The Effect of Temperature Dependency on the Thermo-Electro-Elastic Analysis of Functionally Graded Piezoelectric Spherical Shell

The Effect of Temperature Dependency on the Thermo-Electro-Elastic Analysis of Functionally Graded Piezoelectric Spherical Shell

M. Arefi ^*, M. Takhayor Ardestani

Department of Solid Mechanic, Faculty of Mechanical Engineering, University of Kashan, Kashan 87317-51167, Iran

1. Introduction

Functionally graded materials have been produced for simultaneously bearing thermal and mechanical loads in a structure with high gradient of temperature. The experiments indicate that temperature variation can significantly change the various characteristics (mechanical, thermal and electrical properties) of materials rather than room temperature. This change would lead to significant change of materials behavior and responses of structures made of them subjected to various loadings. A literature review on the previous works related to thermo-electro-elastic analysis of piezoelectric structures indicates that researchers have not comprehensively and analytically investigated the temperature dependency. In this section, a literature review on temperature dependent works and on the thermo-electro-elastic analysis of various structures are presented in order to show the importance of this work.

Teyssedre and Lacabanne [1] studied the behavior of thermal and dielectric P(VDF-TrFE) copolymers in relation with their electroactive properties. Tanigawa et al. [2] presented the transient heat conduction and thermal stress analysis of in-homogeneous materials with temperature dependent material properties. They showed that accounting the thermal conductivity would lead to a nonlinear differential equation. Substitution of temperature distribution using solution of nonlinear differential equation into stress components and then into equations of motion leads to transient thermal stress distribution.

Omote et al. [3] presented the properties of the elastic, dielectric, and piezoelectric of poly (vinylidine fluoride-trifluoroacetylene) shells in the form of graphs with temperature dependency. Wong et al. [4] examined Specific heat and thermal diffusivity of Vinylidene Fluoride/Trifluoroethylene Copolymers. Thermal post-buckling analysis of functionally graded cylindrical thin shell with considering temperature dependency and finite length subjected to uniform temperature distribution was studied by Shen [5]. Nonlinearity was accounted in derivation of governing equations of motion using von Kármán–Donnell-type of kinematic nonlinearity. Buckling temperature and postbuckling load–deflection curves were determined based on a singular perturbation technique.

Liew et al. [6] studied thermal postbuckling analysis of laminated plate made of functionally graded materials. Temperature dependency was accounted for material properties. Kim [7] studied vibration characteristics of functionally graded rectangular plate with initial stress. The plate was assumed to be made of metal and ceramic considering temperature dependency. The material properties were assumed to be graded along the thickness direction based on power law distribution. The Rayleigh–Ritz procedure was applied to obtain the frequency equation. The double Fourier series was employed to examine the effect of material compositions, plate geometry, and temperature fields on the vibration characteristics of functionally graded rectangular plate. Thermo-mechanical post-buckling analysis of cylindrical panels made of functionally graded materials considering temperature dependency was studied by Yang et al. [8] based on classical shell theory with von-Karman–Donnell-type nonlinearity. Differential quadrature method was utilized for solution of the problem. The parametric numerical results were presented to study the effects of temperature-dependent properties, volume fraction index, axial load, initial imperfection, panel geometry and boundary conditions on the thermo-mechanical post-buckling behavior.

Shen [9] studied thermal post-buckling analysis of a functionally graded plate with simply supported boundary condition based on higher-order shear deformation theory subjected to two types of temperature distribution. Temperature dependency and power law function were used for material properties. The buckling and post-buckling temperatures were calculated based on a two-step perturbation technique. It was concluded that considering the temperature dependency would lead to significant improvement of results rather than the case in which the temperature dependency is ignored.

Dynamic buckling analysis of functionally graded temperature dependent cylindrical shell with geometric imperfection and integrated with sensor and actuator layers was studied by Shariyat [10]. The numerical results were presented to investigate the influence of temperature dependency, volume fraction index, load combination and initial geometric imperfections on thermo-electro-mechanical post-buckling behavior. Thermo-electro-elastic analysis of thick-walled functionally graded piezoelectric cylindrical shell was studied by rahimi et al. [11]. Power law function was used for description of functionality for all mechanical, thermal and electrical properties of material. Linear and nonlinear thermo-elastic analysis of functionally graded piezoelectric spherical shell was investigated based on electro-elastic and Maxwell relations by Arefi and Khoshgoftar [12] and also by Arefi and Nahas [13].

