Spherical-tip indentation of viscoelastic material
Introduction
Indentation of a viscoelastic half-space is often used where the constitutive properties of small volumes are required as in a thin coating. Studies on this issue have been concerned with different indenters whose surface profiles are, for example, spherical (Lee and Radok, 1960; Hunter, 1960), arbitrary quadratic (Yang, 1966), or arbitrary smooth axisymmetric (Ting, 1966). The case of indentation by a flat-ended cylindrical punch has been briefly discussed by Hunter (1967) and Findley et al. (1976). However, none of these papers have addressed the particular models and loading configuration needed to interpret nanoindentation tests.
Nanoindentation and microindentation techniques have proved to be efficient tools for probing mechanical properties of coatings on substrates. These techniques have drawn great interest (Cheng, 1996; de Boer and Gerberich, 1996a, de Boer and Gerberich, 1996b; Gerberich et al., 1999) in a wide range of coating industries. Recently, a flat-punch indentation on viscoelastic material has been addressed in detail by Cheng et al. (2000). The analytical solutions in this case have been derived under both creep and load relaxation conditions and can be used to extract viscoelastic material properties from indentation experiments. However, a good alignment between the indenter and material surface is necessary, which thus requires a careful and accurate experimental process. On the other hand, a spherical-tip indenter is less sensitive to alignment and thus would ease the work in both experimental preparation and conduct. Therefore, the problem that we are motivated to solve here is the response to a rigid spherical-tip indenter that is pressed into a viscoelastic semi-infinite solid (Fig. 1). The results find application in probing viscoelastic properties of solidifying/solidified polymeric coatings with nanoindentation or microindentation tests.
The purpose of this paper is to find a viscoelastic solution that can be employed directly in indentation tests for a spherical-tip indenter pressed against a semi-infinite medium under load relaxation and creep conditions. The material is considered as a linear viscoelastic solid and described by a three-element model, i.e., a standard solid model in viscoelasticity. This is a useful approximation to many polymeric materials (Meares, 1965; Ferry, 1980; Sperling, 1986). The solutions presented below are valid not only for incompressible but also for compressible materials. The results include convenient closed-form formulas for approximating both relaxation testing and creep testing. They are important to development of indentation testing as a routine approach.
Section snippets
Method of functional equations
Viscoelastic materials exhibit a complicated time-dependent behavior, including instantaneous elasticity, delayed elasticity, and viscous flow. One way to solve a viscoelastic problem is to remove the time variable in the governing equations and in the boundary conditions by employing the Laplace transformation with respect to time (Lee, 1955). The viscoelastic problem then reduces to an elastic problem called an associated elastic problem. From the solution of the elastic problem, the desired
Results and discussion
The experimental data from relaxation and creep tests on bulk polystyrene (PS) and the analytical results from Eqs. , with the parameters fit to those data are shown in Fig. 8. From Eq. (31) or Eq. (43), the initial values of the parameters E1 and E2 were evaluated at the very start and end of the relaxation or creep tests and η was evaluated at an intermediate stage. The final values were decided through a nonlinear curve fitting based on the initial values. It can be seen that the data fit
Conclusions
Linear viscoelastic analytical solutions of indentation on a semi-infinite solid with a spherical-tip indenter developed by the method of functional equations have been presented in this paper. The derived equations are applied to polymeric materials, which can be described appropriately by the standard three-element viscoelastic model, for both the load relaxation test and creep test. And they can be applied to the response of compressible as well as incompressible coated layers. These
Acknowledgements
The authors would like to gratefully acknowledge the support from Dow Corning and Kodak Corporations, the Center for Interfacial Engineering at the University of Minnesota, and Grant DE-FG02-96ER 45574 for WWG for this research. The computer resources and technical support provided by the University of Minnesota Supercomputing Institute is also gratefully acknowledged.
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