1 Introduction

Various applications of smart structures (Chee et al. 1998; Chopra 2002) in particular actuators and sensors have engrossed the mind of scholars for the last few decades. Researchers have extensively explored the electromechanical transducers based on electrostrictive, piezoelectric, and magnetostrictive actuators (Crawley 1994; Zheng et al. 2004). However, as the new technologies are swiftly moving towards miniaturization these traditional actuation techniques are not able to cope up, due to their various disadvantages. One of the bottlenecks is electrical noise induced due to metallic wires, which will hinder the control signals. On the contrary, recently researchers have started exploring opto-electro-mechanical wireless non-contact actuation of the devices using light stimuli (Ikeda et al. 2007; Uchino 2019; Varadan et al. 2006). The advantages of opto-electro-mechanical actuators are lightweight, compactness, and wireless actuation capabilities (Schnabel 2007). Several material properties can be manipulated using light actuated devices, such as shape, color, and refractive index in photostrictive (Yu et al. 2003), photochromic (Bonora et al. 2010), and photorefractive (Schnabel 2007) devices respectively. The salient feature of photostrictive devices is that it requires neither an electrical wiring nor additional electronic components. Photostrictive materials belong to the family of smart materials, which can convert high-intensity light stimuli of a certain wavelength into a mechanical strain (Tzou and Chou 1996; Uchiho 1997; Shih and Tzou 2007). Photovoltaic effect when combined with converse piezoelectric effect results in photostrictive effect. Due to the photovoltaic effect high intensity of light is converted into electric potential, which is then converted into mechanical strain via converse piezoelectric effect. Therefore, both of these effects should be incorporated into the constitutive equations of photostrictive materials (Tzou and Chou 1996). PLZT is the prominently used material amongst all the photostrictive materials because of its high piezoelectric coefficient and outstanding photovoltaic properties (Tzou and Chou 1996).

The governing equation and analysis of photostrictive actuator were obtained by Tzou and Chou (1996). Photostriction is a multiphysics problem, which includes thermoelectricity, pyroelectricity, piezoelectricity, and photovoltaic effect. The mathematical couplings of these physics for vibration control on plates were explained by Liu and Tzou (1998). Modeling of distributed photostrictive in one and two dimensions was carried out by Shih and Tzou (2000). The term photostriction was coined by Uchino et al. (1982). They have explored the piezoelectricity and bulk photovoltaic effect in PLZT ceramic. Researchers have investigated the optical photostrictive actuators for various fields such as ultra-high vacuum, space technology, and micromechanics. Photophonic devices that produce sound due to mechanical vibrations, when subjected to intermittent light stimuli were discussed by Uchino et al. (2001). Several researchers have explained the relation of irradiated light intensity on the photovoltaic effect (Brody 1975; Uchino 1989; Uchino et al. 1985). Brody (1975) reported that photovoltage and current depends upon irradiation of light intensity in the PZT based ceramics. It was also reported that short circuit photocurrent varies linearly with intensity of light, whereas photovoltage varies linearly at a low light intensity and gets saturated at a higher light intensity. Similar observations were deduced by Uchino et al (1985) in MnO2 doped PLZN ceramic and in single crystal of BaTiO3 by Koch et al. (1975). Shih et al. (2005b) described the general mathematical equations which couples all the physics involved in optopiezothermoelastic actuation. They have demonstrated the static analysis of photostrictive actuator bounded on the flexural beam. This analytical analysis gave an insight into the response of flexural beam deflection when irradiated with specific light intensity. Many researchers have modified the structure design and reiterated the placement of actuator to enhance the deflection response (Barboni et al. 2000; Gaudenzi and Barboni 1999; Main et al. 1994). Recently researchers have explored vibration control in different structures such as cylindrical shells (Shih et al. 2004; He et al. 2013), plate (Shih et al. 2005a; He and Zheng 2013), shallow shell (Shih and Tzou 2007), and parabolic shells (Shih et al. 2009), using PLZT ceramic actuator layers.

Photostrictive ceramic materials, such as PLZT, are very attractive choice for photostrictive sensing and actuation applications. Despite of their interesting properties, their use in shape control applications is limited due to their weight. The composite material can be a better technical solution, which can overcome the several drawbacks of bulk photovoltaic ceramic materials in many applications such as imaging in medical field, damping, and actuators. In the past decade researchers have developed composite materials by combining piezoceramic material with polymers. Extracting benefits out of advantageous properties of both constituent’s materials makes the composite superior to the conventional piezoelectric materials. It is of interest to comprehend the response of local fields in constituting phases and overall optoelectromechanical coupling behavior. Behavior of composite piezoceramics can be characterized by different homogenization techniques. Comprehensive functioning of piezoelectric fiber composite from properties of individual constituents (matrix and fiber) can be derived from micromechanical method by performing analysis over a representative volume element (RVE) or a unit cell. Whereas the advantage of deploying micromechanical approach instead of traditional macromechanical method (replacing the heterogeneous composite with anisotropic properties of homogeneous medium) is not only limited to finding out the global composite properties but it can also be used for crack initiation and propagation analysis. Initially, basic analytical approach was developed which was not able to predict all the elastic moduli (Chan and Unsworth 1989; Smith and Auld 1991). Later, a semianalytical model was explained by Li and Dunn (2001) which proved to be a benefitial tool for theoretical consideration as it was able to predict all elements of material behavior. A single inclusion embedded in an infinity matrix was studied for elecroelasticity by extending the mechanical mean-field type method, which was based on Eshelby type solution (Benveniste 1992; Dunn and Wienecke 1997). Field quatitie’s local variation was difficult to capture using mean field type method. This drawback was tackled when researchers employed the unit cell (microfield) approach, which uses finite element method to solve for field quantities (Gaudenzi 1997). Even though the microfield approach was able to overcome previous bottlenecks, this model’s ability was limited to a few load cases. After all these works, Berger et al. (2006) reported the micromechanical approach which was able to predict the complete behavior i.e., complete tensor of piezoelectric, dielectric, and elastic properties, while considering the local fluctuations of the field quantities. They have applied the finite element approach to analyze the unit cell, which can be periodically repeated to represent a composite. Magouh et al. (2020) reported finite element analysis to report the pyroelectric and dielectric properties of a 0–3 composite and they have experimentally validated their FEA results.

To the best of authors knowledge, there exist no research that enhances the actuation response by enriching the properties of PLZT ceramic if carried out at all. In this paper, 0–3 composite of PLZT and PMN-35PT as inclusions has been proposed in order to improve the photovoltage and converse piezoelectric response of the PLZT ceramics. However, it is worth to focus on the behavior exhibited by photostrictive actuators is different from traditional piezoelectric actuators, as intensity of light irradiation is always positive and the deformation is always positive too.

2 Materials and methods

The current study carries out two finite element analyses. First, to explore all elastic, piezoelectric, dielectric, and pyroelectric properties of transversely isotropic 0–3 composite of PLZT as matrix and PMN-35PT as filler material. Random distribution of spherical inclusions in a cube is generated using MATLAB code, while ensuring there is no overlapping between inclusion-inclusion and inclusion-RVE boundary as shown in Fig. 1.

Secondly, finite element analysis of photostrictive actuation in an unimorph beam subjected to cantilever and simply supported boundary condition. Constitutive materials and their respective material attributes are given in Table 1.

Table 1 Attributes of the constituent materials
Full size table

2.1 Opto-piezoelectricity and opto-pyroelectricity

Photo deformation process is induced due to irradiation of high energy illumination over an optical actuator (Tzou and Chou 1996). The photo deformation effect is the superposition of two effects i.e., the photovoltaic effect and the converse piezoelectric effect (Shih and Tzou 2000). Photostrictive materials can be polarised in two ways: (1) along their length direction \(\tilde{Z }\) or 3 when the electrodes are in \(\tilde{X }- \tilde{Y }\) plane as shown in Fig. 2a and, (2) along its thickness direction \(\tilde{Y }\) or 2 while the electrodes are in the \(\tilde{X }- \tilde{Z }\) plane as shown in Fig. 2b. If photostrictive material is polarized in 3rd direction then one of the electrodes has to be transparent to allow light irradiation to penetrate. In the current study, an actuator is polarised in 3rd direction. Current induced due to light irradiation flows opposite to the polarisation direction, which in turn causes a potential difference between two electrodes, this phenomenan is called as the photovoltaic effect. This induced potential will get converted into the mechanical strain by the virtue of converse piezoelectric effect.

Fig. 1
figure 1

Illustrative diagram of 0-3 composite and its global coordinates

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The electric field El(t) induced due to photovoltaic effect can be expressed as (Shih and Tzou 2000),

$${E}_{l}\left(t\right)= {E}_{l}\left({t}_{j-1}\right)+\left[{E}_{s}-{E}_{l}\left({t}_{j-1}\right)\right]\frac{{\alpha }^{{\prime}}}{{a}_{s}}I\left({t}_{j}\right){e}^{-\frac{{\alpha }^{{\prime}}}{{a}_{s}}I\left({t}_{j}\right)\Delta t}.\Delta t-{E}_{l}\left({t}_{j-1}\right)\beta {e}^{-\beta\Delta t}.\Delta t,$$
(1)

where, as is the aspect ratio defined as the ratio of length to width of the actuator, saturated photovoltaic field is represented as Es and \(\beta\) is a constant for leakage voltage. \({E}_{l}\left({t}_{j}\right)\) refers to the electric field at time tj and I(t) is time dependent function of the light intensity. As I(t) is constant for the step time period \(\Delta t\), electric field at any time \({t}_{j}={t}_{j-1}+\Delta t\) is \({E}_{l}\left({t}_{j}\right)\).

