Abstract The tip of a sharp material inclusion (SMI) embedded in a parent material is considered as the singular stress concentrator. The SMI tip is modelled as a special case of a multi-material junction. The singular stress behaviour is caused by the material mismatch and geometrical discontinuity. The power of singularity is lower in comparison to the case of a crack. The stress field can be described by the asymptotic stress series, in which each term contains generalized stress intensity factor and stress terms exponent. The exponents are determined as a solution of eigenvalue problem. The factors are calculated by combination of analytical and numerical approaches. The terms can be either singular or non-singular depending on the stress term exponent. When approaching the concentrator, the singular terms become unbounded while the non-singular terms vanish. The non-singular terms increase precision of stress description on larger distances from the point of singularity. In some cases they provide the only means to describe the stress field well. Because of the singular stress behaviour near SMI tip, this location is prone to crack initiation. The crack initiation conditions are calculated by the average strain energy density factor (SEDF) criterion. Contrary to the case of a crack, the direction of minimum of SEDF changes with distance from the singular point. Therefore, an averaged value over specific distance is used. If the specific distance is relatively large, the employment of non-singular terms in multi-parameter criterion can significantly improve the critical parameters prediction. Thanks to that e.g. the particle composite design can be optimized.