Abstract
In this paper, a constitutive model is proposed for piezoelectric material solids containing distributed cracks. The model is formulated in a framework of continuum damage mechanics using second rank tensors as internal variables. The Helmhotlz free energy of piezoelectric materials with damage is then expressed as a polynomial including the transformed strains, the electric field vector and the tensorial damage variables by using the integrity bases restricted by the initial orthotropic symmetry of the material. By using the Talreja’s tensor valued internal state damage variables as well as the Helmhotlz free energy of the piezoelectric material, the constitutive relations of piezoelectric materials with damage are derived. The model is applied to a special case of piezoelectric plate with transverse matrix cracks. With the Kirchhoff hypothesis of plate, the free vibration equations of the piezoelectric rectangular plate considering damage is established. By using Galerkin method, the equations are solved. Numerical results show the effect of the damage on the free vibration of the piezoelectric plate under the close-circuit condition, and the present results are compared with those of the three-dimensional theory.
Similar content being viewed by others
References
Moore, R.H., Dillard, D.A.: Time-dependent matrix cracking in cross-ply laminates. Composites Sci. Technol. 39, 1–12 (1990)
Park, S.W., Schapery, R.A.: A viscoelastic constitutive model for particulate composites with growing damage. Int. J. Solids Struct. 34, 931–947 (1997)
Luo, J.J., Daniel, I.M.: A cylinder model for characterization of deformation and damage development in a unidirectional composite. Composites Sci. Technol. 60, 2791–2802 (2000)
Deng, H., Nemat-Nasser, S.: Microcrack arrays in isotropic solids. Mech. Mater. 13, 15–16 (1992)
Zheng, Y.F., Fu, Y.M.: Analysis of the nonlinear vibration for symmetric angle-ply laminated viscoelastic plates with damage. Acta Mech. Sin. 21, 459–466 (2005)
Mindlin, R.D.: Forced thickness-shear and flexural vibration of piezoelectric crystal plates. J. Appl. Phys. 23, 83–88 (1952)
Tiersten, H.F.: Linear Piezoelectric Plate Vibrations. Plenum Press, New York (1978)
Chandrashekhara, K., Thenneti, R.: Termally induced vibration suppression of laminated plates with piezoelectric sensors and actuators. Smart Mater. Struct. 4, 281–290 (1995)
Zhou, X., Chattopadhyay, A., Gu, H.Z.: Dynamic response of smart composites using a coupled thermo-piezoelectric mechanical mode. AIAA J. 38, 1939–1948 (2000)
Wang, B., Rogers, A.: Laminate plate theory for spatially distributed induced strain actuators. J. Composite Mater. 25, 433–452 (1991)
Tzou, H.S., Gadre, M.: Theoretical analysis of a multi-layered thin shell coupled with piezoelectric shell actuators for distributed vibration controls. J. Sound Vib. 132, 433–450 (1989)
Xu, K.M., Noor, A.K., Tang, Y.Y.: Three-dimensional solutions for free vibration of initially stressed thermoelectroelastic multilayered plates. Comput. Methods Appl. Mech. Eng. 141, 125–139 (1997)
Mitchell, J.A., Reddy, J.N.: A refined hybrid plate theory for composite laminates with piezoelectric laminate. Int. J. Solid Struct. 32, 2345–2367 (1996)
Rao, S.S., Sunar, M.: Piezoelectricity and its use in disturbance sensing and control of flexible structures: a survey. Appl. Mech. Rev 47, 113–123 (1994)
Tauchert, T.R.: Piezothermoelastic behavior of a laminated plate. J. Thermal Stress 15, 25–37 (1992)
Zheng, Y.F., Fu, Y.M.: Nonlinear dynamic stability for piezoelectric laminated plate with damage (in Chinese). Chin. J. Theor. Appl. Mech. 38, 570–576 (2006)
Yang, G.S.: Damage Mechanics and Composites Damage (in Chinese). National Defense Industry Press, Beijing (1994)
Spenser: Theory of Invariable (the third volume) (in Chinese). Phoenix Science Press, Nanjing, (1982)
Duan, W.H., Quek, S.T., Wang, Q.: Free vibration analysis of piezoelectric coupled thin and thick annular plate. J. Sound Vib. 28, 119–139 (2005)
Piotr, C.: Three-dimensional natural vibration analysis and energy considerations for a piezoelectric rectangular plate. J. Sound Vib. 283, 1093–1113 (2005)
Author information
Authors and Affiliations
Corresponding author
Additional information
The project supported by the National Natural Science Foundation of China (10572049).
The English text was polished by Keren Wang.
Rights and permissions
About this article
Cite this article
Fu, Y., Wang, X. A continuum damage model for piezoelectric materials. Acta Mech. Sin. 24, 171–179 (2008). https://doi.org/10.1007/s10409-007-0133-y
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10409-007-0133-y