Arefi et al. [16] studied the effect of axially variable thermal and mechanical loads on the 2D thermoelastic response of FG cylindrical shell. Hamida et al. [17] provided a sensitivity analysis for identification of key input parameters affecting energy conversion factor of flexoelectric materials. The numerical results indicated that the flexoelectric constants are the most dominant factors influencing the uncertainties in the energy conversion factor. Ghasemi et al. [18] presented a computational methodology for topology optimization of multi-material-based flexoelectric composites. They provided some numerical examples for two-, three- and four-phase flexoelectric composites to demonstrate the flexibility of the model that can be obtained using multi-material topology optimization for flexoelectric composites. Other related works to optimization and computational methods of flexoelectric and piezoelectric structures were studied by various researchers [19-20].

Literature review on the thermo-elastic analysis of temperature dependent materials and structures was presented in Introduction. Although some important works on the temperature dependent materials and structures have been performed by various researchers, based on author's knowledge, there is no published works on the electro-thermo-elastic analysis of thick-walled spherical shell made of functionally graded temperature dependent materials subjected to thermal, electrical and mechanical loads. Innovative aspects of this research are the study of the dependence of all material properties on temperature, and considering the temperature gradient. It can be seen that considering temperature dependency would lead to significant changes of all material properties and consequently important changes in electro-thermo-elastic results.

In this article, linear temperature dependency is used for all mechanical, thermal and electrical temperature-dependent material properties. In addition, power law function is used for functionally graded piezoelectric materials. The division method is applied for solution of the two second-order governing equations with variable coefficients. The numerical results are presented to show the influence of temperature dependency and functionality of piezoelectric material on the radial displacement, radial and circumferential stress, electric field and temperature distribution.

1. Fromulation

The governing equations of thermo-electro-elastic analysis of functionally graded piezoelectric spherical shell are presented in this section. The spherical shell is subjected to thermal and mechanical loads at inner and outer radii (see Fig. 1).

The heat conduction equation in spherical coordinate system is defined as [12]:

The spherical shell is assumed to be made of functionally graded temperature dependent materials. Based on this assumption, the thermal conductivity coefficient is considered as [14, 15]:

In Eq. (2),  are two constants showing the temperature dependency. Solution of heat conduction equation may be obtained by substitution of thermal conductivity coefficient and two times integration as follows:

The temperature distribution can be obtained by applying the necessary boundary conditions. Applying the boundary conditions  to temperature distribution found in Eq. (3) would yield the integration constants as follows:

where:

In this stage and after completion of thermal analysis, the electro-thermo-elastic relations could be presented. The stress-strain and electric displacement relations by considering the thermal and electrical effects are developed in spherical coordinate system as follows [12]:

in which, are stress, strain, electric field, temperature distribution and electrical displacement. In addition, are stiffness and piezoelectric coefficients. Furthermore,  is the dielectric constant and  is the pyroelectric constant.  could be obtained from [12]:

Due to symmetric distribution of all mechanical and electrical components, the circumferential component of the electrical displacement  , is zero. For a symmetric spherical structure, only nonzero displacement component is radial displacement. Therefore, the strain-displacement and electric field-electric potential relations are defined as [12]:

In which, is the electric potential distribution. Substitution of strain and electric field components into stress and electric displacement relations leads to updated form of these relations as follows:

The equilibrium and Maxwell relations are utilized to derive the governing equations. These relations are presented as:

Before derivation of governing equations, the functionally graded temperature dependent material properties should be expressed. These relations are expressed as [1,3,4]:

The stress components and electric displacement relations are derived as Eq. (11):

Substitution of stress and electric displacement relations into equilibrium and Maxwell relations would result in final governing equations. Presentation of governing equations is not provided due to very large terms.

1. Solution procedure

The solution procedure is developed in this section based on division method. This method is applied for solution of differential equations with variable coefficients.

