Bound Solution for Residual Stress Due to Thermal Mismatch in Particle Reinforced Composites

The residual stress in the particle reinforced composite (both phases isotropic) is studied by both the self-consistent method (SCM) and the generalized self-consistent method (GSCM). Two models of GSCM are employed: Model I is the inclusion phase surrounded by the matrix phase, which is embedded in the effective medium, while in Model II the positions of the inclusion and matrix are interchanged. The results are summarized as follows: (1) SCM yields the residual stress in between the bounds of the residual stress given by the GSCM in the present study. (2) The residual stress from the well known modified equivalent inclusion method (MEIM) and composite sphere model (CSM) reduces to the lower bound of the GSCM (i.e., Model I in the present study) if the inclusion is stiffer than the matrix. If the matrix is stiffer, Model II results in the lower bound, which is equal to the result by MEIM and CSM. (3) The result by Kreher, who expressed the residual stress in terms of the effective bulk modulus (in addition to the elastic properties of the inclusion and matrix) can be reduced to the present bound solutions by GSCM if the bound for the effective bulk modulus is used for the Kreher's work and a proper choice of Model I or Model II of GSCM with a proper choice of the bound for the effective bulk modulus is made.