Master curves for Hertzian indentation on coating/substrate systems
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Pedro Miranda Departamento Electro´nica e Ingenierı´a Electromeca´nica, Escuela de Ingenierı´as Industriales, Universidad de Extremadura, 06071 Badajoz, Spain (Received 13 June 2003; accepted 12 September 2003)
An analytical model was developed to derive an approximate closed-form solution for indenter displacement when a rigid spherical indenter is pressed onto a coating/substrate system. Finite element analyses were also performed to verify the analytical solution. The results showed that the solution could be obtained from the analytical expression for Hertzian indentation on a homogeneous semi-infinite elastic medium multiplied by a modification term. This modification term is a function of two ratios: (i) Young’s modulus ratio between the coating and the substrate and (ii) the ratio between the coating thickness and the contact radius. Based on this modification term, master curves for Hertzian indentation on coating/substrate systems were plotted.
I. INTRODUCTION
In 1882, Hertz published for the first time a solution for the problem of frictionless contact between homogeneous elastic bodies under normal loading.1 Subsequently, progress in the field of contact mechanics has resulted in a considerable amount of literature on the stress analysis of homogeneous elastic bodies subjected to various loading and contact conditions.2 The elastic contact problem of a sphere and a homogeneous semiinfinite medium is now well understood, and indentation of materials with spherical indenters has extensively been used to characterize the mechanical properties (e.g., Young’s moduli) of bulk materials.2–5 However, nowadays monolithic materials are being replaced by coating/ substrate (or multilayer) systems in many microelectronic, optical, and structural applications. Therefore, it is essential to understand Hertzian contact in coating/ substrate systems to characterize their mechanical properties using the indentation technique. Considerable efforts have been devoted to analyzing indentation on coating/substrate systems since the early 1950s. These efforts include the finite element analysis (FEA),6–8 analytical modeling,9–12 and the empirical approach.13 The FEA provides a powerful means to analyze the contact problem in nonuniform systems; however, it suffers from the drawback that it is a case-by-case study, and computation needs to be performed for each change in geometrical parameters and material properties. On the other hand, although analytical modeling starts with analytical expressions of constitutive equations and boundary conditions for indentation on coating/substrate 94
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J. Mater. Res., Vol. 19, No. 1, Jan 2004 Downloaded: 24 Oct 2015
systems, it results in complex integral equations relating the unknown pressure to the displacement over the contact area, and numerical procedures are required to solve these equations which, in turn, result in an obstacle for its practical applications. Finally, for the empirical approach, the most representative work was performed by Waters,1
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