Studies On Some Basic Indentation Problems Of Elastic Materials

In the past decades,advanced materials including thin films,coatings,microelectromechanical systems(MEMS),nanostructured materials,and functionally graded materials are used increasingly.The conventional mechanical tests become ineffective in characterizing the mechanical properties of these advanced materials due to the restriction of the size of specimen and testing equipment.Instrumented indentation,also known as nanoindentation or depth-sensing indentation has been proved successful in the characterization of these advanced materials.Instrumented indentation allows the continuous measurement of load and displacement over a complete loading cycle.Additionally,extremely small applied loads and displacements resolutions of instrument allow the instrumented indentation to be used to characterize micro/nano-scaled materials.Elastic modulus and hardness can be determined from the analysis of loading-unloading curves.Theoretical studies of contact problems play an important role in understanding the mechanism of indentation testing and measurement method of indentation.An alternative method is proposed to solve the problem of a finite-length elastic thin beam indented by a rigid cylindrical indenter(or an elastic thin circular plate indented by a sphere).The present method is based on a Kerr-type model which offers a simple differential relation between the applied pressure and the normal deflection of the pressured surface of elastic beam/plate.It is proved that the Kerr-type differential relation holds both inside and outside the contact zone and makes it possible to analyze both zones by a single governing differential equation.The main works of this thesis are as follows:(1)The axisymmetric and asymmetric indentation problems of an elastic thin beam are studied based on the Kerr-type model.Contact pressure distribution inside the contact zone,the deflection outside the contact zone and the load-displacement relation are obtained in explicit form and illustrated with numerical examples.It is confirmed that the indenter loses contact with the beam in the center and the contact zone becomes two separate symmetric strips when the half width of contact-zone becomes large.The critical contact-zone width and critical loading are investigated.Contact behaviors after separation are illustrated with numerical examples.(2)Similar to the study of an elastic beam,axisymmetric indentation problem of an elastic thin plate are solved based on the Kerr-type differential relation.Explicit relations between the contact-zone radius,indentation displacement and indentation load are derived for different boundary conditions.The separation of the indenter and elastic plate inside the contact zone occurs for relatively large contact radius.(3)Indentation of an elastic soft beam with considering the effect of surface tension is studied,and governing relation between the pressure and deflection of the upper surface of the elastic soft beam are derived based on the Kerr model.Then,contact pressure distribution inside the contact zone,“load-displacement” relation and “contact-zone width-displacement” relation are investigated and illustrated by numerical examples.(4)For the indentation of an anisotropic half-plane or an anisotropic piezoelectric medium with a parabolic boundary,the solutions based on the complex variable method are derived.The main innovative features of the thesis are listed below:(1)Based on the Kerr-type model,a simple method is proposed to solve the indentation problem of an elastic thin beam/plate.“Load-displacement” relation is given in the explicit form.It is proved that the indenter will lose contact with the beam/plate inside the contact zone as the contact zone becomes large.(2)For asymmetric indentation for a supported elastic beam,the two ends of the contact zone are determined as part of the solution.The geometrical non-symmetry of the contact zone is detailed studied.(3)In the context of the Gurtin-Murdoch model,the governing differential relation between applied pressure and normal deflection of the upper surface of an elastic soft beam with surface tension based on Kerr model is derived.Then,the contact pressure distribution inside the contact zone and “load-displacement” relation are determined using beam boundary conditions and continuous conditions at the contact edge.(4)Based on the complex variable method,the elastic field in the anisotropic elastic half-plane indented by a flat/cylindrical indenter is studied.(5)The two-dimensional Green’s functions for an anisotropic piezoelectric medium with a parabolic boundary are derived based on the conformal mapping method and the analytical continuation principle.