Screw Dislocation Equations In A Thin Film And Surface Effects

Due to the large surface to volume ratio,the mechanical properties of thin films can be very different from the bulk counterparts.Even at small scale,the mechanical properties,such as plastic deformation,are dominated by dislocations as well.In this thesis,the fundamental equations of screw dislocations in a thin film are derived for arbitrary boundary conditions and the surface effects are studied quantitatively.According to the central idea of the Peierls-Nabarro(P-N)model,in order to derive the dislocation equation in a thin film,the first thing is to solve the equilibrium problem of the points on the boundary of a thin film,that is the relation between the displacement field on the boundary and the stresses on it.In elastic theory,the equilibrium of a thin film has not been solved yet.The equilibrium equations satisfied by the inner points and the points on the boundary are derived respectively by using the Green's function.And the latter is named as edge equation of a thin film.The displacement field of the inner points is determined by the distribution of displacement field and stress field on the surface.However,the displacement field and stress field on the surface are not independent,the relation between them satisfies the edge equation of a thin film.By using the Fourier transform,it is easy to derive the dislocation equation in wave vector space from the edge equation.The problem is how to find the corresponding dislocation equation in the real space.Once the dislocation equation in a thin film is derived,the stress field on the two sides of the slip plane can be obtained by solving the dislocation equation,and the full displacement field of a thin film with dislocations is determined by equilibrium equation of inner points in a thin film.The screw dislocation equations in a thin film are derived based on the P-N model,with arbitrary boundary conditions: free surface,fixed surface and gradient loading imposed on the surface.According to the modification of interaction between dislocations,boundary conditions can be essentially classified into two types: the stressed boundary(fixed stress on the surface)and the strained boundary(fixed strain on the surface).The interaction between dislocations decreases near stressed boundary and increases near strained boundary.The external stress(strain)field is separated as an independent superimposition with the modification of dislocation interaction.The new equations make it possible to study the modification in dislocation structure and other surface effects.Surface effects on a continuously distributed screw dislocation near a free surface and a fixed surface are quantitatively investigated respectively.The half-space model with a free surface can describe a thin film on substrate,and that with a fixed surface can describe the grain boundary.Free boundary condition is a typical representation of stressed boundary and fixed boundary condition is a typical representation of strained boundary.These two types of boundary have opposite modification on dislocations interaction.It shows that the dislocation becomes narrower near a free surface,and wider near a fixed surface,where the reduction and increase of dislocation width scales proportionally to the reciprocal of the distance between dislocation and surface.The interaction between a dislocation and the boundary includes a long-range image force and a short-range force caused by the variation of misfit energy.In conclusion,the surface effects on wide dislocations spread more deeply from the surface.Because of the two parallel free surfaces,the strain energy of dislocations in a free-standing film decreases further and the dislocation becomes narrower,compared to the dislocation near a free surface of a half-space solid.Correspondingly,as the film gets thinner,the Peierls stress becomes larger,which decreases the mobility of dislocations.The free-standing film is consistent with the supercell in atomistic simulations,which is helpful to directly compare the theoretical result and that of atomistic simulations,and clarify the surface effect on the Peierls stress.The interaction between the external stress field and displacement field on the surface and the dislocation in a thin film is given explicitly.It is realistic in physics to study the effect of surface external loading on the dislocation properties in a thin film.The ability of deformation of dislocations under external loading,which is named as the rigidity of dislocation,is examined by applying a local external stress on the surface.It shows that the shape change of wide dislocations is more obvious.