Bond modulus parameter

The bond modulus is a quite simple parameter. It is the ratio of an atomic (or molecular) energy, and an atomic (or molecular) volume. Essentially all of the volumes of interest are known from crystallography, and the energies are... 

The shear work done for one atomic (molecular) displacement, b is the applied force times the displacement, or xb3. This work must equal the promotion energy 2Eg. Therefore, letting b3 equal the molecular volume, Vm, the required shear stress is approximately 2Eg/Vm. The parameter [Eg/Vm] is called the bond modulus. It has the dimensions of stress (energy per unit volume). The numerator is a measure of the resistance of a crystal to kink movement, while the denominator is proportional to the work done by the applied stress when a kink moves one unit distance. Overall, the bond modulus is a measure of the shear strengths of covalent bonds. 

Most mechanical and civil engineering applications involving elastomers use the elastomer in compression and/or shear. In compression, a parameter known as shape factor (S—the ratio of one loaded area to the total force-free area) is required as well as the material modulus to predict the stress versus strain properties. In most cases, elastomer components are bonded to metal-constraining plates, so that the shape factor S remains essentially constant during and after compression. For example, the compression modulus E. for a squat block will be... 

In this contribution, we report equilibrium modulus and sol fraction measurements on diepoxidet-monoepoxide-diamine networks and polyoxypropylene triol-diisocyanate networks and a comparison with calculated values. A practically zero (epoxides) or low (polyurethanes) Mooney-Rivlin constant C and a low and accounted for wastage of bonds in elastically inactive cycles are the advantages of the systems. Plots of reduced modulus against the gel fraction have been used, because they have been found to minimize the effect of EIC, incompleteness of the reaction, or possible errors in analytical characteristics (16-20). A full account of the work on epoxy and polyurethane networks including the statistical derivation of various structural parameters will be published separately elsewhere. 

A measure of shear strength is the shear modulus. For covalent crystals this correlates quite well with hardness (Gilman, 1973). It also correlates with the hardnesses of metals (Pugh, 1954), as well as with ionic crystals (Chin, 1975). Chin has pointed out that the proportionality number (VHN/C44) depends on the bonding type. This parameter has become known as the Chin-Gilman parameter. 

For interpreting indentation behavior, a useful parameter is the ratio of the hardness number, H to the shear modulus. For cubic crystals the latter is the elastic constant, C44. This ratio was used by Gilman (1973) and was used more generally by Chin (1975) who showed that it varies systematically with the type of chemical bonding in crystals. It has become known as the Chin-Gilman parameter (H/C44). Some average values for the three main classes of cubic crystals are given in Table 2.1. 

What are typical values for the parameters k and kl The bending rigidity of surfactant films is typically of the order of 1-20 kBT at room temperature. Factors that reduce k are short alkyl chains, cosurfactants, double-chain surfactants with unequal chains, and cis-unsaturated bonds. For the saddle-splay modulus only few measurements have been done. It tends to be negative with an amount much smaller than the bending rigidity for the same system. 

The bond graph in Figure 14.2 contains a dissipative coupling between flows A and B, in which only an interacting fraction is involved in the process. Therefore, the linear transducer TD, which converts energy from one form to another, thereby conserving power, is introduced into the bond graph. The operation of transducer is characterized by a modulus r, which may be a function of the parameter of state, such as temperature or concentration, and is independent of flows and forces. 

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