Volume elastic modulus

Thus, the volume elastic modulus of the percolation system near the percolation threshold (p —> pc + 0) may be defined as... 

By using this correspondence principle for a connected set, the complex volume elastic modulus K and the complex shear modulus p at the (k + l)th step can be written... 

Metal-Matrix Composites. A metal-matrix composite (MMC) is comprised of a metal ahoy, less than 50% by volume that is reinforced by one or more constituents with a significantly higher elastic modulus. Reinforcement materials include carbides, oxides, graphite, borides, intermetahics or even polymeric products. These materials can be used in the form of whiskers, continuous or discontinuous fibers, or particles. Matrices can be made from metal ahoys of Mg, Al, Ti, Cu, Ni or Fe. In addition, intermetahic compounds such as titanium and nickel aluminides, Ti Al and Ni Al, respectively, are also used as a matrix material (58,59). P/M MMC can be formed by a variety of full-density hot consolidation processes, including hot pressing, hot isostatic pressing, extmsion, or forging. 

Fig. 11. Modulus inciease as a function of fibei volume fraction alumina fiber-reinforced aluminum—lithium alloy matrix for (a) E (elastic modulus),...

The parameters which characterize the thermodynamic equilibrium of the gel, viz. the swelling degree, swelling pressure, as well as other characteristics of the gel like the elastic modulus, can be substantially changed due to changes in external conditions, i.e., temperature, composition of the solution, pressure and some other factors. The changes in the state of the gel which are visually observed as volume changes can be both continuous and discontinuous [96], In principle, the latter is a transition between the phases of different concentration of the network polymer one of which corresponds to the swollen gel and the other to the collapsed one. 

In order now to evaluate the exponent rp we make recourse to the law of mixtures, given by relation (21), which expresses the elastic modulus of the composite in terms of the moduli and the radii (or volume fractions) of the constituent phases. This relation yields the average elastic modulus for the mesophase Ef. Then, it is valid for the mesophase layer that ... 

Fig. II. The variation of the elastic modulus for the composite Ec, versus the filler-volume fraction, uf, and the mode of variation of the average mesophase modulus, E", as derived from the model...

Since the early 1980s, the study of mechanical properties of materials on the nanometre scale has received much attention, as these properties are size dependent. The nanoindentation and nanoscratch are the important techniques for probing mechanical properties of materials in small volumes. Indentation load-displacement data contain a wealth of information. From the load-displacement data, many mechanical properties such as hardness and elastic modulus can be determined. The nanoindenter has also been used to measure the fracture toughness and fatigue properties of ul-... 

Munns, R., Greenway, H., Stelter, T.L. Kuo, J. (1983). Turgor pressure, volumetric elastic modulus, osmotic volume and ultrastructure of Chlorella emersoni grown at high and low external NaCl. Journal of Experimental Botany, 34,144-55. 

Consider a deformation consisting of repeated sinusoidal oscillations of shear strain. The relation between stress and strain is an ellipse, provided that the strain amplitude is small, and the slope of the line joining points where tangents to the ellipse are vertical represents an effective elastic modulus, termed the storage modulus /r. The area of the ellipse represents energy dissipated in unit volume per cycle of deformation, expressed by the equation... 

In spite of the absence of periodicity, glasses exhibit, among other things, a specific volume, interatomic distances, coordination number, and local elastic modulus comparable to those of crystals. Therefore it has been considered natural to consider amorphous lattices as nearly periodic with the disorder treated as a perturbation, oftentimes in the form of defects, so such a study is not futile. This is indeed a sensible approach, as even the crystals themselves are rarely perfect, and many of their useful mechanical and other properties are determined by the existence and mobility of some sort of defects as well as by interaction between those defects. Nevertheless, a number of low-temperamre phenomena in glasses have persistently evaded a microscopic model-free description along those lines. A more radical revision of the concept of an elementary excitation on top of a unique ground state is necessary [3-5]. 

Recently, Oldfield Ellis (1991) have examined the reinforcement of glass-ionomer cement with alumina (Safil) and carbon fibres. The introduction of only small amounts of carbon fibres (5% to 7-5% by volume) into cements based on MP4 and G-338 glasses was found to increase considerably both the elastic modulus and flexural strength. There was an increase in work of fracture attributable to fibre pull-out. A modulus as high as 12-5 GPa has been attained with the addition of 12% by voliune of fibre into MP4 glass (Bailey et al, 1991). Results using alumina fibre were less promising as there was no fibre pull-out because of the brittle nature of alumina fibres which fractured under load. 

Now, in rheological terminology, our compressibility JT, is our bulk compliance and the bulk elastic modulus K = 1 /Jr- This is not a surprise of course, as the difference in the heat capacities is the rate of change of the pV term with temperature, and pressure is the bulk stress and the relative volume change, the bulk strain. Immediately we can see the relationship between the thermodynamic and rheological expressions. If, for example, we use the equation of state for a perfect gas, substituting pV = RTinto a = /V(dV/dT)p yields a = R/pV = /Tand so for our perfect gas ... 

This is the same as the expression for the viscosity of dilute suspensions (see for example Chapter 3). The expression is however limited to low concentrations of particles, i.e. for tp < 0.05 there are others that can be used throughout the range of volume fractions. Landel26 proposed that the elastic modulus can be described by... 

Kalyanasundaram, Kumar, and Kuloor (K2) found the influence of dispersed phase viscosity on drop formation to be quite appreciable at high rates of flow. The increase in pd results in an increase in drop volume. To account for this, the earlier model was modified by adding an extra resisting force due to the tensile viscosity of the dispersed phase. The tensile viscosity is taken as thrice the shear viscosity of the dispersed phase, in analogy with the extension of an elastic strip where the tensile elastic modulus is represented by thrice the shear elastic modulus for an incompressible material. The actual force resulting from the above is given by 3nRpd v. 

TMA measures the mechanical response of a polymer looking at (1) expansion properties including the coefficient of linear expansion, (2) tension properties such as measurement of shrinkage and expansion under tensile stress, i.e., elastic modulus, (3) volumetric expansion, i.e., specific volume, (4) single-fiber properties, and (5) compression properties such as measuring the softening or penetration under load. 

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