| Inhomogeneity in materials is a hot topic continuously discussed in solid mechanics.The existence of cavities and rigid inclusions interrupts the continuity of materials and inevitably affects the physical and mechanical properties of substrates.Especially for functional materials with brittleness or low mechanical strength,the perturbation field and stress concentration caused by such inclusions have received extensive attention.Therefore,the cavity and rigid inclusion problems,as the extreme cases of the second Eshelby’s problem,are often regarded as the forerunners in the analytical study of inhomogeneity problems.In this paper,the multi-field coupling problems of thermoelastic material,thermoelectric material and soft ferromagnetic material with a smooth cavity or rigid inclusion are studied analytically.The analytical solutions are given using conformal mapping and plane elastic complex variables theory.Explicit expressions including Kolosov-Muskhelishvili(K-M)potentials and rigid-body displacement are solved by a novel tactic of non-positive power series expansion.Compared with the previous reports,the new solutions of K-M potentials can be expressed as compact form with finite terms when the shape of cavity or rigid inclusion is characterized by the finiteterm Laurent polynomial,and the possible rigid-body displacement of the inclusion relative to the matrix is introduced to make the formulation of the problem more complete.In the discussion on rigid-body displacement,flux and stress distribution for different coupling problems,the following main conclusions are drawn:(1)It is necessary to consider the rigid-body displacement.The contribution of rigid-body displacement to deformation coordination and interface force balance cannot be ignored.Among them,rigid-body translation ensures the boundary condition is precisely satisfied,while the purpose of rigid-body rotation is to make the interface resultant moment vanish.In addition,the expression of rigid-body rotation is coupled with K-M potentials so that it affects the stress field.(2)The appearance of rigid-body displacement is closely related to shape parameters,loading type and direction.In the case of small rotation,the rigid-body displacement is distributed symmetrically along the loading direction where the maximum value is obtained,and this symmetry is broken when the finite rotation is considered.(3)Curvature,as a mathematical tool to describe the boundary contour of inclusion,can be used as a reference to judge the severity of flux and stress concentration.As the loading direction changes,the maximum concentration severity at the tip of polygons with the same area is consistent with the maximum curvature relationship between them,and the maximum flux and stress are also proportional to their curvature relations for any two points on the boundary.(4)The maximum flux and stress only appear at the maximum curvature point in a few special loading directions,but in most cases appear near the maximum curvature point,and there is a slight deviation from it.On the one hand,the work developed in this paper provides a more complete mathematical method for solving the problems of cavity and rigid inclusion in microscopic mechanics,and obtains a compact and accurate new form analytical solution for multi-physics coupling problems.On the other hand,it also provides valuable theoretical results for the practical application of materials,some interesting discoveries related to mechanical properties such as stress concentration and rigid-body displacement are of reference significance to the performance optimization and reliability evaluation of composite materials and the design of multifunctional components. |
