Three-Dimensional Thermo-Elastic Solutions Of Elliptical Crack Problem In The One-Dimensional And Two-Dimensional Hexagonal Quasi-Crystals

Quasi-crystal is a solid structure between crystal and amorphous.Compared with traditional materials,due to its unique quasi-periodic structure,it shows many excellent properties,such as low adhesion,poor thermal conductivity,high use intensity,strong resistance to high temperature,etc.,and has a wide application prospect in many scientific and technological fields.However,the brittleness of quasicrystals at room temperature limits their application as structural materials to a large extent.So it is of great scientific value and practical significance to study the crack of quasi-crystals.In this paper,the elliptic plane crack,quasi-crystal fracture and contact are studied by using the generalized potential theory method under the framework of one-dimensional and two-dimensional quasi-crystals thermoelasticity.The analytical solutions obtained will provide theoretical guidance for the application of infrared thermal imaging technology in non-destructive testing of quasi-crystals.The specific research contents are as follows(1)The elliptical crack in one-dimensional hexagonal quasi-crystals is analytically studied.The three groups of uniform generalized loads are applied to the upper surface and lower surface of the cracking plane,including phonon force,phason force and temperature load.According to the symmetry of the cracking plane,the problem is transformed into a mixed boundary value problem in half space.By using the generalized potential theory method and the corresponding general solution,the mixed boundary value problem is solved,and the governing equation expressed as potential function is derived.The thermo-elastic coupling field in the whole three-dimensional space and the exact analytical solution are obtained.In addition,some important physical parameters on the crack plane,including temperature,crack surface displacement,normal stress and stress intensity factor,are given in the form of closed solutions.Some numerical examples are given to verify the correctness of the analytical solution and analyze the distribution of the thermos-elastic coupling field around the crack.The numerical results show that the phason field has a significant effect on the stress intensity factor.(2)The elliptical plane crack problem is studied in the framework of two-dimensional hexagonal quasicrystal thermo-elasticity.The boundary integro-differential equations are derived by methods of generalized potential theory based on three dimensional statics general solutions.By setting suitable potential functions,the basic solutions expressed by elementary functions are derived.For the elliptical crack under temperature loading,the corresponding basic thermo-elastic field variables are obtained precisely by using elementary functions.In addition,the important physical parameters in fracture mechanics,such as crack surface displacement,normal stress and stress intensity factor,are also given.By numerical calculation,the validity of analytical solutions for three-dimensional field variables is discussed,and the coupling effects of thermo-phonon-phason field are characterized.The influence of phonon field and phason field on thermo-elastic coupling field in twodimensional hexagonal quasi-crystals is explained.(3)The analytical solutions of the elliptical crack problem under temperature loading of one-dimensional and two-dimensional hexagonal quasi-crystals are verified numerically.The elliptical cracks are degenerated into circular cracks and compared with the existing literature to verify the validity of the analytical solutions.The contour lines of the dimensionless temperature,the radial displacement,the vertical displacement,the normal stress and the normal stress of the phason field on the crack surface are given.The variation of the dimensionless stress intensity factor with polar angle in phonon field and phason field is plotted.The influence of different ellipse shape on stress intensity factor is discussed.