Indentor spherical

Often, Hertz s work [27] is presented in a very simple form as the solution to the problem of a compliant spherical indentor against a rigid planar substrate. The assumption of the modeling make it clear that this solution is the same as the model of a rigid sphere pressed against a compliant planar substrate. In these cases, the contact radius a is related to the radius of the indentor R, the modulus E, and the Poisson s ratio v of the non-rigid material, and the compressive load P by... 

Implicit in all these solutions is the fact that, when two spherical indentors are made to approach one another, the resulting deformed surface is also spherical and is intermediate in curvature between the shape of the two surfaces. Hertz [27] recognized this concept and used it in the development of his theory, yet the concept is a natural consequence of the superposition method based on Boussinesq and Cerutti s formalisms for integration of points loads. A corollary to this concept is that the displacements are additive so that the compliances can be added for materials of differing elastic properties producing the following expressions common to many solutions... 

Spherical indentor deformability tests of composites with PE synthesized on different fillers have shown [164] that the deformation of PFCM containing high percentages of filler (and polymer concentrations of less than 10% by mass) is substantially elastic and the specimen recovers completely after release of the load. As the polymer content increased to 60% by mass considerable residual deforma-... 

Fig. 2 Typical contact geometries used for scratch testing. The average strain is proportional a to tan9 in the case of conical or pyramidal indentors and b to a/R in the case of spherical indentors...

Figure 16.5. Spherical indentor of radius R. Boundary conditions are time-dependent, and the correspondence principle is not applicable.

SC Hunter. The Hertz problem for a rigid spherical indentor and a viscoelastic half-space. J Mech Phys Solids 8 219, 1960. 

As the rubber hardness is a measurement of almost completely elastic deformation, it could be expected that attempts would be made to relate hardness to elastic modulus. Most rubber hardness tests measure the depth of penetration of an indentor under either a fixed weight or a spring load and when rubber is assumed to be an elastic isotropic medium the indentation obtained at small deformation depends on the elastic modulus, the load applied and the dimensions of the indentor. It is recognised that with a spherical ball indentor the relationship between the indenting force D and the Young s modulus Ei given by ... 

Like the Rockwell test, the Brinell method uses a spherical indentor. but here the diameter of the impression is measured rather than the depth of penetration, the hardnes.s relationship being given by the expression... 

Indentation Test. Indentation hardness is measured by two methods. A spherical or pyramidal indentor is applied to the paint surface for a specified period. The penetration depth under load is measured (ISO 6441, ASTM D1474, Pfund hardness number). In the other method an indentor with a specified load is applied for a... 

Therefore, the load/crack length relations in Equation 3.1 and Equation 3.9 predict P oc for a pyramidal indentor and P oc c / for a spherical indentor. This agrees quite well with experimental results in Eigure 3.2 for ZnS. 

Errors arise when the BrineU test is performed on very hard materials, resulting in low values owing to (a) the spherical shape of the indentor (b) flattening of the ball. The BrineU number is not reliable above 600. 

Because of the multiple variables associated with the chevron notch specimens, a second study has been planned. Standard GAPD flexure bars (1/8 x 1/4 x 2 inch) will be used. Both Vicker s and spherical indentors at various loads will be used to generate cracking. Several tests per bar will be accomplished to generate multiple AE events per bar. This will reduce the error caused by setup variability equipment parameters and machine noise will also be evaluated. After indenting the bars, the crack size will be determined destructively. The AE data will be evaluated as a function of load and crack size. 

This concept is illustrated schematically in Figure 9a. In contact mechanics models like the Hertzian or JKR, a spherical indentor is assumed. There is no way to access the exact shape of the contact zone between tip and sample surface during a SFM experiment. Our assumption of spherical geometry is justified by images of... 

Figure 11. (a) Illustration of polymer indentation by spherical tip with radius of curvature R and contact diameter 2a. (b) Illustration of trench-shaped deformation region swept out by sliding spherical indentor. 

The problem of a moving indentor has been considered by Golden (1982) in an approximate manner, by simply assuming that the contact region is elliptical for an ellipsoidal or spherical punch. The difficulty is that the physical significance of this assumption is not totally apparent. It is valid at both low and high veloci-... 

We will confine ourselves to axisymmetric indentors, noting of course that fairly explicit results can also be given for an ellipsoidal indentor, by virtue of the classical Hertz solution. For a spherical indentor we may approximately take... 

Let us consider in more detail the case where the indentor is spherical and uniform motion at velocity F, under steady-state conditions, prevails. Choosing coordinates moving with the indentor allows us to write (5.4.4) in the form... 

Under a spherical indentor of radius / , the contact region is circular with radius ao, given by (see (5.2.31) with Iq replacing 4) ... 

For low velocities, an expression for /h linear in V can be given, without restriction on the size or nature of viscoelastic effects [Golden (1978)]. Indeed, a complete solution of the problem is possible. We will discuss here the special case of a spherical indentor, though the results may be generalized without difficulty to an ellipsoidal indentor. For steady-state motion in the negative x direction, (5.1.2) may be written in the form... 

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