At any time instant \({t}_{j}\), actuator temperature can be estimated by,

$$\theta \left({t}_{j+1}\right)=\theta \left({t}_{j}\right)+\Delta \theta =\theta \left({t}_{j}\right)+\frac{\left[I\left({t}_{j}\right){P}^{*}-\gamma \theta ({t}_{j-1})\right]\Delta t}{H+\gamma\Delta t},$$
(2)

where \(\Delta \theta\) is the rise in temperature in time \(\Delta t\), actuator’s heat capacity is represented by H, heat absorbed from the light illumination is referred by \({P}^{*}\), heat transfer rate from actuator to structure is represented by \(\gamma\), actuator’s temperature and light illumination at any time instant \({t}_{j}\) is \(\theta \left({t}_{j}\right)\) and \(I\left({t}_{j}\right)\) respectively.

An additional voltage is also generated which is triggered by the pyroelectric effect. Contribution of pyroelectric effect towards the electrical field is determined as,

$${E}_{\theta }\left(t\right)=\frac{{P}_{n}}{\varepsilon }\theta \left(t\right),$$
(3)

where Pn refers to pyroelectric cofficient and \(\varepsilon\) is the dielectric permittivity.

The total electric field generated is the summation of electric field arises due to photovoltaic effect and pyroelectric effect, which can be expressed as,

$$E\left(t\right)={E}_{l}\left(t\right)+{E}_{\theta }\left(t\right).$$
(4)

Actuator’s performance was brought down by the rising temperature due to the reduction in spontaneous polarisation (Fukuda et al. 1993). From the above discussion, strain induced due to light irradiation can be given as,

$$\overline{S }\left(t\right)={d}_{33}\left[{E}_{l}\left(t\right)+{E}_{\theta }\left(t\right)\right]-\lambda \frac{\theta \left(t\right)}{{Y}_{a}},$$
(5)

where \(\lambda\) refers to coefficient of thermal stress, \({Y}_{a}\) is the actuator’s Young’s modulus and \({d}_{33}\) is the piezoelectric strain constant. Electric potential developed in the actuator is,

$${E}_{Z}\left(t\right)=E\left(t\right)\times h,$$
(6)

where h refers to the actuator thickness.

2.2 Constitutive equations of opto-piezo-pyro-elasticity

The constitutive relations of these coupled physics were reported by Tzou and Ye (1994). These constitutive relations comprise three fundamental equations. The first equation represents the contribution of elastic strain, electric field, and temperature rise towards stress (Tij), also known as Duhamel–Neumann’s equation. The second equation gives the net effect of strain, electric field, and temperature towards the electric displacement (Dn). Third and last equation describes the contribution of elastic strain, electric field, and temperature towards entropy density (\(\epsilon\)) (Tzou and Ye 1994; Mindlin 1974).

$${T}_{ij}=C\left(S-{S}_{0}\right)-eE-{\lambda }_{ij}\theta ,$$
(7)
$${D}_{n}=e\left(S-{S}_{0}\right)+\varepsilon E+{p}_{n}\theta ,$$
(8)
$$\epsilon =\lambda \left(S-{S}_{0}\right)+{p}_{n}E+{\alpha }_{v}\theta ,$$
(9)

where C is the elastic modulus, e is the piezoelectric coefficient, \({S}_{0}\) is the initial strain, S is the strain; E is the electric field, \({\lambda }_{ij}\) is the thermal stress coefficient, Pn is the pyroelectric constant and \(\theta\) is the temperature rise from its reference temperature \({\theta }_{0}\). \({\alpha }_{v}\) is a material constant that depends upon factors such as mass density \(\rho\), specific heat at constant volume cv, and reference temperature. \({\alpha }_{v}\) can be written as \({\alpha }_{v}=\rho {c}_{v}{\theta }^{-1}\).

2.3 Pyroelectric coefficient

Spontaneous polarization Ps is generated whenever a polar material is subjected to temperature fluctuations, which in turn induces pyroelectric current I, this effect is known as pyroelectricity. Photostrictive composite, in which both the matrix and inclusions are polarised in the same direction, is subjected to short circuit condition, to calculate the pyroelectric current. The pyroelectric effect arises of the coupling of electric displacement \(\overrightarrow{D}\) and temperature \(\theta\), which is given by (Magouh et al. 2020),

$$\overrightarrow{D}={\varepsilon }_{0}{\varepsilon }_{r}\overrightarrow{E}+{\overrightarrow{P}}_{s},$$
(10)
$$d\overrightarrow{D}=\varepsilon d\overrightarrow{E}+d{\overrightarrow{P}}_{s},$$
(11)
$$d\overrightarrow{D}=\varepsilon d\overrightarrow{E}+pd\theta ,$$
(12)

where p refers to pyroelectric coefficient. For the pyroelectric material, Eq. (12) expresses the relation between dielectric displacement \(\overrightarrow{D}\), electric field \(\overrightarrow{E}\) and temperature \(\theta .\)

When electric field is absent, it can be deduced from Eq. (10), that \(\overrightarrow{D}={\overrightarrow{P}}_{s}\). Therefore, from Eqs. (10), (11), and (12) generalized pyroelectric coefficient can be determined from the indirect method’s equation,

$$\overrightarrow{p}=\frac{d{\overrightarrow{P}}_{s}}{d\theta },$$
(13)
$$p=\frac{1}{A}.\frac{d{Q}_{suf}}{d\theta },$$
(14)

where \({Q}_{suf}\) refers to surface charges.

The current based method, which was reported by Byer and Roundy (1972) uses Eq. (13) and (14) to estimate pyroelectric coefficient, given as:

$$p=\frac{I}{A.\frac{d\theta }{dt}}.$$
(15)

When the composite is subjected to a small variation of temperature rise and fall in small period of time, a pyroelectric coefficient can be determined using Eq. (15).

2.4 Finite element modeling of 0–3 photostrictive composite

Generally, an object under study is a large/macroscopic structure. A unit cell can be created to imitate the macroscopic properties of photostrictive composite, which is also known as a representative volume element. This approach is capable of capturing all the major features of rudimentary microstructure. Homogenization is the best tool to speed up the computational simulation and composite discretization in realistic condition. Strain energy should remain same in the original composite and the identical homogeneous medium.

There are several analytical and numerical models to predict the mechanical and electrostatic attributes of the lead-based piezoelectric composites (Pettermann and Suresh 2000; Li 2001; Pastor 1997; Nelli Silva et al. 1998). These were not successful to calculate overall material properties. It was very difficult to determine the response of composite embedded with complex-shaped particles, therefore finite element method proved to be very useful numerical tool. Kiran et al. (2018) have investigated the RVE approach using finite element method to estimate the overall properties of 0–3 composite comprised of pores and piezoceramic particles in polyvinylidene difluoride (PVDF) matrix.

For a transversely isotropic opto-piezo-pyroelectric solid, the dielectric coefficients, piezoelectric coefficients, stiffness coefficients, pyroelectric coefficients, thermal stress coefficients and \({\alpha }_{v}\) a material constant are all 14 independent coefficients. Consequently, the Eqs. (7), (8), and (9) is written in matrix form as follows,

