Constitutive model of piezoelectric/shape memory polymer composite

The PZT/SMP composite material, combining piezoelectricity and shape memory effects, has emerged as a novel multifunctional smart material and has gained significant attention. The establishment of an effective constitutive model, describing the mechanical properties of the composites, holds tremendous importance in both theoretical and engineering realms. This study aims to develop a model to investigate the mechanical properties of PZT/SMP composites. By incorporating the deformation mechanisms of the PZT/SMP composites and the theories of viscoelasticity, the constitutive equation for PZT/SMP composites has been established. Based on composite material theory and the properties of transversely isotropic materials, combined with the material parameter equations for SMP, the material parameter equations of PZT/SMP composites have been established. Using the polarization direction of the composite material as an example, we conduct simulation analyses on PZT/SMP composites with various volume fractions of PZT particles by using the developed constitutive model. The variations in material parameter performance, piezoelectric performance, and shape memory performance are investigated. The simulation results demonstrate that an increase of PZT content in the PZT/SMP composite enhances both the piezoelectric and mechanical properties of the composite material, which significantly influences the shape memory process.


Introduction
Piezoelectric/shape memory polymer (PZT/SMP) composite materials are multifunctional composites composed of PZT particles and SMP matrices [1].Through the introduction of piezoelectric particles, the composite material exhibits piezoelectric effects, enabling the conversion between mechanical energy and electrical energy.When subjected to external forces or electric fields, polarization occurs within the composite material, resulting in corresponding charges and deformation.Meanwhile, the shape memory polymer matrix imparts shape memory effects to the composite material.This composite material not only possesses the advantages of both piezoelectric materials and shape memory polymer materials, but also demonstrates high strength, high hardness, good flexibility, and excellent mechanical processing performance [2].Additionally, the composite material exhibits characteristics of self-diagnosis, self-adaptation, and self-repair.Due to its superior mechanical properties, the piezoelectric/shape memory polymer composite materials hold great potential for applications in actuators, sensors, medical devices, and other fields [3].
The investigation of the mechanical behavior of PZT/SMP composite materials holds significant importance due to their exceptional mechanical properties.Chen et al. [1,4] successfully prepared PZT/SMP composite materials by combining PZT particles with an SMP matrix and subsequently fabricated thin film actuators by using this composite material.They then conducted a detailed analysis of the mechanical behavior characteristics of the PZT/SMP thin film actuators.Yuan et al. [5] proposed a host thin plate composed of PZT and SMP, denoted as LASMP, and investigated its natural frequencies.Guan et al. [6] proposed a PZT/SMP composite nanofiber by combining PZT particles with an SMPU matrix and investigated the factors influencing the recovery performance.
However, to the best knowledge of the author, theoretical research on PZT/SMP composite materials is relatively limited.Theoretical research plays a crucial role in guiding the practical application of PZT/SMP composite materials.In order to better apply PZT/SMP composite materials in a wider range of fields, conducting theoretical research is indispensable.
In this paper, a novel constitutive model for PZT/SMP composites is proposed to investigate its mechanical properties.The specific structure is as follows: In Section 2, the constitutive equations for the composite material are established by considering the deformation mechanism of PZT/SMP, combining the principles of solid mechanics and viscoelasticity theory.In Section 3, the material parameter equations for PZT/SMP composites are derived based on the composite material theory, transverse isotropy properties of PZT particles, and material parameter equations for the SMP matrix.In Section 4, numerical simulations are conducted by using the established constitutive model to analyze the mechanical performance of PZT/SMP composites.Finally, the primary findings of this study are succinctly summarized.

