SOME CRACK ARREST MODELS FOR PIEZOELECTRIC/PIEZOELECTROMAGNETIC PLATES

Abstract

The technological advancement has lead to the development of material science manifold. Ever search of newer materials or the tailor made materials opened the new field of smart materials. These smart materials have the property (ies) that can be significantly altered in a controlled fashion by external stimuli, such as stress, temperature, moisture, pH, electric and magnetic field. One such material is the piezoelectric ceramic which when heated to a certain characteristic temperature and then cooled the dipoles permanently align with one another giving the material absolute new characteristic. When the poled ceramics are subjected to a small electric field, the dipole responds collectively to produce a macroscopic expansion/contraction and vice-versa. The qualities of piezoelectric materials make them worthy of sensors/actuators/transducers. This lead to the research of other attire of such ceramics. One such attire is the fracture of such materials. The work on the crack problems started way back in 1990 by researchers like Pak, Park, Sosa, McMeeking, Zhang, Shen to quote the few. Study the behavior of the behavior of cracking makes it possible to propose the arrest of the opening of the crack. When the crack rims open on account of the applied mechanical or/and electrical loads the rims of the crack open developing a non-liner region ahead of each tip of the crack which results the crack opening. Under small scale yielding this region is then assumed to lie along the length of the crack. To arrest the further opening of the crack developed strip iv ahead of each tip is subjected to closing loads. This makes the subject matter of the present thesis. ORGANISATION OF THESIS Many crack arrest models are investigated under various loading conditions. > A crack arrest model is proposed for a cracked piezoelectric ceramic plate. The piezoelectric ceramic is electrically more brittle. Consequently under small scale yielding a saturation zone developed in a strip ahead of crack tip. The model is termed as strip saturation model. The strip saturation zone is subjected to linearly varying saturation limit electrical load. It is observed that the saturation zone could be arrested for loads less than saturation limit loads. > The next model considered is the strip saturation model in which to arrest the crack the developed saturation zone is subjected to quadratically varying saturation limit electrical load. It is observed under small scale yielding the developed saturation when subjected to still lesser electrical load is able to arrest the crack opening although for a comparatively smaller zone. > Next model is a combined strip slide and strip saturation model. Both strip slide and saturation zones are arrested by respective yield point mechanical and electrical loads. Three very interesting cases are considered when slide zone is smaller than saturation zone and vice-versa. The third case when both zones are equal is also discussed at length. > The combined strip slide and strip saturation model's are considered in which the developed slide zone rims are subjected to linearly varying yield point shear stress while the saturation zone is subjected to saturation point electrical displacement. It is seen that the crack arrest is possible for the case when slide zone is smaller than saturation zone. > In the next case combined strip slide and saturation zone model the developed slide zone rims are subjected to quadratically varying yield point shear stress. In this case also the slide zone should be still smaller to be able to arrest the crack opening than the saturation zone. > Next case is considered when saturation zone rims are subjected to linearly varying saturation limit in-plane normal electrical displacement, is a combined strip slide and saturation zone model. And the rims of the yield slide zone are subjected to yield point anti-shear stress. To stop the crack opening it is observed that in this case the saturation zone is smaller than the developed slide zone. ➢ The model next considers the case when developed saturation zone is subjected to quadratically varying saturation limit in-plane normal cohesive electrical displacement in a combined yield slide and saturation zone model. For this case also model is valid when saturation zone length is still smaller then the slide zone which is subjected to yield point anti-shear stress. For each model an illustrative case study has been presented for piezoelectric ceramic. ➢ The crack arrest model for a cracked piezoelectromagnetic plate of composite BaTiO3-CoFe2O4 has been investigated for combined strip slide, saturation and magnetic zone model. The developed induction, saturation and slide zones are subjected to in-plane limit point magnetic field, saturation zones vi are subjected to in-plane saturation limit electric displacement and the slide zones are subjected to anti-plane yield point shear stress. In this case it is assumed that induction zone is smallest. Developed saturation zone is bigger than the induction zone. And the slide zone is the biggest. The model suggests that it is possible to arrest the crack opening. > For the second case considered a crack problem for magnetoelectroelastic plate when the plate is subjected to in-plane uniform, constant unidirectional electric and magnetic fields and anti-plane uniform constant shear stress. It is assumed that the crack opens in self-similar fashion and the developed induction zone is smallest, slide zone is bigger than this and saturation zone is the biggest. A quantitative study has been carried out for the composite BaTiO3-CoFe2O4 and reported graphically. All the problems investigated are solved employing either complex variable technique or Fourier integral transform method. For the case study of different ceramics the numerical simulation is done using Mathematica, Analog and self made computer programs.