Durable pyroelectric shell structures for energy scavenging applications

  • Mohammad Khorsand
  • Youhong TangEmail author
Original Paper


In the light of the increasing growth of requests for applicable energy accounts, this study is driven by the desire to determine the energy interactions among the electrical, thermal and mechanical fields in pyroelectric shell structures. The principal purpose of the proposed work is to scrutinize how material properties synthesize new smart systems exhibiting higher levels of electrical energy while possessing considerably lower weight. To achieve such an aim, it is assumed that the material properties of the elastic medium would be graded through the thickness and the mechanical, electrical and thermal fields would have their own independent gradient indexes. If we nominate the grading indexes as design variables, the material elements of the solid should be appropriately functionalized, avoiding any problems stemming from the low electrical outputs of shell structures in energy-harvesting applications. To untangle the complex mathematics governing this problem, a skilfully merged algorithm of an advanced optimization method with two semi-analytical approaches is exploited. When the simulation is implemented, a significant increment in electrical energy in conjunction with a marked reduction in weight is reported. Further, the effects of inhomogeneity and thermal gradient on the electrostatic energy and weight, as well as the stress, displacement, electrical and thermal fields are graphically presented and discussed. With its material development and structural integrity, this study presents an efficient and cost-effective criterion for smart structures for scavenging energy from available thermo-electromechanical energy sources.



Mohammad Khorsand is grateful for the financial support of Postgraduate Research Scholarships (International) for his Ph.D. study at Flinders University.


