Indentation flat punch

Fig. 5 Maximum indentation force and standard deviation as a function of the number of measured silica model aggregates (Xprimary = 50 nm) using a flat punch indenter geometry [39]...

For a cylindrical flat punch indenting an elastic half-space with the flat side parallel to the plane of the half-space, the contact area is simply given by na, where a is the radius of the cylinder. For an applied load P, the indentation 8 (i.e., the change in distance of the center points beyond the first contact of the bodies) of the punch becomes... 

Here, E and v are the Young s modulus and Poisson s ratio of the half space material. The indentation of a flat punch on an elastic half-space is proportional to the applied load. The contact area stays constant independent of load. We obtain a linear relation between load and displacement with an effective spring constant of E a/(l — v ) = E a. The vertical surface displacement Az(r) of the elastic half-space is given by... 

The indentation force increases with a power of 3/2 with indentation depth. The contact area jta increases as and the mean contact pressure Fi/na increases with applied load as In contrast to the flat punch case, the contact does no longer act as a linear spring since Fl is proportional to due to the fact that the contact area... 

Van Landigham et al. reviewed nanoindentation of polymers, [40, 41] including a summary of the most common analyses of load-indentation data. Chief among these methods is an analysis of indentation load-penetration curves according to the Oliver-Pharr method. [42] This method is based on relationships developed by Sneddon for the penetration of a flat elastic half space by different probes with particular axisymmetric shapes (e.g., aflat-ended cylindrical punch, a paraboloid of revolution, or a cone) [43], More recently, Withers and Aston discussed indentation in the context of plasticity and viscoelasticity [44]. 

A very simple explanation of the effect of notching has been given by Orowan [95], For a deep, symmetrical tensile notch, the distribution of stress is identical to that for a flat frictionless punch indenting a plate under conditions of plane strain [102] (Figure 12.31). The compressive stress on the punch required to produce plastic deformation can be shown to be (2 + 7t)K, where K is the shear yield stress. For the Tresca yield criterion the value is l.Sloy and for the von Mises yield criterion the value is 2.82oy, where 0 is the tensile yield stress. Hence for an ideally deep and sharp notch in an infinite solid the plastic constraint raises the yield stress to a value of approximately 2>Oy which leads to the following classification for brittle-ductile behaviour first proposed by Orowan [95] ... 

In 1885 Joseph Boussinesq (6), trying to extend the validity of these results to the case of axi-symetrical rigid convex punches indenting a flat semi-infinite elastic medium, demonstrates that, without an adequate boundary comlition, the size of the contact area is generally unknown. In ordo to overcome this difficulty, he imposes that normal stresses vanish on the border of the contact area. In other words, the profile of the distorted medium must be tangent to the surfiice of the punch on the border of the contact area. Note that this condition is the same as the condition presupposed by the Hertz s theory. With this assumption, the size oh of the contact area and the penetration depth 5h are completely defined (Figure 1). 

The subj t of adhesive contact mechanics may be said to have started when Kendall (//), solving the problem of the adhesive contact of a rigid flat cylinder punch indenting the smooth plane surface of an elastic medium, demonstrated that the border of the contact area can be considered as a crack tip. The more complex problem of a spherical punch was solved in 1971 by Johnson, KendaU and Roberts (72). The JKR theory predicts the existence of contact area greater then that ven by the elastic contact Hertz s theory. The molecular attractive forces are responsible for this increase and, even in the absence of external compressive loading, the contact area has a finite size. Separating the two solids requires the application of an adherence force despite the existence of infinite normal stresses in the border of the contact area. 

Fig. I. Flat, cylindrical punch indenting a compliant layer of arbitrary thickness.

Figure 8.3 Indentation of an elastic half-space by a flat cylindrical punch with contact radius a.

The most basic configuration for an elastic contact is the indentation of an elastic halfspace by a rigid axisymmetric frictionless punch. Frictionless means that we assume that no shear stress can develop between the punch and the half-space. While historically the first solution of such a problem was given by Hertz for the case of a spherical indenter [846], we will start with a flat rigid cylindrical punch (Figure 8.3) that was first worked out by Boussinesq in 1885 [847] and solved in all details by Sneddon in 1946 [848]. 

This is twice the value of that for a flat cylindrical punch with the same contact radius a. The elastic energy (Je stored for a given indentation 6 can be calculated from... 

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