$$\left[\begin{array}{c}{\overline{T} }_{11}\\ {\overline{T} }_{22}\\ {\overline{T} }_{33}\\ {\overline{T} }_{23}\\ {\overline{T} }_{31}\\ {\overline{T} }_{12}\\ {\overline{D} }_{1}\\ {\overline{D} }_{2}\\ {\overline{D} }_{3}\\ {\overline{\epsilon }}_{1}\end{array}\right]=\left[\begin{array}{c}{C}_{11}^{eff}\\ {C}_{12}^{eff}\\ {C}_{13}^{eff}\\ 0\\ 0\\ 0\\ 0\\ 0\\ {e}_{13}^{eff}\\ {\lambda }_{1}^{eff}\end{array} \begin{array}{c}{C}_{12}^{eff}\\ {C}_{11}^{eff}\\ {C}_{13}^{eff}\\ 0\\ 0\\ 0\\ 0\\ 0\\ {e}_{23}^{eff}\\ {\lambda }_{1}^{eff}\end{array} \begin{array}{c}{C}_{13}^{eff}\\ {C}_{13}^{eff}\\ {C}_{33}^{eff}\\ 0\\ 0\\ 0\\ 0\\ 0\\ {e}_{33}^{eff}\\ {\lambda }_{3}^{eff}\end{array} \begin{array}{c}{0}_{ }^{ }\\ {0}_{ }^{ }\\ {0}_{ }^{ }\\ {C}_{44}^{eff}\\ {{0}_{ }^{ }}_{ }^{ }\\ {0}_{ }^{ }\\ {0}_{ }^{ }\\ {e}_{15}^{eff}\\ {0}_{ }^{ }\\ {0}_{ }^{ }\end{array} \begin{array}{c}{0}_{ }^{ }\\ {0}_{ }^{ }\\ {0}_{ }^{ }\\ {0}_{ }^{ }\\ {C}_{44}^{eff}\\ {0}_{ }^{ }\\ {e}_{15}^{eff}\\ {0}_{ }^{ }\\ {0}_{ }^{ }\\ {0}_{ }^{ }\end{array} \begin{array}{c}{0}_{ }^{ }\\ {0}_{ }^{ }\\ {0}_{ }^{ }\\ {0}_{ }^{ }\\ {0}_{ }^{ }\\ {C}_{66}^{eff}\\ {0}_{ }^{ }\\ {0}_{ }^{ }\\ {0}_{ }^{ }\\ {0}_{ }^{ }\end{array} \begin{array}{c}{0}_{ }^{ }\\ {0}_{ }^{ }\\ {0}_{ }^{ }\\ {0}_{ }^{ }\\ {e}_{15}^{eff}\\ {0}_{ }^{ }\\ {\epsilon }_{11}^{eff}\\ {0}_{ }^{ }\\ {0}_{ }^{ }\\ {p}_{ }^{eff}\end{array} \begin{array}{c}{0}_{ }^{ }\\ {0}_{ }^{ }\\ {0}_{ }^{ }\\ {e}_{15}^{eff}\\ {0}_{ }^{ }\\ {0}_{ }^{ }\\ {0}_{ }^{ }\\ {\epsilon }_{11}^{eff}\\ {0}_{ }^{ }\\ {p}_{ }^{eff}\end{array} \begin{array}{c}-{e}_{13}^{eff}\\ -{e}_{23}^{eff}\\ -{e}_{33}^{eff}\\ 0\\ 0\\ 0\\ 0\\ 0\\ {\epsilon }_{33}^{eff}\\ {p}_{ }^{eff}\end{array} \begin{array}{c}{\lambda }_{1}^{eff}\\ {\lambda }_{1}^{eff}\\ {\lambda }_{3}^{eff}\\ 0\\ 0\\ 0\\ {p}_{ }^{eff}\\ {p}_{ }^{eff}\\ {p}_{ }^{eff}\\ {\alpha }_{v}^{eff}\end{array} \right]\left[\begin{array}{c}{\overline{S} }_{11}\\ {\overline{S} }_{22}\\ {\overline{S} }_{33}\\ {\overline{S} }_{23}\\ {\overline{S} }_{31}\\ {\overline{S} }_{12}\\ {\overline{E} }_{1}\\ {\overline{E} }_{2}\\ {\overline{E} }_{3}\\ {\overline{\theta }}\end{array}\right]$$
(16)

where \({\overline{S} }_{ij}\),\({\overline{D} }_{i}\),\({\overline{T} }_{ij}\) and \({\overline{E} }_{i}\) mean values of engineering strain, electric displacement, engineering stress, and electric field respectively.

Since in the micromechanical approach effective properties are calculated using represented volume element (RVE), therefore the correctness of the results mostly depends on the choice of RVE. Bonding between the inclusion and matrix is considered to be prefect. The overall composite is modeled as transversely isotropic material. Both of the matrix and inclusion partials are polarised in the same \(\tilde{Z }(3)\) direction. Assosiaiton of opto-piezo-pyro-mechanical is estabilished by including temperature, electrical potential as degrees of freedom along with displacement degree of freedoms.

2.5 Periodic boundary conditions for the unit cell

Unit cells should be arranged in the periodic order to represent a composite. Hence, a unit cell must be subjected to periodic boundary conditions. These periodic boundary conditions make sure that there is no overlapping between the unit cells after deformation and each unit cell is subjected to the same deformation. The required periodic conditions at boundaries in cartesian coordinates explained by Suquet et al. (1983) and Xia et al. (2003) are as follows,

$${u}_{i}={\overline{S} }_{ij}{x}_{j}+{v}_{i},$$
(17)

In the above Eq. (17) \({\overline{S} }_{ij}\) is the average engineering strain and \({v}_{i}\) is the unknown local fluctuation of the periodic boundary condition, that depends on the loads applied globally. The indices i and j range from 1 to 3, they represent the three global mutually perpendicular directions. From the above general equation, a more suitable form of conditions at boundaries i.e., explicitly for the unit cell can be derived. Displacement on the opposite boundary surfaces of the unit cell as illustrated in Fig. 1, with their normal along \({x}_{j}\) direction can be written as,

$${u}_{i}^{{K}^{+}}={\overline{S} }_{ij}{x}_{i}^{{K}^{+}}+{v}_{i}^{{K}^{+}},$$
(18)
$${u}_{i}^{{K}^{-}}={\overline{S} }_{ij}{x}_{i}^{{K}^{-}}+{v}_{i}^{{K}^{-}},$$
(19)

where the index K+ and K− refers to the positive xj and negative xj direction on the corresponding surfaces A+/ A−, B+/ B− and C+/ C− as shown in Fig. 1. As the unit cell is subjected to periodic boundary condition, local fluctuations \({v}_{i}^{{K}^{+}}\) and \({v}_{i}^{{K}^{-}}\) on the two opposite surfaces are identical around the average microscopic value. Therefore, the condition of applied macroscopic strain is equal to the difference of the above two-equation, which is,

Fig. 2
figure 2

a Polarisation of photostrictive material in 3rd direction, b polarisation of photostrictive material in 2nd direction

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$${u}_{i}^{{K}^{+}}-{u}_{i}^{{K}^{-}}={\overline{S} }_{ij}\left({x}_{i}^{{K}^{+}}-{x}_{i}^{{K}^{-}}\right),$$
(20)

Similarly, the electric potential’s \(\varnothing\) periodic boundary condition can be explained by the condition of macroscopic electric field,

$${\varnothing }^{{K}^{+}}-{\varnothing }^{{K}^{-}}={\overline{E} }_{i}\left({x}_{i}^{{K}^{+}}-{x}_{i}^{{K}^{-}}\right).$$
(21)

It is considered that the mean attributes of a particular composite are equal to the average electrical and mechanical attributes of the unit cell. In a unit cell, average stress and strain is defined as,

$${\overline{S} }_{ij}=\frac{1}{V}{\int }_{V}{S}_{ij}dV,$$
(22)
$${\overline{T} }_{ij}=\frac{1}{V}{\int }_{V}{T}_{ij}dV,$$
(23)

where V refers to the volume of the representative volume element (RVE) or unit cell. Similarly, the average electrical displacement \({\overline{D} }_{i}\) and average electric field \({\overline{E} }_{i}\) are defined as,

$${\overline{E} }_{i}=\frac{1}{V}{\int }_{V}{E}_{i}dV,$$
(24)
$${\overline{D} }_{i}=\frac{1}{V}{\int }_{V}{D}_{i}dV.$$
(25)

To discover the homogenized effective properties of the composite, periodic conditions at boundaries can be implimentd on opposite surfaces of RVE according to Eq. (20) and (21). The boundary conditions are applied such that, only one parameter out of strain, electric field, or temperature is non-zero, while the rest are zero. The size of RVE can be considered as 1 μm, as the effective attributes are influenced only by the volume fraction. One of the corners of RVE is at the origin. Eventually, all the coefficients are calculated by applying the boundary condition given in Table 2.

Table 2 Equations and boundary conditions for effective coefficients
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2.5.1 Determining effective elastic constants (\({C}_{11}^{eff}\) and \({C}_{12}^{eff}\))

To compute the effective elastic constants \({C}_{11}^{eff}\) and \({C}_{12}^{eff}\), boundary condition should be applied in a way that non-zero mechanical strain is in the first direction, while the electric potential gradient and mechanical strain remain zero in other directions. The whole RVE will be maintained at the surrounding temperature. From Eq. (11), we can write,

$${C}_{11}^{eff}={\overline{T} }_{11}/{\overline{S} }_{11},$$
(26)

and

$${{C}_{12}^{eff}=\overline{T} }_{22}/{\overline{S} }_{11}.$$
(27)

2.5.2 Determining effective elastic constants (\({C}_{13}^{eff}\) and \({C}_{33}^{eff}\))

The effective elastic constants \({C}_{13}^{eff}\) and \({C}_{33}^{eff}\) can be estimated by implementing conditions at boundaries of RVE such that strain in the x3 direction is non zero and all other mechanical strains are in other directions. Whereas, the electric field is zero in all directions and the Whole RVE will be at the surrounding temperature. Now, from the third and fourth row of constitutive Eq. (11), we can write,

$${C}_{13}^{eff}={\overline{T} }_{11}/{\overline{S} }_{33},$$
(28)

and

$${C}_{33}^{eff}={\overline{T} }_{33}/{\overline{S} }_{33}.$$
(29)

2.5.3 Determining effective elastic constants (\({C}_{44}^{eff}\) and \({C}_{66}^{eff}\)) and an effective piezoelectric coefficient (\({e}_{15}^{eff}\))

These elastic coefficient (\({C}_{44}^{eff}\) and \({C}_{66}^{eff}\)) depends on the mean shear strain and can be estimated by defining coupling constraint on the two-opposite surface i.e., pure shear state. For example, \({C}_{44}^{eff}\) depends on pure in plane (x1-x2) shear state, the coupling constraint equation on the nodes of opposite faces A−/A+ is given using Eq. (20) as \({u}_{3}^{{A}^{+}}={u}_{3}^{{A}^{-}}+{\overline{S} }_{31}\left({x}_{1}^{{A}^{+}}-{x}_{1}^{{A}^{-}}\right)\). An arbitrary value can be assigned to the fluctuation \({\overline{S} }_{31}\left({x}_{1}^{{A}^{+}}-{x}_{1}^{{A}^{-}}\right)\). The analogous type of constraints must be provided on opposite faces C−/C+. Similarly, \({C}_{66}^{eff}\) and \({e}_{15}^{eff}\) is estimated by enacting suitable conditions at boundaries.