  1. 1.
    Niino, M.: Development of functionally gradient material. J. Jpn. Soc. Powder Powder Metall. 37, 146–241 (1990)CrossRefGoogle Scholar
  2. 2.
    Koizumi, M.: The concept of FGM. Ceram. Trans. 34, 3–10 (1993)Google Scholar
  3. 3.
    Reddy, J.N., Chin, C.D.: Thermomechanical analysis of functionally graded cylinders and plates. J. Therm. Stress. 21, 593–626 (1998)CrossRefGoogle Scholar
  4. 4.
    Zhu, X.H., Meng, Z.Y.: Operational principle, fabrication and displacement characteristics of a functionally gradient piezoelectric ceramic actuator. Sens. Actuator A Phys. 48, 169–176 (1996)CrossRefGoogle Scholar
  5. 5.
    Kieback, B., Neubrand, A., Riedel, H.: Processing techniques for functionally graded materials. Mater. Sci. Eng. A 362, 81–105 (2003)CrossRefGoogle Scholar
  6. 6.
    Wang, J., Shaw, L.L.: Fabrication of functionally graded materials via inkjet color printing. J. Am. Ceram. Soc. 89, 3285–3289 (2006)CrossRefGoogle Scholar
  7. 7.
    Zybala, R., Wojciechowski, K.T.: Anisotropy analysis of thermoelectric properties of Bi2Te2.9Se0.1 prepared by the SPS method. In: Proceedings of the 9th European Conference on Thermoelectrics (ECT), AIP Conference Proceedings, vol. 1, pp. 393–396 (2011)Google Scholar
  8. 8.
    Udupa, G., Rao, S.S., Gangadharan, K.V.: Functionally graded composite materials: an overview. Proc. Mater. Sci. 5, 1291–1299 (2014)CrossRefGoogle Scholar
  9. 9.
    Chmielewski, M., Pietrzak, K.: Metal–ceramic functionally graded materials—manufacturing, characterization, application. Tech. Sci. 64, 151–160 (2016)Google Scholar
  10. 10.
    Tiersten, H.F.: Linear Piezoelectric Plate Vibrations. Plenum Press, New York (1969)CrossRefGoogle Scholar
  11. 11.
    Timoshenko, S.P., Goodier, J.N.: Theory of Elasticity. McGraw-Hill, New York (1970)zbMATHGoogle Scholar
  12. 12.
    Muskhelishvili, N.I.: Some Basic Problems of Mathematical Theory of Elasticity. Noordhoff, Leyden (1963)zbMATHGoogle Scholar
  13. 13.
    Noda, N.: Transient thermal stresses problem in a transversely isotropic finite circular cylinder under three-dimensional temperature field. J. Therm. Stress. 6, 57–71 (1983)CrossRefGoogle Scholar
  14. 14.
    Gomshima, T., Miyao, K.: Transient thermal stresses in a plate with a hole due to rotating heat source. J. Therm. Stress. 13, 43–56 (1990)CrossRefGoogle Scholar
  15. 15.
    Tokovyy, Y., Chyzh, A., Ma, C.C.: An analytical solution to the axisymmetric thermoelasticity problem for a cylinder with arbitrarily varying thermomechanical properties. Acta. Mech. 230, 1469–1485 (2019)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Tanigawa, Y., Morishita, H., Ogaki, S.: Derivation of systems of fundamental equations for a three-dimensional thermoelastic field with nonhomogeneous material properties and its application to a semi-infinite body. J. Therm. Stress. 22, 689–711 (1999)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Pelletier, J.L., Vel, S.S.: An exact solution for the steady-state thermoelastic response of functionally graded orthotropic cylindrical shells. Int. J. Solids Struct. 43, 1131–1158 (2006)CrossRefGoogle Scholar
  18. 18.
    Chen, Y.Z., Lin, X.Y.: An alternative numerical solution of thick-walled cylinders and spheres made of functionally graded materials. Comput. Mater. Sci. 48, 640–647 (2010)CrossRefGoogle Scholar
  19. 19.
    Fallah, F., Taati, E.: On the nonlinear bending and post-buckling behavior of laminated sandwich cylindrical shells with FG or isogrid lattice cores. Acta Mech. 230, 2145–2169 (2019)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Farokhi, H., Ghayesh, M.H.: Modified couple stress theory in orthogonal curvilinear coordinates. Acta. Mech. 230, 851–869 (2019)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Rao, M.N., Schmidt, R., Schröder, K.U.: Static and dynamic FE analysis of piezolaminated composite shells considering electric field nonlinearity under thermo-electro-mechanical loads. Acta. Mech. 229, 5093–5120 (2018)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Vetyukov, Y., Staudigl, E., Krommer, M.: Hybrid asymptotic-direct approach to finite deformations of electromechanically coupled piezoelectric shells. Acta. Mech. 229, 953–974 (2018)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Ravi, S., Zilian, A.: Monolithic modeling and finite element analysis of piezoelectric energy harvesters. Acta. Mech. 228, 2251–2267 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Qin, Q.H.: Fracture Mechanics of Piezoelectric Materials. WIT, Southampton (2001)Google Scholar
