The Research On The Problem Of Cycle Symmetric And The Application Of Quasicrystais And Piezoelectric Materials

The complex variable function method is one of the fundamental methodsto solve the fracture problems, the conformal mapping method is verypowerful in solving problems of complicated defects. By using the techniqueof conformal mapping, map the area in physics plane onto the interior (orexterior) of unit disk in phase plane. Thus reduces the calculation complexity.By using this method, the fracture problem of cycle symmetric in the classicelastic plane and new materials is investigated in this paper, mainly has thefollowing four parts.First, the infinite elastic plane contains two basic cycle symmetrical areasat least. The first fundamental problem of every basic cycle symmetrical areawith only one straight crack is considered. The limits of the stress at the cracktips are obtained. The stress intensity factors at the crack tips are also derived.It is found that the limits of the stress at the crack tips correspondingly inevery basic cycle symmetrical area are the same, and the stress intensityfactors are also identical.Second, by proposing a generalized conformal mapping and using thecomplex variable function method, the problem about a circular hole withasymmetry collinear cracks perpendicular to quasic-periodic direction inone-dimensional orthorhombic quasicrystals was considered. Complex representation of stress was derived in mathematical plane, and the solutionsof the SIFs at the crack tip were obtained. Under the conditions of limitation,the solutions of the SIFs at the crack tip of a circular hole with symmetricalcollinear cracks, a circular hole with one crack and the Griffith crack werefound out.Third, by using above-mentioned method in piezoelectric material, theproblem about a circular hole with asymmetry collinear cracks perpendicularto polarization direction in piezoelectric composite is investigated. Theanalytical solutions of the field intensity factors and mechanical strain energyrelease rate are presented with the assuption that the surtace of the crack iselectrically impermeable. Under the conditions of limitation, the solutions of acircular hole with symmetrical collinear cracks, a circular hole with one crackand the Griffith crack can be obtained. Numerical examples are thenconducted to analyze the influences of the length of cracks and radius on themechanical strain energy release rate and some useful conclusions are drawn.Fourth, by using the complex variable function method and introducing aconformal mapping, the problem about a circular hole with four cracks (a pairof asymmetric collinear cracks and a pair of symmetric collinear cracks) in aninfinite piezoelectric composite material is investigated under remotelyuniform in-plane electric and anti-plane mechanical loadings. The analyticalsolutions of the field intensity factors and the energy release rate are presentedby taking the effect of dielectric permittivity into consideration. Under the conditions of limitation, the solutions of a circular hole with four symmetriccracks, a symmetric cross cracks, mode-T crack and the Griffith crack can beobtained. Numerical examples are then conducted to analyze the influences ofthe geometric parameter of cracks on the energy release rate and thedimensionless intensity and some useful conclusions are drawn.