Multi-scale indentation model of stiff film-compliant substrate structures.

• A multi-scale indentation model (MSI) for the stiff film-compliant substrate structures is established. • Two dimensionless numbers g /(E s −1/3 E f 1/3 t f) and τ 0 /(E s 2/3 E f 1/3 t f) are proposed to evaluate the indentation size effect. • The predicted bending stiffness by the MSI model is consistent with the experimental results. As critical components or platforms, stiff film-compliant substrate structures are widely used in micro-electro-mechanical systems (MEMS) and other devices, in which their indentation behavior plays an important role. In this work, a multi-scale indentation model (MSI) for stiff film-compliant substrate structures is established based on the strain gradient theory and the surface elastic model. With this model, the indentation size effect of the structure is investigated from three aspects: the load-depth relationship, the distribution of bending moments in the film, and the surface morphology. The results show that the indentation size effect is mainly caused by the characteristic length and the surface residual stress of the film, and it is sensitive to the modulus ratio of the film to the substrate but has little relation with Poisson's ratio and the contact radius. Based on the above analysis, two dimensionless numbers are proposed to evaluate the indentation size effect of stiff film-compliant substrate structures. Besides, the predicted bending stiffness by the theoretical model is consistent with the experimental result, which verifies the effectiveness of the MSI model. Our research sheds light on the understanding of the indentation size effect of stiff film-compliant substrate structures and provides theoretical guidance for the design of related devices. [ABSTRACT FROM AUTHOR]