  25. 25.
    Babaei, H., Eslami, M.R.: Thermally induced large deflection of FGM shallow micro-arches with integrated surface piezoelectric layers based on modified couple stress theory. Acta Mech. 230, 2363–2384 (2019)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Li, X.F., Peng, X.L., Lee, K.Y.: Radially polarized functionally graded piezoelectric hollow cylinders as sensors and actuators. Eur. J. Mech. A Solids 29, 704–713 (2010)CrossRefGoogle Scholar
  27. 27.
    Shelley, W.F., Wan, S., Bowman, K.J.: Functionally graded piezoelectric ceramics. Mater. Sci. Forum 308–311, 515–520 (1999)CrossRefGoogle Scholar
  28. 28.
    Zhu, X.H., Zu, J., Meng, Z.Y., Zhu, J.M., Zhou, S.H., Li, Q., Liu, Z., Ming, N.: Microdisplacement characteristics and microstructures of functionally graded piezoelectric ceramic actuator. Mater. Des. 21, 561–566 (2000)CrossRefGoogle Scholar
  29. 29.
    Wu, C.C.M., Kahn, M., Moy, W.: Piezoelectric ceramics with functionally gradients: a new application in material design. J. Am. Ceram. Soc. 79, 809–812 (1996)CrossRefGoogle Scholar
  30. 30.
    Sheng, G.G., Wang, X.: Thermoelastic vibration and buckling analysis of functionally graded piezoelectric cylindrical shells. Appl. Math. Model. 34, 2630–2643 (2010)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Hong, C.C.: Computational approach of piezoelectric shells by the GDQ method. Compos. Struct. 92, 811–816 (2010)CrossRefGoogle Scholar
  32. 32.
    Wu, C.P., Syu, Y.S.: Exact solutions of functionally graded piezoelectric shells under cylindrical bending. Int. J. Solids Struct. 44, 6450–6472 (2007)CrossRefGoogle Scholar
  33. 33.
    Zhu, C.S., Fang, X.Q., Liu, J.X.: Surface energy effect on buckling behaviour of the functionally graded nano-shell covered with piezoelectric nano-layers under torque. Int. J. Mech. Sci. 133, 662–673 (2017)CrossRefGoogle Scholar
  34. 34.
    Chen, C.Q., Shen, Y.P.: Piezothermoelasticity analysis for a circular cylindrical shell under the state of axisymmetric deformation. Int. J. Eng. Sci. 34, 1585–1600 (1996)CrossRefGoogle Scholar
  35. 35.
    Sodano, H.A., Inman, D.J.: Estimation of electric charge output for piezoelectric energy harvesting. Strain J. 40, 49–58 (2004)CrossRefGoogle Scholar
  36. 36.
    Mirjalili, S.A., Mirjalili, S.M., Lewis, A.: Grey wolf optimizer. Adv. Eng. Softw. 69, 46–61 (2014)CrossRefGoogle Scholar
  37. 37.
    Shu, C.: Differential Quadrature and Its Application in Engineering. Springer, London (2000)CrossRefGoogle Scholar
  38. 38.
    Khorsand, M., Tang, Y.: On the qualitative dynamics of rotating disks: thermal shocks and structural integrity. Int. J. Press. Vessel Pip. 166, 35–47 (2018)CrossRefGoogle Scholar
  39. 39.
    Shu, C., Richards, B.E.: Application of generalized differential quadrature to solve two-dimensional incompressible Navier–Stokes equations. Int. J. Numer. Methods Fluids 15, 791–798 (1992)CrossRefGoogle Scholar
  40. 40.
    Khorsand, M., Fu, K., Tang, Y.: Multi-directional functionally graded materials for enhancing the durability of shell structures. Int. J. Press. Vessel Pip. 175, 103926 (2019)CrossRefGoogle Scholar
  41. 41.
    Khorsand, M., Tang, Y.: Thermal analysis and electro-elastic response of multilayered spherical vessels. Int. J. Press. Vessel Pip. 171, 194–206 (2019)CrossRefGoogle Scholar
  42. 42.
    Khorsand, M., Tang, Y.: Design functionally graded rotating disks under thermoelastic loads: weight optimization. Int. J. Press. Vessel Pip. 161, 33–40 (2018)CrossRefGoogle Scholar
  43. 43.
    Kennedy, J., Eberhart, R.C., Shi, Y.: Swarm Intelligence. Morgan Kaufmann, San Francisco (2001)Google Scholar
  44. 44.
    Kennedy, J., Eberhart, R.C.: Particle swarm optimization. In: Proceedings of the 1995 IEEE International Conference on Neural Networks, pp. 1942–1948. IEEE Service Center, Piscataway (1995)Google Scholar
  45. 45.
    Jabbari, M., Sohrabpour, S., Eslamic, M.R.: Mechanical and thermal stresses in a functionally graded hollow cylinder due to radially symmetric loads. Int. J. Press. Vessel Pip. 79, 493–497 (2002)CrossRefGoogle Scholar
  46. 46.
    Eslami, M.R., Babaei, M.H., Poultangari, R.: Thermal and mechanical stresses in a functionally graded thick sphere. Int. J. Press. Vessel Pip. 82, 522–527 (2005)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute for NanoScale Science and Technology, College of Science and EngineeringFlinders UniversityBedford ParkAustralia

Personalised recommendations