Study On Some Anti-plane Problems Of Nano Defects

When the defect size in material is reached to nanometer level,the influence of surface effect to the mechanical properties of materials should not be neglected.But with the increase of defect size gradually,the influence of surface effect would be weakened,so these solutions containing surface effect gradually tend to the solutions without surface effect,namely the classical elastic theory solutions.This article se-lects some anti-plane problems of nano defects as the research object,improves the method in the existing relevant approximate study,and establishes a new method that the exact solutions of these problems are obtained.The main research contents and innovation points are as follows:1.The anti-plane problem of the interaction between a dislocation and a linear nano crack in ordinary elastic solids is investigated.Firstly,based on Gurtin-Murdoch surface/interface theory,the boundary model with surface effect is established.Se-condly,the problem of straight line crack is turned into a unit circle problem by using conformal mapping method,the exact solutions of this problem are obtained via Cauchy integral theorem,and the stress field and image force with and without sur-face effect are formulated.Thirdly,the stress field and image force with surface effect which are obtained compare with the approximate solutions previous solving,which shows the accuracy and advantage of the new method.The numerical results show that when the line crack size is reduced to the nanometer level,surface effect will se-riously affect the numerical value of stress field and image force near the crack tip.The smaller crack size is,the stronger surface effect is,and the greater the numerical changes of stress field and image force are.The larger crack size is,the weaker sur-face effect is,the smaller the numerical changes of stress field and image force are,and finally which can be neglected,namely that are consistent with the classical elas-tic theory solutions.2.The anti-plane problem of the interaction between a piezoelectric dislocation and a linear nano crack is studied.Firstly,this paper introduces the basic equations of piezoelectric material antiplane problems.Then,piezoelectric boundary model that contains residual stress is built,and according to the solving method of the second chapter,the analytic expressions of stress field,electric displacement field and image force which consider or neglect surface effect are derived.The research is shown that the influence of surface effect is depended on the size of linear nano crack in piezoe-lectric materials.Namely that the smaller crack size is,the stronger surface effect is,and the larger crack size is,the weaker surface effect is,and until which can be neg-lected.3.The anti-plane problem of the interaction between a dislocation and a linear nano crack in quasicrystals is analyzed.Firstly,the Gurtin-Murdoch surface/interface theory which be applied to ordinary material successfully be applied to nano crack problems of quasicrystal material,and the boundary model with surface stresses of phonon field and phason field is established.Then,by using conformal mapping and Cauchy integral,the closed-form solution is obtained.Finally,the analytical expres-sions of stress field and image force with or without surface effect are obtained.The results show that when the line crack size in quasicrystal materials is reduced to the nanometer level,surface effect will seriously affects the numerical value of stress field and image force of phonon field and phason field,and further changes mechan-ical properties of quasicrystal materials.Namely that the smaller crack size is,the stronger surface effect is,the greater the numerical changes of stress field and image force of phonon field and phason field,and vice versa.4.The anti-plane problem of the interaction between a dislocation and a nano el-liptical hole in ordinary elastic solids is investigated.Firstly,based on Gur-tin-Murdoch surface/interface theory,the boundary model with surface effect is es-tablished.Secondly,by using conformal mapping method and Cauchy integral theo-rem of complex function,the exact solutions of this problem were obtained.Thirdly,the stress field and image force with and without surface effect were formulated.The research results show that the influence of surface effect is depended on the size of nano elliptical hole.Namely that the smaller crack size is,the stronger surface effect is,and the greater the numerical changes of stress field and image force with surface effect are;the larger crack size is,the weaker surface effect is,the smaller the nu-merical changes of stress field and image force with surface effect are,and until which can be neglected and approximately equal the classical elastic theory solutions.