Study On Dislocation And Inclusion Problems,and Contact Problem In One-dimensional And Two-dimensional Quasicrystals

Since the discovery of icosahedral quasicrystals(QCs)in Al-Mn alloys,the physical and structural properties of QCs have attracted widespread attention.In the process of material manufacturing and development,defects are inevitable,and multiple defects often exist at the same time.Therefore,it is of great practical significance to study the interaction of multiple defects in QCs materials for revealing the fracture characteristics of materials with defects.Because of the multi-field coupling effect of one-dimensional(1D)piezoelectric QCs and two-dimensional(2D)QCs as well as the interaction of multiple defects coexisting,it is very difficult to solve these problems.Therefore,there are few related studies on these problems.In this thesis,the full field solutions of 1D hexagonal piezoelectric QCs with spheroidal inclusion,the anti-plane interference effect of 1D hexagonal piezoelectric QCs with a screw dislocation and an elliptical inclusion,the coupling interaction of a piezoelectric screw dislocation and a circular inclusion in 1D hexagonal piezoelectric QCs bi-materials,and the coupling interaction of an edge dislocation with decagonal QCs bi-materials containing a circular inclusion are studied.Meanwhile,two kinds of contact problems for 2D hexagonal QCs under the action of a rigid flat indenter are also studied.The main contents and results of the thesis are as follows:1.Based on the elastic theory of 1D piezoelectric QCs,the mechanical and electrical properties of 1D hexagonal piezoelectric QCs with a spheroidal inclusion are analyzed.By selecting the appropriate potential functions,the closed form expressions of phonon field stresses,phason field stresses and electric displacement in the matrix and inclusion are obtained under three kinds of far-field uniform mechanical and electrical loading conditions:axisymmetric loading,out-of-plane shear and in-plane shear.The results show that when the spheroidal inclusion reduces to a spheroidal void,only the electric field exists in the void,and the electric field is uniform.When the spheroidal inclusion reduces to a penny-shaped crack,the normal stresses of phonon field and phason field in the z-direction vary with the initial stresses and the crack size,while the electric displacement varies with the initial stresses,the QCs material constants and the crack size.The results are consistent with the classical elastic theory without considering the phason field.2.The anti-plane piezoelectric elastic equations of 1D hexagonal piezoelectric QCs are expressed by matrix.Using the complex analysis and conformal mapping technique,the anti-plane elastic interference effect between a piezoelectric screw dislocation and an elliptical inclusion in infinite matrix of 1D hexagonal piezoelectric QCs is investigated.The general solutions of the anti-plane problem expressed by series for that the dislocation is located in the matrix and the inclusion are obtained,respectively.The special cases such as without considering electric field,without considering dislocation,inclusion reduced to hole,and an elliptical inclusion reduced to a circular inclusion are discussed.The exact solutions of phonon field stresses,phason field stresses,electric fields and electric displacements are obtained.The results show that the phonon field stresses,phason field stresses,electric field intensity and electric displacement are uniform in the elliptical inclusion without considering the dislocation.When the elliptical inclusion reduces to a circular hole,the electric field intensity is not affected by dislocation,but only affected by the far-field equivalent electric field.3.Using conformal mapping and analytical continuation method,the problem of interaction between dislocation and inclusion in bi-materials is transformed into the boundary value problem of two sets of sectionally holomorphic function.The coupling interaction of a piezoelectric screw dislocation and a circular inclusion in 1D hexagonal piezoelectric QCs bi-materials is studied.Assuming that the circular inclusion is located in Material 2,the complex expressions of the corresponding fields that the dislocation is located in Material 1,Material 2 and circular inclusion are obtained,respectively.Three special cases:the same material constants of Material 1,Material 2 and circular inclusion,the same material constants of Material 1 and Material 2,and the same material constants of Material 2 and circular inclusion are analyzed and discussed,respectively,It is found that when the material constants of Material 1,Material 2 and circular inclusion are the same,the obtained results are consistent with those of 1D hexagonal piezoelectric QCs containing a screw dislocation.The results are consistent with the results of the interaction between a screw dislocation and a circular inclusion in 1D hexagonal piezoelectric QCs when the material constants of Material 1 and Material 2 are the same.It is considered that the material 2 and a circular inclusion have the same material constants,the results are consistent with those of the 1D hexagonal piezoelectric QCs bi-materials with a screw dislocation.4.Using the suitable complex potential solution of Green's function with four repeated roots,the complex expressions of the stresses and displacements for the plane problems of decagonal QCs are derived.The coupling interaction of a circular inclusion and an edge dislocation in decagonal QCs bi-materials is studied.Assuming that the circular inclusion is located in Material 2,the general solutions of the corresponding fields that the edge dislocation is located in Material 1,Material 2 and circular inclusion are obtained,respectively.Because the plane final governing equation of pentagonal and 2D hexagonal QCs is the same as that of decagonal QCs,all of them are a quadruple harmonic function.Therefore,this method can provide a theoretical reference for the analysis of the coupling between an edge dislocation and a circular inclusion in the above two kinds of 2D QCs bi-materials.5.By displacement potential function method,the plane final governing equations of 2D hexagonal QCs is reduced to a quadruple harmonic function.Using the complex potential solution of the quadruple harmonic function,the complex expressions of stress and displacement of 2D hexagonal QCs are given.The problems of the finite friction contact and half plane adhesive contact of 2D hexagonal QCs under the action of a rigid flat indenter are studied.The analytical expressions of stress component are obtained.The relationship between the normal load on the indenter and the penetration depth is analyzed.The penetration depth is only related to the normal load of the indenter.The dimensionless stress curves and stress nephogram are given to illustrate the correctness of the analysis method and the effectiveness of the solution.The study in this thesis will be helpful for further analysis and understanding of the interaction of multiple defects in 1D piezoelectric QCs and the coupling mechanism of phonon-phason field in 2D QCs.In addition,the conclusions of this thesis have guidance and reference value for the design and applied security of QCs materials and its structures.