Sneddon model

The elasticity was quantitatively determined by analyzing the recorded force curves with the help of the Hertz model. The Hertz model describes the elastic deformation of two spherical surfaces touching imder the load, which was calculated theoretically in 1882 by Hertz. Other effects, such as adhesion or plastic deformation, were not included in this model. Sneddon extended the calculation to other geometries. For a cone pushing onto a flat sample, the relation between the indentation 5 and the loading force F can be expressed as ... 

The Hertz model, extended by Sneddon to other geometries provides a relation between load force F and the indentation 8 (with Young s modulus E, Poisson ration v, and the half opening angle of the indenting cone a). 

Fig. 4.24 Left AFM force curves at three different maximum loads on three homogeneous polymer samples PC, iPP, and a PPG rubber. Data from measurements with lower maximum loads are depicted with thicker lines. Right Comparison of the elastic moduli obtained by AFM nanoindentation and application of the Sneddon s elastic contact model [43], with macroscopic moduli obtained from bulk tensile testing. (Reprinted with permission from [43]. Copyright 2006. American Chemical Society)...

Analytical models of the aperture variation of a two-dimensional extension fracture subject to a constant overpressure in a homogeneous, isotropic rock show that the fracture opens into a flat ellipse (Sneddon Lowengrub 1969 Valko Econo-mides 1995). For layered rocks, numerical models of the aperture variation of vertical hydrofractures with fluid overpressure as the only loading indicate that the aperture of hydrofractures tends to be... 

As described in the previous section, including plasticity in the modeling of indentation contact is a complex problem and analytical solutions are not easily obtained [32], Fortunately, in most cases, at least the upper part of the unloading curve is elastic, leading to the following modified Sneddon s relation [40] ... 

Tsukruk and co-workers evaluated different classic models of elastic contacts, including Sneddon s, Hertz, and JKR model, to probe the micromechanical properties of elastic polymeric materials (253). A close agreement between absolute values of elastic moduli and bulk data has been observed. With typically available cantilevers with stiffnesses in the range of 0.1-50 N/m, materials moduli... 

Scanning force microscopy (SFM) was used for probing micromechanical properties of polymeric materials. Classic models of elastic contacts, Sneddon s, Hertzian, and JKR, were tested for polyisoprene rubbers, polyurethanes, polystyrene, and polyvinylchloride. Applicability of commercial cantilevers is analyzed and presented as a convenient plot for quick evaluation of optimal spring constants. We demonstrate that both Sneddon s and Hertzian elastic models gave consistent and reliable results, which are close to JKR solution. For all polymeric materials studied, correlation is observed between absolute values of elastic moduli determined by SFM and measured for bulk materials. For rubber, we obtained similar elastic modulus from tensile and compression SFM measurements. 

In the present communication, we report the results from studies of micromechanical properties on polymeric materials interpreted using classic theories of elastic contacts, Sneddon s, Hertzian, and Johnson-Kendall-Roberts (JKR). These models are tested for a set of polymeric materials with known Young s modulus, E, from 1 MPa to 3 GPa. Special attention is paid to the elucidation of applicability of different contact models and optimization of experimental probing procedures. 

After manipulation with Sneddon s equations for this model we obtain ... 

By combining optimal cantilever parameters and experimental conditions one can obtain reliable force distance data which is appropriate for further contact mechanics analysis for a wide selection of polymeric materials. Both Sneddon s and Hertzian models of elastic contact give consistent results in the range of indentation depth up to 100 nm. Close correlation is observed between elastic moduli determined by SFM in compression mode (approaching cycle) and measured values for bulk materials. As shown, for elastic materials force-distance curves can be used for evaluation of tensile elastic moduli from retracing cycle. For rubber material, the latest is in a good agreement with measuremente in compression mode. 

Sneddon in 1965 extended this work by solving for the relation between total applied load and penetration depth for a dass of problem where an axisymmetric Boussinesq punch of arbitrary profile is loaded on an dastic half-space. In 1965, its practical applications were for stainless steel punches of various shapes exerting vertical load on a planar surface. In particular, Sneddon solved the load versus penetration depth relation for a cone and a paraboloid of revolution exerting a normal force on a surface. The model introduced has since been adapted to look at nanoscale systems, since much of the physics still applies, and in the case of Sneddon s work, conical or paraboloidal punch geometry can suffidendy modd the tip interaction. 

This is the normal Hook s Law (1975) where, IQ is the spring constant or the cantilever stiffness. The Ah is the indentation depth, depends on the applied load F(Ah) on the materials. The Hertz model [10] and Sneddon s formula [11] give the relationship between [F(Ah) /Ah)] and composite elastic modulus of axisymmetric indenting tip and sample. 

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