2.5.4 Calculations of effective piezoelectric coefficients (\({e}_{13}^{eff}and {e}_{33}^{eff}\))

The effective piezoelectric coefficients \({e}_{13}^{eff}and {e}_{33}^{eff}\) are determined by enacting non zero electric field in the x3 direction, while electric field is zero in other direction. The mechanical strain is zero in all directions and the RVE is maintained at the surrounding temperature. Now, from first and third row of matrix Eq. (11), we can write,

$${e}_{13}^{eff}=-{\overline{T} }_{11}/{\overline{E} }_{3},$$
(30)

and

$${e}_{33}^{eff}=-{\overline{T} }_{33}/{\overline{E} }_{3}.$$
(31)

2.5.5 Calculations of effective piezoelectric coefficients (\({\epsilon }_{11}^{eff}and {\epsilon }_{33}^{eff}\))

To estimate the effective piezoelectric coefficients \({\epsilon }_{11}^{eff}\) and \({\epsilon }_{33}^{eff},\) boundary conditions should be applied such that only non-zero electric field is in the x1 and x3 direction, while the electric field is zero in all other directions. Mechanical strain remains zero in all directions and the RVE is maintained at the surrounding temperature. Now, from seventh and nineth row of matrix Eq. (11), we can write,

$${\epsilon }_{11}^{eff}={\overline{D} }_{1}/{\overline{E} }_{1},$$
(32)

and

$${\epsilon }_{33}^{eff}={\overline{D} }_{3}/{\overline{E} }_{3}.$$
(33)

2.5.6 Determining effective thermal stress coefficient (\({\lambda }_{1}^{eff} and {\lambda }_{3}^{eff}\))

The effective coefficient of thermal stress can be defined as \({\lambda }^{eff}= {\alpha }^{eff}/\left(1-\upsilon \right)\). Therefore, initially effective thermal expansion coefficient should be explored, by enacting the conditions at boundaries such that the whole unit cell can expand freely. To allow the free expansion, RVE was subjected to unit degree temperature rise while constraining the normal displacement in x3 direction on C− surface as illustrated in Fig. 3. To restrict the rigid motion of body, both of the central axes of the C− surface are constrained to displace in their in-plane normal directions. Thermal expansion coefficient can be calculated as,

$${\alpha }^{eff}={S}_{11}^{eff},$$
(34)
$${\lambda }_{1}^{eff}=\frac{{\alpha }^{eff}}{\left(1-{\upsilon }_{12}\right)},$$
(35)

and

Fig. 3
figure 3

Boundary conditions applied to RVE to find thermal expansion coefficient

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$${\lambda }_{2}^{eff}=\frac{{\alpha }^{eff}}{\left(1-{\upsilon }_{13}\right)}.$$
(36)

2.5.7 Determining effective pyroelectric coefficient (\({p}^{eff}\))

The effective pyroelectric coefficient is estimated from the coupling of current density equation and heat transfer equation. Transient temperature (\({\theta }^{*}={\theta }_{0}\mathrm{sin}wt\)) as a function of time t and spatial coordinates (x1, x2, x3) is applied as a boundary condition at the bottom (C−) surface. Employing the finite element programming, the temperature distribution is obtained in the x3 direction at different time intervals. For the analysis of heat transfer, specific heat cp, conductivity k, density \(\rho\) are required. Convection q relates the surface temperature with the surrounding temperature \({\theta }_{0}\). The equation of non-linear heat transfer can be written as,

$$k\left(\sum_{i=1}^{3}\frac{{\partial }^{2}\theta }{\partial {x}_{i}^{2}}\right)+{Q}_{int}=\rho {c}_{p}\frac{\partial \theta }{\partial t},$$
(37)

where, \({Q}_{int}\) refers to the internal heat source, and the convection at the top surface (C+) is defined as,

$$q=hA\left({\theta }_{surf}-{\theta }_{Ambt}\right),$$
(38)

where A refers to surface area of the composite and h refers to convective coefficient of heat transfer.

Ideal thermal contact conductance is considered between the matrix and the particles. The boundary condition is such that temperature variation \({\theta }^{*}\) is assigned to the C− surface and C+ surface is open for heat convection between composite and surrounding. While keeping the other four surface A+, A−, B+ and B− in perfectly insulated condition, which means no specific heat is applied to these surfaces. As there is no heat generation \({Q}_{int}\) inside the photostrictive composite. Hence, the heat transfer Eq. (37), reduces to,

$$k\left(\frac{{\partial }^{2}\theta }{\partial {x}_{3}^{2}}\right)=\rho {c}_{p}\frac{\partial \theta }{\partial t}.$$
(39)

The above equation allows determining the distribution of temperature and derivative of temperature time-variant throughout the composite. The variation of the current I over the C+ surface is determined as,

$$I={\int }_{S}\overrightarrow{j}.\overrightarrow{n}dS,$$
(40)

where \(\overrightarrow{j}\) refers to current density over the surface. Later the current distribution is used to calculate the pyroelectric coefficient using Eq. (15).

2.6 Finite element modeling of unimorph photostrictive structure

The most effective method for piezolaminated structure is finite element method, most of the researchers have analyzed static and dynamic response of piezostructures using finite element method (Narayanan and Balamurugan 2003; Sharma et al. 2020). The current simulation work uses FEM formulation of a general photostrictive unimorph structure. First order shear deformation theory is employed along with a 4 noded shell element having five-degree freedom at each node. The structure comprises of two layers with substrate layer as a bottom layer and a photostrictive patch as the top layer. It is considered that both the substrate and photostrictive layers are perfectly bonded and the bonding material’s effect is neglected over the structure.

2.6.1 Degenerated shell element

The intention behind this shell element, as shown in Fig. 4, is to reduce the three-dimensional (3D) solid element into the two-dimensional (2D) element, while employing the elasticity theory for 3D solid. At the expense of little accuracy, this element is computationally very cost-effective for finite element analysis. The advantages of degenerated shell are not only limited to cost-effectiveness, but also it is a versatile element that do not depends on any of specific shell theory.

In describing the distorted and undistorted structural geometry isoparametric finite element method (FEM) formulation has been established. The local and the global coordinate system on the midplane of shell element are represented in Fig. 5. The coordinates of any point on the mid plane in terms of global coordinates can be given as (Bathe and Dvorkin 1986) follows,

$${^{l}x}_{i}=\sum_{k=1}^{4}{N}_{k}^{l}{x}_{i}^{k}+\sum_{k=1}^{4}\left[{m}_{k}+\left[\frac{{t}^{n}}{2}{l}_{k}^{n}\right]\right]{N}_{k}^{l}{V}_{ni}^{k},$$
(41)

where, \({x}_{i}^{k}\) is the cartesian coordinates of node \(k={\left\{{x}^{k} {y}^{k}{z}^{k}\right\}}^{T}\); \({m}_{k}=-\left({a}_{k}/2\right)+{\sum }_{i=1}^{n}{l}_{k}^{i}-({l}_{k}^{i}/2),\) Nk refers to shape functions at node k, is equal to \((1/4)(1\pm r)(1\pm s)\), ak refers to the layered shell at node k, measured along the vector \({^{l}V}_{ni}^{k}\), thickness of layered i at node k is represented by \({l}_{k}^{i}\), along the thickness direction natural coordinate is represented by \({t}^{n}\), distance between the mid plane of the complete shell and the mid plane of the nth layer is represented by \({m}_{k}\). Whereas the superscript has value either 0 or 1 for undeformed and deformed structure, respectively.

Fig. 4
figure 4

Schematic of reducing one dimenstion from 3D solid element

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Fig. 5
figure 5

Representation of local and global coordinate system

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In shell formulation, the displacement field in global coordinates is defined as (Cook 2007),

$${u}_{i}=\sum_{k=1}^{4}{N}_{k}{u}_{i}^{k}+\sum_{k=1}^{4}\left[{m}_{k}+\left[\frac{{t}^{n}}{2}{l}_{k}^{n}\right]\right]\left(-{^{0}V}_{2i}^{k}{\alpha }_{k}+{^{0}V}_{1i}^{k}{\beta }_{k}\right),$$
(42)

where at node \(k={\left\{{x}^{k} {y}^{k}{z}^{k}\right\}}^{T}\), midplane displacement is represented by \({u}_{i}^{k}\), the rotation of direction vector \({^{0}V}_{n}^{k}\) about \({^{0}V}_{1}^{k}\) and \({^{0}V}_{2}^{k}\) axes.

