This paper deals with the problem of local buckling caused by uniaxial stretching of an infinite plate with a circular hole or with circular inclusion made of another material. As the Young’s modulus of the inclusion approaches that of the plate, the critical load increases substantially. When these moduli coincide, stability loss is not possible. This paper also shows the difference between them when the inclusion is softer than the plate and when the inclusion is stiffer than the plate. Computational models show that instability modes are different both when the inclusion is softer than the plate and when the inclusion is stiffer than the plate. The case when plate and inclusion have the same modulus of elasticity, but different Poisson’s ratio is investigated too. It is also discussed here the case when a plate with inclusion is under biaxial tension. For each ratio of the modulus of elasticity of plate versus inclusion it’s obtained the range of the load parameters for which the loss of stability is impossible.