Indentation sphere

Furthermore, the almost complete vanishing of the trace of the strain tensor (Fig. 11, lower right panel) showed that most of the irreversible part of the quasi-macroscopic deformation was due to colloidal rearrangements. Within the resolution of the confocal method, only close to the indenting sphere a contribution due to the deformation of individual colloid could not be excluded. 

From the JKR theory, the pull-oflf force or adhesion force of a sphere (radius R) from a flat is >nRy, where y is the interfacial tension between the solid and the indenting sphere. The Derjaguin-Muller-Toporov (DMT) theory predicts that the pull-off force is AnRy, i.e., greater by a factor of 4/3 than the adhesion force predicted by the JKR theory. The two theories thus disagree in their predictions of interfadal tension from force measurements by this constant. However, there are two major differences between the theories that can be tested experimentally. One is the shape of the interface in contact under no load conditions and the second is the contact radius at which the surfaces separate. The JKR theory predicts that when the pull-ofif adhesion force equals 3 tRy, the surfaces separate fi-om a finite area of contact and the radius at pull-off is 0.63 times the radius under no load conditions. On the other hand, the DMT theory predicts that at the instant of separation, the contact radius is 0, i.e., the surfaces separate only at the point where they have achieved their original undistorted shape. 

The radiation and temperature dependent mechanical properties of viscoelastic materials (modulus and loss) are of great interest throughout the plastics, polymer, and rubber from initial design to routine production. There are a number of laboratory research instruments are available to determine these properties. All these hardness tests conducted on polymeric materials involve the penetration of the sample under consideration by loaded spheres or other geometric shapes [1]. Most of these tests are to some extent arbitrary because the penetration of an indenter into viscoelastic material increases with time. For example, standard durometer test (the "Shore A") is widely used to measure the static "hardness" or resistance to indentation. However, it does not measure basic material properties, and its results depend on the specimen geometry (it is difficult to make available the identity of the initial position of the devices on cylinder or spherical surfaces while measuring) and test conditions, and some arbitrary time must be selected to compare different materials. 

The shape is usually either a sphere (Brinell, and Rockwell B or C) a square pyramid with apex angle = 135° (Vickers) a trigonal pyramid (Berkovich) or an elongated four-sided pyramid (Knoop). (See Figure 1.2). For quality control in manufacturing operations, semi-automatic Rockwell machines, and their various indenters, are also useful. 

The deformation of a specimen during indentation consists of two parts, elastic strain and plastic deformation, the former being temporary and the latter permanent. The elastic part is approximately the same as the strain produced by pressing a solid sphere against the surface of the specimen. This is described in detail by the Hertz theory of elastic contact (Timoshenko and Goodier, 1970). 

The flow toward the surface is caused by the pressure under the indenter. It is analogous to the upward flow around a sphere dropped into a liquid. It is also analogous to inverse extrusion. A model of the flow has been proposed by Brown (2007) in terms of rotational slip. This model reproduces some of the observed behavior, but it is a continuum model and does not define the mechanism of rotational slip. 

An approximate model of the ISE can be developed with the aid of Figure 2.9. This figure shows a schematic cylindrical indenter with a conical tip being pushed into a specimen. The plastic zone is approximated by a segment of a sphere, and the diameter of the indent is 2r. The yield stress is Y, and the friction coefficient is a. 

The influence of fillers has been studied mostly at hl volume fractions (40-42). However, in addition, it is instructive to study low volume fractions in order to test conformity with theoretical predictions that certain mechanical properties should increase monotonlcally as the volume fraction of filler is Increased (43). For example, Einstein s treatment of fluids predicts a linear increase in viscosity with an increasing volume fraction of rigid spheres. For glassy materials related comparisons can be made by reference to properties which depend mainly on plastic deformation, such as yield stress or, more conveniently, indentation hardness. Measurements of Vickers hardness number were made after photopolymerization of the BIS-GMA recipe, detailed above, containing varying amounts of a sllanted silicate filler with particles of tens of microns. Contrary to expectation, a minimum value was obtained (44.45). for a volume fraction of 0.03-0.05 (Fig. 4). Subsequently, similar results (46) were obtained with all 5 other fillers tested (Table 1). 

