SOME CRACK ARREST MODELS FOR TRANSVERSELY CRACKED PIEZOELECTRIC STRIPS

Abstract

The un-ending thrusts of human being have developed material science and technology. The new technology is trying to fulfill 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(ies) that can be significantly altered in a controlled fashion by external stimuli, such as stress, temperature, moisture, pH, electrical and magnetic field. The idea of smart material is basically taken from nature, (where these are available in abundance) as to how much one can emulate it. One such smart material is piezoelectric ceramics. At ordinary temperature they do not exhibit any specific property but when heated to a certain characteristic temperature and cooled, the dipoles permanently align with each other giving the material absolute new characteristic. Below this characteristic temperature, called Curie temperature, if the piezoelectric ceramic is subjected to electric field, the dipoles responds collectively to produce a macroscopic expansion or contraction. Inversely if a mechanical pressure is applied the electric field is generated this is called inverse piezoelectric effect. This quality of piezoelectric material makes them worthy of sensor/actuators/transducers. Because of this property they have been widely used in medical, engineering, submarine, aerospace, hi-tech technologies. iv Piezoceramics utilities were noted back in 1940 and at the time some work was done on them. But after observing fifty years lull period it again got impetus in 1990. Many attires of such materials have been studies. One such attire is cracking of such materials. As one would easily appreciate that the fatigue, overloading, cyclic loading and ageing gives rise to weakening of the material. Consequently the crack(s) develop. Much work has been reported on this phenomenon of cracking viz. Cherapanov, Pak, Park, Sun, Sosa, McMeeking, Zhang, Shen, Mai, Ru, to quote few. Going a step further the possibility of crack arrest is also pondered. The models have been developed on the basis of strip-yield model proposed by Dugdale (1960). The present work is a step further in this direction. Many crack arrest models have been proposed under different loading conditions. ORGANISATION OF THESIS The thesis is written from the point of view that any new reader gets the entire material needed to understand the work, is at hand available to the reader, in the thesis. ➢ First two chapters are introductory in nature. Firstly the smart materials are introduced and their properties, utility and applications are discussed at length. One of the smart material namely piezoceramic is discussed then. The concept of linear theory of piezoelectricity is discussed in details. Concepts pertaining to thesis are introduced. The methodology used to investigate the models proposed, is described. An overview of the development in the field of cracks in transversely cracked piezoceramic strip is given. The topics needing the immediate attention and research have been pointed out and the relevance of present research is shown. ➢ Strip-slide yield models Crack arrest strip-slide yield models are proposed for an infinitely long narrow strip. The strip is weakened by a finite, quasi-stationary, hairline straight crack symmetrically situated with respect to the edges of the strip and oriented transversely to these edges. The investigation being carried under the assumption that this strip is mechanically more brittle as compared to electric case. Consequently when the crack opens due to prescribed mechanical and electrical boundary condition, a mechanical singularity is encountered first. It is at this level the investigations are being carried. To arrest the crack from further opening the rims of developed zones are prescribed a mechanical anti-plane shear stress. The model thus obtained is termed as strip-slide yield model. Two cases are investigated: • When developed zones are arrested by linearly varying yield point anti-plane shear stress, • When developed zones are prescribed by a quadratically varying yield point anti-plane shear stress. The results obtained are applied to investigate the crack arrest of transversely cracked piezoceramic strips of each of PZT-4, PZT-5H and BaTiO3. In fact results can be applied to all ceramic exhibiting the symmetry of 6 mm crystal about principal axes. vi ➢ Strip-saturation models A transversely cracked long narrow, poled piezoceramic strip is subjected to combined anti-plane mechanical and in-plane electric boundary conditions. This causes opening of the crack under small-scale yielding both mechanically and electrically. Under the assumption that the strip is more electrically brittle than the mechanical one, an electrical singularity is encountered first. Working on this level saturation zone is encountered first. To arrest the crack opening further the rims of the developed saturation zones are subjected to normal cohesive in-plane displacement. This model is termed as strip-saturation model. Three models are investigated for this case. Strip-saturation model when developed saturation zones are closed by saturation-limit-in-plane normal electrical displacement. Modified strip-saturation model when developed saturation zones rims are closed by linearly varying saturation limit in-plane normal electrical displacement. Generalized strip-saturation model when developed saturation zones rims are prescribed by cohesive normal quadratically varying saturation limit in-plane electrical displacement. For these cases illustrative examples are presented which show that it is possible to arrest the crack in each case under small scale yielding. vii Strip-slide yield and strip-saturation model A poled piezoceramic infinitely long narrow strip is cut along a horizontal, hairline straight crack lying vertical to the edges of the crack. The edges of the strip are stress and electric charge free. The rims of the crack are stress and charge free. Combined electrical and mechanical condition make the crack yields both mechanically and electrically. Consequently a strip-slide yield zone and a strip-saturation zone develop ahead of each tip of the crack. To arrest the crack from further opening the developed strip-slide zone rims are subjected to yield point anti-plane shear stress. And the developed strip-saturation is prescribed by saturation limit in-plane normal cohesive electrical displacement. Three very interesting and practical cases are discussed in details: • In part one the case when saturation zone exceeds the slide-zone is investigated. • The second part the case when saturation zone is smaller than the strip-slide zone is considered. • The third part deals with the case when strip saturation zone and strip-slide zone are of equal length. For each model the case study has been prescribed. Interesting conclusions are obtained. viii