Constitutive equations of pzt/smp composite
In this chapter, we will establish the constitutive equations for PZT/SMP composite materials.To analyze their strain composition, we can separately analyze the SMP matrix and PZT particles in the composite material.Firstly, regarding the SMP matrix, unlike conventional solid materials, the SMP matrix in composite materials is a type of high polymer.Its mechanical behavior is closely related to temperature change rates and loading rates, which are the result of the movement of large polymer chains under different thermodynamic conditions.According to solid mechanics and thermoviscoelasticity theory, we can summarize the factors that affect the mechanical properties of SMP as viscoelasticity and thermal expansion.For the convenience of research, the total strain of the SMP matrix can be decomposed into elastic, viscous, and thermal expansion strains.As for the PZT particles, unlike general elastic solid materials, as a piezoelectric material, its mechanical behavior is influenced not only by elasticity and thermal expansion but also by the electric field.Thus, the total strain of the PZT particles can be decomposed into elastic strain, thermal expansion strain, and electric strain.
In summary, the total strain of the PZT/SMP composite material has four components.The total strain of the PZT/SMP composite material can be expressed as:  denote the elastic strain, viscous strain, piezoelectric strain, and thermal expansion strain tensors, respectively.And the i  can be described as: (2) According to solid mechanics theory, although the SMP matrix is isotropic, the PZT particles are transversely isotropic.Therefore, the PZT/SMP composite material is transversely isotropic.Thus, the elastic strain tensor of the PZT/SMP composite is represented as: e =1, 2, , 6 where ij s represents the elastic compliance coefficient tensor, and j  represents the force stress tensor, which can be described as:  [7], in consideration of the relationship between the viscous strain tensor and the SMP matrix, the representation of the viscous strain tensor can be derived based on the isotropic nature of the SMP matrix and in conjunction with the principles of solid mechanics and viscoelasticity.The resulting expression is as follows: where  and  represent viscosity coefficient and delay time, respectively.Similar to the viscous strain, piezoelectric strain tensor can be derived based on the transversely isotropic nature of the PZT particle.According to the piezoelectric theory, it can be expressed as: where ki d represents piezoelectric strain constant tensor and k E represents electric field tensor.Similar to the elastic strain tensor, according to the theory of solid mechanics, based on the transverse isotropy properties of composite materials, the thermal expansion strain tensor is represented as: where i  represents thermal expansion coefficient tensor, T represents temperature, and 0 T represents initial temperature.
Substituting the four strain tensor expressions, Equation (3), Equation ( 5), Equation ( 6), and Equation (7) into Equation ( 1), the total strain tensor can be described as: To overcome computational difficulties, we differentiate Equation ( 6) with respect to time, , , , , Additionally, when establishing the constitutive equation for PZT/SMP piezoelectric composite materials, the influence of electric displacement should be considered for describing their response and characteristics under the electric field.The electric displacement of PZT/SMP is also composed of two parts: the electric displacements caused by the force stress and the electric field.Thus, the electric displacement of the PZT/SMP composite can then be expressed as: where i D represents the electric displacement tensor, while F i D and E i D represent the force-induced electric displacement tensor and the electric field-induced electric displacement tensor, respectively.They can be expressed as: where lk  represents dielectric constant tensor.By substituting Equation (10) into Equation ( 9), the expression for the electric displacement tensor can be redefined as follows:

Material parametric equations of pzt/smp composite
In this chapter, we will establish the material parametric equations for PZT/SMP composite materials.
According to the analysis of the modeling process in the constitutive equations of composite materials, it can be inferred that the thermal expansion coefficient, piezoelectric strain coefficient, and dielectric coefficient of composite materials can be regarded as transversely isotropic material parameters, while the viscosity coefficient and delay time can be regarded as isotropic material parameters.
According to the properties of transversely isotropic materials, the non-0 elements in the elastic flexibility coefficient tensor are as follows: v denotes isotropic plane Poisson's ratio, 13 v denotes the non-isotropic plane Poisson's ratio, and G denotes the shear modulus of the non-isotropic plane.
In composite materials, there are certain voids presenting between the particles and the matrix.According to Noda and Fujimoto's study [8,9], for composite materials containing voids, their elastic modulus and shear modulus can be described as follows: and as for the remaining material parameters of PZT/SMP composite material, in accordance with the Makoto Utsumi model, they can be described as: where X denotes the various material parameters of the composite material mentioned earlier in the preceding context, l X and p X represent the various material parameters of the PZT particles and SMP matrix, respectively, l c represents the volume fraction of PZT particles, p c represents the SMP matrix volume fraction, and a denotes contact coefficient, reflecting the level of contact between particles and matrix.
For the SMP matrix, as an isotropic material, the following relationships exist in terms of material parameters: Meanwhile, SMP matrix as a polyphase material, the elastic modulus, viscosity coefficient, and delay time vary with temperature changes.According to the study of Zhao et al. [10], they can be described as: where pg X and pr X represent material parameters of SMP matrix in glass and rubber phases, respectively, and   g , h T T is a level-set function which is described as: where g T denotes the glass transition temperature, 0 T represents the temperature corrected parameter and s represents the regularization parameter controlling the interphase interface width of materials.

