Research On The Cracked-hole Problem In The Piezoelectric Materials

During the past century, the properties of piezoelectric materials, such as those of elasticity,piezoelectricity, dielectricity, pyroelectricity, ferroelectricity and photoelectricity, have been found.With the piezoelectric ceramic manufacturing technique beening improved gradually, the application ofpiezoelectric materials has become more and more wide, e.g. they hace benn used in radarcommunications, ultrasound, medical imaging, infrared detection, aerospace, animal bionic, electronicmeasurement and so on.Structures with holes are very common in most practical engineering. Under complicated loadingenvironment, this kind of structures will produce the so-called “stress concentration” phenomenon,which will leads to the damage of the materials. So, it is very important to study the cracked-holeproblems in the peizoelectric materials.In the present paper, the cracked-hole problems in an infinite piezoelectric body are studied byusing the complex variable method and the conformal mapping function, combined with the Cauchyintegral, and the analytical solutions are derived for some classic problems. On the other hand, theboundary element method is used to deal with the abritrary shape cracked-hole problems in a finitepiezoelectric body. The main works can be summarized as follows:(1) By using the complex variable method and the conformal mapping function, the cracked-holeanti-plane problems in the transversely isotropic piezoelectric materials are solved. For the classicproblems, the improved mapping function is obtained, and the explicit and exact expressions forthe complex potentials, field intensity factors and energy release rates are presented respectivelyon the assumption that the surface of the cracks and hole is electrically impermeable. For thearbitrary shape cracked-hole problems, the method of numerical conformal mapping is used toobtain the mapping problem which maps the outside of arbitrary shape with a crack into theoutside of a circular hole. Based on the mapping function, the approximate expressions for thecomplex potentials, field intensity factors and energy release rates are presented, respectively.Numerical analysis is then conducted to discuss the influences of crack length and appliedmechanical/electric loads on the field intensity factors and energy release rates for one and twoedge cracks, respectively.(2) The solution of anti-plane problems for the bimaterials which contain an elliptic hole with twoedge cracks is obtained. The mapping function which maps the ellipse to a line crack is obtained. Based on the assumption of permeable or impermeable crack, the expressions for the complexpotentials and field intensity factors are obtained by using the Stroh formulation, respectively. Thesingularity of the crack tip, the field intensity factors are studied.(3) The cracked-hole problems in the generalized two-dimensional electrical materials is studied,where the heat flow is considered. Firstly, the mapping function based on the elliptic hole has beenobtained, which maps the outer of the cracked-hole into the outer of the circle hole. Secondly, theheat complex function is solved based on the adiabatic condition. Thirdly, the expressions for thecomplex potentials and field intensity factors are presented under the condition that the stress,strain and electric displacement are bounded at infinity, single-valued displacement, theequilibrium of mechanical, and electrical electrically impermeable boundary condition. Andfinally, some numerical analyses are also made to discuss the influences of crack length and heatflux on the electroelastic fields and fields intensity factors, in order to find a way to decrease thefield intensity factors.(4) By using the boundary element method (BEM), the cracked-hole problems in the finitepiezoelectric body is studied. The stress field and the electrical displacement field around thecrack tip are analyzed. For the arbitrary shape of hole, the influence of the geometric dimensionsand the external loads on the field intensity factors are discussed. Since the elliptic hole or thecrack has been considered in the fundamental solution, there is no need to discrete the ellipticalhole boundary or the crack boundary in the discretization process, which avoids the problem ofdiscretization caused by the singularity at the crack tip, and greatly improves the accuracy of thecalculation at the crack tip.