The curve of distribution of grain striking at the surface of the material under test follows a normal distribution, hence the depth of the resultant indentation varies with position of test, being roughly constant at points equidistant from the centre of the indentation. The bottom of the indentation thus formed has the shape of a spherical cup. The radius of the sphere has been determined experimentally, and in measurements with the Mackensen blower it is 3 mm, equal to the radius of the circle on the plane of tested material, within which abrasion proceeds. From examination of the forma-... 

This case is of particular interest because, at least in the case of isotropic materials, there is no end effect since the idealised loading conditions can actually be realized in a practical situation. The problem was first considered by Hertz27 and is treated in greater detail in Love s1 article 138. The general result, applicable for a rigid sphere indenting an elastic body at a plane surface is ... 

More complex expressions arise if the sphere is taken to be elastic, so that the above equation should only apply to soft materials indented by very stiff materials. 

When the indentation hardness is measured, a small sphere or the tip of a pyramid-shaped crystal of a material which is harder than the one to be tested is pressed in the surface. The depth of the imprint is a measure for the hardness. Especially in the case of porous material it is vital that the determination is carried out carefully to avoid that... 

Fig.2. Interaction forces acting in vacuum between a two atoms (f r 7) and b macroscopic particles (e.g., for surface-sphere interaction, F D 2). The tip position at D=0 corresponds to the tip-sample contact, while the range at D<0 corresponds to the sample indentation...

The spheres of the third layer can be in the indentations of the second layer that are not directly above the first layer. These are C positions. Continuation of this sequence ABCABCABC... gives a structure known as cubic close packing (cep). 

The space probe is shaped like a huge sphere. Inside the sphere are a number of smaller, self-contained spheres, each of which holds machinery for specialized tasks. In the biggest of the interior spheres—let s call it the library —are the blueprints for making all the machines in the space probe. These are not ordinary blueprints, however. They can be thought of as blueprints in braille—or perhaps as sheet music for a player piano— where physical indentations in the blueprint cause a master machine to make the machine for which the blueprint codes. 

Indentation hardness is a very common determination in materials testing. In this test a very hard indenter (a hard steel sphere in the Brinell test, a diamond pyramid in the Vickers test) is pressed under a load into the surface of the material. 

From a more fundamental point of view, the selection of different inden-ter geometries and loading conditions offer the possibility of exploring the viscoelastic/viscoplastic response and brittle failure mechanisms over a wide range of strain and strain rates. The relationship between imposed contact strain and indenter geometry has been quite well established for normal indentation. In the case of a conical or pyramidal indenter, the mean contact strain is usually considered to depend on the contact slope, 0 (Fig. 2a). For metals, Tabor [32] has established that the mean strain is about 0.2 tanG, i.e. independent of the indentation depth. A similar relationship seems to hold for polymers although there is some indication that the proportionality could be lower than 0.2 for viscoelastic materials [33,34], In the case of a sphere, an... 

Fig. 4 Brittle failure modes of polystyrene within contacts, a Poly(styrene) film on a poly(methylmethacrylate) substrate. The regular crack pattern is induced by the sliding of a glass sphere under elastic contact conditions, b Poly(styrene) under viscoplastic scratching by a cone indenter (from [40])...

Fig. 17 Contact mechanics analysis of Herztian cracks within brittle materials.a Schematic description of a Hertzian cone crack induced under normal indentation by a rigid sphere, b Reduced plot of JC-field as function of cone crack length and for increasing loads pf < p// < pm during sphere-on-flat normal indentation of brittle materials. Arrowed segments denote stage of stable ring crack extension from Cf to cc (initiation), then unstable to ci at P = P,n (cone-crack pop-in) (From [67]). Branches (1) and (3) correspond to unstable crack propagation (dK/dc > 0), branches (2) and (4) to stable crack propagation (dK/dc < 0)...

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