Revisit of the two-dimensional indentation deformation of an elastic half-space
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Recently, there is a growing interest in two-dimensional (2D) plane indentation as an imprinting method for creating nanostructures. There is also a strong interest in using 2D flat-ended, wedge, and cylindrical indenters for characterizing mechanical properties of materials. In either case the knowledge of load versus displacement of the indenter is important. However, there has been some confusion about the load–displacement relationships for 2D indentation in the literature. Concerning this confusion on the relationship between the indentation load and the indentation depth for 2D elastic indentations, the symmetric indentation of an elastic half-space is studied. Parameters are introduced in determining the semianalytical relation between the indentation load and the indentation depth for flat-ended indenters and in determining the dependence of the indentation depth on the contact size for non-flat-ended indenters. The indentation load is proportional to the indentation depth for the indentation by flat-ended indenters and is a parabolic function of the indentation depth to the first order of approximation for non-flat-ended indenters including the wedge and cylindrical indenters.
I. INTRODUCTION
The study of contact mechanics of two solids has led to the development of a sharp-instrumented indentation technique for determining mechanical properties of thin films and materials of small volume. The principle of the indentation technique relies on the Hertzian contact theory1 and the load–displacement relation for the contact deformation between a rigid, axisymmetrical indenter and an elastic half-space.2 In the last decade, there are numerous studies toward establishing the load– displacement relationship for the contact deformation of multilayer structures and nonuniform elastic materials. Along with the applications of three-dimensional, sharp-instrumented indentation, there are interests in using two-dimensional (2D) line contact between a rigid cylinder and a material to characterize mechanical properties of materials and to create surface morphology of materials such as nanoimprinting.3–5 Although the 2D contact problems have been well studied and summarized by Muskhelishvili6 and a general relation in an integral form is given between the indentation load and
the contact size, there is a lack of analytical and semianalytical relations between the indentation load and the indentation depth for symmetrical indenters. Confusion has arisen in the indentation community. For example, the equation used by Muto and Sakai,7 which was derived by Kunert in 1961,8 diverges when the length of a cylindrical indenter becomes infinite. Giannakopoulos9 stated that “Regarding the plane-strain wedge indentation results, the force–depth relation is indeterminate.” The integral of the load–displacement relation given by Landau and Lifshitz10 also becomes divergent for 2D indentation problems. The purpose of this work is to revisit 2D contact problems and seek semianalytical solutions that can be used for 2D-indentation cha
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