In this method, the variable coefficient differential equations are evaluated at the middle of each layer by substitution of  into governing equations (see Fig. 2). After evaluation of governing equations for each layer, the obtained governing equations are solved. The compatibility relations between two adjacent layers should be written as:

For instance, the governing equations for the first layer are derived as:

Solving the governing equations of layer 1 leads to following one:

Evaluation of solution for each layer (as presented above) and applying the continuity equations (as presented by Eq. 12) gives a system of algebraic relation in terms of . In addition, the boundary conditions for P(VDF-TrFE) copolymers at inner and outer layers are expressed as:

1. Numerical results and discussion

Before presentation of full numerical results, the effect of temperature dependency is studied on the results rather than the cases that temperature in-dependency is assumed. Fig. 3 shows the temperature distribution along the radial direction based on two cases: first for temperature dependent materials and second, for temperature independent materials.

The comparison between temperature distribution for the temperature-dependent and temperature-independent materials indicates that the temperature calculated for temperature-dependent materials is slightly less than the one for temperature-independent materials.

Radial distribution of radial stress is presented in Fig. 4 for two aforementioned cases. It is observed that the applied boundary conditions are satisfied at inner and outer surfaces. In addition, one can conclude that the radial stress for temperature-independent case is greater than the one for temperature-dependent case. Fig. 5 shows the radial distribution of circumferential stress for temperature-dependent/independent material cases. It is observed that considering the temperature-independency, the slope of circumferential stress is reducted and tends to uniform condition along the thickness direction. Fig. 6 shows the radial variation of electric field for temperature-dependent and independent material cases. The similar behavior of Fig. 5 could be observed in the results of Fig. 6.

Fig. 7 shows the radial distribution of electric potential for temperature-dependent and independent material cases. It is observed that the absolute value of the electrical potential for the temperature-dependent properties is lower than that of the independent temperature. Shown in Fig. 8 is the radial distribution of radial displacement in terms of various in-homogeneous index k. it is observed that with increase of in-homogeneous index k, the radial displacement is decreased significantly. One can conclude that with increase of in-homogeneous index k, the stiffness of structure is increased which leads to significant decrease in radial displacements.

Fig. 4. The radial distribution of radial stress based on temperature dependent and independent materials

Fig. 7. The radial distribution of electric potential based on temperature-dependent and independent materials

Fig. 9 shows the variation of electric potential along the thickness direction for various values of in-homogeneous index k. The numerical results indicate the sign of electric potential changes from negative to positive for higher values of in-homogeneous index k. Fig. 10 shows the radial distribution of the electric field along the thickness direction for various values of in-homogeneous index k. Fig. 11 shows the radial distribution of the temperature along the thickness direction for various values of in-homogeneous index k. It could be observed that with increase of in-homogeneous index k, the temperature decreased.

1. Conclusions

The effect of temperature-dependency was studied on the thermo-electro-elastic results of a functionally graded spherical shell subjected to thermal, mechanical and electrical loads. It was assumed that all mechanical and thermal material properties are graded along the thickness direction based on power law function and also temperature dependent. The governing equations were derived based on equilibrium and Maxwell relations considering thermal and electrical effects. The division method was employed for solving the governing equations with variable coefficients. The numerical results were presented to show the influence of temperature-dependency of all mechanical, thermal and electrical loads and in-homogeneous index. The main results of this work are classified as follows:

Fig. 11. The radial distribution of temperature for various values of in-homogeneous index k

The numerical results indicated that considering the temperature-dependency would lead to significant changes of thermo-electro-elastic results. One can conclude that the radial displacement and stress, temperature distribution and electric field for temperature-independent case is significantly greater than that of temperature dependent material. It is concluded that for temperature-independent case, the effect of thermal strains leads to significant increase of deformations and stresses in spherical shell.

Variation of in-homogeneous index leads to significant changes and different behaviors of outputs such as radial displacement, radial and circumferential stresses, temperature distribution and electric field. The numerical results indicate that with increase of in-homogeneous index, the radial displacement and stress, electric potential and temperature would significantly decrease. Furthermore, the behavior of circumferential stress and electric field depend on the radial location study in which at layers near the inner surface, these components decrease with increase of in-homogeneous index while at layers near the outer surface would increase. Temperature, the radial stress and radial displacement for temperature-independent case is greater than one for temperature-dependent case.

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