MATHEMATICAL MODELS FOR MODE-III CRACKS IN PIEZOELECTRIC/PIEZO-ELECTRO-MAGNETIC STRIP

Abstract

The un-ending thrust of human being has developed materials, materials science and their technology. The new technology is trying to ful ll the ever growing demands of mankind in search of material comforts and trying to save out natural resources. This has given rise to the development of smart materials. These smart materials have the property that can be signi cantly altered in a controlled fashion by external stimuli, such as stress, temperature, moisture, pH, electric and magnetic elds. Piezoelectric/piezo-electro-magnetic materials are most popular smart materials and widely used as sensors, actuators or transducers in smart structures, aerospace and automotive etc. However, these piezoelectric/piezo-electro-magnetic materials are mechanically brittle in nature and very susceptible to fracture. Therefore, much attention has been focused by many researchers to study the fracture analysis of such materials. Much work has been reported on cracking of such materials by many researchers viz. Pak, Park, Sosa, McMeeking, Narita, Shindo, Zhang, Gao, Li, Hu, Shen, Ru, Wang, Mai, to quote few. To study the behavior of cracking makes it possible to propose the arrest of opening of the crack. The crack arrest models have been developed on the basis of strip-yield model proposed by Dugdale (1960) and strip-saturation model proposed by Gao et al. (1997). The present problems investigated in this thesis are based on Dugdale and Gao models for cracked piezoelectric/piezo-electro-magnetic strip under Mode-III deformation. The numerical case study is presented for BaTiO3, PZT-5H, PZT-7A iii and PZT-6B piezoelectric ceramics. Based on Dugdale model, a strip-yield model is proposed for longitudinally ori- ented and asymmetrically situated single impermeable crack in piezoelectric strip under four di erent combination of prescribed electro-mechanical loads. It is found that energy release rate is minimum for non-centrically located crack as compare to the centrically located crack. After that strip-yield model is modi ed and strip-electro-mechanical yielding model is proposed for piezoelectric strip weakened by a non-centric crack. This analysis is conducted under the electrically uni ed boundary condition introducing the electric crack condition parameter that describes all electric crack-face boundary conditions: impermeable, semi-permeable and permeable. Fourier integral trans- form technique is used to reduce the problem into two pair of dual integral equa- tions, which are further reduced to Fredholm integral equations of second kind. The analytical expressions are derived for developed slide-yield zone length, saturation zone length and fracture parameters viz. crack sliding displacement, crack opening potential drop, eld intensity factors and energy release rate. In this thesis, the e ect of permittivity of crack gap media is observed for two equal collinear cracks cuts an in nite long and narrow piezoelectric strip under ap- plied electromechanical loading. For this numerical case study is also presented on various fracture parameters. Strip-saturation model and strip-electro-elastic yielding model are also proposed for piezoelectric strip weakened by transversely oriented and symmetrically situ- ated two equal semi-permeable collinear cracks. Fourier series method and integral equation techniques are used to develop these mathematical models. The analytic expressions are derived for developed zones as well as for fracture parameters at both interior and exterior tips of the crack. Numerical case study is presented to study the in uence of electric crack-face boundary conditions, inter-crack distance as well as electro-mechanical loads and piezoelectric material constants on fracture iv parameters. Similar to piezoelectric materials, strip-induction-saturation model (based on Dugdale and Gao models) is also proposed for transversely isotropic piezo-electro- magnetic strip weakened by a semi-permeable non-centric crack. Fourier integral transform technique is used to solve analytically the governing Laplace equations. The Analytical expressions are derived for developed induction and saturation zones; intensity factors for stress, strain, electric eld, electric displacement, magnetic induction and magnetic eld; crack sliding displacement; crack opening potential drop; crack opening induction drop; local and global energy release rates. BaTiO3- CoFe2O3 is chosen for numerical case study to study the combined e ect of pre- scribed mechanical, electrical and magnetic loads on crack opening potential drop, crack opening induction drop, local and global energy release rates. In this model, three di erent possible situations are considered: when developed induction zone is bigger, shorter and equal to developed saturation zone.