2.6.2 Material constitutive laws

The linear constitutive equation of coupled physics is given in Eq. (7), (8), and (9). These equations act as the governing laws for photostrictive actuator patch. However, under the action of external stimuli, the mechanical behavior of the structure can be defined using elastic constitutive law (stress–strain relationship). Stress–strain relationship in terms of local coordinates is defined as follows,

$$\left\{{\sigma }^{{\prime}}\right\}=\left[D\right]\left\{{\varepsilon }^{{\prime}}\right\},$$
(43)

where, \(\left[D\right]\) refers to the material stiffness matrix; \(\{\varepsilon ^{\prime} \}\) refers to the strain in the local coordinate systems and \(\left\{{\sigma }^{{\prime}}\right\}\) refers to the stress in the local coordinate systems. Transforming Eq. (43) from local coordinate system to global coordinate systems, we can write,

$$\left\{{\sigma }\right\}=\left[{D}_{Tr}\right]\left\{\varepsilon \right\},$$
(44)

where, \(\left[{D}_{Tr}\right]\) refers to the transformed material stiffness matrix; \(\{\varepsilon \}\) refers to the strain in the global coordinate systems and \(\left\{{\sigma }\right\}\) refers to the stress in the global coordinate systems.

While applying the above formulation to the thin structure, a considerable amount of inaccuracy was encountered due to artifacts such as membrane locking and shear locking. To avoid such circumstances in calculations, generally, researchers use selective integration technique. Whereas, Bathe (1986) came up with new method known as mixed interpolation of tensorial components (MITC). In current study, MITC4 element is used to imitate the unimorph beam structure. We do not use Eq. (42) to determine the transverse shear strain terms, otherwise the misinterpretation of the transverse shear strain terms would lead to a locking effect. Rather, we separately interpolate the transverse shear strain as illustrated in Fig. 6.

Fig. 6
figure 6

Dipiction of a typical MITC-4 element with its normal vectors in natural coordinates along with the interpolation of transverse shear strain components

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Functions of Bilinear Interpolation are chosen to interpolate the transverse shear strain in an element, as these interpolation functions must provide adequate coupling between the rotation and the displacement. The bilinear interpolation functions can be written as follows,

$${\epsilon }_{13}=\frac{1}{2}\left(1+s\right){\epsilon }_{13}^{A}+\frac{1}{2}\left(1-s\right){\epsilon }_{13}^{C},$$
(45)
$${\epsilon }_{23}=\frac{1}{2}\left(1+r\right){\epsilon }_{23}^{D}+\frac{1}{2}\left(1-r\right){\epsilon }_{13}^{B}.$$
(46)

2.6.3 Equations of motion

To obtain the equation of motion of an element, Hamilton’s principle is used, which can be written as

$${\int }_{{t}_{1}}^{{t}_{2}}\left(\delta T-\delta U+\delta {W}_{ext}\right)dt=0,$$
(47)

where, U, T and \({W}_{ext}\) are the potential energy, kinetic energy, and work done by the external force, respectively. \(\delta\) represents the variational operator. On deriving the potential energy, kinetic energy, and external work using constitutive Eqs. (7), (8) and (9) of photostriction and on clubbing all mechanical terms, electrical term and thermal terms into separate three energy functionals, represented by \({f}_{u}\), \({f}_{\phi }\) and \({f}_{\theta }\) respectively (Tzou and Ye 1994). Therefore, \({f}_{u}\), \({f}_{\phi }\) and \({f}_{\theta }\) can be written as follows,

$${f}_{u}={\int }_{V}\left(\frac{1}{2}c{S}^{t}S-c{S}^{t}{S}_{0}-eE{S}^{t}-\lambda \theta {S}^{t}\right)dV-{\int }_{{S}_{T}}f{u}^{t}dS-{\int }_{V}\left({f}_{b}-\rho \ddot{u}-\widehat{c}\dot{u}\right){u}^{t}dV,$$
(48)
$${f}_{\phi }={\int }_{V}\left[\frac{1}{2}\varepsilon {E}^{t}E+e{E}^{t}\left(S-{S}_{0}\right)+{p}_{n}{E}^{t}\theta \right]dV-{\int }_{{S}_{D}}{\phi }^{t}QdS,$$
(49)
$${f}_{\theta }={\int }_{V}\frac{1}{2}k{g}^{t}gdV-{\int }_{{S}_{qs}}{q}_{s}\theta dS+{\int }_{{S}_{qh}}\frac{1}{2}{h}_{\nu }{\left(\theta +{\theta }_{0}-{\theta }_{\infty }\right)}^{2}dS+{\theta }_{0}{\int }_{V}\dot{\epsilon }\theta dV,$$
(50)

where, \({f}_{b},{u}_{i}\dot{,{u}_{i}} \mathrm{and} \ddot{{u}_{i}}\) represent the body force vector, displacement, velocity, and acceleration, k refers to the coefficient of thermal conductivity, \(\widehat{c}\) is the material damping coefficient, \({h}_{\nu }\) is the coefficient of thermal convection; g represents temperature gradient and \({\theta }_{\infty }\) is the surrounding temperature.

For the photostrictive material, the matrix representation of the three energy functionals \({f}_{u}\), \({f}_{\phi }\) and \({f}_{\theta }\) is given as,

$${f}_{u}={\int }_{V}\frac{1}{2}c{S}^{t}SdV-{\int }_{V}e{S}^{t}EdV+{\int }_{V}{u}^{t}\rho \ddot{u}dV+\alpha {\int }_{V}{u}^{t}\rho \dot{u}dV+\beta {\int }_{V}c{S}^{t}{B}_{u}\dot{UdV-{\int }_{V}\lambda {S}^{t}\theta dV-{\int }_{V}c{S}^{t}{S}_{0}dV-{\int }_{V}{u}^{t}{f}_{b}dV-{\int }_{{S}_{T}}{u}^{t}fdS,}$$
(51)
$${f}_{\phi }={\int }_{V}\frac{1}{2}\varepsilon {E}^{t}EdV+{\int }_{V}e{E}^{t}SdV+{\int }_{V}{E}^{t}{p}_{n}\theta dV-{\int }_{V}{e}^{t}{E}^{t}{S}_{0}dV-{\int }_{{S}_{D}}{\phi }^{t}QdS,$$
(52)
$${f}_{\theta }={\int }_{V}\frac{1}{2}k{g}^{t}gdV+{\int }_{{S}_{qh}}\frac{1}{2}{h}_{\nu }{\theta }^{t}\theta dS+{\theta }_{0}{\int }_{V}\lambda {\theta }^{t}{B}_{u}\dot{u}dV+{\theta }_{0}{\int }_{V}{\theta }^{t}{p}_{n}\left(-{B}_{\phi }\dot{\phi }\right)dV+{\int }_{V}{\theta }_{0}\alpha {\theta }^{t}\left({N}_{\theta }\dot{\theta }\right)dV-{\int }_{{S}_{qs}}qs{\theta }^{t}dS-{\int }_{{S}_{qh}}{h}_{\nu }{\theta }^{t}\left({\theta }_{\infty }-{\theta }_{0}\right)dS.$$
(53)

Field variables can be written in the matrix notations using geometric matrices of shell element and the shape functions, which is given as,

$$\begin{aligned} & u = N_{u} U,\quad \phi^{*} = N_{\phi } \phi ,\quad \theta^{*} = N_{\theta } \theta \\ & S = B_{u} U, \quad E = - B_{\phi } \phi , \quad g = - B_{\theta } \theta \\ \end{aligned}$$
(54)

where, g, E, S, \({\theta }^{*},\) \({\phi }^{*}\) and u is the temperature gradient vector, electric field vector, strain, temperature, electric potential and displacement vector respectively; Matrices of nodal temperature vector \(\theta\), nodal electric potential vector \(\phi ,\) and nodal displacement vector \(U\) are represented by \({N}_{\theta },\) \({N}_{\phi }\) and \({N}_{u}\) respectively; \({B}_{\theta }={L}_{\theta }{N}_{\theta }\), \({B}_{\phi }={L}_{\phi }{N}_{\phi }\) and \({B}_{u}={L}_{u}{N}_{u}\), where L refers to the differential operator. The nodal variables are the internal degrees of freedom. The global functional are obtained by summing up the functional over all the elements, which can be written as,

$${f}_{u}=\sum {f}_{{u}_{n}} , {f}_{\phi }=\sum {f}_{{\phi }_{n}} , \mathrm{and} {f}_{\theta }=\sum {f}_{{\theta }_{n}}.$$
(55)

