Application Of Complex Function Method In Dynamic Inclusion Problem

With the development of advanced composite materials used in aerospace,ships,automobiles and other engineering fields,the study of macroscopic deformation and microscopic failure mechanism of materials has attracted increasingly intensity of attention.As well known,the presence of inclusions（holes,cracks,impurities,etc.）in materials affects or disturbs the elastic field at both local and global scale and thus greatly influences their mechanical and physical properties.Therefore,the inclusion problem has been widely studied because of its fundamental importance in micromechanics of composites.In the long history of studying classic static inclusions,plenty of results have been obtained concerning the shape effect of inclusions,the anisotropy of materials,and the boundary effect of matrices.On the contrary,the dynamic inclusion problems have rarely been visited because of the complexity and corresponding mathematical challenge.At present,the dynamic inclusion problems are often studied using the Green’s function method,and only limited cases are solved analytically,e.g.,homogeneous spherical,ellipsoidal inclusions,and homogeneous continuous fiber inclusions.The Green’s function method has been proved inefficient dealing with bounded matrices in inclusion problems,and therefore hindered the progress in studying dynamic inclusion problems.To overcome this inconvenience,this research attempts to establish a general method with the help of complex functions to find the analytical solution of two-dimensional domain containing a dynamic inclusion of an arbitrary shape.First,a preliminary discussion of the dynamic inclusion problem is carried out based on the information from numerical simulations,which also verifies some assumptions employed in research.Then,the complex potential method is utilized to solve the dynamic inclusion problems based on Sokhotski-Plemelj Theorems,interface continuities and boundary conditions.To deal with the shape effect of an inclusion,the real inclusion is mapped into a virtual unit circle via conformal mapping and the domain of the problem is significantly simplified.Alternatively,to handle a bounded matrix,the matrix is replaced by a conformal imaginary unit circle and the inclusions are approximated by polygons,since simply mapping the inclusion into a unit circle will bring in formidably complicated boundaries.In this research the application of complex potential method is extended to dynamic inclusion problems,and the analytical solutions are shown to be in accordance with their counterparts in static problems in the sense of static limit.In summary,it provides a new idea to handle the dynamic inclusion problems with arbitrary shapes of inclusions and boundaries of matrices,and furthermore,new solutions for many related physics and practical engineering problems.