Forced displacements

If we load a material in compression, the force-displacement curve is simply the reverse of that for tension at small strains, but it becomes different at larger strains. As the specimen squashes down, becoming shorter and fatter to conserve volume, the load needed to keep it flowing rises (Fig. 8.6). No instability such as necking appears, and the specimen can be squashed almost indefinitely, this process only being limited eventually by severe cracking in the specimen or the plastic flow of the compression plates. 

Numerous AFM imaging techniques have been developed and commercialized to monitor topography, friction, mechanical response, capacitance, magnetic properties, etc. However, adhesion measurements require the tip to come into, and out of, contact to measure attractive and adhesion forces. Therefore, other than to select an analysis region, most imaging techniques are not useful for adhesion studies. Instead, measurements are necessarily based on force-displacement curve approaches. 

In a force-displacement curve, the tip and sample surfaces are brought close to one another, and interact via an attractive potential. This potential is governed by intermolecular and surface forces [18] and contains both attractive and repulsive terms. How well the shape of the measured force-displacement curve reproduces the true potential depends largely on the cantilever spring constant and tip radius. If the spring constant is very low (typical), the tip will experience a mechanical instability when the interaction force gradient (dF/dD) exceeds the... 

A series of force-distance curves for various materials pairs examined (gold/ nickel, diamond/graphite, diamond/diamond) are shown in Fig. 4 [39]. For an indentation, the unloading slope (dF/dr) of the force-displacement curve is a measure of the contact stiffness and can be used to determine the modulus if the contact area (A) is known using a variant of Eq. 3 below. 

We have recently been exploring this technique to evaluate the adhesive and mechanical properties of compliant polymers in the form of a nanoscale JKR test. The force and stiffness data from a force-displacement curve can be plotted simultaneously (Fig. 13). For these contacts, the stiffness response appears to follow the true contact stiffness, and the curve was fit (see [70]) to a JKR model. Both the surface energy and modulus can be determined from the curve. Using JKR analyses, the maximum pull off force, surface energy and tip radius are... 

Depth-sensing nanoindentation is one of the primary tools for nanomechanical mechanical properties measurements. Major advantages to this technique over AFM include (1) simultaneous measurement of force and displacement (2) perpendicular tip-sample approach and (3) well-modeled mechanics for dynamic measurements. Also, the ability to quantitatively infer contact area during force-displacement measurements provides a very useful approach to explore adhesion mechanics and models. Disadvantages relative to AFM include lower force resolution, as well as far lower spatial resolution, both from the larger tip radii employed and a lack of sample positioning and imaging capabilities provided by piezoelectric scanners. 

Fig. 2.63 Force displacement behaviour of an elastic cracked plate...

Example 2.19 During tensile tests on 4 mm thick acrylic sheets of the type shown in Fig. 2.63(a), the force-displacement characteristics shown in Fig. 2.64(a) were recorded when the crack lengths were as indicated. If the sheet containing a 12 mm long crack fractured at a force of 330 N, determine the fracture toughness of the acrylic and calculate the applied force necessary to O acture the sheets containing the other crack sizes. 

Vectors are commonly used for description of many physical quantities such as force, displacement, velocity, etc. However, vectors alone are not sufficient to represent all physical quantities of interest. For example, stress, strain, and the stress-strain iaws cannot be represented by vectors, but can be represented with tensors. Tensors are an especially useful generalization of vectors. The key feature of tensors is that they transform, on rotation of coordinates, in special manners. Tsai [A-1] gives a complete treatment of the tensor theory useful in composite materials analysis. What follows are the essential fundamentals. 

Figure 11 Force displacement curves of a-PP, b-nonreac-tive PP-NBR blend, c-reactive blend containing 13 wt% GMA functionalized PP, and d-reactive blend containing 25 wt% GMA functionalized PP. Source Ref. 73.

In the current work a Digital Instmments Dimension 3000 SPM was operated in force-volume mode using a probe with stiffness selected to match the stiffness of the sample. Standard silicon nitride probes with a nominal spring constant of 0.12 or 0.58 N/m were used for recombinant and native resilin samples. These samples were characterized in a PBS bath at a strain rate of 1 Hz. For synthetic rubbers, silicon probes with a nominal spring constant of 50 N/m were used and the material was characterized in air. Typically, at least three force-volume plots (16 X 16 arrays of force-displacement curves taken over a 10 X 10 p.m area) were recorded for each of the samples. 

Fig. 18 A typical force-displacement curve. WF= work done overcoming die wall friction WD, work of elastic recovery Wp/, net work involved in tablet compact formation.

AM Mehta, LL Augsburger. Quantitative evaluation of force displacement curves in an automatic capsule filling machine. Presented to the IPT Section, A.Ph.A. Academy of Pharmaceutical Sciences, 128th Annual A.Ph.A. Meeting, St. Louis, March-April 1981. 

Harrison et al. (27,31) obtained force-displacement profile during extrusion of microcrystalline cellulose (MCC) only formulations using a ram extruder and resolved it into three stages, as seen in Figure 9 compression, steady state, and forced flow. Based on surface smoothness and cohesive strength, a predominant steady state region was found necessary... 

Figure 9 Schematic of a force-displacement profile from an instrumented ram extruder. Source From Ref. 27.

The non-linearity of the force-displacement relationship between a tip and a sample is sketched in Fig. 13.3(a). The force is a non-linear function of the tip-sample displacement, and depends on whether approach or retraction is under way. If the tip and the sample are well separated, at the far right on the graph, there is negligible interaction. During the approach, from right to left on the graph, initially there is an attractive interaction. At closer approach the force becomes repulsive. When reversing the displacement, the tip and the surface adhere until the contact is broken at a certain pull-off distance, which... 

Force-displacement plots to determine the work of compaction and the energy involved... 

There are several uses of tablet press instrumentation in the scale-up process itself. One of these involves obtaining a sample of the scale-up batch and compacting that sample on the pilot-plant or research instrumented tablet press on which the formulation has been previously evaluated. Similarity of the fingerprint or the various research plots (Heckel, force-displacement, radial vs. axial plots) is evidence that the scale-up batch is similar to the previously evaluated research batch [2]. 

Displacement of the crack front—from the start of crack formation until final destruction—takes place at a variable rate. For the crack to overcome impediments (such as macromolecules, chain bundles, super-molecular structure formations, inclusions, and micropores), it needs varying time lengths, and the fissure perimeter takes on a sinuous form. The limit between different formations on the entire perimeter of the crack front is evidently determined by the equilibrium set up between elastic mechanical forces and bond forces displacement of the crack front results from this equilibrium. 

Typically, in compression tests a cylindrical piece of the test sample is compressed between smooth plates using a Material Tester. Assuming constant volume, the stress and strain (Hencky strain) are calculated from the force, displacement data. However,... 

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