Elastic modulus from contact

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 this relation, k. is the cantilever stiffness and ki is the tip-surface effective stiffness given by dFIBS. From the relations (1.1 to 1.3), it can be seen that k depends on the static contact force, the tip geometry and the surface elastic modulus. Knowing the cantilever stiffness, the static contact force and the tip geometry and dimensions, it is thus theoretically possible to determine the surface elastic modulus from the dynamic response. 

Once the contact area is determined from the load-displacement data, the hardness, H, and effective elastic modulus, Egff, follow from ... 

With good experimental technique and careful analysis, the hardness and elastic modulus of many materials can be measured using these methods with accuracies of better than 10 % [59]. There are, however, some materials in which the methodology signihcantly overestimates H and E, spe-cihcally, materials in which a large amount of pile-up forms around the hardness impression. The reason for the overestimation is that Eqs (22) and (24) are derived from a purely elastic contact solution, which accounts for sink-in only [65]. 

The number of cycles of disk rotation required to initiate the wear track correlated positively with the weight percent of the siloxane modifier in the epoxy. However, the initiation times for the ATBN- and CTBN-modified epoxies showed no significant correlation with the percentage of the incorporated modifier. The initiation of the wear track is assumed to result from the fatigue of the epoxy hence initiation time is related to the surface stresses. Because the surface stresses are inversely related to the elastic modulus as predicted by the Hertzian elastic contact theory 52), the initiation time data at ION load were compared to the elastic moduli of the materials in Fig. 16. The initiation times for the siloxane-modified epoxies were negatively correlated with their elastic moduli while samples modified with ATBN and CTBN showed positive correlations with their moduli. At lower loads the initiation times for the siloxane-modified epoxies increased. The effect of load on the CTBN- and ATBN-modified epoxies was too erratic to show any significant trends. 

The essential material property of rubbers is their low elastic modulus, which ensures that the contact deformation remains elastic over a very wide range of contact conditions. The abrasive wear of rubbers is due to either fatigue of the material or tearing by a cutting force from impacts with sharp-edged particles. 

A Models to describe microparticles with a core/shell structure. Diametrical compression has been used to measure the mechanical response of many biological materials. A particular application has been cells, which may be considered to have a core/shell structure. However, until recently testing did not fully integrate experimental results and appropriate numerical models. Initial attempts to extract elastic modulus data from compression testing were based on measuring the contact area between the surface and the cell, the applied force and the principal radii of curvature at the point of contact (Cole, 1932 Hiramoto, 1963). From this it was possible to obtain elastic modulus and surface tension data. The major difficulty with this method was obtaining accurate measurements of the contact area. 

Figure 4 The modified stalk mechanism of membrane fusion and inverted phase formation, (a) planar lamellar (La) phase bilayers (b) the stalk intermediate the stalk is cylindrically-symmetrical about the dashed vertical axis (c) the TMC (trans monolayer contact) or hemifusion structure the TMC can rupture to form a fusion pore, referred to as interlamellar attachment, ILA (d) (e) If ILAs accumulate in large numbers, they can rearrange to form Qn phases, (f) For systems close to the La/H phase boundary, TMCs can also aggregate to form H precursors and assemble Into H domains. The balance between Qn and H phase formation Is dictated by the value of the Gaussian curvature elastic modulus of the bIlayer (reproduced from (25) with permission of the Biophysical Society) The stalk in (b) is structural unit of the rhombohedral phase (b ) electron density distribution for the stalk fragment of the rhombohedral phase, along with a cartoon of a stalk with two lipid monolayers merged to form a hourglass structure (reproduced from (26) with permission of the Biophysical Society).

The elastic modulus can be obtained independently from Eq. (2), which states the relationship between the load, displacement and contact radius [12]. 

The universal hardness , is obtained from the same formula if h is inserted instead of Aplastic- The universal hardness includes both the elastic and plastic deformation. The hnear part of the unloading curve corresponds to the elastic recovery when the diamond pyramid is in a constant area contact with the material. Therefore it represents Hooke s law and allows one to calculate the corresponding elastic modulus E/ — i/) which is a complicated function of the bulk, shear, and tensile moduli is the Poisson ratio). The details of the apparatus, the measuring procedure and possible errors are given in the relevant papers to which we refer here [25-28]. If done correctly, the plastic hardness measured by the indentation agrees within about 10-15% reasonably well with that from the classical Vickers method at least in the range H < 1500kgmm [25]. 

In other words, the elastic modulus of the cubic structured gel, E, should increase with adhesion W, particle elasticity E and Poisson s ratio v, but decrease as the particle diameter D rises. These equations are interesting when we apply them to common gels such as silica, alumina, titania, or zirconia. For example, for 1 (im diameter silica particles, = 70GPa, fF = 0.2Jm because of the hydrated surface and v is 0.3. The contact spot diameter turns out to be 32 nm from Equation (11.1), only 3% of the particle diameter. It is salutary to note that such tenuous adhesive contact between the particles is responsible for the gel behavior. The Young s modulus ofthe gel from Equation (11.7), assuming a cubic structure, is 0.75 GPa. This is close to the measured values for silica gel, which is about a hundred times more compliant than silica itself. Let us now consider the preparation and measurement of such gels. 

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