and significantly decreases the hysteretic strain response. As the matrix becomes stiffer, matrix would experience smaller deformations when an electric field input is applied, resulting in smaller residual stresses^{7} in both fibers and matrix. Although its effect is minimum, the residual stress would affect the overall hysteretic polarization in composites. As also expected compressive stresses applied along the direction of electric field reduce the polarization capability of the composites. We also investigate the effect of frequencies on the overall electro-mechanical responses of hybrid composites. A lower frequency input allows the hybrid piezocomposites to undergo more pronounced time-dependent response, which in this case is shown by broader hysteretic responses. The hysteretic response indicates amount of energy being dissipated, which is converted into heat. It is noted that many applications of active materials would involve cyclic electro-mechan- ical loading, thus the hysteretic response can eventually lead to cyclic failures.

This research is sponsored by the National Science Foundation (NSF) under grant CMMI-1437086.

Chien-HongLin,AnastasiaMuliana, (2016) Nonlinear and Rate-Dependent Hysteretic Responses of Active Hybrid Composites. Materials Sciences and Applications,07,51-72. doi: 10.4236/msa.2016.71006