Analyzing contact problem between a functionally graded plate of finite dimensions and a rigid spherical indenter

• The contact law in an FG plate of finite dimensions is different from Hertz's contact law.
• The contact law is a function of ceramic to metal ratio of moduli of elasticity.
• The contact law is independent of the microstructural interactions of the constituting phases.
• The location of the maximum compressive stress depends on the value of ceramic to metal ratio of moduli of elasticity.
• The plate span does not affect the contact law.

Elastic contact of a functionally graded plate of finite dimensions with continuous variation of material properties and a rigid spherical indenter is studied. The plate is consisted of a ductile (metal) phase at the lower and a brittle (ceramic) phase at the upper surface. The punch acts on the upper surface which is the ceramic richer section of the plate. The contact problem in functionally graded (FG) structures has been studied widely; in such problems the main focus has been on FG structures with infinite dimensions where Hertzian or modified Hertzian contact laws can properly predict the contact parameters such as the size of the contact region and the pressure distribution under the punch. However, due to the finite dimensions of the considered plate in this study, the contact problem needs to be reconsidered. While Hertz's contact law predicts a power equal to 1.5 for the force indentation relation, the results of this study show that for an FG plate the exponent of the contact law depends on the brittle to ductile phases ratio of moduli of elasticity and material properties distribution. In cases in which the brittle phase has a lower modulus of elasticity compared to the ductile phase, the contact law exponent is independent of material properties distribution. In addition, in such cases the maximum compressive contact stress is located directly on the upper surface of the plate. On the other hand, in cases in which the brittle phase is stiffer than the metal phase, the exponent of the contact law is a function of material properties distribution and the location of the maximum compressive contact stress is beneath the upper surface. In addition, in general the contact parameters are independent from the microstructural interactions of the constituting phases. Since several numerical examples are examined here, these findings can be interpreted as the most general rules in the contact problem between an FG plate and a rigid sphere.