Therefore, on substituting Eq. (54) into Eqs. (51), (52), and (53), elemental energy functionals in the matrix form are obtained, which can be written as,

$${f}_{{u}_{n}}=\frac{1}{2}\left\{{U}^{t}\right\}{\int }_{V}^{ }{\left[{B}_{u}\right]}^{t}[C]\left[{B}_{u}\right]dV\left\{U\right\}+\left\{{U}^{t}\right\}{\int }_{V}^{ }{\left[{B}_{u}\right]}^{t}\left[e\right]\left[{B}_{\phi }\right]dV\{\phi \}+\left\{{U}^{t}\right\}{\int }_{V}^{ }{[N]}^{t}\left[\rho \right]\left[N\right]dV\left\{\ddot{U}\right\}+\left\{{U}^{t}\right\}{\int }_{V}^{ }\alpha {[N]}^{t}\left[\rho \right]\left[N\right]dV\left\{\dot{U}\right\}+\left\{{U}^{t}\right\}{\int }_{V}^{ }\beta {\left[{B}_{u}\right]}^{t}\left[C\right]\left[{B}_{u}\right]dV\left\{\dot{U}\right\}-\left\{{U}^{t}\right\}{\int }_{V}^{ }{\left[{B}_{u}\right]}^{t}\left[\lambda \right]\left[{N}_{\theta }\right]dV\left\{\theta \right\}-\left\{{U}^{t}\right\}\left\{{\int }_{V}^{ }{\left[{B}_{u}\right]}^{t}\left[C\right]\left[{S}_{0}\right]dV+{\int }_{V}^{ }{\left[{N}_{u}\right]}^{t}{f}_{b}dV+{\int }_{{S}_{T}}^{ }{\left[{N}_{u}\right]}^{t}{f}_{ }dS\right\},$$
(56)
$${f}_{{\phi }_{n}}=\frac{1}{2}{\left\{\phi \right\}}^{t}{\int }_{V}{\left[{B}_{\phi }\right]}^{t}\left[\varepsilon \right][{B}_{\phi }]dV\left\{\phi \right\}-{\left\{\phi \right\}}^{t}{\int }_{V}{\left[{B}_{\phi }\right]}^{t}\left[e\right]\left[{B}_{u}\right]dV\left\{U\right\}-{\left\{\phi \right\}}^{t}{\int }_{V}{\left[{B}_{\phi }\right]}^{t}\left[{P}_{n}\right]\left[{N}_{\theta }\right]dV\left\{\theta \right\}+{\left\{\phi \right\}}^{t}{\int }_{V}{\left[{B}_{\phi }\right]}^{t}\left[e\right]\left[{S}_{0}\right]dV+{\int }_{{S}_{D}}{[{N}_{\phi }]}^{t}QdS,$$
(57)

and

$${f}_{{\theta }_{n}}=\frac{1}{2}{\left\{\theta \right\}}^{t}{\int }_{V}{\left[{B}_{\theta }\right]}^{t}\left[k\right][{B}_{\theta }]dV\left\{\theta \right\}+\frac{1}{2}{\left\{\theta \right\}}^{t}{\int }_{{S}_{qh}}{\left[{N}_{\theta }\right]}^{t}\left[{h}_{\upsilon }\right][{N}_{\theta }]dS+{\theta }_{0}{\{\theta \}}^{t}{\int }_{V}{\left[{N}_{\theta }\right]}^{t}\left[\lambda \right]\left[{B}_{u}\right]dV\left\{\dot{U}\right\}-{\theta }_{0}{\left\{\theta \right\}}^{t}{\int }_{V}{\left[{N}_{\theta }\right]}^{t}\left[{P}_{n}\right]\left[{B}_{\phi }\right]dV\left\{\dot{\phi }\right\}+{\theta }_{0}{\left\{\theta \right\}}^{t}{\int }_{V}{\alpha }_{v}{\left[{N}_{\theta }\right]}^{t}\left[{N}_{\theta }\right]dV\left\{\dot{\theta }\right\}-{\left\{\theta \right\}}^{t}\left\{{\int }_{{S}_{qs}}{\left[{N}_{\theta }\right]}^{t}{q}_{s}dS+{\int }_{{S}_{qh}}{\left[{N}_{\theta }\right]}^{t}{h}_{\upsilon }\left({\theta }_{\infty }-{\theta }_{0}\right)dS\right\}.$$
(58)

The photostrictive elemental matrices for a thin shell element are as follows,

$$\left[{M}_{uu}\right]={\int }_{V}{[{N}_{u}]}^{t}\rho \left[{N}_{u}\right]dV,$$
(59)
$$\left[{K}_{uu}\right]={\int }_{V}{[{B}_{u}]}^{t}\left[C\right]\left[{B}_{u}\right]dV,$$
(60)
$$\left[{K}_{u\phi }\right]={\int }_{V}{[{B}_{u}]}^{t}\left[e\right]\left[{B}_{\phi }\right]dV,$$
(61)
$$\left[{K}_{\phi u}\right]={\left[{K}_{u\phi }\right]}^{t},$$
(62)
$$\left[{K}_{\phi \phi }\right]={\int }_{V}{[{B}_{\phi }]}^{t}\left[e\right]\left[{B}_{\phi }\right]dV,$$
(63)
$$\left[{K}_{\phi \theta }\right]={\int }_{V}{[{B}_{\phi }]}^{t}\left[{P}_{n}\right]\left[{N}_{\theta }\right]dV,$$
(64)
$$\left[{K}_{\theta \phi }\right]={\left[{K}_{\phi \theta }\right]}^{t},$$
(65)
$$\left[{K}_{u\theta }\right]={\int }_{V}{[{B}_{u}]}^{t}\left[\lambda \right]\left[{N}_{\theta }\right]dV,$$
(66)
$$\left[{K}_{\theta u}\right]={\left[{K}_{u\theta }\right]}^{t},$$
(67)
$$\left[{K}_{\theta \theta }\right]={\int }_{V}{[{B}_{\theta }]}^{t}\left[k\right]\left[{B}_{\theta }\right]dV+{\int }_{{S}_{qh}}{h}_{\upsilon }{\left[{N}_{\theta }\right]}^{t}[{N}_{\theta }]dS,$$
(68)
$$\left[{H}_{\theta \theta }\right]={\int }_{V}{{\theta }_{0}\alpha }_{v}{[{N}_{\theta }]}^{t}\left[{N}_{\theta }\right]dV.$$
(69)

It is assumed that the elemental damping matrix is proportional to the elemental mass matrix and the elemental stiffness matrix, which can be given as,

$$\left[{C}_{uu}\right]={\alpha }^{*}\left[{M}_{uu}\right]+{\beta }^{*}\left[{K}_{uu}\right],$$
(70)

where, \({\beta }^{*}\) and \({\alpha }^{*}\) are known as Rayleigh’s coefficients and damping ratio \({\xi }_{i}\) of mode i is given as,

$${\xi }_{i}=({\alpha }^{*}+{\beta }^{*}{\omega }_{i}^{2})/(2{\omega }_{i}),$$
(71)

where ith natural frequency is represented by \({\omega }_{i}\); \({H}_{\theta \theta }\) and \({K}_{mn} (m \mathrm{and} n=u,\phi ,\theta )\) are defined as the stiffness matrices for temperature fields; \({C}_{uu}\) represents the damping matrix and \({M}_{uu}\) refers to the mass matrix.

The elemental force vectors are as follows,

$$\left\{{F}_{u}\right\}={\int }_{{V}_{n}}{[{B}_{u}]}^{t}\left[C\right]\left[{S}_{0}\right]dV+{\int }_{{V}_{n}}{[{N}_{u}]}^{t}{f}_{b}dV+{\int }_{{S}_{{T}_{n}}}{[{N}_{u}]}^{t}fdS,$$
(72)
$$\left\{{F}_{\phi }\right\}={\int }_{{V}_{n}}{[{B}_{\phi }]}^{t}\left[e\right]\left[{S}_{0}\right]dV-{\int }_{{S}_{{D}_{n}}}{[{N}_{\phi }]}^{t}QdS,$$
(73)
$$\left\{{F}_{\theta }\right\}={\int }_{{S}_{{q}_{sn}}}{[{N}_{\theta }]}^{t}{q}_{s}dS+{\int }_{{S}_{qh}}{\left[{N}_{\theta }\right]}^{t}{h}_{\upsilon }\left({\theta }_{\infty }-{\theta }_{0}\right)dS,$$
(74)

where, \(\left\{{F}_{u}\right\}\), \(\left\{{F}_{\phi }\right\}\) and \(\left\{{F}_{\theta }\right\}\) represent the mechanical excitation vector, electrical excitation vector, and heat excitation vector.

Substituting elemental matrixes from Eqs. (59)–(74), into Eqs. (56), (57), and (58), we can rewrite the three elemental functionals as,

$${f}_{{u}_{n}}=\frac{1}{2}{\{U\}}^{t}\left[{K}_{uu}\right]\{U\}+{\left\{U\right\}}^{t}[{K}_{u\varnothing }]\{\varnothing \}+{\left\{U\right\}}^{t}[{M}_{uu}]\{\ddot{U\}}+\{{U\}}^{t}[{C}_{uu}]\{\dot{U\}}-{\left\{U\right\}}^{t}\left[{K}_{u\theta }\right]-{\left\{U\right\}}^{t}\left[{F}_{u}\right],$$
(75)
$${f}_{{\varnothing }_{n}}=\frac{1}{2}{\left\{\varnothing \right\}}^{t}\left[{K}_{\phi \phi }\right]\left\{\phi \right\}-{\left\{\varnothing \right\}}^{t}{\left[{K}_{u\phi }\right]}^{t}\left\{U\right\}-{\left\{\varnothing \right\}}^{t}{\left[{K}_{\phi \theta }\right]}^{t}\left\{\theta \right\}+{\left\{\phi \right\}}^{t}\left\{{F}_{\phi }\right\},$$
(76)
$${f}_{{\theta }_{n}}=\frac{1}{2}{\{\theta \}}^{t}\left[{K}_{\theta \theta }\right]\left\{\theta \right\}+{\theta }_{0}{\left\{\theta \right\}}^{t}{[{K}_{u\theta }]}^{t}\left\{\dot{U}\right\}-{\theta }_{0}{\left\{\theta \right\}}^{t}{\left[{K}_{\phi \theta }\right]}^{t}\left\{\dot{\phi }\right\}+{\left\{\theta \right\}}^{t}\left[{H}_{\theta \theta }\right]\left\{\dot{\theta }\right\}-{\left\{\theta \right\}}^{t}\left\{{F}_{\theta }\right\}.$$
(77)

