Reconstruction of a 2D stress field around the tip of a sharp material inclusion

Abstract

The stress distribution in the vicinity of a sharp material inclusion (SMI) tip exhibits a singular stress behavior. The strength of the stress singularity depends on material properties and geometry. The SMI is a special case of a general singular stress concentrator (GSSC). The stress field near a GSSC can be analytically described by means of Muskhelishvili plane elasticity based on complex variable function methods. Parameters necessary for the description are the exponents of singularity and generalized stress intensity factors (GSIFs). The stress field in the closest vicinity of an SMI tip is thus characterized by 1 or 2 singular exponents (-1), for which 0<Re()<1, and corresponding GSIFs. In order to describe a stress field further away from an SMI tip, the non-singular exponents, 1<Re(), and factors corresponding to these non-singular exponents have to be taken into account. For given boundary conditions of the SMI, the exponents are calculated as an eigenvalue problem. Then, by formation of corresponding eigenvectors, the stress or displacement angular functions for each stress or displacement series term are constructed. The contribution of each stress or displacement series term function to the total stress and displacement field is given by the corresponding GSIF. The GSIFs are calculated by the over deterministic method (ODM), which finds a solution of an over-determined system of linear equations by the least squares method. On the left-hand side of the system are the displacement series term functions multiplied by unknown GSIFs, while the right-hand is formed by results of finite element analysis (FEA). Thus the results of FEA, namely nodal displacements in the radial and tangential direction, are employed in order to obtain the GSIFs. In the numerical example, the stress field for particular bi-material configurations and geometries is reconstructed using i) singular terms only ii) singular and non-singular terms. The reconstructed stress field polar plots are compared with FEA results.The stress distribution in the vicinity of a sharp material inclusion (SMI) tip exhibits a singular stress behavior. The strength of the stress singularity depends on material properties and geometry. The SMI is a special case of a general singular stress concentrator (GSSC). The stress field near a GSSC can be analytically described by means of Muskhelishvili plane elasticity based on complex variable function methods. Parameters necessary for the description are the exponents of singularity and generalized stress intensity factors (GSIFs). The stress field in the closest vicinity of an SMI tip is thus characterized by 1 or 2 singular exponents (-1), for which 0<Re()<1, and corresponding GSIFs. In order to describe a stress field further away from an SMI tip, the non-singular exponents, 1<Re(), and factors corresponding to these non-singular exponents have to be taken into account. For given boundary conditions of the SMI, the exponents are calculated as an eigenvalue problem. Then, by formation of corresponding eigenvectors, the stress or displacement angular functions for each stress or displacement series term are constructed. The contribution of each stress or displacement series term function to the total stress and displacement field is given by the corresponding GSIF. The GSIFs are calculated by the over deterministic method (ODM), which finds a solution of an over-determined system of linear equations by the least squares method. On the left-hand side of the system are the displacement series term functions multiplied by unknown GSIFs, while the right-hand is formed by results of finite element analysis (FEA). Thus the results of FEA, namely nodal displacements in the radial and tangential direction, are employed in order to obtain the GSIFs. In the numerical example, the stress field for particular bi-material configurations and geometries is reconstructed using i) singular terms only ii) singular and non-singular terms. The reconstructed stress field polar plots are compared with FEA results.