Thermo-mechanical behavior analysis
In this section, we utilize the established constitutive model to conduct numerical simulations.Furthermore, their influence on the material parameter characteristics, piezoelectric behavior, and shape memory behavior is analyzed.According to Zhao et al.'s work [10], the values of the parameters for the PZT particle and SMP matrix can be obtained from Table 1 4.96×10 -10 4.96×10 -10 1 .5 ×1 0 -6 1 .5 × 1 0 -6 2 × 1 0 -6 10 6

Material parameter analysis
In this subsection, we conduct a study and analysis on the material parameters of PZT/SMP composite materials with different PZT particle volume fractions, using elastic modulus 1 Y and viscosity coefficient  as examples.In Figure 1, the plots labeled as (a) and (b) illustrate the temperature-dependent behavior of the elastic modulus and viscosity coefficient for different volume fractions of PZT particles.It is evident that as the temperature increases, both the elastic modulus and viscosity coefficient decrease.This can be attributed to the transition of the SMP matrix from a glassy state to a rubbery state with increasing temperature.Additionally, it can be observed that at the same temperature, the elastic modulus increases while the viscosity coefficient decreases with an increase in PZT particle volume fraction.The behavior can be explained by the reinforcing effect of the PZT particles, which have a higher elastic modulus and a near-zero viscosity coefficient compared to the SMP matrix.The remaining correlation coefficient changes are similar to those of the elastic modulus and viscosity coefficient.

Piezoelectric behavior analysis
In Figure 2, Figure 2(a) and Figure 2(b) show the curves of electric displacement and piezoelectric strain with temperature under different PZT particle volume fractions.It is obvious that with an increase in electric field intensity, observable increases in both electric displacement and electric strain can be observed.Furthermore, experimental observations indicate that at the same electric field intensity, the magnitude of the increase in electric displacement and electric strain gradually increases with an increase in PZT particle volume fraction.It can be explained that the SMP matrix itself does not possess piezoelectric properties, but the presence of PZT particles enables the composite material to exhibit piezoelectric behavior.As the volume fraction of PZT particles increases, the contribution of the piezoelectric properties in the system also increases.Figure 3 illustrates the shape memory effect curves of PZT/SMP composite materials with different volume fractions of PZT particles.Figure 3(a) presents the stress-strain relationship during the shape memory process.It can be observed that under high-temperature loading conditions, the stress-strain behavior of S1 exhibits nonlinearity.This is attributed to the occurrence of both elastic and viscous strains in the composite material with increasing loading time.Moreover, an increase in PZT particle volume fraction leads to an increase in the elastic modulus of composite material, resulting in higher stress for the same strain.
Figure 3(b) displays the stress-temperature curves during the shape memory process.The S2 stage represents the stress freezing process, where it is observed that the stress initially decreases and then increases with decreasing temperature.This is due to the continued increase in viscous strain under external forces while maintaining the total strain constant during the early stages of temperature decrease, leading to stress reduction.However, in the later stages, the elastic modulus and other parameters gradually increase, requiring an increase in external force to maintain the same deformation.Additionally, an increase in PZT particle volume fraction reduces the viscosity coefficient of composite material, resulting in larger viscous strains and more pronounced stress changes.
Figure 3(c) demonstrates the strain-temperature curves.The S4 stage represents the shape recovery process, where it is observed that with increasing temperature, the material parameters begin to recover, and residual strain gradually diminishes.Due to the presence of viscosity, the relationship between strain and temperature exhibits nonlinearity.Furthermore, an increase in PZT particle volume fraction reduces the viscosity coefficient of composite material, leading to more significant variations in strain.

Conclusion
This study has established a novel constitutive model for PZT/SMP composite materials.By integrating the deformation mechanism of PZT/SMP composites with solid mechanics theory and viscoelasticity theory, the constitutive equation for PZT/SMP composites has been derived.Based on the composite material theory and the properties of PZT particles as transversely isotropic materials, the material parameter equation for SMP has been obtained and incorporated into the constitutive equation for PZT/SMP composites.The developed model has been utilized to simulate and investigate the mechanical behavior of the composite material.The results indicate that an increase in the volume fraction of PZT particles leads to an increase in the elastic modulus and coefficients related to the piezoelectric performance while decreasing the viscosity and time delay.As a consequence, the piezoelectric and mechanical properties of the composite material are improved.Furthermore, the reduced viscosity coefficient and time delay have a significant influence on the shape memory effect of the composite material.These parameters, being affected by temperature, result in more pronounced changes during the shape memory process.

Figure 1 .
Figure 1.Curves of (a) elastic modulus and (b) viscosity coefficient with temperature under different PZT particle volume fractions.

Figure 2 .Figure 3 .
Figure 2. Curves of (a) electric displacement and (b) piezoelectric strain with temperature under different PZT particle volume fractions.4.3 Shape memory behavior analysis

Table 1 .
. Material Parameters of PZT Particle and SMP Matrix.