It is considered that heat transfer’s dynamic coupling with the deflection and electric field i.e., the second and third term of Eq. (77) can be neglected as they are small. Thus, a decoupled quasi-static formulation is established. Taking variations of the three functionals \({f}_{{u}_{n}}\),\({ f}_{{\varnothing }_{n}}\) and \({f}_{{\theta }_{n}}\) with respect to field variables U, \(\varnothing\) and \(\theta\) respectively, we will obtain as follows,

$$\left[{M}_{uu}\right]\left\{\ddot{U}\right\}+\left[{C}_{uu}\right]\left\{\dot{U}\right\}+\left[{K}_{uu}\right]\left\{U\right\}+\left[{K}_{u\phi }\right]\left\{\phi \right\}-\left[{K}_{u\theta }\right]\left\{\theta \right\}=\left[{F}_{u}\right],$$
(78)
$${\left[{K}_{u\phi }\right]}^{t}{\left\{U\right\}}-\left[{K}_{\phi \phi }\right]\left\{\phi \right\}+\left[{K}_{\phi \theta }\right]\left\{\theta \right\}=\{{F}_{\phi }\},$$
(79)
$$\left[{K}_{\theta \theta }\right]\left\{\theta \right\}+\left[{H}_{\theta \theta }\right]\left\{\dot{\theta }\right\}=\left\{{F}_{\theta }\right\},$$
(80)

Rewriting the above Eqs. (78), (79), and (80) in the matrix format, as given below,

$$\left[ {\begin{array}{*{20}c} {\left[ {M_{{uu}} } \right]} \\ {\left[ 0 \right]} \\ {\left[ 0 \right]} \\ \end{array} ~~\begin{array}{*{20}c} {\left[ 0 \right]} \\ {\left[ 0 \right]} \\ {\left[ 0 \right]} \\ \end{array} ~~\begin{array}{*{20}c} {\left[ 0 \right]} \\ {\left[ 0 \right]} \\ {\left[ 0 \right]} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\mathop {\left\{ {\ddot{U}} \right\}}\limits^{{}} } \\ {\left\{ {\mathop {\ddot{\phi }}\limits^{{}} } \right\}} \\ {\left\{ {\mathop {\ddot{\theta }}\limits^{{}} } \right\}} \\ \end{array} } \right\} + \left[ {\begin{array}{*{20}c} {\left[ {C_{{uu}} } \right]} \\ {\left[ 0 \right]} \\ {\theta _{0} \left[ {K_{{\theta u}} } \right]} \\ \end{array} ~~\begin{array}{*{20}c} {\left[ 0 \right]} \\ {\left[ 0 \right]} \\ { - \theta _{0} [K_{{\theta \phi }} ]} \\ \end{array} ~~\begin{array}{*{20}c} {\left[ 0 \right]} \\ {\left[ 0 \right]} \\ {[H_{{\theta \theta }} ]} \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\mathop {\left\{ {\dot{U}} \right\}}\limits^{{}} } \\ {\mathop {\left\{ {\dot{\phi }} \right\}}\limits^{{}} } \\ {\mathop {\left\{ {\dot{\theta }} \right\}}\limits^{{}} } \\ \end{array} } \right\} + \left[ {\begin{array}{*{20}c} {[K_{{uu}} ]} \\ {\left[ {K_{{\phi u}} } \right]} \\ {\left[ 0 \right]} \\ \end{array} ~~\begin{array}{*{20}c} {[K_{{u\phi }} ]} \\ { - [K_{{\phi \phi }} ]} \\ {\left[ 0 \right]} \\ \end{array} ~~\begin{array}{*{20}c} { - [K_{{u\theta }} ]} \\ {[K_{{\phi \theta }} ]} \\ {[K_{{\theta \theta ]}} } \\ \end{array} } \right]\left\{ {\begin{array}{*{20}c} {\left\{ U \right\}} \\ {\left\{ \phi \right\}} \\ {\left\{ \theta \right\}} \\ \end{array} } \right\} = \left\{ {\begin{array}{*{20}c} {\left\{ {F_{u} } \right\}} \\ {\left\{ {F_{\phi } } \right\}} \\ {\left\{ {F_{\theta } } \right\}} \\ \end{array} } \right\}.$$
(81)

3 Results and discussion

3.1 FEA results of effective material properties

The results of effective elastic, piezoelectric, and dielectric properties for 0–3 PMN-35PT-PLZT composite, as a function of volume fraction of inclusions, were numerically analyzed using FEM. In the current study, the inclusion’s (PMN-35PT) volume fraction in composite is varied from 5 to 25%. The material properties of the constituents of composite, which are used while calculating the effective properties are mentioned in Table 1. All the material attributes of composite for varied inclusion parentage show significant convergence, for example, the elastic effective stiffness coefficient \({C}_{11}^{eff}\) is shown in Fig. 7, from this figure it is evident that convergence value \(9.27\times {10}^{10}\) Pa of \({C}_{11}^{eff}\) is achieved at 174,138 number of elements.

Fig. 7
figure 7

Representation of convergence of elastic stiffness coefficient \({C}_{11}^{eff}\), volume fraction of PMN-35PT is 15%

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In Fig. 8a–d, the effective elastic properties such as \({C}_{11}^{eff}, {C}_{12}^{eff}\),\({C}_{13}^{eff}\) and \({C}_{33}^{eff}\) show an ascending trend with the rise in the inclusion’s volume fraction, which also satisfies intuition from the rule of mixture. Similar to the effective elastic attributes, effective piezoelectric coupling coefficient \({d}_{31}\) for strain charge form also increases with rise in the inclusion’s volume fraction as shown in Fig. 11a. The value of \({d}_{31}\) increased by 5% when the inclusion’s volume fraction in composite is 25% in comparison to homogneous PLZT material.

Fig. 8
figure 8

Variation of \({C}_{11}^{eff}, {C}_{33}^{eff}\),\({C}_{12}^{eff}\) and \({C}_{13}^{eff}\) as a function of different volume fractions

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To determine the variation of effective permittivity \({\varepsilon }_{11}^{eff}\) and \({\varepsilon }_{33}^{eff}\) as a function of inclusion’s (PMN-35PT) volume fraction, electric field’s distribution is studied in the composite. Figure 9 illustrates the variation of electric field inside the composite sample when subjected to \(1\times {10}^{-6}\) V on the positive z face (C+ surface). Figure 9 illustrates that electric field is different in matrix, in the vicinity of inclusions and inside the inclusions. This exhibits that the variation of dielectric constant at micro scale level has been incorporated. The electric field changes its direction and magnitude at the interface of matrix and inclusion because of the material’s permittivity and the incidence angle to the inclusion’s surface. Another possible reason for the change of electric field is that ceramics are non-conducting material, which leads to the charge generation and hence the electric field. The electric field inside the inclusion is in the same direction, whereas the electric field outside the particle opposes the external electric field (Martínez-Ayuso et al. 2019; Wong et al. 2002).

Fig. 9
figure 9

Variation of electric field in composite when subjected to \(1\times {10}^{-6}\) V electric potential on the positive z face, while constraining the strain in every direction. The volume fraction of PMN-35PT is 15%

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In the numerical method, a dynamic model is used to characterize the pyroelectric property of the photostrictive material. Pyroelectric response of composite is obtained by measuring the current between the top (C+) and bottom surface (C−) of RVE. The surrounding temperature is considered as 25 \(^\circ \mathrm{C}\). A sinusoidal excitation temperature of small frequency 0.05 Hz is applied at the bottom (C−) surface, such a small excitation frequency is employed due to the low conductivity of matrix. For the finite element analysis, temperature for convection at top surface is same as surrounding temperature. The distribution of temperature across the matrix and inclusions (15% volume fraction) is obtained through FEM. Figure 10a shows the temperature variation in the composite when subjected to cooling, as it can be deduced that the lowest temperature is at bottom surface and maximum temperature at the top surface, whereas Fig. 10b shows the temperature variation in the composite when subjected to heating. When the temperature variation is explored at top surface (C+) of RVE for 20 s period, it follows the same sinusoidal trend as expected, which is shown in Fig. 10 c. The maximum temperature is 26 \(^\circ \mathrm{C}\) and the minimum temperature observed is 24 \(^\circ \mathrm{C}\). To obtain the pyroelectric coefficient, it is required to obtain the current due to pyroelectric effect, as shown in Fig. 10d. The pyroelectric coefficient is calculated using Eq. (15), for composite containing 15% volume fraction of inclusions, which is \(6.70\times {10}^{-5}\) C/(m2K). It can be inferred from Fig. 11b that the pyroelectric coefficient escalates with the increment in the inclusion’s volume fraction, this observation strengthens the intuition that pyroelectric coefficient increases with the close packing of the particles.

Fig. 10
figure 10

a Temperature variation in matrix and inclusions, when subjected to cooling, b heating, c temperature variation at the top surface of the RVE when subjected to sinusoidal temperature variation at bottom surface, and d the corresponding pyroelectric current

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Fig. 11
figure 11

a \({d}_{31}^{eff}\), b effective pyroelectric coefficient \({p}^{eff}\), c effective Young’s Modulus \({E}_{1}^{eff}\) and d effective thermal expansion coefficient \({\alpha }^{eff}\) as a function of volume fraction of inclusion in 0–3 composite

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Young’s modulus for the 0–3 composite is calculated using the effective stiffness properties as a function of volume fraction of inclusions, as depicted in Fig. 11c. It can be observed that there is negligible reduction in Young’s modulus as a function of inclusion’s volume fraction. The thermal expansion coefficient of photostrictive composite is also calculated as the function of inclusion’s volume fraction, as shown in Fig. 11d. Other effective material properties of 0–3 composite such as density, specific heat at constant pressure and conductivity are determined using the rule of mixture. All of the estimated effective properties of 0–3 photostrictive composite as a function of inclusion’s volume fraction are enlisted in Table 3.

Table 3 Effective properties of 0–3 composite
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These calculated effective properties of 0–3 photostrictive composite were further engaged to characterize the actuation behavior of the cantilever and simply supported beam.

3.2 Validation study

The results of Shih et.al. (2005) are reproduced to validate the accuracy and applicability of FEM formulation developed in the current study. Domain is discretised using FEM mesh of \(10\times 1\) elements. The actuation results of unimorph cantilever beam bonded with photostrictive patch, is shown in Fig. 12a. A cantilever beam of Length, width, and thickness are taken as 1 m, 3 cm, and 2 mm respectively, whereas the those of the actuator patch are 0.2 m, 3 cm, and 0.3 mm respectively. The center of actuating photostrictive patch is placed at a distance of 0.2 m from the fixed end of the cantilever beam. They had considered the steel beam as their host structure, while the material properties of steel are Young’s modulus E = \(2.1\times {10}^{11}\) N/m2 and density \(\rho =7.8\times {10}^{3}\) Kg/m3. The light illumination intensity was considered as 60 mW/cm2. The photostrictive actuator’s material properties were given in Table 4. The displacement of the cantilever beam actuated by a photostrictive patch is found in good concurrence with that of the literature (Fig. 12b), which evidently proves the validity of current formulation to displacement response of photostrictive actuator.

Fig. 12
figure 12

a Geometric representation of photostrictive actuator on cantilever beam and b comparison of displacement response of current result with Shih et al

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Table 4 Properties of photostrictive actuator
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3.3 Patch test

Two tests are performed to check if the current element assembly satisfies the patch test. Firstly, displacement conditions at boundaries are applied at all the external nodes and then displacement at interior nodes of patch elements are checked to see whether patch is successfully representing the rigid body mode or not. Secondly, the patch is subjected to minimum essential boundary conditions to restrain all rigid body modes and the patch is also subjected to natural boundary conditions (Bathe 2006). After analyzing, all the elements should show a constant stress state to pass the patch test.

Irregular mesh of nine elements, as shown in Fig. 13a is considered for both the patch tests. Four noded quadrilateral element is used to discretize the patch. Length, width, and thickness of the structure for the patch are 1 m, 1 m, and 2 mm, respectively. Young’s modulus and poison ratio is \(2.1\times {10}^{11}\) N/m2 and 0.3.

Fig. 13
figure 13

a Irregular mesh consists of 4 noded element, considered for a patch test and b representation of initial and final position of patch after rigid body motion

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3.3.1 Test A: Rigid body motion test

For such patch test, the unit displacement is prescribed at all the external nodes and in the analysis displacements at all the internal nodes were found to be unity in all the direction as illustrated in Fig. 13b. Hence, the first rigid body motion patch test is passed.

3.3.2 Test B: Constant stress state test

The patch is subjected to normal traction and shear traction. Normal traction of 100 N/m2 and 200 N/m2 is applied in X and Y direction, respectively. Shear traction of 300 N/m2 is applied in XY plane, as shown in Fig. 13a. After the analysis it is found out that at every node in all the element normal stress in X and Y direction is 100 N/m2 and 200 N/m2, whereas the shear stress at all the node is 300 N/m2. This analysis shows that current element assembly passed the constant stress state test.

3.4 Improvement in photostrictive actuation

This study aims to show that using 0–3 photostrictive composite, displacement response can be improved significantly. To demonstrate the improvement in actuation responses, two case studies have been done. Case (1), cantilever beam with the photostrictive patch bonded on the host structure near to the fixed end as shown in Fig. 14a and case (2), simply supported beam with the photostrictive patch bonded at the center of the host structure as illustrated in Fig. 14b. The dimensions and material of the host structure are considered same as in the case of validation problem.

Fig. 14
figure 14

Schematic representation of a cantilever beam and b simply supported beam with a photostrictive actuator

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For the first case study, cantilever beam with photostrictive actuator, a four nodded shell element is used to model the actuator and the host beam. Finite element mesh of \(10\times 1\) elements for cantilever beam and mesh of \(30\times 1\) elements for simply supported beam is used to discretize the domain. For both the cases, the intensity of light is assumed as 60 mW/cm2. The displacement response of both case studies is determined as a function of volume fraction of inclusions in photostrictive composite. For example, when the volume fraction of PMN-35PT was 25% in the PLZT matrix, the displacement response for case (1) and case (2) is shown in Fig. 15a and b respectively.

Fig. 15
figure 15

a Transverse deflection results of cantilever beam using FEM. b Transverse deflection results of simply supported beam using FEM

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It is evident from Fig. 15a, b that the actuation response of both cantilever and simply supported beam significantly enhances with the increase in inclusion’s volume fraction. The maximum deflection of the beam for both the cases is given in Table 5. The maximum tip and center deflection in case of cantilever and simply supported beams occur when the volume fraction of inclusions is 25%.

Table 5 Maximum deflection as a function of inclusion’s volume fraction
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The alteration in the actuation capability of the different composites Fig. 16, can be explained by exploring the effect of various factors influencing the actuation response, such as effective Youngs modulus \({E}_{1}^{eff}\), effective thermal expansion coefficient \({\alpha }_{1}^{eff}\), effective pyroelectric effect \({p}^{eff}\) and effective piezoelectric coupling coefficient \({d}_{31}^{eff}\). As deflection is inversely proportional to Young's modulus, deflection will decreases with increase in Young’s modulus. But in the current study from Fig. 11c, as it can be deduced that there is a marginal increase in the Youngs modulus value, which means in current study, there is a very negligible negative contribution of marginally increased effective Young’s modulus. The thermal expansion coefficient is responsible for the expansion of photostrictive patch, which in turn compels the beam to bend. In the present work, Fig. 11d illustrates that effective thermal coefficient is decreasing as a function of inclusion’s volume fraction, hence it is not assisting the increase in deflection response. Pyroelectric coefficient is responsible for the induction of electric potential in the response of temperature variation. Pyroelectric coefficient has indirect effect on the deflection response, as the improved pyroelectric coefficient instigates a higher electric potential between the actuator’s electrode, which in turn ameliorate the electric field. Now, this enhanced electric field due to converse piezoelectric effect will increase the deflection response. From Fig. 11b it can be easily perceived that the effective pyroelectric coefficient has grown by approximately 4 times, when inclusion’s volume fraction has increased from 0 to 25%. From Fig. 11, it is prudent to say that effective pyroelectric coefficient has a prominent contribution towards enhanced actuation response. Effective piezoelectric coupling coefficient \({d}_{31}^{eff}\) has direct effect on the actuator’s deflection response due to the converse piezoelectric effect. Figure 11a shows that the effective piezoelectric coupling coefficient \({d}_{31}^{eff}\) has increased by approximately 5%, when the inclusion’s volume fraction has increased from 0 to 25%. Therefore, it can be said that effective piezoelectric coupling coefficient has constructive contribution towards enhanced actuation effect of photostrictive composite.

Fig. 16
figure 16

a Comparison of transverse deflection results of cantilever beam actuated by different photo strictive composite using FEM. b Comparison of transverse deflection results of simply supported beam actuated by different photostrictive composite using FEM

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4 Conclusions

In order to explore the effective elastic, dielectric, piezoelectric and pyroelectric properties, FEA of 0–3 composite was extensively carried out as a function of inclusion’s (PMN-35PT) volume fraction in the matrix (PLZT). Suitable periodic boundary conditions for coupled physics were developed and prudently applied to unit cell, to estimate the effective properties of composite. The micromechanical approach was successful in capturing the effect of locally varying constituent’s materials properties. Pyroelectric coefficient is estimated by applying the sinusoidal variation of temperature. Model has accurately predicted the effective properties of composite and shows enhancement in effective properties as expected, with the increase in the inclusion’s volume fraction. These estimated properties were then used to predict the actuation response of photostrictive material bonded to a cantilever beam and a simply supported beam. It was deduced that all the factors, except thermal expansion coefficient and Young’s modulus, which influence the actuation had a constructive effect in the present 0–3 composite. In the current study, the most promising factor which had a maximum contribution in actuation was pyroelectric coefficient. The maximum tip deflection of cantilever beam when actuator (0–3 composite) has 25% inclusion’s volume fraction, is 38% more in comparison to homogeneous PLZT material. It can therefore be inferred that 0–3 composite of photostrictive material having piezoceramic fillers could be seen as an alternative to homogeneous photostrictive material for use in actuation application, where lightweight, wireless and more actuation is required.